A Scanning Tunneling Microscope at the
Milli-Kelvin, High Magnetic Field Frontier
Brian B. Zhou
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Advisor: Ali Yazdani
September 2014
c Copyright by Brian B. Zhou, 2014.
All rights reserved.
Abstract
The ability to access lower temperatures and higher magnetic fields has precipitated
breakthroughs in our understanding of physical matter, revealing novel effects such as superconductivity, the integer and fractional quantum Hall effects, and single spin magnetism.
Extending the scanning tunneling microscope (STM) to the extremity of the B-T phase
space provides unique insight on these phenomena both at the atomic level and with spectroscopic power. In this thesis, I describe the design and operation of a full-featured, dilution
refrigerator-based STM capable of sample preparation in ultra-high vacuum (UHV) and spectroscopic mapping with an electronic temperature of 240 mK in fields up to 14 T. I detail
technical solutions to overcome the stringent requirements on vibration isolation, electronic
noise, and mechanical design necessary to successfully integrate the triad of the STM, UHV,
and dilution refrigeration. Measurements of the heavy fermion superconductor CeCoIn5 (Tc
= 2.3 K) directly leverage the resulting combination of ultra-low temperature and atomic
resolution to identify its Cooper pairing to be of dx2 −y2 symmetry. Spectroscopic and quasiparticle interference measurements isolate a Kondo-hybridized, heavy effective mass band
near the Fermi level, from which nodal superconductivity emerges in CeCoIn5 in coexistence
with an independent pseudogap. Secondly, the versatility of this instrument is demonstrated
through measurements of the three-dimensional Dirac semimetal Cd3 As2 up to the maximum magnetic field. Through high resolution Landau level spectroscopy, the dispersion of
the conduction band is shown to be Dirac-like over an unexpectedly extended regime, and
its two-fold degeneracy to be lifted in field through a combination of orbital and Zeeman
effects. Indeed, these two experiments on CeCoIn5 and Cd3 As2 glimpse the new era of nanoscale materials research, spanning superconductivity, topological properties, and single spin
phenomena, made possible by the advance of STM instrumentation to the milli-Kelvin, high
magnetic field frontier.
iii
Acknowledgements
To begin, I would like to thank my advisor, Prof. Ali Yazdani, for providing me the
opportunity to work on the projects presented in this thesis. Ali’s steady leadership and
motto of focusing on “how far we have come” rather than “how far we need to go” was
one driving force in carrying this long journey through the toughest of times. It has been
amazing to witness and learn from Ali’s special ability to communicate delicate physical
concepts in scientific writing, presentation, and discussion.
No beginning graduate student could undertake alone the job of constructing a dilution
fridge STM (or more colloquially DRSTM), and tremendous credit for this thesis should
go to post-doc Shashank Misra, who mentored me in the experimental aspects over the 5
plus years we spent together debugging the instrument. Shashank’s influence will be long
engrained in the users of DRSTM, from his many clever designs, meticulous protocols, to
even the password on the measurement computer.
I have enjoyed fruitful and instructive collaborations with the groups of Dr. Eric Bauer,
Prof. Bob Cava, and Prof. Ashvin Vishwanath. Staff members in the Princeton physics
department, including Steve Lowe, Bill Dix, Darryl Johnson, Ted Lewis, James Kukon,
Claude Champagne and many others, have facillated the research in this thesis with great
kindness. I would also like to thank Prof. Waseem Bakr for his reading of this thesis, and
Prof. Nai Phuan Ong and Prof. Mariangela Lisanti for serving on my oral committee.
In addition, my long tenure in Ali’s lab has allowed me to bridge two eras of lab members,
whose camaraderie has made the time here fly by. When I first arrived, I looked up to the
elder generation of Kenjiro, Aakash, Anthony, Lukas, Pedram, and Colin, as they set fine
examples in the lab. Concurrent to me were the proverbial “good physicists” of Eduardo,
Jungpil, Pegor, Haim, Andras, Ilya, and Stevan. Nowadays, I applaud the new crew of
Sangjun, Mallika, Yonglong, and Ben for being eager and able to continue the tradition.
Sangjun’s diligent effort and push to explore multiple interpretations of the Cd3 As2 data
iv
was a huge factor in our overall efficiency and led to greater understanding of the basic
model.
Last and foremost, I am deeply grateful for the care, lessons, and encouragement throughout my life from my parents Lingling and Genwen, to whom I dedicate this thesis.
v
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
1 Introduction
1
1.1
Scanning Tunneling Microscopy at the Limit . . . . . . . . . . . . . . . . . .
2
1.2
Heavy Fermions - The Kondo Lattice . . . . . . . . . . . . . . . . . . . . . .
6
1.3
Heavy Fermions - Unconventional Superconductivity . . . . . . . . . . . . .
11
1.4
Three-Dimensional Dirac/Weyl Semimetals . . . . . . . . . . . . . . . . . . .
16
1.5
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
2 The Basics of Scanning Tunneling Microscopy
24
2.1
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2
Quasiparticle Interference Imaging . . . . . . . . . . . . . . . . . . . . . . . .
28
3 The Dilution Refrigerator Scanning Tunneling Microscope
34
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3.2
Ultra-High Vacuum Assembly . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.3
Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.4
The Dilution Refrigerator and Microscope Head . . . . . . . . . . . . . . . .
41
3.5
STM Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
vi
3.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Visualizing d-Wave Heavy Fermion Superconductivity in CeCoIn5
50
52
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.2
Superconductivity on the Two Surfaces of CeCoIn5 . . . . . . . . . . . . . .
53
4.3
Quasiparticle Interference in Normal and Superconducting States of CeCoIn5
57
4.4
Response of Nodal Superconductivity to Potential Scattering . . . . . . . . .
60
4.5
Vortex Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.6
Impurity Bound State: Fingerprint of dx2 −y2 Pairing . . . . . . . . . . . . . .
63
4.7
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
5 The Three Dimensional Dirac Semimetal Cd3 As2
67
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
5.2
Topographic and Spectroscopic Characterization at Zero Field . . . . . . . .
69
5.3
Landau Level Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
5.4
Spatial Homogeneity of Landau Levels . . . . . . . . . . . . . . . . . . . . .
75
5.5
Quasiparticle Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
5.6
Landau Level Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5.7
Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
6 Conclusion
82
A Further Experimental Aspects of DRSTM
88
A.1 X and Z Capacitance Position Sensors . . . . . . . . . . . . . . . . . . . . .
88
A.2 Life on DR: Including Approaching an Sample . . . . . . . . . . . . . . . . .
91
A.3 ‘Joule-Thomson’ 2K Mode Operation . . . . . . . . . . . . . . . . . . . . . .
93
A.4 Dewar Exhaust Management . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
A.5 Electrical Ground Loop Management . . . . . . . . . . . . . . . . . . . . . .
96
vii
B Multipass Spectroscopy: An Alternative to Conventional Conductance
Spectroscopy
B.1 Traditional Conductance Maps
99
. . . . . . . . . . . . . . . . . . . . . . . . .
99
B.2 The Multpass Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
C Comparison of QPI in CeCoIn5 to Other Band Structure Probes and Phenomenological Modeling
105
C.1 Reference to Other Experimental Mappings of the Band Structure . . . . . . 105
C.2 Phenomenological Modeling of Normal State Band Structure . . . . . . . . . 107
C.3 Superconductivity Gapping the Phenomenological Band Structure . . . . . . 109
D Details of Cd3 As2 Landau Level Simulation
112
D.1 Modified Four-Band Kane Hamiltonian . . . . . . . . . . . . . . . . . . . . . 112
D.2 Schematic Demonstration of the Weyl Fermion and the Low Field Regime . . 120
E Charge Ordering in Underdoped Bi2 Sr2 CaCu2 O8+δ in a Magnetic Field
122
E.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
E.2 Energy and Spatially-Resolved Density of States in a Magnetic Field . . . . 125
E.3 Where are the Vortices? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Bibliography
131
viii
List of Tables
C.1 Estimated minimal Q vectors from dHvA measurements of CeCoIn5 . . . . . . 107
D.1 Parameters for the modified four-band model for Cd3 As2 . . . . . . . . . . . . 119
ix
List of Figures
1.1
Goal of Thesis Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Cyrogenic Scanned Probe Instruments . . . . . . . . . . . . . . . . . . . . .
5
1.3
The Heavy Fermion Phase Diagram . . . . . . . . . . . . . . . . . . . . . . .
8
1.4
Tunneling into the Two Surfaces of CeCoIn5 . . . . . . . . . . . . . . . . . .
10
1.5
Pairing Symmetry of Superconductivity . . . . . . . . . . . . . . . . . . . . .
12
1.6
Antiferromagnetic Correlations on a Square Lattice . . . . . . . . . . . . . .
15
1.7
Three Dimensional Dirac Semimetal in a Magnetic Field . . . . . . . . . . .
18
2.1
The Tunneling Current from Density of States . . . . . . . . . . . . . . . . .
26
2.2
Modes of STM Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3
Demonstration of QPI for the Cu(111) Surface State . . . . . . . . . . . . .
30
3.1
General Assemby of DRSTM . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3.2
The Ultra-Quiet Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.3
Vibration Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.4
Microscope Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.5
Aluminum Superconducting Gap . . . . . . . . . . . . . . . . . . . . . . . .
46
3.6
Spectroscopic Mapping Performance . . . . . . . . . . . . . . . . . . . . . . .
47
3.7
Tunneling Current Noise Characteristic . . . . . . . . . . . . . . . . . . . . .
49
4.1
Hybridization, Pseudogap, and Superconductivity . . . . . . . . . . . . . . .
54
4.2
First Order Phase Transition at Hc2 . . . . . . . . . . . . . . . . . . . . . . .
56
x
4.3
Quasiparticle Interference of Heavy Superconducting Electrons . . . . . . . .
58
4.4
Enhancement of Q3 Vector Along (π, π) Nodal Direction . . . . . . . . . . .
59
4.5
Evolution of In-Gap Quasiparticle States Approaching a Step-Edge. . . . . .
61
4.6
Superconducting Gap Approaching an Step-Edge in a s-Wave Superconductor 62
4.7
Vortex Lattice and Anisotropy of Bound State in CeCoIn5 . . . . . . . . . .
63
4.8
Impurity-Bound Quasiparticle Excitations in a dx2 −y2 Superconductor. . . . .
64
4.9
Normalized Spectrum at Center of Impurity. . . . . . . . . . . . . . . . . . .
65
5.1
STM Characterization of Cd3 As2 . . . . . . . . . . . . . . . . . . . . . . . .
71
5.2
Conductance Fluctuation of Defects in As Plane . . . . . . . . . . . . . . . .
72
5.3
Landau Level Spectroscopy of Cd3 As2 . . . . . . . . . . . . . . . . . . . . . .
74
5.4
Quasiparticle Interference in Cd3 As2 . . . . . . . . . . . . . . . . . . . . . .
77
5.5
Landau Level Simulation for Cd3 As2 . . . . . . . . . . . . . . . . . . . . . .
79
6.1
The Phase Diagram of CeCoIn5 with Doping . . . . . . . . . . . . . . . . . .
84
6.2
Surface Atom Ordering in (112) Plane of Cleaved Cd3 As2 . . . . . . . . . . .
86
A.1 X and Z Capacitance Sensors . . . . . . . . . . . . . . . . . . . . . . . . . .
89
A.2 Model Tip Sample Capacitance . . . . . . . . . . . . . . . . . . . . . . . . .
92
A.3 Joule-Thomson Mode Characteristics . . . . . . . . . . . . . . . . . . . . . .
94
A.4 Dewar Exhaust Flow Chart . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
A.5 Electrical Wiring Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
B.1 Traditional Measurement Cycle for Conductance Maps . . . . . . . . . . . . 100
B.2 Schematic of Multipass Technique for Acquiring Conductance Maps . . . . . 101
C.1 Comparison of QPI to Fermi Surface from ARPES and Theoretical Calculation106
C.2 Tight Binding Bandstructure To Experimental Dispersion
. . . . . . . . . . 108
C.3 Superconducting QPI in comparison to simulation of dx2 −y2 and dxy gap symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
xi
D.1 Model Band Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
D.2 Demonstration of Weyl Fermion . . . . . . . . . . . . . . . . . . . . . . . . . 121
E.1 Bi2212 (UD 58 K) in a magnetic field . . . . . . . . . . . . . . . . . . . . . . 124
E.2 Charge Ordering with a Magnetic Field . . . . . . . . . . . . . . . . . . . . . 126
E.3 Fourier Transform of In-Field Spatial Patterns . . . . . . . . . . . . . . . . . 128
E.4 Vortex regions in underdoped Bi2212 . . . . . . . . . . . . . . . . . . . . . . 130
xii
Chapter 1
Introduction
The full splendor of the phases and phenomena of matter cannot be appreciated by our
everyday experience, one that traverses only a small confined area in the phase space of
temperature, magnetic field, and pressure - much like the full richness of a symphony cannot be appreciated from listening to only a restricted section of an orchestra. Throughout
the history of science, the advancement of measurement techniques to lower temperature
and higher magnetic fields has expanded our perception to the far-ranging nuances of this
symphony with mother nature as its maestro. From the liquefaction of helium by Kamerlingh Onnes in 1908 came his discovery of superconductivity in elemental mercury (1911) [1]
and thereafter the discovery of superfluidity in liquid helium-4 (1937) by Pyotr Kapitsa and
John F. Allen [2]. Later experiments on two dimensional electron gases in strong magnetic
fields and at low temperature demonstrated that resistance under these extreme conditions
is quantized in integer and fractional divisions of the fundamental value RH = h/e2 [3]. Indeed, these observations at extreme conditions, which contrast with our ordinary experience,
profoundly advanced our understanding of the details and consequences of quantum physics,
from degenerate quantum wavefunctions, the interplay of localization and dimensionality, to
topological invariance. Guided by this principle, this thesis details an effort to extend the
scanning tunneling microscope (STM) to the extreme limit of milli-Kelvin temperatures and
1
very high magnetic fields. The success of this effort is demonstrated through two experiments
that utilize the new instrument’s access to an expanded phase space to probe unconventional
superconductivity in the heavy fermion material CeCoIn5 and the exotic band structure of
the three-dimensional Dirac semimetal Cd3 As2 . In this introduction, we overview the capabilities of the STM measurement technique and the current state of the technology in this
field. Finally, we introduce the core concepts of heavy fermion materials and Dirac/Weyl
semimetals and preface the experimental findings presented in this thesis.
1.1
Scanning Tunneling Microscopy at the Limit
If a picture is worth a thousand words, then the scanning tunneling microscope has spoken
volumes about the quantum world, illustrating seemingly invisible concepts in plain view,
from the existence of atoms, the wave nature of electrons, to the pairing symmetry of superconductivity. Indeed, Binnig and Rohrer’s invention of the STM at IBM Zurich immediately
clarified a two decade long mystery about the precise nature of the 7x7 reconstruction on
the Si (111) surface [4]. The ability to trace the quantum mechanical tunneling current as
a sharp metal tip is scanned within 10 Å of a clean material surface heralded an new age
of atomic resolution. Moreover, by recording this current while ramping the chemical potential of the sample with respect to the tip, the electronic density of states of the sample,
both above and below the Fermi energy, can be probed with precision, reliability, and in a
magnetic field. While the precise details of STM operation will be discussed in Chapter 2,
it suffices now for the reader to appreciate the power of resolving electronic structure at the
local level over only having information about its bulk behavior. For example, the ability
to correlate electronic inhomogeneity to the local atomic structure, such as strain or chemical substitution, enables the direct connection between structure and function, providing
direction for changing material properties [5]. Second, phase transitions, such as superconductivity, anti-ferromagnetism or charge order, oftentimes nucleate locally before globally,
2
a process which reflects the correlations that drive the phase transition [6]. Finally, the
response of a transitionally-invariant state to local perturbations, such as superconductivity
to a pair-breaking impurity, reveals the fabric of the original state, much like how ripples
from a stone dropped on a lake reveals information about the properties of water [7].
B
T
Figure 1.1: The goal of the project described in this thesis is to extend the powers of the
STM, from visualizing single atoms to the wave nature of electrons, to the lowest temperature
and the highest magnetic fields. The feat to accomplish is to adapt the generically “tabletop” STM (photo courtesy Kenjiro Gomes) and its UHV environment to be compatible with
a dilution refrigerator and superconducting magnet.
Extending these powerful STM measurements to the lowest temperatures, such as that
provided by a dilution refrigerator, and to the highest magnetic fields, provided by a superconducting cryo-magnet, represents the objective of this thesis work. While a long standing
3
goal in the STM community, the marriage of such cryogenic, magnetic, and STM hardware,
however, puts several essential technical requirements in conflict with each other: 1) while
the STM requires sub-picometer vibrational stability, running a dilution refrigerator requires
mechanical pumps which couple to the STM instrument via rigid pumping lines and via the
internal flow of cryogen, 2) the most powerful STM experiments require atomically-clean
surfaces, which must be prepared in ultra-high vacuum (UHV) either through cleaving or
cleaning procedures and be subsequently transferred to the cold STM head without interruption of UHV, and 3) high magnetic fields can induce both static and dynamic picometer
deflections due to any residual magnetism in the STM head. Thus, requirement 2) demands
that the dilution fridge be carefully incorporated into the UHV environment, necessitating
innovative mechanical design to facilitate the sample transfer, while together 2) and 3) necessitate careful choice of UHV-compatible, non-magnetic materials in the construction of
the entire fridge and STM head. However, the first requirement of picometer vibrational
stability while maintaining a closed cycle of cryogen in the presence of an internal liquid
reservoir (the “1K pot”) turns out to be the foremost challenge [8].
This thesis details solutions, some complete, some partial, both in design and implementation, that have overcome the challenges faced to demonstrate high performance scanning
tunneling microscopy at dilution fridge temperatures and high magnetic fields. The instrument was constructed in the lab of Prof. Ali Yazdani at Princeton University over the period
of five years starting from the summer 2008 to the summer of 2013 [10]. The figure of merit
can be taken as the ratio of the magnetic to thermal energy scales µB B/kB T , which can be
regarded as a measure of the spectral resolution for magnetic properties or as a measure of
the accessible area in B ∗ 1/T phase space. As shown in Fig. 1.2, only recently have a handful of high field, cryogenic STMs have come into existence to reach the highest µB B/kB T
metrics. We highlight the dilution fridge-based instruments represented by the data points
labeled “NIST” at the National Institute of Standards and Technology in Maryland [9, 11]
and labeled “Univ. Auton. Madrid” at the Universidad Autnoma de Madrid, Spain [12], as
4
UHV (This Thesis)
True B/Telectron
Princeton
Figure 1.2: The figure of merit µB B/kB T in logarithmic scale for notable cryogenic scanning
probe instruments as a function of year of demonstration, where T is the minimum lattice
temperature and B is the maximum magnetic field of the instrument’s operation. Red circles
denote instruments compatible with ultra-high vacuum, while blue diamonds denote cryovacuum systems. The Princeton UHV dil-fridge-based STM system described in this thesis
is shown as the orange circle, while the the orange circle with black outline denotes for this
system the more operational metric µB B/kB Telectron . Figure adapted from [9].
they have demonstrated scientific measurements in addition to instrument characterization.
However, the instrument at Madrid is restricted to studying only cleaved samples, kept clean
by cryo-vacuum, rather than the full set of metallic samples available by UHV cleaning procedures. The UHV instrument at Princeton constructed during the course of this thesis has
demonstrated scientific measurements at a temperature of 20 mK and a magnetic field of
14 T, yielding a nominal µB B/kB T ∼ 470, where T denotes the lattice temperature. While
using the lattice temperature is the fair comparison to the other data points, we show for
reference on the same graph the more practical metric µB B/kB Telectron ∼ 14 T/240 mK ∼ 40
for the Princeton system. The distinction between electronic and lattice temperature will
be further discussed in Chapter 3. Regardless of the metric, the ultimate value of our instrument lies in the physical insights it can reveal, and we now transition to describing two
experiments that have benefited from its enhanced capabilities.
5
1.2
Heavy Fermions - The Kondo Lattice
A central focus of condensed matter physics is the study of the collective behavior of electrons
in a solid. The interactions of electrons with other electrons and protons in a crystal lattice,
both through charge and spin degrees of freedom, can modify their behavior from the noninteracting, free-space description ε(k) = ~2 k 2 /2m0 , where m0 is the bare electron mass. It
is precisely these interactions that drive the rich variety of emergent solid state behavior,
such as the Mott insulator or ‘heavy fermion’, and lead to technologically useful physical
properties, such as colossal magnetoresistance and high temperature superconductivity. In
the presence of interactions, Landau Fermi liquid theory can describe the new correlated state
in terms of quasiparticles, or new emergent excitations that represent a collective motion of
the many-body system. These excitations can be treated in certain limits as independent
quasi particles which have an renormalized mass m∗ and a renormalized magnetic moment
µ∗ .
The low energy quasiparticles in heavy fermion compounds [13, 14, 15] possess renormalized masses m∗ up to 1000 times the free electron mass m0 as a result of the Kondo effect, to
be described below. Experimentally, since m∗ is proportional to the density of states at the
Fermi level N (0), the physical observables of heavy fermion formation are the corresponding
thousand-fold enhancements of the electronic specific heat (γ) and the Pauli paramagnetic
susceptibility (χ), both proportional to N (0), beneath a characteristic temperature scale
T ∗ . Compounds that display heavy fermion behavior are intermetallics containing partially
filled 4f and 5f elements, predominantly the four elements Ce, Yb, U, and Np. Indeed, as
exemplified by Ce and Yb, which occupy at the extremes of the lanthanide block, possessing
respectively have a single 4f electron and a completely filled 4f shell in the atomic limit,
heavy fermion formation is driven by the instability between localized magnetic moments
and electronic conduction. For these critical f -elements, the balance between the two can be
tipped if the occupation of a single electron switches between a localized, magnetic f shell or
a delocalized, bonding spd shell depending on its chemical environment of the intermetallic
6
compound. In actuality, this critical electron will often occupy a hybridized orbital with both
f - and spd- character. In this sense, heavy fermions lie at the brink of magnetism, where the
enhanced and tunable interplay between electronic and magnetic degrees of freedom produce
remarkable physics such as quantum criticality and unconventional superconductivity.
Formally, we can first consider the propensity of local f -electrons to participate in electronic conduction through the Anderson impurity model for a single magnetic ion dissolved
in a conduction sea:
H=
X
k,σ
|
k nk,σ +
X
k,σ
V (k) c†k,σ fσ + fσ† ck,σ + Ef nf + U nf ↑ nf ↓ ,
|
{z
}
{z
} f Coulomb repulsion
(1.1)
f-c hybridization
where the first set of terms describes the hybridization of the magnetic f -level to the conduction sea via the interaction V (k) and the second set of terms captures the Coulomb physics
of local moment formation for the single f -level [13]. Analysis of this Hamiltonian by Jun
Kondo and others led to the understanding of the celebrated Kondo effect, where it was
shown that a singlet wavefunction, with anti-ferromagnetic coupling of the conduction sea
to the local f moment, results in an energy gain, or equivalently temperature scale TK ,
TK = D exp −
1
,
N (0)|J|
(1.2)
where J ∼ V (k)2 reflects the exchange coupling between the conduction sea and the localized
moment and D reflects the conduction bandwidth. Thus below TK , the conduction electrons
screen the local f moment via hybridization, resulting in a low energy resonance in their
density of states effectively arising from entanglement to the spin degeneracy of the f -ion.
To describe heavy fermion systems, equation 1.1 must be extended from the dilute single
ion limit to the dense limit of a lattice of f -level ions via the so called periodic Anderson
Hamiltonian. In this case, the Kondo temperature TK is renormalized, but maintains a
similar function. Instead of a single Kondo resonance, a heavy fermion band of resonances,
7
a)
b)
Heavy Fermion
T
CeCoIn5
TRKKY
TKondo
“Non-Fermi Liquid”
AF
Order
0
Fermi
Liquid
δ
δc
Superconductivity
Figure 1.3: a) Doniach phase diagram for a generic heavy fermion, showing the competition
between the Kondo and RKKY interactions as a function of a tuning parameter δ, which sets
the strength of hybridization. b) In the CeCoIn5 phase diagram, the rich variety of phases
can be tuned through either pressure, magnetic field, or chemical doping with Cd atoms.
Figure adapted from [16].
incorporating the spin entropy of each f -lattice site, forms at low energies, and produces
the huge density of states at the Fermi level that is the hallmark of heavy fermion properties. In general, the lattice Kondo effect competes with the Ruderman-Kittel-Kasuya-Yosida
(RKKY) interaction, which is also mediated by the conduction electrons, but tends to order
magnetic ions in a lattice antiferromagnetically (AF), rather than compensating the magnetic ions as in the Kondo interaction. The energy gain of the RKKY interaction scales as
a power law in exchange coupling J via
TRKKY = N (0)J 2 .
(1.3)
Hence as shown in Fig. 1.3 a), in the limit of small hybridization J, the AF ordered,
local moment state is favored since the RKKY energy gain dominates the Kondo energy
gain, while at high hybridization, the Kondo-screened Fermi liquid is favored [13]. This
competition underlies the heavy fermion phase diagram, where a quantum critical point
(QCP) separates the AF and Fermi-liquid states at zero temperature and can be explored
8
through tuning the strength of hybridization, via external pressure, applied magnetic field,
and chemical substitution. Moreover, a dome of superconductivity oftentimes occurs near
the QCP where magnetic correlations are the strongest, just as the Kondo and RKKY
interactions balance. In the prototypical heavy fermion CeCoIn5 studied in this thesis, all
three variables of pressure, field, and doping can change the ground state phase, sometimes
even reversibly - for example, Cd-doping transitions the system towards AF, suppressing
superconductivity, but the application of pressure can revert the Cd-doped material back to
superconductivity [17].
STM spectroscopic techniques were first extended to the physics of heavy fermion compounds for URu2 Si2 in Refs. [18] and [19]. In URu2 Si2 , however, the unexplained hidden
order phase with temperature scale THO = 17.5 K [20] interrupts the Kondo physics that
onset eariler around T ∗ ∼ 70 K and thus complicates interpretation of the measurement.
In 2012, through variable temperature experiments, Aynajian and da Silva Neto [21] used
quasiparticle interference (QPI) to definitively visualize the formation of heavy fermions in
CeCoIn5 , a model Kondo lattice system. These authors showed that the weakly dispersing
light band structure is transformed below T ∗ into a strongly dispersing heavy fermion band.
Furthermore, in Ref. [21], it was found that upon cleaving CeCoIn5 , different surface terminations (layers) are exposed, and the local tunneling spectra measured by STM depend on
which surface termination is measured. This effect was explained in terms of the interference
of the two tunneling paths, one into the lighter part of the composite heavy quasiparticle (the
spd-like band) and the other into the heavier, more localized part (the f -resonance). One
of the two atomically ordered surfaces of CeCoIn5 , “surface A” was identified as the Ce-In
surface, and the tunneling spectra show a pronounced gap near the Fermi energy, reflective of
the hybridization gap in the spd-like band. In contrast, on “surface B”, identified as the Co
surface, the spectra show a peak near the Fermi energy, reflective of the accumulation of the
heavy fermion density of states caused by the participation of the f -electrons in conduction.
9
a)
b)
c)
Ce-In
In
Co
Figure 1.4: The pioneering experiments of Ref. [21] showed CeCoIn5 to cleave along [001],
exposing multiple different surfaces, most notably ‘surface’ A (the Ce-In layer) and ‘surface
B’ (the Co layer). Because of different tunneling matrix elements on the two different
surfaces, the measured density of states reflects the hybridization of the band structure in
different manners. On surface A, the density of states become gapped beneath TK , as the
spectral weight of the light electrons (green band in a)) is lost as they hybridize and become
heavy near the Fermi energy. On surface B, the counterpart of the hybridization process is
revealed: the onset of heavy quasiparticles (red band in a)) results in a peak in the density
of states.
The work in this thesis [7] builds previous work by performing higher resolution measurements at the lowest temperature (250 mK) on surface B of CeCoIn5 .1 Here, utilizing the
enhanced tunneling sensitivity to the heavy quasiparticles, these experiments better resolve
the energy-momentum structure of the heavy fermion band to enable quantitative comparisons to band structure calculations. However, even more excitingly, the lower temperature
allows access to the dome of superconductivity, whose proximity to magnetic fluctuations is
thought to mediate unconventional Cooper pairing symmetries. This interplay of magnetism
1
Ref. [21] performed QPI on surface A.
10
and superconductivity ties the heavy fermion compounds to the high temperature cuprate
superconductors, and its understanding may provide the essential link between magnetic
correlations and higher Tc ’s.
1.3
Heavy Fermions - Unconventional Superconductivity
Superconductors, broadly speaking, can be divided into two classes by how its electrons bind
together to form the Cooper pairs that sustain dissipation-less current flow. The first class of
conventional superconductors contains all of the metallic superconductors, such as mercury,
lead, or niobium. Here the attractive potential, or pairing symmetry (represented by a
circle in Fig. 1.5), of the Cooper pair is equally strong for all electrons, and the transition
temperature Tc at which superconductivity emerges is limited to at most 30 K, only one tenth
of room temperature. The second class of unconventional, or extraordinary, superconductors
behave much differently and strikingly can possess transition temperatures in excess of 150
K, half of room temperature. The pairing potential (represented by a four-leaf clover in
Fig. 1.5) now depends sensitively on the direction of the electron’s momentum, with certain
directions (called nodes) having zero pairing strength.
The route to higher temperature superconductors relies on understanding the ingredients
that support strong Cooper pairing. The pairing symmetry (or order parameter) of a superconductor is the key experimental observable that manifests the underlying mechanism which
binds electrons together. The conventional superconductors are well explained by a phononmediated interaction under the Bardeen-Cooper-Schrieffer (BCS) theory that assumes the
simplest k-independent, s-wave order parameter [22, 23]. The conceptual breakthrough of
BCS theory is that the range of the Coulomb repulsion between two spatially separated
electrons can be reduced greatly by screening in metal, to the point that the electron’s joint
interaction with a longer-lived lattice deformation of the ionic cores can create a stronger
11
s-wave ky
d-wave
∆
kx
ky
kx
Figure 1.5: Left: Representation of the pairing symmetry of a conventional superconductor
by a circle. The pairing amplitude is isotropic in direction. Right: Representation of the
pairing symmetry for an unconventional dx2 −y2 superconductor by a clover shape. The
pairing amplitude is zero for certain directions called the nodes, depicted by the dashed
lines.
effective attraction. Thereafter, the Fermi sea is unstable to any arbitrarily weak attraction,
choosing instead to form bound electron pairs of the |k, ↑i and |−k, ↓i states, called Cooper
pairs, which condense into a macroscopic wavefunction at sufficiently low temperature and
give rise to the phenomenal superconducting properties of zero resistance and the Meissner
effect. Indeed, the electron-phonon mechanism proposed by BCS theory is well verified experimentally through the isotope effect, which showed that the Tc ’s of elemental isotopes of
Hg, for example, scaled inversely with the square root of their nuclear mass, a parameter
which only tunes the lattice properties [24].
In addition to the phonon-mediated electron attraction, theorists believe that a magnetically mediated interaction, based on fluctuations in the electron’s spin degree of freedom,
can also provide the attractive force necessary for Cooper pairing and can in contrast produce more exotic, anisotropic, finite angular momentum p- or d-wave order parameters.
These ‘magnetic’ theories trace their origin to the superfluid phase of 3 He, which can also
be considered as the result of Cooper-type pairing of the helium fermions into “extended
molecules”. While the attractive interaction in this case is of van der Waals origin, rather
than spin-mediated, the critical concept is that unconventional, finite angular momentum
12
p-wave pairing (l = 1) can minimize the direct on-site Coulomb repulsion between two helium nuclei to allow the attractive interaction to be effective [25]. Thereafter, these ideas
were extended to explain the anomalous superconducting signatures in the organic and heavy
fermion compounds in the early 1980s. In the heavy fermion compound UPt3 , d-wave pairing
was first proposed as the outcome of a spin density wave instability for the Hubbard model
on a cubic lattice [26]. However, the prominence of such ‘magnetic’ theories exploded after
the shocking discovery of high temperature superconductivity in 1986 in the doped copper
oxides, whose transition temperature ∼100 K eclipsed the previously known superconductors by an order of magnitude. The key distinction of the cuprate superconductors is that
the zero doping parent state is an anti-ferromagnetically ordered Mott insulator, the model
system for the Hubbard Hamiltonian with strong on-site repulsion U . Indeed, both measurements sensitive to the amplitude of the order parameter (angle-resolved photoemission
(ARPES), angle-resolved transport studies) and more powerfully, to the phase (i.e., sign) of
the order parameter (grain boundary Josephson junctions) firmly establish the order parameter symmetry of the cuprate superconductors to be dx2 −y2 , thus lending strong support to
a magnetically-mediated mechanism of superconductivity [27].
It is useful to motivate at least a partial understanding of how unconventional d-wave
pairing can be plausible, particularly in the case of spin fluctuations on a square lattice; however, a rigorous calculation of the origin of this magnetic attractive interaction is nontrivial
and may depend on the details of the particular band and crystal structure. In general, we
can still consider solutions to the BCS gap equation [28]:
∆k = −
∆k0
1 X
Vkk0 p 2
N k0
2 ξk0 + ∆2k0
(1.4)
where Vkk0 is the pairing interaction, ξk is the electronic dispersion referenced with respect to
the Fermi energy, and ∆k is the gap function. For an k-independent, attractive Vkk0 = −V0 ,
an isotropic, BCS-type s-wave ∆k = ∆0 solution to equation 1.4 can be verified to exist.
13
On the other hand, investigations of the Hubbard model on a square lattice showed that
in the limit of large on-site repulsion, the effective interaction can be treated as a spin-spin
interaction between nearest neighbors with an antiferromagnetic J, stemming from virtual
hopping processes, and leads to a ground-state of nearest neighbor AF-ordered spins as shown
in Fig. 1.6. When this ground state is doped with holes, it is plausible that an attractive2
nearest neighbor interaction for the holes can arise, having the generic form
Vkk0 = 2 V1 cos (kx − kx0 )a + cos (ky − ky0 )a
(1.5)
with V1 < 0 and a as the lattice constant for a square lattice. From this one immediately
realizes the BCS gap equation in 1.4 can be satisfied if ∆k maintains its sign (changes
adiabatically) when k−k0 → (0, 0) (and Vkk0 < 0), but flips its sign when k−k0 → (π/a, π/a)
(and Vkk0 > 0). These criteria and the four-fold symmetry of the lattice then motivate the
dx2 −y2 gap function diagrammed in Fig. 1.5
∆k = ∆0 (cos(kx a) − cos(ky a))
(1.6)
which satisfies the the condition of continuity for small displacements along the Fermi surface
and the condition for sign change for parts of the Fermi surface connected by the vector Q =
(π/a, π/a). Moreover, the cuprate Fermi surfaces contain long parallel sections nested by the
vector Q, which maximizes the the advantage of such a dx2 −y2 gap function. In summary, a
nearest neighbor attractive interaction, plausibly mediated by AF spin fluctuations with a
ordering vector ±Q, and a favorable Fermi surface shape can provide the ripe background
for unconventional dx2 −y2 pairing.
The heavy fermion CeCoIn5 has been suggested to share the same pairing symmetry
and thus possibly the same underlying mechanism of superconductivity due to its many
other similarities to the cuprate superconductors, such as its tetragonal lattice structure
2
We have not proven that it is attractive, but will examine the consequences of this assumption.
14
Q
+
U
+
-
t
J = 4t2/U
Figure 1.6: The Hubbard model, characterized by an on site repulsion U and a hopping
energy t, on a square lattice leads to an antiferromagnetic arrangement of spins at half
filling. When holes are doped into the system, remnant antiferrogmagnetic correlations could
lead to an attractive interaction between the now mobile electrons. When this interaction
combines with a Fermi surface with nested (parallel) sections separated by the ordering
vector Q = (π/a, π/a) , dx2 −y2 superconductivity is believed to be favored.
with quasi-2D square lattice planes, its proximity to an AF critical point, and its non-Fermi
liquid normal state.3 Unfortunately, its superconducting phase has largely dodged the same
intense experimental spotlight, as its low transition temperature (Tc = 2.3 K) precludes
access by conventional experimental tools such as ARPES and, until recently, STM. While
angle-resolved thermal transport [29] and neutron scattering experiments [30] have shown
data consistent with dx2 −y2 gap symmetry, no smoking gun identification has yet been reported. In CeCoIn5 , more favorable to experiment, unconventional superconductivity occurs
in the undoped, and thus ultra-clean, samples and can be extinguished by an experimentallyaccessible magnetic field (5 T perpendicular to the c-axis).
This thesis details a comprehensive set of STM experiments performed by Zhou and Misra
[7], spanning spectroscopy and quasiparticle interference, on the superconducting state of
CeCoIn5 at the ultra-low electron temperature of 250 mK. To isolate the salient effects of
3
It is also important to keep in mind the disparities between the heavy fermion and cuprate compounds.
The cuprates are more itinerant, d-electron compounds with strictly two-dimensional conduction and a Mott
insulating parent state. In some sense, the reduced dimensionality and strong Mott repulsion offsets the
greater itinerancy, in comparison to the f -electron, more three dimensional heavy fermion superconductors.
Moreover, the bandwidths t ∼ V in the cuprate superconductors are much larger than in the heavy fermion
compounds (t ∼ 10 mV), but the ratio Tc /t for the two families are comparable.
15
superconductivity, we in fact conducted each experiment first at zero magnetic field, where
superconductivity is strongest, and then repeated each experiment again at 5.7 T, where
superconductivity is extinguished. Unexpectedly, QPI band mapping demonstrates particlehole asymmetric patterns in the superconducting state, reflective of either the more rapid
heavy fermion band dispersion or enhanced impurity effects relative to that for the cuprate
superconductors. However, likewise to the cuprates, a spectroscopic pseudogap appears prior
to the onset of superconductivity and persists above the critical magnetic field. Finally,
through visualizing the spatial symmetry of quasiparticle states bound to atomic defects,
the structure of Cooper pairing in CeCoIn5 is pinpointed to be dx2 −y2 , in parallel with
that of the high-temperature cuprate superconductors. Long term, the ability to study
CeCoIn5 broadens the experimental tool-kit for tackling the questions of unconventional
superconductivity: what role of magnetism, other competing phases, and electron-electron
interaction in the normal state play in making these special superconductors the most robust
of the bunch.
1.4
Three-Dimensional Dirac/Weyl Semimetals
The notion of topologically protected physical effects that manifest without fine tuning and
survive in the presence of disorder is a welcome concept to the experimental physicist, trained
to pay attention to the finest details. Tracing its origins to the integer quantum Hall effect
[3], where exquisitely precise resistance quantization (∼ one part in a billion) literally thrives
because of disorder, this idea that certain effects stem from the intrinsic topology of the system and are thus robust to perturbations that do not violate the original symmetries, has
flourished in recent years with the discovery of topological insulators, whose band structure
guarantees the existence of surface states at its interface [31]. Recently, another example
of a topologically protected phase, this time in a semimetal rather than insulator, has been
proposed: the Weyl semimetal, whose topological band touchings give rise to similarly pro16
tected surface states, known as ‘Fermi’ arcs, and to the condensed matter analogue of the
chiral anomaly, which resolved the paradox of the decay of the π 0 meson in high energy
physics [32].
Weyl semimetals contain discrete points in its Brillouin zone where two non-degenerate
bands touch, in contrast to Dirac semimetals where inversion and time reversal symmetries
guarantee the double-degeneracy of each of the touching bands [33, 34]. Hence, to realize a
Weyl semimetal, either time reversal or inversion symmetry must be broken, for example,
by a magnetic field or by a layered heterostructure scheme, respectively. The reduction
of dimensionality from the 4x4 space used to describe the physics around a Dirac point
to the 2x2 space around a Weyl point presents important consequences for the robustness
and topology of Weyl points. Generically, the low energies dispersions around a Weyl point
is linear in three dimensions and can be described by the most general linearly-coupled
Hamiltonian
H(k0 + q) = v0 · q 1 +
3
X
vi · q σi
(1.7)
i=1
where q is the displacement from the Weyl node at k0 , the identity matrix 1 and the Paul
matrices σi span the space of 2x2 Hermitian matrices, and each of the four three-dimensional
velocities vi linearly couple to the momenta q = (qx , qy , qz ). The simplest reduction of this
Hamiltonian is
H(k0 + q) = ±vF q · σ = ±vF (qx σx + qy σy + qz σz )
(1.8)
where we have taken the velocities of equation 1.7 to be orthogonal and equal to ±vF and
neglected v0 as it only introduces a tilting or asymmetry to the conical dispersion, but does
not affect any of the topological properties. The similarity of Weyl equation 1.8 to the
Hamiltonian for graphene, H(q) = vF (qx σx + qy σy ), is obvious; however, the use of the σz
matrix in the Weyl equation means that no remaining terms can gap out the Weyl node. In
17
graphene, the addition of a term proportional to m σz , such as from the breaking of sublattice
symmetry (e.g., boron nitride), introduces a gap to the Dirac spectrum; whereas, such a term
for the Weyl Hamiltonian merely shifts the Weyl node from k0 to k00 = k0 + (0, 0, m), rather
than gapping it. Thus the Weyl node is robust to arbitrary perturbations that preserve the
original symmetries and can only be annihilated by coupling to another Weyl node of the
opposite chirality, a topological feature we describe below.
a)
b)
3
2
1
En
0
+ Weyl Pt
1
2
3
c)
4
2
0
2
4
3
2
1
En
0
- Weyl Pt
1
2
3
4
2
0
k‖B
2
4
Figure 1.7: a) Schematic band structure showing the linear dispersion around two 3D Dirac
points. In a magnetic field, each Dirac point is split into two Weyl points. b) The Landau
level spectrum for Weyl points of chirality ±χ, represented as a monopole or anti-monopole
of the Berry flux. Notice that the zeroth Landau level is chiral, disperses only in one direction
(i.e. the zeroth Landau level for a Weyl node of chirality χ has dispersion −χvF kz , where kz
is parallel to the field.)
The topological index of a Weyl node, called the chirality χ, can be taken as the handedness of the velocities in equation 1.7
χ = sgn[v1 · (v2 × v3 )] = sgn[det(vij )] = ±1
(1.9)
where vij is the j th component of vi [35]. Perhaps even more elegantly, the chirality χ can
be equivalently computed in terms of the Berry flux F(k) over a two-dimensional surface S
18
enclosing the Weyl node in the Brillouin zone
1
χ=
2π
I
F(k) · dS(k) = ±1,
(1.10)
S
which establishes the Weyl nodes as monopoles of the Berry flux and χ as the Berry ‘magnetic’ charge. As pointed out by Nielsen and Ninomiya [36], the sum of the chiralities for all
the Weyl points must be zero due to the periodicity of the Brillouin zone; thus, there must
be an even number of Weyl nodes in the Brillouin zone, with half of each chirality. Inversion
symmetry then implies that if a Weyl node χ exists at k0 , then one of opposite chirality −χ
must exist at −k0 . Simultaneously, time reversal symmetry implies that for a Weyl node χ
at k0 , a Weyl node of the same chirality must exist at −k0 .4 Thus if both inversion and
time reversal symmetries are intact, Weyl nodes of opposite chiralities must exist at k0 (and
−k0 ) and may annihilate one another, explaining why isolated Weyl nodes require breaking
either inversion or time reversal symmetry.
The conservation of zero total chirality leads to topologically-protected surface states,
even in the gapless system of the Weyl semimetal [34]. As first proposed in the pyrochlore
irridates, these surface states join the projections of two Weyl nodes of opposite chirality
onto the surface Brillouin zone for a particular crystal face, thereby extending the two point
semimetal Fermi surface into a single ‘arc’ (i.e., like a Dirac string joining a monopole
with an anti-monopole). These surface states are guaranteed to exist at the Fermi level by
correspondence to the edge states of quantum hall systems and can disperse in energy away
from the Fermi level, but must exist in portions of the surface Brillouin zone non-overlapping
with the projected bulk band structure. The disjointed ‘arc’-like nature of the Fermi surface
can be rationalized by realizing that as the surface state arc approaches either Weyl node,
its spatial character extends further into the bulk via hybridization with the bulk state at
the Weyl node, and finally it reappears on the other surface, where it can connect to another
4
This is true since in equation 1.8 the momentum q is odd under inversion, odd under time reversal,
while the Pauli matrices σi transform as a pseudovector (even under inversion, odd under time reversal).
19
Fermi arc, before re-entering the bulk again through the other Weyl node and completing
the Fermi surface.
A second novel aspect due to the topology of the Weyl semimetal is the presence of the
chiral anomaly [35, 32]. The Landau levels for a single Weyl node in a magnetic field parallel
to the z-direction are
E0 = −χ~vF qz
p
En = vf sgn(n) 2~|n|eB + (~qz )2 , n = ±1, ±2, ...
(1.11)
where critically, the zeroth Landau level (ZLL) disperses in only one direction depending on
the chirality χ of the Weyl node, as is shown in Fig. 1.7.5 It is useful to keep in mind that
the equations we have presented are only the low energy expansions close to the Weyl point,
and in real materials, Weyl nodes must eventually merge in a Lifshitz transition at some
higher energy scale (since there must be an even number of them). Hence, the dispersion of
the ZLL of a positive chirality Weyl node will link to the dispersion of the ZLL of a negative
chirality Weyl node at some energy scale, somewhere in the Brillouin zone. In this case then,
when a electric field E is applied parallel to the magnetic field B, the electronic states will
drift under semi-classical theory as
~k̇ = −eE
(1.12)
thus depopulating one Weyl node and populating the one of opposite chirality. Thus the
electric field establishes an imbalance of the chiral charge, effectively charging a chiral battery
at a rate proportional to E · B. To see this, let us define the chiral charge Q = e(Nχ −
N−χ ), where Nχ is the number of uncompensated states of chirality χ. In a time dt, the
Fermi momentum of one Weyl node increases by eEdt/~ by equation 1.12, while the Fermi
5
A full derivation is very similar to the Landau levels of graphene, with only the addition of a qz σz term
which is unaffected by the
magnetic vector potential. Therefore, the Hamiltonian in harmonic oscillator
√
qz
2/lb a 0
√
language is H =
and the zeroth Landau level |n=0i
has eigenvalue −qz .
2/l a† −q
b
z
20
momentum of the other Weyl node is reduced by the same amount. The density of states
along the one dimensional ZLL is LB /2π, while the degeneracy of the ZLL (how many copies
of the zeroth chiral Landau band we have) is A⊥ B/Φ0 = eA⊥ B/2π~. Multiplying these
degeneracies, we obtain for the rate of chiral charge accumulation
dQ
eE LB eA⊥ B
= 2e ∗
∗
∗
dt
~
2π
2π~
e3 V E · B
=
2π 2 ~2
(1.13)
where we have taken V = A⊥ LB to be the system volume. This chiral imbalance can
manifest itself through a negative magnetoresistance6 and other exotic magneto-transport
signatures, such as the anomalous hall effect and the chiral magnetic effect, where a pure
magnetic field can induce a non-equilibrium electric current. Moreover, proposals have been
put forth to utilize its nonlocal electronic transport properties or sensitivity to magnetic
fields for practical applications [38].
This thesis describes Landau quantization and quasiparticle interference measurements
performed by Jeon and Zhou on the Weyl semimetal candidate, the ultra-high mobility II-V
semiconductor Cd3 As2 [39]. At zero magnetic field, Cd3 As2 is actually a Dirac semimetal
since inversion and time reversal symmetries are preserved. The band touching points are
formed at the crossing of two doubly-degenerate bands and thus represent two overlapping
Weyl points. As we have mentioned, in this case, the Dirac point is generally susceptible
to gapping, so an additional symmetry is required to preserve the gapless Dirac points in
Cd3 As2 . As discovered in Refs. [40, 41], if the three-dimensional Dirac points occur along
certain high symmetry directions in the Brillouin zone, they are protected by crystalline
space group symmetries. For example, the C4 screw symmetry around the kz axis in Cd3 As2
protects the two Dirac nodes in this direction. Hence, Cd3 As2 can host a Weyl semimetal
phase when time reversal symmetry is broken through a magnetic field applied along the
6
Although in real materials, such as Cd3 As2 , this may be masked by positive contributions to the magnetoresistance [37].
21
c-axis ([001] direction), which still preserves the original rotational symmetry. However, due
to the (112) plane being the natural growth and cleavage plane of Cd3 As2 , we could not
access the Weyl semimetal phase by our experimental restriction of applying a magnetic
field perpendicular to the cleaved sample surface. Nevertheless, our measurements confirm
many aspects of the Dirac semimetal phase, such as the extended linear dispersion away
from the Dirac points and the expected two-fold conduction band degeneracy, resolvable at
high magnetic fields, and reveal for the first time the microscopic details and band structure
regime relevant to the ultra-high mobility seen at the Fermi level [37].
1.5
Thesis Outline
This thesis first begins with Chapter 2 which provides a short exposition of the technique of
scanning tunneling microscopy and a discussion of the quasiparticle interference technique,
the method by which momentum space information can be extracted from a real space STM
measurement. Next in Chapter 3, we describe the design, operation, and performance of
the novel dilution refrigerator-based STM constructed during the course of this thesis and
used exclusively for the experiments presented herein. The first experiment performed on
this instrument, described in Chapter 4, investigated the unconventional superconducting
state in the heavy fermion CeCoIn5 at milli-Kelvin temperatures and proved definitively
that the order parameter is of dx2 −y2 symmetry. In addition, this experiment demonstrated
the onset of a spectroscopic pseudogap prior to superconductivity in the strongly-hybridized
heavy fermion band. In Chapter, 5, we leverage the high field capabilities of the machine to
dissect the intriguing band structure of Cd3 As2 and verify its predicted 3D Dirac dispersion,
which unexpectedly survives to higher energies than originally believed. Here, Landau level
spectroscopy is extended to a three-dimensional band-structure, in distinction to its usual
application in STM to two-dimensional systems, such as surface states and graphene. We
conclude in Chapter 6 with a discussion of future technical improvements and potential
22
extensions to the experiments performed. The appendices provide additional information on
various aspects, including the day-to-day operation of the system, an alternative conductance
mapping technique used for the experiments reported in the thesis, and technical details of
the simulations used to understand the data. Moreover, we briefly discuss preliminary data
on the search for enhanced charge order in underdoped Bi2 Sr2 CaCu2 O8+δ at high magnetic
fields.
It is hoped that others may benefit from the lessons learned and may use this thesis to
push the limits of STM to even colder temperatures, stronger fields, and quieter performance
to behold the marvel of material properties at the atomic scale and at the frontiers of phase
space.
23
Chapter 2
The Basics of Scanning Tunneling
Microscopy
Microscopy, from the Greek words micros, meaning small, and skopos, meaning to observe, is
the development of tools to view and study objects unresolved by the human eye. Among the
many techniques in this field, scanning tunneling microscopy is perhaps the most powerful,
as its resolution is set not by the wavelength of a probe beam of photons or electrons which
scatter from a sample, but by the overlap between the quantum mechanical wavefunctions of
the atoms on a probe and sample. In this chapter, we introduce the theoretical foundations
for this technique and highlight the method of quasiparticle interference imaging, which can
provide momentum space information from an intrinsically real space measurement.
2.1
Theory
A complete history of STM must start with the demonstration of quantum tunneling by Leo
Esaki in semiconductor tunnel diodes (1957) [42] and by Ivar Giaever in superconductorinsulator-superconductor junctions (1960) [43]. These seminal experiments revealed the
quantum nature of matter at reduced length scales, showing that electrons could pass through
a classically forbidden region, if that region, or barrier, is made thin enough. Indeed, the
24
STM is merely an inspired application of this idea, by making one of the tunneling contacts
a sharp metallic wire, called the “tip”, fully positionable over the the other tunneling contact, the sample, with vacuum as the energy barrier in between. Using piezo-electric motors
and tube scanners that enable three dimensional motion with sub-picometer accuracy, the
tip can be placed within ten angstroms from a clean, conductive sample surface, reducing
the barrier length until the tunneling current can be measured as a function of the lateral
position of the tip, as it is scanned across the sample surface.
Mathematically, the sample electron wavefunctions ψs decay across the vacuum barrier
as [44]
ψs (z) = ψs (0)e−κz
(2.1)
where for electrons near the Fermi level Ef ,
√
κ=
2mφ
~
(2.2)
with φ denoting the work function of the material. The tunneling amplitude, proportional
to |ψs (zt )|2 , is exponentially sensitive to the position of the tip zt above the sample surface,
and this extraordinary sensitivity underlies the atomic resolution of the STM. For metals,
φ ∼ 5 eV and the change in tunneling current is an order of magnitude for only a single Å
of tip displacement.
The total tunneling current for a bias voltage −V applied to the sample can be quantitatively calculated via Fermi’s golden rule of time-dependent perturbation theory. As shown
schematically in Fig. 2.1, the density of states (DOS) and occupation level of the sample
EFsample are shifted rigidly up by eV in energy with respect to tip density of states and occupation level EFtip . It is important to distinguish that the action of the bias −V is not to fill
additional levels in the sample DOS, but rather to raise the energy of the originally occupied
states with respect to the tip’s Fermi level such that elastic tunnelling out of the occupied
25
‘
‘
‘
Figure 2.1: Representation of tunneling from sample to tip at −V bias to sample. Conservation of energy implies filled electrons states of the sample tunnel horizontally across to equal
energy, empty states of the tip. Reproduced from [45].
sample states to the corresponding empty tip states can occur. With this understanding, the
correct equations can be written down, accounting for both the dominant current tunneling
from sample to tip and (for completeness) the much smaller, reverse current from tip to
sample
4πe
I(V ) = −
~
Z
∞
(2.3)
f (EF − eV + )(1 − f (EF + ))
{z
}
|
sample to tip
− (1 − f (EF − eV + ))f (EF + ) ρs (EF − eV + )ρt (EF + )|M |2 d
|
{z
}
tip to sample
Z
4πe ∞
=−
f (EF − eV + ) − f (EF + ) ρs (EF − eV + )ρt (EF + )|M |2 d (2.4)
~ −∞
−∞
F
where f () = (1 + exp( −E
))−1 is the Fermi function with the original chemical potential
kB T
EF (of the tip and sample when V = 0), and ρs () and ρt () denote the sample and tip DOS,
respectively. The matrix element |M |2 captures the square amplitude of the overlap between
tip and sample wavefunctions, including their spatial character and decay rate κ across the
barrier. Assuming ρt and |M |2 independent of energy and kB T smaller than features of
26
interest so that Fermi functions are step-like
f (EF − eV + ) − f (EF + ) =




0 for < 0




1 for 0 < < eV






0 for eV < the expression can be simplified to
Z 0
4πe
2
I(V ) = −
ρs (EF + 0 ) d0
|M | ρt
~
−eV
4πe
dI
(V ) = −
|M |2 ρt · ρs (Ef − eV ),
dV
~
(2.5)
where we have redefined 0 = − eV .1 Thus, in theory by performing current-bias sweeps at
a point, the differential conductance
dI
dV
can be numerically computed to reveal the energy-
resolved local DOS of the sample, the key experimental signature of the underlying physics,
from the supeconducting energy gap to the Kondo resonance. Operationally, a fixed position
for the tip zt is first chosen via a setpoint condition I0 (V0 ), the feedback loop is then opened
to maintain zt as the bias voltage is swept, and a small AC modulation is summed on top of
the swept voltage such that a lock-in measurement can be performed to detect the differential
current at the modulation frequency.
In addition to local point spectroscopy, the tip may be scanned across the surface to add
the two lateral real space degrees of freedom x and y to any measurement. This simplest
scanned measurement is called topographic mode, where a feedback loop adjusts the height
of the tip z as it moves across the surface to keep the total tunneling current constant.
Topographic images z(x, y) often reveal the underlying structural features, but in theory
is also sensitive to the local conductivity. However, the most powerful STM measurement
1
The sign convention for these equations is such that +V tunnels electrons out of the sample. Generally
in STM, one applies bias to the sample, such that experimentally a negative voltage −V is applied to the
sample to tunnel electrons out of the sample. For this setup, we generally just remember that −V probes
filled states of sample, +V probes empty states of sample.
27
Point Spectra - LDOS
Topographic Mode
Spectroscopic Imaging
dI/dV (nS)
0.8
0.6
0.4
0.2
0
−150 −100
−50
0
50
Bias (mV)
100
150
Figure 2.2: The three modes of STM operation exemplified on the high-Tc superconductor
Bi2 Sr2 CaCu2 O8+δ , perhaps the single one material that proved the power of STM as a condensed matter probe. Local dI/dV measurements reveal the unconventional superconducting
gap spectrum (contrast with s-wave superconducting gap in Fig. 3.5). Constant current topographic (z) image of the BiO plane shows the individual Bi atoms and a stripe-like bulk
supermodulation. Finally, by plotting the differential conductance at a particular voltage
(dI/dV (V = 22 mV )) over the real space view, the STM directly visualizes the ordering of
the electronic states into a ‘checkerboard’ pattern.
is called spectroscopic imaging. Here, it is differential conductance measurement
is performed on a grid of points, and the resulting maps of
dI
dV
dI
dV
that
(x, y, V ) ≡ C(x, y, V ) at a
particular energy V reveal the inhomogeneity of electronic states, such as the ‘checkerboard’
charge ordering in the high-Tc superconductor Bi2 Sr2 CaCu2 O8+δ as shown in Fig. 2.2, and
be analyzed for quasiparticle interference, as described below. Such conductance maps often
take several days to complete; hence, it is the ultimate test of the stability of the STM, to
both long time scale drifts and instantaneous vibration performance.
2.2
Quasiparticle Interference Imaging
Electronic inhomogeneity on metal surfaces can arise from many sources, from random fluctuations due to chemical doping or periodic order due to charge density waves. Perhaps of
the most generic source is the effect known as quasiparticle interference (QPI), whereby the
breaking of translational symmetry due to defects on the surface of a crystal, such as an
atomic step edge or localized point disorder, mixes the Bloch states of the translationally
28
invariant system and introduces (Friedel) oscillations in the local charge density. When the
plane wave state Ψ1 = eik1 ·r interferes with a second plane wave state Ψ2 = eik2 ·r due to a
scattering potential, the resulting charge density
ρ ∝ |eik1 ·r + eik2 ·r |2 = 2(1 + cos(q · r))
(2.6)
acquires a modulation at the wavevector q = k1 − k2 , with an amplitude that decays away
from the scattering center. A STM conductance map C(x, y, V ) visualizes these modulations
of the charge density, ideally over a large field of view where many defects contribute to many
quasi-independent modulations.2 The wavelength and direction of the modulations can then
be determined by taking the two-dimensional Fourier transform of this map, once the length
scale of the piezo scanner is calibrated from the atomic Bragg peaks. A typical set of QPI
data is shown for the prototypical example of “surface waves” on Cu(111) in Fig. 2.3.
To discuss how the wavevectors q relate to the Fermi surface of the underlying material,
we note that scattering processes conserve energy so that the set of possible q vectors span the
vectors that connect two points on the Fermi surface. For now let us assume that the Fermi
surface is two-dimensional, such as in the case for surface states of Cu(111) or topological
insulators, as the three-dimensional Fermi surface requires additional assumptions. Then
heuristically, we might expect the Fourier transform map Ĉ(qx , qy , V ) = F{C(x, y, V )} to be
proportional to the auto-convolution of the Fermi surface intensity I(k) in two-dimensions
Z
Ĉ(q, V ) ∝
I(k, V )I(k + q, V )d2 k.
(2.7)
BZ
2
Experimentally, the range L of the real space field of view determines the momentum q space resolution
2π/L of the Fourier transform map; while the real space resolution L/N, where N is the number of pixels of
the map, determines q space range ±πN/L.
29
Energy
kf
Q
ki
100 Å
ky
kx
(1.0, 1.0) Å-1
-220 mV
-20 mV
180 mV
Figure 2.3: The surface state of Cu(111) displays quasiparticle interference caused from
the scattering from carbon monoxide molecules and step edges on the surface. The energyresolved spectroscopic maps display wave modulations, whose wavelength decreases with
increasing energy (the overlaid triangle shows that the real space scale does not change
between images) The 2D Fourier transform of such real images reveal a ring of wavevectors
whose radius Q is equal to twice the k vector of the contour of constant energy of the
parabolic surface state band structure, shown in the left schematic. Q = 2k reflects the
relative importance of backscattering, which in the limit of delta function potentials and
infinite lifetime should be the only wavevector existing [46, 47].
This is the so-called joint density of states (JDOS) approximation.3 The crucial insight is
that the Fourier transform of a real space image, due to interference modulations, can be used
to extract momentum space information about the Fermi surface shape, although generally
the convolution is not exactly invertible, especially in the presence of noise. However, for
many simple Fermi surface geometries, such as a circle or a square, the auto-convolution of
the Fermi surface, measured by the QPI technique, remains dominated by the same shape
(i.e., square or circle) except for the scaling q = 2 k. This is most clearly demonstrated
3
The JDOS approximation is a model for QPI and should not be treated as an exact equation. JDOS
in general underestimates the intensity of 2kF backscattering, which should dominate hard wall and delta
function potentials.
30
in Fig. 2.3 for the circular Fermi surface contours of the parabolic surface state band of
Cu(111).
More generically, we can define a matrix element that modulates the scattering contribution depending on the initial and final states such that the equation is modified to
Z
Ĉ(q, V ) ∝
I(k, V )T (k, q)I(k + q, V )d2 k.
(2.8)
BZ
Such a matrix element might arise due to the spin projection of the initial and final states
of spin-momentum locked surface states in topological insulators or due to superconducting
coherence factors for quasiparticles in d-wave superconductors. The next question arises
as to what to use for the intensity of the Fermi surface I(k). Generally, we can use the
experimental ARPES intensity for I(k) when it can be shared with us by our ARPES
collaborators. However, when ARPES data is not available, such as for the heavy fermion
experiments described in this thesis, I(k) can be taken as the Green’s function for some
model band dispersion (k)
I(k, ω) ∝ Ĝ(k, ω) =
1
,
ω + iΓ + (k)
(2.9)
where we now imagine trivially computing the JDOS at arbitrary energy ω (i.e., bias voltage
V), and the lifetime Γ can be chosen to suitably broaden the resulting features in accordance
with experimental data. When the Green’s function is used for I(k), equation 2.7 is precisely
the first term of the Born scattering series for an impurity. These equations are operationally
computed via matrix Fourier transforms that take advantage of the convolution theorem.4
We extend the discussion to QPI in the case of three dimensional band structures. Formally, since the Fermi surface must be considered in the three dimensional Brillouin zone
4
In general, to take advantage of the convolution theorem, the integrand in equation 2.8 must be factorized
into the product of two terms, one involving k and one involving k + q. For example, T (k, q) in the case of
spin projection takes the form cos (θ1 − θ2 )/2 , where θ1 (k) and θ2 (k + q) are the orientations of the spinor
at the initial and final states. This is computed by expanding the expression into 1/2 ∗ (1 + cos(θ1 ) cos(θ2 ) +
sin(θ1 ) sin(θ2 )) and thereafter summing the three separate convolutions by FFT.
31
and the STM can only measure the component of the modulation projected onto the surface
qk , the joint density of states becomes the integral over 3D Fermi surface and over the qz
component of the scattering vector
Z Z
Ĉ(qk , V ) ∝
I(k, V )I(k + q, V )d3 k dqz ,
(2.10)
3D BZ
which is significantly more difficult to compute because of the added dimensionality. Accordingly, the simplifying assumption one makes is that the QPI signal for a 3D band structure
is a sum, perhaps weighted sum, of the individual 2D QPI from slices of the Fermi surface
for fixed kz , where z is perpendicular to the surface.
Ĉ(qk , V ) ∝
X
kz
Z
I(kk , kz )I(kk + qk , kz )d2 kk .
w(kz )
(2.11)
2D BZ for fixed kz
This less general equation restricts q to have only zero qz component, which is approximately
correct in limit of strong backscattering for a Fermi surface with symmetry about the kz = 0
plane. In general, we can even further simplify by saying the the sum in 2.11 is dominated
by one or two particular values of kz , such as kz = 0 or kz = π as in the case of the CeCoIn5
measurements to be discussed in this thesis.
Despite some of these inconveniences, one could wonder why QPI as a technique would be
favored over conventional ARPES which can measure the momentum structure of materials
directly and quickly. Certainly in a direct head-to-head match, ARPES is strongly favored
to QPI. However, QPI is applicable in many situations when ARPES is not, such as above
the Fermi level, in a magnetic field, and when high energies resolution is needed. This is
particularly true for the rapid dispersions of heavy fermion materials, where the current thesis
used ∼ 80 uV energy resolutions to resolve the “flat” bands that emerge within millivolts of
the Fermi level. Finally, we argue that sometimes it useful to measure scattering (the effect)
directly, rather than just the Fermi surface (the cause), because it is ultimately scattering
than determines the transport properties of the material. This is beautifully illustrated in
32
the experiments on topological insulators by Roushan [48], which directly demonstrated the
absence of backscattering, the key consequence of spin-momentum locking in those materials.
33
Chapter 3
The Dilution Refrigerator Scanning
Tunneling Microscope
This chapter is based upon the publication:
Misra S., Zhou B. B. et al.,“Design and performance of an ultra-high vacuum scanning tunneling microscope operating at dilution refrigerator temperatures and high magnetic field.”
Review of Scientific Instruments 84, 103903 (2013) [10].
We describe the construction and performance of a scanning tunneling microscope (STM)
capable of taking maps of the tunneling density of states with sub-atomic spatial resolution
at dilution refrigerator temperatures and high (14 T) magnetic fields [10]. The fully ultrahigh vacuum system features visual access to a two-sample microscope stage at the end of
a bottom-loading dilution refrigerator, which facilitates the transfer of in situ prepared tips
and samples. The two-sample stage enables location of the best area of the sample under
study and extends the experiment lifetime. The successful thermal anchoring of the microscope, described in detail, is confirmed through a base temperature reading of 20 mK, along
with a measured electron temperature of 240 mK. Atomically-resolved images, along with
complementary vibration measurements, are presented to confirm the effectiveness of the vi34
bration isolation scheme in this instrument. Finally, we demonstrate that the microscope is
capable of the same level of performance as typical machines with more modest refrigeration
by measuring spectroscopic maps at base temperature both at zero field and in an applied
magnetic field.
3.1
Introduction
Scanning tunneling microscopy (STM), since its development almost 30 years ago, has become a powerful technique in condensed matter physics, providing not only structural information about surfaces, but also spectroscopic measurements of the electronic density of
states at the atomic length scale. However, most instruments operate at temperatures above
1 K, limiting access to exotic electronic phases and quantum effects expected at lower temperatures, which are studied as a matter of routine by other techniques. Generally, very little
spectroscopic information about the electronic density of states is known at dilution refrigerator temperatures, usually being limited to what can be learned using either point contact
spectroscopy or planar tunnel junctions. Moreover, STM can make such measurements on
the atomic length scale, allowing it to probe systems, such as single spins and atomic chains,
which are not directly accessible any other way.
While the integration of STM with a dilution refrigerator can be conceptually reduced
to simply attaching the microscope to the end of a mixing chamber in lieu of some other
cryogenic refrigerator, the technical requirements for sub-Angstrom positioning of an STM
tip above an atomically clean surface are often at odds with those for cooling a sample to
milli-Kelvin temperatures. For example, when attaching the microscope to the refrigerator,
the former would favor the use of a soft mechanical joint using springs, which would isolate
vibrations, while the latter would favor the use of a rigid mechanical joint with a metal rod,
which would provide a strong thermal contact. Nevertheless, a number of STM instruments
have been developed that cool the sample using a dilution refrigerator [49, 50, 51, 52, 9, 53,
35
54, 55]. However, among these, few feature ultra-high vacuum (UHV) environments, which
would facilitate the in situ preparation of tips and samples, a crucial step in preparing many
samples and functionalizing STM tips [55, 51, 9]. Moreover, few have the level of stability
and performance required to measure spectroscopic maps of the electronic density of states
with atomic spatial resolution, crucial to obtaining detailed information about the electronic
state of the compound under study [50, 54, 51, 9]. Here, we describe the construction and
performance of a home-built STM designed specifically to extend the level of functionality
and stability common in higher temperature systems to dilution refrigerator temperatures.
3.2
Ultra-High Vacuum Assembly
The successful integration of a dilution refrigerator into an ultra-high vacuum environment
has the unmeasurable benefit for scanning tunneling microscopy that the full range of samples
available to the technique could be studied, and standard techniques for the in situ preparation of tips and samples could be used without alteration. Although standard dilution
refrigerators contain materials, such as nylon, brass and soft solder, which are anathema to
ultra high vacuum, substitution by UHV-compatible materials (PEEK, OFHC copper, and
high temperature solder) and the adoption of proper cleaning methods has been successfully
implemented in a number of systems [56, 57, 58]. The remaining difficulty lies in devising a
scheme to transfer tips and samples between various UHV stages and the microscope.
Toward this end, we mount a specially designed bottom-loading dilution refrigerator
insert onto a z-manipulator which can translate the insert and attached microscope between
the measurement and sample access positions within contiguous UHV space (Fig. 3.1b). This
UHV space of the instrument extends up from the chambers via an inner vacuum can (IVC)
with a flexible bellow up to a head flange on the insert (Fig.
3.1a). With the exception
of the wiring interface, which is through a chamber at the top of the insert and connects
to the head flange through a series of tubes, the head flange is the terminus of the UHV
36
a
d
Insert
Manipulator
b
Manipulator
Platform
& Track
3.3 c
m
Cryostat
Manipulator
Bellows
OVC
e
Measurement
Position
Sample Access
Position
Insert Head
Flange
IVC
Bellows
Chambers
2.16 m
Helium
Space
c
Preparation
Chamber
UHV
Space
IVC
Bottom
Flange
Transfer
Chamber
Cryostat
Bottom
Flange
2.71 m
Figure 3.1:
(a) This 3D CAD drawing shows a zoomed-in cross-section of the insert,
manipulator and cryostat, whose connection to the UHV chambers is shown in (b). When
the manipulator platform moves down its track (yellow), the manipulator and IVC bellows
(red) contract, and the insert (brown) moves out the end of a UHV neck at the bottom
of the cryostat into the chambers below, translating the attached microscope between the
measurement and sample access positions labeled in (b). The relevant Conflat flanges which
interface the insert to the IVC (IVC head flange), the IVC to the cryostat (IVC bottom
flange), and the cryostat to the vacuum chambers (cryostat bottom flange) are shown in
pink. (c) The UHV chambers beneath the cryostat include a transfer chamber, a preparation
chamber, and a load lock (green). The wobble sticks and manipulators used to transfer
samples are shown in red. The focal point of the transfer chamber can accommodate a
number of evaporation sources (two are shown in purple). The left-hand focal point of the
preparation chamber has a resistive heater stage for samples or tips (not shown), and can
have up to four evaporators pointed at it (two are shown in cyan). The right-hand focal
point of the preparation chamber has an e-beam heater stage with an Ar ion sputter gun
pointed at it (dark blue). (d) This drawing shows a close-up of the e-beam heater stage, with
the sample holder shown in green. Alumina pieces are shown in light blue, and the filament
in red. (e) Photograph of the transfer and preparation chambers attached to the cryostat.
37
space. The top of the rigid insert is secured to a heavy duty (non-UHV) z-manipulator.
This manipulator lowers the entire insert down 65 cm, collapsing the IVC bellows. This, in
turn, moves the microscope, which normally sits at the center of a 14 T magnet (103 mm
bore diameter) when the manipulator is up, through the bottom neck of the cryostat, and
produces it at the center of the UHV chamber below. After opening a rotary door on the
radiation shield of the refrigerator using a multi-motion wobble stick, we have direct visual
access to the microscope itself.
The UHV utilities in the three chambers attached to the cryostat have been specifically
designed to enable the implementation of the full suite of recipes for in situ preparation of
spin-polarized STM tips and samples[59]. New tips and samples are introduced into the UHV
chambers through a standard load lock attached to the preparation chamber (Fig. 3.1c). The
preparation chamber contains two points which each lie at the focus of multiple ports of the
vacuum chamber, one of which has an e-beam heater (Figure 3.1d) and sputter gun to clean
tips and metal samples, and the other of which has a resistive heating stage and evaporators
which can be used to grow thin metal films on them. To allow for evaporation onto a cold
sample, the transfer position of the dilution refrigerator insert sits at the focal point of three
ports of the transfer chamber, to which standard evaporators can also be attached. Finally,
cleavable samples can be both cleaved and stored in the preparation chamber.
The operating base pressure of the system is ∼ 10−10 torr. The UHV chambers can
achieve this level of vacuum simply by baking to 130 C for two days. The insert has a pair of
heaters located near the microscope, but can only be baked to 60 C. Despite this limitation,
after cooling to liquid helium temperatures, the insert does not change the level of vacuum
in the transfer chamber, even when swapping samples or tips. As shown in the last section,
this level of vacuum is more than sufficient to leave prepared surfaces clean for examination
with the microscope.
38
3.3
Vibration Isolation
The quality of the data taken by any STM is largely determined by its ability to limit the level
of vibrations in the tip-sample junction, to which the tunnel current is exponentially sensitive.
For our microscope, the need for strong thermal coupling to the dilution refrigerator precludes
the use of springs at the microscope itself, a common technique which is remarkably effective.
Instead, as we outline in Figure 3.2, we have isolated the entire instrument shown in Figure
3.1b from external acoustic and floor-borne sources of noise. Vibrations present in the floor of
the laboratory are first attenuated down to very low frequencies (∼ 1Hz) by a set of six passive
pneumatic legs which float a 30 ton concrete plinth, as shown in Figure 3.3. A passively
isolated 4 ton granite slab sits on top of this concrete plinth, providing an additional layer of
isolation from floor-borne noise. To realize this low level of vibrations in the microscope itself,
the instrument (cryostat and chambers) is secured to a dissipative heavy duty aluminum
frame, whose only rigid attachment is to this vibration-isolated granite slab above. A similar
two-tiered scheme was realized to attenuate acoustic vibrations. Two layers of acoustic
shielding are realized by surrounding the plinth first by an acoustic enclosure, and then
surrounding that enclosure by a second room built from grout-filled concrete blocks.
This scheme to isolate vibrations can be rendered useless unless proper care is taken in
handling the large pumps and pumping lines required to run a dilution refrigerator. The
pumps generate a lot of noise, and the pumping lines not only transmit these vibrations, but
their stiffness can mechanically short the pneumatic isolation stages together. To attenuate
high frequency vibrations, the four gas lines, which includes the still and 1K pot pumping
lines, the condenser and the cryostat exhaust, are cast in a 0.5 ton concrete block located
on the lab floor (Fig. 3.2). To attenuate low frequency vibrations and prevent mechanical
shorting of subsequent pneumatic stages, the 1K pot pumping line, the condenser and the
dewar exhaust have long looped sections of formed bellows between the lab floor and the
plinth. To achieve the same effect on the much larger diameter (and thus much stiffer) still
pumping line, a double gimbal based on the design of Ref. [60] (left inset of Fig.
39
3.2,
(7)
+y
(6)
+z
59 cm
+x
(3)
(8)
(9)
(9)
(1)
(4)
(5)
(8)
(2)
Figure 3.2: The instrument (1) is mounted to a custom-made aluminum support structure
(2), both of which are surrounded by a radio frequency (RF) isolation enclosure (3). In order
to facilitate the mating of the insert to the cryostat, which requires moving the chambers
out of the way, the chambers are mounted to an aluminum table, which is attached to
the aluminum frame by way of special kinematic mounts. This entire support structure is
suspended freely by securely bolting to a 4 ton granite table (4) above. This table floats
on a set of 6 passive isolators, and serves as one isolated stage. This, in turn, rests on a 30
ton concrete plinth (5), which itself sits on a set of another 6 heavy passive isolators, and
defines a second isolated stage. An acoustic enclosure (6) surrounds the entire structure.
The plinth, the acoustic enclosure, and an external grout-filled concrete wall (7) connect
only through the floor of the basement lab. A concrete block (8) and gimbal (9) is used to
isolate vibrations which might be transmitted by the pumping lines between the floor and
the plinth, and again between the plinth and the granite table (pumping lines delineated by
yellow line). (left inset) This shows a 3D CAD drawing of our double gimbal, whose purpose
is to decouple the motion of the two flanges (green), which are each attached to rigid pipes.
This is accomplished by attenuating the motion of either of the flanges with respect to the
central elbow (pink) through the use of two sets of edge welded bellows (red). The position
of this elbow is determined by vacuum forces balanced by the tension on wire ropes (black)
connected to the arms (brown). If the pipe attached to the bottom flange of the assembly
were to move in +z (+x), then the horizontal (vertical) arm would twist upwards (right) on
an axial bearing (light blue). Similarly, if it were to move in +y, the base plate (purple)
would twist to accommodate the motion. (right inset) Aerial photograph of the instrument
with detachable RF room top moved aside.
40
3
Velocity SD (nm/(s ⋅√Hz))
10
Lab Floor
Plinth
Granite
Noise Floor
2
10
1
10
0
10
−1
10
0
100
200
300
Frequency (Hz)
400
Figure 3.3: This plot shows the velocity spectral density, as measured by a Wilcoxen 731A,
present on the lab floor (blue), on top of the plinth (red), and on top of the granite slab
(green). The combined baseline sensitivity of the accelerometer and spectrum analyzer is
shown as the noise floor (black). The data were taken while running the dilution refrigerator.
manufactured by Energy Beams, Inc.) is used to bridge the gap between the lab floor and
the plinth. This combination of concrete block and gimbal is repeated again when going from
the plinth to the granite table. As shown in Fig. 3.3, the end result of our isolation scheme
√
establishes sub-nm/(s· Hz) vibration levels approaching the noise floor of our accelerometer
for a wide range of frequencies even with all pumps attached and running. Having created
a suitably low vibration environment, the STM, which we describe in the next section, must
be made as rigid as possible to realize the level of performance required to take low noise
measurements, which we describe in the last section.
3.4
The Dilution Refrigerator and Microscope Head
The last challenge is the conceptually simple step of attaching an STM head to a dilution
refrigerator, with the goal of attaining the lowest possible temperature while retaining the
maximum amount of functionality in the STM head. Thermally, the insert on our system
is a fairly standard design, but with four notable exceptions (Figure 3.4a) to accommodate
41
Microscope
Shutters
69 cm
e
2.5 cm
f
13 cm
Filters
Silver Rods
Washboards
136 cm
1 K Shield
Rotary
Door
d
2.0 cm
56 cm
Tail
Wiring Interface
DR
Strike
Plate
c
3.3 cm
Accessible
by 1K pot
1K
Pot
Mixing
Chamber
b
a
3.7 cm
3.5 cm
2.0 cm
g
2.5 cm
Figure 3.4: (a) This 3D CAD cross-section of the general assembly highlights the nonstandard features of the insert. The oval cam and strike plate used to pre-cool the microscope
are shown in yellow. The approximate volume of the main bath accessible to the 1K pot
pickup line is shown in purple. The rotary shutter on the 1K radiation shield is shown in
cyan. The two sets of radiation baffles on the cryostat, which sit at roughly 77 K and 4K, are
shown in green. The location of the 1K pot (purple), the 1K radiation shield, the tailpiece
(tail), and dilution refrigerator (DR) are also labeled. (b) This is a 3D CAD drawing of STM
tailpiece, highlighting the washboards (cyan), the tip and sample RF filters (purple), and the
silver rods (yellow). (c) This shows a 3D CAD drawing of the microscope, with the z-motor
shown in blue, the x-motor shown in red, and the sample cubby shown in green. A radiation
shield which shields the tip and the sample is hidden from view. (d) A cross-section of the
z-motor is shown here, with the prism in purple, the scan piezo in red, piezo stacks in blue,
the bottom plug in dark green, and the capacitance sensor in light green. (e) A cross-section
of the x-motor, with the prism in purple, the piezo stacks in blue, the arms which hold the
sample cubby in dark green and the capacitance sensor plates in light green. (f) This is a 3D
CAD model of the sample cubby, showing the samples (green) and the PEEK lid (white).
This lid, when pushed down, compresses two BeCu springs on the backs of the samples. The
two dovetail pieces (red) ride on tracks, and can lock the lid into place. (g) Photograph
of the microscope with the milli-Kelvin radiation shield pulled down to reveal access to the
empty tip receptacle and sample cubby (the CAD in (f) shows a newer version of the cubby
than the photograph does).
42
the UHV compatibility of the system. First, given that the IVC is a UHV space, exchange
gas cannot be used to cool the insert from high temperatures down to 4 K, either for the
initial cooldown of the system, or when transferring samples. Instead, we have an ovalshaped mechanical heat switch, operated using a rotary feedthrough, which allows the mixing
chamber to be thermally shorted to the 1K shield via mechanical contact. This allows us
to cool the system from room temperature to 2 K in around 40 hours during the initial
cooldown, and from around 40 K to 2 K in around 6 hours when transferring samples.
Second, because the insert needs to be moved up and down and the 1K pot has a fixed length
pickup tube, about half the helium in the main bath is accessible to the pot, resulting in
shortened time between refills of the main bath. That the cryostat has two satellite necks (to
accommodate the demountable magnet current lead and cryogenic services) exacerbates this
by increasing the helium consumption. Still, the helium consumption rate with the dilution
refrigerator running is around 18 liters a day, resulting in a hold time for the main bath of
4 days. Finally, in order to have access to the microscope inside UHV, the radiation shields
in the cryostat and the 1K shield need to be able to be opened inside the vacuum space,
which could compromise their performance. These appear not to introduce any unexpected
radiative heat load, as the dilution refrigerator was measured to achieve a base temperature
of 8 mK (measured using a cobalt-60 nuclear orientation thermometer) and had a cooling
power of 400 µW at 100 mK with only the thermometry installed.
To cool the microscope to milli-Kelvin temperatures, we borrow standard techniques used
for sample-in-vacuum dilution refrigerator instruments (Figure 3.4b). The primary heat load
added when installing an instrument on the mixing chamber comes from the wiring. Both the
shielded and unshielded lines extending down from room temperature are thermally anchored
first at 4K, then at the 1K pot using 3 cm long washboards, and then connected to either
flexible stainless/ NbTi coaxial cable (custom from New England Wire Technology) or NbTi
wire. Because NbTi superconducts at 1K pot temperatures, these effectively act as a thermal
break between the 1K pot and the mixing chamber. Between the mixing chamber and the
43
microscope, however, we would like to maximize the thermal conductivity, and accordingly
we use silver-coated copper coaxial lines and wires that are anchored using 10 cm long
washboards screwed tightly into a copper stub that is attached to the mixing chamber. In
addition, the tip and bias lines are fed through a lumped element RF filter (VLFX-1050+
from Mini Circuits) at the mixing chamber to filter out unwanted high frequency noise in the
tunnel junction. To efficiently cool the body of the microscope itself, which is 42 cm away
from the mixing chamber, we link the two with silver rods (3N5 purity, 50mm2 in crosssection, from ESPI Metals, Inc.) which have been annealed [61] to enhance their thermal
conductivity. These fit in a PEEK frame which houses electrical connectors and serves as a
secure mechanical attachment point for the microscope. With the microscope installed, the
base temperature of both the mixing chamber and the microscope was measured to be 20
mK using a ruthenium oxide thermometer anchored at the microscope. The mixing chamber
now retains 260 µW of cooling power at 100 mK, leaving sufficient flexibility for adding more
lines to the system when more elaborate experiments need to be done.
The microscope (Fig. 3.4c) contains three functional blocks: a z-axis piezo motor, an xaxis piezo motor, and a two-sample cubby. Both the motors are Pan walkers [62], in which a
triangular sapphire prism (custom, from Swiss Jewel Company) can be moved along a single
axis of motion using three pairs of piezoelectric shear stacks (Model P121.01H from Physik
Instrumente L. P.) in a slip-stick motion. The bodies of both the motors are made of OFHC
copper, coin silver, and PEEK pieces held together using non-magnetic silicon bronze screws
(custom, from Swiss Screw Products, Inc.). The z-motor (Fig. 3.4d), which provides a total
of 4 mm of vertical motion, is used to approach and withdraw a scan piezo (3 Å/V sensitivity
in z, 9.5 Å/V sensitivity in x/y, EBL #4 material from EBL Products, Inc.) which is glued
to the inside of a cylindrical cavity in the prism. This motor reliably produces 80 nm-sized
approach steps at low temperatures with a drive voltage of 375V supplied by a Nanonis
PMD4 piezo motor driver. Its absolute position can be tracked using a cylindrical capacitor
formed by a metal end-plug in the prism and a corresponding piece which is part of the
44
(static) microscope body. The x-motor provides a total of 7 mm of horizontal motion in 280
nm-sized steps to the attached sample cubby (Fig. 3.4e). A pair of capacitors, each formed
by the arms that carry the sample cubby and two internal (static) plates on the microscope
body, allow us to track the absolute horizontal position of the sample. The x-motor has been
designed to have such a large offset range specifically to allow us to move the tip between
two samples (Fig. 3.4f) in the cubby. In addition to being designed to accommodate two
samples, the cubby has a PEEK lid, which, when opened using the wobble stick, allows
samples to be easily slid into the stage, and, when closed using the wobble stick, compresses
two BeCu spring contacts firmly into the samples and locks into place. This mechanism
provides a more solid mechanical and thermal joint between the sample holders and the
sample cubby than would otherwise be possible. In addition to the two samples, the STM
tip can also be swapped in situ by plugging it with a wobble stick into a BeCu collet. While
the ability to offset the sample over a large range provides considerable conveniences as
discussed below, the lowest resonance frequencies of the microscope is likely associated with
the pendulum modes of the two arms (molybdenum) attaching the sample cubby to the
x-motor (shown in Fig. 3.4e). By exciting the the x-motor piezo stacks with a drive voltage
of swept frequency and measuring the response in the current, we determine the strongest
resonance of the x-motor and sample cubby assembly to be at ∼900 Hz at room temperature,
with an additional weaker response at ∼700 Hz.
The combination of the two-sample holder cubby, the reliability of the motion of the
motors, and the repeatability and precision of the capacitive position sensors provide a
critical function when studying samples which cleave poorly. Approaches onto bad areas of
the cleavable sample which change the tip do not terminate the experiment, but rather can
be simply corrected by field emitting and checking the tip on a simple metal sample in the
other slot. After field emission, using the previous position register, we can return to the
same spot (macroscopically) on the cleavable sample to continue looking for an acceptable
area. Most importantly, even when an acceptable area is found, the microscope can be
45
1.8
1.6
1.4
dI/dV (au)
1.2
1
0.8
0.6
0.4
0.2
0
−0.6
∆ = 176 uV
T = 240 mK
−0.4
−0.2
0 pm
0
Bias (mV)
0.2
3 pm
0.4
0.6
Figure 3.5: This plot shows the differential tunneling conductance of a superconducting
Al(100) sample (Tc = 1.16 K) measured with a normal PtIr tip, using a lockin amplifier
with an ac modulation of 5 µV at 865 Hz, along with a fit to the thermally broadened BCS
density of states with Telectron = 240 mK (green). The spectrum was acquired at a setpoint
of 75 pA at -0.6 mV. (Inset) Unfiltered topographic image at base temperature over 30 Å
at a setpoint current of 1 nA and bias of -1 mV of the same Al sample showing ∼2 pm
amplitude atomic modulations.
used to look for an even better area, or for areas with rare surface terminations, with the
knowledge that the sensors and motors are reliable enough to allow the microscope to return
to the original area should another good area not be found. The ability to systematically
search for the best area, or a very rare surface, on a cleavable sample greatly reduces both
the number of samples and the time it takes to do an experiment when compared to being
limited to examining representative areas, as is the case in most STM instruments.
3.5
STM Performance
As we have already described, the instrument provides a similar level of functionality to
that present in higher temperature STMs. In this last section, we show that it also provides
a comparable level of performance, but at significantly lower temperatures, and in high
46
a
b
0T
123 nS
5.7 T
135 nS
Figure 3.6: This figure shows two spatial maps of the tunneling conductance, each recorded
at a bias of +2 mV, over a field of view of 67 nm square on the heavy fermion superconductor
CeCoIn5 [7]. The maps were taken at base temperature on the same area at (a) zero and
(b) 5.7 T applied field using an ac modulation of 66 µV at 1.11 kHz.
magnetic fields. All data presented are measured using a Nanonis SPM controller and a
Femto LCA current preamplifier with 1 kHz bandwidth and 5 · 109 V/A gain. In Figure
3.5, we show data taken on a Al(100) surface prepared in situ using ion sputtering and
annealing, and measured at base temperature and zero magnetic field. The topograph in the
inset shows well-resolved atoms, even on a challenging material where the atomic corrugation
is very small (< 5 pm). Moreover, the differential conductance measured on Al provides
a direct measure of the electron temperature of samples placed in our microscope. This
temperature can exceed the measured lattice temperature (20 mK) due to heating from
unthermalized electromagnetic radiation transmitted from room temperature to the sample
along the electrical line delivering the sample bias [63, 64]. Fitting our tunneling density of
states on Al to the thermally broadened density of states for a Bardeen-Cooper-Schrieffer
superconductor ρBCS (E) ∝
√
|E|
,
E 2 −∆2
where ∆ is the size of the gap, we find the size of the gap
to be ∆ = 176 µV , in agreement with that expected for Al(100) tunnel junctions [65], and
the electron temperature of our instrument to be 240 mK. Ideas for reducing this electron
temperature are discussed in the conclusion.
47
More importantly, the overall level of noise is low enough to enable us to measure one of
the most demanding, but most powerful, kinds of data typically taken using STM: spectral
maps, in which the STM is used to visualize spatial patterns in the density of states at a
fixed energy with sub-Angstrom spatial resolution. In Figure 3.6, we show two such maps
taken at base temperature on the heavy fermion superconductor CeCoIn5 [7], a system in
which superconductivity develops below 2.3 K and can be extinguished upon application of
a 5 T field at low temperature. The maps are taken at an energy outside the superconducting gap (measured to be 0.5 mV), and demonstrate that the standing waves created
by the interference of scattered quasiparticles, which bear the fingerprint of the underlying
band structure, are not altered by the application of a field large enough to suppress superconductivity in this material. These maps also indirectly demonstrate the stability of the
microscope, which, over the course of several weeks, was able to take two dozen spectral
maps at different energies, both at zero field and in applied field, on the same area of the
sample.
The data in Figures 3.5 and 3.6 are qualitative proof of the level of performance of this
instrument. A more quantitative figure of merit for any STM, one which can be used to
compare the relative levels of performance of different instruments, or different modes of operation, is the integral of the spectral density of the current over low frequencies. The limiting
noise in STM measurements is invariably low-frequency noise because it is time-consuming
to average out and shows up in both topographic and spectroscopic measurements. In Figure
3.7, we compare the spectral noise density of the tunneling current for this instrument and
another STM in the lab which uses springs and magnetic damping right at the microscope
head itself to isolate vibrations. Notably as shown in Figure 3.7a, this instrument has a
comparable amount of low frequency noise (in orange; integral over 125 Hz is 0.49 pA2 ) to
the more conventional design (in green; 0.28 pA2 over 125 Hz) when the system runs with the
1K pot in single-shot mode and the dilution refrigerator off. The amount of low-frequency
noise when also running the dilution refrigerator continuously, while considerably larger (in
48
a
Current SD (A/√Hz)
10
10
10
10
Base Temperature
1K Mode (T ~ 2 K)
Tip Retracted
Compare: STM after Ref. 17
−12
−13
−14
−15
0
25
50
75
Frequency (Hz)
100
125
1000
1250
b
Current SD (A/√Hz)
10
10
10
10
Base Temperature
1K Mode (T ~ 2 K)
Tip Retracted
Compare: STM after Ref. 17
−12
−13
−14
−15
0
250
500
750
Frequency (Hz)
Figure 3.7: These are plots of the open feedback noise on the tunneling current over two
frequency ranges (125 Hz in (a), 1250 Hz in (b)) with the 1K pot running in single shot-mode
(orange) and the dilution refrigerator circulating with the 1K pot in single-shot mode (blue).
Also shown is the open feedback noise on an STM based on the design of Ref. [66] (green).
The data were taken with a DC tunneling current of 100 pA and a bias of -200 mV on a
clean Cu (111) surface. The black curve is the noise on the current with the tip outside of
tunneling range, and serves as a measure of the bare amplifier noise due to stray capacitance.
49
blue; 2.5 pA2 over 125 Hz), is directly comparable to published measurements from the UHV
dilution refrigerator STM of Ref. [9] (∼ 3 pA2 over 100 Hz, using Fig. 19). However, in our
case, the system must be run with the 1K pot in single-shot mode, which means it must
be refilled every 8 hours, in order to achieve this level of noise. While the spectral noise
density contains many seemingly deleterious resonances at higher frequencies, as shown in
Figure 3.7b, these have a minimal impact on our measurements in practice. Most occur
at frequencies far above the bandwidth of the feedback loop, and hence don’t appear in
topographs or as a set-point error in spectroscopic measurements. Moreover, the lockin oscillator frequency can still be set to frequencies where the noise spectrum is no worse than
the amplifier background (near ∼1100 Hz, for example). Evidence suggests that much of
the relevant noise originates in the 1K pot itself. The level of noise with both the dilution
refrigerator and 1K pot running single-shot is comparable to that when running only the 1K
pot single-shot with the dilution refrigerator off. Conversely, the level of noise when running
the 1K pot single-shot and dilution refrigerator continuously is comparable to that when
running the 1K pot continuously and leaving the dilution refrigerator off.
3.6
Conclusion
The recent development of this and other dilution refrigerator STMs opens the door to
studying exotic electronic phases and quantum phenomena which only occur at milli-Kelvin
temperatures with a spectroscopic tool and at the atomic scale. We have described the
construction of an instrument which extends both the functionality and level of performance
present in higher temperature STMs down to dilution refrigerator temperatures. As the
current measured electron temperature of 240 mK is an order of magnitude larger than
the measured lattice temperature of 20 mK, further improvements to thermalizing the tip
and sample electron temperature can be made. For example, increasing the cooling power
delivered to the sample cubby itself (currently thermally anchored through only a thin strip
50
of silver foil 0.4 mm2 in cross-section to facilitate sample motion)1 and adding more stages
of low temperature RF filters [63, 64] on all of the electrical lines are planned. Only the tip
and bias lines are currently filtered at the mixing chamber; all other lines to the microscope
are filtered externally at room temperature, potentially causing heating via cross-radiation
at the microscope. Moreover, for ease of access, the system has never been operated with
the RF enclosure fully closed, which may be necessary for the lowest electron temperatures.
Finally, a reduction of the noise likely created inside the 1K pot would reduce the time it
takes to acquire the high resolution data shown here, which is limited to being taken in 8 hour
intervals due to the single-shot lifetime of the 1K pot. Ideally, a set of two dozen spectral
maps could be taken in a handful of days, as is common in more conventional instruments,
instead of a couple of weeks.
1
Recently, the foil strip was upgraded from 0.300”x0.002” 4N silver to 0.300”x0.004” 5N gold, the sample
arms were changed to non-superconducting BeCu, and the 1K rotary door was upgraded to overlap (shield)
more when closed. These changes resulted in a minimal lowering the electron temperature from 250 mK
to 240 mK, pointing to another factor as the dominant source of the elevated temperature. Inadequate RF
filtering and the poor thermal conductivity of the 0.001” thick Kapton strip that electrically isolates the
sample cubby thermal anchor are now thought to be the remaining culprits.
51
Chapter 4
Visualizing d-Wave Heavy Fermion
Superconductivity in CeCoIn5
This chapter is based upon the publication:
Zhou B. B., Misra S. et al., “Visualizing nodal heavy fermion superconductivity in CeCoIn5 .”
Nature Phys. 9, 474 (2013) [7].
Understanding the origin of superconductivity in strongly correlated electron systems
continues to be at the forefront of unsolved problems in all of physics [67]. Among the heavy
f -electron systems, CeCoIn5 is one of the most fascinating, as it shares many of the characteristics of correlated d-electron high-Tc cuprate and pnictide superconductors [68, 69, 70],
including the competition between antiferromagnetism and superconductivity [71]. While
there has been evidence for unconventional pairing in this compound [29, 30, 72, 73, 74, 75],
high-resolution spectroscopic measurements of the superconducting state have been lacking.
Previously, we have used high-resolution scanning tunneling microscopy (STM) techniques
to visualize the emergence of heavy-fermion excitations in CeCoIn5 and demonstrate the
composite nature of these excitations well above Tc [21]. Here we extend these techniques
to much lower temperatures to investigate how superconductivity develops within a strongly
52
correlated band of composite excitations. We find the spectrum of heavy excitations to
be strongly modified just prior to the onset of superconductivity by a suppression of the
spectral weight near the Fermi energy (EF ), reminiscent of the pseudogap state [76, 77]
in the cuprates. By measuring the response of superconductivity to various perturbations,
through both quasiparticle interference and local pair-breaking experiments, we demonstrate
the nodal d-wave character of superconducting pairing in CeCoIn5 .
4.1
Introduction
CeCoIn5 undergoes a superconducting transition at 2.3 K. Despite evidence of unconventional
pairing, consensus on the mechanism of pairing and direct experimental verification of the
order parameter symmetry are still lacking [29, 30, 72, 73, 75]. Moreover, experiments have
suggested that superconductivity in this compound emerges from a state of unconventional
quasiparticle excitations with a pseudogap phase similar to that found in underdoped highTc cuprates [78, 79, 80]. Previously, we demonstrated that STM spectroscopic techniques
can be used to directly visualize the emergence of heavy fermion excitations in CeCoIn5
and their quantum critical nature [21]. Through these measurements, we also demonstrated
the composite nature of heavy quasiparticles and showed their band formation as the f electrons hybridize with the spd-electrons starting at 70 K, well above Tc [21]. This previous
breakthrough, together with our recent development of a high-resolution milli-Kelvin STM,
offers a unique opportunity to measure how superconductivity emerges in a heavy electron
system.
4.2
Superconductivity on the Two Surfaces of CeCoIn5
Figure 4.1 shows STM topographs of the two commonly observed atomically ordered surfaces
of CeCoIn5 produced after the cleaving of single crystals in situ in the ultra-high vacuum
environment of our milli-Kelvin STM. We have previously shown through experiments and
53
a
Surface A
b
Surface B
c
a
b
G(V)/GS
a
b
0.8
0.6
0.4 T = 245 mK
0.2
e
0.9
∆SC
0.7
0.6
−30 −20 −10
0
10
Energy (mV)
20
30
0.5
1
1.5
∆SC
1
0.8
Surface B
15
1.2
2.5 K
Gs = 1 nS
245 mK
Gs = 5 nS
0
f
1.4
0.8 H = 0 T
−1 −0.5
Energy (mV)
G(V) (nS)
G(V)/GS
1
∆PG
1.6
G(V)/G(V = -30 mV)
Surface A
Surface B, 0 T, Gs = 125 nS
Surface B, 5.7 T, Gs = 85 nS
Surface A, 0 T, Gs = 63 nS
Surface B
1.8
1.1 ∆HG
0
−1.5
5 nm
5 nm
d
∆SC
1
∆PG
13
∆SC
11
5.7 T
H=0T
7.2 K
1.7 K
5.3 K
245 mK
0.6
−30 −20 −10
0
10
Energy (mV)
9
−10
0T
−5
T = 245 mK
0
5
10
Energy (mV)
20
30
Figure 4.1: Topographic image measured on surface A (a) and on surface B (b) of CeCoIn5
at 245 mK. Insets in (a) and (b) zoom in on 12 by 12 nm2 regions on their respective
surfaces. The arrows in the figure indicate the in-plane crystallographic a and b directions.
(d,e) Corresponding conductance spectra G(V ), proportional to the local electronic density
of states on surface A and B carried out at temperatures above and below Tc , showing the
evolution of the different energy scales (∆HG : hybridization gap; ∆P G : pseudogap; ∆SC :
superconducting gap) with temperature. Spectra are offset for clarity in (e). (c,f) Blow up
of the superconducting gap energy scale showing the destruction of the superconducting gap
in a magnetic field of H = 5.7 T > Hc2 while the pseudogap feature is preserved. The spectra
G(V ) in (c) and (d) are normalized by their corresponding junction impedances GS .
54
theoretical modeling that different surface terminations change the coupling between the
tunneling electrons and the composite heavy fermion excitations in this compound [21].
Tunneling into such composite states can be influenced not only by the coupling of the tip
to spd- or f -like component of such states but also by the interference between these two
tunneling processes. On surface A, tunneling measurements are more sensitive to the lighter
component of the composite band structure, and accordingly, the spectra show evidence
for a hybridization gap centered at +9 mV, as shown in Fig. 4.1 d). At temperatures
below Tc , this hybridization gap is modified by the onset of an energy gap associated with
superconductivity (Fig. 4.1 c,d), as further confirmed by its suppression with the application
of a magnetic field larger than the bulk upper critical field (Hc2 = 5.0 T perpendicular to
the basal plane of this tetragonal system) of CeCoIn5 (see Fig. 4.2).
Instead of focusing on measurements of surface A, where the tunneling is dominated by
the lighter part of the composite band, we turn to measurements of surface B. On this surface
tunneling directly probes narrow bands of heavy excitations which result in a peak in the
density of states near EF (Fig. 4.1 e). Lowering the temperature from 7.2 K to 5.3 K, above
Tc , we find that this peak is modified by the onset of a pseudogap-like feature at a smaller
energy scale. Further cooling shows the onset of a distinct superconducting gap below Tc
inside the pseudogap. Measurements in a magnetic field corroborate our finding that the
lowest energy scale on surface B (∼ ±500 µV, as shown in Fig. 4.1 c) is indeed associated with
pairing, as it disappears above Hc2 , while the intermediate energy scale pseudogap remains
present at low temperature in the absence of superconductivity at high magnetic field (Fig.
4.1 f). This behavior is reminiscent of the pseudogap found in underdoped cuprates, where
the superconducting gap opens inside an energy scale describing strong correlations that
onset above Tc . However, unlike cuprates, here we clearly distinguish between the two
energy scales by performing high-resolution spectroscopy in a magnetic field large enough
to fully suppress superconductivity. Detailed measurements of changes in the spectra with
the magnetic field also confirm that the transition out of the superconducting state at Hc2
55
is first order (Fig. 4.2), showing that our measurements are consistent with the bulk phase
diagram of CeCoIn5 .
a
b
Figure 4.2: a) The conductance spectra show a sudden shift in their zero bias conductances at Hc2 , resembling a 1st order phase transition from the superconducting into the
normal state, consistent with bulk transport measurements in CeCoIn5 . b) The zero bias
conductance G(V = 0) as a function of magnetic field across Hc2
The spectroscopic measurements suggest that electronic or magnetic correlations alter
the spectrum of heavy excitations by producing a pseudogap within which pairing takes
place. These measurements also show the shapes of the spectra at the lowest temperature to
be most consistent with a d-wave superconducting gap, as they have a nearly linear density
of states near zero energy (Fig. 4.1 c). However, measurements on all surfaces and on several
samples reveal that this d-wave gap (with a magnitude of 535±35 µV, consistent with that
extracted from point contact data [81, 82] is filled (40%) with low energy excitations - a
feature that cannot be explained by simple thermal broadening (determined to be 245 mK
from measurements on a single-crystal Al sample). The complex multiband structure of
CeCoIn5 could involve different gaps on different Fermi surface sheets, and there is the
possibility that some remain ungapped even at temperatures well below Tc [83]. Another
contribution to the in-gap density of states could come from surface impurities, since even
non-magnetic impurities perturb a nodal superconductor, as we demonstrate below. Before
56
we address the nature of the in-gap excitations, we first demonstrate in more detail the
connection between pairing and the heavy fermionic states of CeCoIn5 .
4.3
Quasiparticle Interference in Normal and Superconducting States of CeCoIn5
Energy-resolved spectroscopic mapping with the STM can be used to measure the interference of quasiparticles (QPI) in order to examine the heavy Fermi surface. As shown in Fig.
4.3 a)-d), features in the discrete Fourier transform (DFT) of these maps show wavevectors
related to the elastic momentum transfer Q(E), connecting the initial and the final momentum states on the contours of constant energy. Previous theoretical calculations, quantum oscillation, and angle resolved photoemission spectroscopy measurements have shown
CeCoIn5 to have a complex three-dimensional band structure, with the α and β bands being
the most relevant near EF (Fig. 4.3 e) [84, 85, 86]. Our previous QPI measurements on
surface A show features that are most consistent with 2kF scattering originating from the α
band. The QPI measurements presented here on surface B (outside of the superconducting
energy scale) display scattering wavevectors originating from a larger Fermi surface volume
and are more consistent with scattering involving the β band. Since QPI does not probe
the Fermi surface directly, inferring a unique Fermi surface in a three-dimensional, multiband material without making large number of assumptions is not possible. Nevertheless,
the results of QPI measurements (Fig. 4.3 a-d) together with spectroscopic measurements
(Fig. 4.3 e) demonstrate that the superconducting instability occurs within a correlated
heavy quasiparticle band of CeCoIn5 with a large density of states at the Fermi energy (for
additional details see Appendix C).
We focus our discussion next on the momentum structure of the superconducting gap,
first by examining the conductance maps within this smaller energy window on the same
area of the sample (with the same tip) in the normal (H > Hc2 ) and superconducting
57
Q2
b
10 nm
a
+8 nS
2
1.5 mV c
3.0 mV
PSD
0
d
Q2 Q3
α
β Q3
Q2
Q1
0.1 0.2 0.3
Q1,2,3 (rlu)
Q1
e
Q3
f
g
h
i
j
k
l
m
n
o
Normal
H > Hc
Superconducting
H=0
-8 nS
4
Q1
b
Energy (mV)
Q3
a
-255 µV
-85 µV
0 µV
+85 µV
+255 µV
Figure 4.3: Real space conductance map (a) and its DFT (b) at a bias of 1.5 mV measured
at T = 245 mK on surface B. Colorbar in (a) denotes deviation from the mean. Q1 , Q2 , Q3
correspond to the different quasiparticle scattering vectors. (c) DFT at V = 3 mV. Axes in (c)
denote the Bragg orientation for all DFTs and for the schematic (e). (d) Energy-momentum
structure of Q1 , Q2 , Q3 showing rapid dispersions reflective of mass enhancements m* =
34 m0 , 29 m0 , 23 m0 respectively. Error bars are derived from the width of the peaks in
the DFTs. (e) Schematic of the band structure in the first Brillouin zone derived from Refs.
[84, 85, 86] showing the α (magenta), β (blue) and small (orange) Fermi surfaces in the kz = 0
(solid) and kz = π (dashed) planes. The measured Q1 , Q2 & Q3 QPI scattering vectors are
drawn to scale for comparison. DFTs for selected energies in the superconducting (f-j) and
normal (k-o) states. The Q-space range of the DFTs in (b,c,f-o) is ±0.5 rlu, where 1 rlu =
2π/a0 = 2π/4.6 Å.
58
Figure 4.4: While the overall differential conductance at zero bias is reduced due to the
superconducting gap, the QPI strength at Q3 (labeled in Fig. 4.3), in contrast, is enhanced.
Subtraction of the peak power spectral density of the Q3 feature in the normal state from
the superconducting state shows enhancement centered at zero energy, consistent with the
confinement of the Fermi surface around the (π, π) nodes by a dx2 −y2 superconducting gap.
(H = 0) states of CeCoIn5 . As the data in Fig. 4.3 f)-o) demonstrate, we observe clear
differences between the DFT maps in the superconducting (H = 0) and normal states (H =
5.7 T). Typically, quasiparticle interference at low energies in a superconductor is associated
with the scattering of Bogoliubov-de Gennes (BdG) excitations and is often analyzed to
obtain information about the momentum structure of the superconducting gap [45, 87, 88].
In particular, contrasting the zero-energy DFTs in the superconducting (Fig. 4.3 h) and
normal (Fig. 4.3 m) states, we see an enhancement of quasiparticle interference at wavevector
Q3 (see also Fig. 4.4), suggestive of nodal BdG quasiparticles in a dx2 −y2 superconductor.
However, if such features were only due to BdG-QPI, then they should display a particle-hole
symmetric dispersion in their energy-momentum structure away from the nodes, as seen for
example in similar measurements of high-Tc cuprates [45]. The absence of such particlehole symmetry in our data (Fig. 4.3 f-j) together with the large zero-bias density of states
(40%, see Fig. 4.1 c) suggests that such QPI measurements are complicated by an ungapped
portion of the Fermi surface or by in-gap impurity-induced states, which are expected to
have a particle-hole asymmetric structure (see measurements & discussion below). These
59
complications together with complex three-dimensional nature of the Fermi surface of this
compound makes extraction of the gap function from such QPI measurements unreliable (see
Appendix C).
4.4
Response of Nodal Superconductivity to Potential
Scattering
In contrast, using the power of STM to probe the real space structure of electronic states, it
is still possible to find direct spatial signatures of the nodal character of superconductivity in
CeCoIn5 that do not require multi-parameter modeling or ad hoc assumptions to interpret.
The first such signature can be found by examining the response of low-energy excitations to
extended potential defects such as atomic step edges. Spectroscopic mapping with the STM
upon approaching such steps shows direct evidence for the suppression of superconductivity
in their immediate vicinity (Fig. 4.5 a-b). This suppression is consistent with the expected
response of a nodal superconductor to non-magnetic scattering (Fig. 4.5 c), analogous to
similar observations in the cuprates [89], and in marked contrast with our step-edge measurements of the conventional s-wave superconductor Al (see Fig. 4.6). The data in Fig.
4.5 d) provide a direct measure of the Bardeen-Cooper-Schriefer (BCS) coherence length
ξBCS = 56 ± 10 Å, in agreement with ξBCS ∼ ~vF /π∆ ∼ 60 Å using the gap observed in
Fig. 4.1 (0.5 meV) and the Fermi velocity extracted from Fig. 4.3 (1.5 · 106 cm/s) [90].
4.5
Vortex Anisotropy
Application of a magnetic field can also be used to probe the local suppression of heavyfermion superconductivity in CeCoIn5 due to the presence of vortices and the Abrikosov
lattice. As shown in Fig. 4.7 a-b, STM conductance maps can be used to directly visualize
the vortex lattice in this compound, which can have different structures depending on the
60
c
d
+
_
_
+
G(V) - G(V, r = 153 Å) (nS)
5 nm
G0(r) - G0(r = ∞) (nS)
Lower
Step
b
b
a
a
0Å
25 Å
51 Å
76 Å
102 Å
10
5
0
-5
-2
-1
0
1
Energy (mV)
2
y = a*exp(-x/ξ)
ξBCS = 56 � 10 Å
10
1
0
50
100
Distance (Å)
150
Figure 4.5: Topographic image (V = -100 mV, I = 100 pA) of surface A showing a single
unit-cell step-edge oriented at 45◦ to the atomic lattice. The arrows in the figure indicate
the in-plane crystallographic a and b directions (b) Evolution of the spectra near the stepedge: G(V ) subtracted by the spectrum far away from the step-edge G(V, r = 153Å). The
locations of the spectra in (b) are plotted on (a). (c) Schematic representation of nodal
superconducting quasiparticles scattering off a step-edge. (d) Zero-bias conductance G0 (r)
subtracted by the extrapolated G0 (r = ∞) as a function of distance from the step edge. Line
represents an exponential fit to the data, where error bars denote the standard deviation on
the averaged spectra. ξBCS denotes the characteristic decay length obtained from the fit in
(d), which is a measure of the BCS coherence length.
.
magnetic field. Such structural changes of the vortex lattice (transition between rhombic
and square lattices) have been previously studied in neutron scattering experiments [92] and
various theoretical models [93]. Complementing these efforts, the STM can be used to probe
the electronic states within the vortex core directly, as shown in Fig. 4.7 d, to demonstrate
the presence of a zero-energy vortex bound state. Analysis of this core state demonstrates
61
a
b
Figure 4.6: In contrast, the gaps measured on Al(100) single crystal show no sensitivity
to the proximity to the step edge, consistent with the s-wave nature of the superconducting
gap, which is robust against potential scattering by Anderson’s theorem [91].
the anisotropic decay of the vortex bound state (Fig. 4.7 c,e), the angular average (Fig.
4.7 e) of which determines the Ginzburg-Landau coherence length scale ξGL = 48 ± 4 Å,
consistent with an independent estimate from |dHc2 /dT |(T =Tc ) [94].1 While observation of
such anisotropy is consistent with the nodal character of pairing, an understanding of the
role of the underlying Fermi surface symmetry and vortex-vortex interactions is required to
model the STM data in more detail.2
62
85 nS
135 nS
a
d
40 - 65 nS
θ = 45º to
b-axis
b
a
b
]
0
10
b[
84
100
(r, θ)
200 nm x 120 nm
e
G0(θ)
c
0
75
80
76 [100] [010]
0
90 180
270
y = a*exp(-x/ξ)
ξGL = 48 � 4 Å
80
θ (º)
50
25
r ‖ [110]
(Å)
20
40
G0(r) - G0(r = ∞) (nS)
a
a
60
b
10 nm
−2
−1
0
1
Energy (mV)
2
80
0
20
40
60
r (Å)
Figure 4.7: Zero-bias conductance maps both taken at H = 1 T (separate field dials) and
at T = 245 mK show the vortex lattice structure expected below (a) and above (b) the
transition seen at this field by neutron scattering in Ref. [92]. The arrows in the figure
indicate the in-plane crystallographic a and b directions (c) Close-up zero-bias map of the
vortex lattice on surface B showing an anisotropic square vortex core (H = 1.5 T). (d) Linecut of spectra starting from the center of a vortex and moving radially outward at 45 degrees
to the b-axis showing the evolution of the bound state inside the superconducting gap (H
= 0.5 T). (e) Radial dependence of the angularly averaged zero-bias conductance G0 for a
single vortex core at H = 1 T. Error bars (estimated from the standard deviation in the
analyzed map) are smaller than the marker size in (e). Inset shows the angular dependence
of the radially averaged conductance showing the four-fold anisotropy of a single vortex
with higher conductance extending along the a- and b-directions directions. ξGL denotes the
characteristic decay length obtained from the fit in (e), which is a measure of the angularly
averaged Ginzburg-Landau coherence length.
4.6
Impurity Bound State: Fingerprint of dx2−y2 Pairing
A more spectacular demonstration of the nodal pairing character in CeCoIn5 can be obtained
from examining the spatial structure of in-gap states associated with defects on the surface
of cleaved samples. The spatial structure of impurity quasi-bound states, which are mixtures
1
One can relate both quantities mentioned in this sentence to the estimated orbital critical field Hc2,orbital .
2
We estimate Hc2,orbital = Φ0 /(2πξGL
) ∼ 14.5 T from our measurement. Accordingly, for a BCS superconductor in the clean limit, Hc2,orbital (T = 0) = −0.73|dHc2 /dT |(T =Tc ) ∼ 15 T, using the value reported in
Ref. [94]. It is an open question why the BCS formalism still applies rather well for a d-wave superconductor
such as CeCoIn5 .
2
Specifically, the extent of the vortex bound state farther along the crystallographic a, b directions is inconsistent with the naive expectation based on purely gap symmetry considerations for a dx2 −y2 superconductor
[93], where the state at zero energy is expected to leak out along the diagonal nodes. This demonstrates
that the other effects of band structure and vortex-vortex interactions must be more important in CeCoIn5 .
63
5 pm
a
4 a0
b
Calculation
1
hole-like
0.51
∆ / ∆max
a
G(±V) / [G(+V)+G(-V)]
0 pm
0.50
0.49
h
electron-like
0
90
180
270
Angle from a axis (degrees)
H > Hc2
H=0
0
360
- 195 µV
d
f
c
e
g
+ 195 µV
b
low
high -7 nS
+7 nS
Figure 4.8: (a) Topographic image of an impurity on surface B. (b) Model calculation
for the real space structure (roughly 10 Fermi wavelengths across) of the hole-like part of
the impurity bound state in a dx2 −y2 superconductor, reproduced from Ref. [95] (Copyright
(2000) by the American Physical Society). (c) Electron-like state for the same impurity in
(b). (d-g) Local density of states obtained on the same field-of-view as (a) at ±195 µV in the
normal (H > Hc2 ) and superconducting (H = 0) states as indicated on the figure. Colorbar
in (d-g) denotes deviation from the mean. (h) Radial average of the density of states across
the lobes measured in (d,e), normalized to their sum, as a function of angle from the b axis.
Data at negative (positive) energy is shown in blue (red) symbols; the lines are guides to the
eye. A dx2 −y2 gap is shown in yellow.
of electron-like and hole-like states, can be a direct probe of the order parameter symmetry
[90, 95]. Figure 4.8 shows an extended defect with a four-fold symmetric structure, which
perturbs the low energy excitations of CeCoIn5 by inducing an in-gap state. As shown in
64
Norm. Conductance (a.u.)
hole
electron
1.1
1
0.9
−1
−0.5
0
0.5
Energy (mV)
1
Figure 4.9: The spectrum on top of the impurity in CeCoIn5 (cyan dot in Fig. 4.8 a),
normalized by the spectrum far away from the impurity, shows that the impurity attracts
the electron component of the Cooper pair as its spectrum shows an enhancement at positive
energies.
Fig. 4.9, this particular impurity attracts the electron part of the BdG quasi-particle, and
hence its hole counterpart, linked by the backdrop of Cooper pairing, will revolve around
the impurity with the symmetry of the pairing potential. The spatial distribution of the
hole (electron) state can be imaged directly with STM by applying a negative (positive)
bias. Indeed, probing the spatial structure of these impurity states, we not only find their
expected electron-hole asymmetry, but also find that their orientation is consistent with
that predicted for a dx2 −y2 superconductor (Fig. 4.8 b-e) [95]. The minima (maxima) in
the oscillations for hole-like (electron-like) states identify the nodes of the d-wave order
parameter to occur at 45◦ to the atomic axes (Fig. 4.8 h). In fact, these features in the STM
conductance maps are identical to those associated with Ni impurities in high-Tc cuprates
[90, 96]. However, in contrast to measurements in the cuprates, we are able to determine
the spatial structure that such impurities induce on the normal state by suppressing pairing
at high magnetic fields. Such measurements allow us to exclude the influences of the normal
state band structure, of the impurity shape, or of the tunneling matrix element [90] on the
spatial symmetries of the impurity bound state in the superconducting state. Contrasting
such measurements for H > Hc2 (in Fig. 4.8 f-g) with measurements on the same impurity
65
for H = 0 (Fig. 4.8 d-e) we directly visualize how nodal superconductivity in CeCoIn5 breaks
the symmetry of the normal electronic states in the vicinity of a single atomic defect.
4.7
Outlook
The appearance of a pseudogap and the direct evidence for dx2 −y2 superconductivity reported
here together with previous observations of the competition between anti-ferromagnetism
and superconductivity closely ties the phenomenology of the Ce-115 system to that of the
high-temperature cuprate superconductors. An important next step in extending this phenomenology would be to explore how the competition between anti-ferromagnetism and
superconductivity manifests itself on the atomic scale in STM measurements. Similarly, extending our studies of the electronic structure in magnetic vortices could be used to examine
the competition between different types of ordering in the mixed state, and the possible development of the Fulde-Ferrell-Larkin-Ovchinnikov state in this Pauli-limited superconductor
[92, 97, 98].
Note added in proof of dissertation: The ‘pseudogap’ feature observed in the spectra of
CeCoIn5 could alternatively be a manifestation of the Fermi surface singularity seen in the
Gutzwiller projection of a strongly-interacting metal, as pointed out by Phil Anderson in
Nature Phys. 2, 626-630 (2006). Hence, this feature may not necessarily require proximity
to competing order at a quantum critical point.
66
Chapter 5
The Three Dimensional Dirac
Semimetal Cd3As2
This chapter is based upon the publication:
Jeon S., Zhou B. B. et al., “Landau quantization and quasiparticle interference in the
three-dimensional Dirac semimetal Cd3 As2 .” Accepted for publication in Nature Mater.
http://dx.doi.org/10.1038/nmat4023 (2014) [39].
Condensed matter systems provide a rich setting to realize Dirac [99] and Majorana [100]
fermionic excitations and the possibility to manipulate them in materials for potential applications [101, 102]. Recently, it has been proposed that Weyl fermions, which are chiral,
massless particles, can emerge in certain bulk materials [34, 103] or in topological insulator
multilayers [104] and can produce unusual transport properties, such as charge pumping
driven by a chiral anomaly [35]. A pair of Weyl fermions protected by crystalline symmetry [33], effectively forming a massless Dirac fermion, has been predicted to appear as low
energy excitations in a number of candidate materials termed three-dimensional (3D) Dirac
semimetals [33, 40, 41]. Here we report scanning tunneling microscopy (STM) measurements
at sub-Kelvin temperatures and high magnetic fields on one promising host material, the
67
II-V semiconductor Cd3 As2 . Our study provides the first atomic scale probe of Cd3 As2 ,
showing that defects mostly influence the valence band, consistent with the observation of
ultra-high mobility carriers in the conduction band. By combining Landau level spectroscopy
and quasiparticle interference (QPI), we distinguish a large spin-splitting of the conduction
band in a magnetic field and its extended Dirac-like dispersion above the expected regime.
A model band structure consistent with our experimental findings suggests that for a specific
orientation of the applied magnetic field, Weyl fermions are the low-energy excitations in
Cd3 As2 .
5.1
Introduction
One starting point for accessing the novel phenomena of Weyl fermions is the identification
of bulk materials with 3D Dirac points near which the electronic dispersion is linear in all
three dimensions [105] in analogy to 2D Dirac points observed in graphene [99] or topological
insulators [31]. With time reversal and inversion symmetries preserved, 3D Dirac points can
be formed at the crossing of two doubly degenerate bands and constitute two overlapping
Weyl points. However, 3D Dirac points are generally not robust to gapping unless they
occur along special high symmetry directions in the Brillouin zone, where the band crossing
is protected by crystalline point group symmetry [33, 40, 41]. In these 3D Dirac semimetals,
individual Weyl nodes can be isolated only by breaking either time reversal or inversion
symmetry. Since Weyl nodes are topological objects of definite helicity, acting as either a
source or sink of the Berry curvature, they are robust against external perturbation and are
predicted to harbor exotic effects, such as Fermi arc surface states [34] and chiral, anomalous
magneto-transport [35, 106]. These unusual transport phenomena of Weyl fermions have
been proposed as the basis for novel electronic applications [38, 107].
Several candidate materials, including Na3 Bi and Cd3 As2 , were recently predicted [40, 41]
to exhibit a bulk 3D Dirac semimetal phase with two Dirac points along the kz axis, stabilized
68
by discrete rotational symmetry. While photoemission measurements [108, 109, 110, 111] indeed observed conical dispersions away from certain points in the Brillouin zone of these materials, high energy resolution, atomically-resolved spectroscopic measurements are needed
to isolate the physics near the Dirac point and clarify the effect of material inhomogeneity
on the low-energy Dirac behavior. Low-temperature scanning tunneling microscopy experiments are therefore ideally suited to address these crucial details. Previously, Cd3 As2 has
drawn attention for device applications due to its extremely high room temperature electron
mobility [112] (15,000 cm2 /V s), small optical band gap [112], and magnetoresistive properties [113]. The recent recognition that inverted band ordering driven by spin-orbit coupling
can foster nontrivial band topology renewed interest in Cd3 As2 , which is the only II3 -V2
semiconductor believed to have inverted bands. Updated ab initio calculations predict 3D
Dirac points formed by shallow band inversion between the conduction s-states, of mainly
Cd-5s character, and the heavy hole p-states, of mainly As-4p character [41, 114]. However,
the large unit cell of Cd3 As2 with up to 160 atoms due to Cd ordering in a distorted antifluorite structure [114] present complications to first-principles calculations, which must be
corroborated by careful experimental measurement of the band structure.
5.2
Topographic and Spectroscopic Characterization
at Zero Field
To probe the unique electronic structure of Cd3 As2 , we perform measurements in a homebuilt low-temperature STM [10] capable of operating in magnetic fields up to 14 T. Single
crystal Cd3 As2 samples are cleaved in ultra-high vacuum and cooled to an electron temperature of 400 mK, where all spectroscopic measurements described here are performed.
Figure 5.1 a) and its inset show an atomically-ordered topography of a cleaved surface and
its associated discrete Fourier transform (DFT). The pseudo-hexagonal Bragg peaks, circled
in red, reveal a nearest-neighbor atomic spacing of 4.4±0.15 Å. Their magnitude and orien69
tation precisely match the As-As or Cd-Cd spacing in the (112) plane of this structure [114],
schematically illustrated in Fig. 5.1 b), and identify this facet as a natural cleavage plane
for Cd3 As2 . Because we image atoms at ∼96% of the sites in the pseudo-hexagonal lattice,
we further attribute the cleaved surface to an As layer, since any Cd layer would contain
∼25% empty sites in this projection.
We present in Fig. 5.1 c) the tunneling differential conductance (proportional to the
local DOS) measured at B = 0 T along a line spanning 30 nm. Photoemission measurements
[108, 109] locate the Dirac point (EDirac ) for naturally grown Cd3 As2 at -200±20 mV, corresponding to a carrier concentration ne ∼ 2 · 1018 cm−3 . In agreement, the STM conductance
spectra show a depression near this energy, and the measured DOS rises as (E − EDirac )2
away from it as expected for 3D Dirac points [115]. The conductance near the Dirac point is
nonzero and smooth, representative of a semi-metallic band crossing rather than a band gap.
While the presence of surface states can mask a bulk gap [48], we rule out this possibility
by performing QPI measurements, shown below, that do not resolve a strong surface state
signal near EDirac . The absence of a gap, particularly at the low temperature of our measurement, is consistent with the proposed theoretical description shown in the inset of Fig.
5.1 c) and d), which illustrate a shallow inversion between the valence and conduction bands.
Additionally, the zero-field spectra in Fig. 5.1 c) display significant spatial fluctuation for
energies below EDirac , while in contrast, they are highly homogeneous for energies above
EDirac . Since the carrier concentration in as-grown Cd3 As2 is attributed to As vacancies
[116], these lattice defects would be expected to primarily impact the valence band rather
than the conduction band. In Fig. 5.2, we show that a common, clustered defect in the As
plane (visible as the dark depressions) produces strong fluctuations in the conductance of
the valence band, but is virtually invisible at the Fermi level. This microscopic information
may explain the broad valence band seen in photoemission measurements [108, 109] and the
high mobility at the Fermi level [112], and suggests routes for further materials optimization.
70
a
b
c
[001] direction
4
dI/dV (nS)
d
e
Dirac Points
3
Dirac Point
2
1
0
−300
−200
−100
0
Energy (meV)
100
200
Figure 5.1: Crystal and band structures of Cd3 As2 (112) cleaved crystal. a) Atomicallyordered topographic image (I=50 pA, V=-250 mV) of the Cd3 As2 (112) surface. Inset shows
its 2D Fourier transform. Red circles are associated with Bragg peaks and blue circles with
reconstruction peaks. b) Schematic of the Cd3 As2 unit cell along the (112) plane (red).
Cd atoms and As atoms both make a pseudo-hexagonal lattice. c) Differential conductance
spectra (I=300pA, V=250mV) taken at 90 spatial positions over a line spanning 30 nm. The
blue curves show the individual spectra and the red curve is the spatial average. Spatial
variation in the local density of states is especially pronounced below the Dirac point. The
inset shows the schematic band dispersion along the [001] direction passing the Γ point. d)
Schematic band structure of Cd3 As2 based on ab initio calculations. Two 3D Dirac points
−
+
marked as kD
, kD
are located along the [001] direction and are evenly separated from the Γ
point. The k⊥ direction refers to any axis perpendicular to the kz direction. e) Schematic of
the Fermi surfaces above (red) and below (blue) the Lifshitz transition. The overlaid solid
curves represent the extremal cross-sections parallel to the (112) plane, showing two pockets
merging into a single ellipsoidal contour.
71
Figure S2
a
0
5Å
b
0
dI/dv(r)/mean
3
c
0
dI/dv(r)/mean
3
100 Å
E = -500 mV
E = 0 mV
(μ = 154 pS, σ = 52 pS, σ/μ = 0.34)
(μ = 49 pS, σ = 10 pS, σ/μ = 0.21)
Figure 5.2: Stronger conductance variation is seen at energies in the valence band b), than
at energies in the conduction band c). Surface defects in the As plane imaged as depressions
in the topography a) localize strong enhancements in conductance in the hole-like band,
while they negligibly impact the conductance at the Fermi level.
5.3
Landau Level Spectroscopy
Landau level spectroscopy with STM has previously been applied to extract precise band
structure information for graphene [117, 11], semiconductor 2D electron gases [118], and
topological insulator surface states [119, 120]. Here in distinction, we extend this technique
to quantify the bulk 3D dispersion of Cd3 As2 by applying a magnetic field perpendicular to
the cleaved (112) surface of the sample. The 3D band structure is quantized by the magnetic field into effectively 1D Landau bands that disperse along the momentum k3 parallel
to the field. The projected bulk DOS measured by STM is an integration over all k3 and
accordingly displays peaks at the minimum or maximum energies of these Landau bands,
which contribute inverse square root divergences to the DOS. Semi-classically, these extrema
describe Landau orbits along the constant energy contours of the band structure with extremal cross-sectional area perpendicular to the magnetic field. In Fig. 5.1 e), we illustrate
the extremal contours parallel to the (112) plane in Cd3 As2 for energies above and below the
72
Lifshitz transition, demonstrating the merging of two Dirac pockets into a single ellipsoidal
contour.
Figure 5.3 a) illustrates the Landau level fan diagram for Cd3 As2 assembled from spectra
measured from 0 to 14 T at a single fixed location on the sample surface. Four aspects
are immediately striking. First, the Landau levels emanate from a point slightly below
-200 mV, revealing the presence of a band extremum in the vicinity of the Dirac point
determined by photoemission. This suggests that the band inversion is small, consistent
with ab initio predictions. Second, all prominent Landau levels are electron-like, dispersing
towards positive energies with increasing field. The observation of hole-like levels in the
valence band is apparently hindered by their electronic disorder, as demonstrated in Fig.
5.1 c), and by their lower band velocity. The data also reveal that the spacing of the
Landau levels decreases with Landau level index n, indicating a non-parabolic conduction
band. Finally, satellite peaks for the dominant Landau level peaks are resolved at high field,
revealing the lifting of a degeneracy with increasing field. Figure 5.3 b) shows individual
spectra for the higher fields which resolve a double peak structure for up to the first 8 pairs
of levels (e.g. 12 T).
We first extract information about the band structure of Cd3 As2 from Landau level
spectroscopy measurements using a model-independent method. The semi-classical LifshitzOnsager relation specifies that the extremal area Sn in reciprocal space for the Landau level
n occurring at energy En must quantize as Sn = 2πe(n + γ)B/~, where γ is the phase offset
of the quantum oscillations [121]. As verified by QPI measurements presented later, the
constant energy contours in the (112) plane are nearly circular; hence, we can take Sn = πkn2 ,
where kn is the geometric mean of the high symmetric axes of the Fermi surface contour in the
(112) plane. We use γ = 1/2 and adopt an intuitive assignment of the index n to the peaks,
labeling every two with the same index starting with n = 0 as shown in Fig. 5.3 b). In Fig.
5.3 c), the average peak position En and its associated kn for various B fields trace out an
effective dispersion relation. Remarkably, the entire set of peaks collapses onto a single Dirac73
a
c
n=0
1
3
2
5
4
6
300
14
200
10
Energy (meV)
8
6
4
2
0
−250
dI/dV (ns)
−200
−150
0
1.2
−100
−50
0
Energy (mV)
b
2.4
50
100
k (1/A )
0
−0.05
0.05
0.1
LL Index
0 1 2 3 ....
16
vF =
9.4 × 105 (m/ s)
100
0
n=4
n=3
−100
n=2
n=1
−200
n=0
−300
1
2
1
2
3
4
4
5
6
7
8
d
13.0 T
12.5 T
e
Distance (nm)
13.5 T
12.0 T
11.5 T
11.0 T
f
Distance (nm)
10.5 T
10.0 T
−300 −250 −200 −150 −100
Energy (mV)
−50
0
50
100
40
-200
0
-100
100
4
12.25 T
20
0
0
1
2
1
3
4
5
6
dI/dV (nS)
0
2
2
0
0
40
4
12.25 T
dI/dV (nS)
0
Distance (nm)
14.0 T
3
20
0
40
2
-20
0
20
12.75 T
20
0
0
4
dI/dV (nS)
Magnetic Field (T)
12
dI/dV (arb.unit)
−0.1
...
Fig. 2
2
-20
0
Energy (mV)
20
0
Figure 5.3: a) Landau level fan diagram measured at 400 mK, consisting of point spectra in
1 T increments. b) The point spectra, obtained from 10 T to 14 T in 0.5 T increments, show
a doublet Landau peak structure whose separation decreases at high index. Plots are shifted
vertically and a smooth background is subtracted based on the 2 T data. c) Effective band
dispersion in the (112) plane formulated from the Lifshitz-Onsager quantization condition.
Sixteen Landau levels for each magnetic field are plotted, where the average energy is used
for indexes with two split peaks. d) Spatial variation of Landau Levels at 12.25 T. The green
curve is the spatial average. e), f) Spatial variation of Landau Levels around the Fermi level
at 12.25 T and 12.75 T. The spectra in d), e), and f) were all taken along the same line cut.
74
like
p
(n + γ)B ∝ |k| scaling for a wide energy range, revealing the strong linearity of the
conduction band. The linear dispersion with very high Fermi velocity vF = 9.4 ± 0.15 · 105
m/s extends to at least 0.5 V above EDirac , far beyond the expected Lifshitz transition
where the two Dirac cones merge. While this extended linearity is not guaranteed by the
Dirac physics around the band inversion, it presents important consequences for transport
properties of samples with similar carrier concentration. For example, under the assumption
of scattering from a screened Coulomb potential, the mobility for a 3D linear dispersion scales
1/3
as vF2 ne /ni , in stark contrast to the ne /(m∗2 ni ) scaling for a 3D quadratic dispersion, where
m∗ is the effective mass, ne is the carrier density, and ni is the concentration of scattering
centers1 . This contrasting physical regime for Cd3 As2 , which cannot be considered as the
limit of normal band structures, may be critical to understanding the ultrahigh mobility and
large magnetoresistance reported in a recent transport experiment [37]. Finally, we observe
that the extrapolated crossing point from the high energy dispersion occurs at -300 mV,
below EDirac , and that the effective velocities of the n = 0, 1 levels become increasingly
small relative to the high energy behavior. We will explain below in detailed modeling that
this deviation is a consequence of our sensitivity to the band minimum in the kz dispersion.
5.4
Spatial Homogeneity of Landau Levels
Next, we discuss the spatial homogeneity of the Landau levels. In Fig. 5.3 d), we verify that
the dominant peak positions are homogeneous in space, with exception of fine features which
occur near the Fermi energy. In Fig. 5.3 e) and f), we show the n = 4 and n = 5 Landau levels
for the respective fields when they approach and pass the Fermi level. Remarkably, in certain
locations, we resolve a four-peak structure in the n = 5 level and weaker hints of splitting
of the n = 4 state. Because this fine structure occurs in the vicinity of the Fermi level,
we speculate that it may arise from band structure effects (states at different momenta but
The relationship between linear and quadratic dispersions can be see via the substitution vF → k/m∗
and noting that ne ∝ k 1/3
1
75
the same energy) that become resolvable near the Fermi level due to the extended electron
lifetime, or from many body effects [120]. Since the four-fold structure shifts together with
increased field as shown in Fig. 5.3 e) and f), we rule out half-filling of the Landau levels
[11].2 As we are above the Lifshitz transition in this energy range, the additional splitting
should also not be interpreted as the lifting of the valley degeneracy of the two Dirac points
[11].
5.5
Quasiparticle Interference
Moreover, the spatial resolution of STM enables independent confirmation of the band structure derived from our Landau level spectroscopy measurements. The Fourier transform of
spatial modulations in the local DOS mapped by STM provides information about quasiparticle interference (QPI) caused by elastic scattering wavevectors that connect points on the
constant energy contour. QPI for a 3D band structure can be approximated as the integration of the 2D QPI intensities for the Fermi surface planes at fixed k3 perpendicular to the
sample surface. Therefore, QPI for the nearly spherical constant energy surfaces of Cd3 As2
should represent the weighted sum of 2D QPI patterns for contours of “latitude” at fixed k3 .
Each ring of latitude of radius k contributes a QPI ring of radius 2k; thus, the sum will fill up
a disc-like pattern. The regions near the equator of the Fermi surface (maximal kmax ) will be
weighted more due to the slower dispersion of the radius of the ring of latitude, while regions
near the pole (k → 0) will be weighted less due to the faster dispersion. Consistently, the
experimental QPI data contains significant contributions inside intensity up to the maximal
radius 2kmax , as is exemplified for E = 150 mV shown in Fig. 5.4 e).
In Fig. 5.4 a-c), we display spectroscopic maps measured at B = 0 T and T = 2 K
for three different energies that display vivid wave-like features. The evolution of the QPI
maps from E = 450 mV to E = 150 mV shows the length of the scattering wavevectors to
2
Half-filled Landau levels splits an single Landau peak into two, resulting in peak on either side of the
Fermi level. Hence it would generally result in an odd number of observed peaks.
76
a
dI/dV(pS)
501
817
b
dI/dV(pS)
90
272
c
10
g
55
-200 mV
150 mV
450 mV
dI/dV(pS)
600
500
150 A
d
dI/dV(pS)
0
2.7
450 mV
e
150 mV
dI/dV(pS)
0
0.7
f
dI/dV(pS)
-200 mV
0
0.1
Energy (mV)
400
300
200
100
0
−100
Landau levels
QPI 6 band
QPI second band
−200
−0.2
0.3 / A
0
k (1/A )
0.2
Figure 5.4: a), b), c) Spectroscopic maps of Cd3 As2 at 450 meV, 150 meV, and -200
meV, respectively. d), e), f) 2D discrete Fourier transforms (2D-DFTs) of a), b), and c),
respectively. The red dashed circles show the scattering of the electron-like conduction band,
and the cyan dashed circle shows that of a second band which emerges at higher energy. g)
Plot of QPI peaks and reproduced Landau level peaks. The red (cyan) momentum vectors
are obtained from the radius of the QPI feature. Blue and green curves are guides to the
eye. The orange circles reproduce the Landau level data shown in Fig. 5.3 c).
increase with decreasing energy. At E = -200 mV, this interference signal can no longer be
resolved as the diverging wavelength near EDirac overlaps with the background electronic
puddling. The interference patterns seen in the spectroscopic maps and their DFTs (see Fig.
5.4 d-f)) distinguish the shape of the extremal Fermi contour above the Lifshitz transition as
quasi-circular, justifying our previous assumption. Figure 5.4 g) summarizes the extracted
dispersion from the QPI and Landau level measurements, which together reinforce the consistent picture of a conduction band that onsets near -200 mV and disperses linearly at
high energies. Above 500 mV, the linear dispersion becomes flatter and a second scattering
vector, likely from another bulk band, is resolved in the QPI data.
77
5.6
Landau Level Simulation
To gain further insight into the nontrivial Landau level structure and to determine when Weyl
fermions appear as the low energy excitations of Cd3 As2 , we introduce a band structure
model that captures the salient features of our data. Following previous work, the lowenergy dispersion around the point for Cd3 As2 can be described by an inverted HgTe-type
band model using a minimal 4-band basis of the states S1/2 , 1/2 , P3/2 , 3/2 , S1/2 , −1/2 ,
P3/2 , −3/2 [41, 122]:

Ak+
0
0
 M (k)

 Ak− −M (k)
0
0

Hef f (k) = 0 (k) + 
 0
0
M (k) −Ak−


0
0
−Ak+ −M (k)









(5.1)
where 0 (k) and M (k) encode the band structure and k± = kx ± iky (details are described in
Appendix D). Landau quantization in the (112) plane reflects both the kx -ky and kz dispersions, where the latter could not be precisely determined from photoemission measurements
on samples with (001) cleavage planes [108, 109]. As these previous studies demonstrated
linear kx -ky dispersion, the linearity in Fig. 5.3 c) implies that the kz dispersion is also linear
at high energies. To capture this trend, we modify the original parabolic kz dispersion in
M (k) to be hyperbolic. This simple modification maintains all qualitative aspects of the
low energy band inversion and is essential for modeling the extended energy range of the
data. When a magnetic field is applied, we transform the momentum k → k − e/~A via
Peierls substitution of the magnetic vector potential A and include a Zeeman term in the
total Hamiltonian H(k) = Hef f (k) + HZeeman (k).
In Fig. 5.5 a), we show the results of numerical Landau level simulations using band
structure parameters consistent with the kx − ky dispersion measured by photoemission and
with the presence of band inversion indicated by our zero field spectra. Although a precise
78
Fig. 4
b
a
n=2
Data
Energy (mV)
•••
ELifschitz, lower
0
n=0
n=3
n=3
−50
n=1
40
30
20
10
n=2
n=2
−100
g* (∆En/μBB)
EDirac
0
30
60
90
θ from [001] (°)
d
n= 0 1 2 3 4 5 6 7
50
n=1
ELifschitz, upper
50
c
10.5 T
12.0 T
13.5 T
0
−200 −150 −100 −50
0
Energy (mV)
e
50
100
n=1
n=0
−200
Energy
n=0
Energy
−150
EDirac
EDirac
−250
0
2
4
6
8
10
12
14
[1 1 2]
B⊥(T)
Γ
k3
[1 1 2]
[0 0 1]
Γ
k3
[0 0 1]
Figure 5.5: a) Simulation of Landau levels and their splitting (peak positions of 10T to 14T
Landau level spectra are plotted as red circles). The electron-like (blue curves) and hole-like
levels (red curves) are derived from the extrema of the Landau level bands at the Γ point.
The Dirac and Lifshitz points at zero field are marked on the vertical axis. b) Theoretical
angle-dependent orbital splitting of the Landau levels. The measurements reported here
were performed at =54.7, denoted by the yellow bar. c) Effective total g-factor g* extracted
from the experimental data as a function of energy and magnetic field. d), e) Calculated
Landau level bands for a magnetic field along the [112] direction and [001] direction. The
corresponding calculated density of states is shown for the [112] directed field. The inset in
(d) and (e) zooms in on the crossing point between the lowest electron- and hole-like bands,
showing the opening of a gap in (d) due to broken C4 symmetry.
79
determination of the size of the inversion is not possible (20 mV is used in Fig. 5.5), the
data are more consistent with shallower band inversions. Nevertheless, the model illustrates
the essential physical origin for the observed Landau level structure. At high fields, the
DOS singularities observed in the data correspond to the energies of the Landau level band
minima at the Γ point (see Appendix D for discussion of the low field regime where additional
extrema may occur inside the two Dirac cones). Hence, the deviation from Dirac scaling for
the lowest levels in Fig. 5.3 c) reflects the parabolic (massive) band minimum in the kz
dispersion, which is probed by the tilted magnetic field.
More importantly, the agreement of our data with the model calculations suggests that
the Landau level doublet structure arises from a combination of orbital and Zeeman splitting
of the spin-degenerate conduction band. Orbital splitting depends on the shape of the band
structure and diminishes away from EDirac . In Fig. 5.5 b, we theoretically illustrate this
evolution of the Landau levels due to orbital effects as the angle of the field is tilted away
from the c axis (for clarity we have set the Zeeman term to zero here as it introduces only
an additional nearly constant splitting). For our data, measured at the intermediate angle
denoted by the yellow bar, it is natural to adopt the assignment scheme n shown on the right
side of Fig. 5.5 b) such that the pairs of levels closest in energy have the same index. In
Fig. 5.5 c), we extract an effective total g* from the experimental Landau level splitting for
each index at several different magnetic fields. We find that g*=37±2 for the lowest level
and that g* decreases with increasing energy from EDirac , consistent with theoretical models
based on prior Shubnikov-de-Haas measurements [123].
5.7
Outlook
In the case of a magnetic field titled from the c-axis, calculations based on our model band
structure show that the Weyl nodes are eliminated by small gaps at the Dirac points caused
by the broken rotational (C4) symmetry (Fig. 5.5 d). Therefore, to observe Weyl fermions
80
in Cd3 As2 , application of a magnetic field along [001] is required to break time reversal
symmetry while maintaining C4 symmetry (Fig. 5.5 e). Moreover, the direction of the
magnetic field is shown here to tune the orbital and orbital-independent splitting in this
material. Exploration of that phase space in next-generation samples with lower carrier
concentration opens the possibility of engineering and observing topological states in 3D
Dirac materials.
81
Chapter 6
Conclusion
This dissertation described the construction of a fully-featured, ultra-high vacuum STM capable of operating at 240 mK and at magnetic fields up to 14 T. Indeed, the rewards of this
long effort have come into fruition through two experiments on unconventional heavy superconductivity in CeCoIn5 and the topological band structure of the Dirac semimetal Cd3 As2 .
In the former experiment, strong parallels were established between (low temperature) superconductivity in CeCoIn5 and the high-Tc cuprate superconductors, most fundamentally in
the dx2 −y2 symmetry of Cooper pairing. Moreover, the appearance of a pseudogap independent of, yet in close proximity to, the superconducting gap in CeCoIn5 represented another
commonality between this heavy fermion superconductor and the cuprates, where such commonalities may be expected from the similar phase diagrams dominated by the presence of
anti-ferromagnetism. In the latter experiment, the magnetic field was used to resolve the
astonishing dispersion of the conduction band of Cd3 As2 , linear in three momentum dimensions over nearly a volt range and with two-fold spin degeneracy split by a g-factor as high
as ∼40. Naturally, the performance of this singular instrument has motivated a long list of
future experiments, both to extend and answer the questions raised by the initial CeCoIn5
and Cd3 As2 experiments and to span new fields, such as single spin dynamics in semiconductors and the quest for Majorana fermions in 1D superconductor-nanowire systems. As the
82
list is as endless as one’s curiosity, this conclusion will modestly only address future avenues
in heavy fermion and topological semimetal research.
The demonstration of the close relation between superconductivity in CeCoIn5 and the
high-Tc cuprates opens a new avenue to understanding magnetically-mediated superconductivity. Moreover, this new avenue can even be advantageous as the stoichiometric CeCoIn5
samples are ultra-clean, with coherence lengths ∼50 Å, three to four times longer than that
in the cuprates, enabling real space effects visualized by STM such as vortex physics and
impurity quasiparticle states to appear particularly striking as we have shown. The natural
first order of business should be to explore the phase diagram of CeCoIn5 through doping.
Hole doping with Cd (equivalently Hg) on the In sites has been shown to induce antiferromagnetism (AF) - above 0.75% Cd, AF coexists with superconductivity, which gives way to
only AF above 2% Cd. The QPI and impurity state experiments should be repeated both in
the coexistence regime and in the purely AF regime to contrast with the results of undoped
CeCoIn5 (pure superconductivity). Does the size or depth of the pseudogap scale with the
Neel temperature TN , which would connect the pseudogap to magnetic fluctuations if so? It
is further believed from nuclear magnetic resonance studies that AF nucleates locally around
single Cd impurities before establishing long range order. The STM can ideally address this
hypothesis by tracking the local spectroscopic signature around Cd impurities as a function
of the density of Cd and imaging the nucleation and growth of this transition. Moreover,
electron doping CeCoIn5 , such as with Sn for In, or doping onto the f -electron site, such as
with Yb for Ce, would also be interesting. Both these types of dopants (particularly Yb) are
believed to less drastically impact superconductivity than Cd does; hence, understanding
the microscopic reason for this may have implications for engineering more robust, higher
temperature magnetically-mediated superconductors.
Finally, studying CeCoIn5 in non-natural cleaveage planes, such as (100) plane for example, would enable application of the magnetic field in the ab plane of the sample, due to
our experimental restriction of being able to only apply the magnetic field along the sample
83
Figure 6.1: The interplay of doping and superconductivity in CeCoIn5 . What happens on
the local level when hole doping with Cd introduces AFM, while electron doping with Sn
much less strongly suppress superconductivity? Figure courtesy Eric Bauer.
normal. A (100)-oriented magnetic field would allow access to exotic high field phases in
CeCoIn5 , such as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase (spatially modulated
order parameter with Cooper pairs between |k, ↑i and |−k + q, ↓i, comprising a center of
gravity of finite linear momentum) and the so-called “Q” phase (superconductivity coexisting with a spin density wave). What are the spectroscopic signatures of such phases?
Generically, a non-spin polarized tip is not sensitive to the spin channel; however, perhaps
spin modes may couple to the superconductivity creating a ‘wiggle’ in the spectrum outside
the gap at E = ∆ + ω similar to electron-phonon coupling seen in BCS superconductors
by McMillan and Rowell [124]. Alternatively, both phases predict some spatially modulated
gap strength; however, very careful data acquisition and analysis will be necessary to isolate
the true effect on the already much diminished gap strength (due to the high magnetic field)
from the unrelated, but coincident effects of vortex physics and STM setpoint. These are
84
difficult, but pioneering experiments at the frontiers of this heavy fermion superconductivity
field.1
With regards to Cd3 As2 , our experiments demonstrated that the band inversion that
forms the Weyl nodes is shallow, likely < 50 mV. Although this may seem small, in actuality
it is quite reasonable since even 25 mV corresponds to approximately room temperature, 273
K. The obstacle, therefore, in Cd3 As2 is a materials one - that the Dirac point, predicted
to be exactly at the Fermi level for the stoichiometric compound, is nevertheless buried
by 200 mV below the Fermi energy by native defects introduced during imperfect growth,
particularly As vacancies. The buried Dirac point and its inhomogeneity in space introduce
strong lifetime and disorder broadening of the lowest, chiral Landau level, and therefore
mask the nontrivial effects due to it. To raise the Dirac point up to the chemical potential is
a challenge for our chemistry colleagues and will likely require less defected, intrinsic samples,
rather than the current (intrinsically) electron-doped samples to be compensated with hole
dopants, which would introduce more disorder. Another avenue, long term, may be to pursue
thin film samples grown by molecular beam epitaxy, where the addition of a back gate
may be used to tune the carrier concentration. Very recently, Ref. [126] reported ARPES
measurements on Cd3 As2 samples with Fermi energies pinned to the Dirac point, where such
low carrier concentration samples have been traditionally very hard to synthesize.2 However,
this paper also claimed the absence of Cd ordering (random Cd vacancies) in the surface
layers, attributed to diffusion after the cleaving procedure. Our atomic resolution STM
images show any reconstruction or supercell of the surface atoms to be regular (at least
over O(50 nm) sized patches); hence, suggesting that the Cd ordering on or near the surface
is periodic, rather than random. However, the supercell, although similar, is difficult to
precisely match 1-to-1 with the atomic structure presented in [114] for the bulk of Cd3 As2 .
1
Very recently (Dec. 2013), Ref. [125] discovered the Q-phase in Ce0.95 Nd0.05 In5 in zero magnetic field
(and coexisting with superconductivity (Tc = 1.85 K, TN = 0.9 K)). This would enable the study of the
Q-phase on any cleavage plane of CeCoIn5 and absent of the effects of vortex physics.
2
This is at least my impression from reading papers from the 1970s, the last period of strong basic research
in Cd3 As2 . Admittedly, growth techniques may have advanced significantly in the intervening 40 years.
85
Thus, in the fast moving field of topological semimetals, it perhaps prudent to take a breath
to carefully resolve the nuances of the crystal structure and growth, as the properties of the
band structure hinge directly on these basic assumptions.
290 x 290 Å
54 x 54 Å
Unit Cell: 16 atoms; 4 displaced
Figure 6.2: One of the developing controversies in study of Cd3 As2 is the direction of
the Dirac points on the (112) cleaved surface, which is determined by the symmetries of
the crystal lattice. In Ref. [126], it was claimed that upon cleaving, Cd atoms diffuse in
the surface layers to randomize their ordering pattern and therefore shift the Dirac points
from the bulk [001] axis to the [112] axis. Our STM measurements on cleaved Cd3 As2 show
a zigzag ordering of the surface atoms likely reflective of the Cd ordering, either on the
surface layer (if this surface layer is Cd) or in the layer underneath (if this surface layer is
As). Regardless, this pattern has a regular supercell of 16 atoms, with 4 atoms significantly
distorted from their ideal hex lattice position. This fraction of distortion (25%) strongly
suggests it is reflective of the Cd ordering (25% vacancy). Hence, our measurements suggest
that the Cd is indeed ordered on the surface layers.
Once the materials issues are addressed, further STM studies should be performed to
reveal both the electron- and hole-like Landau levels and pin down the details near the Dirac
point. Study of both the (112) cleave, discussed in this thesis, and the (001) cleave, as
claimed by ARPES experiments, will allow the search for the topological effects predicted
for this Dirac material. For the (112) plane, Fermi arc surface states, with long wavelength
due to the small separation between the Dirac points, may be revealed once the background
charge inhomogeneity is reduced. For the (001) plane, where applying a magnetic field
preserves the C4 symmetry, the addition of an electric field from a back gate may enable the
search for E · B chiral current effects. In any case, the ability to study two cleavage planes
86
necessarily tunes the direction of the magnetic field and it may be worthwhile to verify the
angular dependence of the Landau level splitting presented in Fig. 5.5 to fully justify the
underlying model.
In conclusion, a promising future for materials research at high fields and ultra-low temperatures awaits. However, the technical advance achieved in this dissertation should be
considered only the first step in a continual effort to push the boundaries of experimental
capability. Indeed, plans are already underway to design a new dilution insert that can be
substituted into the cryogenic and vibration support structures of this current instrument
and that upgrades upon its performance and convenience. The capability to run continuous measurements with a thermally anchored fixed impedance to continuously feed the 1
K pot would drastically change the measurement duty cycle of the instrument. Moreover,
careful planning and implementation of multi-stage RF filtering for all electrical lines down
to the microscope will aim for lower electron temperature. The STM landscape is rapidly
evolving as in the next decade, high performance dilution refrigerator-based systems will become commonplace, even commercially available. Hence, continual innovation at labs such
as Princeton will be required to push and stay at the forefront of the technique.
87
Appendix A
Further Experimental Aspects of
DRSTM
This appendix describes further details of DRSTM operation and design to support the more
general, global overview given in Chapter 3.
A.1
X and Z Capacitance Position Sensors
The two sample design of DRSTM is a wonderful institution in that if the tip ever ‘fails’,
due to acquiring a bad tip shape or an insulating spectra, the operator can simply walk
over to the auxiliary Cu sample, reprep the tip by field emission, and come back to the
sample under study with a fresh tip. However, on samples that do not cleave well, such as
in particular the CeCoIn5 samples studied in this thesis, it is exceedingly rare to approach
a large atomically terraced area and hence the advantage of the two sample holder design
would be somewhat minimized if the tip is damaged reapproaching ‘bad’ areas.1 Hence it
is imperative that once an decent region is located, its location be tracked in situ with a
sensor of the x-walker position. Here, I describe an innovative, small form factor differential
1
Usually a fresh tip can withstand a few (<10 for the persistent) bad approaches on metallic samples. On
samples such as Bi2 Sr2 CaCu2 O8+δ with an insulating top layer, one bad approach is game over; however,
Bi2 Sr2 CaCu2 O8+δ cleaves like the Great Plains and the success rate of approach in >90%.
88
capacitance sensor that crucially maintains high sensitivity for the x-walker position on both
the left and right extremes of its motion.
a)
c)
L
CL
CL
CR
CR
VL
R
5
L
d)
Data
Fit
5.2
L
R
0
−5
−10
−15
0
R
2
4
X Position (mm)
CZ
R2
R1
10
C −C (uV)
VR
Z Capacitance (V)
b)
 Ctip/sample
Vosc
5
4.8
Approach (45 mV/1K)
Retract (65 mV/1K)
4.6
0
6
5
10
Steps (Thousands)
15
Figure A.1: “The Eyes of the STM”: Capacitance sensors for X (a) and Z (c) Pan Walkers.
(b) The X walker metric is a differential lock-in measurement performed at 617 Hz with an
oscillator amplitude of 0.25 V (0.5 V into the divide by 2 box we use conventionally; R1 ∼ 5
kOhm). (d) The Z walker metric is performed applying the same 0.5 V oscillator to one of the
z walker capacitance electrodes, amplifying the current pickup on the other electrode with
an Ithaco preamp at 10− 5 gain, and reading the voltage on the lock-in at 1 mV sensitivity.
Note that the inner metal part of the z capacitance sensor has a ∼10 kOhm short to ground,
which fortunately only negligibly impacts the measurement as the wire resistance is ∼100
Ohm.
The principle of the x-walker capacitance sensor is shown in Fig. A.1 a). Two additional
floating pieces of metal (shown as the orange L’s) are placed on the inside of the sample
carriage arms (shown in blue). The orange pieces are fixed to the microscope body and do
not move, while the blue arms are translated by connection to the sapphire prism on the
horizontal Pan x-walker (above). Hence, the separation of the blue arms is constant, and
the arms move relative to the orange pieces. Thus, a metric of the position of the walker is
the difference in capacitance CL − CR , where CL (CR ) is the capacitance between the left
89
(right) sample arm and the left (right) capacitance plate. Notice that critically this metric
maintains high sensitivity when the walker is close to either the left and right sides (where
the tip is over the samples), but is less sensitive in the middle (where conveniently this
is the gap between the two samples). Readout of this capacitance is performed via lock-in
detection through the bridge circuit shown, where an oscillator is applied to the sample arms
(i.e., the bias of the STM) and the voltage pickup VL and VR are differentially detected by
the lock-in (VL − VR ∝ CL − CR by simple analysis), thus canceling common mode noise.
One of the two resistors R1 can be tuned to cancel any slight offset such that the reading is
0 V when the walker is exactly centered in its range.
In Fig. A.1 b), we show the experimental capacitance trace as the walker is moved over
(the greater part) of its ∼7 mm range of motion. The red line is a fit to the equation:
C(x) = C0
1
1
−
,
x x0 − x
(A.1)
representing the difference between two parallel place capacitors. We note that the instantaneous error in the capacitance reading is ±0.003 uV, but the reproducibility (backlash) after
walking away and back is probably of order 0.02 uV. Hence, near the center of the samples,2
the backlash error can be converted to a effective positional reproducibility of order 5 um.
In addition to the x capacitance sensor, DRSTM possesses a linear z capacitance motion sensor, which was useful during the testing of the walking of the z-walker in a 4K
dipstick, however in daily operation now has been supplanted by measuring the tip-sample
capacitance, as will be described in the following section. As shown in Fig. A.1 c), the
z-capacitance sensor design consists of two concentric cylinders, the inner cylinder which
is attached to the bottom of the scanner tube (which is walked by the Z Pan walker) and
another outer cylinder which is stationary. Testing of the z-walker at room temperature
using a Nanonis waveform of amplitude 125 V (hence bipolar peak-to-peak of 250 V) results
2
For historical purpose, we note that the CeCoIn5 approach that engender the published QPI and impurity
maps was at x = +5.83 uV.
90
in a linear position signal which is shown in Fig. A.1 d). The important aspect of the figure
is the ratio of the backward (retract, with gravity) to forward (approach, against gravity)
speed (= 65/45 ∼ 1.5) which is a good diagnosis of how tightly the Pan walker should be
tuned at room temperature. If this ratio is too large, the walker (BeCu spring plate) is too
loose and walking up is inefficient; if this ratio is too small, the walker is too tight and may
seize at low temperature.
A.2
Life on DR: Including Approaching an Sample
After the sample has been demonstrated to possess well-cleaved areas, the experimental
lifetime is essentially infinite on DRSTM. The same area of the sample can be tracked from
0 to 14 T. In some situations, a few steps must be taken in z, in order to keep the sample at
the safe z-range for refilling. However, the trained operator can easily track the same area
even with walking a few steps due to the reliability of the walkers. Hence, the measurement
will continue until the user is unhappy with the tip, at which point the microscope is moved
to field emission on copper, and afterwards moved back over the sample using the same
registry of the x-capacitance sensor, described above. The next step is to approach the
sample, that is to say, to safely position the tip within 10 Å of the surface, starting from
a macroscopic millimeters away, without ever having the tip be in direct contact with the
sample. This of course is the whole marvel of STM and its solution relies on the utilization
of piezoelectrics. With a combination of a piezoelectric coarse stepper motor and a piezo
tube scanner, the problem is reduced to using the stepper motor to place the STM tip
within the range of the piezo tube scanner (∼150 nm). In principle, this can be achieved
by checking for the tunneling current after each step of the motor, if the range of the piezo
tube scanner (how much its length changes given the full range of applied voltage) is larger
than the step size. In ‘safe approach’ or ‘tip retract mode approach’, the tip is first retracted
(compressed). Then a step is taken, and the tip is released slowly (extended) to check for
91
tunneling current, which as soon as it finds it, the tip is retracted and no more steps are
taken. Since the motor step is smaller than the range of the tube scanner, this checking for
tunneling current is redundant, and the sample surface will be found before any step towards
it crashes the retracted tip into the sample.
Tip/Sample Reading (mV)
78
76
117
Data
Approached
Linear Background
after 656 steps
Background + Sample
Start “safe’
approach
74
114
111
108
72
106
70
103
68
100
66
97
64
62
0
True Capacitance (fF)
80
94
Commence approach after FE
5
10
15
Z Steps (Thousands)
91
20
Figure A.2: The tip-sample capacitance versus steps taken during the approach. The
experimental trace is fit to the empirical form Ctip/sample = C0 + C1 x + C2 /(x0 − x). The
total approach took ∼ 20,000 steps, where only the only during last 656 did we check for
the tunneling current after each step. It is intriguing that the tip-sample capacitance in real
units is 100 fF when the sharp metallic tip is within ∼100 nm (fully retracted scanner tube)
of the sample (infinite plane).
Generally the first part of this process (moving to, performing, and moving from the
field emission) takes about 2 hours. Thereafter, the approach can be easily done in under
90 minutes if the tip-sample approach capacitance (the capacitance when we are within
the range of the scanner piezo) is known. This value is actually relatively insensitive to
the sample since the sample always looks like an infinite plane, but is more sensitive to
the z-height (how thick of a sample holder you use) due to background geometric factors.3
Here we apply a oscillator on top of the bias of the sample4 and measure the resulting
capacitative pickup on the tip using the same current preamplifier for the tunneling current.
Of course, the ‘safe approach’ mode is extremely time consuming compared to simply taking
3
4
May also be slightly sensitive to x-location of approach.
Generally we apply, 1V divided by 20 = 50 mV at a frequency of 434.3 Hz.
92
steps without checking the current, and one only launches ’safe approach’ mode once one is
sufficiently close to the sample. To know when we are sufficiently close to the sample, we
monitor the tip-sample capacitance, which is displayed for a prototypical approach as shown
in Fig. A.2. Empirically the Ctip/sample is linear far away from the sample (we can consider
this linear term as background, or as a geometric effect from the sides of the tip rather
than the apex), but near the sample it is dominated by a 1/z divergence. Safe approach is
only launched when the derivative in Ctip/sample exceeds a certain threshold. In Fig. A.2,
we required 656 steps after launching safe approach, which at a frequency of ∼11 steps per
minute for settings that I like to use, took almost exactly 1 hour of approaching. This is a
reasonable, if not conservative time, and the daring or impatient can reduce even further.
A.3
‘Joule-Thomson’ 2K Mode Operation
One of the downsides of the current operation of DRSTM is the necessity to single shot the
1K pot for optimal noise performance, resulting in a run time around 8 hours. As a result,
conductance maps are taken one energy at a time, which can be frustrating if the tip changes
in between. Oftentimes it is more useful to have the complete set of energies over a partial
area of the map (if the tip has to change), rather than the incomplete set of energies, but a
few energies finished completely, over the full area. This was precisely the scenario during
the high field Bi2 Sr2 CaCu2 O8+δ experiments completed during the summer of 2013. The
necessity to run 72 hour maps during that time spurred operation of the system in so-called
‘Joule-Thomson’ (JT) mode, where the pot is cooled from a completely empty state and the
needle valve is cracked incrementally until the 1K pot barely cools to a minimum of ∼2.1 K.
If the NV is opened further (T1K ∼ 2.0 K), then liquid will begin to accumulate inside the
belly of the pot and noise performance will slowly deteriorate as the liquid level rises. It the
pot is not started from a completely empty state, the influx of liquid corresponding to the
same NV setting as JT mode would also introduce excessive noise to the the turbulence of
93
the mixing of the existing superfluid liquid inside the belly with the influx of ∼4 K liquid.
Hence, it is imperative that the belly of the pot be empty and the incoming liquid be just
enough to cool the pot through evaporative cooling as it turns to gas.5 Likely, the gas
expansion theory is correct for JT mode operation between 3 K and above. For the lower
temperatures near 2.1 K, it is believed that the influx of liquid is only enough to fill up the
smaller capillary tube that feeds the pot and wraps around the bottom of the pot, leaving
the bulk of the pot still empty. Here, we may have some 1K pot action, but for an extremely
small diameter tube, rather than for the pot directly, and is empirically determined to be less
noisy. The key point is that once JT mode is established, it is continuous and the system
run is limited only be the refill of the main dewar, which is 3 and a half days. Moreover, a
further advantage of JT mode is relative absence of pot ‘pings’.
a)
b)
NV 2.8
NV 2.9
T 1K Pot
NV 3.0
T MxChb
Figure A.3: a) Engaging Joule-Thomson mode by slightly cracking the needle valve. The
mixing chamber temperature has not had enough time to equilibrate. b) Joule-Thomson
mode closed feedback noise performance for a setpoint of 100 pA at +400 mV on Cd3 As2 .
Fig. A.3 a) shows the initialization of JT mode from an empty pot state. Each additional
cracking open of the pot results in approximately 0.3 K cooling. Notice that if the system had
liquid in the pot, each opening of the NV would result in warming of the pot temperature,
since the pot has to cool the incoming liquid. This trace shows that we can in principle
stabilize at all temperatures between ∼10 K and 2.1 K in approximately 0.5 K increments.
5
Hence, the analogy to Joule-Thomson cooling, where a closed cycle of expansion of pressurized gas
produces the cooling power. Here we expand from atmosphere to vacuum.
94
In general, the temperature will drift once the main dewar liquid level falls below 25%. The
closed feedback noise performance is displayed in Fig. A.3 b) for a setpoint current of 100
pA at +400 mV on Cd3 As2 , where all QPI maps on Cd3 As2 for were taken in this mode.
Comparison to the performance of single shot pot, fridge not running (Fig. 3.7 b) orange
trace6 ), shows worsened noise at frequencies between 850 Hz and 1250 Hz; however, the
noise is still much better than single shot running the fridge, hence is quite capable of taking
excellent data in conjunction with the multipass technique.
A.4
Dewar Exhaust Management
Figure A.4 diagrams the connection of our main bath boil-off to the Princeton recovery
system. Proper management of this boil-off is necessary to allow the dewar to stabilize
after refill, prevent dewar ‘pings’, and reduce the sensitivity of our system to fluctuations in
the recovery back-pressure. The boiloff from the main bath is connected out of the dewar
through its most central port (thus believed to better guide the boiloff through the vapor
shields on the insert) with a rubber hose with NW16 tube fittings. Rubber is chosen for
its flexibility in order to minimize transfer of vibrations to the granite table, and the small
diameter tube chosen in order to minimize the transfer of acoustic noise into the dewar. The
NW16 rubber hose is adapted to NW 40 PVC hose at a concrete block (in which all pumping
lines are cast) sitting at the edge of the floating plinth. Thereafter, the boiloff runs through
a Hayward PVC ball check valve, which defends against back-flow. A piece of polyurethane
foam is stuck in the line to block acoustic noise from entering the dewar, as the exhaust lines
reach the recovery inside the noisy pump room. Together the foam and Hayward check valve
“constipate” the line, increasing the time constant for any pressure changes in the main bath
which is good for stability. The boiloff then continues to an Alicat pressure controller which
controls the pressure in the line. The Alicat is generally set for a setpoint of 780 Torr, with
proportional gain 15000, and differential gain 1500. The gain should not be set so high as to
6
The trace in Fig. 3.7 is open feedback, but the feedback loop does not affect frequencies above ∼200 Hz
95
V
Vent
V
To Building
Recovery
V
Foam
V
V



To Yazdani
Bag
Alicat PD
Pressure
Controller
NW40 everywhere else
Hayward
Ball Check Valve
NW16
Rubber
Hose
Main Bath
Figure A.4: Flow chart for the boil-off from the experimental main bath. The ’V’ symbol
denotes ball-valves that can be either opened or closed. The Alicat pressure controller is
model PCR-30PSIA-D/5P LIN,5IN Range:800Torr.
cause “thudding” of the pressure controller. During quiet measurement, the boiloff is routed
after the Alicat to Yazdani lab’s dedicated bag, which has its own compressor and acts as a
buffer before going into the building recovery. Connecting directly into the building recovery
is problematic, as the building recovery’s pressure is sometimes higher than the setpoint
780 Torr and is highly sensitive to refills that occur in other labs. A series of bypass valves
is in place to enable a straight shot to atmosphere or recovery during high dewar pressure
situations, such as during refill and during the movement of the insert up and down for
sample transfer.
A.5
Electrical Ground Loop Management
While certainly not as deleterious to STM performance as vibrational noise, which scales
with the tunneling current, minimizing noise from 60 Hz and its harmonics due to electrical
ground loops is a workman-esque way to gain the slightest bit of an advantage. While
certain principles, such as cutting ground loops when possible and minimizing its length
and resistance when not) can guide in the reduction in 60 Hz noise, the ultimate test is
empirical: try out various configurations and settle on the one with the cleanest tip-retracted
96
Single Grey
Power Outlet
Computer
Single White
Power Outlet Preamp strap
Extremely Thick
Copper Braid
Ion Pump
‘Deadman’ Controllers
Preamp
Grounding
Bar
Ion Pump
Chasis Ground
Rack
System
(Insert and
Chambers)
IGH
Black Power
208 V 3-Phase
Rack ground
through cable
shields, etc.
Figure A.5: Schematic for the electrical wiring of the DRSTM instrument. The key is to have
all power touching the system (preamp, ion pump, Nanonis controller, other measurement
electronics) emanate from a single white power outlet.
noise spectrum (out of tunneling range). Figure A.5 shows the configuration of the DRSTM
electronics which produced the best 60 Hz characteristics.
In measurement condition, the only electronics touching the system that must be powered
are the Femto preamplifier and the ion pumps. We make sure that these are supplied with
white (filtered and backed up) power from the same power outlet. Furthermore to eliminate
the ground loop to the ion pump, the ground plug of the power outlet is broken to the ion
pump via a “deadman” connector (i.e., the ion pump only gets neutral and hot). Instead,
chassis ground for the ion pump is provided directly from the chambers via a ground strap
(star distribution of ground with the chamber at its center). The chambers and insert
themselves are grounded to the copper grounding bar at the back of the room with the
thickest copper braid that we could purchase. We then wire power to the electronics rack,
containing Nanonis SPM controller, SRS lock-in, oscilloscope, helium level meter, from the
same white power outlet. Additionally, the rack obtains ground from the insert via the
shields of the cabling (scanner, walker, etc.) that touch both the insert and rack. One would
97
think that one should use the same white power for the computer that communicates with
the SPM controller; however, the computer is actually quite noisy and having it off white
power, on its own separate “grey” power was more advantageous. The remaining electrical
parts that must touch the system are the thermometer cabling and needle valve control
cabling from the Oxford IGH, which is powered from a mixture of black power and 208V
three phase power. The thermometer cabling is not an issue. Although minimal, the needle
valve motor represents the biggest source of 60 Hz noise for our system, and we oftentimes
manually disconnect it during measurement (although likely does not matter). Finally all
metallic pumping lines that touch the system are electrically broken somewhere farther away
from the insert with plastic o-rings and plastic clamps (we don’t do this at the insert, since
it would represent a ‘hole’ for electromagnetic radiation to enter and heat the inside of the
system). Finally, to solidify the BNC ground connection between the preamp and the insert,
a strap is used to clamp the outside of the BNC on the preamp to the barrel of the insert
port.
98
Appendix B
Multipass Spectroscopy: An
Alternative to Conventional
Conductance Spectroscopy
In this appendix, I describe a novel technique developed for acquiring the energy-resolved
conductance maps in spectroscopic imaging STM. The so-called multipass technique is in
principle equivalent to the traditional method of conductance maps, but has been implemented in DRSTM for both the CeCoIn5 and Cd3 As2 experiments for its better noise performance and time efficiency when taking single energies.
B.1
Traditional Conductance Maps
Figure B.1 delineates the measurement cycle for obtaining the energy-resolved spatial density
of states maps using the traditional method, called “conductance mapping”. Here, one can
think of the individual measurement unit as the ‘point’ in that a series of N 2 repeated
measurements must be made at each point on a N by N real space grid. At each point,
first the setpoint condition (the particular height of the tip z that establishes the setpoint
tunneling current at a particular setpoint voltage) is reached and then the feedback loop
99
is opened and the bias is ramped over the increments of voltage, recording the differential
conductance for each specified voltage. Crucially here, one must wait at least three times the
lock-in time constant Tlockin between each voltage, otherwise the history from the previous
voltage will affect the next voltage’s measurement. In general, the traditional method is
rather ‘jerky’ (i.e., stop and go between each point, between each voltage). In other words,
the fundamental measurement outcome that is analyzed - the conductance at a particular
energy over the entire field of view - is not taken in a continuous manner, but must be
reconstituted over measurements separated in time. This technique therefore randomizes
the effect of noise that is coherent over short time scales (the usual damped vibrational
resonance oscillation), distributing it between the different energies. As a consequence, the
FFTs of the conductance maps contain “white noise” or “salt-and-pepper” noise, which is
spread throughout Q-space.
Figure B.1: Tradtional conductance map duty cycle. A ramped dI/dV measurement is
performed at each point on a N by N grid, with a delay between each voltage to allow the
lock-in to settle. Schematic courtesy Kenjiro Gomes.
100
B.2
The Multpass Technique
In contrast to traditional conductance mapping, where the measurement unit is the point,
the measurement unit for the multipass technique is the “line”. The essence of this fundamental restructuring of the total spectroscopic imaging measurement is the following: the
traditional conductance mapping technique prioritizes continuity in energy, while the multipass technique prioritizes continuity in space. For a particular energy, we think of taking the
individual N lines of a N by N grid one at a time, where in general adding to the number
of pixels of the line (e.g. taking a N by M grid) does not necessarily affect the total measurement time. For each energy, a new pass over the line must in principle be accomplished.
Historically, the multipass technique was developed under conditions where the measurement
time was limited by the single-shot hold time of the 1 K pot (at base temperature), such that
only a single energy of the conductance map could be taken at once using the traditional
method. For simplicity, let us first describe the multipass technique for a single energy.
FWD1: Scan the line at setpoint conditions with feedback closed at T1 = M * (Time per Pixel).
Record line of setpoint heights z.
BWD1: Scan back to start of line at TB, some fast speed ~ 1s.
FWD2: Ramp bias to V1. “Play back” recorded z signal from FWD1 with same time T1 and take
simultaneous dI/dV data.
BWD2: Scan back to start of line at TB, some fast speed ~ 1s. (Technically feedback is also off
here for bias ramping. We actually play BWD2.)
Repeat for all energies before moving on to next line.
On/Closed
Feedback
Off/Open
Z Position
VSetpoint
Bias (V)
V1
FWD1
BWD1FWD2
BWD2 FWD1
V2
BWD1FWD2
BWD2 Time






Grid N lines by M pixels per line
Energy V1
Energy V2
Figure B.2: The multipass techique divides the grid into a series of lines. The setpoint values
for an particular line is first recorded with the feedback loop engaged. The bias voltage is
ramped, and the tip is then asked to retrace the same line with the recorded setpoints while
the feedback loop is disengaged. The simultaneous dI/dV due to a small summed oscillator
is measured.
101
In the multipass technique, the tip is first scanned over a line (slowly) at the setpoint
condition, and the z-positions (i.e. setpoint values) for the entire M pixels of the line are
recorded and memorized. The tip is then rastered back (quickly) to the original starting
point of the line, and the feedback loop is opened. At this point, the bias is ramped to the
particular energy of interest, and the tip is asked to retrace the memorized z-positions over
the same line with the feedback open. This is indeed a blind retracing and hence requires
some stability of the STM instrument, but we have found it to be no more stringent than
that demanded by the traditional conductance map, as in general line times are of order
20s, and this is a normal time for the feedback loop to be open for spectroscopic dI/dV
measurements. But to reemphasize, here the feedback being open does not mean the tip
is stationary, but rather than the tip is asked to retrace a particular x, y, z line a second
time without regard to the setpoint current. During this retracing line, the small oscillator
voltage is turned on, and we spend on average 6 to 8 lock-in time constants per pixel such
that the lock-in records the simultaneous differential conductance as the line is traced. If
faster settings are used, then the lock-in effectively performs a hardware Gaussian-smooth
on the conductance map, where this is already common practice in post-processing of maps
taken by the traditional method.1 In some sense, the multipass technique is similar to the
method of simultaneous dI/dV mapping with topography; however, the crucial difference is
that for the former method, we are able to use the same setpoint condition (i.e the same
setpoint voltage V0 ) for all energies because of the adaptation of tracing a line feed-back
open, rather than using using a different setpoint voltage for each energy map for the latter
method. In this sense, in the absence of drift, the multipass technique should perfectly
simulate the conductance map for a particular energy taken by the traditional technique.
The described technique can be implemented using the “record and play” feature of the
1
For the CeCoIn5 experiments, we used 28 s per line over 272 pixels with a 10 mS lock-in time constant.
Here the oscillator was 66 uV (very small) hence required longer averaging. For Cd3 As2 , we used faster
settings at 14.9s per line over 208 pixels with 10 mS time constant and a whopping 7 mV oscillator. Interestingly, the two experiments performed in this thesis span the range of the fastest dispersion (heavy fermion)
to slowest dispersion (light-like), as reflected by the range of oscillator amplitudes used.
102
“multipass” module of the Nanonis control system, calling a user written VI for greater
convenience and for bias ramping, rather large single-stepped bias changes.2
Let us now discuss the pros and cons of the multipass technique. For a single energy, the
main pro is that the bias is not ramped at each point (thus you don’t need to wait the lockin settling time at each point). The bias is only ramped once at the start of the line! The
lock-in settling time thus can be used as integration time, enabling much faster, or quieter
maps. This is the most critical benefit. A second benefit is the that noise correlated on short
time scales is now imprinted onto the image as in general, smooth coherent background
modulations (of a set wavelength and direction depending on the line by line timing). Of
course, the scale of the noise that survives the integration time should be much weaker than
the signal of interest and can be made at a different frequency that the real waves by suitable
choice of timing. Effectively, this concentrates the effect of the noise into a single peak or
region of the FFT, instead of having it as white noise over the entire FFT (due to its more
stochastic nature if it is distributed among different energies for the traditional method,
resulting in lack of spatial coherence). The idea is to keep the noise coherent in space rather
than white, such that it can be effectively filtered by the FFT. Another trick one can then
play is to set the integration time per pixel to an exact integer multiple of the dominant
noise period.3
The con of the multipass technique described above is that each energy is taken separately
and hence the maps at different energies will not line up as perfectly (the delay between
energies is on order of 10 s rather than 0.1 s). Also generally, we need to take the full set
of energies with the same tip, and you run the risk of having the tip change in between
energies and not being able to crop the full map down in size to salvage the set of data. The
extension of the multipass technique to multiple biases was implemented during the high field
Bi2212 experiments. The conceptual extension is simple, after finishing a particular line at
2
There are some details here relating to writing your own VI, in particular to how bias changes are
handled depending on whether the feedback loop is on or off for a “record” (on) or “play” (off) line that the
operator should be careful of.
3
The so-called ‘Delft’ technique courtesy Stevan Nadj-Perge.
103
one bias, just repeat the same line at a different bias, and only move onto the next line
when all biases are finished. The question then becomes whether to use the same recorded
setpoint trace for the second/or third bias, or to re-record the setpoint line again. The
author chose to be conservative and recorded the setpoint line each time for each different
bias.4 This multiple recording of the setpoint z for each bias negates the time advantage
(from not having to wait lock-in settle at every point) of the multipass technique when the
number of energies becomes large. However, even on Bi2212, much higher quality data was
obtained with multipass technique than with the regular conductance mapping technique
for the same overall amount of time, testifying to the better noise performance. However,
a perhaps crucial con is that the multipass technique requires tracing over the same area
many times (4*NBias if fact), where the traditional technique traces over it twice (forward
and back). On hostile materials such as Bi2212, the more you scan an area the more likely
the tip is to change.
B.3
Conclusion
In summary, we have developed an alternative to the traditional conductance mapping technique that has its advantages in particular when very high quality data is needed for a small
set of energies. Most importantly, the ability afforded by this technique to isolate noise
in Q-space makes high quality data possible even in intrinsically more difficult systems to
operate, such as the dilution fridge system in this thesis.
4
With more advanced drift correction code, its very likely the setpoint line can be predicted for later
lines, rather than measured again.
104
Appendix C
Comparison of QPI in CeCoIn5 to
Other Band Structure Probes and
Phenomenological Modeling
This appendix discusses further details of the quasiparticle interference patterns measured
in CeCoIn5 , whose rather complex three dimensional band structure necessitate particular
care in interpreting the data.
C.1
Reference to Other Experimental Mappings of the
Band Structure
Since QPI measures the momentum transfer vectors (Q) which connect two points on the
Fermi surface (FS), rather than the k-vectors of the FS directly, inferring a unique FS
from QPI in a three-dimensional, multi-band material without making a large number of
assumptions is not possible. As schematically illustrated in Fig. C.1, three bands cross the
Fermi level of CeCoIn5 (identified previously by various theoretical and experimental efforts,
including quantum oscillation and ARPES): band 135 (i.e., “α band” with cylindrical Fermi
105
surfaces around the M points), band 133 (i.e., “β band” with a large, complicated Fermi
surface), and finally band 131 with small Fermi surfaces [127, 85, 86]. Because of the expected
light mass for band 131, inconsistent with the rapid dispersions seen in our Q vectors, and its
small size, we focus instead on the α and β bands for the origin of our Q vectors (specifically
to only the surfaces of α and β seen in quantum oscillation experiments). The decrease in
length of our Q vectors with increasing energy restricts us to look for scattering between two
disconnected surfaces of the α and β bands rather than for scattering within a single closed
surface. By the same dispersion argument, interband scattering between concentric α and β
sheets can also be excluded as a possibility since in general β disperses faster with increasing
energy than α does (lengthening the Q with increasing energy).
a
c
Q3
Q2
Q1
b α
β Q3
Q2
Q1
Figure C.1: a) Labeled Q vectors from QPI in comparison to Fermi surface suggested
by theory b) and measured by ARPES c). Panel c) is reproduced from Ref. [86]. The
QPI features seen likely originate from inside the yellow dashed box in either the Γ or Z
planes. The key insight from QPI is the resolution of two features along (π, π) that may
indicate additional folding of the β band Fermi surface or slight mismatch of sizes as the kz
momentum is changed.
In Table C.1,we convert the measured de Haas-van Alphen (dHvA) frequencies to Fermi
surface areas in units of 1 Brillouin zone. The extremal areas of each band then give lower
106
limits on the length of possible connecting Q vectors, which we estimate by assuming simple
FS shapes consistent with theory. It is immediately apparent that the (π, π) Q vectors
(Q2,3 = (0.24, 0.24), (0.27, 0.27)) can only come from the β band as the α band cylinders are
too far separated in that direction. However, Q1 along (π, 0) may originate from either the
α or β band as the measured Q1 = (0.29, 0) can originate on the zone edge where the two
bands are close together.
Table C.1: Estimated minimal Q vectors from dHvA measurements of CeCoIn5 .
CeCoIn5
α
β
C.2
dHvA Areas
Up to 28% of BZ
Up to 61% of BZ
dHvA Mass
8-18
40-50
Q’s along (π, π)
>(.47,.47) rlu
>(.22,.22) rlu
Q’s along (π, 0)
>(.3,0) rlu
>(.22,0) rlu
Phenomenological Modeling of Normal State
Band Structure
To speculate on the qualitative features of QPI in the superconducting state, we first capture
the energy dispersions of the normal state in an over-simplified 2-dimensional (2D) model.
For concreteness, we identify two 2D surfaces (corresponding for example to two different kz
cuts of the band) whose energy dispersions k and χk are given by
(kx , ky ) = µ + t1 (cos (kx ) + cos (ky )) + t2 cos (kx ) cos (ky )
+ t3 (cos (2kx ) + cos(2ky ))
χ(kx , ky ) = ν + s1 (cos (kx ) + cos (ky )) + s2 cos (kx ) cos (ky )
(C.1)
+ s3 (cos (2kx ) + cos (2ky ))
(µ, t1 , t2 , t3 ) = (−31.3, 67.2, −124.6, 13.2) mV
(ν, s1 , s2 , s3 ) = (−54.6, 36.3, 5.0, −9.2) mV.
107
By suitable adjustment of the hopping parameters, the 2kf scatterings within this model can
be made to reproduce the dispersions and general QPI pattern measured in the experiment
as shown in Fig. C.2.
Figure C.2: (a,b) show the experimental QPI peaks dispersing along (π, 0) and (π, π)
directions, respectively, overlaid with the dispersion of the appropriate 2kf scatterings calculated from the parameterized 2D (k) and χ(k) surfaces. The in-field H = 5.7 T data
(non-superconducting) was substituted for the linecut at 0 energy. The temperature of the
measurement was 245 mK. (c) shows the typical constant energy contour for (k) (blue solid)
and χ(k) (blue dashed). The simulated Born scattering QPI pattern for 1.5 mV is shown in
(d).
To calculate the QPI patterns, we applied the Born scattering approximation:
2
2πe X dI(q, ω)
=
Nt
t̂N̂ (q, ω)t̂ i,j
S(q, ω) =
dV
~
i,j=1
Z 2
dk
1
Ĝ(k, ω)Û Ĝ(k + q, ω)
N̂ (q, ω) = − Im
π
(2)2
(C.2)
(C.3)
where S(q, ω) reflects experimental Fourier transform of the differential conductance. We
take the density of states of the STM tip Nt = 1, t̂ = t0 t0χ with t = −1 and tχ = 0.7
G0 0 denoting the propensity to tunneling into the and χ surfaces, and Ĝ = 0 G0χ encoding
108
the full Greens functions G0 (k, ω) = (ω + iΓ − (k))−1 and G0χ (k, ω) = (ω + iΓχ − χ(k))−1 .
The lifetimes Γ and Γχ are taken to be 0.1 mV for the normal state, and the scattering
U U 1 0.3 .
=
matrix Û = Uχ Uχ
.3
.7
χ
Finally, we note that the broad feature Q1 along (π, 0) seen in experiment may come
from an overlap of intra-surface scattering from both the k and χk surfaces, and would be
more precisely captured in a full 3D model that considers kz dispersion.
C.3
Superconductivity Gapping the Phenomenological
Band Structure
We investigate how superconductivity qualitatively changes the QPI patterns by applying
both a dx2 −y2 and dxy gap function on our model normal state band structure k and χk .
In the presence of superconductivity, the Greens functions for k and χk acquire particle/hole channels given by the 2x2 matrices GS (k, ω) = ((ω + iΓ )I − (k)σ3 − ∆(k)σ1 )−1 and
GSχ (k, ω) = ((ω + iΓχ )I − χ(k)σ3 − ∆(k)σ1 )−1 , where σi are the Pauli matrices. The above
equations for S(q, ω) and N̂ (q, ω) still hold with now

(GS )11
(GS )11
0
0



0
(GSχ )11
0
(GSχ )12

Ĝ = 
 (GS )
0
(GS )22
0

21

0
(GSχ )21
0
(GSχ )22 ,


 −t 0



 0 −tχ



 , t̂ = 
 0

0




0
0
0

0 

0 0 


t 0 


0 tχ ,
(C.4)
and considering potential scattering

0
0
 U Uχ

 Uχ Uχ
0
0

Û = 
 0
0
−U −Uχ


0
0 −Uχ −Uχ
109





.



(C.5)
Using identical model parameters as in the normal state calculation (with the exception of
Γ = Γχ = 0.05 mV), we simulate the experimental QPI at three energies for a dx2 −y2 gap
on the k and χk surfaces (Fig. C.3 b)
,χ
∆ (kx , ky ) =
∆,χ
x2 −y 2
2
(cos (kx ) − cos (ky ))
(C.6)
with ∆x2 −y2 = 0.67 mV and ∆χx2 −y2 = 2.55 mV such that the maximum gaps on and χ are
0.5 mV. The assumption of equal gaps on both and χ is an arbitrary feature of our model.
Since our high resolution QPI data show subtle changes between the superconducting and
normal states throughout Q-space (see Fig. 4.3 of main text), no unambiguous identification
of completely ungapped portions of the FS can be made; likewise, determining the size of
the gap on different surfaces based on these subtle features would be purely speculative.
Alternatively, we can also consider a dxy gap (Fig. C.3 c)
∆,χ (kx , ky ) = ∆,χ
xy (sin (kx ) ∗ sin (ky ))
(C.7)
with ∆xy = 0.67 mV and ∆χxy = 1.09mV , maintaining a maximal 0.5 mV gap on both
surfaces.
Comparison of the panels shows that the experimental data cannot be reconciled with
a dxy gap on the surface, and is qualitatively most consistent with a dx2 −y2 gap on both
surfaces. However, such analysis cannot reproduce the strong electron-hole asymmetry displayed by the data in Fig. 4.3 f)-o) of the main text, whose explanation may require additional assumptions about impurity effects or ungapped regions of the Fermi surface. These
assumptions, absent independent experimental justification, together with the complex 3D,
multi-band Fermi surface of CeCoIn5 , make extraction of the superconducting gap from QPI
data ambiguous.
110
Figure C.3: Panel (a) shows the experimental data, while the bottom two panels show the
resulting QPI pattern with application of a dx2 −y2 gap symmetry (b) and dxy gap symmetry
(c) on our phenomenological band structure model. The measured data is more consistent
with a dx2 −y2 gap.
111
Appendix D
Details of Cd3As2 Landau Level
Simulation
In this appendix, we describe the four-band model Hamiltonian used in this thesis to capture
the Landau level spectra of Cd3 As2 within the framework of prior ab initio calculations and
ARPES measurements.
D.1
Modified Four-Band Kane Hamiltonian
An eight band Kane model is generally used to describe the low-energy band structure of
small band gap semiconductors such as InSb and HgTe. Near the charge neutrality point,
the description of the 3D Dirac semimetal phase can be further reduced to a Hamiltonian
using the four-band basis of the S1/2 , 1/2 , P3/2 , 3/2 , S1/2 , −1/2 , P3/2 , −3/2 states.
The total Hamiltonian is composed of two terms
H = Hef f + HZeeman ,
112
(D.1)
where Hef f models the band structure and HZeeman is the Zeeman energy. The effective
Hamiltonian Hef f has the form

∗
Ak+
Dk−
B (k)
 M (k)

 Ak− −M (k) B ∗ (k)
0

Hef f (k) = 0 (k) + 
 Dk
B(k) M (k) −Ak−

+

B(k)
0
−Ak+ −M (k)









(D.2)
where k± = kx ± iky , and the terms A, 0 (k), M (k) encode the electronic dispersion. We
ignore the terms involving D, which describe any possible inversion symmetry breaking in the
crystal structure, and B(k), which contains higher order terms allowed by crystal symmetry,
since they are expected to produce only higher order corrections. The direction of the
momentum kx , ky , and kz are associated with the a, b, and c axes of the crystal, respectively.
However in the presence of a magnetic field, it is more convenient to describe the Hamiltonian
in terms of the magnetic axis (k3 ) parallel to the field, because the momentums k1 and k2
perpendicular to the magnetic axis are quantized by the field. Accordingly, k1 and k2 can
be transformed into the sum of ladder operators:
k1 = √
1
i
(a† − a), k2 = √ (a† + a)
2lB
2lB
(D.3)
p
(~/eB) is the magnetic length. The raising (lowering) operators a† (a) for the
√
√
Landau levels |ni obey the usual relations: a† |ni = n + 1 |n + 1i, a |ni = n |n − 1i. We
where lB =
use a 3D rotation matrix U to transform the vectors (k1 , k2 , k3 ) of the magnetic frame into
113
the (kx , ky , kz ) of the crystal frame,1






 kx 
 k1 
 k1 






 k  = U  k  = Rz (45◦ )Ry (φ = 54.7◦ )Rz (−45◦ )  k 
 y 
 2 
 2 






kz
k3
k3
(D.4)
and numerically diagonalize the Hamiltonian in the |ni basis to compute the Landau level
energy spectrum. In the above, we explicitly give the form of the matrix U to rotate [001]
into [112] while keeping [100] and [010] symmetric about the [110] line.
The semi-classical Landau quantization analysis shown in Fig. 5.3 c) demonstrates that
electronic dispersion in the tilted (112) plane for the conduction band of Cd3 As2 is linear in
an extended energy range from -100 mV to 300 mV. The original Hamiltonian proposed in
Ref. [41] allows for both linear (via Ak± ) and quadratic terms in the dispersion in the kx or
ky directions, but only a quadratic term in the kz direction. To reflect the linear dispersion in
a wide energy range in all three momentum directions, we introduce a hyperbolic dispersion
along the kz direction.
0 (k) = C0 + C1 kz2 + C2 (kx2 + ky2 )
q
M (k) = M0 + M32 + M1 kz2 + M2 (kx2 + ky2 )
(D.5)
(D.6)
We remark that this modified Hamiltonian is not the unique description of the data, but
is the minimal modification that is consistent with the high energy linear dispersion and
the quadratic band minimum at low energy. With the proper parameters, Hef f (k) results
in inverted bands with two Dirac points located along the [001] direction, evenly separated
from the Γ point and dispersing linearly in 3D momentum space away from the two Dirac
points.
1
All rotation matrices in a series of compound rotation matrices still rotate with respect to the original
frame of axes, not about the intermediate frame. The Rz and Ry matrices are the basic right hand rule
rotation matrices.
114
The Zeeman term HZeeman has the form

HZeeman (k) =

µB
 gs 0 
(σ · B) ⊗ 

2
0 gp
(D.7)
where µB is the Bohr magneton, σ are the Pauli matrices, and g(s(p)) is the effective g-factor
for the S(P) band. The effect of gp on the electron-like Landau levels is negligible away from
the band minimum where the S and P bands are well separated in energy. Hence, we can only
reliably estimate gs from the data and take gp = 2. We note that the effective Hamiltonian
Hef f (k) implies an induced orbital angular momentum which breaks the degeneracy of the
bands in the presence of magnetic field. Landau level splitting caused by this orbital angular
momentum strongly depends on the angle of magnetic field as shown in Fig. 5.5 b), while
splitting caused by Zeeman term depends less on the angle.
The numerical implementation of the above description is given in the source code below,
the original version written by Itamar Kimchi and Andrew Potter of University of California,
Berkeley. Here it has been streamlined to output a matrix of simulated energies (specifically
at the Γ point, or k3 = 0) for a vector of magnetic fields B. Using below and by comparing
with the experimental data matrix, we can minimize the error by least squares and determine
a good set of parameters to reproduce the data. Generally, the inability of the experimental
data to constrain details near the Dirac point due broadening leads to local minima in
the least squares fit, stemming from the flexibility of parameters related to the low energy
features. However, the well-defined vF given by the semiclassical analysis must be still
reproduced in this model calculation, so the combined effect of the kx and kz dispersions
at high energy is well constrained. Moreover, splitting of the levels in a B-field naturally
arises. In short, the model illustrates the features of the data are consistent with the original
theoretical model of inverted bands.
Listing D.1: Appendix2/LL Cd3As2.m: Landau Level Simulation Code
115
% % Computes LL ’ s of Cd3As2 and DOS , using linearized low - energy
Hamiltonian
function [ out all_levels ] = LL_Cd3As2 ( param , x , levels , cutoff , q )
% This function can be used with lsqcurvefit to produce a matrix of
% simulated levels " out " , where the columns are the B fields and the rows
% are the levels .
%
%
%
%
%
%
param - parameters for the hyperbolic Hamiltonian
x - one column vector of the required B fields
levels - an integer specifying how many levels you want in the output
cutoff - start counting levels above this minimum energy
q - regularization coefficient , needed to move spurious levels away from
energy range of interest
N_max = 36; % number of oscillator levels to keep ( truncate Hilbert space
at High LL index ) .
N_vals = 4*( N_max ) ; % if full diagonalization
% 3 - dispersion ( dispersion along B field direction )
% Here we are only interested in the minimum of the Landau band at k3 = 0.
N_k3 = 1;
k3s = 0;
dk3 = 0;
% % band structure parameters
M0 = param (1) ;
M1 = param (2) ;
M2 = param (3) ;
inversion = param (4) ;
M3 = -M0 - inversion ;
E_dirac = param (5) ;
C1 = param (6) ;
C2 = param (7) ;
A = param (8) ;
g_e = param (9) ;
phi_degrees = param (11) ;
g_h = param (10) ;
% %%%%%%%%%% Constants for now %%%%%%%%%%%%
Delta = 0; % Bulk inversion asymmetry
% %%% Compute C0 to keep Dirac Point at 0 energy . kd is set by above .
kd = real ( sqrt (( M0 ^2 - M3 ^2) / - M1 ) ) ;
C0 = C1 * kd ^2;
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % Rotating crystal coordinates x ,y , z to magnetic field coordinates 1 ,2 ,3
% ( magnetic field is along \ hat {3})
phi = pi /180* phi_degrees ;
% the following rotation matrix rotates (0 ,0 ,1) towards (1 ,1 ,2) :
116
% essentially Ry (45) Rz ( phi ) Ry ( -45)
U = ...
[1/2*(1+ cos ( phi ) )
1/2*( -1+ cos ( phi ) )
sin ( phi ) / sqrt (2) ;...
1/2*( -1+ cos ( phi ) )
1/2*(1+ cos ( phi ) )
sin ( phi ) / sqrt (2) ;...
-( sin ( phi ) / sqrt (2) ) -( sin ( phi ) / sqrt (2) ) cos ( phi ) ];
% % LL Raising and Lowering Operators
adag = sparse ((2: N_max ) ,[1:( N_max -1) ] , sqrt (1: N_max -1) , N_max , N_max ) ;
a = sparse ([1:( N_max -1) ] ,(2: N_max ) , sqrt (1: N_max -1) , N_max , N_max ) ;
n_hat = adag * a ;
% truncate landau level tower by coupling weyl nodes above a large n :
id_k = speye ( N_max ) ;
large_n_reg = n_hat -(( N_max -2) .* id_k ) ;
% % Construct Hamiltonian
% generic pauli matrices :
pauli_z = sparse ([1 0; 0 -1]) ;
pauli_x = sparse ([0 1; 1 0]) ;
pauli_y = sparse ([0 -1 i ; 1 i 0]) ;
pauli_0 = speye (2) ;
% Pauli Matrices ( orbital )
mu_z = pauli_z ;
mu_x = pauli_x ;
mu_y = pauli_y ;
mu_0 = speye (2) ;
mu_up = ( mu_0 + mu_z ) ./2;
mu_down = ( mu_0 - mu_z ) ./2;
% orbital g - factor
% Note that g_h is defined as a multiplier on g_e inorder to keep them
% the same sign when using lsqcurvefit .
mu_g = g_e .* mu_up + ( g_h * g_e ) .* mu_down ;
% Pauli Matrices ( spin )
s_z = pauli_z ;
s_x = pauli_x ;
s_y = pauli_y ;
s_0 = speye (2) ;
% % Construct Hamiltonian and Compute Spectrum for each k3 eigenvalue
Evals = zeros ( N_k3 , N_vals ) ;
H_reg = q * sqrt (2* N_max ) * kron ( kron ( s_x , mu_x ) , heaviside ( large_n_reg -0.5) ) ;
% H_reg should regularize the theory by coupling the two weyl nodes at
% large landau level .
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculate all terms outside of the B field loop and scale by B field
only
% inside the loop for speed . Below are the " zero - field " terms .
k2_0 = 1/ sqrt (2) .*( adag + a ) ; % k2 - momentum operator ( guiding center
shifted to x =0)
117
k1_0 = 1 i / sqrt (2) .*( adag - a ) ; % k1 - momentum operator
% Rotate axes from along field to along crystal axes
kx_0 = U (1 ,1) * k1_0 + U (1 ,2) * k2_0 ;
ky_0 = U (2 ,1) * k1_0 + U (2 ,2) * k2_0 ;
kz_0 = U (3 ,1) * k1_0 + U (3 ,2) * k2_0 ;
H_Z_0 = 1/2*(5.8 e -5) *...
kron ( kron ( U (1 ,3) .* s_x + U (2 ,3) .* s_y + U (3 ,3) .* s_z , mu_g ) , id_k ) ;
H_A_0 = A .* kron ( kron ( s_z , mu_x ) , kx_0 ) - A .* kron ( kron ( s_0 , mu_y ) , ky_0 ) ;
out = zeros ( size ( levels ,1) , size (x ,1) ) ;
all_levels = zeros ( N_vals , size (x ,1) ) ;
% Loop over B - field .
for i_B = 1: size (x ,1)
B = x ( i_B ,1) ;
% LL Lowering and raising op ’ s
magnetic_length = (256.6/ sqrt ( B ) ) ; % 257 A / sqrt ( T ) converted to
unit - cell size
% Rotate axes from along field to along crystal axes
kx = (1/ magnetic_length ) * kx_0 ;
ky = (1/ magnetic_length ) * ky_0 ;
kz = (1/ magnetic_length ) * kz_0 ;
% construct the Hamiltonian
E_k = kron ( kron ( s_0 , mu_0 ) , C0 * id_k - C1 * kz ^2 - C2 *( kx ^2+ ky ^2) ) ;
H_z = kron ( kron ( s_0 , mu_z ) , M0 * id_k + sqrtm ( full ( M3 ^2* id_k - M1 * kz ^2) )
- M2 *( kx ^2+ ky ^2) ) ;
H_A = (1/ magnetic_length ) * H_A_0 ;
H_Z = B * H_Z_0 ;
H_BIA = Delta .*( kron ( kron ( s_y , mu_y ) , id_k ) ) ;
H = ( H_z + H_A + E_k + H_Z + H_BIA + H_reg ) ;
Evals = sort ( real ( eig ( full ( H ) ) ) ) + E_dirac ;
all_levels (: , i_B ) = Evals ’;
% We want to compare with only the levels that we have data for . In
% other words the first ( N = levels ) levels above some cutoff energy .
ind = find ( Evals > cutoff ,1 , ’ first ’) ;
levels_out = ( ind -1) + levels ;
out (: , i_B ) = Evals ( levels_out ,1) ’;
end
To find the proper parameters for our model, we first fix the value of A, M2 , and C2
(related to kx -ky dispersion) to agree with the photoemission data measured in the (001)
118
b
0.2
0.2
0
0
Energy (eV)
Energy (eV)
a
-0.2
-0.4
-0.4
-0.2
-0.2
0
-0.05
0.2
(1/A)
0
0.05
(1/A)
Figure D.1: a), b), The band dispersion along kx axis (a) and kz axis (b) passing the Dirac
point for parameters used to simulate the Landau level peaks. The Fermi velocity along the
kx direction is 5.1 eV Å(= 7.6 · 105 m/s), in agreement with prior ARPES measurements.
plane of Cd3 As2 samples [108] from the same sample grower. The remaining parameters
(summarized in Table 1) are chosen to reproduce the observed Landau level peaks. These
parameters are used for the plots in Fig. 5.5 a, b, d, and e. Due to the absence of the
valence band Landau levels, we cannot obtain a precise determination of the size of the band
inversion (20 mV is used in our model). However, the general behavior of the Landau level
structure, such as the diminishing two-fold splitting and linear high energy dispersion, is
independent of this quantitative detail. Figure D.1 a) and b) show the band dispersion for
the two axes along kx and kz passing the Dirac points. The band dispersion measured by
angle resolved photoemission is well reproduced in this model with the above parameters as
shown. The band inversion and two 3D Dirac points are revealed.
Table D.1: Parameters for the modified four-band model for Cd3 As2 .
C0 (eV) = -0.219
C1 (eV Å2 ) = -30
C2 (eV Å2 )
A (eV Å) = 2.75
gs = 18.6
M0 (eV) = -0.060
M1 (eV2 Å2 ) = 96
M2 (eV Å2 ) = 18
M3 (eV) = 0.050
gh = 2
119
D.2
Schematic Demonstration of the Weyl Fermion
and the Low Field Regime
As discussed in the main text, the Weyl fermion can be realized in Cd3 As2 when the magnetic
field is applied along the [001] direction. To schematically demonstrate the Weyl fermion,
we simulate the Landau levels using the Hamiltonian and parameters provided in the prior
calculation paper. The degenerate Weyl fermions with two different chiralities are separated
in momentum space when the magnetic field is applied along the [001] direction as shown
Fig. D.2 a-b). The crossing points are shifted away from the original Dirac points as field
strength is increased (Fig. D.2 b).
Landau levels of non-zero index (n > 1) are formed below the Lifshitz energy for a
relatively small magnetic field (Fig. D.2 a). For these levels, extrema in the energy dispersion
of the Landau level band lead to two singularities in the DOS: one from maximum at the
gamma point and another from minima near the Dirac points. However, the high index
Landau levels formed above the Lifshitz transition have only one DOS singularity from the
gamma point. In the measured spectra, all the peaks originate from the singularity at the
gamma point due to the shallow band inversion as modeled in Fig. 5.5 e). For the purpose
of comparison, calculated Landau levels with non-zero field angle are plotted in D.2 c-d).
Moreover, the lowest Landau levels open a gap at the crossing points due to the broken C4
symmetry.
120
= 54.7
(1/A)
0
−10
−0.05
0
(1/A)
0
−10
−0.05
0.05
d
0.05
Weyl points
E Dirac
Weyl points
20
E Lifshitz
=0
B= 12T ,
= 54.7
0
0.05
0
0.05
(1/A)
E Lifshitz
B= 1T ,
0
B= 12T ,
E Dirac
20
Energy (meV)
E Lifshitz
E Dirac
Weyl points nearly
overlapping at Dirac points
E Lifshitz
Energy (meV)
20
0
−10
−0.05
c
b
=0
E Dirac
Energy (meV)
20
B= 1T ,
Energy (meV)
a
0
−10
−0.05
(1/A)
Figure D.2: a), b), Simulated Landau levels with the magnetic field applied along the [001]
direction for B = 1 T (a) and B = 12 T (b). The Weyl points and the energy of the Lifshitz
transition are marked. c), d), Simulated Landau levels with the magnetic field applied along
the [112] direction for B = 1 T (c) and B = 12 T (d). The lowest Landau levels have a gap
at the Dirac points due the C4 symmetry breaking. The momentum k is along the magnetic
field direction.
121
Appendix E
Charge Ordering in Underdoped
Bi2Sr2CaCu2O8+δ in a Magnetic Field
An overwhelming amount of experimental evidence, dating from the discovery of stripelike spin and charge order in coincidence with the suppression of Tc in the La2−x Srx CuO4
and La2−x Bax CuO4 family [128], has now established that superconductivity in the high-Tc
copper-oxides competes with an alternative electronic order, such as spin or charge density
wave. Most recently, charge order was discovered in the YBa2 Cu3 O6+x (YBCO) compounds
near 1/8 doping using resonant x-ray techniques [129]. In Ref. [130], the intensity of this
order was shown to be strongly magnetic field dependent below Tc , with an enhancement
by a factor of 3 in a modest magnetic field of 15 T. Subsequently, Refs.[131] and [132]
extended the ubiquity of this ordering phenomena to the Bi-based cuprate family, in bilayer
Bi2 Sr2 CaCu2 O8+δ (Bi2212) and single layer Bi2 Sr2 CuO6+δ (Bi2201), respectively. Therefore,
a natural question to raise is whether the magnetic field enhancement of charge order in
the YBCO family similarly affects charge order in the Bi-based family. Combined with the
tantalizing evidence that charge modulations of wavelength 4 a0 are enhanced inside of the
vortex cores of slightly overdoped (Tc = 89 K) Bi2212 at 7 T [133], the presumptive answer
might be a resounding ‘yes’. If so, as the field is increased, does the charge order spread in
122
space through the extent and number of vortex cores, or does the intensity of the existing
charge order at zero-field simply become enhanced? In the summer of 2013, the author, with
the help of Mallika Randeria, attempted to answer these questions in DRSTM, applying a
maximal field of 14 T.
E.1
Experimental Results
Two underdoped samples of Bi2 Sr2 CaCu2 O8+δ were used in this study: GG0904 with Tc = 76
K and GG0959 with Tc = 58 K. Inverting the empirical relation between Tc and hole doping
δ,
Tc = Tc,max · (1 − 82.6(δ − 0.16)2 )
(E.1)
with Tc,max of Genda Gu’s optimally-doped samples estimated to be 90 K, we obtain δ = 0.12
for the Tc = 76 K sample, and δ = 0.094 for the Tc = 58 K sample. Hence, we are studying
samples near 1/8 doping and thus can directly compare with the YBCO studies.1 The
highest quality data was obtained for the Tc = 58 K sample, but the qualitative results are
similar for both samples.
In Fig. E.1, we show an area on the UD 58 K sample over 20 b-axis supermodulations
wide, and of length 75% of the width. This area of the sample was tracked from 1 T to 14
T with the same tip at a temperature of 2.5 K.2 The spatially averaged spectra show the
robustness of the high-Tc materials to a magnetic field. Indeed, since the upper critical fields
for Bi2212 are in excess of 150 T, the modest 14 T field applied by our superconducting
magnet barely changes the spectra. Division of the 1 T spectrum by the 14 T spectrum
reveals more clearly the resolvable changes. First, lets us address the change in the larger,
1
The magnetic field enhancement effect is not expected to be restricted to 1/8 doping, but just perhaps
strongest there.
2
We took data at 1 T to take advantage of slightly better noise performance due to magnetic damping;
however 1 T is not expected to be different than 0 T.
123
UD 58 K
PG
e-b
coupling?
530 x 400 Å
SC
Figure E.1: The same area of a Bi2212 (UD 58 K) sample tracked from 1 T to 14 T. The
spatially averaged spectra show suppression of the pseudogap coherence peaks (labeled in
cyan) at high field. Moreover, the depth of the SC gap (labeled in purple) is filled in at high
fields. Finally, more speculative features (orange) outside of the pseudogap, reminiscent of
the electron-boson coupling ‘wiggle’, is reduced in amplitude at high field.
dominant gap in spectrum with energy scale ±50 mV, known as the ‘pseudogap’ in underdoped cuprates. We see that its ‘coherence’ peaks are broadened by the field. Second, the
zero bias conductance and the conductance in vicinity near zero bias (the Fermi level), set
by the superconducting gap, are enhanced by up to ∼15% at 14 T, showing a increase in
the number of nodal quasiparticles in the d-wave superconductor due to vortex/field line
penetration. Finally, we notice a wiggle in ratio of the spectra at large positive biases outside the pseudogap. Traditionally, wiggles in the spectra outside the gap for optimally and
overdoped Bi2212 samples have been attributed to electron-boson coupling, where the boson
is of electronic, rather than lattice origin [134]. Such features have not been extensively
studied on the underdoped side; however, it does appear that the magnetic field also damps
such features and may tune the underlying electron-boson coupling. One criticism of these
124
interpretations is that ‘magnetic’ broadening due to Zeeman-type physics naturally broadens
any feature in the spectra. However, an estimation with g = 2 gives an energy scale of ∼1.4
mV at 14 T, and it appears that the experimental changes in the spectra with field occur
over much larger ∼10 mV scales; hence, these changes may not be directly attributable to
simple broadening. Of course, this crude argument needs to be further vetted in quantitative
simulation.
E.2
Energy and Spatially-Resolved Density of States
in a Magnetic Field
The next natural question to pose is to what extent the energy-resolved spatial modulations
change in a field, either through the introduction of identifiable vortex cores or the widespread
enhancement of charge order. Figure E.2 shows several the real space conductance maps
taken from -49.5 mV to +49.5 mV in 5.5 mV increments. One immediately notices the
almost identical features in the 1 T and 14 T maps; that if the maps were mixed up, one
would be hard pressed to sort out which one was taken at 1 T and which one taken at
14 T. Indeed, it appears that the dramatic 300% magnetic field enhancement of charge
order seen in YBCO at 15 T is not so obviously present in Bi2212 at 14 T, at least on the
surface, visualized by STM measurement. From first glance, any changes resolved by more
quantitative analysis would likely be restricted in magnitude to the order of the change in
the spectra, shown in Fig. E.1, that is to say around ∼10 %. Moreover, as the conductance
maps at 1 T and 14 T match almost feature to feature,3 it does not appear that charge order
spreads in the sample, but that the existing charge ordering at 1 T is amplified in intensity
(especially for high biases).
3
Upon revisiting this data more carefully during the writing of this appendix, it appears that the correlation between the maps at 1 T and 14 T is weakest at low energy scales, namely between ±5.5 mV.
125
B = 1T
B = 14T
E = 22 mV
E = 11 mV
E =0 mV
B = 14T
E = 33 mV
E = -11 mV
E = -22 mV
E = -33 mV
E = -44 mV
B = 1T
Figure E.2: Real space resolved spectroscopic mapping of the electronic states in Bi2212
UD 58K at 1 T and 14 T. The same color scale is used for both fields. The nearly identical
features at all energies for both fields show that charge order does not in general spread in the
sample due to the increased magnetic field. Rather, there does appear to exist small patches
(e.g., circled in white for E = +22 mV) where the existing charge ordering appears stronger
in a magnetic field; however, it is difficult to definitively rule out very minute differences
in tip and noise condition that may affect the outcome of the measurement. At low biases,
(e.g. E = 0 mV), there does appear to be some areas where charge modulations appear
from previously absent areas. In any case, Fourier transforms of these maps can give a more
quantitative estimate of the enhancement.
Comparison of the Fourier transforms of the energy-resolved spectroscopic maps provides
a more quantitative estimate of the enhancement of charge ordering in a magnetic field. In
Fig. E.3, we show the cut along the Bragg direction (π, 0), denoted by the red arrow in the
FFT of E = 22 mV, as a function of energy. It is apparent that the charge ordering is a
little stronger in a magnetic field, as the subtraction of the two panels shows enhancement
126
near E = 22 mV, circled in magenta. The linecut at E = 22 mV shows the intensity of
the charge ordering wave vector Q∗ = (0.25, 0) · 2π/a0 to be enhanced by approximately
11%, which seems reasonable as this is on the order of magnitude of the changes that we
see in the spectra. Thus, the magnetic field likely does not precipitate an extraordinary or
disproportional response in Bi2212, but produces a more benign response proportional to the
weakening of superconductivity (recall that the zero bias conductance was also elevated by
∼15%). This contrasts with the case of YBCO, where a 300% enhancement of the charge
order was seen for fields that hardly affect superconductivity in the sample.
A further complication to the interpretation of the results presented here is the presence
of the QPI wave vector Q1 , which is nearly coincident with Q∗ . Hanaguri’s seminal measurements on another cuprate family superconductor, the oxychloride Ca2−x Nax CuO2 Cl2 ,
showed the vector Q1 to be enhanced in a magnetic field due to coherence factor effects for
magnetic scattering [135]. To elaborate, QPI wavevectors that connect parts of the Fermi
surface with the same sign of the gap function (Q1 , Q4 , Q5 ) are enhanced by scattering off
magnetic vortices, while QPI wavectors that connect parts of the Fermi surface with opposite sign of the gap function (Q2 , Q3 , Q6 , Q7 ) are alternatively weakened. Thus, to prove
that charge order is enhanced, one must separate the effect on Q∗ from that on Q1 , which
is also expected to be enhanced by a magnetic field. Ideally, one must show that degree
of enhancement in the vicinity of Q1 /Q∗ is significantly stronger than the enhancement of
Q4 , Q5 ; however, the data does not clearly show the other Qi for such underdoped samples.
In this case, one can only comment that the enhancement of Q1 /Q∗ region appears to be
particle-hole asymmetric, favoring the particle side, which suggests that it is indeed Q∗ that
is enhanced. Enhancement of Q1 is in principle particle-hole symmetric, while the charge
order Q∗ is centered at positive energies, around positive 20 mV.
A final thought on the difference between the results obtained in YBCO and in Bi2212 is
whether it could be attributed to bulk vs. surface probes. Could charge order be enhanced in
the bulk, but not on the surface? Only time will tell whether this magnetic field phenomena is
127
1T
a)
1T
14 T
c)
14 T
14 T - 1 T
d)
f)
Hanaguri’s Result on
Ca2-xNaxCuO2Cl2
Q* or Q1?
14 T - 1 T
b)
e)
Peak Height
364
328
Figure E.3:
a) The wavevector intensity cuts along (π, 0) as a function of energy for
1 T and 14 T. b) Difference of the wavevectors cuts showing an enhancement of the Q∗
charge ordering feature centered around 20 mV. c) Typical FFTs of the spatial patterns
shown in Fig. E.2 at 1 T and 14 T. d) Difference of the FFTs showing enhancement in
the region near Q∗ /Q1 . Unfortunately the other octet model Qi ’s are not well resolved for
such underdoped samples. e) The single energy cut along (π, 0) at E = 22 mV showing an
approximately 11% enhancement at 14 T over 1 T. f) Hanaguri’s data [135] on an oxychloride
superconductor showing a phase-sensitive enhancement of the octet model Qi ’s. Note that
the map shown in his data is a so called Z-map. His paper makes no mention of whether
the strong checkerboard charge order in that material is enhanced in a magnetic field.
truly ‘ubiquitous’ across the cuprate landscape, since these x-ray experiments in a magnetic
field should be readily doable on the Bi2212 samples.
128
E.3
Where are the Vortices?
The initial apparent observation was made that no static vortices could be imaged in underdoped Bi2212. In the process of looking at the data more carefully during the writing of
this thesis, there probably exist static vortices, with at least one definitive identification, although they are very difficult to identify. In overdoped and optimally doped Bi2212 samples,
vortices pinned by strong disorder (sometimes from intentional Zn dopants) could be imaged
clearly from the conductance map at Ekink ∼ ±7 mV [62]. Whether the vortices are static
is an important experimental question, necessary to reconcile with the fact that resistivity
measurements show these samples to have zero resistance at the temperatures and fields of
the STM measurement, suggesting pinned vortices.
At a given field B, we expect B/Φ0 vortices per unit area, where Φ0 = 2.1 · 10−15 T m2 .
In other words, at 14 T, one expects a single vortex to occupy 15000 Å2 (or a square of side
length 122 Å). Hence in our field of view of 530 Å by 400 Å, we expect approximately 14
vortices at 14 T, and accordingly approximately one vortex at 1 T. As mentioned before the
STM conductance maps show the strongest change between 1 T and 14 T for energies near
the Fermi energy, i.e. E = -5.5, 0, and 5.5 mV (near Ekink ). Summing these conductance
maps to gain the strongest signal, we can compare the two fields in Fig. E.4 to identify several
regions of enhanced conductance, also containing new charge modulations. We tentatively
identify these regions as vortices. However, the number is insufficient to account for all of
the approximately 13 extra vortices that are expected to thread the field of view as the
field is dialed up from 1 T to 14 T. More careful analysis is currently being investigated to
drift correct the maps and enable pixel-to-pixel real space subtraction; however, if the simple
‘eye’ test cannot pickout the vortices, they likely they are of fundamentally different character
than the vortices in near optimally doped Bi2212. Qualitatively, our result is similar to Ref.
[133] in that 4 a0 charge ordering accompanies the vortex regions. However, the analysis by
Fourier transform nevertheless does not reveal a strong quantitative enhancement. Perhaps
this suggests that the metric for STM is somehow not directly comparable to the metric
129
of x-ray scattering; or that the background charge modulation for underdoped Bi2212 is
already too dense at zero field, leaving little room for charge order to spread on the surface.
1 T ~ 1 vortex
14 T ~ 14 vortices
Figure E.4: These maps are the sum of the conductance maps at the three lowest energies:
-5.5 mV, 0 mV, 5.5 mV. The higher energy maps show much less variation with field. In
this sized area of the sample, an additional 13 vortices should penetrate at the higher 14
T field. We can identify at least 3 regions where the spatial electronic patterns change
significantly. The lowest one near the center of the field of view is strongly suggestive of an
vortex core, since an area with previously low conductance now possesses high conductance
in field. Moreover, charge modulations appear inside this presumptive vortex core. However,
the number and intensity of the changes over the entire field of view seems to be a bit lacking.
Therefore, if indeed additional vortices exist in this field of view, it appears they are well
‘camouflaged’ in the background charge order and require exquisitely low noise measurement
and drift correction to see.
130
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