Inelastic X-Ray Scattering Studies of Broken
Symmetry in BSCCO
by
Craig John Bonnoit
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
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A uthor ................
-. ... V............
Department of Physics
April 29, 2013
Certified by.................
.j
Young S. Lee
Professor
Thesis Supervisor
Accepted by ....
. . . .7.. ...... ..
... . . . . . ..
................. ...%.
John Belcher
Professor, Associ e Department Head for Education
2
Inelastic X-Ray Scattering Studies of Broken Symmetry in
BSCCO
by
Craig John Bonnoit
Submitted to the Department of Physics
on April 29, 2013, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Physics
Abstract
Interactions between charge and lattice degrees of freedom play a critical role in determining the properties of canonical BCS superconductors where integration out of
the phonon subsystem results in an effective pairing interaction between electrons. In
the study of high temperature superconductors the importance of phonons is less well
understood and charge ordering properties vary between the families of high-Tc cuperates. While superconductivity in these materials is not believed to originate from
phonon excitations, there is evidence for strong electron-phonon coupling from significant electron dispersion renormalization and the observation of increased breadth
in optical Cu-O bond modulating phonons.
Here we present measurements of acoustic phonons in single and double layer
BSCCO which show several effects: broadening of the longitudinal acoustic in correspondence to approximately period-four ordering tendencies and signatures of timereversal and inversion symmetry breaking. Measurement of these anomalous properties is feasible due to renormalization of the lattice propagator by strong interactions
with underlying symmetry-breaking electronic states. These symmetries are broken at
room temperature for all materials in the 'strange metal' state above the pseudogap,
but are enhanced, particularly around the period four intercell ordering wavevector,
as the system is cooled into the pseudogap state. In-plane acoustic phonons are a
probe of the electron physics localized on the Cu-O plane due to the residual eigenvector components in this plane. These phonon measurements then present a picture
of BSCCO in which charge correlations stay dynamic with a pronounced tendency toward ordering at a specific wavevector and an underlying symmetry-breaking ground
state.
Thesis Supervisor: Young S. Lee
Title: Professor
3
4
Acknowledgments
I would like to first thank my advisor Young Lee for his guidance throughout my
time at MIT. I am very grateful to him for his deep engagement and support during
these experiments. The value Young placed on this work and on helping us become
better scientists became quite clear to me during the frequent midnight phone conversations I had with him during experiments when he worked his schedule around
our experimental time frame. I would also like to thank him for the confidence he
had in his students in entrusting us with the responsibility to comprehend of our data
and find the meaning. I am also grateful for Young's open but discriminating mind
to help filter through a large series of (often incorrect) ideas we have had to explain
the effects presented in this data.
I also thank my co-workers and friends, whose company and conversation has
made my time at graduate school enjoyable.
Firstly, I thank Dillon Gardner for
coming with me to Argonne National Lab to help with these experiments. These
trips, especially the early ones while Sector 30 were new and the beam was dumped,
every couple days were very challenging and exhausting, and I am very thankful for
his cheerful help through this often sleepless toil, without which this data could not
have been taken. Every student I have talked to has brought a different perspective
of physics to me, and I thank Robin Chisnell, Harry Han, Joel Helton, and Kittiwit
Matan for discussing my projects and theirs with me and helping teach me how to be
a scatterer. From my time outside of the lab I thank my friends Ankur Moitra, Mike
Petr, Matt Johnson, Emilio Nanni, and the MIT Ultimate Frisbee Team, particularly
Brian Yutko and Joel Brooks. Lastly, I thank my parents Albert and Marsha and
my sister Alyssa, whose support and confidence in me has always been absolute, even
when unmerited.
5
6
Contents
1
2
3
17
Introduction
1.1
The Hubbard Model ..............
. . . . . .
20
1.2
High T, Phenomenology . . . . . . . . . .
. . . . . .
23
1.2.1
La 2 -2SrCuO 4 and La 2-,BaCuO 4
. . . . . .
27
1.2.2
YBa 2 Cu3
. . . . . . . . . . . .
. . . . . .
30
1.3
Details of the BSCCO Family . . . . . . .
. . . . . .
32
1.4
Thesis Outline . . . . . . . . . . . . . . . .
. . . . . .
38
6 +6
39
X-Ray Scattering
2.1
Crystal Structure . . . . . . .
44
2.2
Scattering Cross Section . . .
46
2.3
Experimental Configuration
.
52
55
Measurements on Bi2201 and Bi2212
3.1
3.2
3.3
Elastic Scattering . . . . . . . . . . . . . . . . . . .
. . . . . . . .
56
3.1.1
Resolution Function
. . . . . . . . . . . . .
. . . . . . . .
56
3.1.2
Elastic Scans for Charge Ordering . . . . . .
. . . . . . . .
58
3.1.3
Main Bragg Peaks
. . . . . . . . . . . . . .
. . . . . . . .
61
3.1.4
Superstructure
. . . . . . . . . . . . . . . .
. . . . . . . .
63
Inelastic Scattering . . . . . . . . . . . . . . . . . .
. . . . . . . .
66
3.2.1
Longitudinal Phonons Along (110)
. . . . .
. . . . . . . .
69
3.2.2
Transverse Phonons Along (110) . . . . . . .
. . . . . . . .
73
Absense of Supermodulation Associated Phonons in (11( ))Cut . . . .
7
73
3.4
3.5
3.6
3.3.1
Comparison Between Longitudinal and Transverse Scans
75
3.3.2
Scattering in (020) and (200) Zones
. . . . . . . . . . .
80
3.3.3
Comparison with Hg 3 -6AsF 6
. . .
. . . . . . . . . . .
83
Damped Harmonic Oscillator Fitting . . .
. . . . . . . . . . .
85
3.4.1
Effect of Resolution Convolution . .
. . . . . . . . . . .
87
3.4.2
Phonon Sea Background . . . . . .
. . . . . . . . . . .
88
3.4.3
Optic Mode Fitting . . . . . . . . .
. . . . . . . . . . .
. 91
3.4.4
Phonons Along (010) . . . . . . . .
. . . . . . . . . . .
92
Fitted Parameters . . . . . . . . . . . . . .
. . . . . . . . . . .
97
3.5.1
Doping Dependence . . . . . . . . .
. . . . . . . . . . .
101
352
Renormali7pd Dispersin
1 6
3.5.3
Finite Domain Model and Coherence Off High Symmetry Cut
108
3.5.4
Asymmetry in Scattering Between (2 ± c, 2 ± 6, 0) . . . .
113
.
Shell Model Calculations . . . . . . . . . . . . . . . . . . . . . . . . .
119
3.6.1
121
Comparison with Optic Mode . . . . . . . . . . . . . . . . . .
3.7
Absence of Temperature Dependence in Phonon Broadening
. . . . .
121
3.8
Implications of Phonons Coupling to Electron Structure . . . . . . . .
125
4 Symmetry Breaking
4.1
x"
4.2
Longitudinal Phonons
131
as Probe of Time-Reversal and Inversion Symmetry . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
4.2.1
Effect of Resolution Function
4.2.2
Dependence on Momentum and Temperature
. . . . . . . . . . . . . . .
133
139
142
. . . . . .
147
4.3
Transverse Phonons . . . . . . . . . . . . . . . . . . . . . . . . .
150
4.4
Implications of Phonon Signatures of Symmetry Breaking . . . .
154
4.5
Comments on Other Evidence for Broken Symmetry . . . . . . .
155
4.5.1
Tim e Reversal . . . . . . . . . . . . . . . . . . . . . . . .
155
4.5.2
Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . .
161
4.5.3
Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . .
162
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
163
4.6
8
List of Figures
1-1
Phase diagram of hole-doped High T, materials. . . . . . . . . . . . .
1-2
Tetragonal BSCCO Fermi surface nesting, dashed red lines indicating
25
a smaller number of (hole) dopants and hence a larger qCDw nesting
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
2-1
Single crystal monochromater . . . . . . . . . . . . .. . . . . . . . . .
41
2-2
Sector 30 layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2-3
HERIX analyzer arm . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2-4
Bi2201 and Bi2212 unit cells . . . . . . . . . . . . . . . . . . . . . . .
45
2-5
Single crystal Bi2201 UD31K. Scale bar is 2mm . . . . . . . . . . . .
46
2-6
Symmetry of tetragonal 14/mmm, orthorhombic Cccm, and enlarged
w avevector.
supercell C 222[1]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
2-7
Eulerian cradle of four-circle diffractometer . . . . . . . . . . . . . . .
53
3-1
Inelastic scans through a block of Plexiglas, taken with Analyzer 5
prior to measurement of UD50K Bi2212 and OD70K Bi2212 . . . . .
3-2
57
Elastic scans searching for CDW signal in Pb,Bi2201 UD32K (A,D) and
UD50K Bi2212 (B,C). Data sets are color coded based on temperatures
of 300K (red), 150K (purple), and 6K (blue). . . . . . . . . . . . . . .
3-3
60
Scans through three orthogonal directions through the (220) Bragg
peak in the UD50K sample at 300K.
9
. . . . . . . . . . . . . . . . . .
62
3-4
(A) Cuts through the superstructural modulation along b* at L = 1 in
all samples. (B) Transverse cuts through Bi2201 UD25. (C) Evidence
of four symmetry-related first order peaks in Bi2201 UD25. (D) Second
harmonic peaks in Bi2201 UD25.
3-5
. . . . . . . . . . . . . . . . . . . .
First Feynman diagram renormalizing phonon propagator in the presence of electron-phonon interaction . . . . . . . . . . . . . . . . . . .
3-6
65
68
Sample longitudinal phonon scans from all samples measured. The vertical axis is intensity (counts/sec) and the horizontal is energy transfer
(meV ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-7
Sample transverse phonon scans from all samples measured. Axes are
counts per second along y and energy transfer w along x. . . . . . . .
3-8
71
Reciprocal space, showing projection of L E [-1, 1].
74
Solid circles:
Bragg peaks. Stars: superstructure peaks. Blue arrow: Scans measuring longitudinal phonons. Green Arrow: Scans measuring transverse
phonons. Blue ellipse: area of observed broadening. . . . . . . . . . .
3-9
76
The K-L plane for H=O, decorated by Bragg and superstructure peaks;
hollow stars are second harmonic superstructure peaks. There is no interaction between these and orthogonal H zones; this picture is translation invariant along a* except for the obvious removal of Bragg structures. 79
3-10 Scans taken for Bi2212 UD50K for (2+6, 0+c, 0) (blue), (0+ , 2+ , 0)
(purple), and interpolated from (2 + E, 2 + E,0) via Eq 3.3 (gold)
. .
82
3-11 Scans taken for Bi2212 UD50K for (2+E, 0+E, 0) (blue), (0+E, 2+6, 0)
(purple) highlighting role of elastic scatter. . . . . . . . . . . . . . . .
83
3-12 Induced breadth due to the momentum resolution, compared with that
from the energy component of the ellipsoid.
10
. . . . . . . . . . . . . .
88
3-13 Bi2212 data taken at (2.25, 2.25, 0) as a function of energy at several
temperatures, highlighting role of background. (Top) shows scans at
300K for w > 0 and w < 0, while bottom shows scans for w > 0 for
T=300K,150K, 60K. A single phenomenological fit line (thick red) fits
all data well when Bose scaled to induce the other fit lines for negative
energy transfer and for varying temperature. . . . . . . . . . . . . . .
90
3-14 Scans near zone center showing presence of optic mode, fit to a resolutionlimited lineshape corresponding to the convolution of a flat dispersion
at 6(W
-
w2 ) over the full resolution ellipsoid . . . . . . . . . . . . . .
91
3-15 Individual phonon scans along the direction of the supermodulation,
(0, 2 +
6,
0), Bi2201 UD31K and Bi2212 UD50K . . . . . . . . . . . .
93
3-16 Scans taken for Bi2201 UD31K and Bi2212 UD50K for (2 + E, 0). A
number of phonon modes equal to the number of dots visible was fit,
with the low-lying mode pinned to a resolution-limited width. The two
sets of lines represent simple theory models with phenomenologically
generated sinusoidal dispersions . . . . . . . . . . . . . . . . . . . . .
3-17 DHO fit parameters for the longitudinal cut (2 + E, 2 +
94
0) . . . . . .
98
3-18 DHO fit parameters for the transverse cut (2 - E, 2 + E, 0) . . . . . . .
99
6,
3-19 Momentum dependence as a function of doping for DHO fit parameters
for the longitudinal cut (2 + e, 2 + E, 0) Legend as per 3-17
. . . . . . 103
3-20 Momentum dependence as a function of doping for DHO fit parameters
for the longitudinal cut (2 + E, 2 + E, 0) . . . . . . . . . . . . . . . . .
107
3-21 (A-F) Cuts away from the high symmetry direction (E, E, 0) passing
through (2.25,2.25,0).
Top right inset: Reciprocal space diagram of
scan directions (blue and purple arrows) and Bragg structure (solid
circle: (220) Bragg peak, 'X': out-of plane superstructure). Similarities
between blue and purple scans highlight features obeying a tetragonal
reflection symmetry contrasted to differences associated with symmetryreducing superstructure associated modes. . . . . . . . . . . . . . . . 110
11
3-22 Fit parameters from the cut a shown in Figure 3-21. Plotted is the w,
and F parameters describing the main (broadened) phonon respecting
tetragonal symmetry and a second resolution limited phonon of amplitude n 2 at energy w2; for comparison, the amplitude of the tetragonal
phonon ranges from 4-6 on the same au scale. Dashed line shows a
linear extrapolation leading to a predicted value of w2
=
2.3 meV used
to check the robustness of the primary phonon parameters
3-23 Cuts at +E (purple) and -E
. . . . . . 112
(blue) for Bi2212 UD. . . . . . . . . . . .
116
3-24 Results of simultaneous fitting longitudinal acoustic mode at tE, shown
in hollow squares, compared with fits to a single mode from Section
3.2.1 in filled dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118
3-25 Fit curves (solid lines) for Bi2201 (black) and Bi2212 (orange) superimposed over phonon fit data
. . . . . . . . . . . . . . . . . . . . . .
122
3-26 Temperature dependent w, and F data for several scans; filled squares
are data at room temperature, hollow are at reduced temperature
. .
123
3-27 Ratio of phonon frequency over self-energy, w/F for acoustic BSCCO
samples (this work) and optic Cu-O bond stretching modes ([2, 3])
measured along the orthorhombic (110) direction
4-1
Conversion of raw data (left panels) to
Ix"I
. . . . . . . . . . . 128
(right panels). Left panels
highlight elastic lineshape, subtracted before conversion. Data for W <
0 (purple) and w > 0 (blue) superimposed.
From top to bottom,
data from UD31K Bi2201 at 300K, UD31K Bi2201 at 100K, UD50K
at 300K, and UD50K at 60K. Inserts: Asymmetry scans repeated for
equivalent
cycle)
Q on
a separate experimental run (after a warming/cooling
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
140
4-2
Example showing the process by which time-reversal or inversion symmetry obeying data is renormalized by the resolution function and
converted to Am. In the last panel, the integrated difference appropriately normalized is Am
=
0.004539. This can be compared with the
calculated value of 0.004537 from Equation 4.23.
4-3
. . . . . . . . . . .
146
Temperature dependent asymmetry for Bi2201 and Bi2212. Dashed
lines are guides to the eye drawn kinked at T* for each sample as listed
in Table 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-4
Momentum dependent asymmetry A(Q) as defined in Equation 4.11
for Bi2212 UD50K (top) and Bi2201 UD31K (bottom)
4-5
147
Example transverse symmetry-breaking scans
13
. . . . . . . .
149
. . . . . . . . . . . . .
152
14
List of Tables
2.1
Characteristic measures of the HERIX spectrometer at Sector 30, APS
40
2.2
List of samples measured . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.1
Summary of crystallographic, mosaic, and instrumental parameters. .
61
3.2
Momentum dependence of the peak position in c of anomalous properties as a function of doping. (*) Different background lineshape used
as explained in text . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
15
16
Chapter 1
Introduction
One of the open problems in the study of strongly correlated electron physics is understanding the nature of the high-temperature superconductors. The first detection of a
superconducting state was in simple metals such as Hg and Pb at low temperature[4],
and the explanation for superconductivity in these systems was provided by Bardeen,
Cooper, and Schrieffer[5].
The phonon system interacts with the free electrons to
produce an effective Hamiltonian consisting of negatively charged electrons with an
effective attractive pairing interaction induced by renormalization out of phonon degrees of freedom. The existence of a pairing interaction between electrons is sufficient
in the BCS framework to generate superconductivity at temperature scales below the
pairing energy. In conventional superconductors, pairing of electrons with opposite
momentum and spin then resulted in a ground state with gapped quasiparticle (Bogoliubov) excitations. As the electrons condensate into a coherent paired state, the
low lying excitational degrees of freedom are gapped out resulting in an absence of
gapless states for electrons to scatter in and zero resistance. Similarly, the existence
of a paired ground state in which equal numbers of spin are up and down implies
a macroscopic energy scale for the introduction of a field into the bulk, leading to
perfect screening of external magnetic fields and diamagnetism. The strong-coupling
limit of phonon pairing related superconductivity then predicted a temperature bound
of T, ~ 40K from the McMillan equation [6].
The discovery of superconductivity in LBCO and related copper-oxide materi17
als quickly surpassed this bound, showing superconductivity as high as 135K in
HBCCO[7, 8].
Superconductivity was seen in Cl, F, Pb, La, Bi, Y, Hg, and TI
based copper oxide compounds with similar environments, where the superconducting transition temperature generally increases with the number of Cu-O layers per
unit cell between n=1, n=2, and n=3, with saturation seen with increasing number of
layers above that. Commonalities between these families of materials are the presence
of octahedral oxygen cages around in-plane copper atoms and the layered stacking
perpendicular to the copper-oxygen plane. Additional charge carriers are provided
by chemical doping, which includes either substitution of different valence atoms or
insertion of additional oxygen atoms. These charge carriers result in conduction occurring through the Cu-O plane[9].
In the hole doped case, the Cu-O bond then
hybridizes into a Zhang-Rice singlet[10] while in the electron-doped case hopping is
dominated by the motion of d'0 electrons. We focus on the Bi-based materials with a
chemical formula Bi 2 Sr 2 Ca_1CunO 2 n + 2 +6 . This material is chosen because the gap
between successive Bi-O layers is large and the held together by weak Van der Waals
forces, which leads it to easily cleave perpendicular to the Cu-O plane. The ease
of cleaving means that a large number of surface-sensitive measurements have been
taken on this family with high precision, highlighting the role of charge correlation.
These materials are also characterized by strong chemical substitution disorder and
the presence of a superstructural modulation, as will be described in Chapter 2.
Aside from the presence of superconductivity at remarkably high temperatures,
several other features common to the high-Tc materials indicate strong and unusual
electron-electron correlation. A simple probe of interaction strength is measurement
of the spin wave excitation spectra by inelastic neutron scattering, which reveals an
usually large exchange constant J - 140 meV in parent LCO [11]. Electron quasiparticle excitations out of the superconducting state are seen to be broad in ARPES
measurements on CCOC
(~ 300
meV)[12] and the lineshape measured suggests the
system is in the polaronic regime with strong electron-phonon coupling as well. A
few of the universal emergent measures of the strongly correlated state are electrical
and optical conductivity. At temperatures above the superconducting transition, the
18
resistivity is seen to increase linearly at high temperature for optimally doped materials, in contrast with the T 4 scaling expected from phonon scattering. The Hall angle
indicates an inconsistency between the low temperature scaling of the anisotropy of
the optical conductivity
havior,
COt(OH)
cot( 6 H)
oC T 2 and what is expected for a Drude metal be-
oc p oc T [13]. While de Haas van Alphen shows existence of a Fermi
surface, ARPES measurements show loss of weight of the coherent residue peak in
the spectral function in the underdoped case. These materials are in general characterized by an upturn in coherence upon transitioning to the superconducting state,
seen in many probes of quasiparticle behavior - ARPES, thermal Hall, and microwave
surface impedance. [14, 15, 16].
Conceptually the issue of loss of coherence is thought about in terms of two things:
firstly a Cooper pair decay rate which increases with field , saturating at Hc2 when the
superconducting state is destroyed; secondly, the constituent single particle excitation
gains self energy (or exhibits increased scattering rate) with increasing temperature,
with a critical temperature dependence set by the coherence temperature Tcoh[17].
The exact microscopic scattering mechanism for these decohering transitions is not
established, but some experimental support for short range density-wave fluctuations
has been presented[18].
Fluctuations can play an important role at intermediate
temperatures in the absence of a coherent condensate[19]. Supercoductivity is present
when both the single particle excitations are coherent and d-wave singlet pairing forms
a condensate[20].
Here we focus on charge ordering; while the issue of superconductivity is nominally
separate from that of coexisting order parameters, charge ordering is common to many
of the High T, materials, as will be presented in Section 1.2. To understand the phase
diagram in which superconductivity lives, we then look to understand the coexisting
or competing ordering phases. Much of the classification work was done by surface
sensitive probes due to the difficulty of growing large samples. Here we take inelastic
x-ray scattering measurements on n=1 and n=2 BSCCO to probe the lattice dynamics
in the bulk to search for signatures of the orderings seen at the surface. Revealing
the differences between surface and bulk behavior regarding formation of a density
19
wave state by measurement of the elastic and inelastic contributions of the density
response function is a critical contribution of this work.
1.1
The Hubbard Model
The parent compound of the high T, materials is an antiferromagnet insulator where
electrons are localized on copper sites and the dynamics are primarily in the spin
sector. A representation of this physics, and the evolution of it under doping, is presented by the Hubbard model. The idea behind this approach is to write an effective
field theory in terms of the second quantized electron creation and annihilation operators, where the interaction is naturally represented in real-space localized terms.
This Mott-Hubbard Hamiltonian is then
=
-t
c
' (f)c,(r) + cl(r')c,(f) + U
nt(-)n(r)
(1.1)
The first term of Eq 1.1 is the hopping term accounting for the kinetic energy of
electrons as they move between lattice sites. The second term is called the Hubbard
U term, where n,.(r) = ct(rF)c,(F) is the number of spin o electrons on the site ',
and the constant U represents the energetic cost of overcoming Coulomb repulsion
to doubly occupy a localized orbital. The sum <
, r'I > refers to summing over
all nearest neighbors. An interesting rephrasing of the U term is accomplished by
expanding the spin operator in terms of the Pauli matrices
SC(V)
=
ht
2ct (r)roc(r)
',
(1.2)
which allows writing the square of the spin operator in terms of the electron
number operators (h = 1),
($r))
n(r) - -nT(fln (r)
20
(1.3)
implying the Hamiltonian in Eq 1.1 is equivalently
=
-t
c
(
(ca(?)+ c ( )ca(i?))
-
~E($())
2
(1.4)
+ NU
where N is the total number of electrons. The case of the high T, cuperates is that
of this model around half filling. At exactly half filling, considering t a perturbation
around U, each site will be singly occupied to minimize the U term in 1.1, or by 1.4
the state will maximize S for each site, which implies single occupation since double
occupation produces a spin 0 state by Pauli's exclusion rule. Satisfying the large U
term then leaves a degeneracy between the space of all
2N
spin configurations such
that each site is singly occupied. A careful treatment by degenerate perturbation
theory results in
=-t
c
,()c,(i)
+ c!(i)ca(r) + J
S(i) -(
)
(1.5)
where J = f'U' and the Hamiltonian is understood to be over the restricted Hilbert
space of single occupation in the exact half filling case, and in the case of some slight
hole doping, it is taken over the restricted space of 0, 1 occupied sites. Intuition for
this state in the half-occupied case is provided by noting that among the space of
all single occupation states, one in which for each site all its neighbors are of the
opposite spin allows a maximal amount of hopping. This degeneracy of low energy
modes available from this state is the origin in the calculation of the energetic gain
favoring this state as a ground state in the limit t/U < 1.
The above result for strong coupling can be complemented with an expansion for
low coupling in which the main Hamiltonian is the hopping term and the Hubbard
term is viewed as a perturbation on this ground state. Diagonalizing the hopping
term results in unperturbed eigenstates produced by Fourier transformed free electrons, c,(r-)
=
>
enc,(k),
resulting in a final Hamiltonian expressed in Fourier
21
transformed second quantization operators of
=
2) 2E
2J(
c,(k)ca(k)+U
J
,k
k(22
k1+k2-k3 k4 c+(k1)cT(k2)cj{k3)c (k4)
(1.6)
where e
= -4t(cos k. + cos k,) and the 6 constraint is enforced modulo G,
the reciprocal lattice vector. At half filling in the limit of dominant t and at zero
temperature, all states contained in the convex hull of points (tir,
0), (0, +w) will
be filled with both spin states, and all states outside this will be unoccupied. This
sets up perfect nesting of the entire Fermi surface along the wavevectors (7, 7r) and
(-7r, 7). The critical insight here is that particles of opposite spin can be exchanged
between opposite ends of the Fermi surface, connected by the aforementioned nesting
vectors. This exchange has zero energy cost as it is invariant under 1.6, and hence
in represents an unstable perturbation. This leads to the formation of a spin density
wave with wavevector (+w, 7r). This is a spin sector analog of the formation of an
electronic charge density wave in the presence of nesting of the Fermi surface, the
Peierls transition.
The argument for formation of a charge density wave is simpler to make. The origin of this transition is that a small displacement in lattice translation commensurate
with the Fermi surface has a cost quadratic in the displacement, while the induced
gain in electronic energy is linear in that displacement.
Hence a one dimensional
metal will always be unstable toward formation of a charge density wave, and the
degree of nesting plays a critical role in deciding the energetics of this. In higher dimension, exact nesting is lost and a static charge ordering will not. exist. However, the
susceptibility of a free electron gas in 2D still shows logarithmic diversion (the Lindhardt equation), and in the presence of impurities at the surface, charge oscillations
are seen in the statics with a periodicity of twice the Fermi surface wavevector kF.
These are referred to as Freidel oscillations, and are believed to persist even when
a metal transitions to a Mott insulator under increasing U, where the role of the
charge Fermi surface is played by a ghost charge-zero spinon Fermi surface[21], which
then though emergent spin-charge interaction can yield oscillations of electron density
22
near impurities even though the Landau quasiparticles that degenerately nested near
Fermi surface to produced this effect in the free electron case are absent.
In the high temperature cuperates, many of the materials are seen to evidence
spin or charge density wave behavior in some doping configurations, most commonly
the far underdoped regime. These are often more complicated than the simple t - J
model or Fermi surface nesting picture described above, often depending on complications such as oxygen ordering that is neglected in these simple conduction-electron
only models. However, the underlying interactions of the Hubbard model and band
splitting that explain the simple case highlight the tendency toward such order, and
the issues of coherence, pinning in the statics, and long range ordering are then complexities subject to the detailed energetic scales and temperature. We now begin a
categorization of the experimental observation of the cuperates, with an eye toward
symmetry breaking phases which we will later comment on by use of inelastic x-ray
scattering as a probe.
1.2
High T, Phenomenology
The phase diagram common to the hole-doped cuperates is shown in Figure 1-1. In
the absence of doping these materials are Mott insulators due to the strong electronelectron repulsion on the copper sites. In the presence of doping to either add or
remove electrons, the Hubbard term is overcome and hopping conductivity dominates.
As would be expected from the Hubbard model on a square lattice at half filling, the
undoped parent material shows long range antiferromagnetic order with wavevector
(7r,7
r), resulting in unit cell doubling. In the presence of vacancies, the kinetic energy
term favors delocalization which then competes with the energtic gain from the static
antiferromagnetic order. The local moment is then an order parameter for a standard
phase transition as at high temperature the overall long range order is lost and only
short range spin-spin correlation remains, since on a local scale there is still energetic
preference for anti-alignment by the sign of J driving antiferromagnetic order. This
is an explanation for the low doping region of the phase diagram, covering the AF
23
insulator, spin glass, and loss of long range order of these states at high temperature
into the pseudogap state.
At intermediate doping low temperatures bring the onset of superconductivity at
doping x > 0.05. A critical point of interest motivated by the structure of the phase
diagram that arises at this point is if proximity to spin ordering is a competing or coexisting order parameter; is superconductivity suppressed by such ordering, as might
be expected in a classical BCS model in which the localized spin ordered states could
not form free electron Cooper pairs, or is the proximity to AF order and associated
fluctuations in the spin degrees of freedom which then induces a resonant valence
bond state enabling the superconducting condensate? Coexistence of superconductivity and magnetism is rare, but exists in the so-called Heavy Fermion compounds in
which flat band structure and strong spin-orbit coupling produce highly renormalized
free electron band masses. In the cuperates, the superconductor is characterized by
an absence of static magnetic order and exclusion of applied field by the 'dirty' type II
mechanism in which vortices are generated at intermediate field strength. Energetic
favorability of this is produced by the small coherence length of the superconductors,
which permits disruption of the order parameter on smaller scale to enable passage
of flux lines.
The superconducting order parameter is seen to have predominantly d-wave symmetry by measurement in ARPES of the gap structure. [22, 23] Although loss of Landau quasiparticles in the antinodal region complicates this description, repeated measurement of the gap function across several families of material have indicated this observation is universal. Confirmation of the d-wave nature came from phases-sensitive
SQUID measurements on YBCO using superconducting junctions to determine the
nature of the pairing.[24, 25] Similar Josephson tunneling measurements on Bi2212,
TBCCO, and LSCO supported d-wave symmetry in these materials as well[26, 27].
Canonical electron-phonon driven superconductors (ala BCS) are expected to favor
s-wave pairing, while electron-electron repulsion produces d-wave pairing[28].
At
elevated temperatures the superconducting state is destroyed by thermal induced
disorder, and the material looks like a strange metal, with clear residue quasiparticle
24
spectral weight indicating Fermi liquid properties, unusual power law correlations of
thermodynamic probes indicating strong correlations.
350
300
C
250
T
',
N
normal" state
T
coh
200
( &150
1pseudogap
'
E.
.2
Ferni
liquid
100 E
T
50
spiii
C (superconductor
0
0
0.05
0.1
0.15 0.2
Hole Doping (x)
0.25
0.3
Figure 1-1: Phase diagram of hole-doped High T, materials.
For large dopings and at elevated temperature, the normal metal characteristics
return; the Fermi surface closes at the antinodal region in agreement with quantum
oscillation measurements, and the resistivity shows upturn at high temperature as
T 4 . The underdoped regime and the pseudogap phases are then the regimes in which
the state of understanding is weakest and we will focus on. Many proposals have been
suggested for this state, such as quantum criticality[29], localized current loops[30],
and d-wave pairing with loss of single particle coherence[20]. The idea behind these
states is to describe the fluctuating state in the pseudogap and explain the loss of
coherent excitations at the Fermi surface. To do so, they propose a new order parameter or behavior of such fluctuations which the system transitions into upon entering
the pseudogap regime; these fluctuations are then suppressed at large doping when
the kinetic energy of the Zhang-Rice carriers is large enough to overcome the local25
izing pinning effects of antiferromagnetic correlation, and the material behaves as a
normal metal with well defined Fermi liquid like quasiparticles. Our focus is on the
antinodal electron regime as measured by probes sensitive directly to the single and
two-particle response function.
An important consideration in the puzzle of high-Tc materials is the change in
Fermi surface topology under cooling through the pseudogap transition. Luttinger's
theorem states that as interactions are turned on, the volume of the Fermi surface
should be invariant. At high temperature, a normal Fermi surface is seen with large
volume from ARPES; cooling appears to gap this out, and the remnant arcs do not
obviously form a closed surface. Some ARPES measurements suggest closure of the
remnant arcs in the shape of bananas in the nodal regime[31, 32], however, this is not
conclusive and is subject to confusion due to zone folding due to superstructure[33].
Arguments have even been made in ARPES that the arcs that are seen are not true
Fermi surface sheets, but rather a propogation of spectral weight into the gapped
region due to scattering, where the temperature dependence corresponds with the
scale at which the scattering rate is comparable to the d-wave gap and thus shows
quasiparticle-like weight along the entire arc[34]. This is a reflection of a pair-breaking
process such as fluctuating density-wave pairs since spectral weight is shifted away
from the Fermi surface[35].
Quantum oscillation measurements on YBCO [36] support existence of a large
Fermi surface in the overdoped materials, while in the underdoped case there is evidence at low temperature and high field for a small Fermi surface. This is consistent
with explanations in which zone folding induces a Fermi surface reconfiguration and
provides much of the motivation for looking for CDW/SDW features in these materials, since this anomalously small Fermi surface suggests different physics is relavent
than in the overdoped case. However, it is possible such charge order states are also
enhanced by the large field needed for these measurements and do not reflect the true
zero field Fermi surface. In terms of Luttinger's theorem, the relevant observation is
that in underdoped YBCO the carriers are given by the hole doping level, not the
total electron number, and the temperature dependence (using the Knight shift in
26
this case as a probe) is calculated in terms of a dopon theory which is a quantum
(non-Fermi) liquid composed of a resonant valence bond state and a Fermi liquid[37].
1.2.1
La 2 -. SrCuO4 and La 2 -zBaCuO 4
Initial exploration of the phase diagram in La2-BaCuO 4 (LBCO) revealed a sharp
anomaly near x = 1/8 in which superconductivity was strongly suppressed. This
is suggestive of charge or spin ordering since the filling fraction is a commensurate
fraction. The proposed order is alternating stripes of Cu atoms with a single electron residing on them and antiferromagnetic (7r, 7) order and rivers of reduced charge
with no spin order. In the commensurate case, this worked out to filling three consecutive rows of atoms, then having the river of charge with every other d10 electron
missing. This configuration, enabled by the commensurability of the 1/8 filling, allowed a large number of Cu orbitals to satisfy AF exchange, while maximizing the
kinetic energy induced by the absent holes by delocalizing the electron over a 1D
river. Experimental evidence for this static charge and spin order was first seen in
Lai. 6 _xNd. 4 BaxCuO 4 [38] , where the substitution of Nd was used to pin the oxygen
ordering into a low temperature tetragonal phase in which in plane oxygen atoms
are successively tilted above and below the plane to form stripes along one of the
tetragonal directions. This chemical instability then helped pin charge stripes, which
were seen as static order peaks near (ir, 7r) in neutron scattering. The need to use a
dopant to pin the structure is a first suggestion that static order, while energetically
favored in the cuperates, is delicate and pinning into the statics is determined by
weak shifts in the oxygen positioning.
Two important details emerged from this order: first, the wavevector was seen
to be commensurate with the doping, x = 0.12 in this case, and secondly, separate
temperature scales were resolved, where with decreasing temperature the material
first transitioned to low temperature tetragonal, then developed charge order with
characteristic wavevector 2 x, and finally magnetic order with wavevector proportional
to x, subject to appropriate choice of zone center to ensure nontrivial form factor for
these scattering mechanisms. The scaling of wavevector with doping indicates the
27
correct framework to view these ordering transition is as a spin and charge density
wave. In this case, the charge segregates into alternating filled stripes and a single
row of half-filled carriers, which disrupts the exact (ir, 7) magnetic ordering vector,
splitting it into four (7irx, 7i±x) sites instead. While the driving forces are described
here in terms of real space energetics in which the charge degrees of freedom are
represented in a localized second quantized basis, the same physics drives reciprocal
space ordering characterized by the doping wavevector.
Similar order was later reported in La 2-xBaxCuO
for x
=
4
for x = 1/8[39], and Lai.s_Eu. 2 SrxCuO 4 [40]
0.125 and x = 0.15. Spin ordering was again seen with a wavevector increas-
ing with doping, and was enhanced around a spin ordering temperature
x
=
$ 80K for
0.125 and a 60K x = 0.15, revealing a strengthening of the interactions favor-
ing spin ordering further into the underdoped regime. However, the ordering seen
in LESCO was observed using resonant soft X-ray diffraction, which has a distinctly
shorter penetration depth, on the order of nanometers, compared with standard hard
x-ray or neutron diffraction as was used in other reports of ordering. It is possible
then that the order in La 1 .s_-Euo. 2 SrxCuO 4 is not present in the bulk material. Static
order in LBCO for x a 1/8 was seen to continue to favor a CDW origin as qco
=
26
for several dopings measured near 1/8, where application of a 10T field was sufficient
to significantly enhance the measured signal[41] when suppressing superconductivity
as for x k 1/8.
A combined measurement using resonant soft X-ray and hard X-ray diffraction on
La 2 -xSrxCuO
4
[42] revealed an interesting surface-bulk dichotomy. Surface sensitive
probes saw a signal of charge-stripe formation, but the bulk measurement failed to
observe the correspondent order to within the detection limit of a factor of 10-
of the
Bragg intensity. The surface indicated a similar temperature scale of charge ordering
of approximately 60K, significantly above the temperature scale of superconductivity,
which onsets at 33K. While the surface sensitive resonant probe also benefited from a
strong increase in resolution due to directly probing the electronic configuration of the
oxygen or copper atoms while hard X-ray diffraction looks only for a commensurate
distortion in the lattice, it seems likely that there are no charge stripes in the bulk,
28
and the induced temperature scale is due to surface physics.
A general summary of the state of charge and spin ordering in these materials is
then that segregation into stripes by a charge density wave is energetically favored
as is the local (7r, 7r) spin ordering within these stripes. However, to pin this in the
statics generally require some aid from change in structure, either the low temperature
tetragonal phase and/or reduced dimensionality at the surface. Measurements of short
length scale C4 symmetric order as suggested by several surface-sensitive probes[43,
18] is then indicative of an energetic tendency toward such ordering that is likely
common to the bulk, but not a definitive test of existence of this order without the
local pinning effects shown to be relevant for the LSCO family.
The spin dynamics at low doping contain significant weight associated with fluctuations of the order parameter of the static T=0 magnetic order. Neutron scattering
measurements on La 2 -,Ba.CuO 4 as a function of doping revealed that while 3D static
commensurate order is lost quickly around x = 0.02, 2D static order persists up to
the highest measured concentrations x = 0.12, although the order parameter is seen
to rotate direction 450 at a doping x = 0.05[44]. The fluctuations above this are then
interpreted as arising from a quantum critical point associated with the T=0 magnetic transition of rotation of the static order parameter by 45'. Significant inelastic
magnetic fluctuations are seen to persist up to a high temperature with a pseudogap
type scale.
Measurement of phonons in LSCO has focused on the optical bond-stretching
branch involving Cu-O bonds. This excitation matches the energy of the kink seen
in ARPES studies, and density-functional theory predictions indicates this couples
in to the free electron strongly. Measurement of the phonon spectra at x = 0.15[45]
showed a huge softening of this phonon around halfway from the zone center to
zone boundary. While a softening was predicted by DFT[46], the observed phonon
width and strength of the contribution were unmatched. It was hypothesized that
electron-electron correlations in the underdoped state are responsible for the enhanced
electron-phonon coupling. [47] It is possible the same energetics that drove static order
at the surface are then responsible for dynamic screening in the bulk. The issue of
29
relation between CDW formation and phonon softening will be revisited in 3.2.
1.2.2
YBa 2 Cu3 0+6
The YBCO family of compounds, like LCO, show a series of structural phase transitions concerning oxygen ordering. Instead of detailing all phases, we summarize the
classes of ordering arguments for which existence in YBCO have been made. Recent hard X-ray measurements of 6
=
0.67 YBCO displayed incommensurate charge
ordering signatures[48] that were enhanced significantly with application of a magnetic field (17T), and furthermore persist up to a temperature approximately 150K.
Similar field enhanced (but present at H
=
0) signals were seen in several other
oxygen stagings[49]; in net this description as a function of doping imply that the
incommensurate wavevector for spin and charge order move in different directions
with doping, suggesting differing orders rather than the locked in qco
=
2qso seen
for early LNSCO work; the charge ordering wavevector seen here instead supports a
band structure/Fermi surface origin distinct from spin ordering. This description of
charge density wave is notably different from that of LSCO, in which the presence of
a magnetic field would compete with the existing spin density wave order; here, local
probes show no strong magnetic order as is seen in LSCO[50], and instead the addition of a magnetic field suppresses superconductivity, which competes with charge
ordering. Unlike the long range order in LBCO, the CDW has a coherence length
4 30 unit cells, and is seen along both axes, with a slight difference in the ordering
wavevector.
ARPES measurements on YBCO can be taken, although obtaining a good surface is difficult and requires the deposition of potassium after cleaving. Tracking
the change in the band structure with doping in these materials reveals a strong
Fermi surface in the overdoped regime which with decreasing doping gets gapped out
around the antinodal (7r, 0) region over a scale which grows with decreasing doping.
This loss of Fermi liquid behavior and strong damping away of the Landau quasiparticle spectral weight is accompanied with a changing character of the gap above the
superconducting transition temperature from a clear sharp d-wave which is nodal at
30
a single point to a more parabolic structure in the pseudogap state. The ARPES
description shows that not only is the coherent peak of the carriers lost, but so is
the net spectral weight in this whole region, indicating a breakdown of the Fermi
surface and potentially the conservation of the size of the surface under interaction
(Luttinger's rule). This gapping out of carriers in the antinodal region of the Fermi
surface is consistant with coupling of electrons to a local density-wave state, as seen
by the aforementioned photon scattering measurements.
Measurements in YBCO of phonon anomalies have focused on the 'half-breathing'
mode[51] and buckling mode [52]. These both correspond with Cu-O bond motions,
and are seen to soften with decreasing temperatures. The breathing mode indicate
softening at qj= (0, 0.3) while the corresponding (0.3, 0) phonon is unchanged. These
softenings set in around 1OOK; this tendency to nematic electron order is in agreement
with the nematic nature of spin ordering and strong orthorhombic crystallographic
asymmetry. Instead, this is interpreted as tendency in the dynamics toward formation
of a one-dimensional CDW and gap opening at the Fermi surface.
For the half-
breathing mode, the phonon is seen to dip along both tetragonal axis, but soften
upon cooling only along b*, consistent with the measurements of the buckling mode.
While in the phonon response a strong difference is seen between the axis, the Fermi
surface is still predominately d-wave like[53], although obtaining a good surface is
difficult [54].
Interestingly, while elastic measurements show lattice distortions along both directions and phonon measurements only show coupling along b*, neutron scattering
data shows formation of a static SDW along a*. [55] This state is seen to be enhanced
upon application of a 15 T field; Fermi surface reconstruction driven by this SDW
then is a possible explanation for the observation in quantum oscillation of a Fermi
surface at -60 T. While enhanced at lower doping concentration, there is a region
of coexistence of the SDW and superconductivity. This effect was first reported as
staying dynamic[56], but with superior energy resolution is seen to be static within a
scale of 1 pLteV. The dynamics are seen to be broad along the a* axis on which at low
energy the SDW is formed; this scatter then plays the role of low energy spin waves
31
from this spin ordering peak.
A final neutron scattering study on YBCO used polarized neutron scattering to
resolve spin flip events coincident with the Bragg peaks[57]. Corresponding to magnetic ordering that did not break lattice translation, this is referred to as 'intracell'
or 'Q
0'
0 scattering. This scattering has a large component out of plane, but is also
also strong in plane, with a canting angle approximately splitting the two. The presence of such scattering is consistent with in plane magnetic moments, such as would
be produced from either Varma's current loop proposal or decoration of the oxygen
with static moments, however, the strong in plane component is surprising. Scattered
intensity decays with temperature consistent with the pseudogap, and most importantly, is absent in the overdoped case. This measurement, repeated in several other
members of the cuperate family, is one of the critical probes of the order parameter
associated with the pseudogap itself rather than the coincident formation of density
wave order. Such ordering maintains lattice translation, but breaks time reversal and
inversion symmetry[58].
1.3
Details of the BSCCO Family
BSCCO naturally cleaves easily between layers of Bi-O bonds, resulting in accessible
surfaces for ARPES and STM measurement, but rendering growth of large single
crystals difficult. Recently, neutron scattering measurements of spin waves have been
taken on Bi 2 +,Sr2 -,CuO
6 +6
[59].
While no significant elastic peaks were resolved,
a similar pattern of fourfold magnetic peaks in the inelastic scattering was seen,
comparable to the description of LSCO. This indicates spin and charge separation of
a similar sort, where local antiferromagnetism persists in stripes of singly occupied
sites, separated by hole-doped rivers of charge. Critically, the wavevector of these
incommensurate spin peaks scale with increasing doping, q Oc p for q the displacement
of the peak off the (7r, 7r) ordering wavevector and p the hole concentration. While the
observation is for magnetic order, it is believed the charge order obeys similar scaling
due to analog with the scattering in LSCO in which the relation qc0
32
=
2qm, is well
established between charge and magnetic order respectively. At x = 0.4, ordering
was only seen along (7r, 7r), absent at (-wr,pi), where in this case the orthorhombic
superstructure lies along (110). As expounded on in Section 2.1, this is very close to
a structural Bragg peak due to the second harmonic of the superstructure, and the
assignment of magnetic ordering to excitations originating here is weaker than those
found along the (110) direction absent superstructure.
The measurement in neutron scattering is contrasted with prior work from STM
suggesting formation of a charge density wave in the underdoped state of PbyBi 2 -y-xLaxCu0
[60]. By computing the Fourier transform of the density of states as a function of
tunneling bias, a non-dispersive feature was observed that scaled with doping in the
opposite direction as the doping dependence of typical coupled spin charge segregated
SDW/CDW structures a seen in LSCO; instead, at higher doping, the wavevector of
this feature scaled to lower momentum. This is consistent with Fermi surface nesting
producing this oscillation, since as the filling fraction increases the parallel segments
near the antinodal regime get closer, as illustrated in Figure
1-2, taken from [60].
Tight binding calculations, on which this is figure is based, suggest that while the
antinodal segments are not parallel, there is a reasonable component close to parallel. The wavevector that couples these Fermi surface segments is equivalent within
modeling error to that for which STM observes charge oscillations, and hence it is
expected the underlying lattice distorts in accord.
Since the nesting wavevector is set by doping, as the number of holes varies on
a microscopic scale, so too does the nesting wavevector.
By carefully examining
the lineshape of dI/dV curves taken from STM and masking the image to segregate
regions of similar hole concentration, follow-up STM work showed that beyond sample
dependence, the wavevector was in fact driven by local hole concentrations, which had
average coherence size on the order of several nm[43]. Since the doping is chemically
determined on a local scale by the presence of La impurities as annealed at high
temperature, there should be no coherence between the pockets of similar doping,
hence similar wavevector, between layers. Thus in the 3D material, charge ordering of
this form should present rod-like scatter, incoherent (or possibly short-range ordered)
33
6
(gc/a0, K/a0O)
(x /aO)
.
qcow
I
Figure 1-2: Tetragonal BSCCO Fermi surface nesting, dashed red lines indicating a
smaller number of (hole) dopants and hence a larger qCDw nesting wavevector.
along the c*-axis
The same checkerboard order was seen in STM on n=2 BSCCO [61]. Here, and
as later confirmed with better resolution across a wide range of doping concentration
[62], the same non-dispersive excitation is seen around a wavevector (±27r/4.5,0)
and (0, ±27r/4.5). However, the predominant feature noticed in Bi2212 is dispersive
quasiparticle excitations; these are well explained by the 'octet'[63] model in which
dispersion is due to interference from the 8 symmetry-related points of maximal density of states. This observation implies the existence of segments of the Fermi surface
in the highly underdoped regime in which well defined electron quasiparticles can
scatter; in contrast, the excitational spectra from the antinodal regime shows loss
of coherence of this structure. The temperature dependence of the charge order in
STM is in agreement with the pseudogap temperature scale [64], and furthermore
34
shows strong enhancement at the 1/8 doping fraction. The similarity to doped LSCO
in which 1/8 is a special doping concentration particularly likely to form static order suggested the role of stripes in this material; in general, the issue of stripes or
checkerboard ordering is not well resolved from STM measurements since the presence
of locally coherent domains implies that the transform of a multidomain window ~
600A will result in approximate fourfold symmetry regardless if the underlying state
is twofold or fourfold symmetric. A final important discovery of this work was that
proceeding to lower dopings further into the Mott state did not enhance the CDW;
while the pseudogap is a prerequisite of charge ordering, it is not determined solely
by the magnitude of this charge ordering vector as resolved at the surface. Examining
the electron dynamics as a function of tunneling bias shows that the observed lineshapes are consistent with charge-density modulations over a range ±40 meV, and
quasiparticle scattering off impurities cannot reproduce these features[65].
The electronic state is also seen to break C4 symmetry; by comparing the ratio
the Fourier transform of dI/dV at positive and negative transfer at reciprocal lattice
points (27r, 0) and (0, 27r), a difference is observed absent from the topography [66].
The coherence length of this nematicity is approximately 4 nm at low energy transfer,
similar to the size of the pseudogap binning mask used in Bi2201 to select regions of
similar local doping. The correspondence between nematicity and the intrinsic low
energy scale confirms the absence of topographic features such as the superstructural
modulation from producing this effect; any symmetry breaking due to this mechanism
would be expected to be single domain across the sample for all energies. At energy
transfer comparable to the gap, the domain size of such nematic order becomes large
compared with the size of the localized charge (single domain over a window of ~ 100
nm). This indicates that while the Fermi surface is varying on a scale of several nm,
coherent features can emerge from this disorder to persist across a range of dopant
concentrations. This order does not break lattice translation in contrast with the
previous STM density-wave features and seemingly originates from a different mechanism. This feature is thought to be in correspondence with the polarized neutron
scattering measurement of magnetic order in YBCO[57].
35
Another source of intracell ordering is the observation of inversion symmetry
breaking in the 2D plane by displacement of the Bi atoms at the surface in n=2
BSCCO[67].
However, this distortion was seen to persist independent of field and
doping. By measuring overdoped samples, it is seen that inversion symmetry is broken at the surface even outside of the pseudogap regime, indicating this transition is
uncorrelated with the pseudogap regime. A final structural comment from STM measurements is that the value of the superconducting gap is modulated by the phase of
the superstructural modulation[68]. The implications of this are that the superstructure, while widely ignored, plays an important role and is responsible for driving the
amplitude of the superconducting gap by 10%. Our work will later show significant
differences in the presence and absence of the superconducting gap; by measuring
locally and comparing phase of electronic phenomena with the superstructure, this
work was one the first probes to show that an effective field theory ignoring this feature is a significant oversimplification; electron-phonon interaction and the effect of
structure is generally strongly relevant and nontrivial.
Photoemission results are largely dominated by similar sharp features to what is
already mentioned from STM; the dominant excitations are quasiparticle like around
the nodal region, and the antinodal quasiparticle excitations lose coherence and get
gapped out. This is seen for both n=1[69, 70] and n=2[71]. The primary character
of the superconducting region is as a d-wave gap [72, 73]. Two gaps can be resolved
by considering the temperature dependence; the superconducting d-wave component
looses character with Tc, while the antinodal structure reverts to that of a free Fermi
liquid at a higher temperature, approximately the pseudogap temperature T*[18].
The gapping of the Fermi structure in the antinodal regime is consistent with the
formation of charge order such as a CDW. Importantly, the maxima in the dispersion
of the gapped out low temperature electron coherent response peak is shifted from the
location of such maxima at the Fermi surface at at high temperature to a different
wavevector, indicating the presence of an additional length scale in the problem,
which is inconsistent with a purely quantum critical or electron-electron driven loss
of electron quasiparticles. This is given the name of broken particle-hole symmetry,
36
and is clearly understood as introduction of an additional length scale upon cooling
into the pseudogap. The falloff in the antinodal region of the quasiparticle excitation
energy at the Fermi surface vector determined from high temperature is in exact
agreement with polar Kerr measurements[74].
The polar Kerr effect implies time
reversal symmetry breaking or formation of gyrotropic order[75]; that such a measure
is enhanced exactly with the ARPES signal implies the same order parameter is
responsible for both loss of electron coherence and gapping out the antinodal charge
degrees of freedom, and breaking time reversal symmetry. A simple state that breaks
time reversal symmetry is that of a ferromagnet, in which a single spin orientation
is selected.
However, neutron measurements searching for static spin order have
in general been unsuccessful, and the typical splitting due to local field effects' is
absent in local probes[76]. Recent NMR measurements even suggested formation of
charge order in YBCO absent corresponding spin order[77]. The current loop proposal
previously mentioned features this characteristic absense of local moments on the
Cu sites[58]. In either case, this is indicative of simultaneous charge and magnetic
ordering transitions, which is surprising emergent behavior since these degrees of
freedom might have been expected to separate.
This measurement is however, controversial. Both ARPES and the polar Kerr
effect are surface sensitive. Taking simultaneous LEED and ARPES data sets as
a function of temperature, it was shown that at the surface there is a modulation
along the orthorhombic b* axis with varying period as a function of temperature
which shares the same temperature dependence as the ARPES measurement of the
decay in the electron coherence peak in the antinodal region upon cooling. Using
hard X-ray on the same sample this modulation was seen in the statics to represent
rod-like ordering. with period ~ 8, which was found to be invariant as a function of
temperature [78]. This was interpreted as suggesting the temperature dependence seen
in ARPES and the gapping out of electron degrees of freedom in the antinodal regime
is a surface artifact driven or coexisting with a lattice modulation that in the bulk
has no temperature dependence, suggesting some measurements of the pseudogap are
only sensitive to surface effects.
37
1.4
Thesis Outline
In Chapter 2, we describe the experimental setup responsible for producing the inelastic X-ray scattering data which this thesis is based on. A brief theory interlude
is presented to explain the scattering cross section and how phonon excitations are
formed and picked up by the photon.
Measurements of phonons are presented in Chapter 3, detailing one of our primary
results of anomalous broadening of the longitudinal acoustic phonon. Comments are
made on the polarization of such scatter and presence of additional modes, and the
lack of inversion symmetry in the data shown.
In Chapter 4 we discuss measures of broken symmetry from asymmetry in phonon
scattering at positive and negative energy transfer. Implications for this in the ground
state under discussion in literature are presented.
38
Chapter 2
X-Ray Scattering
The first measurement of crystal structure taken with X-ray photons was done by
Max von Laue in 1912. The insight leading up to this was that photons interact
with electrons in the charge density providing a mechanism for sensitivity of light to
the periodicity of matter if the light was of the correct periodicity. The conversion
between length scale and energy is expressed as hw = hck = hc27r/A, implying the
X-ray regime of order 10 keV covers wavelengths of approximately A length scales
for which scattering at atomic sites separated by typical crystal bond lengths can
contain constructive scattering terms. Early instrumentation reaching this regime
applied a high voltage between a cathode and anode end of a tube, with the voltage
driving both emission of an electron from the surface and subsequent accelerated of
the electron across the tube, emitting photons via Bremsstrahlung. This technique
was surpassed by second generation light sources, which employed a synchrotron ring
to maintain a constant traveling current, which as it is bent into a circular path,
emits extremely bright radiation. To generate light specifically at a beamline station,
the path of the electrons can be perturbed by application of a varying magnetic field
to introduce local sinusoidal oscillations in the path, resulting in locally enhanced
photon emission; the devices that are responsible for this are referred to as wigglers.
The scattering data presented in this work is from the Advanced Photon Source
at Argonne National Laboratory. The steady state ring current is 100 mA, delivering
103 photons/sec to each of 30 beamline sectors. Each sector is designed to operate
39
Incident Energy
Flux
Momentum Res
Energy Res
Momentum Range
23.7 keV
2*109 photons/sec
0.065 A-i
1.5 meV
0-+ 7.5 A- 1
Energy Range
-200
Beam Size
Num. Detectors
35 pm*10 pm
9
-+
200 meV
Table 2.1: Characteristic measures of the HERIX spectrometer at Sector 30, APS
in separate energy and momentum resolution regimes, providing a strong degree of
flexibility to facilitate experiments spanning soft condensed matter and biophysics to
the energy-resolved hard x-ray regime and resonant scattering. The HERIX spectrometer at Sector 30, the beamline site from the data in this thesis was taken, is
characterized by the description in Table 2.1.
The critical technical feat of this spectrometer is to obtain energy resolution of
a part in 107; this is done by a series of temperature-controlled monochromaters. It
is worth comparing this scattering with two other photon probes: energy unresolved
x-ray diffractometers, and Raman scattering. Typical X-ray diffractometers such as
used in powder diffraction to determine the structure of a crystal integrate differ in
several ways: they are a several orders of magnitude less bright, the sample is not
mounted on an Eulerian cradle with full configurational degrees of freedom, and the
final energy is not resolved. While this tool is excellent for structural determination
for which intensity and resolution demands are not that high since the scattering
under study is the strong coherent elastic Bragg scattering from the static crystal
structure, this measurement cannot resolve phonon structure. On the other hand,
while Raman scattering measures in the the optical regime and is capable of measuring
phonons (and other excitations capable of changing the photon energy) with superior
energy resolution (sub-meV) to that available at Sector 30, these measurements are
exclusively at a momentum transfer very close to zero, since the wavelength of visible
light is hundreds of nanometers, corresponding to a wavevector that is small compared
with the inverse angstrom length scale over which the Brilloin zone spans. Reaching
40
both the wavevectors of interest for condensed matter crystal physics and resolving
dynamics on the quanta excitational scale of meV requires use of inelastic neutron or
x-ray scattering. X-ray scattering has the additional benefit that the small beam spot
allows high quality data to be obtained from a small single crystal, motivating its use
in this work in which large single crystals as a function of doping are not available.
Figure 2-1: Single crystal monochromater
The features of Sector 30 that permits these unique measurement are a series
of extremely well controlled monochromating crystals. As hinted at by the name, a
monochromator is a set of crystals that pass light after several Bragg reflections; however, the light that exits is tightly controlled in angle by the geometric constraints
that it reflects from the aligned single crystal faces and exits the monochromator.
An example of a monochromator is shown in Figure 2-1. Hence by controlling the
monochromator structure carefully, photons outside a certain energy range can be
selectively excluded from transmission. In Sector 30, there are two sets of monochromating crystals, corresponding to progressively tighter windows of energy resolution.
A full setup of the beamline is shown in 2-2. The motivation behind having these
seperate is that it is important in getting meV resolution to have control of the lattice parameters of the monochromator crystal to the order of 0.01 K. It is difficult
to control temperature this finely while discarding all but one in 10' photons, as
done in Sector 30 to obtain the desired sharply peaked resolution function. The first
41
monochromater, denoted as the high heat load monochromator, is water cooled and
produces a energy resolution of 6E ~ 1.6 eV; the HERIX monochromater then pares
this further to 6E
=
1.1 meV using nitrogen cooling to fix a temperature of 123K.
The analyzer after the sample is set up in a backscattering geometry as will be motivated below; this produces an energy resolution on final X-rays of roughly 1.6 meV.
A focusing mirror in the incident optics enables illumination of small samples, which
also functions to exclude background intensity from the sample holder since the incident light can be tightly controlled and spatially separated from such externalities.
Lastly, we note that nine such analyzer crystals are set up, each measuring a 20 value
separated by 0.650 in the scattering plane.
CdTe detector
Bimorph focusing mirror
Beam size (V x H)=
15 p~m x 35 gm
High-heat-load
monochromator C (
AE
~
1.6 eV
monochromator
AE= 1.1 meV
S Sample ~High-resolution
working at T= 123 K
oo11)
Be cmnd
refractive lens
Si (12 12 12)
analyzer
Figure 2-2: Sector 30 layout
A schematic of the interior of the analyzer arm at the far right of 2-2 is shown
below in Figure 2-3. This diagram depicts scattered light incident from the sample
in green passing through the nine pinhole windows analyzer and proceeding down
the analyzer arm before being backscattered at the Si (12 12 12) reflection. Light
after this backscattering event is shown in red, which is then measured by a bank
of detectors immediately past the pinholes, shows as the terminal point of the red
columns in the innermost inset.
The backscattering geometry used in this beamline is necessary to reduce the width
in energy of photons scattered off a given Bragg reflection, under careful consideration
of the error. To compute the energy acceptance, we first write Braggs law for elastic
42
Figure 2-3: HERIX analyzer arm
scattering,
A = dsinO
(2.1)
where d is the inter-plane spacing of the relevant coherent scattering planes, determined in this case by the Si (12 12 12) indices, and 0 is the angle from the surface at
which light is diffracted. The width in energy produced by this backscattering event
can be determined by computing the derivative of this form,
6AO
A
__
6d
6E
-=-+50cot0
d
E
(2.2)
For HERIX, the exact energy used was 23.7253 keV, which corresponded to
backscattering at an angle of 0 = 89.590.
With a minimal achievable angular ac-
ceptance determined by rocking scans of 60 ~~200prad[79], we then have a minimal
energy tolerance of the backscattering analyzer of 1.6 meV. In design of a spectrometer, it makes sense to have a comparable width of incident and final energy resolution,
since the total instrument resolution is in effect determined by the convolution of these
43
terms. In practice, the final resolution determined is around 1.5 meV, indicated the
angular mosaic of these crystals is slightly superior to 200 prad. This angular dependence is due to two terms, the intrinsic Darwin width due to the effect of absorption
and interference between multiple scattering events, and a finite size crystal effect
due to cutting the analyzer crystals into relatively small domains so as to spherically
focus light into the detectors.
2.1
Crystal Structure
The crystal structure of Bi2201 and Bi2212 are shown in Figure
2-4. While the
literature is almost exclusively in terms of the tetragonal zone - eg the antiferromagnetic instability is at (a*e + b* t)/2, we here switch to the cryptographically correct
orthorhombic notation so as to facilitate careful discussion of symmetry. For the rest
of this thesis, coordinates in momentum space will be given in terms of orthorhombic
relative reciprocal lattice vectors, and this antiferromagnetic instability is now at aorth
or brth. The conversion is given by
a*=a
atet
orth
± bthr
We now discard the orth subscript. The perturbation to this structure is the
presence of an incommensurate superlattice modulation with reciprocal lattice vector
=
+0.22b* ± c*. All four sign combinations are observed, as are mixed second
harmonics showing that the relevant Fourier decomposition for a single crystal domain
contains both modulation directions; evidence for this is shown in Section 3.1.4. This
modulation is thought to be driven by the mismatch of bond lengths between the Bi-O
layer and the Cu-O layer; the solution is that the crystal buckles such that the Bi-O
sites are perturbed by to reside out of a fixed c plane, effectively increasing the average
bond length. This also allows for incorporation of an additional oxygen atom[1] into
the Bi-O layer, where the additional oxygen is in a bridging configuration between
44
BiO
SrO
*,
-
Bio
CuQ 2
Ca
SrO
CuO 2
CuO
w
(N
SrO
SrO
RiO
---
... .m. r---
BiO
*
II
U
--
*
0
*0
BiO
Bro
II
*
-
SrO
-
-
CuOQ
Ca
SBO
CU02
BiO
--
,
~BiO
h
a=
.
5.4A
Figure 2-4: Bi2201 and Bi2212 unit cells
successive rocksalt-like BiO layers. This modulation is single domain over the entire
crystal. The modulation is of equivalent wavevector between Bi2201 and Bi2212;
further discussion of the exact position of shifted atoms is continued in Section 3.1.4.
An example of a single crystal measured in this work is shown in Figure 2-5. The
cleaving axis is along the crystallographic a* b* axes (perpendicular to c*); looking
closely at the surface, lines running parallel to the superstructure direction b* are
visible by eye. All samples measured were visibly single crystal as shown.
In Figure 2-6 we show the loss of symmetry under successive distortions, from
the tetragonal to orthorhombic to the addition of the superstructure. Note that the
45
AL
Figure 2-5: Single crystal Bi2201 UD31K. Scale bar is 2mm
X-ray refinement[1] of the superstructure indicates this removes the crystallographic
inversion center, as well as breaking the mirror plane along b* and leaving only a
series of twofold rotations (ellipses in the figure) and screw axes (ellipse with wavy
screw lines). For the orthorhombic case there is neither a fourfold rotation about the
origin nor a reflection plane along (110) hence even at this level of crystallographic
complication this is not an exact high symmetry direction.
2.2
Scattering Cross Section
It is convenient to introduce notation concerning the change in momentum and energy
of the photon (for the sample, these changes are of the opposite sign),
hQ
-
hkf - hki
hw = Ef - E
(2.4)
Now via Fermi's golden rule (with h = 1), we can write the probability to transition
from the initial (Ai) to final (Af) system state while the incident x-ray changes from
46
Space Group 66, Cccm
Space Group 139, 14/mmm
C j C ii
K
4
k
144
U
-f
4
4
t4
%
Space Group 21, C222
A222
t
P
t_P
4
t
-67
a
C'4*
--
Figure 2-6: Symmetry of tetragonal 14/ mmm, orthorhombic Ccem, and enlarged
supercell C222[1]
ki -+ kf and E, -* Ef,
PAf,kfAj,k
jkfAf)
-- 27r (kj A
12 6(Ef
-
Ej + w)
(2.5)
Let n index the electrons, localized at positions 'a.
(kjjY|kf) =
V (f
n,T
dr-V(-) e
) e~-;k-
(2.6)
'
Since the total probability of a photon scattered into a final solid angle Q and
47
energy Ef is the sum of scattering over all final sample states, we can write
Pkf kA
Af)(Aje-iQ"|nAi)6(Ef - Ei + w)
= 27rlV(Q)12 ZAile
(2.7)
Af
We can rewrite the 6 function as
6(Ef - Ej + w) =
dt ei(E-Ei+)t
(2.8)
Representing each position operator R,, as a Heisenberg operator, , we can now
use the transformation on the 6 to write
Pk,_As
(2.9)
()-)
= 27rlV(Q)2
The interaction between electrons and photons is given by the Thompson cross
section,
V(Q
V(Q)
=e
9
-
mc
27rhc
yr
V
kc
(2.10)
W
The partial differential cross section d- for scattering photons into a solid angle
dQ with energy within a window dEf of Ef is given by
(d~dEf
d
2
(2.11)
Pkf,--kp(Ef)
I0
where the density of states p is given by
Vw 2
p(E 1 ) = 8 2 hc3
(2.12)
Thus in total, we have
( dQdEf)
d
E
2
m2c4 (
)
ew
27ezejWt
* fdt
n(o)e-Qrm(t))\
(2.13)
n,m
where for these experiments the incident beam is unpolarized and there is no
selectivity of the final polarization. Since the predominant scatter is from the electrons localized at atom positions which move together, we replace the indexing over
48
electrons r, with that over atoms, R,, where n now runs over unit cells and
T
over
the basis atoms in such a cell, where each atom is then located at a position
RnT.
Let there be N, electrons on the atom at site R, 7 . The sum over all these electrons
localized around R, 7 can be written as
f (Q)eQR
=
-
J
(2.14)
drea'ap(r)
N,
Where by p and rN, we refer to the localized electrons in a cloud around the atom
centered at R, 7 . The form factor
fO(Q)
is close to the number of electrons and is
relatively unchanging in the energy range used for this work. Far from the absorption
edges, absorption is a relatively small cross section and we thus drop the labels 0,
Q
to write fT which produces the scattering equation
d2 U
.d~dEfJ
-
m
2
c4
f o dt
2
_Jo 2ir
ffe
f-fn(e
EI
ln(O)6
i(tM
(2.15)
(e
n,r,m,?7
The two exponentials can be combined by the identity (eAeB)
-
e((A 2 +2AB+B 2 ))/2,
where this is true for all operators A, B that are linear in creation and annihilation
operators (such that the higher order commutators vanish). The terms a e-x 4i-r2
thermal average to produce the Debye-Waller factor, leaving the final scattering form
(d~dEf
Tm2C4
[-00
C
2,
n,r,MJr/
We next series expand the exponential, which amounts to an expansion in number
of phonon creation or annihilation event. Let us write ]n, (t)
=
n+b,
+ un7 (t) where
the motion is given by the local displacement UnT. The zeroth order terms are elastic
scattering:
49
e 4 e-2wN
d2a.
d
d2a.
e4 e- 2WN
2,r
y
fwt
fdei-.+N-e-N-en)
eiwt
0
-
fdEf M2C4
nrm,77
eQ( -- 60)f, f
6()6(Q - G)
(2.17)
Single phonon scattering is given by the form
dE
2U4
dQdEf/
m 2 c4
e 2W
]
2reiWt
ffei-+t~,b-mb)(Q
-un,-(0) * Q Umn(t))
(2.18)
We now take a brief aside to derive the photon eigenstructure. Given a crystal
in the previous notation, every atom at zero temperature is in an energetic minima,
hence the derivative of the overall energy with respect to each of the position coordinates is locally zero, and the energy can be written to second order in displacement
as
E = Eo(0) + 2
Dnvamv/aIUnvaUmvIn
(2.19)
where the subscript a denotes polarization eg {x, y, z}. We next write the force
equation on each atom, then search for a solution that is periodic, eg
MUnva
Unve~M
-MvW2 Av,c
mE' D&
= A , (q',w) e i
=
E
vaUMVIa
(2.20)
"-Wt)
(2.21)
Dnvce,mvtafe-'4 f-r")ALi,
(2.22)
The final equation of these is recognizable as a secular equation for A; nontrivial
solutions result only for solutions to the inhomogeneous determinant equation
||Dva'ain(q) --
Mvw 2 6aa 6vVI| = 0
50
(2.23)
This is the formation of eigenvectors A(q, p) characterized by eigenfrequency Wo,.
Given a complete expansion, we can then write the position of an atom in terms of
these as
Unr
=(
NM
,
±
&_
_' )
(2.24)
This result can then be substituted in Eq 2.18, where in the Heisenberg representation the creation and annihilation operators pick up time dependence ehiw(0).
After simplification, this yields
/( ~
Sd2dEf/
Tn 2 C44 e 2W E
m c
frfOQaQ,3
nmrn
(eflw(q-
AaqA~p3
qjp
1 6(w
27M7Mnwp(q-)
w(q')) +
+
-
6(w + w(q')) .25)
This concludes the review of scattering formalism. Note the first term in parenthesis corresponds to photon energy gain, while the second to photon energy loss;
hence the first corresponds to annihilation of a phonon in the sample, while the second creation. The amplitude of the creation event is larger, as would be expected.
The term ei((+T("-Fm) implies the momentum added to the phonon system is equal
to the momentum lost from the photon system, up to a reciprocal space lattice vector
G. The last insight to be gained from study of this equation is to note that the product of QaAo, implies phonons with eigenvectors along the direction of momentum
transfer are selectively picked up.
This thesis concerns phonons measured along high symmetry directions. Along
each high symmetry direction, the phonon eigensystem can be decomposed into longitudinal modes p' such that q' 1
,, and transverse such that
I_
,,, where for the
cuts of interest along (110) the transverse modes will be of separate c* polarized or
in plane transverse character. Given such a decomposition, we will refer to phonons
more simply as polarized along a direction r since this decomposition implies such
eigenvectors can be chosen. The mathematical reasoning that implies this is that
symmetry operators which reflect about these axis commute with the Hamiltonian,
51
hence the dynamic matrix is piecewise diagonalizable, and the resulting eigensystem
will have such separate character. In the presence of orthorhombic distortion, some
of the tetragonal symmetry operators are lost and certain high symmetry directions
cease to be; however, such effects are likely higher order and it is expected the modes
retain predominant character as described by the symmetry operators of the tetragonal structure. More on mode polarization will be discussed in Section 3.3.2.
While it is more convenient to motivate the cross section in terms of the photon
and its energy loss or gain, we will now switch convention to considering energy and
momentum transferred into the sample, which is the opposite sign of the convention
in Eq 2.4. Thus w > 0 will now correspond to a phonon creation event.
2.3
Experimental Configuration
The samples are mounted on a displex, hung in the center of a 4-circle goniameter
that is also aligned with the incident photon beam. Three of the angular degrees
of freedom originate from the Eulerian cradle, shown in Figure 2-7. Each sample
was mounted in transmission geometry, and attempt was made to align the crystals
such that the Eulerian cradle was at
#
0, X = 0,6 = 20/2 for all Bragg peaks in
the (HHO) zone. The 9 = 0 angle is defined such that the x circle is normal to the
incident beam; the X = 0 angle is defined such that the displex arm from which the
sample was held from is vertical, pointing down from the top. Motors were movable in
a range 0 E [-15", 150], q E [-604, 600], X E [-200, 600], 29 E [0, 33.5], the bounds of
which were set such that the displex and 6 arm did not collide with other components
of the diffractometer.
Six total samples were measured, with characteristics listed in Table
2.2. Es-
timates for the value of T* are taken from the literature [80, 81, 82, 83]. Samples
were generally of size 0.5 mm x 2 mm x 0.05 mm, where the penetration depth of the
23.7253 keV X-rays used was calculated to be 35pm [84]. These were chosen to span
a range of doping in the presence and absence of lead doping, and to match thickness
slighly greater than one penetration depth. The beam was aligned on the center of
52
Figure 2-7: Eulerian cradle of four-circle diffractometer
the sample by translation of the entire goniometer/sample table while monitoring the
direct beam transmission. at 0, 20 = 0. The key understanding is that transmission
through air leaves the beam unattenuated while the sample cuts the scatter down by
about 2/3 and the copper post the sample is affixed to fully blocks the direct beam.
In horizontal scans we then look to center the sample on the region of a 1/3 dip,
while in vertical scans we look for the midpoint of the 1/3 region between the full
beam and complete attenuation. This process puts the beam in the center where the
crystal quality is likely the highest and further from possible strain effects near the
mounting region.
To mount in the (HHO) plane then, assuming one of the orthorhombic axis is
vertical and the back is flush with a sample post, this crystal was rotated 450 about
c* prior to being affixed to a copper post (mounting done with GE varnish). This
results in the long edge shown in 2-5 protruding at a 450 angle from the horizontal
plane, implying the sample is aligned in the (HHL) zone. This sample and post was
53
Name
Bi2201 UD25K
Bi2201 UD31K
Pb,Bi2201 UD27K
Pb,Bi2201 UD33K
Bi2212 UD50K
Bi2212 OD70K
Composition
Bi 2 Sr 1 .6 La 0 .4 CuO 6 +6
Bi 2 Sr 1 .6 5 La0.35 CuO 6 +3
Bij. 35 Pbo. 8 5Sr 1.4 La 0.45CuO6 +6
Bii.35Pbo.8 5Sr 1 .4 7 La 0 .38 CuO6 +6
Bi 2 Sr 2 CaCu 2 0 8 +6
Bi 2 Sr 2 CaCu 2 0 8 +6
Tc
25K
31K
27K
33K
50K
70K
T*
250
150
200
150
250
-
K
K
K
K
K
Doping
Slightly underdoped
Underdoped
Optimally doped
Optimally doped
Underdoped
Overdoped
Table 2.2: List of samples measured
then attached to the end of a displex-cooled cold finger, with thermocouples attached
with Teflon tape to the copper mount. The sample was enclosed in three successive
Be domes, the innermost chamber of which helium gas was sealed inside for thermal
exchange, while the outer pair was maintained at vacuum as a heat shield. This
apparatus could vary the temperature between 350K and 6K; Beryllium domes were
used to minimize background scatter.
54
Chapter 3
Measurements on Bi2201 and
Bi2212
In this chapter we discuss scattering measurements taken on the samples. Motivated
by evidence from STM, ARPES, and elastic studies of other members of the superconducting cuperates, we first search for evidence of formation of a CDW around
(.25, .25, 0), or evidence in the dynamics of softening due to fluctuations tending toward this order. The X-ray cross section is predominantly sensitive to high atomic
number sites, implying these measurements are most directly a probe of the position
of Bismuth atoms (Z=81 for Bi, compared with Z=38 for Sr, Z=29 for Cu, Z=20 for
Ca, Z=8 for 0). Measurements taken in the statics are expected to be sensitive to
this since a static electron-density wave, if present in this material, should result in a
modulation of all atomic positions with equivalent wavevector. In the inelastic regime,
we primarily probe acoustic phonons due to their origin as a Goldstone mode around
the F point associated with bulk translation of all atoms; this implies the phonon
character at the F point is uniform motion for all atoms; throughout the zone, this
character evolves but by continuity this mode has the strongest tendency to retain
characteristic of all atoms moving in phase. This implies that the presence of a Fermi
surface nesting driven softening such as a Kohn anomaly[85], even through produced
by in this case electrons localized in the Cu-0 layer would affect the energetics of the
acoustic mode, resolved primarily through the motion of Bi atoms in the Bi-O layer.
55
This chapter proceeds as follows: in the first section, we discuss elastic scans, commenting on the strength of the superstructure in the bulk in presence and absence
of lead doping. We then present elastic measurements through several zones at high
temperature looking for signatures of charge density wave ordering. No such features
are detected, however, temperature-independent features are seen to grow with beam
exposure; these are discussed, and form one of the prime motivations for searches
in the inelastic, where subtle structural modulations have less dramatic effect. Inelastic scattering data shows the presence of anomalously broadened phonons along
the (110) direction with longitudinal polarization around the (.25, .25, 0) wavevector
which is interpreted in terms of effective self-energy increase due to renormalization
of the propagator by electronic degrees of freedom. Different methods of analysis are
presented, as are temperature dependent phonon scans. We conclude by examining
the polarization by measuring in several zones, and present calculations showing significant deviations from the shell model, highlighting the role of interactions beyond
simple local coulomb.
3.1
Elastic Scattering
Measurements taken at Ef = Ej with the HERIX spectrometer integrate over a
relatively narrow band of energies near w = 0 due to the tight monochromators previously described. Formally, the intensity observed is a convolution over 3 dimensions
of momentum resolution and one of energy,
I(Q, W)=
3.1.1
dqdr b(qg(r) d~df (Q -
w
-
'r)
(3.1)
Resolution Function
2
Elastic intensity contains terms for which dQdEf
d ,
oc 6(w); were
an inelastic scan to be
weea1nlstcsa.ob
taken of a pure elastic scatterer then, what would be seen in the resulting intensity
scan is the resolution function g(w). This is the motivation behind scans of Plexiglas
taken for several of the samples as an resolution function scan. Scatter from Plexi56
glas disordered is broad in momentum due to lack of long range order between the
molecules and sharp in energy at w = 0 since molecular motion is weak compared
with the broad elastic scattering. A scan is shown in Figure 3-1, which is fitted to
two phenomenological models: first, in orange, and as used for analysis of scans,
g(w) = A
2 /y
+21
r( 2 + 4W2)
-02
21
-, p
ILog[2]
v
7r
+C
(3.2)
60)
1
-20
-10
20
10
4)
20
15
10
5
-20
III I
10
-10
I
-
I
20
Figure 3-1: Inelastic scans through a block of Plexiglas, taken with Analyzer 5 prior
to measurement of UD50K Bi2212 and OD70K Bi2212
Alternately, and with slightly better fidelity to the data, additional high energy
spectral weight can be added by modifying the first term of this equation to be a
57
Lorentzian 2 / 3 , as shown in the purple dashed curve. We use the first labeled equation
in work here due to its prevalence in the literature; as the sum of a Lorentzian and
Gaussian term with relative weight w, it is called the Pseudo-Voight lineshape. Note
the resolution function scan does not contain significant weight from scattering from
Plexiglas dynamics (the disordered remnant of phonons, motion on a THz or meV
scale), as can be seen by noting the scatter for w > 0 is equal to that for W < 0. If
phonon scattering were relevant, it should have some sort of scaling between positive
and negative transfer corresponding to detailed balance, or the Bose occupation number (nrB(w) + 1) on the positive side vs nB on the negative side. The tails should be
particularly sensitive to this effect since the ratio of scattering between positive and
negative transfer diverges as nB
-
0; however, fitting the entire lineshape reveals no
regions of significant phonon scatter. This method of determining resolution function is used whenever available. Since the resolution changes between experimental
runs due to replacement of the analyzers with improved crystals, for this method to
be applicable the Plexiglas scan must be taken during the same run, which is not
always available. If no Plexiglas scans were taken, to estimate the lineshape with
good accuracy the data from each phonon measurement at least 0. 15rlu away from
the zone center and at energy transfer close to w = 0 (defined as the intensity falloff
at most 2/3 of the peak data) was fit to a Pseudo-Voight lineshape, and the resultant
parameters were averaged from all such phonon scans, which were in good agreement
with a momentum-independent constant. The energy resolution determined for each
sample (dependent on the analyzer crystals mounted at the time) are listed in Table
3.1.3.
3.1.2
Elastic Scans for Charge Ordering
An example of a elastic scans at is shown in Figure 3-2. Part A and B show data
from Pb,Bi2201 UD32K and Bi2212 UD50K respectively. We can see there is no
clear ordering peak emerging in the elastic. The background counts, approximately
107
times weaker than the main Bragg intensity, are predominantly due to crystal
disorder. If dopants or chemical disorder is present, the exact collapse that enabled
58
the formation of coherent Bragg scatter is disallowed; since the lattice ordering stays
static, the total intensity at w = 0 must be conserved and hence in the presence
of static disorder it must be the case that background picks up the spectral weight
removed from the Bragg peaks due to the random nature of the form factor due to
substitution disorder. In these crystals, the predominant form of disorder is substitution of Bi/Sr/Pb atoms in adjacent layers [86], which is estimated at approximately
5% in absence of intentional lead doping.
Let us turn here to study of the elastic scattering near the (220) Bragg peak.
Several cuts along the (110) and (110) direction were taken to look for evidence of a
static modulation peak. Sample scans are shown below in Figure 3-2. (A) and (B)
show respectively no signature of order in Pb,Bi2201 UD32K and Bi2212 UD50K; the
temperature dependence is in general weak and smooth through the proposed CDW
ordering wavevector of E ~ 0.25[60]. The weak peak seen in Bi2212 at E = 0.5 in the
longitudinal cut is reproduced in the scan shown in (C) taken near the same Bragg
peak but rotated by 90'; these scans are shown on equivalent y-axis scales and indicate
this order is anisotropic; however, this is seen to show no temperature dependence
and is not pursued further. Figure (D) displays an example cut through another zone
center, demonstrating no CDW peak near either the tetragonal forbidden (330) peak
or on the lower side of a (440) peak. E > 1 is not accessible here due to the limits on 20;
a weak Bragg peak is seen around (330) due to the orthorhombic distortion. Further
scans taken (not shown) confirm the absence of temperature changes at L = 0.5, L
=
1
for several samples.
Unlike prior work using higher flux diffractometers, the narrow resolution function
g(w) greatly reduces the contribution of phonon scatter to the measured intensity.
Typical diffractometers accept all final energy X-rays, integrating contributions from
all acoustic and optic phonons as well as the elastic scattering. The acoustic phonon
is of approximately half the intensity of the elastic signal, hence eliminating that
contribution alone provides a significant reduction in background. However, it appears
that either there is no systemic elastic CDW signal, or it is below the detection
threshold here (approximately 10' times weaker than the main Bragg peak).
59
(2+E,2+ E,Q)
A
5
0K
4
30
15 0K
6K
If
it
3
2
1
( 2 +E,
B
2
E
0.5
0.0
-0.5
+E,Q
20
15
10
5
-
C
40[
0.5
0.5
D
10)
(3+E,3+E,Q)
r-~
5
4
30[
3
20[
10
I6
0.2
0.4
0.6
-.
-0.5
00
0.5
1
1.0
Figure 3-2: Elastic scans searching for CDW signal in Pb,Bi2201 UD32K (A,D) and
UD50K Bi2212 (B,C). Data sets are color coded based on temperatures of 300K (red),
150K (purple), and 6K (blue).
60
Name
Pb,Bi2201 UD33K
Bi2201 UD25K
Bi2201 UD31K(OOK)
Pb,Bi2201 UD27K( 3 00K)
Bi2212 UD50K
Bi2212 OD70K(OOK)
a
5.319
5.386
5.404
5.399
5.401
5.399
c
24.17
24.08
24.69
24.61
30.757
30.658
Long. FWHM
0.032
0.024
0.020
0.045
0.010
0.016
Energy FWHM
2.3 meV
2.5 meV
1.5 meV
1.5 meV
1.3 meV
1.3 meV
Table 3.1: Summary of crystallographic, mosaic, and instrumental parameters.
3.1.3
Main Bragg Peaks
The momentum resolution of this spectrometer, unlike that of most spectrometers
designed for elastic measurement, is relatively broad, making this instrument a poor
determinant of lattice constants and crystal structure. Since the measurements of
concern are of phonons for which the eigenstructure is smooth in momentum space, we
do not make attempt to separately resolve orthorhombic lattice constants, and align
the sample at (220) and (115), facilitating determination of a ~~b and c. Example
scans through the (220) elastic peak are shown in Figure 3-3; note the mosaic is
tighter in the longitudinal than transverse direction, indicating that bond length
disorder is less relevant than disorder corresponding to relative rotation of the (110)
axis.
The resolution function and lattice constants for the six samples measured is
described here in terms of the full width at half maximum of the energy resolution
function and the longitudinal momentum cut through (220). The component of the
resolution function in the other two directions has only a second order effect on
phonon measurements along high symmetry directions since the eigenspectra obeys
reflection, implying that the derivative of the dispersion on both the off symmetry
direction is zero, and integrating over these components of the resolution ellipsoid
does to first order, nothing. Complete cuts through the crystal are shown in Fig 3-3.
All measurements are presented at base temperature unless otherwise noted.
The listing in Table 3.1.3 is in order of experimental runs conducted. The improvement in energy resolution is due to replacement of analyzer crystals as fabrication
61
UD50K Bi2212 (220) Bragg Peak
Intensity (2+e,2-c,0)
2.5
2.0
d1.f
1.0
I'T
U
a-
±
-0.04
0.00
-0.02
2
0.04
0.02
2
Intensity ( -E, +E,O)
2.5
2.0
f
i
f
1.0
0.5
I
EU
-0.04
-0.02
0.00
0.02
0.04
Intensity (2,2,f)
2.5r
2.0
1I
II~
1.
1.01
0.5
II
I'
I
-0.04
I
I
-0.02
0.00
U.~UUk
0.02
I.-
0.0
Figure 3-3: Scans through three orthogonal directions through the (220) Bragg peak
in the UD50K sample at 300K.
62
technique improved, resulting in improved performance of the HERIX spectrometer.
Samples of Bi2212 were seen to be of superior crystallographic quality as measured
by the mosaic spread. For all crystals, count rates on the order of 800 counts per
second with four attenuators on (corresponding to 8 * 106 unattenuated counts) were
measured at the (220) Bragg peak and on the order of 400 counts per second were
observed at the (115) Bragg peak. The (115) is nominally a more intense Bragg peak,
but the alignment of the crystals in a transmission geometry implied the scattering
path of the (220) reflection passed through a lesser length of sample than (115), hence
resulting in less attenuation.
3.1.4
Superstructure
The superstructural modulation is associated with a buckling of the layers to accommodate energetic preferences for bond length differences between layers and incorporate additional oxygen. A similar modulation is seen in both Bi2212 and Bi2201,
characterized by a wavevector qj = t0.22b* ± c*. Early work characterizing this by
refining X-ray diffraction patterns showed that the presence of lead substantially effects the atomic positions inside the cell, corresponding to displacements of Bi and Sr
atoms of up to 0.4A and Cu atoms by 0.3A[87]. Compared with ~ 2A bond lengths,
these shifts are very substantial even if distributed throughout the modulation period, although they are generally neglected in theoretical treatments. The structure in
the presence of modulation has been refined to a 3+1-dimensional Fourier model[86].
BiioSr 15 FeIOO46 is a structural analog, isostructural to Bi2212 except that the modulation is commensurate with the new unit cell elongated along b by a factor of 5,
This material was grown and characterized by X-ray scattering[1]; the refinement of
structural parameters indicated good agreement in terms of the locations of positions
of the Bi atoms with the prior work on the incommensurate crystal. The presence of
lead is known to suppress the superstructural modulation; our measurements show
this is between a factor of 10-100 reduction in amplitude to scatter into the elastic
superstructure peak q,. The use of lead is common in surface-sensitive electron spectroscopy, where the superstructural modulation is responsible for the visibility of a
63
host of 'Shadow' Fermi surfaces corresponding to the electron dispersion translated
by a superstructural modulation wavevector[33].
The superstructural modulation results in shifts in Cu-O bond lengths, and correspondingly affects the amplitude of the gap. By simultaneously tracking the superstructural modulation and the gap value, scanning tunneling microscopy work[68]
has been able to assign a 10% variance of the magnitude of the gap to structural
effects. Suppressing the superstructural modulation with lead undoubtedly effects
the energetics even in the Cu-O plane, but in an effort to reduce the high T, problem to a two dimensional carrier/temperature phase diagram, this effect is typically
neglected since comparable superconducting temperatures can be achieved under different amounts of donor substitution, allowing renormalization of this effect on T,
into one of effective carrier number.
STM measurements have been taken on Bi2201 and Bi2212 in the presence and absence of lead dopants. FT-STM continues to show modulation of the electron density
of states along the Cu-O bond direction [881. Interestingly, while the overall strength
of superstructure scattering is suppressed in the presence of lead doping at the surface
as well, it remains visible as occasional modulation lines along the orthorhombic b*
direction, spaced by large regions in which no evidence of the modulation is visible.
X-ray scattering scans through the superstructural modulation from the samples
studied here are shown in Figure
3-4. In A, a comparison is made between all
samples measured. Two features are evident from this comparison: first, generally,
the scattering on the side +b* is stronger than that of -b* (A only shows this for fixed
L = 1, but as indicated in C, this is true for L
=
-1
as well). This effect is stronger
for lead-free samples; the amplitude is almost equivalent within error for both leaddoped samples measured. Secondly, the wavevector of this modulation is not the
same for all samples. This observation is consistent with the periodicity being in
part set by incorporation of additional oxygen sites, the number of which is expected
to change between samples. Figure 3-4(B) shows cuts along (2 + , 2.24, 1) in blue,
(2, 2.24, 1 + E) in purple. These cuts through superstructure peaks in Bi2201 UD25,
indicate the modulation is inherently two dimensional. (C) shows cuts for L=t1,
64
(2,K,1)
100
A
+(2,2.24,O)
1000-
B
PbBi2201 UD32K
PbBi2201 UD27K *0.1
Bi2201 UD31K *0.01
Bi2201 UD25K *0.01
Bi2212 OD70K *0 01
Bi2212 UD50K *0.01
(100)
(001)
800
00200-
1.6
1.8
~2002.2
-0.10
2.4
(2,K,± 1)
C
E
0.10
D
D(2,KL)
if
700-
L=1
L=-1
0.05
0.00
-0.05
1000
600'
500-
800-
400-
600-
L=2
L=0
I
300I400-
200
1%
1001.7
1.8
1.9
20
2.1
2.2
-K
2.1
2.3
2 .2
2.3
2.4
2.5
2.6
Figure 3-4: (A) Cuts through the superstructural modulation along b* at L = 1 in all
samples. (B) Transverse cuts through Bi2201 UD25. (C) Evidence of four symmetryrelated first order peaks in Bi2201 UD25. (D) Second harmonic peaks in Bi2201
UD25.
indicating the superstructural modulation is not equivalent under reflection about
the a*c* plane through (220), though reflection about a*c* stays a good symmetry.
In (D) two second harmonics are shown; the blue data shows the second harmonic of
the (0, 0.24, 1) modulation, while the purple data shows the presence of a reflection
present by closure of (0, 0.24, 1) and (0, 0.24, -1),
indicating the two modulations
coexist rather than existing in domains. Translation of the entire sample by large
distances on the order of 0.5 mm was seen to leave the superstructure essentially
unchanged, indicating the modulation is single domain over the entire sample.
There have been few general comparisons of properties of BSCCO as a function
of lead doping. Crystallography revealed changed lattice constants by approximately
1%[89] accompanying the change in intracell atom location produced by removing the
65
modulation. Raman measurements on the Big phonon in Bi2212 show the narrowing
of a mode around 37 meV under doping of lead; the loss of intrinsic breadth is
associated with relaxation of the structure due to the addition of lead dopants[90].
This argument will later re-emerge in our measurement of acoustic phonons, which
will be seen to be similarly narrower in the presence of lead. Most calculations of
the electron-phonon coupling constant attempt to do so for dynamics in the Cu-O
plane, neglecting both the effect of atoms out of this plane and the modulation on the
Cu-O bonds; these calculations may hint at anisotropic coupling in different regions,
but cannot capture the differences between samples depending on the strength of the
superstructural modulation[91].
3.2
Inelastic Scattering
Inelastic measurement using X-rays is a relatively new technique, and the importance
and complexity of the instrumentation developed at the National labs to support this
cannot be overstated. At the Advanced Photon Source there are two sectors where
inelastic measurements are possible: Sector 30, where this work was conducted, has
been operational since 2008; and Sector 4, operational since 2000. Outside the US,
inelastic measurements are possible at Japan on SPring-8 and in the ESRF in France.
Of these, Sector 30 is the newest, featuring a superior energy resolution by at least a
factor of 2 compared with these competitors. The low energy resolution of Sector 30
optimizes it to measure low-energy excitations such as acoustic phonons in crystals
and slow dynamics in liquids. Measurement of optic modes is feasible, but intensity
rather than resolution is often more important in such studies, and this instrument
is then sub-optimal.
In this thesis we measure predominantly along (110), focusing particularly around
(0.25, 0.25, 0). Examining Figure 1-2, this wavevector corresponds to the tetragonal
(0.25, 0, 0) which links parallel sides of the Fermi surface in the antinodal regime.
A basic intuition for the motivation of this wavevector can be seen by drawing the
first correction to the phonon Green's function, shown in Figure
66
3-5.
The bare
phonon propagator is renormalized by scattering over all events in which the phonon
of momentum q is annihilated and a forward electron of momentum k + q is created,
with annihilation of an electron of k. For q = 2kF, for electrons near the Fermi
surface, the energy of such electronic excitations is gapless, hence the electron Greens
function diverges. This is the origin of an increase in response when the phonon
has enough momentum to couple sides of the Fermi surface. The phonon we want
examine then is not that located in the antinodal region of momentum space, but that
which couples antinodal Fermi surface arcs. The evidence for gapping in ARPES and
charge density wave order in STM are both consistent with static order here, while
our data will later show only a dynamic tendency.
To put this in slightly more formal terms, we note that the sum over k states is
taken such that at each vertex both energy and momentum are conserved. Conservation of energy implies what is actually coupled are states for which the difference
in energy between electron states described by momentum q + k' and k' is equal to
the phonon energy, which is very low on a scale of the electron band, (on the order of
the meV compared with band energies eV). Thus it is not exactly sites at the Fermi
surface we couple with this exact phonon, but there is a range over which there is
a relatively large weight of states accessible that satisfy both conditions. Reduced
dimensionality dramatically increases the number of available states satisfying this
momentum and energy matching condition by flattening the dispersion along c* from
a canonical parabolic dispersion in all directions; the layered nature of the cuperates is
expected to produce relatively flat dispersions along c* while the antinodal segments
are nearly parallel. ARPES by virtue of insensitivity to the component of momentum
along the surface normal always integrates scattered electrons that have picked up
this off axis component, also widening the number of contributing electrons. Note
this form of interaction Hamiltonian, shown as a green dot in 3-5, is of the Fr6lich
form, eg oc ctc(a + at) and couples electron density to atomic position.
Softening of the longitudinal phonon was first seen by neutron scattering of phonons
in single crystal lead, where the longitudinal phonon along the (111) direction was
observed to show a
-
0.4 meV kink in the dispersion at a nesting condition
67
[92].
q+
k
Figure 3-5: First Feynman diagram renormalizing phonon propagator in the presence
of electron-phonon interaction
In the case of lead, the mostly spherical Fermi surface was perturbed by the crystal
potential, resulting in band splitting from degenerate perturbation theory between
electron momentum eigenstates separated by a reciprocal lattice vector; this resulted
in formation of some significant length segments that were close to parallel. The
energetics in the case of lead are increased due to the monatomic basis, which implies
all atoms contribute conduction electrons which thus feel screening under the phonon
eigenvector.
The classic modern system for electron-phonon driven softening is that of 2HTaSe 2 and 2H-NbSe 2 , studied by neutron scattering first in 1975[93, 85]. Both these
materials have a commensurate crystal structure at room temperature, which undergo
transitions to incommensurate structures driven by formation of a charge density wave
at temperatures 122K and 33K respectively. Interestingly, at high temperature before
formation of a charge density wave, the longitudinal phonon polarized along (100) is
seen to show a kink in its dispersion corresponding to a softening at the nesting
position; the magnitude of this dip is ~ 2 meV in TaSe 2 and 1 meV in NbSe 2 .
Later work on NbSe 2 using inelastic X-ray scattering showed as temperature was
cooled, the longitudinal acoustic phonon smoothly became gapless at the CDW nesting point below the transition[94]. However, this effect was seen to be broad, span68
ning 0.2 rlu along the (100) cut through the CDW ordering vector. The authors
interpreted this as suggesting Fermi surface nesting played a less significant role than
an anisotropic electron-phonon coupling constant. ARPES measurements of the electron spectral function showed pronounced kinks and a band gap characterized by
strong electron-phonon coupling[95] (A/TCDW = 11, to be compared with a the
weak-coupling theory prediction for this ratio of 3.5). Subsequent density functional
theory calculations of the charge susceptibility show no divergence in the response at
the Fermi surface nesting condition, indicating the predominant driver of the modulation is electron-phonon coupling[96].
In fact, phonon softening can be due purely to anomalous increases in the electronphonon coupling. The phonon propagator is renormalized regardless of nesting, and
either increase in the vertex strength or the measure of density of states that satisfy
the coupling energy-momentum conservation rule can be responsible. Inelastic X-ray
scattering work on Chromium has shown this to be the case, observing
-
1 meV
softening of the acoustic phonons in several narrow momentum ranges[97], where
DFT calculations indicate no Fermi surface nesting spans these vectors.
The issue of decoupling effects driven by electron-phonon coupling and Fermi
surface nesting is subtle. The arguments presented so far from the literature rely on
claims of breadth in momentum space of such feature and detailed band structure
calculation.
In the cuperates, the underlying crystals are far more complex, and
these become difficult. We will examine crystals as a function of doping in an effort
to drive changes in the Fermi surface and its nesting condition. It is always possible
the underlying band structure and electron-phonon coupling constant changes as well,
but it is likely that an effect that moves in the direction of the nesting condition is
in fact driven by such electron excitation physics.
3.2.1
Longitudinal Phonons Along (110)
The principle cut of study in this work is along (2+c, 2-+ E, 0). The orthorhombic (220)
peak is one of the strongest accessible Bragg reflection, the reflections (110) and (330)
being forbidden in the tetragonal symmetry, and the maximal range of the 20 pre69
venting this spectrometer from accessing (440). Measurements presented here, unless
otherwise noted, are from scattering in Analyzer 5 with the other analyzers shielded
with lead tape. This had the effect or removing the possibility of Bragg scatter into
the aperture in which the other analyzer crystals were intended to backscatter, which
was observed to occasionally cause background in all analyzer channels. Measurements were collected over a varying energy between -25 and 70 meV; the phonon
modes that were observed are the acoustic and, near the zone boundary, sometimes
the first optic mode. Due to the dependence of the cross section on Z, photons preferentially scatter off phonons involving heavy atom motion, which tend to be lower
lying in energy.
Phonon scans are shown in Figure 3-6 for E = 0.15, E = 0.25, E = 0.4 At first
pass, the clear indication is that the lineshape at (2.25, 2.25, 0) is broad compared
with those at (2.15, 2.15, 0) and (2.5, 2.5, 0). This signal is driven by coupling of
the phonon to electronic degrees of freedom; the same energetics that at the surface
drive this phenomena to the static here are seen to result in increased coupling and a
dynamic tendency toward formation of a density wave state. An important alternate
interpretation would be that the broad phonon is due to the simultaneous measurement of multiple phonons close in energy to each other, for which the amplitude of
the eigenvector changes throughout the zone so as to only show up prominently at
(2.25, 2.25, 0). Careful analysis of the intensity along different cuts and other analysis
regarding the momentum position of maximal broadening and polarization analysis
suggest this is not the case, as will be developed in 3.3. Additional evidence comes
from fitting, where attempting to introduce a second fitted phonon mode generally
results in only a marginal reduction in linewidth comparable to fitted linewidth error,
and no clear evidence of mode splitting further along the zone is seen; this suggest
a second phonon mode, if present, contributes almost negligibly and the phonon
linewidth is intrinsic.
Since STM measurements [43] indicate local variations of the Fermi surface, it
is possible that the observed breadth is due to averaging over an ensemble of CDW
wavevectors in the range E E [0.15, 0.3]. For each of these, the measurement taken at
70
(2.15,2.15,0)
(2.25, 2.25, 0)
(2.5, 2.5, 0)
0.
PbBi2201
UD32
-15-10-5
0 5 10 15 -15-10-5
0 5 10 15
0.
0
-15-10-5 0 5 1015
Bi2201
UD25K
- 15- 10- 5 0 5115- 15- 10- 5 0 5 10 15
Bi2201
UD31K
0
0
-15-10-5 0 5 10 15 -15-10-5 0 5 10 15 -15-10-5 0 5 10 15
Pb,Bi2201
UD27K
-15-10-5 0 5 10 15 -15-10-5 0 5 10 15 -15-10-5 0 5 10 15
Bi2212
UD50K
-15-10-5
Bi2212K.
OD70K
-15-10-5
0 5 10 15 -15-10-5
0 5 10 15 -15-10-5
0 5 1015
0 5 10 15 -15-10-5
0 5 10 15 -15-10-5
0 5 10 15
8
Figure 3-6: Sample longitudinal phonon scans from all samples measured. The vertical axis is intensity (counts/sec) and the horizontal is energy transfer (meV).
71
6
= 0.25 would be a corresponding different length away from the new CDW elastic
peak, and one might imagine picking up components of acoustic modes originating
from this new incommensurate modulation, which could produce a signal appearing
to be broad where the width was not due to intrinsic lifetime, but disorder. Regardless
of the origin as intrinsic breadth or shift in dispersion, this measurement indicates
significant electron-phonon coupling inducing renormalization at room temperature.
The important caveat is that these effects are not seen in lead doped samples,
either PbBi2201 UD32K or Pb,Bi2201 UD27K. Assuming the width is intrinsic, this
is evidence for increased coupling driven by the structural change in the absense of
lead. The simplest classical picture explaining increased electron-phonon coupling in
absence of lead is that the presence of a sinusoidal modulation is smeared out by relaxing each bond; the superstructure is also a measure of high internal energy of the
bonds, and substitution of lead commensurately reduces electron-phonon coupling
along with the lattice energy. Alternately, it is possible the electron band eigenstructure is changed by hybridization with the superstructure, and it is the band structure
change that induces increased electron-phonon coupling. These explanations are of
similar orgin, since the electron phonon coupling constant is a function of the electronic band structure.
To determine if shadow Fermi surfaces contributed directly to increased nesting
due to zone folding and hybridization of the electron Fermi surface and shadow Fermi
surface bands, the single electron quasiparticle response along the c* direction would
have to be measured; however, this axis is inaccessible to ARPES due to the resolution only of momentum transfer parallel to the surface. Theoretical calculations
on the effect of superstructure are difficult due to the requirement of large numbers
of atoms, but early LDA work on the effect of including orthorhombic refinement
driven by the unit cell averaged superstructure produced a substantial shift in the
band structure around the Fermi surface if not the location of the surface itself, supporting the explanation that the difference measured in the presence of lead is a
result of a transformed band structure[98]. This is explanation is consistent with the
aforementioned narrowing of a Raman active optic phonon mode under doping with
72
lead[90].
3.2.2
Transverse Phonons Along (110)
Measuring transverse phonons is more complicated than the case of longitudinal
phonons. Consider decomposition of the momentum transfer Q into G + q' the transfer to a zone center plus the relative momentum transfer. For a periodic crystal,
the phonon structure is equivalent modulo G, so measurement picks up phonons at
q polarized along Q
=
G + q. To measure transverse phonons, G + q' must be per-
pendicular to q. An essential simplification is that for these measurements, G
in this case G
=
>
q;
(220) while q is at most (0.5, 0.5, 0). Thus this problem essentially
reduces to that of measuring such that C I
.
There are two ways to do this, which
we will illustrate in the case of the cut of interest for our sample. Firstly, one could
measure phonons at (2 + E, -2 + E, 0). Since (2,-2,0) is a good zone center, the structure here would be equivalent to that measured at (220), and we would be measuring
essentially transverse phonons since (2, -2, 0) is perpendicular to (E, E, 0). Secondly,
we can make use of the reflection argument that the phonon eigenspectra at (E,E, 0) is
equal to that at (-E, E, 0) since there is a mirror plane spanning b*, c* in the averaged
orthorhombic cell (see Figure 2-6). Thus measurement at (2 - E, 2 + E, 0),
which is
sensitive to transverse phonons since (220) is perpendicular to (-E, E, 0) , measures
an equivalent structure to what exists in the transverse at (E, E, 0). This argument is
what is used here, and the resultant data is shown in Figure 3-7.
3.3
Absense of Supermodulation Associated Phonons
in (110) Cut
An important consideration in this work is disentangling intrinsic breadth due to
electron-phonon driven self-energy from other scattering mechanisms such as secondary phonon modes or instrumentation effects. The general effect of resolution
convolution will be covered in Section 3.4.1; we here address the possibility of ad73
(1.75, 2.25, 0)
(1.9,2.1,0)
(1.5, 2.5, 0)
1
Bi2201
UD25K
2.
10
4
-15-10-5
1
0 5 10 15
-15-10-5
0 5 10 15 -15-10-5
12
0 5 10 15
2.
Bi2201
UD31KJ
4
2
-15-10-5
0 5 10 15 -15-10-5 0 5 10 15 -15-10-5
0 5 10 15
0 5 10 15 -15-10-5 0 5 10 15 -15-10-5
0 5 10 15
Bi2212
UD50K
-15-10-5
Bi2212
OD70K
6
-15-10-5 0 5 10 15 -15-10-5 0 5 10 15 -15-10-5
0 5 10 15
Figure 3-7: Sample transverse phonon scans from all samples measured. Axes are
counts per second along y and energy transfer w along x.
ditional scatter due to the introduction of optic modes by the superstructural modulation. There is little improvement in quality of fit upon introducing a secondary
mode, but the breadth can be reduced some. The well fit single mode characteristic
of the transverse scan is thus the single greatest point of evidence suggesting the
observed breadth in the longitudinal is in fact intrinsic. The essence of the appeal of
this explanation is that provides a consistent mechanism for the observed breadth in
the lead free sample and absence in the lead-doped case. However, it appears that
this is coincidental and electron-phonon interaction changes are the primary drivers
74
of the difference between the presence and absence of lead, as we will develop below. Raman scattering has revealed additional low-lying optic modes at 20 cm41 cm-
1
for Bi2201 and 28 cm
1
and
and 47 cm- 1 for Bi2212[99] that are ascribed to
superstructure-associated amplitude modes. The essential component of this is that
the enlargement of the unit cell has resulted in a change in the phonon eigenstructure
which results in some scatter at low frequency.
The study of incommensurate modulations is complex in part because formally the
strict collapse of -sums of the form E. einrQ oc N6(Q - G) for R,
the sum over all
atoms does not hold true since exact periodicity of the translation is broken. However,
to good approximation the physics should be equivalent to that of a commensurate
translation of a near periodicity; to approximate the qsi wavevector, often a period
5 supercell is used[1]. In this approximation the traditional zone centers stay good
zone centers and the additional modes can be viewed as resulting from zone folding
(then hybridization as a perturbation due to the distorted crystal structure). While
there is no single test to confirm the absence of significant scattering from a second
(or larger) number of modes in the longitudinal scattering, we present arguments
relating several scans which indicate this does not significantly contribute to the
inferred breadth. Reciprocal space, decorated with features corresponding to Bragg
peaks, superlattice peaks, and the observed broadening and scans are shown in Figure
3-8.
3.3.1
Comparison Between Longitudinal and Transverse Scans
The orthorhombic distortion breaks several symmetries, most prominently C4 rotation
of the tetragonal crystal. Rotation by 1800 about b* is conserved, as shown in Figure
2-6, which maps (x, y, z)
-
(-x, y, -z). This can equally be considered as a product
operator of reflection about b* and inversion. This rotation operator is self inverse,
hence the structure at (H, K, L) is equivalent to that at (-H,K, -L), and specifically,
the L = 0 subspace reflection along a* leaves the structure of the subspace invariant.
The details of the symmetry operators of these groups are shown in 3-4. Thus even
under consideration of the reduced symmetry due to the superstructural modulation,
75
K
3
2
.I
0
3 H
1
Figure 3-8: Reciprocal space, showing projection of L E [-1, 1]. Solid circles: Bragg
peaks. Stars: superstructure peaks. Blue arrow: Scans measuring longitudinal
phonons. Green Arrow: Scans measuring transverse phonons. Blue ellipse: area
of observed broadening.
measurement along (2 - 6, 2 + c, 0) remains a good probe of the transverse phonons
along (E, c, 0).
Since the eigenstructure is invariant under reflection, for each eigenmode at (6, 6, 0)
a corresponding mode exists at (-E, 6,0) with eigenvector reflected about a*, c*. This
maps scattering which would show up as longitudinal at (E, c, 0) to longitudinal along
(-C, E, 0) and similarly for the transverse. This is equivalent to the action of fourfold
rotational symmetry in transforming between these zones, although fourfold rotation
would extend higher symmetry constraints off axis which are removed under the intro76
duction of the superstructural modulation. Conceptually, modes originating from the
superstructure at the Bragg peaks associated with either (2, 2.22, ±1) or (2, 2.44, 0)
should not be exclusively (110) polarized at (2.25, 2.25, 0), and hence if such scatter
existed to be measured in the longitudinal scan at (2.25,2.25,0) there must exist a
corresponding second scattering channel at (1.75,2.25,0) to be seen at the transverse.
The intuition for this is that as the longitudinal zone cut is taken the zone center at
(220) is replaced by those at (2, 2.22, +1)
and (2, 2.45, 0) in determining the reduced
wavevector; with respect to these zones, the cut is no longer along a high symmetry
(110) direction and likely there are superstructure modes with a component along b*
which would be visible in the transverse.
In fact, one might expect modes from the superstructure to retain character of
(010) polarization since near the Gamma point there is an additional Goldstone mode
in the case of a sinusoidal incommensurate modulation associated with translating
the superstructure sublattice along the direction of the modulation. If the underlying
displacements of the modulation are sinusoidal, all neighboring crystal environments
are seen equally by each atom due to the incommensurate nature, and hence this
translation is gapless in the long wavelength limit and represents essentially a noninteracting subsystem Goldstone mode. In a real crystal this mode is gapped out by
energetic preferences to lock in a 3D crystal environment and at low energy the second
superstructure Goldstone mode is gapped out on a scale of a fraction of a meV as
seen by low-Q inelastic neutron scattering along this cut[100]. If predominately (010)
polarized scatter from this superstructure extended to (2.25,2.25,0) it would show up
with exactly equal amplitude at (1.75,2.25,0), which is clearly not the case from the
data. Further measurements in Section 3.4.4 describe use of scans in different zones
to selectively access polarized scatter to reject the hypothesis of off (110) polarized
scatter due to the superstructure.
Any superstructure phonon effect is expected that these effects are weak as the
superstructure is 101 -102
times weaker than the main Bragg peaks. The significance
of this structure on reducing symmetry can be examined by comparison of the phonon
structure along (c, 6, 0) for ±c; as we will see in Section 3.5.4, there is a signature of
77
inversion symmetry breaking in the longitudinal, while the corresponding transverse
shows no effect.
The critical physical observation from Figure 3-7 is that the transverse phonon
at (2.25, 2.25, 0) is not broadened as the longitudinal mode is. This dependence on
phonon polarization is consistent with the CDW origin as a bond-stretching deformation; longitudinal phonons renormalize nearest neighbor bond lengths along (110)
which include the Cu-O bonds while transverse modes only effect this bond length at
second order in the displacement. Complete analysis of the fitted form shows that the
transverse phonon is in fact close to resolution limited throughout the dispersion cut
along (110), evidence for which can be seen by noticing the relatively sharp lineshape
present in all scans, comparable with the energy resolution-limited scatter at W = 0.
Additional zone centers and zone folding is shown for BSCCO in Figure 3.3.1.
Green squares highlight the result of zone folding from an initially broad (filled square)
phonon position in terms of what additional scatter is potentially picked up. Due to
the out of plane nature of the superstructure, L=0 scatter is actually only shifted
a relatively small distance for K comparable to the K of the superstructure. This
implies the broadening effect of the superstructure in terms of the spread of the eigenspectra subject to sensitivity is actually near a minima at the observed broadening
point. This effect is expected to be similarly relevant for measurements at (6, E, 0)
and (-E, e,0) since are both are equivalently located in K, and the zone folding argument is invariant under H translation. The second effect of sensitivity to additional
acoustic-like phonons originating from the superstructure is also expected to be equivalent between longitudinal and transverse cuts, since the phonon eigenstructure from
the superstructure should be even under reflection through a* as well; were an effect
to be present in the longitudinal cut at (f, 6, 0), a similar mode would need to exist
to scatter into the transverse measured at (-E, E, 0).
This polarization analysis is the primary point of evidence indicating the observed
breadth is intrinsic due to charge ordering, and not due to direct changes in the bare
phonon dispersion due to the superstructure. However, the data presented in Figure
3-6 indicates the superstructure is necessary to see this effect, indicating that it is
78
L
1
I\
l
\
I\
I\
\/
0
I
'4
I
\
\-
'4\ /
'41
\
\
4
I
1.5
2
2.5
K
Figure 3-9: The K-L plane for H=O, decorated by Bragg and superstructure peaks;
hollow stars are second harmonic superstructure peaks. There is no interaction between these and orthogonal H zones; this picture is translation invariant along a*
except for the obvious removal of Bragg structures.
changes in electronic properties due to the structural differences which then drive a
electron-phonon coupling enhancement as observed in the longitudinal mode. As was
seen in lead[92], interaction of the bare bands with the crystal lattice potential can
result in flattening of the Fermi surface and enhance the density of states available
electron states for coupling in the antinodal regime. The insensitivity of ARPES and
STM to this effect as considered since both see signatures of charge ordering may be
compensated for by the reduction of dimensionality at the surface.
79
3.3.2
Scattering in (020) and (200) Zones
An important consideration in understanding the observed broadening near the zone
center is understanding the polarization of such scatter. In particular, if part of
the scatter was due to modes associated with zone folding or excitations associated
with the superstructure, one would expect the cut along (2 + E, 2 + E,0) contained
some scatter polarized other than along (110), such as possibly that along (010)
associated with eigenvectors corresponding to translating the superstructure along
its incommensurate modulation axis. To gain information about the polarization,
we take measurements at (0 + E, 2 + E, 0) and (2 + E, 0 + c, 0). These scans contain
information about both the longitudinal and transverse modes.
phonon scattering,
The equation for
2.25, indicates that exact comparison between the intensity of
phonon scatter between two zones is possible only if (a) the form factor is equivalent
at these Q and (b) the compared zone centers G1 , G 2 satisfy eiGl. b
_ eiG2br for all
basis atoms r. Constraint (a) is approximately satisfied since in this high energy
range,
f
~ Z independent of Q. In BSCCO constraint (b) is satisfied at (220), (200),
and (020) for all atoms except oxygens up to the detail of the orthorhombic distortion
since the in plane component of the positions of each atom are one of the set (0,0),
(0.5,0), (0,0.5), (0.5,0.5), all of which are in phase at these zone centers. Since oxygen
contributes essentially negligibly to the scatter due to the
f2
~ Z 2 dependence, a
comparison between scans taken in these zones is to good approximation associated
with dependence only on the polarization dependence and Q2 terms contained in Q -7.
Formally we write
2
I(Q = G +
2m;2c
1e
(nB(Wp)6(w + w(j)
fq± (riB(Wp) + 1)6(W
-
Wp(q))
Noting A,-P has separable dependence along a as the polarization ' is either
longitudinal or transverse, we can pull this dependence out as a product to defining
80
all terms save for
Q,
-
as I,(c). As previously mentioned, all significant scatterers
(except for oxygen) at the relevant Bragg peaks satisfy the condition that G -b, are
in phase, hence there is only dependence on q', not G, and we can write
I(7)=
(Q -1ong)
2
Iiong (C)+
(77
-itans) 2 ltrans(E)
where E parameterizes the cut along (110). To calculate a comparison with the data
measured at (2 +
6,
2 + E, 0), let us note the following
I(2+,2+E,O)
2(2 +
I(2-E,2+EO)
2E2 Iiong(E)
I(O+c,2+c,O)
0.5(2 + 26)262 Ihong (E) + 2 Itran(6)
(2 + 26)2 _ 2
(4+ 26)2 I(2+E,2+E,O) + 4I(2-,2+,O)
E)2 hong(E)
+ 8 Itran(E)
(3.3)
I(2+E,O+c,O)
Since in this calculation this we assumed (110) was a high symmetry direction,
were there to exist scatter along a* or b* along the (2 + E, 2 +
6,
0) cut that did
not satisfy this constraint, it would not follow this scaling and would instead show
up most strongly in near the corresponding (200) or (020) scan; if both terms are
present with equal amplitude, the lines for (200) and (020) will be equivalent but an
additional mode structure should be visible in comparison with the interpolated data
from (220).
Figure 3-10 shows the result of this comparison throughout the zone. Except for
close to the IF point, the agreement between blue and purple data is good, suggesting
that any additional modes present along the (E, E, 0) are either polarized longitudinally or present with equal amplitude along the (100) and (010) direction. This
confirms the intuition that (010) scattering associated with the superstructure is not
relevant in terms of producing the broad phonon lineshapes seen. This comparison
is done without any adjustable parameters, but for each scan the elastic line was
81
E0.15
E= 0.1
E = 0.2
f
fi
-15
-10
0
-5
E=
5
10-
15
-15
-10
-5
0.25
3.0
0
5
10
15
-15
-10
-5
E = 0.3
E= 0.4
3.0
2.0
2.5
2.5
-10
-5
0
5
10
15
5
10
15
to
1.5
2.0
1
15
0
1.0
5
10
15
-15
-10
-5
0
5
10
15
-15
-10
-5
0
Figure 3-10: Scans taken for Bi2212 UD50K for (2+ , 0 + c, 0) (blue), (0 + E, 2+
(purple), and interpolated from (2 + c, 2 + E, 0) via Eq 3.3 (gold)
subtracted off since this intensity is not expected to obey
(Q -r7)2
, 0)
scaling (and in fact,
a substantial increase is seen around w = 0 near (020) relative to (200), likely due
to C4 symmetry breaking in the elastic by the superstructure). From this figure, we
note that the intensity calculated from the (220) longitudinal and transverse phonon
scans is generally slightly weaker than that intrinsically measured at (200) and (020).
There are two components which contribute to this: first, these calculations neglect
the decay of the electron form factor with increasing
Q. Secondly, the
presence of a
superstructure causes site disorder with respect to the reciprocal lattice vector; displacement of all sites by a qc implies the sum e-G is no longer exactly in phase (to
the order of the strength of the displacement caused by the superstructure), and this
may cause a weakening in intensity at elevated
Q.
The intuition of this term is that
it is the same factor as present in the Bragg peak sum, and since elastic intensity has
moved to the superstructure peaks and total scattering is conserved under increasing
modulation amplitude, this additional elastic scatter is compensated for exactly by
a decrease in Bragg intensity in the unmodulated tetragonal peaks. Critically, the
structure of these lines appear to be the same between all three scans; no functional
82
dependence indicating presence of additional scattering modes is seen. This indicates
that it does not appear to be the case that there is significant component of (100)
and (010) polarized scatter through this cut, most specifically near f = 0.25 where
anomalous broadening is seen. This confirms the increased width seen in Figure 3-6 is
due to an increased range in energy over which nontrivial intrinsically (110) polarized
scatter is seen.
E =0.15
E = 0.1
-15-10-5
20
1
20
10
10
1
5 10 15
-15-10-5
Figure 3-11: Scans taken for Bi2212 UD50K for (2+
(purple) highlighting role of elastic scatter.
5
10 15
, 0 + c, 0) (blue), (0 + c, 2 + E, 0)
The additional scatter at c = 0.1, and to a lesser degree c = 0.15 present near
(200) compared with (020) is an interesting compliment to the difference in elastic
intensity. Without subtraction, the data looks as shown in Figure 3-11. The intensity
of the superstructure at (0,2.2,1) is 2 * 106 c/s, while that of (2,0.2,1) is 3 * 104 c/s;
the Bragg peaks (020) and (200) both have comparable intensity 2 * 10' c/s. It is
then likely that the suppression of dynamic intensity at (0.1, .2.1, 0) and is due to
shifting of spectral weight to w ~ 0 static order or slow fluctuations near regions of
significant elastic weight.
3.3.3
Comparison with Hg 3- 6AsF 6
Hg 3 6 AsF 6 is an incommensurate modulated system for which the phonon eigenvectors associated with both the primary lattice and incommensurate distortion are well
studied [101, 1021; this provides a useful reference for discussing these effects. This
83
system features 1D chains of Hg atoms embedded in an AsF6 lattice; the Hg atoms are
observed to behave as a liquid at high temperatures, but lock in an incommensurate
modulation at 120K. The phonon dynamics of this composite crystal are then best
considered as consisting of two excitation classes: originating as a Goldstone mode of
translation of the entire system, and those associated with translation of the Hg sublattice. This second class is gapless since the observed modulation is incommensurate
and every Hg atom is exposed to the distribution of crystal environments; translation
of the subsystem then leaves the energetics invariant. Note this does not need to
be the case and it is possible to have an incommensurate modulation with a dispersion other than a sinusoidal modulation by a locking in to the local environment to
the AsF 6 sublattice that maintains the incommensurate wavevector as a fundamental harmonic of the deformation. In this case not all neighboring interactions could
be seen and the interaction with the AsF 6 sublattice would gap out the Goldstone
mode of the Hg sublattice sliding. For every static (non-glassy) crystal, at some low
energy scale this is the case; however, for energies on the order of meV transfer the
gap is often far lower and the systems often essentially seem noninteracting with an
additional longitudinal acoustic phonon.
The form factor for this mode is such that around AsF 6 Bragg peaks one predominantly sees the standard (AsF6 sliding) phonons while around superstructure
peaks it is primarily the Hg sliding mode that is observed. Formally, however, one
should consider the eigenspectra everywhere to be a superposition of the two acoustic
modes, and consider the amplitude of scatter into each. In the case of Hg 3-6 AsF 6
the superstructure (Hg) mode is harder than the bulk AsF 6 mode, while in the case
of BSCCO, it appears the opposite is true and the superstructure-associated mode is
softer (as shown in Figure 3-16). This picture of spectral weight evolving from gapless at the superstructure Bragg points to a dynamic phonon weight associated with
Hg chain motion to back to gapless at the next Bragg peak is then representative of a
zone folding argument in which the full scattering cross section involves this spectra
replaced (or backfolded) from each superlattice zone center. This reflection is likely
slightly hybridized by the crystal potential at the new zone boundaries, but other84
wise looks like the unperturbed superposition of Hg and AsF 6 modes each associated
with a separate distinct lattice. The eigenstructure is dependent on all modes in
this reduced zone with amplitude depending on the specifics, but as per the intuitive
perspective, expected to gain character of the nearest Bragg peaks. In BSCCO, we
expect the weight of superstructure-associated phonon modes to be primarily strong
around first order superstructure Bragg peaks such as those at (2, 2 + 0.22, ±1) and
decay elsewhere (the second harmonic Bragg peaks are a factor of ~ 102 counts/sec
weaker than the first order peaks as shown in Figure 3-4).
3.4
Damped Harmonic Oscillator Fitting
The scans shown in Figures
3-6 and
3-7 show a need for fitting in terms of a
quasielastic line and a broad phonon lineshape; some scans (especially longitudinal at
c = 0.5) show need for a second optic phonon. A background term is required as well
to fit the tails in energy substantially away from the phonon lines. A minimal set of
parameters then is one elastic amplitude (A), one phonon amplitude (n), a phonon
frequency (w,), phonon width (F), and background c, with some scans requiring
at least a second phonon amplitude and frequency,
n2
and w2 respectively. The
signature of nontrivial breadth (self-energy) of the phonon lineshape motivates the
use of a damped harmonic oscillator lineshape given below to model phonons in the
presence of strong coupling, as used in prior work on systems with strong electronphonon effects[103, 94].
In particular, as will be motivated in Section
3.4.2, the
background for each scan is fit to a known fixed detector dark rate (0.01 c/s) and a free
parameter (cph) corresponding to a flat phonon intensity for positive energy transfer,
Bose scaled on negative transfer. This phenomenological model is the simplest 1parameter approximation that is a valid extension to tw; in principle, some sort of
amplitude decaying in w for w > 0 should be used, as will be commented on in 3.4.2,
however, describing this would require multiple parameters which are not justified due
to the insensitivity of the physically relevant (wO, F) quantities on the exact details of
this background lineshape. Considering the convolution over the resolution function
85
as well, we are left with a fitting form as follows:
I(Q, w)
=
A g(w) + c(w)
IDHO(P1W 0
d7rdA g(T)b(A)IDHO
+
1
F,n)
,=
c(w)
=
1
_e-eLu/kBT
C, + Cph
2
(w
j
(
-
T, wo(Q)
-
--
A, (Q) n(Q)
2wnF
W2 )2 + w2 F2
w>O
1-ew/(kBT)
1-e-hw/(kBT)
VW < 0
Where in this equation g(w) is the energy resolution and b(A) is the longitudinal
momentum resolution, given in Table 3.1.3. In Equation 3.4 several assumptions
are made. Firstly, the momentum component of the resolution ellipsoid was only
considered along the direction of the scan dispersion. This is justified since by existence of a (tetragonal) reflection plane operator along (110) in the plane mapping
between points on either side of the dispersion, the eigenstructure must be at a local
minima or maxima with respect to deviation off the high symmetry cut, hence the
local slope along the direction off the symmetry cut of all parameters is expected to
be zero. Thus to first order in the width of the ellipsoid along these off symmetry
direction, the intensity is unaffected by such components. Here the resolution function is small, of order 0.03 rlu in FWHM in the transverse in plane direction; this is
indeed small and the effect of neglecting this resolution component should be second
order. Secondly, we only consider the effect of broadening on the bare frequency, eg
there is not a similar term F(Q)
-
(8w))
A and correspondingly with n. The
primary motivation for this assumption is simplicity, so the model need only be done
self-consistently in as few degrees of freedom as possible. Justification and implication of the convolution is discussed in section 3.4.1. The primary result of this work
is existence of nontrivial broadening, which can not be induced by neglecting this
effect (as could be the case for neglecting the slope in wo) since the canonical phonon
has a flat (zero) linewidth and hence a zero slope of the linewidth. Thus while a
slight renormalization and sharpening of the lineshape may be induced by including
86
(3.4)
dispersion convolution over this parameter, the observed results are a lower bound
on the presence of anomalous broadening. Elaboration on the background term is
provided in 3.4.2, and addition of an optic mode is-discussed in 3.4.3.
3.4.1
Effect of Resolution Convolution
The damped harmonic oscillator lineshape near the bare frequency can be well approximated by a Gaussian lineshape, described by a mean p=2
deviation a =
.
2
and a standard
2-
We can similarly approximate the momentum resolution
22Log(2)
b(A) and energy resolution g(T) with Gaussians centered at zero with widths ab and
ag respectively. This enables closed form calculation of the final intensity including
the effect of the dispersion and energy terms, where we write VQ --
I(Q, w) oc
dTdAe
(W - (t+v A))
2
:
2
e
2o
q
2
e
=
e
2
2(a2+a
-2)
(3.5)
thus the new standard deviation is the quadrature sum of the intrinsic breadth,
the standard deviation of the energy resolution, and the product of the standard
deviation in the momentum cut times the slope of the dispersion. All these lead to
an effect of reducing the intrinsic breadth; the utility of fitting is then testing the null
hypothesis that all the observed broadening is due to canonical resolution function
effects; it turns out the data cannot be explained without a large component of a,
the intrinsic phonon breadth. Given a value of
9b =
0.02 used for Bi2201, we see
the momentum component produces a breadth 1.5 meV comparable to that of the
resolution function only when the slope of the dispersion is 75 meV/rlu. Using the
values of the dispersion later found in Section 3.5, we can produce Figure 3.4.1
below showing the contribution VQOb for a representative Bi2201 and Bi2212 sample,
incorporating the narrower crystallographic resolution in Bi2212.
As seen from the figure, by E = 0.1, the momentum component of the resolution
ellipsoid is dominated by the pure energy component, becoming essentially irrelevant
by E
=
0.5; however, in fitting scans close to the zone center, this component can
become important.
In section
3.5, we find a phonon F ~ 4 over a range from
87
V(ecobo- Bi2201
2.0
V(EEO)Ob Bi2212
og Energy
1.5
1.0
0.5
0.0
0.1
0.3
0.2
0.4
0.5
E
Figure 3-12: Induced breadth due to the momentum resolution, compared with that
from the energy component of the ellipsoid.
c = 0.2 -- 0.4. One possibility is that the dispersion is actually kinked, and what
appeared as breadth in Figure 3-6 is actually a fast dispersion. To account for F = 4
without intrinsic breadth requires abVQ =
79 meV/rlu.
1.6, corresponding to a dispersion slope
To persist this over a range of 0.2 rlu implies either a net deviation
in absolute value of 16 meV, without shifting the center position. This requires a
sawtooth function in the dispersion with at least 5 oscillations across the dispersion;
such seems unphysical, and we instead propose the width is due to intrinsic lifetime.
3.4.2
Phonon Sea Background
An important consideration in this analysis is detailing the background signal. The
primary component of scatter for energy w
elastic lineshape, previously discussed.
> w, is the resolution function from the
To examine possible phonon contributions,
we subtract off the elastic resolution-limited scatter and plot the resultant intensity
as a function of energy, overlaying positive and negative energy transfer as shown
in Figure
3-13.
This highlights the consistently lower intensity seen on negative
88
transfer, indicating phonon amplitude plays a significant role. The red line shows a
phenomenological w- 2 and constant decay term used to approximate the scatter in this
range of up to 25 meV; in green, the Bose scaling nB(w)/(nB(w) + 1) is used to scale
to intensity at negative transfer; the result is a good fit. The temperature dependence
shown below also is in good agreement; the purple and blue lines generated at 150 and
60K are generated in a parameter-free manner from scaling from the red line. This all
indicates the majority of scatter in the range 10-25 meV is consistent with a smooth
background term (seen here to best fit to w-2 + const) of phonon weight; the negative
transfer then should be calculated by appropriate Bose scaling of whatever functional
form is postulated from the positive side. In many of the data sets collected, the full
range out to 25 meV was not collected; as shown in 3-6, many scans only contain
data out to approximately 12-15 meV. Fitting to a term of the form w-2 + const takes
three parameters, since the divergence of w2 must be cut off at some short energy
scale
to prevent a divergence; this is clearly overfitting a background for which often
only a small handful of points are visible. To this end we fit instead with a constant
function on the positive side, recognizing that the fits will be driven to an average of
a probably decaying form.
The argument presented does not require that all of the fit line is from this phonon
sea; undoubtedly, some of the scatter originates from the tail of the normal acoustic
phonon, and in some scans a clear longitudinal optic mode is visible. However, as
long as the intensity is of a phonon origin it should be fit by a heuristic form as best
described by the data on the positive side and appropriate scaled, as done in Eq 3.4.
The origin of a phonon sea background of this form is likely due to zone folding,
which produces an enormous formal number of modes (the commensurate crystal of
BeFeSrO synthesized had a unit cell of nearly 300 atoms). The net effect of weakly
scattering into a huge sea is this slow decay in energy, which looks locally flat in any
smaller range. Lastly, to estimate detector dark counts or other sources of background
that do not decay as a phonon Bose factor, we counted intensity at energy transfer
of -100 meV at 300K, where any phonon should extremely heavily suppressed. This
resulted in an upper bound for the background count rate of co < 0.0069 + 0.0040;
89
(2.25,2.25,0)
2.5
w>0
2.0-
w<.
1.5
1.0
0.5
0
5
10
15
20
25
(2.25,2.25,0)
0.6 -
300K
150K
60K
0.5
0.40.3
0.2
0.1
0.0
0
5
15
10
20
25
Figure 3-13: Bi2212 data taken at (2.25, 2.25, 0) as a function of energy at several
temperatures, highlighting role of background. (Top) shows scans at 300K for W > 0
and w < 0, while bottom shows scans for w > 0 for T=300K,150K, 60K. A single
phenomenological fit line (thick red) fits all data well when Bose scaled to induce the
other fit lines for negative energy transfer and for varying temperature.
in this work, we used the value c, = 0.01 for all analysis for simplicity. This value
is extremely small and the details of this choice do not affect the results; phonon
scattering is generally on the order of a few counts per second, several hundred times
stronger. A value of 0.01 is consistent with the published value of dark count rates
90
for these detectors[104].
3.4.3
Optic Mode Fitting
We can see from the longitudinal cuts near E
-
0.5 the presence of an optical mode
around 13 meV contributing to scattering. This excitation can be tracked locally
and remains visible for c > 0.4 where data is available. While fitting the optic mode
is not essential to understand the acoustic mode behavior, we will later see when
comparing with shell calculations that while the acoustic mode is strongly affected by
interactions, the optic mode is not, and remains a good check for model calculations.
B12201 UD31K (25,25,0)
Bi2201 UD31K (24,24,
1.Y
0
if.2-
Q43
-20
-10
10
20
W
-20
-10
10
20
Gj
UD50K Bi2212(25,25,C)
UD50K Bi2212(245,245,Q
1.
1q1IL6.
4
0.2-20
-10
Co
10
20
-20
-10
Co
10
20
Figure 3-14: Scans near zone center showing presence of optic mode, fit to a resolutionlimited lineshape corresponding to the convolution of a flat dispersion at 6(w - W2 )
over the full resolution ellipsoid.
The fits shown in Figure
3-14 show resolution-limited lineshapes capture the
width of the optic mode. To limit the number of parameters then, when necessary
to obtain a good fit, a second optic mode will be fit with a resolution limited peak,
appropriately Bose scaled between ±w2 . The error in these fits is that the phonon sea
91
actually decays faster than a single constant term as previously remarked on; fitting
these scans to a truncated decay with a w- 2 falloff corrects this for Bi2212 where
this calculation is possible due to the increased scan range, but only shifts the peak
position by 0.1 meV, which is comparable to statistical noise and irrelevant within
the scales of the effects under study here.
3.4.4
Phonons Along (010)
An interesting comparison with the measured phonon properties is seen upon taking
the dispersion cut along the direction of the superstructural modulation (010). As
shown in the raw data visible in Figure 3-15, close to the F point only a single mode is
obvious, but beginning at E = 0.3 a second mode can be fit at low energy. In the case
of Bi2201, the elastic scatter from the proximity to the superstructure was so strong
that at both c = 0.4 and E = 0.5 no phonon peaks could be resolved; at E = 0.6 a two
peaked structure is again visible. Note that the tetragonal zone boundary is at (300)
and it is expected that the phonon structure continues effectively dispersing to that
point, although formally the orthorhombic distortion is sufficient to break symmetry,
increase the unit cell, and cause via zone folding an amplitude for measuring the
modes beyond E = 0.5 at the corresponding zone folded 1 - E location. This contrasts
with the case of (220), where the zone boundary is at (2.5,2.5,0) due to the presence
of nearby Bragg peaks at (3, 3, ±1).
The resulting dispersion parameters are plotted in Figure 3-16. In this figure, we
sketch with dashed lines twoalternative models for the multiple mode characteristics
seen, associating the low lying modes with (top) scatter from the peak at (0,0.44,0)
and (bottom) from (0,0,0). In both figures, by the dashed black line we refer to the
primary longitudinal acoustic mode, originating from the undistorted crystal structure
and presumably reverting to that form in the case of lead doping of the crystal, and by
the grey line we postulate additional scatter from the superstructure peaks located
at E = n * 0.44 Vn E Z. By the formal zone folding, the dispersion is the same
and all dispersion possibilities are present from all zone centers; the amplitude is of
course subject to different factors in different zones, responsible in the limit of the
92
Bi2201
E=0.15
E=0.1
D31K
0
-10-5
..
5 10
0
-10-5
0
5 10
E=0.25
2.
22
1
.50
-10-5
5 10
5
0
-10-5
10
0
-10-5
E=0.4
IE=0.5
E=0.6
4
2
0.
0.
0.
11
5 10
5
-10-5
0
5
10
-10-5
5 10
0
5
10
5
10
E=0.5
2.
E=0.35
Bi2212 E=0.15
UD50K
1
0
-10-5
1.
1
2
-10-5
5
10
-10-5
5
10
-10-5
Figure 3-15: Individual phonon scans along the direction of the supermodulation,
(0, 2 + E, 0), Bi2201 UD31K and Bi2212 UD50K
superstructure modulation amplitude going to zero to reverting the picture to the
unfolded form.
Assuming equivalent dispersion for the two samples as a heuristic, the data for
the Bi2201 and Bi2212 samples shows similarity within a meV, below the degree of
explanation attempted in this picture such that we will attempt simultaneous fitting
of the phonon peaks of Bi2201 and Bi2212 with a single model curve. The phonon
93
data analyzed in Section 3.5 will show that this is reasonably justified in that Bi2212
and Bi2201 have slightly different speeds of sound for the bulk phonon but otherwise
similar functional behavior through the measured cuts.
at (O,2 +E,O)
WJO,WJ2
12
10
Bi22O1 UD31K
%
Bi2212 UD50K
%~
'
8
-
6
-
~
/
N
N
.
N/A
x
%
U
N
N
4
4f
2
0.0
0.2
12
0.6
0.4
W OW2
-
at (0,2+
0.8
E,0)
1.0
10,0
000-400,-
10
00-
"
N0
;O o-
N;
8
Z
N
-0
N
6
0 0
N-
-
000
N-.
-
0
N
U
N
--
100
N0
2
00
--
0
N0
4
--
000
Z
U0~
-
UNN-
C
0.0
0.2
0.4
0.6
0.8
1.0
Figure 3-16: Scans taken for Bi2201 UD31K and Bi2212 UD50K for (2 + E, 0). A
number of phonon modes equal to the number of dots visible was fit, with the lowlying mode pinned to a resolution-limited width. The two sets of lines represent
simple theory models with phenomenologically generated sinusoidal dispersions
94
The driving factor in a dispersion as in the top panel of Figure 3-16 is the attempt to match the observed low-lying phonon frequency at e
=
0.35,0.6. Postulating
that the amplitude seen in these scans is due to this scatter from near the first order
superstructure peaks implies this mode is strong, hence it is denoted in bold. This
has the added advantage that the folding locating this mode at (020) also matches
the dispersion well in the range near c = 0.1. This, however, is not critical since the
sinusoidal curve shown in black is only a simple model and should not be viewed as
a strength of this postulate; it is entirely plausible in this single valued region the
fundamental curve the phonon is harder at low q than predicted from a sinusoidal
dispersion. The alternate model, shown in the bottom panel, matches a sloped line
through the four low-lying data points e
=
0.3, 0.35,0.5, 0.6. The difference between
these models is best examined near the superstructure, where the top model is expected to go gapless while the second predicts a medium energy mode around 4 meV.
Unfortunately, the presence of the superstructure makes this measurement difficult;
however, by eye, it appears the second model where the data for E = 0.3, 0.35, 0.5,0.6
is matched with an acoustic-like mode originating from the (020) zone center produces superior agreement. This suggests that unlike the case of Hg 3-6AsF 6 , it is the
folded mode originating from the undistorted crystal zone center that describes the
character of the scatter rather than that of the replicated (thin gray dashed) mode
around (0,2.44,0). This distinction is consistant with proportionality of the scatter
to the strength of the Bragg peak, resulting in suppression of the modes originating
from (0,2.44,0) and also a loss of weight upon moving off the superstructure peaks
responsible for the unimportance of scatter from neighboring peaks at (0, 2.22, +1)
over the comparable modes originating from (0,2,0).
Further support for the for model postulated in the bottom panel of Figure 3-16
comes from measurement at very low energy and momentum in Bi2201 and Bi2212
of phonons by neutron scattering[100, 105]. In this work, phonons polarized along
b* were measured by along this longitudinal cut near both the original (020) zone
center and additionally near the incommensurate shifted peak (0, 2.2, 0.42). Finding
the superstructure peak at L
=
0.42 is surprising; scans in our samples through this
95
L reveal no ordering, while the superstructure is found at L = +1 for all measured
samples. Nonetheless, assuming that difference in crystallography does not affect
the dynamics among the nominally identical materials, this group found evidence
of two longitudinal acoustic modes at and above an energy scale 0.5 meV; the first
characterized by speed of sound 4150 m/s in Bi2201 and 5900 m/s in Bi2212, and
the second by speed 2460 m/s and 2400 m/s in Bi2201 and Bi2212, respectively.
For comparison, the superstructure mode in the bottom of Figure 3-16 is drawn
with this speed of sound, 2460 m/s. This produces good agreement with the data,
which is a strong parameter-free confirmation of the assignment of intensity to modes
originating from the superstructure at (020). Surprisingly, while the soft superstructure acoustic modes measured in neutron were observed to lose intensity quickly
away from zone centers with intensity vanishing by 0.1 rlu, the scatter observed here
is strong throughout the zone. The implication of this for the cuts of interest is that
scatter should be thought of as originating from additional Goldstone-like modes at
the main tetragonal zone center; the branch is best thought of as evolving from b*
polarized scatter along the (HOO) cut. The evolution of this mode along K off this
axis is then symmetric under K-+-K by the previously mentioned 180' rotation operator along H in the L = 0 subspace. Since the superstructure reduces the symmetry
by increasing the unit cell, (110) is no longer a high symmetry cut with respect to
the superstructural modulation, and hence the 2D incommensurate modulation along
K,L evolves anisotropically along H, with character of eigenvector modulation along
b* likely persisting even up to the formerly high symmetry (E, E,0) q positions.
The X-ray refinement of atomic position[1] indicates the modulation is not purely
sinusoidal; it is likely that while the atoms are modulated with an incommensurate
period, they are locally trapped by interaction with their neighbors. Thus while as
an ensemble, modulations associated with additional oxygen and bridging regions are
equally likely to occur at any atom due to the incommensurate nature, at low enough
energies the additional acoustic mode should not be gapless. The studies of [100, 105]
suggest this interaction energy scale on which this mode is gapped is under 0.5 meV.
Thus while we draw an additional gapless mode, from a formal standpoint the only
96
true Goldstone excitation is total crystal translation. This is not the case if there
truly were no interaction, such as would be the case for an interpenetrating lattice
with a displacement given by a single moded sinusoidal variance. Hg 3 6 AsF6 is close
to this limit, but for any real physical system, it is likely there is a sufficiently low
energetic scale that gaps out this second acoustic mode.
3.5
Fitted Parameters
We now turn to presenting the results of fitting the data of the sort shown in Figure
3-6 and 3-7 to the functional form 3.4. Plots presenting all observed fit parameters
for all data sets are shown below in 3-17 and 3-18. Let us go through each set
of observables for longitudinal and transverse phonons before proceeding to more
detailed analysis.
The A parameter describes the quasielastic amplitude, and is in agreement with
the measurements that would result in a fixed energy elastic scan through momentum,
resolved with very high precision due to counting several points in the quasielastic
line. The longitudinal phonon shows a sudden falloff with increasing cEJ as would be
expected from departure from the Bragg peak; the presence of as strong scatter as
is seen is a reflection of the disordered crystal. No static CDW signature is visible
in this data set, which would show up as a q-dependent peak in the quasielastic
A parameter. The transverse shows an upturn near c = 0.5; all measured samples
showed some degree of Bragg structure at (1.5,2.5,0).
This was not observed to
be temperature dependent, and consequently was not revisited for further study.
Additional future work could study the implications of this structure in terms of the
proposed crystallography, however, this is outside the domain of this thesis, which
focuses on signatures in the dynamics from electronicly driven physics.
The n parameter in the longitudinal cut shows an overall decay, albeit more slowly
than the Bragg peak. In the case of lead-free samples, some peak is seen, most
visibly in strongly underdoped Bi2201 UD25K. A break from monotonic decrease is
clearly confirmed in Bi2201 UD31K as well; this occurs at a smaller wavevector than
97
A
n
80
60
"
A
A
40
A
* I
*05f!*
-0.4
-0.2
8
0
0
I 0
5
S
A
0 A
3
A
0
I
4
A
U
E
$4
U
-0.4
-0.2
S
a
U U
0 A 0 S S
A * A A A 3
A
I
0.0
F
5F
*
E
T
T
I
3.
2
I
2
-0.2
0.4
0.2
4
0
-0.4
+L
1
0.4
5
6
0
J
,S
0.2
0.0
I
i
0
A
0
A
4
31
*a
A
6- 11
5
4 **
1
0.2
0.0
0.4
.5
Ix'
E
-0.4
I
-0.2
0.0
j
I
0.2
0.4
E
Cph
4
+
* UD27K Pb,Bi2201
* UD25K Bi2201
A OD70K Bi2212
0.3
*
+4
.
-0.4
4
0.2 -t
AL
-0.2
I, ~
I
0.1
0.0
0.2
0.4
m UD32K Pb,Bi2201
e UD31K Bi2201
A UD50K Bi2212
U
4
E
Figure 3-17: DHO fit parameters for the longitudinal cut (2 + e, 2 + 6, 0)
that of the UD31K sample; this dependence will later be analyzer in Section
3.5.3.
In the context of the clear signatures seen for Bi2201, one can analyze the falloff
line in Bi2212 to suggest similar humped structures coexisting with the falloff in
intensity. The signature in OD70K is here slightly weaker in amplitude but is more
98
A
50
40
n
A
5
.
.,
0
A
- .
0
0
-0.4
-0.2
I
A
10
*0
0.0
0.2
A
A
54
0 3
2
0
.
A
gO
0
Al
0 0
1
0
0 A A
0
0
2.0
A
1.5
I
-0.2
0.0
t
0.4
0.2
*
0
C
I
1.0
0.5
1
-0.4
0.
0
&
*Ii I
1
. . . . . . . . . . . . . . . . . . .0.2
0.4
-0.4
-0.2
0.0
E
0.4
0
6
0
2
*
7
0
03
4
I
*A 1
I
0
0
20-
I'
14
I
I
30
-I
I
I
I
'4
It
I 1 I'
-0.4
-0.2
0.0
0.2
E
-
0.4
E
Cph
0.7
0.6
0.5
$
* UD25K Bi2201
A OD70K Bi2212
0.3
I
-0.4
0.2
0.1
4
-0.2
0.0
* UD31K Bi2201
A UD50K Bi2212
*E
0.2
0.4
Figure 3-18: DHO fit parameters for the transverse cut (2 - c, 2 + c, 0)
clearly analyzed than the broad feature ascribed to the structure in UD50K. The
lead-doped samples here provide references curves for which no such falloff is seen. In
the transverse measurements, similar overall falloff is seen, although the amplitude
to scatter stays high toward the zone center.
99
The bare phonon frequency shows a surprising amount of scatter between measurements in Bi2201, with the maximal energy recorded at the peak of the dispersion
varying from almost 8 meV in Bi2201 UD25K to only marginally over 6 meV in
Bi2201 UD31K. We will show later in Section 3.5.2 that these curves are very close
to self-similar except for an overall stiffness factor. The side c < 0 shows a surprisingly
different dispersion than that for e > 0, where around c = -0.3 the observed scatter
is primarily from a sharp mode at a very low energy 4 meV. This data is a reflection
of broken inversion symmetry in the dynamics, which causes different scattering amplitudes for phonon modes to be observed at tE dependent on the eigenvector phases.
A toy model reproducing this phenomena is introduced in Section 3.5.4 and use to
interpret this data in terms of inversion symmetry breaking. The transverse mode is
not similarly asymmetric, and shows a quite normal dispersion for all samples, albeit
with an again quite strong difference in stiffness between samples.
The F parameter is a measure of the full width at half maximum, and is proportional to the phonon self-energy, inversely proportional to the phonon lifetime. A
high value indicates strong coupling to states which can decay the phonon; an example process by which this occurs is shown in the Feynman diagram in Fig 3-5. All
lead-free samples are seen to exhibit an anomalous increase in the self-energy along
the longitudinal cut that is not matched in the transverse, which seems to remain
relatively low, although increases toward the zone boundary. The broadening of the
transverse at the zone boundary is statistically significant, although it is more delicate than that found in the longitudinal since the resolution function contributes a
comparable broadening
-
1.5 meV, and as previously remarked, such terms add as
a quadrature sum, and we are relatively less sensitive to intrinsic lifetimes below the
width of the resolution function and refrain from remarking on the physical process
of relevance here. In the longitudinal, the doping dependence is very pronounced,
suggesting further underdoped systems are described by a stronger coupling vertex
than those closer to optimally doping which yields a maximal effect at a larger E in
the underdoped relative to optimally doped case. Lead again is seen to eliminate
this effect; some trace of additional breadth is present near 6 = 0.3, but this is not
100
statistically significant outside of modeling error.
Lastly, the phonon sea amplitude is seen to generally take values between 0.1 and
0.2 c/s. As previously remarked, this is an approximation of a likely decaying functional form as the phonon sea is seen to be suppressed at high energy transfer; the
magnitude of
Cph
is then in part a reflection of the scan range, and physical inter-
pretation of this parameter should be refrained from. Different background models
strongly changed this value, but left the rest of the resulting fit parameters essentially
unaffected within error.
For all scans, an important caveat is that resolution of fit parameters close to the IF
point becomes unreliable as the phonon lineshape merges with the resolution function;
for all scans of e > 0.1 no such troubles are present, but data taken at E = 0.07 can
show fitting artifacts due to this proximity. The region of principle interest E = 0.25
is far from these effects and all terms in the lineshape are well separated in domain
of relevance; the functional space is not over parameterized.
3.5.1
Doping Dependence
The critical new observation in FT-STM in Bi2201[60] was that the observed electron
density of state maxima scaled with doping, consistent with being driven by Fermi
surface nesting. This comment is in reference to the bare bands; the actual excitation
spectra measured in ARPES in the antinodal region shows no clear Fermi surface or
quasiparticles, with instead a gapped out spectra and broad predominantly incoherent
spectral weight [72]. Interpreting this data as evidence for existence of a charge density
wave at least at the surface, ARPES measurements then are consistent with strong
electron coupling to this resulting in, and the loss of quasiparticle integrity in the
antinodal region is ascribed to short correlation length CDW domains.
We observe evidence in Bi2201 UD25K and Bi2201 UDIK of a similar trend in
doping dependence of the anomalies in the longitudinal acoustic phonon, suggesting
the observed breath is due to a similar origin as seen in FT-STM, consistent with coupling to a Fermi-surface driven phenomena. This is also supportive of the argument
previously developed that the breadth is not to be associated with the superstructure
101
or modes emanating from these Bragg peaks; the superlattice ordering vector qaj is
identical between these samples, and physics originating from that scatter should not
shift as is seen for the center location of the peaks in n, F. This all suggests that
electron excitation degrees of freedom near the Fermi surface are responsible for coupling to produce the observed broadening rather than an anomalous electron-phonon
coupling, which would more strongly pair all electrons and be, to first order, insensitive to the location of the Fermi surface. This is not to suggest the electron-phonon
coupling constant is low, but rather that the change in the momentum space peak of
the features suggests that electron-phonon anisotropy as strongly seen, for example,
the oxygen bond-stretching and bond-bending modes[91] does not drive the acoustic
phonon anomaly. This is consistent with what is seen on the surface by STM; had
electron phonon coupling been responsible, the observed doping dependence of the
peak in FT-STM would likely have not been observed.
We can quantitatively analyze the data several ways to pull out momentum dependence. We consider three here: first, we fit F to a Gaussian, and compare center
location and amplitude between samples. A second Gaussian fixed at the origin is
added to better describe the upturn seen in some cases near the gamma point, as in
Bi2201 UD25K and Bi2212 OD70K. This feature is believed to be associated with
difficulties describing the lineshape at low dispersion due to the mixed parameter
space of the Pseudo-Voight elastic line and the DHO phonon. A similar process is
done with the longitudinal n parameter; here we append to the fitting form a linear
term to approximate the background. Lastly, we calculate the inferred phonon weight
in the limit e -+ 0, that is, the dynamic contribution to the intensity at w
=
0 due to
slow fluctuations, and fit this to a sum of two Gaussians, representing a falloff near
the gamma point due to the strong structure factor, picked up again near the nesting
wavevector due to strong coupling.
All of these probes show doping dependence in the functional dependence of the
momentum associated with the anomaly, as presented below in Table
3.2.
The
linear background used in fitting n works reasonably well for Bi2201 UD25K, Bi2201
UD31K, and Bi2212 UD50K, but does not capture the nonlinearities of the trend in
102
6-
1.2
5
1.0
4
0.81
3-
0.6
2
0.4
1
0.2
0.
0.1
0.2
0.3
0.4
0.
E
0.
0.0
0.1
0.2
0.5
0.4
0.3
n(
OQ
7
6-
I ~
5-
70.1
43-
0.2
T
T
0.3
0.4
-20L
i
2S0
1
0.1
0.2
0.3
0.4
-40 1
0.5 -60
Figure 3-19: Momentum dependence as a function of doping for DHO fit parameters
for the longitudinal cut (2 + c, 2 + e, 0) Legend as per 3-17
Bi2212 OD70K. To obtain a satisfactory fit, we use a second gaussian lineshape at
the origin in place of a linear approximation. The comparison in the location of the
second (free) Gaussian tabled in Table 3.2 with that obtained from Bi2212 UD50K
with a linear background term is not exact then due to the different lineshape used,
which is denoted by a (*) next to the result for Bi2212 OD70K.
To check the dependence in this data set, we turn to an alternate method of
examination; plotted in the bottom right panel of Figure
3-19 is the numerical
derivative of n(Q). Estimating the maximal point in the range 0.1-0.3 then is a
measure of approximation for where the peak of an underlying function may lie,
absent a monotonically decreasing overall dependence of n. We can see this matches
the
Q dependence
seen for n in Bi2201, with UD25K clearly peaked further to the right
than UD31K; in Bi2212, note OD70K hits positive slope at e = 0.175 (this is using
103
Peak MaximaE
Bi2201 UD25K
Bi2201 UD31K
Bi2212 UD50K
Bi2212 OD70K
S(Q,0)
IF
0.298
0.263
0.252
0.237
±
±
+
+
0.009
0.004
0.004
0.003
0.257
0.223
0.182
0.174
t
t
±
t
0.012
0.010
0.055
0.014
n
0.306
0.222
0.232
0.188
+
+
±
±
0.018
0.007
0.004
0.005*
Table 3.2: Momentum dependence of the peak position in E of anomalous properties
as a function of doping. (*) Different background lineshape used as explained in text
data at E = 0.15 and E = 0.2 to calculate the slope), while UD50K is maximal further
to the right at E
=
0.225. Thus this method confirms the doping dependence listed
in Table 3.2 calculated by fitting a Gaussian and background lineshape. In terms
of all the parameters, the doping dependence is Bi2212 is only weakly statistically
significant while that in Bi2201 is substantially stronger.
This result is in good agreement with the doping dependence in the STM literature. In the seminal work, Wise et al. found an associated wavevector of 0.215 for the
Pb,Bi2201 UD32K sample in use in that study, and 0.236 for the Pb,Bi2201 UD25K
in use. Results in Bi2212 have confirmed the presence, but have not shown as strong
a doping dependence as in Bi2201[62]. The good agreement in both directionality
and magnitude of the shift with doping between both Bi2201 and Bi2212 suggests
that the same underlying physics that drive formation of FT-STM intensity are responsible for the observed broadening and spectral weight here. While many studies
on phonons in high T, materials have been conducted as will be reviewed at the
end of this chapter and plenty of effects have been seen to be doping dependent and
enhanced in the underdoped regime, this work is the first to track a specific phonon
feature in momentum space as a function of doping, and offers the most conclusive
signature of coupling to Fermi-surface driven effects.
STM measurements and our own measurements of n(Q) and S(Q, w -
0) both
measure slightly skewed peak lineshapes even if the underlying anomaly was pure
Gaussian due to the presence of complications in terms of functionally unknown and
sloped backgrounds. At fixed eigenvector, the scattering amplitude for a phonon falls
off as 1/w,; here, as we know the frequency of the phonon actually decreases for all
104
c > 0.3, the observed decline of amplitude suggest other factors such as the exact
nature of the eigenvector play a role suppressing phonon scattering closer to the zone
boundary; this dependence is then observed in addition to any increase in scattering
amplitude close to where tendency is to charge density waves. The measurement of
' is then the most pure probe of the phonon self-energy and coupling effects, and is
only directly observable in energy-resolved scattering experiments.
The measurement of S(Q, w -+ 0) bears some comment on interpretation. To
clarify, by this we refer to the fitted functional form, but absent the convolution
over the resolution ellipse; the data is fitted including this convolution, but then
the intrinsic w -+ 0 scatter is computed by examining the fitted lineshape prior to
convolution, and hence this excludes the w = 0 static lineshape. In this sample,
there is large and broad elastic scatter from disorder; both random chemical dopants
and the closure of incommensurate periods cause the exact Bragg translation to be
lost, and shift static scattering to incoherent momentum transfer positions. This
lineshape is found to decay fast enough around the Bragg peak that no peak in A(Q)
is observed through the cut (2 + E, 2 + ,0).
The calculation of S(Q, W
-
0) as
extrapolated from fit parameters is essentially dominated by scatter in the range 2-10
meV and allows a probe of the component of slow dynamics in this scale free from
the contaminating signal of incoherent elastic scatter. This allows observation of a
peak in intensity at w = 0 as a function of Q; note this is not a peak as a function of
w such as a central peak found in the case of slow dynamics in SrTiO 3 [106]; in that
case spectral weight is condensing into an elastic (Bragg) peak as the temperature
was varied, while in this case we note only that the tendency of phonons to be close
to overdamped drives significantly more weight to low energy at certain momenta,
producing a peak in the dynamic susceptibility after subtraction of the purely elastic
scatter as a function of Q at fixed w = 0. Were impurities or other pinning sites to
then condense out spectral weight into the elastic, it is feasible for this maxima in
S(Q, w
-+
0) to induce momenta dependence in the observed density of states. It is
this which we suggest is the observation in STM and ARPES in the nodal regime,
where the reduced dimensionality of the surface is sufficient aid to pinning to also
105
overwhelm the subtleties observed between lead-doped and undoped Bi2201.
3.5.2
Renormalized Dispersion
The observed variance in dispersion of the longitudinal and transverse acoustic phonons
between doping levels of Bi2201 highlight the difficulty of accurately modeling the
phonon properties; in the absence of disordered dopants, these materials should be
equivalent; the - 2 meV observed difference on w,, is then a measure of the strength
the interaction of lattice vibrations with dopant atoms. To first order, much of the
variance can be collapsed by an overall scale factor between samples, as shown in
Figure 3-20. This scaling is computed by renormalizing each scan by the slope at
low E is 1.
Explicitly, to estimate the slope close to the Gamma point we take the average
between slopes inferred at E= 0.1, 0.15 in the longitudinal measurements and C =
0.1, 0.2 in the transverse. These values were chosen since they are the lowest pair
of momentum points available in all data sets, and represent a compromise between
low E where the slope of the dispersion is closer to the speed of sound and higher E
where the fit parameters are better resolved; averaging the two values also reduces
statistical error. The transverse shows strong agreement between the four data sets
for all c, while in the longitudinal the lineshapes diverge around c = 0.2. The observed
maximal spread is now 10% of the value, where in the comparable raw scan shown in
Figure 3-17, 3-18 exhibits a variance 30% of the value (w,(2.3, 2.3, 0) - [5.5, 8] meV).
The transverse, for most of the dispersion, is collapsed to a singular value within 5%.
If almost all of the observed difference if phonon measurements can be explained
by differences in material stiffness and an overall scale factor, the resulting picture is
consistent with doping-driven softening. In the longitudinal cut, the black and purple
data corresponding to Pb,Bi2201 UD27K and Bi2201 UD25K are seen to disperse
to the largest value in terms of the renormalized dispersion. If we then postulate a
universal curve for all data in absence of electron-phonon interaction, this observation
is consistent with a softening in a doping-dependent manner in which softening once
sufficient momentum transfer is reached so as to couple electrons on opposite sides
106
Longitudinal w0
@(2 +E,2 +E,O),
300K
0.5 F
4
0.4
I
0.3
I
.II
0
I
*
B
0.1
-0.4
-0.2
0.4
0.2
0.0
E
Transverse wo @(2 -E,2 +E,O), 300K
A
0.5
0
0* aA
A
A
0.4
I
2I
a
0.3
I
0.2
0.1
I
I
-0.4
-0.2
I
0.2
0.0
0.4
E
Figure 3-20: Momentum dependence as a function of doping for DHO fit parameters
for the longitudinal cut (2+ , 2 + E, 0)
of the Fermi arcs, resulting in phonon softening for all c greater than this minimal
nesting satisfying condition. Thus the lower maximal dispersion seen in Pb,Bi2201
UD32K and Bi2201 UD31K are in this explanation due to softening occurring at
107
a lower wavevector than in Pb,Bi2201 UD27K and Bi2201 UD25K, resulting in the
renormalized dispersion not reaching as high of a maximal value. In fact, by eye one
can observe that for Bi2201 UD31K for example, the phonon dispersion seems to have
flattened out after E
=
=
0.2, while for Bi2201 UD25K this does not occur till a later
0.3.
Interpretation of the bare phonon frequency in these terms is less definitive of a
case than the direct readout of doping-dependence in the momenta transfer associated
with increased broadening, but in the context of those established measurements, such
analysis of the dispersion seems feasible. Thus the total picture of the phonon dispersion, which initially was seen to vary widely between samples, can be summarized in
terms of two sample dependent factors: an overall stiffness parameter showing a complex dependence (maximal for higher dopings in the presence of lead or lower dopings
in the absence of lead) and a doping-driven screening factor resulting in softening of
the dispersion due to electron-phonon effects.
3.5.3
Finite Domain Model and Coherence Off High Symmetry Cut
The observed range over which the phonon is observed to be broad in Figure 3-6
is quite broad; from this, we can construct a measure of the correlation length in
real space of the effects which drive this physics by use of the Finite Domain model.
This model simply calculates the width of the intensity of scatter near a Bragg peak
computed from a small crystal; the observed breadth in momentum space of the
lineshape is then a direct measure of the size of the crystal domain.
Averaging
of such domains has no coherence and hence the lineshape is not modified by the
postulate that space is tiled with such domains in a disordered manner. Specifically,
the calculation is that for m atoms ordered with a spacing a, the intensity is
2
sin 2 M(aQ)/2
sin 2 aQ/2
(3.6)
The first zero of this function away from maxima at aQ = 27rn for n an integer
108
is at
Q
=
27r/(Ma), so the FWHM is approximately 27r/(Ma) as well. In fitting
the linewidth in Section
,
we obtained FWHM values of 0.179 ± 0.01, 0.154 ± 0.01,
0.249 ± 0.01, and 0.188 ± 0.01 for Bi2201 UD25K, Bi2201 UD31K, Bi2212 UD50K,
and Bi2212 UD70K, all measured in rlu, which in turn correspond to domains of
size 25A, 351A, 22A, and 29A respectively, with error ±2A. This size domain is in
agreement with the coherence length used to model ARPES data[72] of 38 A and
the region over which the pseudogap was found to be singular valued in STM[43] of
around 20A and indicates local doping drives nanoscale variations in the electronic
filling fraction and is responsible for the breadth of the observed effect.
We can examine the loss of breadth in a transverse cut through the observed peak,
eg (2.25 +
,
2.25 -
,
0). However, the difficulty here is that the eigenmode structure
also changes and the exact decomposition into longitudinal and transverse modes no
longer holds; we expect to see up to three modes for each measured . In the figure
below, we compare cuts along two directions :
, which offers direct measurement of
the falloff in width of the lineshape in the cut direction orthogonal to the longitudinal,
and a, which compares the effect of the superstructure between scans either passing
directly adjacent to it or far removed.
Both of these cuts make use of the tetragonal symmetry mirror plane described
by a normal vector of -!(a*- b*).
This symmetry is broken by the orthorhombic
distortion and the superstructure, and comparison between these scans highlights
the measure by which these effects influence the phonon dynamics. The intrinsic
electron-phonon driven coupling is expected to respect the tetragonal symmetry, and
we expect the comparison can be used to highlight this term, since the component
of the phonon spectra that is varying between the compared blue and purple scans
is that due to additional non-fundamental (tetragonal symmetry breaking) effects,
while that that is the same is the traditional continuous extension of the longitudinal
acoustic mode through reciprocal space.
Scans along
that are denoted in Figure 3-21 as at (
(, 2.25 + (, 0) and (2.25 +
, 2.25 -
0.1 show data at (2.25 -
, 0) in blue and purple respectively, as shown in
the insert. Along a the comparison is (2.25 - a, 2.25, 0) and (2.25, 2.25 - a, 0) in blue
109
e=0
A
=01
.05
B
2.5
.0
5
.5
5
-15-10-5
a=e=0
2.5-
C
H
5
a=0.1
10 15
E
3.5
3.0
2.5
a=0.5
6
5
4
3
2
-15-10-5
10 15 -15-10-5
D
5
-15-10-5
_
3.0
2.
4[J
5
10 15 -15-10 -5
G
a=0.4
10 15 -15-10-5
F
8
5
6
4
5
10 15 -15-10 -5
10 15 -15-10 -5
5
10 15
5
10 15
6 =0.3
7
6
5
=0.2
67
5-
Figure 3-21: (A-F) Cuts away from the high symmetry direction (E E, 0) passing
through (2.25,2.25,0). Top right inset: Reciprocal space diagram of scan directions
(blue and purple arrows) and Bragg structure (solid circle: (220) Bragg peak, 'X':
out-of plane superstructure). Similarities between blue and purple scans highlight
features obeying a tetragonal reflection symmetry contrasted to differences associated
with symmetry-reducing superstructure associated modes.
and purple respectively. We first consider the scans along the az cut. For blue scans
well separated from the superstructure peaks along K, the lineshape is well fit by a
single peaked function. Formally, we expect that the scatter at O = 0.25 along this
direction decomposes along the high symmetry directions (100) and (010), for which
we should be equally sensitive to the transverse and longitudinal mode. However,
within resolution a single lineshape is observed, suggesting either one of these sources
dominates (along (110) the transverse reaches a much greater maximal amplitude) or
they are close in energy. Regardless, no evidence in this is seen for a second mode of
any sort, and furthermore the breadth is greatest near a = 0, suggesting the multiple
110
modes which may appear near a = 0.25 are not driving significant width.
This
suggests we are tracking the evolution of the mode that is longitudinal at (2.25, 2.25,
0) along this cut as its character changes.
In contrast, the scans taken along a shown in purple down the K axis reveal
the presence of an additional peak. The lineshapes fit as shown in Figure
3-21
are constrained to contain a resolution limited phonon peak at w2 and a phonon
peak of finite lifetime characterized by the same wo, F values as in the corresponding
(blue) scan along H with equivalent a. This hypothesis fits well and supports the
decomposition into a tetragonal symmetry obeying mode common to the cuts along H
and K and a second supermodulation-induced peak that does not respect tetragonal
symmetry and is most strongly observed upon approaching the superlattice. These
fit parameters are summarized in Figure 3-22. Any evidence of tetragonal symmetry
breaking along the a cut along H is outside our detection limit here.
By approximating a smooth curve to the functional dependence of w2 along a,
we can test the robustness of the assumption that there is no additional scatter at
(2.25,2.25,0) from this mode. Fitting to an additional phonon amplitude with this
fixed energy (as shown by a hollow circle in Figure 3-22) results in the amplitude
of this mode being driven to zero. Fitting confirms the intuition from the raw data
that the scatter at (2.25,2.25,0) is well fit by a single mode, and the additional modes
resulting from tetragonal symmetry breaking induced by the superstructure are absent
at this momenta. This assumption is reasonably robust; assuming a larger w2 = 3
meV results in a nontrivial. n 2 = 0.059 ± 0.028, compared with n ~ 3 for the main
phonon as shown in Figure 3-17.
We can now calculate the correlation length off axis of the phonon breadth parameter F using the value computed along the H cut of a or the (110) component of ,
since these terms have been shown to have no measurable phonon contribution from
the superstructure. This results in an inferred FWHM along the transverse direction
of the broadening of 0.192 ± 0.015. This indicates a transverse to bond modulation
correlation length of 30 At 2 A, corresponding to, within error, isotropic broadening.
This is consistent with the observed breadth over which the momentum is observed
111
(meV)
U2 (meV)6
r (meV)
Lo
n 2 (au)
5
0-2
*
0.0
0.1
S
0.2
0.3
0.4
Figure 3-22: Fit parameters from the cut a shown in Figure
0.5
3-21. Plotted is the
wo and F parameters describing the main (broadened) phonon respecting tetragonal
symmetry and a second resolution limited phonon of amplitude n 2 at energy w 2 ; for
comparison, the amplitude of the tetragonal phonon ranges from 4-6 on the same
au scale. Dashed line shows a linear extrapolation leading to a predicted value of
W2 = 2.3 meV used to check the robustness of the primary phonon parameters
to be broad being driven by finite size domains over which the doping changes; this
sizing constraint is naturally roughly isotropic, as seen in STM
[43],
unlike what
might be observed from a broadening driven by anomalous electron-phonon coupling,
which could drive the response function in an anisotropic manner with respect to the
longitudinal or transverse to bond stretching cuts.
A final comment about polarization bears mention - as suggested by the ordering
of scans in Figure 3-21, there is an equivalence generated by the 180" rotation about b*
resulting in formal equivalence of the eigenspectra in the L = 0 subspace at a, 0.5
-
a,
which relates scans C/H, D/G. and E/F . This suggests that what is seen in Figure
3-22 can in fact be compressed by inversion about a
=
0.25, and the modes measured
below 3 meV at a < 0.25 must formally exist at the corresponding 0.5 - a, and vice
112
versa for the modes at a > 0.25 above 3 meV, even though no direct evidence in these
scans is seen for this. We also know from the scans along (0, 2+, 0) that a low energy
(2-3) meV mode is present at a = 0.25, likely polarized predominately along (010),
which could be responsible for evolving into the dispersion in W2 seen. Insofar as the
structure of Figure 3-22 is self-similar along reflection about a = 0.25, this suggests
character of (010) scatter persists; as the scatter becomes (110) or (110) polarized, it
gains different amplitude to scatter in the different zones and potentially results in
differences in w 2 for these values of a. Seeing as the W2(a) plot is relatively flat, this
suggests that the superstructure induced phonon retains predominately symmetric
(010) polarized character.
3.5.4
Asymmetry in Scattering Between (2 ± c, 2 ± c, 0)
The equation derived in 2.20 indicates the phonon amplitudes are the solution to
a secular eigenvalue problem. The eigenspectra will in fact always be symmetric
under q
-+ -q,
as we show below by proving the characteristic matrix governing the
secular equation is Hermitian. Given a Hermitian matrix, the complex conjugate of
the matrix, which in this case is the result of q' -+ -q, is satisfied by an equivalent
set of eigenvalues, with complex conjugated eigenvectors.
13 Dnv/'
,mvaePt
ce
Dmnz-"o'
-rm
E
m
epS
m)
Dnva,(2n-m)va~eitnm
Dnvamvace iftmFn)
m
(S:
Dnva,mvaeiT(Fn-Fm))*
(3.7)
m
where we have made use in. the first line of symmetry of the force constants under
exchange of a pair of atoms; in the second, invariance of the force under translation
under equal distance of the two unit cells; in the third, redefined the sum over m,
113
and in the fourth, explicitly simplified the sum of position vectors. This implies that
the matrix D expressed in the subspace v 0 a is Hermitian, since exchange of the
arguments is equal to complex conjugation. Now we note that the formal eigenvalues
will always be equivalent under inversion through the origin, but the amplitude to
scatter into the mode may change.
We investigate here the possibility for this explanation to produce the asymmetry
seen in Figure 3-17 between w,(tc). Under this inversion, the phonon amplitude is
related by
2
fIZ Arpa(q) ei(G+q-.r
I(G + q)oci
pAa WP(J
__
T
2
A * (q-J
1l
-1v--
-
where we have neglected overall constants and the energy dependence and assumed
the phonon propogator is equivalent at ±q. This indicates that the intensity for
longitudinal scattering at
q'is equivalent up to
Q2 factors
if
ei0*br =
1 for all T (and
the form factor is real and is equivalent at G t q). At (220) this is satisfied for all
atoms in the high symmetry tetragonal representation, but this equivalence is lost
under the presence of sucessive distortions.
Considering the isostructural BiSrFeO
compound synthesized[1], we project the atom positions into the orthorhombic cell
to find that the Bi atoms in this compound reside at a E [0.225, 0.236] U [0.764, 0.775]
while b E [0.685,0.8215] U [0.179,0.327] in relative lattice coordinates.
Thus due
to the presence of superstructure, one expects shifts of the Bi atom from centering
conditions of up to 0.15 rlu along the direction of the superstructure. This implies
that modes at tq measured here are equivalent in bare frequency and polarization
direction, but are scattered into with substantially different selection rules.
In BSCCO, what appears to be the case is that the presence of an incommensurate
modulation results in a low-lying optic mode which features a linear (accoustic-like)
dispersion at energies above 0.5 meV, similar to what is seen along the (010) cut as
measured in neutron[105] and described in Section
114
3.4.4. The measured inelastic
intensity is then a sum of scattering from two distinct modes. We can search for this
by simultaneously optimizing fits for (2+
, 2 + E, 0) and (2 - e, 2 - c, 0). Let us denote
the position of a second mode by w2 and characterize its amplitude by n 2 ; we fix
the mode to be resolution-limited to constrain the number of parameters; the results
presented below will confirm the results show good fidelity to the data and variable
width is not justified.
We perform the following procedure: Given a point in the space (wi, w2 , F) let
us induce a measure X2
X2
+
X2 whereby x 2 we refer to the quadrature sum of
the residuals from the fit value produced at (w1 , w2 , F) normalized by the standard
deviation, divided by the number of degrees of freedom (essentially normalizing to
the number of data points). The fit is produced by the functional lineshape given
by the parameter space (argmini,n
2 ,A X2.(n,, n 2 , A, w 1 ,w 2 , F))0(wi,w 2 , F) where the
constant term is used from the prior fitting described in 3.5. The implication of this
is that for each of tE, given a fixed set (w1i, w 2, F) we minimize X 2 for each of tE
by searching the parameter space of (ni, n 2 , A); the fit values of (wi, w 2 , F) that are
then ascribed to the data are those that best simultaneously satisfy tE, assuming the
scans are independent measures.
The motivation behind this procedure is that given a fixed value (w1 , w2 , F) we
can perform the computation of the fitted lineshape involving a computationally
expensive numerical convolution over the resolution function once, then scale this
lineshape appropriately with addition of the quasielastic specified by the parameter
set (ni,, n 2, A); this computation is then easy, involving a simple quadrature sum of
order 100 terms with prefactors in terms of three variables; argmax is then efficient to
implement, and we have used the computational redundancy of lineshape maximally
in searching fit space. The parameter space (ni ,n 2 , A) for fixed (w1 , w2 , F) is also
very featureless since the correlation between variables is essentially removed when
the peak positions are fixed; the parameters have well defined easily measurable
values, and numerical optimization is simple and should proceed quickly due to the
reduced topological complexity of X2 space. The more difficult search of (wi, w2 , F)
is simply grid searched over with a resolution of 0.2 meV, sufficient for our analysis
115
and resulting in approximately a thousand grid points for each e; since the underlying
convolution is only performed once though for each grid point, this is computationally
feasible.
Bi2201 UD31K: E=±O. 2
2
Bi2201 UD31 K :=± .25
1
2
2
T
0.8.6-
0-
0.4.5-1
0.
w0
-iu
-05
5
2
-11)
10
5
- n
Bi2201 UD31K : c=*0.35
Bi2201 UD31K: c=*0.3
1
1
0
8
6
6
.4
-10
-5
5
10
-10
Bi2212 UD5 OK E=±O.2
4
-10
-5
(L)
10
.4
-'5.
5
Bi2212 UD50K
3
5
2
.0
1
0.5
53
-10
1U
Figure 3-23: Cuts at +c (purple) and -e
-5
10..
to)
E=±QL3
if
5
10
to
(blue) for Bi2212 UD.
The scans shown in Figure 3-23 show the resulting optimal fits in terms of quadrature summed x2 . Each pair of fits shown in the figure is described by equivalent
dispersion parameters, but variable amplitude representing the different scattering
116
form factors responsible for intensity. In the high symmetry undistorted tetragonal
crystal symmetry, the difference in overall scattering cross section for the two modes
is of order of a factor of 1/2, eg for f = 0.3 the ratio is 1.72/2.32 ~~0.55. Clearly,
the scaling between the corresponding blue and purple scans in Figure 3-23 does not
agree up to this scale factor; however, the equivalence that remains is that between
the intrinsic phonon parameters. We can see clearly at e = ±0.2, 0.25 in Bi2201
UD31K that there is symmetry under inversion in c between the position in energy
that a second peak around 3 meV appears at. These scans are the principle evidence
motivating this analysis; all other fit parameters and all the scans taken in La31K are
fairly well described by a single mode; as seen, the amplitude for this second mode is
small, even in these maximally visible regions, on the positive e side the amplitude of
the second modes is only 1/ 3 0th that of the primary mode; however, on the negative
side, this secondary mode becomes the most significant source of scattering, as seen
±0.3 for which the amplitude of the second resolution limited mode is stronger
at c
for E
=
-0.3 than the harder damped mode.
Fitting in this manner requires presence of both data sets from ±E, since the
E > 0 data is sensitive only to a small degree to w2 scattering and there is likely more
residual weight due to statistical error rather than sensitivity to this phonon mode,
leading to poorly defined pinning of the location of this peak to the optic-like phonon
frequency in absence of corresponding data at -c.
In Bi2201 UD31K, data was taken
for E E [-.1, -. 4] with .05 rlu spacing, but in Bi2212 UD50K it was only taken over
the same range with 0.1 rlu spacing. We can slightly extend the range over which
such fitting can be performed by linearly extrapolating between the 0.1 rlu spacing
implied values of w2 and then performing the grid search over only (w, I'), minimizing
only the
X2
term. This produces the two plots shown below, highlighting the change
in fit parameters due to the introduction of a secondary scattering mode at lower
energy.
Before discussing the implications, a caveat is worth mentioning, which is that at
both large c ~~0.4 and small e ~ 0.1 the fit results are less reliable for two reasons: at
large E, the scatter at -E is very weak and the pinning of w2 is not as strongly enforced.
117
Bi2201 UD31K
0W2
El
I
a
6
I
5
El
4
EI I
3
1
I
.1 .
2.1
(0
6
&2
2.2
Bi2212 UD50K
E
El
U
El
5
2.5
2.4
2.3
I
El
31
N
11
0
4
3
2
1
a
2.1
Ii
2.3
2.2
E
2.4
2.5
Figure 3-24: Results of simultaneous fitting longitudinal acoustic mode at ±, shown
in hollow squares, compared with fits to a single mode from Section 3.2.1 in filled
dots
At small c ~ 0.1, the convolution over the momentum component of the resolution
ellipsoid, neglected here due to numerical computational complexity, plays a significant role and is likely responsible for the observed uptick in F seen in Bi2201 UD31K.
In the medium region, e E [0.15, 0.35], these calculations are likely an improvement
118
over the single mode computation previously established. However, we note that this
effects the fit parameters relatively little; the bare frequency w, is shifted by generally
within 0.2 meV, the error due to finite step size in two mode X2 optimization, while F
is decreased outside of error bar, but leaving the conclusion of anomalous broadening
around E = 0.25 invariant; for Bi2212 UD50K, all the differences are within error save
for at E = 0.25, where the self-energy is decreased by - 0.5 meV.
These samples are the only ones on which this computation can be performed,
but as evident from the raw data shown in Figure 3-6, in Bi2212 UD25K no evidence
for a second mode is visible by eye. Differences in samples causing increase of this
scattering are not obvious, but since the focus of this work is on electron-phonon
driven interaction rather than effects of incommensurate structures, we proceed now
that the physically relevant conclusion of interaction-driven broadening have been
shown to be model invariant under addition of this second mode. No evidence for
further structure is seen.
3.6
Shell Model Calculations
We now turn to calculating the phonon dispersion from a shell model.
Given a
set of interactions between all the atoms, the formal process of defining the matrix
of force constants and diagonalization as a function of q is trivial. The difficulty
in this process is creating a good set of phenomenological forces which reproduces
the observed crystal as an equilibrium solution and matches the bulk observables, eg
stiffness and the dielectric constant in the high and low frequency limits. In particular,
the number of free parameters in a force model is on the order of three for each bond
type, producing on the order of 45 degrees of freedom, while the set of observables is
of comparable size corresponding to the three Cartesian coordinates for each atom.
Fortunately for our studies, extensive prior work in BSCCO[107] has been conducted to parameterize the bond interaction parameters and search for a part of
parameter space that well reproduces the known crystal. This work was focused on
matching Raman data concerning c* axis polarized phonons at q ~ 0 with the calcu119
lation making use of Buckingham potentials to correspond with the shell interaction.
We implement this in GULP[108], a program capable of locally relaxing atomic positions to find energy minima and calculating phonon dispersions and eigenvectors
while making use of symmetry operators to block diagonalize the system.
The interactions parameters are fully described in[107] and are not explicitly repeated here; the essential components are that each atom is represented as a positively charged massive core with a massless negatively bound electron cloud held to
the core with a spring constant. Coulomb interactions are retained, as are explicitly parameterized shell-shell interactions of the Buckingham form, described by a
screening length on the order ~ 0.2A for the exponential term. A critical test of
the parameters is positivity of the phonon frequencies along all dispersion cuts; negative eigenvalues are signals that the interaction parameter is unstable with respect
to a symmetry-lowering distortion along that wavevector, and hence the model is a
poor choice. Interestingly, while it is quite easy to perturb the force constants or
atomic positions such that a region of negative eigenvalues can be found, the higher
eigenvectors corresponding to the original acoustic and optic modes seem generally
unperturbed, subject to the addition of a new low energy mode (of course, since
mode number of conserved, such mode originates somewhere in the 'spaghetti' of the
42 optic modes spanning 5-85 meV).
The results of this calculation for the prediction of the longitudinal acoustic modes
are shown below in Figure 3-25. As seen, a simple shell model does not produce a
particularly good fit for the data, especially toward the zone boundary. However, the
dispersion at small 6 is in good agreement with the data. We can read off a calculated
speed of sound of 4310 m/s and 4620 m/s for the longitudinal acoustic for Bi2201
and Bi2212 respectively, while the speed of sound of the transverse is 2910 m/s and
2780 m/s. This is in reasonably good agreement with the measured vales of 4220 and
3690 m/s for the longitudinal mode and 2540 and 2570 m/s for the transverse, where
in all cases the values are listed with Bi2201 first, Bi2212 second. The calculation
values are read off from Bi2201 UD31K and Bi2212 UD50K.
120
3.6.1
Comparison with Optic Mode
While the acoustic phonon is significantly softer in measurement than calculated
from the shell model, the optic data shows reasonable agreement with the first optic
modes measured, as shown in Figure 3-25. Here a second mode w2 corresponding to
a resolution-limited phonon is fit to the data for c > 0.4 in the cases of sufficient data
for this to be well supported; the resultant frequency at the zone boundary is near
12 meV for both samples.
This picture is similar to the result of shell calculations in TiOCl, where several
optic modes were well matched, but significant electron-phonon coupling induced interaction in the longitudinal acoustic ensured that model calculations predicted an
incorrectly hard mode[103]. In TiOCl, the mode would become gapless as a function
of temperature as the system condensed into a incommensurate spin-Peierls state, but
even well above the transition a shell-interaction based computation showed significant discrepancies in the predicted energetics. It is interesting that optic modes seem
to be more resilient with properties more simply determined by the shell interactions
than acoustic modes; this suggests in both systems, interactions more strongly renormalizes the acoustic phonon than the optic; this in turn implies a significant role for
the acoustic phonon in the electron propagator, as will be later discussed along with
ARPES results in Section 3.8.
3.7
Absence of Temperature Dependence in Phonon
Broadening
To this point, all phonon data presented in this thesis has been taken at 300K.
Measurement at high temperature results in substantial increase in phonon scattering
via the Bose scaling factor, roughly a factor of 4 scattering increase over what is seen
at low temperature T
<
w, ~ 70 K, which has allowed us to collect a substantially
larger data set than would be available at low temperature.
Work presented so
far clearly indicates effects of interaction between the electron and phonon systems,
121
A Longitudinal Wo
@(2+E,2+E,0),
300K
14
12
-0.4
U
e
4
-0.2
0.0
B Transverse wo
£
1
0.2
0.4
@(2 -e, 2 +E,O),
300K
12
10
8
4p
Ip
I
6
;iiaA
4
2
K
L
-0.4
-0.2
0.0
0.2
0.4
UD32K Pb-Bi2201 UD31K Bi2201OD70K Bi2212
UD27K Pb-Bi2201 UD25K Bi2201 UD50K Bi2212
Figure 3-25: Fit curves (solid lines) for Bi2201 (black) and Bi2212 (orange) superimposed over phonon fit data
most visibly in terms of an anomalous increase in the self-energy near the nesting
wavevector (1/4,1/4,0). Elastic measurements as a function of temperature showed
122
that this spectral weight does not condense out in the bulk to form an elastic CDW
peak in the intensity, or if such a feature is present, it is extremely small. As a check,
we present here an overview of the temperature dependence of phonon fit parameters
among several scans taken at reduced temperature.
too
8r
I
3
.
a
U
"
U
I
0
U
4
U
U
6
U
Pb,Bi2201 UD32K
300K= a
10K = o
U
Bi2201 UD31K
300K= a
100K= o
Bi2212 UD50K
300K= @
150K= o
2V
0.0
0.1
0.2
C.3
0.4
IE
0.5
F
4-
I
I
32
1
~t
0.0
0.1
I
I M
0.2
0.3
0.4
I _P
0.5
E
Figure 3-26: Temperature dependent w, and F data for several scans; filled squares
are data at room temperature, hollow are at reduced temperature
By comparing the filled and empty data points in Figure 3-26, we notice that the
123
essential phonon description parameters of w, and ' are essentially unchanged outside
of error for all samples at lowered temperature. This is consistent with the anomalies
reflecting, in the bulk, an increase in self-energy due to coupling to dynamic degrees
of freedom but an absence of a formation of a macroscopic condensate of modes such
as a static ordering peak. The measured momentum dependence of the self-energy
peak as a function of doping indicates dependence on electron quasiparticles near
the Fermi surface at the antinodal region. In ARPES coherent spectral weight is
lost from this region below a temperature scale T* associated with the pseudogap
transition[72], and the Landau quasiparticle residue decays into a continuum in the
spectral function; however, the temperature independence observed here suggests
that either this transformation leaves the coupling to phonon modes invariant, or this
temperature dependence is not characteristic of the bulk.
An extensive study combining ARPES results with in-situ LEED (low-energy electron diffraction) and hard X-ray scattering was conducted on Bi2201[78]. ARPES
results revealed several shadow bands, corresponding to closure of the Fermi surface
under two modulation wavevectors, one corresponding to the usual superlattice qs
and the other, found to be temperature dependent with periodicity qe,, e [0.08, 0.12].
This measurement was confirmed in LEED, where as a function of temperature it was
seen that qj is unchanged as a function of temperature as well, while the temperature
dependence in LEED of an additional elastic periodicity exactly matched that seen in
ARPES as a shadow band. In bulk X-Ray on the same samples, the q, modulation
is easily found, but the qnew peak apparent in the 2D probes becomes instead a rod
of scatter along L representing incoherent ordering along that direction with fixed
b* = 0.12, the low temperature value seen at the surface. This implies the surface
structural physics is fundamentally different than the bulk; there is a static crystallographic order peak that is a function of temperature with a critical temperature
dependence comparable to the pseudogap temperature, and this phenomena is different in the bulk where no temperature dependence is observed and scattering is
smeared along c*. This raises the worrisome prospect that APRES measurements
are seeing a temperature dependence induced not by condensation of the bulk into
124
a charge-density or otherwise correlated electron ground state, but instead due to
crystallographic changes at the surface. In particular, the temperature dependence
of this structural feature produces ARPES spectra that, if not interpreted in terms of
an additional shadow mode [109] that is very comparable to the energy drop (gapping
out) of the residual Fermi surface in the antinodal region as was argued [110] to be a
fundamental probe of the pseudogap.
3.8
Implications of Phonons Coupling to Electron
Structure
The most direct measurements of electronic structure, STM and ARPES, both see
evidence at the surface of formation of a CDW, which we believe is pinned out from
dynamics that remain broad and temperature-independent in the bulk. The method
taken here was to examine the acoustic phonons, which are seen to couple at a comparable wavevector as that associated with the phenomena of STM and ARPES. If
electron-phonon renormalization is sufficient to drive the self-energy of the longitudinal mode, it is reasonable to imagine similar anomalies might exist in other modes as
well, particularly those for which the eigenvector strongly couples the phonon to the
Cu-O plane in which the conduction electrons are primarily delocalized.
This work is the first in the literature to conduct detailed study of the acoustic
mode as a function of doping through the zone, but a large family of literature exists
on the Cu-O bond stretching and bond bending modes, located at higher energy.
Generally, these are seen to show anomalous properties around q1 / 4
- (1/4,1/4, 0) in
orthorhombic notation. In YBa 2 Cu3 O 7 , several temperature dependent optic phonons
are seen. First, the half breathing mode is seen to lower from 60 meV to 52 meV
around q1 /4 [51] along the tetragonal b* axis, corresponding in terms of the Cu-O plane
to one of the a* ± b* axes in this work. The eigenvector assigned to this mode corresponds to in-plane motion of the oxygen in the square Cu-O layer, and c axis polarized
motion of the adjacent Ba-O layer oxygens. The assumption is that the in-plane Cu125
o
motion is what couples to the electrons state, and the doping-dependent phonon
renormalization anisotropic along the tetragonal axis is due to electron-phonon coupling, as explained in[111]. Examining the in-plane bond-stretching (Cu-O in plane
motions of Cu) and bending (c axis motion of Cu in Cu-O plane), further work[47]
observed a sensitivity to T, in the phonon amplitude for the b* stretching mode at
high temperature which is suppressed down to 10K. Finally, the tetragonal b* polarized buckling mode is observed to soften between 120K and 5K, while the a* mode
is temperature invariant[52]. Explanation of these phenomena are all associated with
soften driven by formation of a gap due to stripe charge ordering with a wavevector
corresponding to the observed kinks. Some weak evidence for this is seen in hard
X-ray scattering, where analysis of the tails of the Bragg peaks is consistent with the
presence of a second charge order peak with a short correlation length[112].
In HgBa 2 CuO 4 +5,
the principle work[113] has focused on the bond-stretching
phonon, which is seen to disperse steeply along the tetragonal a* axis - this crystal is
truly tetragonal, and presumably the same effect would be seen along the tetragonal
b*. Shell model based calculations in general do a reasonable job reproducing most
of the observed mode spectra below 50 meV, but this bond-stretching mode deviates
strongly from the predictions. This is interpreted as a reflection of significant electronphonon coupling which is absent from shell model calculations. Note measurements
are only taken at relatively low temperatures of 55K. Critically, this work suggests
that the observation of a kink in this mode as seen in YBCO, LSCO, and BSCCO is
not due to material-specific orthorhombic type lattice distortions since these are all
absent from the HBCO system.
Extensive work on the bond stretching phonon in LSCO and LBCO [45, 3] showed
a sharp dispersion with pronounced kink near q1 / 4 . The exact form was seen to vary
as a function of dispersion, with both strongly underdoped and strongly overdoped
samples showing no kink but instead a steep dispersion similar to HBCO, while near
optimally doped samples show a sharp kink. For all samples except for overdamped
ones, the phonon was observed to be anomalously broad near q1/ 4 ; this includes the
case of near 1/8 doping in which static stripes are seen. Thus proximity to the stripe
126
ordered phase in doping seemed to enhance the tendency to form strip ordering.
The bond-stretching phonon in Bi2201 is similarly seen in prior work to broaden
and possibly mix mode character with a second lower-lying mode near 65 meV[2].
This exhaustive summary shows that almost all of the high-temperature cuperates
see phonon anomalies in dispersion or lifetime around halfway between the zone center
and zone boundary along the Cu-O bond direction, eg at qi/ 4 . The feature analyzed
in this work in the acoustic mode is then in accord with this established special
wavevector for electron-phonon coupling. The breadth in the Cu-O bond stretching
phonon near qi/ 4 is proposed to be associated with an observed kink in the ARPES
electron dispersion near 63 meV, which occurs for Fermi arc segments spanned by a
comparable wavevector to q1/4.
The energy resolution of traditional ARPES does not permit resolution of kinks in
the electron dispersion down at the energies relevant for coupling to acoustic phonons
of < 10 meV. Such measurements are possible with laser ARPES, but the resultant
small energy and momentum transfer reduces the accessibility of the antinodal regime.
Probes along the nodal cut show evidence of a kink in the electron dispersion around
10 meV below the Fermi energy[114, 115], consistent with significant renormalization
of the electron propagator due to coupling with acoustic phonons. If phonons are
significant enough in the nodal region to effect the electron dispersion, it is likely that
the same is true in the antinodal regime. In fact, electron phonon coupling is expected
to be several times stronger along the orthorhombic (110) along the Cu-O bond
direction than then (100) as measured by the shorter relaxation time of time-resolved
electron diffraction using a pump-probe approach to dissipate heat through phonon
dynamics[116]. This kink is important in understanding the relevant interactions on
the electron system, but is believed to account for the superconducting glue[117].
To show the combined effect of phonon renormalization in the different high temperature cuperates, we overlay below the measured linewidths of the optic modes
previously mentioned with that measured in the longitudinal acoustic for BSCCO,
on a scale of F/w of the relative self-energy to bare frequency, shown below in Figure
3-27. The anomaly for BSCCO in the acoustic phonon is stronger by a factor of ~ 3
127
on this dimensionless scale.
Accoustic
1
0.7
0.6
B12201 UD25K
B12201 UD31K
Bi2212 UD50K
Bi2212 OD70K
4
9B12201
0.5
LNSCO
0.30.2 0.1
0
0.0
0.1
0.2
0.3
0.4
0.5
Figure 3-27: Ratio of phonon frequency over self-energy, w/IF for acoustic BSCCO
samples (this work) and optic Cu-O bond stretching modes ([2, 3]) measured along
the orthorhombic (110) direction
Electron-phonon coupling is believed to be prevalent and strong in all the cuperates from evidence in ARPES[118] suggesting coupling between a mode around 80
meV and the electron spectral function, causing a kink in the electron dispersion.
While the origin of this Bosonic mode between magnetic and phonon arguments has
long been debated, the consensus among the literature seems to be moving toward
a phonon origin, supported by the ubiquitous previously discussed phonon anomalies at matching wavevectors and energies. This correspondence strongly suggests
important role of electron-phonon coupling which is not well captured in ab initio
calculations [46].
Further signatures of the role of phonons coupling into the electronic state comes
from isotope substitution of "80 for
160
in Bi2212[119]. This work identified a kink
in the peak of the real part of the electron self-energy (equivalent to the kink location) near 70 meV in the normal
160
material, that shifted down by 3.4 meV in the
presence of doping to 180, a shift corresponding to roughly a square root mass depen128
dence as expected. The magnitude of the shift was used as a probe to indicate the
coupling responsible for the kink in ARPES response was due to the breathing mode.
Critically for our work, this shows the character of the Bosonic mode responsible for
coupling at 70 meV is phononic, and is consistent with the case made here for breadth
and universality of strong electron-phonon coupling in which multiple phonon modes
renormalize the electron dispersion or couple in to the formation of electron order
states in materials for which this is in the statics.
129
130
Chapter 4
Symmetry Breaking
The prior chapters of this thesis focused on translational symmetry breaking; as was
argued in Chapter 1, many materials in the high-temperature cuperate family show
evidence toward formation of such a state. The work presented in Chapter 3 argued
in the case of BSCCO the manifestation of this phenomena was existence of phonon
spectral weight at low energies corresponding to a heavily damped phonon due to
electron-phonon coupling. This is not a static CDW as seen in canonical systems
such as the diselenides, but indicates a fluctuations tending toward broken symmetry
with a preferential vector near (1/4,1/4,0) corresponding to intercell ordering. This
ordering wavevector is seen to scale in correspondence with that seen in STM as a
FT-DOS peak, consistent with a doping dependence to the electron response function
driven by Fermi surface population which results in the formation of ~ 30 A domains
over which the local environment controls the effective doping and hence momenta
dependence of this effect. The pseudogap phase has long been associated with broken
symmetry, most notable rotational[120], inversion [121], and time-reversal[122, 57,
123]. However, these measurements have always been somewhat controversial, due
to both instrumentation difficulties since the effect of these broken symmetries is
generally quite small, and potential material dependences[124].
Further comments
on these measurements will be presented in Section 4.5.
Several possible order parameters have been discussed in the literature to explain these broken symmetry phases. Firstly, Neel order is present in the parent
131
antiferromagnet, transitioning into incommensurate SDW order[125] at finite doping,
This in some systems coexists with CDW order[38], for which evidence also exists in
other systems without clear SDW ordering signatures[72, 43]. Theoretical support for
CDW ordering was developed in the large-N expansion[126]. Translational symmetry
is also broken in theoretical computations suggesting a Peierls state in which a subset of squares of Cu-O plaquettes feature increased electron density[127]. Lastly, an
order parameter corresponding to orbital currents running in loops among the Cu-O
bonds was proposed [58], resulting in maintaining lattice translation symmetry but
broken inversion and time-reversal. Resonant X-ray scattering experiments claim to
have directly observed this order parameter in bulk Cu-O[128], and in general there
has recently been recently a growing experimental literature analyzed in terms of this
model, particularly in regards to spin-polarized neutron[57, 129, 130] support for the
relevance of this model. Serious experimental issues with the loop model persist most importantly, such a ordering should result in an observable NMR signal corresponding to a local magnetic field of 260 G[131] which is not seen[132]. Absence of
local fields is also confirmed in muon spin relaxation[133], but theoretical explanations
for this even in the presence of loop current have been suggested[134].
Some aspects of the phase diagram support parts of all these order parameters.
In Bi2201 and Bi2212, ARPES[72] and STM[60] results support the role of CDW
physics, while optical Kerr and spin-polarized neutron [110, 130] measurements suggests time reversal and optical reciprocity results[121] indicates broken inversion.
However, which, if any, of these states is the appropriate order parameter of the
pseudogap is less clear; some may be enhanced by formation of a pseudogap, but the
pseudogap transition may be best described in terms of a different order parameter,
perhaps even one not yet discovered. The critical issue with several of these symmetry breaking results is that it is not clear these effects resolve the true pseudogap
puzzle of gapping out the antinodal region as seen in ARPES and maintaining a sharp
Fermi surface as seen in quantum oscillation[135]. There is not yet a consensus on
the pseudogap order parameter of BSCCO or microscopic description of time reversal
and inversion symmetry breaking.
132
The work presented in this chapter addresses three things: firstly, it confirms the
existence of broken time-reversal and inversion symmetry in BSCCO; secondly, it
makes specific comment on the role of special momenta points in these transitions,
and thirdly, it describes a new experimental way of searching for broken symmetry
in strong electron-phonon coupled materials.
4.1
X"
as Probe of Time-Reversal and Inversion
Symmetry
The X-ray scattering data presented in Chapter 3 is a direct probe of S(Q, w), which
is a Fourier transform of the charge-charge correlation function. Labeling the formal
eigenstates of the system as In), we can write a finite temperature form for the
scattering by converting the trace to identity to over a density matrix to write
S(-w )
=
Z- 1 Ze-Emp-)InPtq,
(4.1)
)
mn
we can exchange the indices to write
S(qw)
=
Z-
1
Z e-(EnEm)-#EmV4)mn16(W
+ Wnm)
(4.2)
mn
(4.3)
= eI8S(-q, -w)
where Z = tr{e-OW}. This is called the principle of detailed balance, and always holds
for a system in thermal equilibrium; any interaction, regardless if between Bosonic
or Fermionic modes will not disrupt this identity since the factor eW comes from
Boltzmann weighting an ensemble of states. If the eigenstates of the Hamiltonian
are time-reversal invariant, then S(-q, -w)
= S(q, -w)
since the reversal maps mo-
mentum states to their conjugate but leaves energy invariant; the same is true for
spatial inversion which also maps q'-
-q'. Note in the literature spatial inversion is
often referred to as a Parity operator acting on the state. Intution for the principle of
detailed balance is that for each excitation of the system which adds momenta q and
133
energy w to the system, there is a (Boltzmann weighted) amplitude for the system
to initially be in this excited state, in which case we know there is a state which is
separated by excitation adding momenta -q and energy -w corresponding to the
initial state of the transition first considered; it is this equivalence mapping which
produces detailed balance.
We can work out a trivial examples for the response function - if the density
fluctuation is long wavelength motion such as for phonons, the density operator pj
-
(At+ A-4) up to some constants, and then
S(qw)
Z-
=
1e-Em*
m
(mIAyAtm)J(w - wy) + (mjA
4
- wy))
((ng + 1)6(w - wy) + ngj(w + way))
-
where ng =
TA_gym)6(w
(4.4)
. In a pure harmonic phonon Hamiltonian (including those aris-
ing from inversion asymmetric crystals), the phonon propogator AAt is the thermal
expectation as simplified above, and furthermore w4
=
satisfy Equation
Interactions are necessary to
4.3 and obey S(-q, w)
=
S(q, w).
w-,, and thus system will
transform the phonon propogator into a term with that is asymmetric under q -+ -q.
If the electron propogator contains asymmetry between tq, this can couple in to
the renormalized phonon propogator to result in an asymmetry between the phonon
propogators at +q.
We can write S(q, w) in terms of the imaginary part of the density-density response, x"(q, w) by noting that
x"(q-,w)
lim 9ZZ1e-Em (4)
=
=
-++(w
77 40+
-rZ-
12
2nm(4)
+ ir)2 Wm
m,n
1 Z:e-Em
(4)mn
2
(6(W
-
Wnm) -
(45)
n
6(W +
Wnm))
(4-6)
m,n
S-7r
(S(, w) - S(, -w))
134
(4.7)
In the presence of time-reversal or inversion symmetry, we can then make use of
the detailed balance condition to write
X"(q, W)
(4.8)
- r (1 - e-w) S(, w)
The measurements presented below are taken at fixed q and we will compare the
intensity at +w. The intuition of this is that such a comparison is in both cases a process which transfers momentum q to the sample, but the case of positive transfer to
the sample, this process corresponds to creating a phonon of momentum q described
by energy wy and in the case of negative energy transfer a phonon must be destroyed
since the energy is bounded below by zero, and thus the process corresponds to annihilating a phonon of momentum -q described by energy w_,. The asymmetry in the
response function is a measure of the asymmetry of the phonon propagator between
these momenta which is induced by condensation of the interacting component into
a broken-symmetry state. If the system is time-reversal or inversion invariant, then
symmetry of the ground state between ±q ensures the density response is equivalent, but if both these symmetries are broken, an asymmetry can be seen in S(q, w)
between +w at fixed q.
This comparison is then facilitated by the data already collected in X-ray scattering presented so far, where extensive measurements at fixed q' of the transfer at tW
have been cataloged. Critically, in X-ray scattering unlike neutron scattering, the resolution function is unchanged between in constant qscans since kf
>
w (by six orders
of magnitude) and hence the instrument need not rotate the scattering angles to scan
energy resulting in an equivalent cross section for all non-intrinsic processes along the
w scan. In neutrons scattering, this effect, if present, will be strongly convoluted with
the changing instrument resolution function, preventing detailed analysis. Only one
side of the energy transfer side in neutron scattering will be in a focused condition
while the other is defocused due to the significant tilt of the resolution ellipse in the
q - w plane.
Use of this measure of comparing x"(q, ±w) is not well established in the general
135
literature, although the theoretical justifications are solid. In x-ray scattering only
materials with magnetic (time-reversal breaking) order can produce this effect, but
these need furthermore exhibit strong spin-phonon coupling such that the phonon
propogator contains significant weight from the asymmetry in the underlying spin
response since the x-ray is to first order not a direct probe of magnetic response.
Neutron scattering studies of chiral magnets have seen asymmetries in the spin excitation at ±q[136] which would presumably correspond to asymmetries in x", but
due to significant resolution ellipsoid issues this method of analysis has not been developed. This work is the first to conduct detailed analysis at tW in scattering as a
method of probing broken symmetry in the bulk. Furthermore, the X-ray work here is
unique in that it is sensitive to both intercell charge orderings which break lattice periodicity and intracell time reversal and inversion symmetries; this is in fact then, one
of the most general probes of broken symmetry, subject to the important caveat that
this measurement is only successful when phonons couple strongly to these symmetry
breaking states.
We present data here in terms of X"(, w) by making use of the conversion implied
in Equation
4.8, which will be used as evidence for time-reversion and inversion
symmetry breaking. In the presence of these symmetry breaking terms, the derivation
arriving at x" from S is invalid; however, use of the ratio as an indicator of broken
symmetry remains valid. In any time-reversal or inversion-symmetric state then,
-1
_
= -eB
S(,w)
S (q' - W)
--
(1 - e-&) S(q, W)
(1 - e13w) S(qj, -W)
q"(_,)
-W)(49
_
"(
=x"G(q'
(4.9)
This implies any region in which the X-ray scattering data on positive and negative energy transfer are not equivalent after conversion to x" using the time-reversal
or inversion implied formula implies the sample must exhibit a broken symmetry
ground state that violates S(q, w) = S(-j, w), or equivalently, both time reversal and inversion symmetry. To improve statistical resolution, we will here construct integrals of x" over a finite window in energy. This is still a valid check;
f'
dwx"(q, w) inequal to - f_' dwX"(q, w) implies existence of at least one energy
136
such that x"(q, w) # -x"(q -w), and thus consequently a broken symmetry ground
state.
Relevance of Absorption and Thermal Equilibrium
The invariance in the presence of time reversal or inversion symmetry between the
intrinsic structure factor (and imaginary component of the dynamic susceptibility)
as described above is absolute up to only the restriction that the system is in good
thermal equilibrium such that the state can be well represented as a thermal weighted
trace. This assumption should be well met, as the system is held at constant temperature scale for order half an hour before such measurements were taken. More
importantly, scattering measurements do not directly probe the charge structure
pj-
e
O
, but instead measure a form-factor weighted average of such, p(
Z fje'Q'i where fh
is the form factor of atom
j, which may in general
-
contain imagi-
nary components to account for the absorption cross section, where absorption is due
to coupling of the photon magnetic field to the spin and orbital angular momentum
degrees of freedom of the electron.
First, let us
The effects measured here are invariant under this modification.
address the presence of time reversal; we will show below that in the presence of
absorption but an underlying time reversal symmetric state the measured intensity
is constrained to be equivalent at constant
Q under the mapping
w
-+
-W. Denoting
the formal eigenstates by quantum number m, we write states as im) characterized
by energy Em, and let the difference of energy between two states be defined as
Wmn =
Em - En to write
dQdE a Z-eOW mne- Em
(mn
iv
iQr
v
2n)16(w
± Wnm)
(4.10)
Wnm)
(4.11)
which under conjugation can be written as
=
Z-le-wE
mn
eEm I(n I
eiQrivf In) 16(W
iv
137
+
inserting TTt = 1 on both sides and acting on Tin) = linT),
SZ-le-w
S
e-,Em
Q-r f*ITmT) 2 6(W
(nT
mn
(4.12)
nm)
iv
we now act by conjugation on the interior of the matrix element, noting the operator
r is even under time reversal to write
da
dQdE c Z-le
mn
e-E (nj
) 12 6(W + Wnm)
iQiv
(4.13)
iv
where we have also noted that for all states n there exists a state nT with equal
energy (time reversal) that is part of the sum according to the Boltzmann population
factor (thermal equilibrium), and have relabeled nT
-+
n. But this line is just the
first equation, with n +-+ m, which is equivalent to a relabeling and
Wmn
=
-Wnm,
-w, and we can write
hence this is equivalent to mapping w
da EQ~-
d_
-
d (QM-w)
d~dE (Q W) edwd~dE'
(4.14)
Which is the desired claim. Next, we address inversion symmetry by noting that
for all atoms at a position 'ii, there exists a conjugate atom i' with i
-rt,
hence
2 6(W
En)I ± Wnm)
(4.15)
=
we can write the sum as
da
d
1
Z-e'W
c0dE
mn
e-,EmI(mI
eiQ-iv + eiQ-ri') f
iv
/
But this term in parenthesis is real, which means that since the norm is invariant
under conjugation, the measured intensity is invariant under the mapping f*
-+
f,, and hence the conjugation present in Equation 4.11 can be undone to result
immediately with the same conclusion in Equation 4.14 as implied now by inversion
and thermal equilibrium.
Thus we have shown that either inversion or time reversal symmetry is sufficient
to require equivalence of the measured intensity (appropriately weighted by the Bose
factor) between positive and negative energy transfer, and thus measuring an asym138
metry in the scattered intensity is sufficient to show both symmetries are broken in
the material. This conclusion can also be reached by examining time reversal invariance of the sample-probe interaction via the principle of reciprocity[137]. Note this
statement is entirely distinct from that of Friedels law, which is a comment about
w = 0 intensity at ±Q, and in this microscopic notation can be observed as noting in
Equation 4.11 the intensity is equivalent to that in Equation 4.10 under Q -+ -Q if
and only if f, is real, or if there exists an isomorphism between matrix elements and
their conjugate, which is equivalent to the presence of inversion symmetry.
4.2
Longitudinal Phonons
In calculating X"(q, w) we first subtract the elastic signal from S(q, w) then perform
the scaling as (1 - e--W). The reason for doing this is that the scaling above truly
applies for the fundamental
da,,
dQdEf'
not for the resolution-function convolved intensity
we measure here. The observed intensity due to the convolution of a delta function at
w
=
0 over the resolution function fallaciously induces additional scatter at negative
energy transfer relative to positive in terms of the scaled X"; by subtracting off this
a
=
0 resolution lineshape, we mitigate the effect of the resolution convolution.
Further comments on the effect of convolution of the resolution function are developed
in Section 4.2.1 below, where the case is made that these effects are not significant.
We can see the basic effect in the raw data of Figure 4-1. In both Bi2201 and
Bi2212 it appears as an asymmetry in the intensity of the scatter at tW occuring
at respective q values of (2.25,2.25,0) and (2.2,2.2,0). Conversion from intensity to
X" makes this asymmetry more clear; upon lowering temperature, we clearly have
X"(q, w) # X"(q, -w)
in the data shown where the effect is maximal for 1OOK and
60K for Bi2201 and Bi2212 respectively. In an interaction-free system X" should be
temperature independent. In the plots of X"(q, w) we see the asymmetry is primarily
found on the higher energy side but as a component of the acoustic phonon peak. We
will define as a measure the integral of X" as the integral between 3
-+
9 meV. This
is seen to capture the spectral weight of the acoustic phonon without generating as
139
lilt
1
4
0.25
300K
JI
0.20
I
0.15
0.10
4
0.05
Bi2201
UD31K
-15
-10
-5
5
10
0
15
i~
t
4
2
6
8
10
14
11,1
Int
100K
S11
0.3(
0.21
0.2(
0.1'
0.1(
0.05
T
.4
-15
-10
5
-5
10
15
Int
-
-15
-10
-5
5
10
15
int
4
6
j{
8
10
12
14
'U
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
1W1
12
1.2
14
1
1
60K
2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1
300K
0
1i f
Ix"I
1
Bi2212
UD50K
12
1.0
0.8
0.6
0.4
0.2
-15
-10
-5
5
10
15
1w?
Figure 4-1: Conversion of raw data (left panels) to IX"I (right panels). Left panels
highlight elastic lineshape, subtracted before conversion. Data for W < 0 (purple)
and w > 0 (blue) superimposed. From top to bottom, data from UD31K Bi2201
at 300K, UD31K Bi2201 at 100K, UD50K at 300K, and UD50K at 60K. Inserts:
Asymmetry scans repeated for equivalent Q on a separate experimental run (after a
warming/cooling cycle)
140
much sensitivity to terms outside that range of the w
=
0 signal on the low side and
background on the positive side. By the lower cutoff of 3 meV we further exclude
the 6(w) scatter from the elastic structure for which the resolution convolution can
inappropriately induce asymmetry in the conversion to X" from measured data.
This calculation is model independent; the issue of if there is a second phonon
at a low energy scale w2 %~ 3 meV is irrelevant, as is the exact decomposition of
the phonon into a phonon sea background. As long as the system is in thermal
equilibrium, detailed balance holds and the disagreement between X" on the sides ±W
is a direct probe of broken time reversal and inversion symmetry. Integrating over a
window of x" improves the statistics, but for broken symmetry we need only a specific
q, w pair for which inequality on ±w is satisfied outside error; the test of the integral
is a sufficient check of this. After subtraction of the elastic scatter (which actually
only marginally effects the data, given this range of integration), this analysis is free
from any fitted parameter.
Performing the integral as described, we define the following terms:
AG(j
X"(q, w)dw
jX±(q)
(4.16)
+ Ix±(ql
jX_(q)|+ IX+(q|
(4.17)
ix-(qll
For the data sets shown in Figure 4-1, this value is (0.0730 ± 0.031, 0.252 t
0.039, -0.002 t 0.020, 0.145 ± 0.034 ) for Bi2201 UD25K at 300/100K and Bi2212
at 300/60K respectively. Further discussion about the temperature dependence will
be continued in Section
4.2.2, but at this point we can clearly assert that both
Bi2201 and Bi2212 show evidence in this measure of a broken symmetry state at
these lower temperatures of 100K and 60K. As a point of clarification about the
numerical approximation of these integrals, the data was binned with step size 0.5
meV for Bi2212 and 0.75 for Bi2201 as a uniform Riemann partition, where this size
corresponded to the step size in the raw data collected. Formally then, including the
binning boundaries, data from 2.75 meV to 9.25 meV is included in this sum.
An important consideration in this work is the potential for spurious background
141
scatter to lead to overestimation of the asymmetry by inducing a thermal scalingviolating constant count rate on both sides of energy transfer. In some scans, analyzers
closed from the sample by a lead screening were seen to exhibit charging and produce
a count rate of order 0.1 c/s. To account for this, the error was enlarged to cover the
family of possible background count rates consistent with the data. To best estimate
this effect, three values of asymmetry as detailed above were calculated: using a
minimal, maximal, and best estimate of the flat background. The reported data
points presented beyond this point (eg in Fig 4-3,4-4) are the best estimate value
with an error bar extended to be the larger of the statistical error from the calculation
of A and the deviation in value from the best and minimal/maximal data sets. This
enlargement of the error captures model error and signifies that our conclusions are
robust to this spurious count channel.
The minimal estimate of background is provided by finding the smallest constant
consistent with (a) positivity, (b)
Ix"(w
< -15)1
does not diverge (formally, that
this is sum is equal, within error, to that for x"(w > 15) - in this range, far from
the phonon weight, this value is essentially zero), and (c) X"(W > 15) > 0 within
statistical error of one standard deviation. The maximal value is the minimal value
of measured intensity after subtraction of the elastic lineshape for IwI > 0. The best
estimate is arrived at by fitting a 1/w
2 +const
term to the tails of the data (IwI > 10)
and taking the smaller of either of the const values provided they are positive and
at least one standard deviation in fit value above zero. If in any of these estimates
a sufficient range of data is not available, a value of 0 is used consistent with the
expectation that the presence of high anomalous background is rare.
4.2.1
Effect of Resolution Function
An important consideration before proceeding is to examine the effect of the resolution
function. The effect of the momentum component is exactly zero because simultaneous measurement of phonons at a range of momentum transfer has no power to induce
asymmetry in the structure factor; the same ensemble is measured at the opposite
energy transfer. The energy component however, as we will see below, can induce
142
an artificial shift in the asymmetry, although we will show it to be negligibly small.
Let us define some additional notation for use in this section only; in concluding, we
will present a measure of the effect due to resolution function and argue it is small,
justifying the reversion to the previous notation. For this section, we write Sm(q, w)
for the measured intensity, and let S(T, w) be the underlying dynamic structure factor. It is this S(q, w) and X"(q, w) obtained from this by multiplication of (1 - e--w )
which obeys the equality in Equation 4.9. We in this section address the deviation
of this ratio induced by comparison in which S
+
Sm.
We everywhere in this section suppress the argument
(q) for notational
simplicity.
We compute the normalized difference of X"(±w), which must be zero for a timereversal or inversion symmetric ground state in the presence of a delta-like resolution
function. In this section, we compute the induced asymmetry by the presence of
a nonzero convolution, assuming time-reversal or inversion still holds. Thus when
necessary, we write S(w) = S(-w)eO' as per Equation 4.3, and suppress the
7-+ -q
transformation since under the time-reversal or inversion symmetric assumption, this
can be mapped back to -q -4 q. Finding a value of Am not equal to zero outside of
the induced error by the resolution function is then a good test of broken symmetry,
indicating this assumption step must be incorrect.
Consider the comparison defined by our asymmetry measure
A =
f dw (f S(w
-
M-J -XMP
x"(-w) + x" (w)
T)g(T)dT (1 - e-Uw) + f S(-w -
f dw (f S(w - r)g(T)dT(1 - e-w) - f S(-w
-
(1 - e 3w))
T)g(T)dr (1 - e10))
T)g(T)dT
(4.18)
Where by x'' (w) we refer to x" as induced by the measured data which includes a
convolution over g(w). We can simplify the terms inside the integral in the numerator
as
143
S(w - T)g(r)dT (1 - e-"w) +
SJS(W -
T)g(T)dr (1 - e-,w) +
= (1 - e-lw)
= (e-+
-
J
1)
S(-w - T)g(T)dT (1 - e"W)
Je-(w-)S(w
J
S(w - T)g(T)dT + (e-"w - 1)
(
±
2)
S(w
-
- T)g(T)dT
e 3 w)
(1 -
e lS(w - r)g(r)dT
(4.19)
T)g(T)dT + JO(34T)g(T)dr
Using our previous result, we can expand this to quadratic order in
f3r
(before
the prefactor in g(T) responsible for the truncation) and then integrate each term to
obtain Am. This results in
f m(e-
) S(W - T)g(T)drdw
- 1) f (r +
3
2 f (1 - e-O ) f S(W - T)g(T)drdw
(4.20)
Where by the integral over w we refer to the integral over the region on the positive
transfer side in which there is nontrivial spectral weight of the acoustic phonon, found
here to be 3 meV to 9 meV. To simplify this, we assume S(w) is a Gaussian described
by a mean p and a standard deviation o-,, then extend the integral over w to between
±oc.
Within our assumption that the region of truncation includes the spectral
weight, this extension adds no spectral weight. The motivation for this extension is
that a direct expansion of S(w) is difficult since for some scans the function is broad
and a series expansion in terms of powers of
A
is convergent, while in other scans
it is resolution-limited and a delta-function like expansion is more appropriate. By
extending the range of integration and assuming a Gaussian in which u-, can run
between the large (smoothly varying) and small (delta function) limit, we can get a
single analytic form that is valid for all data sets. This simplification results in
144
)2
(L --
f
(e-"
1)
-
e
(,2a2+2(,2+02)3
2(,2+)3~
Am
(
2-(a2+2(1+3w))+
5
2(0,2+a2) /
+a2 (2+w2)))
dw
1
e
2 f(1 -eJOw)
(4.21)
Since the exponentials induce a peak around w
=
p, we can series expand the Bose
factors as
(e-4w
-
1)
(-1 + e-op)
-
e ,3(w
P) + O((W - A)2)
-
(4.22)
and this collapses remarkably as
Am =
In this data,
#
-e-"
e20
(1 + eI3p) VT0 2 -3
(1 + eMp)
-e-Mt(--1+e#il)V/7o-
2 2
4(-1+eOt)
(4.23)
E [0.0387,0.1933] meV- 1 corresponding to temperatures between
300K and 60K, o is 0.637 meV is the energy resolution function standard deviation
associated with a FWHM of 1.5 meV, and y ~ 5 meV for the phonon peak, resulting
in of magnitude between A A E [0.0016,0.0084], with larger values approached at
lower temperature. This is small compared with the values of A being measured of
on the order 0.1, and is dominated by statistical error from the numerical integrals
on the order of 0.03. Hence we will neglect this term.
It is important to note that this process is particularly robust because we are
both away from the w = 0 line and integrating over the full phonon spectra. For
a pure delta-function scatter, we can generate the explanatory graphs shown in 4-2.
Critically, the calculation reproduces the induced asymmetry to 5 digits of precision,
suggesting the series expansions are well justified and this effect is well understood
and incapable of producing a calculated asymmetry value comparable with what is
observed.
145
S(w)
1.2
w>0
w<0
1.0
Resolution
0.8
0.6
0.4
0.2
2
4
6
8
10
2
4
6
8
10
Sm(w)
3.05-
A'm(s)
0.07
0.06
0.05
0.04
0.030.020.01
2
8
46
10
Iwi
Figure 4-2: Example showing the process by which time-reversal or inversion symme-
try obeying data is renormalized by the resolution function and converted to Am. In
the last panel, the integrated difference appropriately normalized is Am = 0.004539.
This can be compared with the calculated value of 0.004537 from Equation 4.23.
146
4.2.2
Dependence on Momentum and Temperature
Some temperature dependence was suggested in the selection of data presented in
Figure
4-1. We can make this dependence more explicit by plotting the value of
asymmetry calculated for a series of temperature steps measured on the same sample
at the same momentum. This data is all processed as shown in Figure 4-1 to convert
from raw data to an accurate measure of X", then integrated to improve statistical
resolution (and also, as seen previously, to reduce the effect of the convolution over
the energy resolution ellipsoid).
A(Q)
0.25
1
(2.25,2.25,0) Bi2201 UD31K
(2.20,2.20,0) Bi2212 UD50K
(2.25,2.25,0) Bi2212 OD70K
0.20
0.15
0.10
0.05-50
1Q0
150
-0.05-
200
250
Temperature (K)
Figure 4-3: Temperature dependent asymmetry for Bi2201 and Bi2212. Dashed lines
are guides to the eye drawn kinked at T* for each sample as listed in Table 2.2
In the temperature dependence of both Bi2201 UD31K and Bi2212 UD50K an
enhancement of this signal is seen at low temperatures. The detailed balance scaling
factor 1- e-3' has a similar increase at low temperature, but this asymmetry measurement is not affected directly by this canonical behavior; this measurement indicates
a temperature dependence to the difference in x", which is absent such a temperature factor in simple models of the lattice density-density response. Enhanced signal
at lower temperature suggests that as temperature as lowered, the acoustic phonon
is more strongly affected by time-reversal and inversion breaking state. There are
147
two distinct possibilities responsible for this effect: it is possible the coupling to the
acoustic phonon decays on temperature scales greater than the phonon energy as the
thermalisation disrupts this correlation, or that the underlying order parameter is enhanced at low temperature. Measurements in Bi2201 OD70K indicate that in the case
of an overdoped sample, it appears as no statistically significant signal is obtained at
lower temperature. This indicates the response in A is controlled by a dependence
on the order parameter rather than the phonon energy scale, and furthermore that
the overdoped case does not (at least in the regions measured) indicate any broken
symmetry. More comprehensive measurement is needed, and of course the absence of
any symmetry breaking signal does not automatically imply the symmetry is maintained, but the evidence we have here suggests the phonon couples to the symmetry
broken order parameter and is a good probe of broken symmetry, suggesting such
order parameter is small if nonzero in the overdoped case.
In Bi2201, it appears that the signal does not revert to zero at a scale
-
150
K, but instead approaches a constant value around 0.03. This indicates that timereversal symmetry is formally broken at a higher temperature above the pseudogap
temperature, but as the system is cooled into this state an enhancement of the order
parameter results in increased sensitivity of the acoustic phonon to the broken symmetry state. Some prior work on YBCO suggests a training effect to be discussed
further, indicating that in that material there is a broken time-reversal state at a
temperature scale above the pseudogap temperature, which then is enhanced sharply
upon cooling[122].
The momentum dependence we measure is shown in 4-4. Critically, we can see
that the greatest asymmetry is seen for low temperature Bi2201 at (2.25, 2.25,0) and
for Bi2212 at (2.2, 2.2, 0). Formally, statements of broken inversion and time reversal
concern the system as a whole, not any specific q. The observation of anisotropy in
the asymmetry indicates differences in the response different phonons couple to. The
high values of this effect around the 1/4 nesting wavevector suggest that the timereversal symmetry breaking order parameter couples to the, same electronic states
that drive dynamic softening toward density wave order.
148
A(q,q,O)
Bi2212 UD50K
300K
0.3
150K
100K
60K
0.2
0.1-
1.6
1.8
2.2
2
2.2
24
-0.1-0.2-
Bi2201 UD31K
0.30.2
0.1
1.6
1.8
-0.2
Figure 4-4: Momentum dependent asymmetry A(Q) as defined in Equation 4.11 for
Bi2212 UD50K (top) and Bi2201 UD31K (bottom)
Detailed balance requires that S(q, w) = ewS(-q, -w). Thus the asymmetry seen
at, for example (2.25, 2.25, 0) in Bi2201 UD31K implies that had we measured at
(-2.25, -2.25, 0) we necessarily would have had to seen an equal and opposite sign
of the asymmetry A(-Q). The issue of if closure under reciprocal lattice translation
preserves this is more subtle. In fact, our measurements suggest A(q
149
74 A(q+
G),
since we measure for Bi2212 UD50K a positive signal at both (1.8, 1.8, 0) and (2.2, 2.2,
0). The most likely explanation for this is that the character of the modes probed
is different for q + G and -q + G for G = 0. Due to the lack of crystallographic
inversion symmetry, the phonon cross section produces different amplitudes to scatter
into modes as for tq as outlined in 3.5.4; it specifically suggests that for E < -0.25
the phonon gains primary character of a secondary low lying mode, while for E >
0.25 the measurement continues to be a good probe of the canonical longitudinal
acoustic phonon. This is consistent with the enhancement of error in measurements
for E < 0 corresponding to reduced scattering amplitude, and furthermore with the
zero (within error) measure at (1.75,1.75,0) in Bi2201 UD31K. The nonzero value
probed at (1.8,1.8,0) in Bi2212 UD50K in part potentially a reflection of the increase
amplitude to scatter into the longitudinal acoustic mode at lower E compared with the
zero (within error) measurement at (1.75, 1.75, 0) in Bi2201 UD31K as reproduced
in the toy model of 3.5.4.
Choice of mounting direction is arbitrary. While we measure the same sign of
A(Q) near (220) for both samples, the odds seeing the same sign in the two samples
are 50%. If further work is conducted in this vein, confirming that the asymmetry
measured flips sign under crystal rotation and that for some samples it is measured
to be negative seem like prudent checks. Note that flipping the sample to measure at
-Q can change the absorption factor, but as argued previously, the ratio of scattering
at tw is unaffected by this factor which is equivalent for both sides, and hence the
A(-Q) = -A(Q)
equality should hold since A is normalized in a manner that will
cancel out all overall scale factors such as the absorption correction.
4.3
Transverse Phonons
Transverse phonons were only measured at room temperature. Most scans through
the zone such as those at (1.75, 2.25, 0) show no asymmetry outside of error; however,
no statement can be made with the data collected so far as to if this is due to
an absence of such signature from the form of the coupling of the phonon to the
150
broken symmetry state or simply because the phonon was not measured at reduced
temperatures in which the order parameter is expected to be enhanced. However,
anomalous properties were measured in several samples near the IF point, as visualized
in the raw scans extracted below.
Two features are particularly surprising about these scans: firstly, for Bi2201
UD25K an unusual signature is seen wherein spectral weight appears to move from
the peak at ~3 meV to a second feature around ~5 meV. As seen as the example
scans for the longitudinal scans, this feature is unique to the transverse. Secondly,
the asymmetry between Bi2212 OD at (1.9,2.1,0) and (2.1,1.9,0) appears to satisfy
inversion about the crystallographic unit cell; the A(Q) values calculated here are
equal and opposite within error.
The raw data in the transverse presents an even more direct compelling case than
seen in the longitudinal. The absence of a peak in the spectra of Bi2201 UD25K
at (1.93, 2.07,0), and the signature of higher intensity on negative energy transfer
than positive in Bi2212 OD70K at (1.9,2.1,0) cannot be an artifact of any sort of
spurious background for example. It is also reassuring that in these measurements
A(Q) < 0 is seen in several cases, supporting the well founded intuition that the two
positive values seen in Bi2201 UD31K and Bi2212 UD50K are coincidences rather
than a fundamentally always positive measure.
To convert these low-energy transverse modes to asymmetry values, we integrate
from 2.5 to 3.5 meV (corresponding to data in [2.25,3.75] meV); this still excludes the
resolution function, but captures the majority of the phonon spectral weight. The
values produced in this process are as follows: A(1.93, 2.07, 0) = -0.186 ± 0.042 for
Bi2201 UD25K, A(1.9, 2.1, 0) = -0.048 t 0.020 for Bi2212 OD70K, A(2.1, 1.9, 0)
=
-0.052 t 0.021 for Bi2212 OD70K.
Fundamentally different physics is seen in the transverse compared with the longitudinal measures presented in the previous section; this signature is seen at room
temperature, and in the overdoped case. This signature quickly dissipates at momentum transfers away from the Gamma point; in Bi2212 OD70K no evidence is found
for the next points measured at (2.2,1.8,0) (A = -0.004 t 0.031) or (1.8,2.2,0) (A =
151
Int
Bi2201 UD25K
Resolution
20 -
w>I
1. 5
1.93,2.07,0)
w<O
1. 0
0. 5
5-15
-10
-5
5
10
15
i t
2
to
2
4
6
8
10
4
6
8
10
4
6
8
10
Int
20F
Bi 2212 OD70K
(1 .9,2.1,0)
f
2.0
15-
1.5
1
i
1.0
0.5
I
-15
-10
-5
5
10
15
15
I
-as
to
2
Int
Bi2212 OD70K
f
lXi
.5
1.5
(2.1,1.9,0)
I
1
1.0
0.5
-15
-10
-5
5
10
15
W
2
Figure 4-5: Example transverse symmetry-breaking scans
-0.006 t 0.023), where the range in these scans has been changed to [3.25,5.75] meV
to account for the shift in the acoustic phonon. In Bi2201 UD25K the next measurement along the transverse for which both transfers were measured was (1.85,2.15,0),
for which again no difference outside error (A = -0.046 ± 0.071 for a range [2.25,3.75]
meV) was seen between appropriately scaled scattering at positive and negative transfer. Unfortunately data could not be taken much closer to the Gamma point, since
the quasielastic lineshape overlaps the phonon rendering separation of these scattering
152
mechanisms difficult.
Similar measurements were taken on Bi2201 UD31K at (2.1,1.9,0), Pb-Bi2201
UD25K at (2.07,1.93,0), and Bi2212 UD70K at (2.1,1.9,0). While none of these were
as statistically significant as the data points shown in Figure 4-5, the values obtained
were as follows: 0.037+0.021,0.034±0.033,0.035+0.018 for the samples in the order
listed above, respectively. Unfortunately this is the complete set of data collected for
applicable scans, and no temperature dependence was measured. The scan of UD25K
at (1.93, 2.07, 0) was repeated several times to confirm the striking asymmetry was
not a spurious instrumentation effect such as the sample slipping during the scan;
no evidence for any instrumentation artifact which could produce this signature was
seen during any of the experimental runs.
Given the existence of this measurement in several samples well outside the pseudogap temperature scale, it seems likely this observation is universal to at least the
(220) Bragg peak; the existance near the zone center suggests a lack of intercell ordering for this symmetry, suggesting the presence coincident with all Bragg peaks.
Further., in consideration of the reduced amplitude further from the Gamma point
and the relative strength of the signal seen in 4-5 for the measure closest to q= 0, it
seems likely the special momenta point around which this feature is peaked is in fact
q= 0. This order parameter seems finally to respects inversion about the zone center,
eg A(q
= A(q+
are taken.
G) for the sample in which such symmetry related measurements
Further experimental work is needed to confirm these postulates; the
measurements presented here are formally sufficient only to show existence of broken
symmetry in these measured samples at a small j near (220). Nonetheless, this signal
is strikingly strong and clearly maximal around the phonon peak in inelastic transfer
spectrum, suggesting this measure is robust.
153
4.4
Implications of Phonon Signatures of Symmetry Breaking
The conclusions we have inferred from the phonon measurements above are as follows:
firstly, crystallographic inversion symmetry is broken for all temperature (3.5.4); secondly, all measured samples support a q = 0 signature of coupling of the phonon to
a time-reversal and inversion symmetry breaking excitation (4.3), and thirdly, upon
cooling to a temperature scale of roughly the pseudogap T*, a signature of broken
T,I symmetry occurs in the longitudinal phonon at E = 0.25 (4.2.2).
The physical implications of T, I breaking at room temperature are a comment
about the stange metal state of the high-Tc materials. Since these symmetries are not
broken upon cooling into the pseudogap state but rather above it, this suggests that
the pseudogap transition is not intrinsically a broken symmetry transition, but rather
reflects a different order parameter which couples into these broken symmetry states
to explain the symmetry broken signal seen with temperature dependence in Kerr
and polarized neutron experiments.
While our probe of phonon spectra suggests
no coupling to a thermal transition into a density wave state (supporting instead
existence of fluctuations of this parameter over long time scales at all temperature),
if higher temperature induces a loss of coherence of this fluctuating CDW, it is possible
that there is, with respect to some time scale, a thermal transition into a CDW which
the phonon propogator is not sensitive to. In this case, the measurement of asymmetry
at the (longitudinal) CDW wavevector is consistent with this order parameter riding
on the transition that is truly driven by charge order, with some nontrivial coupling
inducing a replica of the asymmetry from q = 0 onto the new CDW wavevector due
to coupling of these order parameters.
This hypothesis explains the temperature
and momenta dependence seen in A measurements as a function of temperature and
Q. The exact nature of the q = 0 symmetry breaking order parameter which then
extends into the strange metal state is outside the scope of this work; the current loop
proposal[58] is consistent with this, as any (potentially small) intracell spin ordering.
The alternative is to suggest two separate (uncoupled) T, I breaking order param154
eters, one of which results in broken symmetry above the pseudogap near the CDW
wavevector for which the transverse phonon is sensitive, and a second transition to
induce sensitivity to the CDW ordering wavevector. In absence of a CDW based
explanation as above wherein periodicity at the (1/4,1/4, 0) wavevector originates
from q'= 0 dependence induced at the new static order wavevector, one must posit
a order parameter for which this momenta is special, such as formation of a chiral
in-plane period 4 ordering upon cooling. Arguments for a chiral out of plane period
4 order parameter have been made[138], but the experimental support for this model
is weak. Most critically, this seems to require an entirely unnecessary re-symmetry
breaking (considering that we present evidence this symmetry is already broken at
room temperature, as confirmed by the training effect in Kerr measurements) that is
unrelated to the fluctuation physics of the pseudogap.
4.5
Comments on Other Evidence for Broken Symmetry
We began this thesis searching for evidence of formation of charge density-wave order and broken translational symmetry; wide range of literature works supporting
this were presented in Chapter 1, and in Chapter 3 we related these to our own
measurements showed a dynamic tendency toward softening at a special wavevector (1/4,1/4,0). In this section we present the corresponding literature that makes
comment on broken symmetry with an eye toward explaining the symmetry of the
pseudogap state. We discuss time reversal, inversion, and nematic (rotational) symmetry breaking.
4.5.1
Time Reversal
Spin-Polarized Neutron
The magnetic moment of a neutron interacts with the magnetic properties of a system in two ways: firstly, a localized spin can act like a delta-function scattering
155
center and induce differences in the cross section for different polarizations, and secondly, the neutron can scatter off a local magnetic field as ' - B. In consideration
of the predicted magnetic order signal at Q = 0 from the current-loop model[30],
spin polarized neutron measurements were taken of several materials. Spin polarized neutron measurements are sensitive to this order parameter due to scattering
from a local field[139, 140]. The first material in which such an effect was measured
was YBCO[57, 141, 142], for which polarized measurements at the tetragonal (101)
Bragg peak displayed temperature-dependent increases in the spin flip ratio at lower
temperatures. The temperature scale at which such scattering increased was in accord with resistivity measurements which are the baseline definition of the pseudogap
transition[143]. This confirms that there is a strong correlation between the strength
of the spin-flip signal and the order parameter of the pseudogap, indicating that if it
is not a direct probe of this transition itself, magnetic scattering is strongly related
to formation of the pseudogap order parameter.
Additional measurements taken in HBCO[129, 144] show similar spin-polarized
scatter coincident with Bragg peaks and enhanced in dropping below the pseudogap
temperature. This measurement is particularly critical in confirming the universality
of this effect since YBCO is highly orthorhombically distorted, with additional Cu-O
chains along one of the axes. This suggests that time-reversal symmetry breaking is
common to many of the materials. In fact, similar measurements showed the same
effect present in Bi2212[130]. The presence of this puts our evidence here of broken
time-reversal and inversion symmetry on solid experimental footing, since this is a
new method of detecting broken symmetry. A significant unexplained complication of
these spin-polarized measurements is that polarization measurements with the polarization in or out of the scattering plane results in the observation that the magnetic
moment appears to contain significant out-of-plane component. This implies the simple current loop model is not a complete explanation of the observed phenomena; it
is possible that current loops involve apical oxygen as well, as supported by some
calculations[145]. Theoretically, even if the predominant component of the order parameter is as loop current, this must also be accompanied by some small in-plane spin
156
order[146].
In LSCO the signature[147] in spin polarized neutron scattering is seen to be
peaked at the same intercell ordering peak locations coincident with the structural
Bragg peaks, however, the peaks display a short correlation length
-
10A indicating
the order is two-dimensional. The loss of coherence in c* is in correspondence with
the weaker signature of the pseudogap in LSCO[148], and the appearance of this
scattering is at a lower temperature T=120K than suggested by other probes of the
pseudogap temperature.
The intracell ordering characterized by Q = 0 spin-flip
scattering is accompanied on an equivalent temperature scale within error of a shift
and slight enhancement of the scattering at the incommensurate SDW AF ordering
at Q = (7r, 7r)
(6, 0) or (0, 6). This suggests a relation between intercell and intracell
ordering and the electronic energetics that drive both transitions.
While all these experimental results were interpreted by the authors as in confirmation of the current loop proposal, the key feature is that the observed ordering
is at q = 0, which indicates intercell ordering, and the current loop order parameter
does not break translation symmetry. However, this feature is also a source of experimental difficulty, since the effect being detected is a very small signature coincident
with a large Bragg feature, and particularly susceptible to spurious effects such as
trapping of flux vortexes. Our measurements support existence of q' -+ 0 symmetry
breaking, but suggest that the real temperature scale of this symmetry breaking is
in fact higher than the pseudogap. If this interpretation is correct, the effect seen in
spin-polarized neutron is a product of the coupled order parameter (or spurious due
to field trapping), and reflects perhaps the formation of a SDW commensurate with
the CDW coherence as theoretically expected.
Nernst Effect
Subjecting a sample of LBCO (x
=
1/8) to a thermal gradient along the Cu-O bond
direction (a* here) and a magnetic field along the stacking axis c*, a Nernst electric
field is seen along b*. While typical Nernst signals are odd under inversion of the magnetic field, a nontrivial even contribution is seen to develop below
157
-
50K, comparable
to the charge ordering temperature [149]. The implication of this is that time reversal
symmetry is broken; a proposed model is for spontaneous polarization of a trapped
vortex density of definite handedness appearing, which is in-plane entropically costly,
but potentially favored due to increased interlayer coherence.
The definite handedness of the material was not able to be overcome by application
of a field up to ±14T applied during cooling from 290K. This suggests the handedness
is preformed at a higher temperature such that application of a field can not reverse
the directionality; this is consistent with the inability of an applied field to reverse
the sign of the signature seen in the Kerr effect presented below.
ARPES
Further evidence for time-reversal symmetry breaking comes from circularly polarized
ARPES data[123]. In this work, a difference in the photocurrent in between incident
light of different circular polarization was seen. This signal similarly depended on
temperature and doping and was only observed in what corresponds roughly to the
pseudogap phase. Further theoretical work confirmed foundations on which current
loops [30] could couple to this permitting observation of a time-reversal symmetry
broken state without breaking translational symmetry[150]. Interestingly, breaking
time reversal results in a superconducting state d+ip polarized, which is helical [151].
Our intuition is that scattering from a helical order parameters is what generates
asymmetry in the response function, as seen in the spin operators in earlier work on
spiral magnets [5].
However, ARPES results on circular dichrosim were controversial. The primary
avenue of disagreement was the absence of this effect in a lead-doped sample[124],
which was theoretically supported by work on the distorted structure, suggesting
structural effects could produce this form of artifact [152], where the temperature
dependence observed is proposed to be due to a structural shift of the oxygen in the
Bi-O layer. This case was responded to by the original ARPES authors, arguing the
predicted effects do not capture the momentum dependence and the domain averaging of the structural distortion[153]. While arguments persist about this specific
158
measurement[154], in light of uncontroversial bulk evidence from spin polarized neutron and that presented here using x-ray scattering, it seems likely these symmetries
are broken in this material, and the initial ARPES signal was likely sensitive to this.
Kerr Effect
Nonzero optical Kerr effect was first seen in YBCO[122]; the signature appeared upon
cooling through a transition temperature obeying similar scaling to the pseudogap
transition, ranging from ~ 50K in the near-optimally doped case to ~ 200K for
underdoped samples. Interestingly, the sign of this effect is determined by the directionality of a 4T field applied at room temperature, which is then removed during
cooling, and the measurement is taken upon warming. This applies the presence of
broken time-reversal symmetry at room temperature, which is then enhanced upon
cooling through the pseudogap transition, consistent with our nonzero measurements
of broken symmetry in the transverse scans close to the Gamma point and the longitudinal scans in Bi2201 near the 1/4 nesting wavevector at 300K. The nature of this
hysteretic signal is unknown, but in YBCO it is hypothesized the signature is similar
to that of having a small ferromagnetic moment present at high temperature that
accompanies the transition; the magnitude of this moment is estimated at less than
or equal to 10-5
I/Cu.
A similar effect was also observed in Bi2201[74], where the temperature dependence of the Kerr effect was shown to be functionally equivalent to that of the drop
of the energy of the antinodal quasiparticle peak upon cooling. This correspondence
between the temperature dependence of gapping out the antinodal electronic excitations and enhanced time-reversal symmetry suggests the formation of a single order
parameter responsible for both features. If antinodal electrons are involved, the response for these quasiparticles suggests coupling to the nesting wavevector may play
an important role, which is in agreement with our observation of momentum dependence in the effect as measured in phonons, since phonons of non-nesting wavevector
do not couple Fermi surface arcs that are gapped out at lower temperature.
Both these Kerr effect measurements are then in good agreement with the mea159
surements of phonon asymmetry in BSCO, and strongly suggest from an electronic
standpoint that time reversal symmetry is broken upon cooling, in agreement with
the temperature dependence measured by our calculation of x"(±w). If the conclusion reached by ARPES is correct that time-reversal breaking accompanies a 0.05 eV
shift in the electron bands, it seems entirely feasible the phonon structure is sensitive
to this asymmetry.
Recently a new interpretation of the Kerr effect has been proposed in terms of
optical gyrotropy instead of time reversal symmetry breaking[75, 155].
This is in
effort to reproduce the experimental observation that the Kerr effect is invariant
under rotation of the sample by 180', in contrast to the expectation for a magnetic
order parameter based symmetry breaking effect, which would be expected to be odd
under such rotation. This explanation relies on sensitivity of the Kerr effect to a linear
term in the dielectric response, which can formally be present if the system breaks
chiral symmetry. To break such symmetry, proposals were constructed making use
of chiral tilings of CDW ordering[75]. While these proposals in literature all rely on
c axis coherence, we remark that chiral ordering is possible in a simpler model in
which two in-plane directional vectors are created by (1) a CDW-type amplitude of
period
-
4 that is neither exactly bond nor cite centered (as used in [75]), and (2)
the crystallographic shift along b seen by for example, STM [67]. In defense of his
current loop, Varma has proposed a chiral based effect originating from a particular
current loop configuration where time reversal was broken but not the driving factor
of the gyrotropic-like signal that is invariant under flipping the sample[156].
This interpretation is consistent with our proposed interpretation of the phonon
symmetry breaking signatures measured here, which we have interpreted in terms
of T, I breaking in the pseudogap, but. which are then enhanced by coherence of
CDW-type fluctuations at the pseudogap scale. Formation of such a CDW amplitude
can then generate necessary chiral symmetry to produce optical gyrotropy. While we
believe T is still broken at room temperature, the implication from NMR is that this
moment is very small[131], which supports that the enhanced Kerr signal is not due
to direct coupling to this order parameter but instead a chiral symmetry breaking.
160
4.5.2
Inversion
X-ray Crystallography
First, there is a wide range of literature suggesting broken crystallographic inversion symmetry in Bi2201 and Bi2212[1, 86, 157].
All these measurements display
some weak peaks breaking the centering condition, and crystallographic refinements
are then extended accordingly. The primary component of the noncentrosymmetry
refinement is a displacement of the Bi-O layer atoms, which we are similarly most
sensitive to in inelastic scattering. The magnitude of the proposed shift is up to 0.02
rlu along the crystallographic b axis[86] in the plane.
STM
Measurement taken by STM[67] confirmed the presence of this shift locally at the
surface layer, measured here to be a 1-2% of the position of the Bi atoms. This feature
was observed to be invariant as a function of doping, temperature, and field, likely in
correspondence with the bulk X-ray results previously mentioned. Interestingly, while
the bulk shows a single domain of the orthorhombic distortion and directionality of
the Bi modulation, at the surface a domain wall was found in STM, indicating a
multidomain feature that must vanish at some surface length scale.
X-Ray Circular Dichrosim
X-ray optical activity measurements on Bi2212[121] provided a further probe of broken inversion symmetry. These measurements, taken at the Cu K absorption edge,
reveal a difference in the transmission rate of circularly polarized light of different
polarizations. While the prior two results show existence of relatively plain temperature independent inversion symmetry primarily associated with Bi atom shifts, this
measurement, sensitive to Cu atomic motion, revealed a temperature dependence to
the inversion symmetry breaking with this signal vanishing around 225 K, consistent
with the pseudogap and other symmetry breaking temperatures. This symmetry then
joins the family for which there is evidence symmetry is broken at high temperature,
161
but signals are enhanced upon cooling. These measurements were questioned due
to the similarity between the lineshape of the signal obtained and that produced
by a potential instrumentation error of misaligning the energy of the left and right
circularly polarized light[158]; however, numerical estimates of dichrosim for realistic assumptions of inversion-breaking chiral charge order predict similar magnitude
effects as are measured [159].
Surprisingly, comparison of measurements of linear dichrosim (difference in transmission between linear polarized lights) suggest that time-reversal symmetry is maintained. From the previously mentioned measurements there is conclusive evidence
that time reversal is broken in these materials, and good evidence that this feature
contains signature in the Cu atoms for which this probe should be sensitive. Arguments of domain averaging seem unlikely since the focused spot was of order 0.1 x 0.1
mm, substantially smaller than the samples used in polarized neutron probes which
detected this order. This is consistent with the similar measurement of the Kerr effect
wherein interpretation is made in terms of a chiral symmetry breaking rather than
sensitivity to time-reversal breaking. Further theoretical work is needed to illuminate
the direct sensitivity to this probe to various suggested order parameters to estimate
how hard this bound of maintained time-reversal symmetry is.
4.5.3
Rotation
STM
Measurement of the ratio of the density of states at positive and negative transfers in
STM are constructed to remove topographical features. In Bi2212 this measurement
observed [66] a difference between the Fourier transformation of such a map at the
local tetragonal (1,0) and (0,1) Bragg peaks. This feature corresponds to intra-unit
cell electronic nematicity, that is, a Q = 0 signal indicating broken rotational symmetry in the electronic excitation spectra. Finite size domains on order 4 nm were
found; unfortunately, due to the requirement of low vibration noise, STM measurements could not be taken as a function of temperature allowing determination of the
162
temperature scale of this nematicity. Chiral type ordering as has been proposed will
clearly break C4 symmetry and could be responsible for this signature. These measurements were recently called into question due to the potential for tip anisotropy
to introduce energy dependent features that were not appropriately controlled for by
the topography[160].
Inelastic Neutron
Inelastic neutron scattering on YBCO[56] indicated the spin sector shows a similar
anisotropy breaking C4 rotational symmetry at low energy transfers, to be restored at
higher transfer. Examining the inelastic excitations around the AF ordering peak, a
longer spatial correlation length was observed along the crystallographic b* direction
in which additional Cu-O chains are found. While the underlying structure lacked
pure C4 symmetry, as did the Bi2212 structure in the previously discussed STM
measurements, the nontrivial result is that this symmetry breaking is strongly coupled
to the excitational quasiparticles that describe the low energy degrees of freedom;
in particular, the asymmetry is greatest for the lowest energy transfer close to the
incommensurate peaks. As a function of temperature the incommensurability is seen
to vanish by 150K, comparable to the temperature scale of the anisotropy in electronic
resistivity.
4.6
Conclusions
The high temperature cuperates are characterized by a wide range of broken symmetries.
Substantial evidence is present for electronic or spin density-wave order
breaking translation symmetry, as is for broken time-reversal, inversion, and rotation
symmetry. Signatures of these are all relevant in the low-energy excitations, and
mostly show some enhancement on a temperature scale around that of the pseudogap transition. A critical question is understanding how these different symmetries
are related and describing the evolution of the hierarchy of broken symmetry upon
cooling.
163
Our measurements make several comments on these questions. Firstly, we have
shown that the phonon spectra indicates a tendency toward dynamic order in the
phonon spectra in which strong electron-phonon coupling drives significant spectral
weight to low energy/long time scale fluctuations, but a true static or central peak
does not appear. This suggests that the effects seen indicating a static CDW are
likely a surface effect or at least quite small in the bulk, but the energetics breaking
translation symmetry in the dynamics are relevant in the bulk resulting in this anomalous softening. We see evidence for time-reversal and inversion symmetry breaking at
room temperature in multiple samples near Q
00 in the phonon spectra in transverse
measurements; this signature becomes visible at Q 4 (0.25, 0.25, 0) and is enhanced
upon cooling into a temperature scale comparable to the pseudogap scale. A consistent explanation for this is proposed that the temperature scale of T* is a coherence
transition for CDW fluctuations, which when effectively static replicate the Q = 0
symmetry breaking signal around the (0.25,0.25,0) wavevector and are responsible for
inducing this temperature dependent asymmetry.
This description of coupled degrees of freedom between electronic transitions offers a detailed description of the formation of both intracell symmetry breaking at
higher temperatures and enhanced intercell signature at lower temperature. Support
for all features presented here exists in separate literature work, but the techniques
developed for searching for symmetry breaking using differences in the dynamic susceptibility at positive and negative transfer and analyzing finding slow fluctuations
in the limiting process of a dynamic phonon propagator at low transfer to exclude
spectral weight of disordered quasielastic scatter are new methodologies we hope will
be useful in other works using phonon spectra to probe coupling to electronic features.
164
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