UNIVERSITY OF CALIFORNIA, SAN DIEGO
Investigation of the Terahertz, Infrared, Optical
and Magneto-Optical Properties of Novel
Materials.
A dissertation submitted in partial satisfaction of the
requirements for the degree of Doctor of Philosophy in Physics
by
Willie John Padilla
Committee in charge:
Dimitri N. Basov, Chair
Sunil Sinha
David R. Smith
Sia Nemat-Nasser
Ed Yu
2004
Copyright
Willie John Padilla, 2004
All rights reserved.
The dissertation of Willie John Padilla is approved and it is acceptable in
quality and form for publication on microfilm.
Chairman
University of California, San Diego
2004
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The mathematical sciences particularly exhibit order, symmetry, and
limitation; and these are the greatest forms of the beautiful.
-Aristotle Metaphysica, 3-1078b.
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Contents
Signature Page . . . . . . . . . . . .
Dedication . . . . . . . . . . . . . . .
Epigraph . . . . . . . . . . . . . . . .
Contents . . . . . . . . . . . . . . . .
List of Figures . . . . . . . . . . . . .
List of Tables . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . .
Vita, Publications, Presentations and
Abstract . . . . . . . . . . . . . . . .
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1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
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Composite Medium with Simultaneously Negative Permeability and
Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3
Constant Effective Mass Mott Insulator Transition in the Cuprates
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4
THz Magnetic Response from Artificial Materials . . . . . . . . . .
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5
Searching for the Slater Transition in the Pyrochlore Cd2 Os2 O7 with
Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . .
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An Infrared Investigation of Phonons in a series of Detwined Single
Crystal of La2−x Srx CuO4 . . . . . . . . . . . . . . . . . . . . . . .
59
Broadband multi-interferometer spectroscopy in high magnetic fields:
from THz to Optical . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.2
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Study
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SYSTEM DESIGN AND COMPONENTS .
7.2.1 Optics . . . . . . . . . . . . . . . . .
7.2.2 Interferometers . . . . . . . . . . . .
7.2.3 Evacuated light guide and light pipes
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7.2.4 Superconducting magnet . . . . . . .
7.3 ZERO FIELD CHARACTERIZATION . . .
7.4 MAGNETO-OPTICAL MEASUREMENTS
7.4.1 Reflectance Mode . . . . . . . . . . .
7.4.2 Transmission Mode . . . . . . . . . .
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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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List of Figures
2.1
2.2
2.3
Resonance curve of an actual copper split ring resonator (SRR).
c=0.8 mm, d=0.2 mm, and r=1.5 mm. The SRR has its resonance
at about 4.845 GHz, and the quality factor has been measured to
be Q=f0 /∆f3dB ¿600, consistent with numerical simulations. . . .
8
(a) Dispersion curve for the parallel polarization H|| . The lines
with the solid circles correspond to the split ring resonators only.
The inset shows the orientation of the split ring with respect to
the incident radiation. The horizontal axis is the phase advance
per unit cell, or kd , where k is the wave number. (b) Dispersion
curve for the perpendicular polarization H⊥ . The lines with the
solid circles correspond to the split ring resonators only. The inset
shows the orientation of the split ring with respect to the incident
radiation. (c) Expanded view of the dispersion curve shown in
(a). The dashed line corresponds to the split ring resonators with
wires placed uniformly between split rings. (d) Expanded view of
the dispersion curve shown in (b). The dashed line corresponds
to the split ring resonators with wires placed uniformly between
split rings. The insets to (c) and (d) show the orientations of the
split rings with respect to the wires. . . . . . . . . . . . . . . . .
12
A transmission experiment for the case of H|| . The upper curve
(solid line) is that of the SRR array with lattice parameter a=8.0
mm. By adding wires uniformly between split rings, a passband
occurs where µ and are both negative (dashed curve). The transmitted power of the wires alone is coincident with that of the
instrumental noise floor (-52 dB). . . . . . . . . . . . . . . . . .
16
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3.1
3.2
Spectra of the optical conductivity and of the loss function probed
for La2−x Srx CuO4 and YBa2 Cu3 Oy single crystals. The YBCO
data were obtained in the polarization E||a and therefore represent
the genuine response of the CuO2 planes without a contribution of
the Cu-O chains running along the b-axis. All spectra are at 10K
except for results generated at T∼Tc for superconducting crystals.
The response of weakly doped materials (x ≤ 0.06; y=6.3-6.43) are
multi-component and are comprised of a ”Drude-like” contribution
at low energies (red segments) followed by a mid-IR resonance
(blue segments). As doping increases the two contributions merge
and the separate absorption channels can no longer identified. For
σ1 (ω) fits are shown for the low energies and MIR terms. The
green area is σ1 (ω) with the ”Drude-like” term subtracted and
thus represents other MIR contributions. . . . . . . . . . . . . .
22
Transport and spectroscopic data characterizing the response for
the electric field within the CuO2 plane of 3% doped LSCO single crystal. Top panel: the temperature dependence of the Hall
coefficient RH . Middle panel: the T -dependence of the DC resistivity (black solid line) and spectra of the optical resistivity
ρ(ω) = 1/σ1 (ω) at 13K (red and blue). The solid black line shows
ρ(T ) data at constant pressure; the dashed line has been corrected
for thermal expansion following ref. [18]. Bottom panel: the frequency dependence of the effective spectral weight Nef f (ω) defined
in the text. The absolute value of Nef f (ω) is proportional to the
number of carriers participating in the optical absorption below
ωc . In order to convert Nef f (ω) to a hole number (holes/Cu) we
used the effective masses plotted in Fig. 3.3. The red segments
of the spectra in the middle and bottom panels represent the low
energy response while the blue indicates the MIR terms. . . . .
25
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3.3
4.1
4.2
4.3
5.1
Bottom panels: carrier density extracted from the Hall data
at 300K (solid red circles) and at 800K (solid blue circles) for
La2−x Srx CuO4 and YBa2 Cu3 Oy systems. The effective spectral
was obtained from integration of the conductivity with a cut-off
frequency of 600 cm−1 (open red squares) and 10000 cm−1 (open
blue squares). Top panels: the effective mass of conducting holes
determined from the ratios detailed in Table 3.1. Remarkably,
both approaches produce m∗ values virtually independent of doping over broad regions of the phase diagram and also agree with
the m∗ values inferred from the extended Drude analysis of the
data obtained at higher dopings (open black circles). . . . . . .
28
Illustration depicting the orientation of the 30 degree ellipsometry
experiment. The polarization shown is S-polarization, or transverse electric (TE), for excitation of the magnetic response and
P-polarization, transverse magnetic (TM), is also measured. Image shown in the inset is a secondary ion image of sample D1 taken
by focused ion beam microscopy. . . . . . . . . . . . . . . . . .
36
The top panel shows the ratio of the magnetic to electric response
(described in the text) for three different artificial magnetic structures D1, D2, and D3, (shown as the red, black and blue solid
curves respectively) in the terahertz frequency range. Theoretical
magnetic response as determined by simulation for each SRR, as
described in the text, is displayed in the bottom panel. . . . . .
39
Ratio of the magnetic to electric response repeated in the top
panel from Fig. 4.2. Bottom panel shows the real (µ1 ) and imaginary (µ2 ) magnetic permeability functions as simulated by HFSS
for samples D1 (red), D2 (black), and D3 (blue). Inset to the
bottom panel depicts the dimensions of an individual SRR. . . .
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Reflectance of single crystal Cd2 Os2 O7 at various temperatures
from 40 cm−1 to 1000 cm−1 . Inset shows the entire energy range
characterized for room temperature and 25K on a log scale. . . .
49
x
5.2
5.3
5.4
6.1
Real part of the optical conductivity as obtained by KramersKronig analysis on a log scale. A gap can be seen to develop
continuously as the temperature is reduced. The reduction in
spectral weight occurring in the infrared region is compensated
by a shift to higher energies. The inset shows the hardening Real
part of the optical conductivity as obtained by Kramers-Kronig
analysis on a log scale. A gap can be seen to develop continuously
as the temperature is reduced. The reduction in spectral weight
occurring in the infrared region is compensated by a shift to higher
energies. The inset shows the hardening of the 347 cm−1 phonon
at low T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
Infrared region of the real conductivity at T < TM IT . The expected theoretical frequency dependence of the gap edge (σ1 (ω) ∼
ω 1/2 ) is shown for each temperature as a dashed line. A linear
fit for the region below the gap edge is also shown. The intersection of the two fits can be taken as a measure of the energy
gap 2∆. The inset shows the optical energy gap as determined
by the method described above (open circles), and the expected
theoretical dependence (solid curve). . . . . . . . . . . . . . . .
53
Effective spectral weight vs. cutoff frequency for temperatures as
indicated. The top axis is in energy units normalized by twice the
gap energy. The upper inset shows the temperature dependence
of the 347 cm−1 phonon and the BCS gap function is also plotted,
(solid curve). The lower inset shows the temperature dependence
of all seven phonons. . . . . . . . . . . . . . . . . . . . . . . . .
55
The infrared portion of the real conductivity for undoped x=0.00,
top panel, and x=0.01, 0.03, 0.04, 0.06, and 0.008 in the bottom
panel. In the undoped compound (top) the Aortho axis (gray) displays the expected four infrared active phonons and the Bortho
axis (black) the expected 7 IR active phonons. These theoretical expected in-plane phonons can be observed as one progresses
across the phase diagram all the way to the superconducting x=6%
cyrstal, but importantly also seen in the twinned superconducting
8% sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6.2
6.3
6.4
Infrared region of σ1 (ω) for the 10K, room temperature, and the
HTT phase (T=473K) of 4% LSCO. The phonons for the HTT
phase are indicated by arrows with the center frequencies given in
the text. The bottom panel list, in order from left to right, the
assignment of each of these modes from low to high frequency as
depicted by the CuO6 octahedra. The temperature evolution of
these modes can be tracked into the LTO phase displayed in the
top two panels. Both the A-axis (grey) and B-axis (black) in the
LTO phase are plotted. . . . . . . . . . . . . . . . . . . . . . . .
63
Low temperature infrared spectrum of all three crystallographic
axis for the LTO phase of x=4% Sr doped LSCO. The A and Baxis (thick gray and thick black lines respectively) are shown in
the main panel and their units correspond to the left coordinate.
In order to display all phonons, the C-axis (thin black line) is
displayed at 4X magnification and corresponds to the right coordinate. The inset shows the C-axis phonon at 244 cm−1 and the
A and B-axis all on the same scale. . . . . . . . . . . . . . . . .
65
Temperature dependence of all in-plane phonons for 0% (left
pane) from room temperature to 10K and 4% doped LSCO (right
pane) from 500K to 10K. All A-axis phonons (open circles) and
B-axis phonons (filled squares) are found except one mode in 4%
at the highest temperatures. Notice that for phonons split from
those in the HTT phase (shown as triangles), the A-axis phonons
always occur lower in frequency. . . . . . . . . . . . . . . . . . .
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7.1
7.2
Schematic and optical layout of an apparatus for absolute measurements of R(ω) and T(ω) in magnetic field. Collimated light
input from the Michelson interferometer is reflected from a 45◦
off-axis parabola with a 8.89 cm focal length. Light then passes
through an intermediate focus and through two flat mirrors before reflecting from the final focusing mirror and into the cryostat.
From here the light may then reflect from the sample back into the
reflection unit, or continue through to the transmission unit. In
both cases the light strikes a focusing mirror then off a flat mirror
before impinging upon the detector, here depicted as bolometers.
The Martin-Puplett interferrometer is coupled by 1/2 I.D. brass
pipe depicted as the dashed line segments. The reference channel
goes directly to the detector while the sample channel is guided
into the same optics used by the Michelson setup. This design allows for a quick change between interferometers by simple removal
of the off-axis parabola. The inset shows two different sample holders used for absolute measurements. In reflection both the sample
and a reference mirror are fixed on two cones mounted 180◦ apart.
In transmission mode the sample is located behind an aperture
and reference measurements are obtained by vertical translation
to an open channel. Both methods allow for absolute measurements and are controlled by a stepping motor with encoder feed
back. The rotational resolution in reflection mode is 0.02 degrees
and the step size in vertical translation is 0.001 mm. The rotational or translational alignment are both highly reproducible and
any error in misalignment and within the signal to noise of the
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
Photograph of the 9 Tesla superconducting magnet connected to
the Michelson interferometer. The units are connected by 750 cm
of evacuated light guide. The platform is where the detector would
sit for reflection experiments. A similar box sits on the opposite
side of the magnet for transmission (not shown). . . . . . . . . .
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7.3
7.4
7.5
7.6
Power spectrum, configuration, and 100% lines in reflectance
mode. The left panels depict the intensity of several configurations plotted vs. frequency while the middle panels show the configuration used for these spectra, and the right panels display the
corresponding 100% lines. The top two rows are for the MP setup
and the next three left panels are for the far-infrared range, below
is a spectrum from the infrared portion and the bottom panel displays the mid infrared spectrum. The corresponding 100% lines
for each spectrum are presented in the right panels. The particular
components used for these measurements are listed in the middle
panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
Top panel: Reflectance of graphite (HOPG) at 0 Tesla and 7
Tesla, both at 10K. Bottom panel: Real part of the optical conductivity σ1 (ω) of HOPG graphite obtained by Kramers-Kronig
transformation of the reflectance data in the top panel. The bottom inset shows the detailed peak structures of σ1 (ω) in the MIR
region, which are due to the interband transitions between Landau
levels in magnetic fields. The reflectance at non-zero field values
was obtained by the Martin-Puplett Spectrometer, with a 320mK
bolometer from 5cm−1 to 50cm−1 and with a 1.6K bolometer from
20cm−1 to 110cm−1 . The Michelson interferometer, with a 4.2K
bolometer, was used from 70cm-1 to 700cm-1, and with a photoconductor detector from 400cm−1 to 3,000cm−1 . The data in NIR
region above 3,000cm−1 were measured by Michelson interferometer with an InSb detector and a grating spectrometer in the UV
region up to 50,000cm−1 . . . . . . . . . . . . . . . . . . . . . . .
89
Reflectance spectra of y=6.65 YBCO. Two different protocols in
application of the 7 T field are used. The zero-field cooling process
means the samples is cooled down to 5 K in zero field and then
the field is changed to 7 T magnetic field for the measurement.
In the field cooling process the sample is cooled from T > Tc to
T=5K with a 7T magnetic field applied. . . . . . . . . . . . . .
91
Top panel: ratio of the transmission in the superconducting state
at zero field to that of the transmission at 5 Tesla. Bottom panel:
ratio of the superconducting transmission at 3.5K to that at 15K.
93
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List of Tables
3.1
4.1
Middle left column shows nH (T ) both for the low energy ”Drudelike” contribution and the ”total” intra-gap response. Middle right
column displays Nef f (ω) for the same two energies. Far right
column shows m∗ as calculated from Nef f and nH for both energy
regimes. This legend is used in Fig. 3.3. . . . . . . . . . . . . .
26
Dimensions corresponding to the samples characterized in this
study (see the inset to Fig. 4.3 ). The four geometrical parameters
describing the SRRs are: the gap between the inner and outer ring
(G), the width of the metal lines (W), the length of the outer ring
(L), and the lattice parameter (P). . . . . . . . . . . . . . . . .
42
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Acknowledgments
First and foremost I would like to thank my advisor Prof. D.N. Basov without
whom this thesis would not exist. I also owe him most of my knowledge of
condensed matter physics.
Special thanks to Sheldon Schultz and Saul Oseroff without whom I would
not have pursued a Ph.D.
I acknowledge the help of all my collaborators who contributed to this dissertation. I would also like to thank Jason Singley, Sasa Dordevic, Syrus NematNasser, David Schurig, Mladen Barbic, Keith Clark and David R. Smith for
numerous enlightening discussions.
Finally I would like to dedicate this thesis to my family, for without their
support none of this would have been possible.
The text of chapters two and five are in part, reprints of material as it appears in Phys. Rev. Lett. 84, 4184 (2000) and Phys. Rev. B 66, 035120 (2002)
respectively. Chapter 4 is, in part, a reprint of material as it appears in Science
303, 1742 (2004). The text of chapters 3, 6, and 7 are, in part, a reprint of
material submitted for publication in Science, Phys. Rev. Lett., and Review
of Scientific Instruments, respectively. The dissertation author was the primary
researcher and author in these publications. The co-authors listed in these publications directed and/or collaborated on the research which forms the basis for
these chapters.
xvi
Vita
1996
B.S., Physics, San Diego State University.
Senior thesis advisor: Saul Oseroff
1997 - 1999 Research Associate, Department of Physics
University of California, San Diego
2002
M.S., Physics, University of California, San Diego.
1999 - 2004 Graduate Student Researcher, Department of Physics
University of California, San Diego
2004
Ph.D., Physics, University of California, San Diego.
Publications
1. W.J. Padilla, Ta-Jen Yen, Nicholas Fang, D.N. Basov, D. Vier, D.R. Smith,
J.B. Pendry, and Xiang Zhang. ”THz Magnetic Response from Artificial
Materials” Accepted for publication, Science 2004.
2. W.J. Padilla, Y.S. Lee, M. Dumm, D.N. Basov, G. Blumberg, S. Ono,
Kouji Segawa, Seiki Komiy, and Yoichi Ando. ”Mott transition and hole
dynamics in high-Tc cuprates” Submitted to Nature.
3. D.N. Basov, S.V. Dordevic, E.J. Singley, W.J. Padilla, K. Burch, J.E.
Elenewski, L.H. Greene, J. Morris and R. Schickling. ”Subterahertz Spectroscopy at He-3 temperatures” Review of Scientific Instruments 74, 4703
(2003).
4. Dongmin Wu, Nicholas Fang, Cheng Sun, Xiang Zhang, Willie J. Padilla,
Dimitri N. Basov, David R. Smith, and Sheldon Schultz. ”Terahertz plasmonic high pass filter” Appl. Phys. Lett. 83, 201 (2003).
5. A. G. Flores, M. Zazo, J. iguez, V. Raposo, C. de Francisco, J. M. Muoz and
W.J. Padilla ”Ferromagnetic resonance in double perovskite Ba2 FeMoO6 ”
J. Magn. Magn. Mater. 254, 583 (2003).
xvii
6. W. J. Padilla, D. Mandrus, D. N. Basov ”Searching for the Slater transition
in the pyrochlore Cd2 Os2 O7 with infrared spectroscopy” Phys. Rev. B 66,
035120 (2002).
7. D. R. Smith, W. J. Padilla, D. C. Vier, R. Shelby, S. C. Nemat-Nasser, N.
Kroll and S. Schultz ”Photonic Crystals and Light Localization”, ed. C.
M. Soukoulis (Klumer, Netherlands, 2000).
8. D.R. Smith, W. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz ”Negative permeability from split ring resonator arrays”. Conference Digest.
2000 Conference on Lasers and Electro-Optics Europe, Nice, France, 10-15
Sept. 2000. Piscataway, NJ, USA: IEEE, 2000. p.1 pp. xii+394 pp.
9. D. R. Smith, Willie J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz
”Composite Medium with Simultaneously Negative Permeability and Permittivity” Phys. Rev. Lett. 84, 4184 (2000).
10. D.R. Smith, D.C. Vier, Willie Padilla, Syrus C. Nemat-Nasser, S. Schultz
”Loop-Wire Medium for Investigating Plasmons at Microwave Frequencies”
Appl. Phys. Lett. 75, 1425 (1999).
11. C. Rettori, J. Singley, D. Kidwell, W. Padilla, K. Clark, S.B. Oseroff ”Temperature dependence of the ESR and magnetization in the paramagnetic
phase of R1−x Bx MnO3+d (R=La, Pr; B=Ca, Sr, K)” Intl. Workshop on
Non-Crystalline Sol., July 2-5 (1997).
Presentations
1. ”THz Magnetic Response from Artificial Materials”, Invited talk at the
Meeting of the American Physical Society, Montreal, Quebec, Canada
March 2004.
2. ”Infrared Spectroscopy of THz Metamaterials”, Invited talk at the Progress
in Electromagnetics Research Symposium, Honolulu, HI October 2003.
3. ”Artificial Magnetic Response at THz Frequencies”, Invited talk at the
Workshop on Infrared Microscopy and Spectroscopy, Lake Tahoe, CA July
2003.
xviii
4. ”Unusual Anisotropic Electromagnetic Response in Lightly Doped
La2−x Srx CuO4 ”, Contributed talk at the Meeting of the American Physical
Society, Austin, TX. March 2003.
5. ”Infrared Spectroscopy of Metamaterials”, Contributed talk at the Department of Defense MURI MetaMaterials and Devices, Coronado, CA. June
2002.
6. ”Searching for the Slater Transition in the Pyrochlore Cd2 Os2 O7 ”, Contributed talk at the Meeting of the American Physical Society, Indianapolis,
IN. March 2002.
Fields of Study
Major Field: Physics
Novel Artificial and Metamaterial Systems.
Discovery of first Left-handed material.
Discovery of first THz magnetic artificial material.
Superconductivity
Discovery of the constant mass Mott insulator transition in cuprate high temperature superconductors.
Systematic doping and temperature investigation of phonons in cuprate superconductors.
Technical Research Experience
Investigation of the THz, infrared, optical and magneto-optic properties of novel
materials utilizing various spectroscopic methods, including Fourier transform
spectroscopy and ellipsometry. Past instrumentation projects have included:
Design and fabrication of a reflectance and transmission unit capable of broadband frequency domain magneto-optical spectroscopy in the range 6 cm−1 to
25000 cm−1 .
Design of a program for real time Kramer-Kronig transformation of reflectance
data.
xix
ABSTRACT OF THE DISSERTATION
Investigation of the Terahertz, Infrared, Optical
and Magneto-Optical Properties of Novel
Materials.
by
Willie John Padilla
Doctor of Philosophy in Physics
University of California, San Diego, 2004
Professor Dimitri N. Basov, Chair
In this thesis we study a variety of condensed matter and artificial systems
with various spectroscopic methods as a function of doping, temperature, and
magnetic field. The specific condensed matter systems include density wave
materials, Mott-Hubbard insulators, high-Tc superconductors and strongly correlated electron systems. Artificial materials which exhibit electric and/or magnetic effects are also studied. Infrared spectroscopy has proven to be a powerful
tool for studying such systems. The latter experimental technique probes all
excitations is solids that have a dipole moment associated with them, such as
gap excitations, interband transitions, phonons, polarons, magnons etc.
xx
Chapter 1
Introduction
This thesis is concerned with the studies of various systems including strongly
correlated electrons, density wave materials, artificial metamaterials, and high-Tc
superconducting cuprates. Infrared spectroscopy is used to probe the electronic
and lattice degrees of freedom of natural materials, and the magnetic resonant
response due to artificial materials at terahertz frequencies. The first demonstration of the so-called ”left handed” materials were measured utilizing microwave
spectroscopy.
In chapter 2 we present measurements of artificial materials at microwave
frequencies. The measurements are performed in a X-band 2D microwave scattering chamber which allow us to study uniaxial response. We prove the concept
of negative magnetic response and demonstrate, for the first time, the properties
of left handed materials. We discuss the implications of this discovery.
In chapter 3 we report the results of optical studies on a series of strongly cor-
1
2
related electron systems know as the cuprates. These compounds, La2−x Srx CuO4
and YBa2 Cu3 Oy , display high temperature superconductivity when holes are
doped into the systems. We show an evolution from a two component to a single
component conduction occurs as one proceeds from the insulating state. The
Mott insulating state in these compounds happens in the limit of vanishing carrier density and constant mass, which is in contrast to that which is expected
theoretically.
Chapter 4 deals with measurements of micron sized artificial materials at terahertz (THz) frequencies. The response from a single layer of split ring resonators
(SRR) is observed to exhibit a strong resonant magnetic response and is tunable
throughout the THz frequency regime. An ellipsometric method is utilized at
oblique incidence to characterize the resonant response. The SRR structures are
simulated using HFSS(TM) and the material parameters are extracted from Sparameter transmission and reflection coefficients. We find excellent agreement
between the experiments and simulation results. We suggest these structures
may be important for future devices due to the scarcity of natural magnetic
materials at THz and higher frequencies.
In chapter 5, we report on the metal insulator transition (MIT) observed
in a pyrochlore Cd2 Os2 O7 . The transition to the insulating state is studied as
a function of temperature and it found that the conductivity shows the clear
opening of a gap with 2∆ = 5.2kB TM IT . All infrared active phonons are tracked
3
as a function of temperature and it is determined that charge ordering is not
responsible for the MIT observed in this compound. The continuous nature of
the opening of the gap and the temperature independence of the phonons suggest
the transition is second order. All experimental data are best described as that
due to a Slater transition. This is the first known observation of a 3D Slater
transition.
Chapter 6 describes a systematic study of phonons in the high temperature
superconducting series La2−x Srx CuO4 . High quality detwinned single crystals
are utilized for this study where the phonons are studied both as a function of
temperature and doping. Optical spectra of the entire underdoped region of the
phase diagram from the undoped Mott-Hubbard insulator to near optimal doping
is presented. We find all expected group theoretical infrared active phonon modes
in all three crystallographic axes.
In chapter 7 we present a system capable of broad band frequency domain
spectroscopy in the range 6 cm−1 to 25000 cm−1 at cryogenic temperatures.
The apparatus couples two different commercial interferometers to a 9 Tesla
superconducting split coil magnet and is designed to work with various detectors,
including thermal bolometers and semiconducting detectors. The optical layout
utilizes an intermediate focus while preserving optical f/#’s throughout enabling
DC magnetic field measurements of small crystals with polarized light in both
Voight and Faraday geometries. For transparent samples, simultaneous reflection
4
and transmission can be performed.
Chapter 2
Composite Medium with
Simultaneously Negative
Permeability and Permittivity
We demonstrate a composite medium, based on a periodic array of interspaced
conducting nonmagnetic split ring resonators and continuous wires, that exhibits
a frequency region in the microwave regime with simultaneously negative values
of effective permeability µef f (ω) and permittivity ef f (ω) . This structure forms
a ”left-handed” medium, for which it has been predicted that such phenomena
as the Doppler effect, Cherenkov radiation, and even Snell’s law are inverted. It
is now possible through microwave experiments to test for these effects using this
new metamaterial.
5
6
A periodic array of conducting elements can behave as an effective medium
for electromagnetic scattering when the wavelength is much longer than both
the element dimension and lattice spacing. What makes the resulting media
special is that the effective permittivity ef f (ω) and permeability µef f (ω) can
have values not observed in ordinary materials. An example medium is a threedimensional array of intersecting thin straight wires, for which propagating modes
follow a dispersion relation analogous to that of a neutral plasma. This medium
was initially used to study microwave propagation through the ionosphere[1, 2]
(which exhibits a negative ef f (ω) below the plasma frequency), and has since
been carefully reexamined and found to have further striking electromagnetic
properties, including longitudinal plasmon modes.[3, 4, 5]
In recent work[6], Pendry et al. have extended the range of electromagnetic
properties of effective media by introducing a periodic array of nonmagnetic
conducting units whose dominant behavior can be interpreted as having an effective magnetic permeability. By making the constituent units resonant, the
magnitude of µef f (ω) is enhanced considerably, leading to large positive effective µef f (ω) near the low frequency side of the resonance and, most strikingly,
negative µef f (ω) near the high frequency side of the resonance.
The concept of negative µef f (ω) is of particular interest, not only because
this is a regime not observed in ordinary materials, but also because such a
medium can be combined with a negative ef f (ω) to form a “left-handed” mate-
7
rial (i.e., E × H lies along the direction of −k for propagating plane waves).
In 1968, Veselago[7] theoretically investigated the electrodynamic consequences
of a medium having both and µ negative and concluded that such a medium
would have dramatically different propagation characteristics stemming from the
sign change of the group velocity, including reversal of both the Doppler shift
and Cherenkov radiation, anomalous refraction, and even reversal of radiation
pressure to radiation tension. However, these effects could not be experimentally
verified since, as Veselago pointed out, substances with µ < 0 were not available.
Negative µef f (ω) has been shown to be possible when a polariton resonance exists in the permeability, such as in the antiferromagnets[8, 9] MnF2 and FeF2 , or
certain insulating ferromagnets.[10] However, a negative permeability with low
losses coexisting with a negative has not been demonstrated. The split ring
resonator (SRR) medium recently introduced by Pendry et al.[6] has now given
us the opportunity to make a material with negative permeability, from which a
left-handed medium can be constructed, as we demonstrate below.
The system presented here is anisotropic to simplify the analysis; the isotropic
medium, constructed by adding elements to increase the symmetry of a unit cell,
will be presented elsewhere. A single copper SRR is shown in Fig. 2.1 (inset)
with the dimensions indicated. A time varying magnetic field applied parallel to
the axis of the rings induces currents that, depending on the resonant properties of the unit, produce a magnetic field that may either oppose or enhance the
8
Figure 2.1: Resonance curve of an actual copper split ring resonator (SRR). c=0.8 mm,
d=0.2 mm, and r=1.5 mm. The SRR has its resonance at about 4.845 GHz, and the
quality factor has been measured to be Q=f0 /∆f3dB ¿600, consistent with numerical
simulations.
9
incident field. The associated magnetic field pattern from the SRR is dipolar.
By having splits in the rings, the SRR unit can be made resonant at wavelengths
much larger than the diameter of the rings; that is, there is no half-wavelength
requirement for resonance, as would be the case if the rings were closed. The
purpose of the second split ring, inside and whose split is oriented opposite to the
first, is to generate a large capacitance in the small gap region between the rings,
lowering the resonant frequency considerably and concentrating the electric field.
The individual SRR shown in Fig. 2.1 (inset) has its resonance peak at 4.845
GHz. The corresponding resonance curve is shown in Fig. 2.1. Because the
dimensions of the units are so much smaller than the free space wavelength, the
radiative losses are small, and the Q’s are relatively large.
By combining the split ring resonators into a periodic medium such that
there is strong (magnetic) coupling between the resonators, unique properties
emerge from the composite. In particular, because these resonators respond to
the incident magnetic field, the medium can be viewed as having an effective
permeability, µef f (ω). The general form of the permeability has been studied by
Pendry et al.[6], where the following expression was derived:
µef f = 1 −
F ω2
πr2 /a2
=
1
−
1 − 3l/π 2 µ0 ω 2 Cr3 + i (2lρ/ωrµ0 )
ω 2 − ω02 + iωΓ
(2.1)
Here ρ is the resistance per unit length of the rings measured around the circumference, ω is the frequency of incident radiation, l is the distance between
10
layers, a is the lattice parameter, r is defined in Fig. 2.1 , F is the fractional
area of the unit cell occupied by the interior of the split ring, Γ is the dissipation
factor, and C is the capacitance associated with the gaps between the rings. The
expressions for ω0 and Γ can be found by comparing the terms in Eq. 2.1.
While the expression for the capacitance of the SRR may be complicated
in the actual structure, the general form of the resonant permeability shown in
Eq. 2.1 leads to a generic dispersion curve, such as that shown in Fig. 2.2(a).
There is a region of propagation from zero frequency up to a lower band edge,
followed by a gap, and then an upper passband. We note, however, that there is
a symmetry between the dielectric and permeability functions in the dispersion
relation ω = ck/ (ω)µ(ω), where c is the velocity of light in vacuum. The gap
corresponds to a region where either ef f (ω) or µef f (ω) is negative. If we assume
that there is a resonance in µef f (ω) as suggested by Eq. 2.1 and that ef f (ω)
is positive and slowly varying, the presence of a gap in the dispersion relation
implies a region of negative µef f (ω). One cannot uniquely determine via only a
simple measurement, or even the measurement of the dispersion relation itself,
whether the gap is due to a resonance in the ef f (ω) with constant µef f (ω) or
due to a resonance in µef f (ω) with constant ef f (ω), which we have indicated is
the case. We discuss below the method we use to demonstrate that there is a
region of negative µef f (ω).
Using MAFIA[11], a commercial electromagnetic mode solver, dispersion
11
curves were generated for the periodic infinite metallic structure consisting of the
split ring resonators. There are two incident polarizations of interest: magnetic
field polarized along the split ring axes (H|| , Fig. 2.2(a), inset), and perpendicular to the split ring axes (H⊥ , Fig. 2.2(b), inset). In both cases, the electric field
is in the plane of the rings. As shown by the curves in Figs. 2.2(a) and 2.2(b), a
band gap is found in either case, although we will show that we can interpret the
H|| gap of Fig. 2.2(a) as being due to negative µef f (ω) and the H⊥ gap of Fig.
2.2(b) as being due to negative ef f (ω). The negative permeability region for the
H|| modes begins at 4.2 GHz and ends at 4.6 GHz, spanning a band of about
400 MHz. Not evident from the figure, but consistent with the model indicated
in Eq. 2.1, µef f (ω) switches to a large negative value at the lower band edge,
decreasing in magnitude (but still negative) for increasing frequency through the
gap. At the upper band edge, µef f (ω) = 0, and a longitudinal mode exists (not
shown), identified as the magnetic plasmon mode by Pendry et al.[6] For the
dielectric gap shown in Fig. 2.2(b), the same behavior is observed, but with the
roles of ef f (ω) and µef f (ω) reversed.
Short of performing an intricate reflection and transmission measurement to
fit the material constants, there is no simple means to deduce whether the split
ring medium is responding electrically or magnetically. We can, however, use a
physical approach and alter the dielectric function of the surrounding medium,
creating scattering properties that can distinguish whether the band gaps are
12
Figure 2.2: (a) Dispersion curve for the parallel polarization H|| . The lines with the
solid circles correspond to the split ring resonators only. The inset shows the orientation
of the split ring with respect to the incident radiation. The horizontal axis is the phase
advance per unit cell, or kd , where k is the wave number. (b) Dispersion curve
for the perpendicular polarization H⊥ . The lines with the solid circles correspond to
the split ring resonators only. The inset shows the orientation of the split ring with
respect to the incident radiation. (c) Expanded view of the dispersion curve shown
in (a). The dashed line corresponds to the split ring resonators with wires placed
uniformly between split rings. (d) Expanded view of the dispersion curve shown in (b).
The dashed line corresponds to the split ring resonators with wires placed uniformly
between split rings. The insets to (c) and (d) show the orientations of the split rings
with respect to the wires.
13
due to either the µef f (ω) or ef f (ω) of the SRR being negative.
In a 2D medium composed of periodically placed conducting straight wires,
there is a single gap in propagation up to a cutoff frequency, ωp , for modes with
the electric field polarized along the axis of the wires.[1, 2, 3, 4, 5] This onset
of propagation has been identified with an effective plasma frequency dependent
on the wire radius and spacing, with the effective dielectric function following
the form ef f (ω) = 1 − ωp2 /ω 2 . A reduction in ωp can be achieved by restricting
the current density to the thin wires, which also increases the self-inductance per
unit length, L.[3, 4, 5] When the conductivity of the wires is large, the plasma
−1/2
frequency has been shown to have the general form ωp = (d2 L0 )
, and thin
wire structures can easily be made to have ωp at microwave or lower frequencies.
Combining the SRR medium having a frequency band gap due to a negative
permeability with a thin wire medium produces a resultant left-handed material
in the region where both µef f (ω) and ef f (ω) have negative values.
Numerical simulations were carried out in which parallel wires of radius 0.8
mm were added in between the split rings, in a direction parallel to the incident
electric field as shown in the inset of Fig. 2.2(c). The results of these simulations
are shown as dashed lines in Figs. 2.2(c) and 2.2(d). For the wires alone, a
gap extends from zero frequency to ωp , at 13 GHz. When wires are added
symmetrically between the split rings, for the H|| case a passband occurs within
the previously forbidden band of the split ring dispersion curves of Fig. 2.2(a).
14
That this passband (the dashed line in Fig. 2.2(c)) occurs within a previously
forbidden region indicates that the negative ef f (ω) for this region has combined
with the negative µef f (ω) to allow propagation, as predicted.
By combining the ideal frequency dependence for the wire medium[3] with
Eq. 2.1 for the permeability of split rings, we can derive the following expression
for the dispersion relation of the combined medium:
ω 2 − ωp2 (ω 2 − ωb2 )
k =
C2
(ω 2 − ω02 )
2
(2.2)
This equation shows that the range of the propagation band (k real) extends
√
from ω0 to ωb = ω0 / 1 − F . This was formerly the region of the gap of the
SRR structure in the absence of the wires. Note that the dispersion relation
leads to a band with negative group velocity everywhere, and a bandwidth that
is independent of the plasma frequency (provided ωp > ωb ).
The behavior of the magnetic gap can be contrasted with that occurring
for the H⊥ case, which we have identified as a dielectric gap. Because H is
parallel to the plane of the SRR, we assume magnetic effects are small, and
that µef f (ω) is small, positive, and slowly varying. As shown in Fig. 2.2(d),
a passband again occurs, but now outside of the forbidden region, and within
a narrow range that ends abruptly at the band edge of the lowest propagation
band. The passband in this case occurs where the effective dielectric function
of the split rings exceeds the negative dielectric function of the wire medium.
As the dispersion curves calculated do not include losses, there will always be
15
a range of passband frequencies, however narrow, when the resonant dielectric
medium of split rings is combined with the negative dielectric medium of wires.
Once again, we can describe the behavior of the dielectric gap by an approximate
dispersion relation,
2
2
2
2
ω
−
ω
ω
−
ω
f
p
k2 (2.3)
C2
(ω 2 − ω02 )
where ωf2 = ω02 ωp2 / ω02 + ωp2 . In deriving Eq. 2.3, we have neglected the differ
ence between ω0 and ωb , as ωb does not play an essential role here, and assumed
ωp ω0 . The propagation band in this case extends from ωf to ω0 , with a bandwidth strongly dependent on the plasma frequency. As the plasma frequency
is lowered, the lower edge of the propagation band lowers, increasing the overall bandwidth. The group velocity of this band is always positive. Both Eqs.
2.2 and 2.3 neglect material losses (i.e., Γ = 0). The contrast between the two
propagation bands in the H|| and H⊥ cases illustrates the difference between the
magnetic and dielectric responses of the split ring medium.
The SRRs such as that shown in Fig. 2.1 (inset) were fabricated using a
commercially available printed circuit board. In order to test the results of the
simulations, square arrays of SRRs were constructed with a lattice spacing of 8.0
mm between elements. The resonant mode of an individual element resembles a
magnetic dipole, with electric fields strongly confined to the gap region between
the rings, and the magnetic field circulating around the rings as in a wire loop.
As the magnetic flux generated by the SRR is required to return within the unit
16
Figure 2.3: A transmission experiment for the case of H|| . The upper curve (solid line)
is that of the SRR array with lattice parameter a=8.0 mm. By adding wires uniformly
between split rings, a passband occurs where µ and are both negative (dashed curve).
The transmitted power of the wires alone is coincident with that of the instrumental
noise floor (-52 dB).
cell, the fractional area F is the critical parameter for the enhancement of the
permeability.
Microwave scattering experiments were performed on the fabricated SRR
medium, and the combined SRR/metal wire medium. In order to ease the required size of the structure, we utilized a two-dimensional microwave scattering
chamber, discussed previously in detail.[12] The scattering chamber is made out
of aluminum, with a grid pattern of holes in the top plate to allow source and
probe antenna coupling. Microwave absorber material placed around the periphery of the chamber minimized reflection back into the scattering region.
For the H|| polarization we constructed 17 rows of split rings in the H direction
(eight elements deep in the propagation direction), oriented as in Fig. 2.2(a),
17
inset. Figure 2.3 shows the results of transmission experiments on split rings
alone (solid curve), and split rings with wires placed uniformly between (dashed
curve). The square array of metal wires alone had a plasma frequency of 12
GHz; the region of negative below this frequency attenuated the transmitted
power to below the noise floor of the microwave detector (-52 dBm). When split
rings were added to the wire array, a passband occurred, consistent with the
propagation region indicated by the simulation (Fig. 2.2(c)).
We have demonstrated both by numerical simulation and experiment that
an effective medium of only conducting elements responds predominantly to the
magnetic field of incident electromagnetic fields. Remarkably, there is a band of
frequencies for which µef f (ω) can be negative, here manifested as a region of attenuation in scattering from a finite section of material. We have combined this
material with a negative ef f (ω) material to form a “left-handed” medium, forming a propagation band with negative group velocity where previously there was
only attenuation. We are now in a position to further investigate the fascinating
electrodynamic effects anticipated for such composite metamaterials.
18
The text of this chapter is, in part, a reprint of the material as it appears
in Phys. Rev. Lett 84, 4184 (2000). The dissertation author was the primary
researcher and author in this publication.
Chapter 3
Constant Effective Mass Mott
Insulator Transition in the
Cuprates
We have investigated the hole dynamics in two prototypical doped Mott-Hubbard
(MH) systems: La2−x Srx CuO4 and YBa2 Cu3 Oy using a combination of DC transport and infrared spectroscopy. By exploring the effective spectral weight data
obtained with optics in conjunction with the DC Hall results we find that the
Mott transition in these systems to be of the ”vanishing carrier number” type as
we observe no mass divergence as one proceeds to undoped phases.
For over half a century a significant portion of the condensed matter community has been captivated by the exceptionally rich physics of doped MH insula-
19
20
tors. Unconventional universal patterns in heavily doped MH systems, discovered with transport and spectroscopic probes[13], encapsulate a notion of ”non
Fermi liquid” which epitomizes the inabilities of standard solid-state theories to
describe both spin and charge excitations in this class of solids. Some exceptionally intriguing MH insulators, the Cuprates, display superconductivity at high
temperatures when sufficiently doped. To elucidate the high-Tc superconductivity, it is imperative to understand the nature of electronic excitations of doped
MH phases. This calls for the systematic studies of the dynamical characteristics of conducting carriers in the proximity to the “Mott transition” from the
undoped insulator to a non Fermi liquid conductor.
Here we present a study of the in-plane optical and transport properties in
two prototypical doped MH systems: La2−x Srx CuO4 (LSCO) and YBa2 Cu3 Oy
(YBCO). This work encompasses the underdoped regime of the phase diagram
from long range antiferromagnetically (AF) ordered insulators, to high-Tc superconducting compounds. The optical measurements were carried out from 10
cm−1 to 50,000 cm−1 in near normal reflectance geometry. Optical constants: the
complex conductivity σ1 (ω) + iσ2 (ω) and the dielectric function ε1 (ω) + iε2 (ω)
were obtained by performing the Kramers-Kronig transformation of the raw data
and were found to be consistent with direct ellipsometric measurements in the
visible range. Both the LSCO and YBCO samples are high quality single crystals,
the preparation of which was described previously in detail.[14]
21
We start by analyzing the absorption features found directly in the AC response of all weakly doped single crystals. It has been established that the lowest
electronic excitation in un-doped crystals is associated with a charge transfer gap
at 12000cm−1 = 1.5eV . The data presented in Fig. 3.1 reveals the development of two distinct features in the intra-gap response upon carrier doping, in
accord with the earlier results.[15, 16] We represent the dissipative portion of the
conductivity composed of two types of contributions as,
σ1 (ω) =
2
ωpD
γD
γM ω 2
2
+ ωpM 2
2
2
ω 2 + γD
(ωM − ω 2 )2 + γM
ω2
(3.1)
where ωp is the plasma frequency for each term, ωM is the center frequency of the
MIR term, γD and γM are the Drude and MIR damping terms respectively. This
first term in Eq. 3.1 provides an accurate description of the conductivity data
below 600 cm−1 displayed in red. A subtraction of the Drude contribution from
the experimental data yields the dashed lines in Fig. 3.1. The general character
of this latter contribution is described by a broad Lorentzian (2nd term in Eq.
3.1) where ω0 is the center frequency, γM the scattering rate, and ωM p is the
center frequency, (we adopt a color scheme, used throughout, in which red colors
signify a ”Drude-like” term[17] and blue an mid infrared term).
The two channels are evident in the data over much of the phase diagram
<
from symmetry broken AF-ordered samples (x ≤ 0.02; y ∼ 6.4) to under-doped
superconducting crystals (0.08 ≤ x ≤ 0.125; 6.4 < y ≤ 6.65). Characteristic of
the lightly doped group of samples is the observation of two distinct peaks in
22
-1
-1
Frequency (cm )
0
2000
4000
Frequency (cm )
0
2000
4000
6000
2000
2000
6.65
0.125
1500
1500
1000
1000
−1
0
0
0.06
6.43
1000
1000
500
500
0.03
6.4
0.01
6.3
0
600
300
0
400
200
0
0
La 2-xSr xCuO 4
YBa 2Cu 3O y
0.4
0.4
6.65
12.5%
0.3
0.5
0.3
0.2
6.43
6%
0.1
3%
6.3
0.2
0.1
Im(-1/ε)
Im(-1/ε)
0.5
−1
0
600
300
0
400
200
−1
σ1(ω) Ω cm
500
σ1(ω) Ω cm
−1
500
0.0
0.0
0
4000
8000
-1
Frequency (cm )
0
4000
8000
-1
Frequency (cm )
Figure 3.1: Spectra of the optical conductivity and of the loss function probed for
La2−x Srx CuO4 and YBa2 Cu3 Oy single crystals. The YBCO data were obtained in
the polarization E||a and therefore represent the genuine response of the CuO2 planes
without a contribution of the Cu-O chains running along the b-axis. All spectra are at
10K except for results generated at T∼Tc for superconducting crystals. The response of
weakly doped materials (x ≤ 0.06; y=6.3-6.43) are multi-component and are comprised
of a ”Drude-like” contribution at low energies (red segments) followed by a mid-IR
resonance (blue segments). As doping increases the two contributions merge and the
separate absorption channels can no longer identified. For σ1 (ω) fits are shown for
the low energies and MIR terms. The green area is σ1 (ω) with the ”Drude-like” term
subtracted and thus represents other MIR contributions.
23
the loss function spectra Im(−1/ε): one associated with the Drude-like coherent
response, and the other one directly attributable to a mid-IR resonance at higher
energies. As carrier density increases beyond a critical value the two components
of the AC response merge and can no longer be unambiguously distinguished
either in σ1 (ω) or in the Im(−1/ε) spectra.
The evolution of σ1 (ω) and Im(−1/ε) is, as mentioned, generic however the
implications of these effects for transport properties have remained obscured.
New data (resistivity and Hall) measured to 1000 K (Fig. 3.2) uncover the pivotal role of multiple components seen in optics. Examining the temperature dependence of the DC resistivity ρ(T ) for the weakly doped x=0.03 La2−x Srx CuO4
crystal, (black curves in the middle panel of Fig. 3.2, reveals 3 characteristic
regimes: (i) localized behavior for T<80K where ρ(T ) shows a sharp increase
as T→0; (ii) metallic behavior in the range 80<T<300K where dρ(T )/dT > 0;
(iii) and finally at T>300K, a crossover in the curvature of ρ(T) from positive
to negative, thus signaling an onset to resistivity saturation. This trend is even
more obvious if the data set is corrected for thermal expansion of the crystal
(dashed line).[18] All three regimes are also clearly visible in the spectrum of the
AC resistivity defined as ρ(ω) = 1/σ1 (ω, T = 13K) displayed in the middle panel
of Fig. 3.2 as the red and blue solid curve. An agreement between AC and DC results is unmistakable both in terms of the temperature/frequency dependence of
the two data sets but also importantly in terms of the absolute values of the two
24
resistivities.[19] One interpretation of the saturation observed in ρ(T ) and ρ(ω) is
that additional conductivity channels become accessible at higher energies thus
preventing both DC and AC resistivity from continuous growth with increasing
temperature or frequency.[20] Interestingly, the effect of resistivity saturation
is also realized in all weakly doped and moderately doped crystals (x<0.08).[21]
This has to be contrasted with the nearly linear dependence of ρDC (T ) extending
up to high temperatures in many optimally doped cuprate superconductors.[22]
Additional support for the multi-channel transport picture is provided by
the Hall data RH displayed in the top panel of Fig. 3.2. Comparing the Tdependences of ρDC and RH in Fig. 3.2 one notices that a crossover to saturated
resistivity occurs concomitantly with a drop in the Hall number. The Hall effect
in cuprates has been notoriously difficult to interpret in particular for the heavily
doped superconducting phases. Nevertheless the T dependence of the Hall number with two plateau regions, found in these weakly doped compounds, suggests
a surprisingly simple picture. At T<300K the magnitude of RH is independent
of temperature (apart from the region below 80 K governed by localization) and
yields approximately 0.035 holes/Cu. As T rises above 300 K the Hall coefficient
sharply drops and then begins to flatten out to a lower level corresponding to
much higher carrier density of approximately 0.22 holes/Cu. This suggests that
additional conduction channels become available at higher energies: a conjecture
directly supported by the form of the conductivity spectra as well as by the onset
25
La1.97Sr0.03CuO4
RH
0.035
0.055
10
0.220
Resistivity ( Ω cm)
0
0
0.008 0
250
250
750
750
1000
1000
T (K)
0.006
0.004
0.002
0.000
0
300
-1
900
(cm )
1.50
0.200
0.75
0.100
6
(10 Ω
-2
10000
-1
600
1000
0.00
10
100
meV
1000
holes / Cu
cm )
100
Neff(ω)
500
500
holes / Cu
0.000
20
0.000
Figure 3.2: Transport and spectroscopic data characterizing the response for the electric field within the CuO2 plane of 3% doped LSCO single crystal. Top panel: the temperature dependence of the Hall coefficient RH . Middle panel: the T -dependence of the
DC resistivity (black solid line) and spectra of the optical resistivity ρ(ω) = 1/σ1 (ω)
at 13K (red and blue). The solid black line shows ρ(T ) data at constant pressure; the
dashed line has been corrected for thermal expansion following ref. [18]. Bottom panel:
the frequency dependence of the effective spectral weight Nef f (ω) defined in the text.
The absolute value of Nef f (ω) is proportional to the number of carriers participating
in the optical absorption below ωc . In order to convert Nef f (ω) to a hole number
(holes/Cu) we used the effective masses plotted in Fig. 3.3. The red segments of the
spectra in the middle and bottom panels represent the low energy response while the
blue indicates the MIR terms.
26
LSCO 3%
Low Energy
(ωc=600 cm-1,T=300K)
Total
(ωc=1.1 eV, T=800K)
22
nH(Τ) (10
Neff(ωc) (106Ω-1cm-2)
cm )
n
Neff ~
m*
-3
m*
0.035 =
0.218 =
~ 4.1me =
0.220 =
1.403 =
~ 4.2me =
Low Energy
Total
~
1
6.3
~
1
6.4
Table 3.1: Middle left column shows nH (T ) both for the low energy ”Drude-like”
contribution and the ”total” intra-gap response. Middle right column displays Nef f (ω)
for the same two energies. Far right column shows m∗ as calculated from Nef f and
nH for both energy regimes. This legend is used in Fig. 3.3.
of saturation in ρ (T ). These extra conduction channels appear to deprive the DC
and AC resistivity of unrestricted growth with increasing T and ω respectively.
The multi-channel scenario we advocated here provides a consistent quantitative account of both AC and DC data. To show this we define the electronic
spectral weight Nef f (ω) =
∞
0
dω σ1 (ω ) proportional to the number of carri-
ers participating in absorption processes at energies below ω. The spectra of
Nef f (ω) display the ”availability” of conducting carriers for electronic transport. It is therefore instructive to analyze the energy dependence of Nef f in
conjunction with the T -dependence of the Hall data since the latter quantifies
the availability of holes as T progresses to higher values. The Nef f spectrum
plotted in the bottom frame of Fig. 3.2 (red and blue solid curve) shows that
the spectral weight associated with the coherent contribution to the conductivity
Nef f (600cm−1 ) amounts to approximately 1/6 of the total Nef f (1.1eV ) induced
by doping within the charge-transfer gap.[23] Remarkably, exactly the same fac-
27
tor of 1/6 is obtained from the ratio of carrier densities inferred from the Hall
data at 300 K and 800 K. The quantitative agreement between the two estimates conclusively supports the notion of the multi-band transport at least in
lightly doped high-Tc crystals. The doping trends of the nH and Nef f data are
displayed in Fig. 3.3. Inspecting these results one notices that both the low
and high energy ratios are close to 1/6 throughout most of the phase diagram.
Interestingly this is the region where two distinct components can be identified
in the AC response.
Excellent agreement between the AC and DC data emerging from the multicomponent analysis provides the rationale to extract the effective masses of conducting carriers from a combination of the Hall and optics data. According to
the oscillator strength sum rule, the magnitude of Nef f is given by the n/m∗
ratio where n is the carrier density. In the spirit of a multi-component picture,
an estimate of m∗ can be found in two complimentary ways: the low-T low-ω
regime, as well as the high-T high-ω range in which MIR conduction becomes
relevant (calculated for 3% LSCO in Table 3.1). The doping dependence of m∗
extracted thereby is plotted in the top panel of Fig. 3.3. Both approaches produce consistent results. Since YBCO high-T transport is inaccessible, only m∗
as calculated in the low energy regime is displayed.
The effective mass analysis unravels a number of unexpected trends. First,
both in LSCO and YBCO effective masses are only weakly dependent on doping
28
Doping y
0.00 0.04 0.08 0.12 0.16 0.20
10
8
6
4
2
0
La 2-xSr xCuO 4
1.2
0.4
2
6.6
6.8
7.0
10
8
6
4
2
0
1.2
1.0
0.8
4
0.6
0.4
2
0.2
nH(10 22cm -3)
4
0.6
6.4
6
N eff(10 6Ω-1cm -2)
0.8
6.2
YBa 2Cu 3O y
6
1.0
nH(10 22 cm -3)
6.0
m*/m e
m*/m e
Doping x
0.2
0.0
0
0.00 0.04 0.08 0.12 0.16 0.20
Doping x
0
6.0
6.2
6.4
6.6
6.8
0.0
7.0
Doping y
Figure 3.3: Bottom panels: carrier density extracted from the Hall data at 300K
(solid red circles) and at 800K (solid blue circles) for La2−x Srx CuO4 and YBa2 Cu3 Oy
systems. The effective spectral was obtained from integration of the conductivity with
a cut-off frequency of 600 cm−1 (open red squares) and 10000 cm−1 (open blue squares).
Top panels: the effective mass of conducting holes determined from the ratios detailed
in Table 3.1. Remarkably, both approaches produce m∗ values virtually independent
of doping over broad regions of the phase diagram and also agree with the m∗ values
inferred from the extended Drude analysis of the data obtained at higher dopings (open
black circles).
29
and show no noticeable enhancement even in immediate proximity to undoped
phases. This observation is at odds with a conventional picture of the Mott
transition in which it is expected to occur through the divergence of m∗ . The
particular type of Mott transition reported here occurs in the regime of vanishing
n and constant m∗ , and has been discussed theoretically[13] but to the best
of our knowledge has not been previously detected. Second, the lack of mass
enhancement in the AF-ordered phase (x ≤ 0.02) challenges the dominant role
of spin fluctuations in governing hole dynamics in cuprates. Indeed, one would
naturally expect a stronger impact of spin fluctuations on the renormalization of
m∗ near the AF boundary of the phase diagram. However, this latter expectation
is not validated by the data in Fig. 3.3, also putting into question the role of
spin fluctuations in mediating superconducting pairing. In fact, our results do
not reveal any unambiguous indications for m∗ renormalization over the band
mass preset by the electronic structure. Lower m∗ values consistently detected
in YBCO phases compared to that of LSCO compounds may be reflecting the
band structure effect as well. Obviously, stronger dispersion of the electronic
bands, and therefore smaller mass values, are expected to occur in YBCO which
has two CuO2 planes per unit cell provided the two parts of the bi-layer are
interacting.
Several other aspects of multi-component analysis are worthy of attention. As
pointed out above, in weakly doped phases the low energy contribution of fairly
30
light quasiparticles (m∗ = 2 − 4me ) is energetically separated from additional
channels with the latter becoming relevant to conductivity processes only beyond
a certain energy threshold. At doping levels higher than x > 0.1 − 0.12 in
LSCO and y > 6.6 in YBCO an obvious separation is no longer found in the
data. At these higher dopings the low-energy absorption is merged with the
mid-IR resonance. The absence of a well-defined ”barrier” between multiple
components causes a rather peculiar situation: as T or ω increase a continuously
escalating number of conduction channels becomes available for hole transport
in moderately doped phases. The ever-growing number of channels feeding the
conduction processes may account for the T dependence of RH and a much
slower drop of σ1 (ω) than expected by the power law prescribed by the Drude
model. The optical response near optimal doping is often parameterized within
the so-called ”single-component” model, permitting frequency dependence of the
scattering rate 1/τ and m∗ within the Drude formula.[24] Notably, the effective
masses extracted from this latter approach are fully consistent with those inferred
from the multi-component analysis described above, and depicted as the black
open circles in Fig. 3.3. This consistency supports the notion of a continuous
transition from a multi-channel to single-component description of the optical
constants.
It is prudent to inquire into the microscopic origins of the incoherent channel.
A detailed analysis of the electronic structure of doped Mott-Hubbard insulators
31
predicts several forms of bound states within the MH and/or charge transfer
gap.[25, 26] Inter-band transitions involving these states may give rise to the
observed effects. The role of the interband transitions in transport properties of
ordinary metals has been discussed by Allen[20] specifically in connection to ρ(T )
saturation. An important distinction of cuprates from simple metallic systems
in this context is related to the remarkably strong scattering associated with the
mid-IR contribution.[27] A multi-component response can be derived from real
space electronic inhomogeneities occurring, for example, due to embedding of
metallic particles in an insulating host.[28, 29] Near the percolation threshold
the optical conductivity of such a system shows a Drude response at low energies
followed by a flat nearly featureless background.[30] One difficulty in applying
this approach is that the multi-component response is observed over a large
fraction of the phase diagram with overall carrier density changing by more than
an order of magnitude.
In conclusion, by tracking the evolution of the multi-channel features through
the underdoped region of the phase diagram we have been able to gain valuable
insights into the nature of the Mott transition in undoped counterparts as well
as effects underlying the ”non-Fermi liquid” transport at moderate and high
doping levels. Several approaches to extract m∗ produce consistent results and
indicate that the transition to the Mott insulating state occurs with constant effective mass and vanishing carrier density. Surprisingly, despite a drastic change
32
in the carrier density, the masses appear to be fairly light even at dopings in
immediate proximity to the Mott insulating state. This suggests that the transport is governed by excitations topologically compatible with an antiferromagnetic background. Specific schemes permitting a constant effective mass Mott
insulator transition include spin-charge separation[31, 32] and electronic phase
separation.[33, 34] These scenarios propose explanations of high-Tc superconductivity which departs from standard BCS theory. The dynamical hole behavior
discussed here is common to both LSCO and YBCO thus unveiling the generic
properties of doped MH systems independent of the details of a particular host
material.
Chapter 4
THz Magnetic Response from
Artificial Materials
Artificial materials with tailored electric and magnetic response have been the
subject of a vast amount of interest recently, due in large part to the experimental
demonstration of negative refraction at microwave frequencies. Of particular
interest is the realization of a magnetic response in an inherently non-magnetic
material (i.e., no net spin component). We show here that magnetic response
at terahertz (THz) frequencies can in fact be achieved in a planar structure
composed of non-magnetic conductive scattering elements. The novel effect is
realized over a large bandwidth and is tunable throughout the THz frequency
regime. Due to the rarity of magnetic phenomena at higher frequencies, we
suggest that artificial magnetic structures, or hybrid structures that combine
33
34
natural and artificial magnetic materials, will play a key role in new THz devices.
The range of electromagnetic material response found in nature represents
only a small subset of that which is theoretically possible. To realize intriguing
properties not available in naturally occurring materials, artificially structured
materials have been developed over the years. A dramatic illustration of this
can be found in the recent experimental demonstrations of negative refractive
media at microwave frequencies,[35] for which both the electric permittivity and
the magnetic permeability must be simultaneously negative. The enabling factor for negative refraction was the realization that a metamaterial comprised of
nonmagnetic conductive resonant elements[6] could exhibit a strong response to
the magnetic component of an electromagnetic field.
Real materials which exhibit magnetic response are far less common in nature than materials that exhibit electric response, and particularly rare at THz
and optical frequencies. The reason for this imbalance is fundamental in origin:
there are no magnetic monopoles in nature and therefore magnetic polarization
in materials follows indirectly either from the flow of orbital currents, or from
the spin of the electron. Thus, materials that respond to magnetostatic fields
necessarily derive their properties from the spin — a quantum mechanical degree
of freedom — associated with the unpaired electrons of the constituent atoms or
molecules.
In fact materials in general rarely achieve resonant response at THz frequen-
35
cies. In condensed matter systems resonant phenomena usually occur in the
infrared frequency regime and higher such as: phonons, collective modes, inter and intra-band absorption, and so on. The energy of the response is set
by fundamental parameters like interatomic spacing and atomic forces, thus in
general materials universally do not display resonant behavior at THz frequencies. It is only very exotic materials that are able to achieve such a response in
the so called “THz Gap”.[36] This makes any THz resonant response, especially
magnetic, exceptionally intriguing.
Magnetic response of materials at THz and higher frequencies is particularly important for implementation of devices such as compact cavities, adaptive lenses, tunable mirrors, isolators and converters. Although rare, a few natural magnetic materials that respond above microwave frequencies have been reported. For example certain ferromagnetic and antiferromagnetic systems exhibit
a magnetic response over a frequency range of a several hundred gigahertz[37, 38,
39] and even higher[9, 40]. But the magnetic effects in these natural materials
are typically weak and often exhibit narrow bands[41] which hinder applications
for THz devices. The realization of magnetism at THz and higher frequencies
will significantly impact THz optics and their applications.[36]
From a classical point of view, a magnetic moment can be viewed as being
generated by a microscopic current flowing in a circular path. Such solenoidal
currents can be induced, for example, by a time-varying magnetic field. While
36
H
-E
K
30
O
Figure 4.1: Illustration depicting the orientation of the 30 degree ellipsometry experiment. The polarization shown is S-polarization, or transverse electric (TE), for
excitation of the magnetic response and P-polarization, transverse magnetic (TM), is
also measured. Image shown in the inset is a secondary ion image of sample D1 taken
by focused ion beam microscopy.
this magnetic response is typically weak, the introduction of a resonance into
the effective circuit about which the current flows can dramatically enhance the
response. Resonant solenoidal circuits were proposed early on as the basis for
artificially structured magnetic materials,[42] although primarily intended for
lower radio-frequency applications.
With recent advances in metamaterials it has become possible to construct a
system with desired electric[1, 2] or magnetic properties or even both.[35],[43, 44,
45] Although these materials have generated a vast amount of interest they have
only been demonstrated in the microwave regime and have yet to be pursued at
higher frequencies.
Metamaterials promise to push the limits of magnetism, as they can be designed to work at arbitrarily high frequencies, can attain negative values of the
37
magnetic permeability (a feat not observed in natural materials), are broad band,
tunable, and both surface and bulk modes can exist.[46] A particular interesting
metamaterial comprised of nonmagnetic conducting units, called split ring resonators (SRR), (inset to Fig. 4.3), was proposed in 1999 by J.B. Pendry.[6] The
unit consists of two concentric annuluses of conducting material, each with a gap
situated oppositely. The gap allows the structure to be resonant at wavelengths
much larger than its physical dimensions and the combination of many SRRs
into a periodic array allow the material to behave as a medium with an effective
magnetic permeability µef f (ω). The origin of the effective permeability enhancement stems from the coupling between the structure’s internal inductance and
capacitance, which reduces the impedance necessary to generate a resonance.
The effective permeability can be expressed in the form,[47]
µef f (ω) = 1 −
F ω2
= µef f (ω) + iµef f (ω)
ω 2 − ω02 + iΓω
(4.1)
where F is a geometrical factor, ω0 is the resonant frequency and Γ the resistive loss in the resonating SRR. In the quasi-static limit, the qualitative picture
of this magnetic response is straightforward: the external magnetic field with a
varying flux normal to the metallic loop will induce a current flow, which in turn
results in a local magnetic dipole moment. Below the resonance frequency ω0 ,
the strength of magnetic dipole increases with frequency, and this dipole stays
in-phase with the excitation field, i.e. a paramagnetic response. As the frequency
38
of the incident field further increases, the currents generated in the loop can no
longer keep up with the external field, and they begin to lag. At frequencies
above ω0 , the induced dipole moment continues to lag more and more until it
is completely out-of-phase with the excitation field, which results in a magnetic
permeability smaller than unity, i.e. diamagnetic response, and even with values
less than zero. In contrast to conventional ferromagnetism, the magnetic activity
associated with these conductive elements is completely devoid of any permanent
magnetic moment.
The dimensions of the SRR can be calculated in order to obtain resonant
behavior in the THz range from Eq.4.1. We have designed and constructed
three different SRR samples on a 400 µm quartz substrate. The SRRs are made
from copper and are 3 µm thick. The periodicity of the SRR is typically subwavelength (λ/7 in our samples) which allows the composite to behave as an effective medium to external THz radiation (λ= 300 µm= 1 THz). The SRRs were
fabricated using a unique self-aligned microfabrication technique called photoproliferate-process (PPP).[48] This fabrication approach allows a well defined
shape with sharp edges, and a very high filling density with high yield. The inset
to Fig. 4.1 shows a focused ion beam (FIB) image of a magnetic metamaterial
with elements consisting of SRRs.
Most reported work on microwave metamaterials have focused on characterizing bulk one and two-dimensional structures, where waveguide configurations
39
Frequency (THz)
tan-2(Ψ) = |rs/rp|2
0.6
1.6
0.8
1.0
1.2
1.4
1.6
1.8
1.4
1.2
tan-2(Ψ) = |rs/rp|2
1.0
1.4
1.2
1.0
20
30
40
50
-1
60
Frequency (cm )
Figure 4.2: The top panel shows the ratio of the magnetic to electric response (described in the text) for three different artificial magnetic structures D1, D2, and D3,
(shown as the red, black and blue solid curves respectively) in the terahertz frequency
range. Theoretical magnetic response as determined by simulation for each SRR, as
described in the text, is displayed in the bottom panel.
40
are frequently used. However in the THz frequency region we gain from the convergence of beams, and geometrical optics become more feasible. This allows us
to study a monolayer of SRRs in free space. The measurements presented here
were performed at UCSD utilizing ellipsometry at oblique incidence. A setup
based on a Michelson interferometer adapted for S-polarized (Fig. 4.1) and Ppolarized light from 0.6 THz to 1.8 THz was used for the measurements with a
1.6 K silicon bolometer as the detector. The sample is placed within an evacuated compartment and light from a mercury arc lamp source is focused on the
substrate at an angle of 30 degrees from the surface normal. This geometry was
chosen because it permits the total dimensions of the array to remain small while
still allowing coupling to the magnetic resonance with the incident radiation.
The results of the frequency dependent ellipsometry measurements are presented in Fig. 4.2. The parameter plotted tan−2 Ψ represents the inverse absolute
square of the ellipsometric parameter ρ (ω) = tan (Ψ) exp (i∆) which displays the
reflectance ratio of two polarizations. The SRRs are expected to respond magnetically when the magnetic field penetrates the rings, i.e. S-polarization, (depicted
in Fig. 4.1) and no magnetic response when the magnetic field is parallel to the
plane of the SRR, P-polarization. As a result, the reflectance ratios presented in
Fig. 4.2 are the natural function to use since this parameter provides the ratio
of the magnetic to electric response from the SRRs. As shown in Fig. 4.2, the
solid red curve (sample D1) exhibits a peak, centered at approximately 1.25 THz
41
Frequency (THz)
|rs/rp|
2
0.6
1.6
0.8
1.0
1.2
1.4
1.6
1.8
1.4
1.2
µ 1(ω)
1.0
4
2
0
µ ω
-2
G
4
W
2
0
0.6
L
0.8
1.0
1.2
1.4
1.6
1.8
Frequency (THz)
Figure 4.3: Ratio of the magnetic to electric response repeated in the top panel from
Fig. 4.2. Bottom panel shows the real (µ1 ) and imaginary (µ2 ) magnetic permeability
functions as simulated by HFSS for samples D1 (red), D2 (black), and D3 (blue). Inset
to the bottom panel depicts the dimensions of an individual SRR.
in the spectrum. The feature peaks above a frequency independent background
and the resonance in the reflectance is broad, nearly 30% bandwidth of its center
frequency.
42
Sample G (µm) W (µm) L (µm) P (µm)
D1
2
4
26
36
D2
3
4
32
44
D3
3
6
36
50
Table 4.1: Dimensions corresponding to the samples characterized in this study (see
the inset to Fig. 4.3 ). The four geometrical parameters describing the SRRs are: the
gap between the inner and outer ring (G), the width of the metal lines (W), the length
of the outer ring (L), and the lattice parameter (P).
If indeed the magnetic response seen in Fig. 4.2 is due to the constituent
SRRs, then we would expect the resonance to scale with dimensions in accordance
with Maxwell’s equations. In order to elucidate our findings, two more SRRs
with different dimensions have been characterized. The dimensions of all three
SRRs are listed in Table 4.1 and depicted in the inset to Fig. 4.3. These SRRs
exhibit a similar magnetic mode and their resonant frequencies occur between
0.8-1.0 THz. We find a monotonic red-shifting of resonant frequencies as the
dimensions of SRRs are scaled up. Additionally, this magnetic response has a
large bandwidth which can be tuned by adjusting the parameters of the SRR
element.
As a further verification that the peaks displayed in Fig. 4.2 are due to the
magnetic response of the SRRs, we perform a numerical simulation using HFSS,
a commercial electromagnetic mode solver. Dispersion curves were calculated
for the periodic infinite metallic structure corresponding to D1, D2, and D3 all
upon a quartz substrate. The electrical conductivity used for the copper elements
43
was σ=5.8 x 107 S/m. The magnetic permeability µ (ω) can be extracted from
the dispersion curve simulation results[49] and used to calculate the expected
theoretical reflectance ratio tan−2 Ψ (bottom panel of Fig. 4.2).[50] The results
are shown for each sample characterized and are displayed in the bottom panel of
Fig. 4.2. Good agreement between the results of the simulation and experimental
curves is evident. In Fig. 4.3 we display the simulated real and imaginary
portions of the magnetic permeability corresponding to D1, D2, and D3. For
sample D1 the imaginary permeability peaks at approximately 1.15 THz which
corresponds well with the onset of the peak located near 1.25 THz in tan−2 Ψ.
The difference in peak location is to be expected since tan−2 Ψ consists of the
ratios of absolute values, i.e. the width observed is dependent on the strength
of the oscillator. Thus it is important when considering tan−2 Ψ and µ (ω), to
compare the onset of the resonances.
In conclusion, we have demonstrated a strong magnetic response at THz frequencies from a non-magnetic, conducting microfabricated metamaterial. The
THz magnetic effect is broad band and tunable by scaling the SRR dimensions.
The scalability of these magnetic metamaterials throughout the THz range and
potentially into optical frequencies promises many exciting applications such as
biological[51] and security imaging, biomolecular fingerprinting, remote sensing
and guidance in zero visibility weather conditions. Additionally the effect is
nearly an order of magnitude larger than that obtained from natural magnetic
44
materials.[52] Structures with a negative magnetic response, when combined with
plasmonic wires that exhibit negative electrical permittivity[53, 54, 3, 4], have
the potential to produce a negative refractive index material at these very high
frequencies, enabling the realization of devices that are needed to fill the gap of
the terahertz regime. These new materials extend the frequency range of metamaterials by over two orders of magnitude from previous microwave frequency
results. This novel result sets a new upper frequency bound to the extension of
artificial magnetic and natural magnetic materials alike.
45
The text of this chapter is, in part, a reprint of the material as it appears
in Science, to be published (2004). The dissertation author was the primary
researcher and author in this publication.
Chapter 5
Searching for the Slater
Transition in the Pyrochlore
Cd2Os2O7 with Infrared
Spectroscopy
Infrared reflectance measurements were made on the single crystal pyrochlore
Cd2 Os2 O7 in order to examine the transformations of the electronic structure and
crystal lattice across the boundary of the metal insulator transition at TM IT =
226K. All predicted IR active phonons are observed in the conductivity over all
temperatures and the oscillator strength is found to be temperature independent.
These results indicate that charge ordering plays only a minor role in the MIT
46
47
and that the transition is strictly electronic in nature. The conductivity shows
the clear opening of a gap with 2∆ = 5.2kB TM IT . The gap opens continuously,
with a temperature dependence similar to that of BCS superconductors, and the
gap edge having a distinct σ(ω) ∼ ω 1/2 dependence. All of these observables
support the suggestion of a Slater transition in Cd2 Os2 O7 .
Mott’s classic paper[55] published over half a century ago has triggered extensive research on correlated electron systems which undergo a metal insulator
transition (MIT). Both Mott and Hubbard[56] have suggested that for systems
at half filling, the Coulomb repulsion between electrons could split the band, thus
producing an insulator. Alternatively, Slater in 1951 suggested that antiferromagnetic order alone could produce an insulator by a doubling of the magnetic
unit cell[57]. While there are numerous examples of Slater / spin-density-wave
(SDW) insulators in the realm of 1-dimensional (1D) conductors[58, 59], the
experimental situation at higher dimensions is less clear. Impossibility of the
Slater state in the 2D regime has been recently argued based on dynamical cluster approximation calculations[60]. For 3D solids, spin ordering alone usually
generates an energy gap corrupting only a fraction of the Fermi surface so that
metallic conductivity persists[61]. In this context the metal-insulator transition
in the 3D pyrochlore Cd2 Os2 O7 is exceptionally intriguing since transport and
magnetic properties across the MIT boundary appear to be in accord with the
Slater mechanism[62]. In this paper we report on the first spectroscopic studies
48
of the MIT in Cd2 Os2 O7 . Our analysis of the infrared data reveals that the
transition into the insulating state is driven solely by the electronic interactions
without significant involvement of the crystal lattice. We discuss new facets of
the spin-driven insulating state in a 3D material.
The pyrochlore Cd2 Os2 O7 was first characterized by Sleight et al.[63]. The
metal-insulator transition in the resistivity has been found to occur at the same
temperature 226 K as the antiferromagnetic transition in susceptibility measurements. No evidence of concurrent structural changes were detected through
X-ray diffraction (XRD) analysis. Thorough examination of transport and magnetism in Cd2 Os2 O7 has been recently reported by Mandrus et al.[62] with a
Slater picture delivering a coherent interpretation of all experimental data. This
particular mechanism in Cd2 Os2 O7 may be favored by the fact that Os5+ is in
the 5d3 configuration so that t2g band is near half filling.
The Slater transition is characterized by several hallmarks in the frequency
domain which so far remained unexplored in Cd2 Os2 O7 . Among them, the specific temperature dependence of the energy gap as well as the frequency dependence of the dissipative response at energies above the gap edge that are distinct
from the Mott-Hubbard case[64]. Also the analysis of the IR active phonon
modes allows one to verify if a development of charge ordering is concomitant
with spin ordering. Despite the fact that IR spectroscopy is ideally suited for
directly studying the nature of the MIT in solids, the extremely small size of
49
!
!
ω
Figure 5.1: Reflectance of single crystal Cd2 Os2 O7 at various temperatures from 40
cm−1 to 1000 cm−1 . Inset shows the entire energy range characterized for room temperature and 25K on a log scale.
Cd2 Os2 O7 single crystals so far has rendered these measurements impossible.
Spectroscopic tools available in our lab at UCSD are tailored for investigations
of microcrystals,[65] allowing us to fill voids in the experimental picture of the
insulating state in Cd2 Os2 O7 .
The near normal reflectance R(ω) of Cd2 Os2 O7 was measured from 40 cm−1
to 14000 cm−1 using a Fourier transform spectrometer and from 12000 cm−1
to 35000 cm−1 using a grating monochromator. A test involving polarized light
displayed no signs of anisotropy, thus unpolarized radiation was used for a de-
50
tailed study of the temperature dependence. Samples were coated in situ with
gold or aluminum and spectra measured from the coated surface were used as a
reference. This method, discussed previously in detail[66], allows one to reliably
obtain the absolute value of the reflectance by minimizing the errors associated
with non-specular reflection and small sample size. We inferred the complex
conductivity σ1 (ω) + iσ2 (ω) by use of Kramers-Kronig (KK) analysis after extrapolating data to ω → 0 and ω → ∞. Various low frequency extrapolations
(Drude, Hagen-Rubens) were used; however the data did not significantly depend
on the particular method of low-ω extrapolation. We employed the usual ω −4
dependence for the high-energy extension of the data[67].
Fig.5.1 shows the infrared reflectance taken between 25 K and room temperature from an unpolished facet of Cd2 Os2 O7 crystal with dimensions less
than 0.5x0.7 mm2 . The reflectance at room temperature is high and is metallic
in nature. R(ω) decreases with decreasing temperature in the infrared region.
Although the reflectance decreases monotonically, it changes little near room
temperature and at low temperatures. Depression of R(ω) is most significant in
the vicinity of the MIT: between 225K and 150K. The absolute value of the reflectance for the 80K data is, within experimental accuracy, equivalent to that at
25K. As temperature is lowered the free electron screening is gradually reduced
unveiling several strong phonons in the infrared region. Notably, all phonons are
still visible in the room temperature data. In the inset to Fig.5.1 we plot the
51
!
σ1 Ω
" # $ "
Figure 5.2: Real part of the optical conductivity as obtained by Kramers-Kronig analysis on a log scale. A gap can be seen to develop continuously as the temperature is
reduced. The reduction in spectral weight occurring in the infrared region is compensated by a shift to higher energies. The inset shows the hardening Real part of the
optical conductivity as obtained by Kramers-Kronig analysis on a log scale. A gap
can be seen to develop continuously as the temperature is reduced. The reduction in
spectral weight occurring in the infrared region is compensated by a shift to higher
energies. The inset shows the hardening of the 347 cm−1 phonon at low T.
reflectance over the entire range characterized for room temperature and 25K.
Fig.5.2 shows the real part of the conductivity σ1 (ω) at different temperatures.
The room temperature spectrum can be adequately described with a simple
Drude model σ1 (ω) = σ0 /(1 + ω 2 τ 2 ) at least for ω < 1000 cm−1 where σ0 is
the DC conductivity and τ is the relaxation time. This behavior is followed by
a somewhat slower decrease than that prescribed by the Drude form, and an
52
interband feature at 22000 cm−1 . Several strong phonons are visible in the IR
region. As temperature is lowered one witnesses the continuous opening of a
gap with a drastic reduction of σ1 (ω) in the infrared. In the 25 K spectrum the
intragap conductivity is frequency independent (apart from the sharp phonon
peaks). At ω above the gap edge the spectrum reveals the ω 1/2 dependence
expected for a Slater transition[64, 68]. An intersection between the two segments
at 818 cm−1 can be chosen as a quantitative measure of the energy gap 2∆. The
ω 1/2 behavior can be recognized at higher temperatures as well. In the latter
spectra (Fig.5.3) the intragap region is best described with a linear dependence.
The temperature dependence of the intersection between the σ1 (ω) ∝ ω and
σ1 (ω) ∝ ω 1/2 regions is plotted in the inset of Fig.5.3.
It is instructive to characterize the development of the energy gap in Cd2 Os2 O7
through the spectra of the effective spectral weight Nef f (ω) =
120
π
ω
0
σ1 (ω )dω .
The magnitude of Nef f (ω) depicted in Fig. 5.4, is proportional to the number of
carriers participating in the optical absorption up to a cutoff frequency ω, and
has the dimension of frequency squared. The significant reduction in spectral
weight occurring in the intragap region is transferred to the energy region above
3∆. Interestingly, the spectral weight does not become completely exhausted
until 16000 cm−1 implying that the energy range as broad as 40∆ 104kB TM IT
is involved in the metal insulator transition.
Important insights into the origin of the insulating state in Cd2 Os2 O7 may
53
"
!
%& #
$
$
$!
$
$
$
∆ / ∆%'
Figure 5.3: Infrared region of the real conductivity at T < TM IT . The expected
theoretical frequency dependence of the gap edge (σ1 (ω) ∼ ω 1/2 ) is shown for each
temperature as a dashed line. A linear fit for the region below the gap edge is also
shown. The intersection of the two fits can be taken as a measure of the energy gap
2∆. The inset shows the optical energy gap as determined by the method described
above (open circles), and the expected theoretical dependence (solid curve).
54
be reached through the analysis of the phonon spectra. The pyrochlore structure
belongs to the space group Fd3̄m and reveals seven IR active modes[69, 70]. We
observe phonon peaks at 86 cm−1 , 108 cm−1 , 200 cm−1 , 347 cm−1 , 371 cm−1 ,
440 cm−1 , and 615 cm−1 in the 25K spectrum[71]. Both the frequency position
and the oscillator strength of all phonons (with the exception of the 347 cm−1
resonance) are independent of temperature (lower inset of Fig. 5.4). The 347
cm−1 mode assigned to the OII − Os − OII bend[69] shows weak hardening
at T < TM IT . The key outcome of the examination of the phonon spectra in
Figs.5.1,5.2, is that no new phonon modes appear at T = TM IT and the low-T
data for Cd2 Os2 O7 displays only the 7 modes expected for the ideal pyrochlore
structure.
The experimental evidence presented above suggests that the development of
the insulating state in Cd2 Os2 O7 occurs without visible signs of charge ordering.
Examination of the IR-active phonons has proven to be one of the most sensitive tests for the charge-ordered state. Quite commonly additional lattice modes
appear in the phonon spectra or a dramatic redistribution of the spectral weight
between several resonances takes place provided the system reduces its symmetry
in the charge-ordered regime[72]. We failed to detect any of these effects. Furthermore, a detailed analysis of the x-ray diffraction data did not produce any
indications for structural changes at T < TM IT [62]. These observations allow
us to conclude that the metal-insulator transition in Cd2 Os2 O7 is driven solely
55
∆ !"#$ ω
ω
!"#$ Figure 5.4: Effective spectral weight vs. cutoff frequency for temperatures as indicated.
The top axis is in energy units normalized by twice the gap energy. The upper inset
shows the temperature dependence of the 347 cm−1 phonon and the BCS gap function
is also plotted, (solid curve). The lower inset shows the temperature dependence of all
seven phonons.
56
by electronic processes without noticeable indications for involvement of the lattice. Another important fact pertaining to the nature of the insulating state is
the continuous development of the energy gap in the electronic conductivity displayed in Fig.5.2 which is consistent with the Slater picture of the MIT[73, 74].
The second order transition is in accord with earlier specific heat and magnetic
susceptibility data[62].
Further support for the Slater hypothesis in the context of the Cd2 Os2 O7 data
is provided by the electronic conductivity. We first note that our data reveals several hallmarks of the Bardeen-Cooper-Schrieffer (BCS) electrodynamics expected
for systems with spin density waves[59], including the ratio of 2∆/kB TM IT 5.1
(expected in a modified BCS theory that takes into account the scattering of
electrons by phonons, as for the canonical SDW Chromium)[75, 76] as well as
a characteristic decline of the gap value at non-zero temperatures [73, 74]. The
general form of the conductivity spectra is also consistent with the BCS picture where type 2 coherence factors lead to an overshoot between the data at
T TM IT and T > TM IT at frequencies above the gap. As pointed out above,
the behavior of the low-T spectra above the gap edge are adequately described
with the σ1 (ω) ∼ ω 1/2 dependence, as is expected for a Slater transition. This
finding is important because the ω 1/2 dependence is distinct from the ω 3/2 dependence observed in the Hubbard limit[64].
Given the experimental evidence discussed above, the Slater mechanism emerges
57
as a viable model of the MIT in Cd2 Os2 O7 . Therefore, this compound may be
the first well documented case of a three-dimensional SDW material[77], subject
to further direct verification of the spin structure. An unexpected feature of
the antiferromagnetically-driven MIT is a mismatch between the TM IT 200K
and the frequency range involved in the redistribution of the spectral weight
in the insulating state Ω 20000K. Similar mismatch is commonly found
throughout the spectroscopic studies of the so-called pseudogap state in highTc superconductors[78], in which antiferromagnetic fluctuations are perceived as
a likely cause of the pseudogap state. Finally, it is worth mentioning that other
pyrochlore compounds reveal very different properties at the verge of the metalinsulator transition. For instance, the transition to the insulating regime in the
closely related Tl2 Ru2 O7 system appears to be of first order and additionally is
accompanied by charge ordering effects judging from the transformations of the
phonon spectra[79]. It is yet to be determined what microscopic factors define
the peculiar character of the MIT in Cd2 Os2 O7 .
58
The text of this chapter is, in part, a reprint of the material as it appears
in Phys. Rev. B 66, 035120 (2002). The dissertation author was the primary
researcher and author in this publication.
Chapter 6
An Infrared Investigation of
Phonons in a series of Detwined
Single Crystal of La2−xSrxCuO4
Using infrared spectroscopy we show that the expected group theoretical phonons
for all axes and crystallographic phase are observed in La2−x Srx CuO4 . The inplane phonons are tracked throughout the phase diagram as Sr is doped for La
in this system. This previously unknown result may be important for electron
phonon coupling in high temperature superconductors.
Phonons are one of the most fundamental excitations in all of condensed
matter physics. In classical superconductivity, phonons are particularly importance and their involvement has been know for over fifty years. Even before the
59
60
theory of Bardeen Cooper and Shieffer (BCS), it had been experimentally discovered that electron-lattice coupling was intimately related to Tc [80, 81]. The
theory of BCS[82], proposed over four decades ago, put this empirical knowledge into a firm understanding as the mechanism that was responsible for superconductivity itself through the lattice mediated formation of Cooper pairs.
With the discovery of high Tc superconductivity, the phonon assisted mechanism came under question as it was observed that there were deviations from the
expected values prescribed by BCS[83, 84]. Theories that include anharmonic
lattice vibrations could account for some aberent aspects[85], and more recent
theories of high temperature superconductivity still display the importance of
the electron-phonon mechanism in superconductors by way of charge and or
spin stripe inhomogeneities[86]. Many recent experiments also stress the importance of the phonon, for example: electron phonon coupling[87], anharmonicity
of phonons[88], and temperature dependent softening of phonons[89].
Thus the ubiquitous phonon finds itself a participant common to the majority of recent theories of superconductivity and the subject of a vast amount of
research attempting to elucidate its connection. However the majority of spectroscopic measurements have been unable to resolve all expected modes in many
superconducting systems, and this may be problematic for many theories. So despite the fact that there is strong evidence for the importance of lattice vibrations
and although there have been numerous studies of high temperature supercon-
61
meV
0
20
60
80
0%
A-axis
B-axis
T=10K
1
0
2
1%
A-axis
B-axis
3%
A-axis
B-axis
4%
A-axis
B-axis
6%
A-axis
B-axis
1
Real Conductivity (10
3
Ω -1 cm -1)
2
40
0
2
1
0
2
1
0
2
1
0
2
8%
AB-plane
1
0
0
100 200 300 400 500 600 700
-1
Frequency (cm )
Figure 6.1: The infrared portion of the real conductivity for undoped x=0.00, top panel,
and x=0.01, 0.03, 0.04, 0.06, and 0.008 in the bottom panel. In the undoped compound
(top) the Aortho axis (gray) displays the expected four infrared active phonons and the
Bortho axis (black) the expected 7 IR active phonons. These theoretical expected inplane phonons can be observed as one progresses across the phase diagram all the
way to the superconducting x=6% cyrstal, but importantly also seen in the twinned
superconducting 8% sample.
ductors, there has been little account for the lack of the expected phonons in
these systems[94], and the role of the phonon has yet to be determined.
Detwinned single crystals of La2−x Srx CuO4 (LSCO) have been available for
over a decade[98], however there has yet to be a systematic investigation of in-
62
plane anisotropy and studies of the low temperature phase. A common problem
of existing studies is the lack of observation of all of the theoretically expected
phonons in the LTO (low temperature orthorhombic) phase of LSCO. Some
infrared studies of LSCO have supposed the reason for this was due to the fact
that the transition to orthorhombicity was small[99] and had negligible effects
on phonon/electronic structure, or that the presence of screening currents due
to carriers obstructed their view[100].
We present a systematic in-plane spectroscopic investigation on a series of detwinned (0 ≤ x ≤ 0.06) and twinned single crystals (x = 0.08) of La2−x Srx CuO4 .
From the parent Mott insulator to the strontium doped superconducting samples (x = 0.06, 0.08) the in plane electrodynamics are characterized by reflectance
measurements using polarized light. Infrared spectroscopy has proven to be one
of the most sensitive methods to observe: phonons, electron-lattice coupling[95,
96], anharmonicity[97], and charge ordering[59], among other important electrodynamic behavior. The samples are high quality single crystals grown by
the traveling-solvent floating-zone technique[14], and are cut into rectangular
platelets with edges along the orthorhombic axes, (typical size of 1.5 x 0.5 x 0.1
mm3 ), where the C-axis is perpendicular to the platelets within an accuracy of
1◦ , as determined by x-ray Laue analysis.
LSCO has two distinct crystallographic phases, a high temperature tetragonal
(HTT) phase, and a low temperature orthorhombic (LTO) phase. More gener-
63
-1
Real Conductivity ( Ω -1 cm -1)
Frequency (cm )
3000
0
100 200 300 400 500 600 700
La 1.96 Sr 0.04 CuO 4
2000
T=10K
A-axis
B-axis
1000
0
T=293K
A-axis
B-axis
500
0
T=473K
AB-plane
500
250
0
External
Mode
Apical Bending
Mode
Bending
Mode
Streching
Mode
Figure 6.2: Infrared region of σ1 (ω) for the 10K, room temperature, and the HTT phase
(T=473K) of 4% LSCO. The phonons for the HTT phase are indicated by arrows with
the center frequencies given in the text. The bottom panel list, in order from left to
right, the assignment of each of these modes from low to high frequency as depicted by
the CuO6 octahedra. The temperature evolution of these modes can be tracked into
the LTO phase displayed in the top two panels. Both the A-axis (grey) and B-axis
(black) in the LTO phase are plotted.
64
ally LSCO belongs to a family of structures related to the K2 NiF4 with space
group I4/mmm (HTT) where the HTT phase can evolve into any of its Landau
subgroups Bmab (LTO), P ccn (low-T tetragonal, LTT) and P 42 /ncm (low-T
orthorhombic 2, LTO-2)[90]. In LSCO the HTT→LTO transition is known to
be of second order and happens as a result from the bond length mismatch between the CuO2 planes and the La2 O2 bilayers. This mismatch is relieved by
a buckling of the CuO2 plane and a rotation of the CuO6 octahedra, (depicted
in the lower panel Fig. 6.2), around the Aortho axis which is the reason for the
structural change. Additionally since the CuO6 octahedron are tilted along the
longer Bortho axis, a reduction of symmetry compared to the HTT phase occurs
and group theory predicts 17 infrared active phonons, ΓIR = 6B1u + 4B2u + 7B3u ,
i.e. 7 modes along the B-axis, 4 modes along the A-axis and 6 along the C-axis. In
contrast the HTT phase has ΓIR = 3A2u + 4Eu , 4 modes parallel to the AB-plane
and 3 perpendicular to it.
In Fig. 6.1 we show the low temperature infrared portion of the frequency
dependent real conductivity obtained by a Kramers-Kronig transformation of
the raw reflectance data[91, 67]. In each panel the strontium doping is indicated, with the undoped parent compound La2 CuO4 (La214) in the top panel
and its superconducting counterparts x=6%, (Tc =8K) and x=8%, (Tc =14K) in
the lower panels. The undoped Mott insulating compound exhibits low values
of the conductivity indicating its insulating behavior, however the spectrum is
meV
0
20
40
60
80
La 1.96 Sr 0.04 CuO 4
σ 1(ω) (103Ω-1 cm -1)
3000
2500
2000
1500
T=10K
1.5
A-axis
B-axis
C-axis
1.0
200
250
-1
(cm )
300
300
150
500
0
600
450
0.5
0.0
1000
750
0
100 200 300 400 500 600 700
0
Real Conductivity ( Ω -1 cm -1)
Real Conductivity ( Ω -1 cm -1)
65
-1
Frequency (cm )
Figure 6.3: Low temperature infrared spectrum of all three crystallographic axis for
the LTO phase of x=4% Sr doped LSCO. The A and B-axis (thick gray and thick black
lines respectively) are shown in the main panel and their units correspond to the left
coordinate. In order to display all phonons, the C-axis (thin black line) is displayed at
4X magnification and corresponds to the right coordinate. The inset shows the C-axis
phonon at 244 cm−1 and the A and B-axis all on the same scale.
66
punctuated by several strong phonons visible in both the A and B-axis, (the
notation A-axis and B-axis will indicate the orthorhombic axis unless specified
otherwise). As expected by point group analysis we find 4 phonons in the A-axis
(103, 173, 351, and 679 cm−1 ) and 7 in the B-axis (120, 155, 168, 191, 367, 399,
and 684 cm−1 ). As holes are added to the system by increasing the strontium
doping the conductivity increases and evolves smoothly and monotonically in
both axes. All expected phonons can be realized in all detwinned samples studied and even found in twinned superconducting crystals (bottom panel of Fig.
6.1).
Also of great interest is the observation of electronic anisotropy evident in
Fig. 6.1. In the low frequency low temperature limit the A-axis displays greater
values of σ1 (ω) for all crystals in the range x ≤ 4%. This is in accord with
recent transport measurements on detwinned crystals of LSCO[92] which show
this anisotropy implies that stripes are inherently conducting. As mentioned, this
anisotropy is realized for all detwinned cyrstals except for the x = 6% sample
which displays a reversal of this trend, (bottom panel of Fig. 6.1). However as
determined by nuetron measurements, the stripes are known to undergo a 45◦
rotation at a strontium doping of x = 5.5%, above which the stripes lie parallel
to the Cu-O bond direction[93].
In Fig. 6.2 the infrared region of La1.96 Sr0.04 CuO4 at 10K, and 292K along
with the spectrum from the HTT phase is shown. The HTT phase, as stated
67
above, has four in-plane infrared active phonons and their normal modes are
depicted in the lower portion of Fig. 6.2, while in the panel above, each mode
is indicated by an arrow with their corresponding center frequencies at 129, 165,
355, 666 cm−1 . It can be seen that some of these modes in the HTT phase split
nearly symmetrically from their location into phonons in the A and B-axis of the
LTO phase. For example the phonon at 355 cm−1 splits into the 352 cm−1 and
367 cm−1 phonons in the A and B-axis respectively. The high frequency phonon
at 666 cm−1 hardens and splits into the phonons at 680 cm−1 (A-axis) and 684
cm−1 (B-axis). The splitting is such that the A-axis phonon always lies lower in
frequency compared to the B-axis phonon in all dopings and temperature ranges
characterized, (also see the right pane of Fig. 6.4).
For completeness the C-axis been measured for one particular doping (x=4%),
and the results are shown in Fig. 6.3 along with the A and B-axis at low temperature. As previously mentioned one would expect 6 IR active modes in the
C-axis and this is realized as we find phonons at 137, 151, 244, 317, 349, 501
cm−1 . With this full characterization of all crystallographic orthorhombic axes,
we can understand all features seen for the in-plane measurements. For example the features near 250 cm−1 and 500cm−1 in both the A and B-axis can be
explained as C-axis leakage as they occur in proximity to C-axis phonons. One
can also observe leakage from the A to the B-axis and vice versa, i.e. the triplet
of phonons centered near 170 cm−1 in the B-axis can be seen in the A-axis data.
68
Temperature (K)
700
0
150 300 450
La 1.96 Sr 0.04 CuO 4
La2CuO4
690
700
690
680
680
670
670
400
400
A-axis
B-axis
375
375
350
350
175
175
150
150
125
125
100
100
0
150 300 450 0
ω0 (cm -1)
ω0 (cm -1)
150 300 4500
150 300 450
Temperature (K)
Figure 6.4: Temperature dependence of all in-plane phonons for 0% (left pane) from
room temperature to 10K and 4% doped LSCO (right pane) from 500K to 10K. All
A-axis phonons (open circles) and B-axis phonons (filled squares) are found except
one mode in 4% at the highest temperatures. Notice that for phonons split from those
in the HTT phase (shown as triangles), the A-axis phonons always occur lower in
frequency.
69
These effects can happen for a number of reasons including misscut crystals,
non-normal reflectance measurements, and inhomogeneous samples.
In Fig. 6.4 we show the temperature dependence of all in-plane phonons for
both the parent Mott-Hubbard insulator and for 4% LSCO. Many phonons are
seen to be temperature independent with a few exceptions. The lowest frequency
A-axis phonon near 120 cm−1 is seen to soften with decreasing temperature by
nearly 15 cm−1 , while the B-axis phonon near 176 cm−1 hardens by the same
amount. Phonons from the A and B-axis of the HTT bending mode at 355
cm−1 split symmetrically and soften and harden, respectively, by 4 cm−1 with
the A-axis phonon lying lower in frequency. The high frequency A and B-axis
phonons from the HTT stretch mode are seen to harden by about 7 cm−1 , but
interestingly they also exhibit a significant doping dependence, as they both
harden by 10 cm−1 from undoped compound to the superconducting 6% sample.
A salient feature of all data presented is that despite the fact that the infrared conductivity changes by over an order of magnitude from La214 to the
x=8% LSCO samples (Fig. 6.1) all predicted phonons are observed over all
temperatures and dopings in both the HTT and LTO phases and along all crystallographic axes. These observations evince the quality of the single crystal and
the importance of this study. A signature of electron-phonon coupling is the
occurrence asymmetric lineshapes, known as Fano resonances[101], observable in
many spectroscopies including infrared. Asymmetric phonons are observable in
70
Fig. 6.1 at 680 cm−1 for all dopings except for the undoped parent compound.
This observation may be important for models of superconductivity and will be
published elsewhere.
In summary we have observed all expected infrared active modes in a series
of strontium doped La2−x Srx CuO4 from undoped crystals across the phase diagram into the superconducting regime. All of the aforementioned dependence
of phonons happens smoothly and monotonically as a function of temperature
and doping. This suggests that the phase transitions from the antiferromagnetic
insulator to the metallic regime and also into the superconducting regime are of
second order. The observation of electronic anisotropy for all crystal dopings in
the regime 0% ≤ x ≤ 4% support the interpretation that spin stripes are intrinsically conducting and confirm the observation of the spin stripe re-orientation
across the normal-superconducting transition.
Chapter 7
Broadband multi-interferometer
spectroscopy in high magnetic
fields: from THz to Optical
7.1
Introduction
Spectroscopy is one of the most fundamental and ubiquitous experimental techniques for scientific investigations. For instance spectroscopy in the infrared
regime (3-30 THz, 100-1000 cm−1 ) is particularly informative due to numerous intrinsic material response associated with both intra-band electronic transitions, collective modes, and the interaction of radiation with lattice vibrations,
i.e. phonons. Many-body electronic effects including density waves[59] and su-
71
72
perconductivity also give rise to spectroscopic signatures in the infrared range.
There are equally important ranges at lower and higher energies from the infrared
region. Currently there is great interest in the lower energy terahertz (THz) range
as there are natural absorption bands, due to bio-molecular vibrations,[102, 103]
bio and chemical agents. As one proceeds higher in energy to the mid and near
infrared, there are other various and important contributions associated with
vibrational states in organic material and inter-band transitions in both organic
and inorganic solids. A quantitative analysis of optical phenomena in solids relies
on a determination of the optical constants.[104] A broad band instrument which
covers the above specified ranges is a versatile and invaluable tool. Couple this
instrument to a cryostat for temperature dependent measurements and to a high
field magnet for magneto-optic measurements and the phase space of properties
one can investigate is enormous.
Although investigation of the optical constants in magnetic field is far from
new, it is also true that magneto-optical measurements are far from commonplace. Spectroscopy in magnetic fields is technically challenging and has yet to
establish itself as a standard measured quantity for scientific exploration. The
importance of using magnetic field as a thermodynamically tunable parameter
for scientific investigation can be summarized by listing some of the phenomena
which has been studied. Historically magneto-optical measurements were performed in order to elucidate phenomena such as cyclotron resonance,[105, 106]
73
BCS superconductivity,[107] 2D-gas/ plasmon-phonon interaction, and antiferomagnetic (AF) resonances.[108, 109] The importance of magneto-optical measurements has not diminished over the years and currently there is strong interest in: magnetic semiconductors,[110] metamaterials, strongly correlated systems (field-induced stripes[111], AF correlations), quantum critical phenomena,
electron spin resonance[112], and high-Tc superconductors. With a demand for
devices such as, adaptive lenses, tunable mirrors, isolators, and converters[113]
operating at THz frequencies, an ever increasing important characterization of
materials is their magnetic properties. As mentioned, a striking example is the
recent explosion of interest in Diluted Magnetic Semiconductors (DMS), which
has produced great scientific and technological attention in recent years. While
these materials hold great technological promise[114], their broadband magnetooptical response has yet to be fully characterized as appropriate facilities are not
numerous.
The National Magnet Laboratories in Tallahassee and Los Alamos are excellent facilities for magneto-optical measurements, however the magnets are often
solenoidal in design and thus measurements are often confined to simple transmission (sometimes reflection) measurements in one field orientation. Additionally
the sizes of samples one can measure are constrained to be large, as the optics are
not suitable to handle small crystals. With advances in superconducting magnets it has become feasible to set up large magnetic fields for magneto-optical
74
measurements within an ordinary educational laboratory environment.[115]
In this article we describe the design and implementation of a magneto-optical
spectroscopic system capable of broad-band high field low temperature measurements. A novelty of the system is the ability to perform absolute measurements
of small samples with a good signal to noise. Our research program is aimed
at detailed investigations of strongly correlated electron systems, semiconductor
superlattices, heavy electron systems and novel superconductors among other
magneto-electrodynamic behavior. It is often necessary to go to low temperature and high magnetic fields to elucidate the exotic behavior exhibited by these
novel systems.
7.2
7.2.1
SYSTEM DESIGN AND COMPONENTS
Optics
The heart of the system is a reflectance unit which couples two interferometers to
a superconducting magnet. This novel design enables the following capabilities:
ability to perform measurements of micro-sized crystals with polarized light, two
channel acquisition, determination of absolute values, intermediate focus, and
ability to perform broadband spectroscopy. This combination is unique and to
the best of our knowledge can be conveniently implemented only with a magnet
with a split-coil design.
75
While the benefit of most of the above are clear, the advantage of the intermediate focus may be easily overlook as, for example, commercial systems lack
this particular ingredient. A focus within the reflection unit allows one to achieve
superb image quality on the sample thus permitting the characterization of small
samples. This is an important distinction because many novel and exotic samples
tend to be small when first discovered. Yet another advantage of the intermediate
focus is the ability to measure the magneto-optical permittivities thus permitting determination of the non-diagonal components of the conductivity.[116] The
converging beam allows easy placement of a polarizer near the focus and in front
of the detector, thus permitting crossed polarized measurements.
A layout of the optics utilized for the reflection and transmission unit coupled
to both the Michelson and the Martin-Puplett instruments is depicted in Fig.
7.1. The optics are permanently arranged on bread-boards which sit within the
reflection and transmission units and have been designed to work with both
interferometers without changing components. This is a big advantage in the
set-up time for different experiments. The bread boards are custom made and
rest on kinetic mounts.
As mentioned a novelty of the system is the ability to perform referenced
measurements utilizing dual channel acquisition. The optical layout depicted in
Fig. 7.1 details how this is accomplished when the unit is connected to both
interferometers. The input from the Martin-Puplett interferometer arrives as
76
Split Coil Superconducting Magnet
Polarizer
Dual Channel
320mK Bolometer
Bolometer
Polarizer
Reflection
Transmission
From Michelson
Interferometer
From Martin-Puplett
Interferometer
Figure 7.1: Schematic and optical layout of an apparatus for absolute measurements of
R(ω) and T(ω) in magnetic field. Collimated light input from the Michelson interferometer is reflected from a 45◦ off-axis parabola with a 8.89 cm focal length. Light then
passes through an intermediate focus and through two flat mirrors before reflecting
from the final focusing mirror and into the cryostat. From here the light may then
reflect from the sample back into the reflection unit, or continue through to the transmission unit. In both cases the light strikes a focusing mirror then off a flat mirror
before impinging upon the detector, here depicted as bolometers. The Martin-Puplett
interferrometer is coupled by 1/2 I.D. brass pipe depicted as the dashed line segments.
The reference channel goes directly to the detector while the sample channel is guided
into the same optics used by the Michelson setup. This design allows for a quick
change between interferometers by simple removal of the off-axis parabola. The inset
shows two different sample holders used for absolute measurements. In reflection both
the sample and a reference mirror are fixed on two cones mounted 180◦ apart. In
transmission mode the sample is located behind an aperture and reference measurements are obtained by vertical translation to an open channel. Both methods allow
for absolute measurements and are controlled by a stepping motor with encoder feed
back. The rotational resolution in reflection mode is 0.02 degrees and the step size in
vertical translation is 0.001 mm. The rotational or translational alignment are both
highly reproducible and any error in misalignment and within the signal to noise of
the system.
77
both a sample channel and a reference channel. The reference channel is then
simply directed to the reference channel on the detector. For referenced measurements using the Michelson interferometer, a reference beam is taken from
the main input beam and directed to the reference channel of the detector. This
is accomplished utilizing a polarizer aligned at 45◦ to the beam direction and
thus no additional loss is incurred for polarized measurements. The advantage of
referenced measurement is the ability to compensate for: magneto-optical effects
on the interferometer or source, light source instability, system drift, and long
acquisition times. These issues are primarily relevant for spectroscopy in the FIR
regime and thus a light pipe is suitable for the reference channel.
After light traverses from each interferometer through either the light
pipe or light guide, it then exits into the reflection unit. Both the reflection and
transmission units have been constructed from a solid piece of aluminum and
fabricated on a digital mill. The cost of such a procedure is approximately 10%
more than had the units been constructed from welded flat aluminum plates.
However the advantage of this design is that it yields a more robust unit with
relatively thin walls, which is capable of being evacuated. The boxes sit on
three legs and have various flanges and ports built in on all sides for versatility.
A black hard anodized finish is applied to the boxes ensuring a long lifetime
for parts which are frequently bolted and unbolted, and additionally minimizes
reflections. A lid sits atop the boxes and seals with an O-ring gland.
78
7.2.2
Interferometers
The system couples the output of two interferometers, with overlapping frequency coverage, to the superconducting magnet. Interferometers utilized are a
Fourier transform (Michelson) interferometer and a Martin-Puplett interferometer. The Michelson interferometer is a commercially available Fourier transform
unit (Bruker IFS 66v/S Vacuum FT-IR) capable of measurements in a broad
range (10cm−1 -25000cm−1 , 0.3THz-750THz, 1.24meV-3eV). This interferometer
utilizes light sources such as a mercury (Hg) arc lamp, globar, and tungsten
lamp. Mercury is utilized primarily in the THz and infrared region, globar in the
infrared and MIR regimes, and tungsten in the MIR and NIR frequency range.
Several different beam splitters are used for various ranges and include: 100µm
thick mylar, 50µm thick mylar, 6µm thick mylar coated with germanium (Ge),
potassium bromide (KBr), and quartz. The ranges of these beam splitters are
listed in the middle panels of Fig. 7.3, expect for that of quartz, which the usable
range is approximately 3000cm−1 -30,000cm−1 .
The Martin-Puplett interferometer is designed to work in the lower sub-THz
and mm-wave region of the spectrum (1cm−1 -100cm−1 ) and is based on a Sciencetech 200 instrument. A 150 watt Mercury-Xenon arc lamp (Hamamatsu
photonics L2482) is used for a source. The Martin-Puplett design utilizes a wire
grid polarizer rather than beam splitters as in the Fourier transform setup. This
allows for nearly perfect beamsplitter efficiency at frequencies as low as 20 GHz.
79
All components of the interferometer are assembled on an optical breadboard
and are housed in a vacuum compartment to avoid strong absorption in THz
range at ambient conditions. The Martin-Puplett interferometer is a step-scan
instrument and thus the acquisition times can be quite long, ∼5-20min. Thus
the outgoing polarizer is also used as a beam splitter to split off a reference beam
from the main channel. This allows one to collect accurate measurements by use
of a second channel used in conjunction with a two channel detector, described
previously in detail.[117]
7.2.3
Evacuated light guide and light pipes
The coupling of both the Martin-Puplett and the Michelson interferometer
to the reflection unit is performed by utilizing both evacuated light guides and
by conventional light pipes. Light pipe segments are ideal and easily handled
however the multiple reflections and properties of the light pipe do not permit
their use at higher frequencies.
In the case of the Michelson interferometer, collimated light is internally reflected from a computer controlled motorized mirror to a port located on the
side of the instrument. From here the light passes through an evacuated compartment (light guide) to the reflection unit. This method is preferred since
the light makes no internal reflections within the light guide, and loss is minimized thus permitting measurement up to optical frequencies. The coupling to
80
the Michelson interferometer is performed through a standard ASA fittings, i.e.
a cylindrical segment with O.D. 3”. The utilization of these standard O-ring
sealed components allows different segments to be used to vary the distance of
the Michelson interferometer to the magnet, which may be important to minimize magnetic field effects on the spectrometer. We have found that a light guide
length of 750cm−1 is suitable to minimize magnetic effects on the spectrometer,
while preserving the quality of the collimated beam.
For the Martin-Puplett instrument conventional light pipes are used, since
the light at these frequencies is extremely long wavelength. New brass pipe, ( 12 ”
I.D.), readily available form any hardware store is used where the interior is not
polished or plated. Light from the sample and reference channel on the MartinPuplett interferometer are focused into the opening of the light pipe. Due to the
particular arrangement of equipment it may be necessary for the light pipes to
make one or more 90◦ bends before entering the reflection unit. This has been
accomplished by constructing 90◦ elbows that attach to the light pipe and are
sealed with O-ring flanges. The ideal geometry to use for such a turn is in-fact an
ellipsoidal mirror with the light pipe terminating at a specific distance (depending
on the f/# of the instrument) beyond the focus of the ellipse. However the 90◦
elbows are very simple and easy to implement and end up reducing the intensity
of light by ∼20% per elbow.[118]
81
Figure 7.2: Photograph of the 9 Tesla superconducting magnet connected to the
Michelson interferometer. The units are connected by 750 cm of evacuated light guide.
The platform is where the detector would sit for reflection experiments. A similar box
sits on the opposite side of the magnet for transmission (not shown).
82
7.2.4
Superconducting magnet
The magnet used here is a 9 Tesla superconducting split coil magnet made
by Oxford (Spectromag SM-9000) Fig. 7.2. The particular arrangement of the
coils is depicted in Fig. 7.1. The split coil design allows both measurements with
the magnetic field parallel to the propagation vector (
k) called Faraday geometry
and perpendicular to k Voight geometry.
The design of the magnet requires two windows for the light to pass
through before making it to the sample. A room temperature window, and a
liquid helium window (T=4.2K) mounted on the variable temperature insert
(VTI). Both the 300K and LHe windows are made from 50µm thick transparent polypropylene. A big advantage of the polypropylene utilized here is the
transparency of the windows to visible light which is ideal for alignment of the
samples.
Two custom sample holders have been designed to attach to the supplied sample tube, one for reflectance and the other for transmission measurements. The reflectance sample holder is constructed such that two samples can
be mounted, thus a reference mirror can be mounted 180◦ from the sample and
absolute reflectance values can be obtained. The sample and the reference are
mounted on a cone which is attached to the sample mount. The cones allow one
to study a sample physically smaller that the spot focus since only light which
is specularly reflected from the surface of the sample makes it to the detector
83
and light reflected by the cone does not. The transmission custom sample holder
allows for referenced measurements by vertical translation to an open channel,
(aperture). The repositioning of both the reflection and transmission sample
holders is accomplished with a stepping motor attached to the top of the sample
tube with encoder feedback. Both of these custom sample holders are depicted
in the inset to Fig. 7.1.
7.3
ZERO FIELD CHARACTERIZATION
We have characterized the system performance in a number of different configurations. In each particular setup the components are optimized for a particular
frequency range and the electronic gain is identical for bolometers. One component common to all data presented is the windows used on the superconducting
magnet, which is polypropylene in this case. The results from these tests are
displayed in Fig. 7.3.
An indicator of the quality of a spectroscopic system is encapsulated within
the power spectrum. This is depicted in the left panels of Fig. 7.3 for each configuration. The optical cut-offs are determined by cold (T=77K, and T=4.2K)
filters installed within the bolometer detectors. The frequency dependent fluctuations are intrinsic to each arrangement and are due to interference fringes
related to the thickness of the beam splitter. For example the MIR spectrum exhibits significant frequency dependence and the intensity has several minimums.
84
15
10
5
0
InterSource
ferometer
1.5
MP
10
20
0
30
1
50
2
25
50
0
75
1
10
20
0
Hg
1.0
1.5
Polarizer
30
1
40
2
50
3
Bolo
1.6K
cm-1
3 THz
2
0.5
Michelson
Hg
Michelson
Hg
50µm
Bolo
320mK
Michelson
Hg
50µm
Bolo
1.6K
0
4
2
0
1.50
0
25
50
0
75
100
10
20
cm-1
50
75
1
2
100
3
0.00
0.050
0
0
0
200
20
400
40
600
Ge on
6µ Mylar
50
100
20
0
KBr
200
20
400
2000
3000
-1
Frequency (cm )
1.02
40
cm-1
80 THz
600
60
1.02
Photo
Cond.
1.00
0.000
1000
cm-1
THz
1.00
0
Michelson Globar
0
1.05
0.98
cm-1
0.025
75
Bolo
4.2K
80 THz
60
25
10
Michelson Globar
cm-1
THz
1.00
0.95
1.05
1.00
0.95
1.05
1.00
0.95
THz
0.75
1.05
0.95
25
0
2
0.95
cm-1
THz
1.00
0
0
Bolo
100µm
320mK
1.05
1.00
0
100
2
Frequency (THz)
Bolo
Polarizer
320mK
cm-1
3 THz
40
MP
0
Hg
Beam
Detector
Splitter
0.0
100% Lines
Intensity (arb. units)
1.0
100% Lines
Intensity (arb. units)
4
0.5
100% Lines
Intensity (arb. units)
Frequency (THz)
0.0
6
4
2
0
0
0
1000
0.98
3000
2000
-1
Frequency (cm )
Figure 7.3: Power spectrum, configuration, and 100% lines in reflectance mode. The
left panels depict the intensity of several configurations plotted vs. frequency while
the middle panels show the configuration used for these spectra, and the right panels
display the corresponding 100% lines. The top two rows are for the MP setup and
the next three left panels are for the far-infrared range, below is a spectrum from
the infrared portion and the bottom panel displays the mid infrared spectrum. The
corresponding 100% lines for each spectrum are presented in the right panels. The
particular components used for these measurements are listed in the middle panels.
85
These are associated with five absorption lines due to the polypropylene windows
used for all experiments. The reduced intensity associated with these lines is very
manageable as the bandwidth is narrow. The advantage of using the polypropylene windows is the large frequency range over which it can be used. Usually
this is of little concern as the frequency dependence of samples characterized in
transmission or reflection within this range is small and thus this is a suitable
tradeoff.
Another typical measure of a spectroscopic system is a so called 100% line.
This gauge is obtained by measuring either a transmission or reflectance spectra and then dividing by a subsequent spectra. Thus the division of these two
should be a straight line at 100% and deviations from this represent the noise
and instability of the system. 100% lines for numerous configurations are displayed in the right column of Fig. 7.3. This parameter displays the noise as a
function of frequency and thus the relative error bars for all energy ranges can
be characterized.
These gauges of the zero field performance have been done in reflectance
mode. The performance of the unit in transmission mode is not shown, but
typically both the power spectrum and 100% lines are significantly enhanced.
Another relevant comparison is to that of the zero field interferometer performance when connected directly to the various detectors. We have found that
the magneto-optical system presented here has a reduction in intensity of ap-
86
proximately 50% compared to that of the original systems. This is a reasonable
reduction considering the added optical length and addition of extra optical components.
7.4
7.4.1
MAGNETO-OPTICAL MEASUREMENTS
Reflectance Mode
As a gauge of the magneto-optical system in reflection mode we have chosen graphite and YBa2 Cu3 Oy (YBCO) as two canonical samples to measure.
Although there has been a large amount of work done on graphite[119], there
has yet to be a systematic investigation of the optical constants at various magnetic fields and temperatures. Additionally the development of intraband and
interband transitions in magnetic fields is unexplored. As for high-Tc superconductors, one of the most important unresolved issues is the exact role of the
spin resonance in superconducting properties. It is also not clear how the spin
resonance is relevant to the optical conductivity. YBCO is one of the most appropriate systems for this issue because of the many systematic studies on the spin
resonance. However, there are not many studies of the magneto-optical properties of YBCO, possibly due to the limiting factor imposed by small sample
size.
The particular graphite sample characterized is a so-called Highly Oriented
87
Pyrolytic Graphite (HOPG). The dimensions of the ab plane surface characterized in reflectance measurements are ∼6 x 6 mm2 . With our unique instrument,
we are able to measure the reflectance of graphite at various temperatures and
magnetic fields up to MIR frequencies. The magneto-optical measurements were
performed from 5cm−1 to 3,000cm−1 at temperatures between 4.2K and room
temperature. Also measurements were completed from 5cm−1 to 50,000cm−1 at
zero field.
In order to obtain the optical constants of graphite, we combined reflectance
data in magnetic fields below 3,000cm−1 and zero field reflectance data above
3,000cm−1 . This is warranted since the reflectance spectra have no observable
magnetic field dependence as one approaches ∼3000 cm−1 . The data was then
extrapolated to low frequency using the standard Drude form and to higher frequencies using the free electron model. A Kramers-Kronig (KK) transformation
of the data then allows us to obtain the optical constants. Various different extrapolations to low frequency were used and verify that the particular method
produced no noticeable effects of σ1 (ω) over the measured frequency range.[67]
Therefore, by KK transformation of the reflectance data, we are able to obtain
the real part of the optical conductivity of graphite at all fields and temperatures
up to about 3,000cm−1 , with good accuracy.
In the zero field case, a conventional KK analysis assumes time reversal symmetry. However in the presence of magnetic fields, time reversal symmetry may
88
be broken and a modified form of the KK relations has to be used.[120] This
form includes non-zero off-diagonal components of the response function tensor,
e.g. σxy = 0. A knowledge of the zero field optical constants of graphite, enables
us to estimate the contribution of the off-diagonal components which, in the case
of graphite, are negligible. Moreover, we measured the magneto reflectance up
to 8T with two crossed polarizers (depicted in Fig. 7.1) and no effect was found
within the entire infrared region. This directly verifies the validity of neglecting
the off-diagonal components, and therefore, the optical constants in magnetic
fields can be obtained by conventional a KK transformation directly.
As an example, Fig. 7.4 shows the reflectance spectrum R(ω) and the corresponding σ1 (ω) spectrum of graphite at H=0T and H=7T, both at 10K. In
σ1 (ω) at zero field (grey curve bottom panel of Fig. 7.4), a Drude component is
observed below 100cm−1 , which is due to the intraband transition in graphite. In
the MIR region (from 100cm−1 to 1,000cm−1 ), there is a broad absorption peak
due to the interband transition of the π bands. Notably at fields of 7 Tesla field
there is no Drude component centered at zero frequency. Instead, the free carrier
absorption is shifted to a finite energy resonance (from 100cm−1 to 300cm−1 ),
i.e., cyclotron resonance, by the magnetic field. The cyclotron mode is broadened
because it is a combination of resonance due to holes and electrons. Holes and
electrons in graphite have slightly different masses (mh 0.04m0 , me 0.06m0 )
and the corresponding cyclotron frequencies are separated. The energy bands of
89
1.0
1
10
100
T=10K
0.8
0T
7T
4500
3000
1000
σ1
-1
0.6
-1
σ1(ω)(Ω cm ) Reflectance
Frequency (THz)
500
1000
1500
1500
ω (cm-1)
0
100
1000
-1
Frequency (cm )
Figure 7.4: Top panel: Reflectance of graphite (HOPG) at 0 Tesla and 7 Tesla, both
at 10K. Bottom panel: Real part of the optical conductivity σ1 (ω) of HOPG graphite
obtained by Kramers-Kronig transformation of the reflectance data in the top panel.
The bottom inset shows the detailed peak structures of σ1 (ω) in the MIR region,
which are due to the interband transitions between Landau levels in magnetic fields.
The reflectance at non-zero field values was obtained by the Martin-Puplett Spectrometer, with a 320mK bolometer from 5cm−1 to 50cm−1 and with a 1.6K bolometer
from 20cm−1 to 110cm−1 . The Michelson interferometer, with a 4.2K bolometer, was
used from 70cm-1 to 700cm-1, and with a photoconductor detector from 400cm−1 to
3,000cm−1 . The data in NIR region above 3,000cm−1 were measured by Michelson
interferometer with an InSb detector and a grating spectrometer in the UV region up
to 50,000cm−1 .
90
graphite are split into different Landau levels and the peaks of σ1 (ω) in the MIR
region at 7 Tesla are a signature of the interband transitions between these levels. It is worth pointing out that the spectra of graphite show field dependences
up to energies as high as ∼2500cm−1 =310meV. This unique instrument permits
measurements in large magnetic fields continuously over frequencies from 5cm−1
up to 3,000cm−1 , thus enabling us to systematically investigate these interesting
high energy phenomena.
An experimental challenge of the YBCO crystal measurements is their relatively small size 1.0 x 1.2 mm2 compared to the large graphite crystals. Thus
a characterization of YBCO was undertaken also to verify the sensitivity of the
instrument for small samples. The samples were measured and referenced with
respect to a perfect reflector, i.e. a mirror. The samples were then coated with
gold and then the referenced measurements were repeated. Then a ratio of the
two of these measurements (double ratio) allows a determination of absolute
values for small sample sizes, and has been described previously in detail.[66]
YBCO is a high temperature superconductor with a maximum Tc ∼95K
at the so-called “optimal doping” of y=6.95. For underdoped YBCO, the system exhibits a spin resonance at approximately 34 meV.[121] Additionaly the
reflectivity is nearly unity due to the superconducting response and/or high conductivity. Thus a characterization of the magnetic field dependence of this effect
is challenging.
91
YBa2Cu3O6.65
1.00
Reflectance
T = 5K
0.98
0.96
0.94
0.92
100
7 T, Field cooling (7 T)
7 T, Zero-field cooling
200
300
400
500
600
-1
Frequency (cm )
Figure 7.5: Reflectance spectra of y=6.65 YBCO. Two different protocols in application
of the 7 T field are used. The zero-field cooling process means the samples is cooled
down to 5 K in zero field and then the field is changed to 7 T magnetic field for the
measurement. In the field cooling process the sample is cooled from T > Tc to T=5K
with a 7T magnetic field applied.
92
In Fig. 7.5 we show two reflectance spectra of y=6.65 YBCO which has a
critical superconducting temperature of Tc ∼60K. The figure is a detail of the
reflectance spectra with a span of 10%. The abrupt decrease near 500 cm−1 is
referred to as the “knee”,[122] and has been considered to be relevant to the spin
resonance.
According to neutron scattering experiments, the intensity of the spin resonance should be reduced by about 30 % at H=6.8 T.[121] Thus one can expect
that the knee structure should be suppressed with magnetic field. However,
there is no observable effect at fields up to 7 T. The small changes observed in
the spectra are within the noise level of the instrument, ∼1% in the FIR regime
for small crystals. So, it appears that the optical conductivity does not couple
to the spin resonance, the effect lies lower in frequency, or the effect is smaller
than expected.
7.4.2
Transmission Mode
Niobium (Nb) is a BCS superconductor with a bulk superconducting temperature of Tc =9.2K and a critical field of Hc =0.2T. For thin films of Nb the
dependence of Tc vs. film thickness has been characterized and found to be
strongly reduced for thicknesses below about 100 A. Figure 7.5 shows transmission spectra of a Nb thin film with a thickness of 120A on a 50 µm sapphire
substrate grown at the University of California, San Diego. Bulk Nb has a Tc of
93
Frequency (THz)
Tn(T=15K)
Ts(H=0T)
Ts(T=3.5K)
Tn(H=5T)
0.0
0.5
1.0
1.5
1.5
T=3.5K
1.0
0.5
1.5
H=0T
1.0
0.5
0
20
40
60
-1
Frequency (cm )
Figure 7.6: Top panel: ratio of the transmission in the superconducting state at zero
field to that of the transmission at 5 Tesla. Bottom panel: ratio of the superconducting
transmission at 3.5K to that at 15K.
9.2K and the measurements were performed with the Martin-Puplett interferometer at a resolution of 3 cm−1 and a scan time of four minutes. The interferometer
was coupled to the reflection unit through 10 feet of 1/2” brass light pipe and undergoes one 90◦ bend. After light passes through the focusing optics it impinges
upon the sample and then on to the transmission box and finally to the detector, (see Fig. 7.1). The detector used for collection of the data was a thermal
bolometer designed for use at 1.6 K.
The data is displayed as transmission ratios of the superconducting state
divided by the normal state, Ts (ω)/Tn (ω). In the top panel the normal state
is obtained by increasing the magnetic field beyond the critical value HC, while
in the bottom panel the normal state is obtained by raising the temperature
above the critical temperature Tc . In both curves a strong enhancement of the
transmission spectra occur do to superconducting state. In the theory of MattisBardeen, the spectra should peak at an energy of 2∆=kB Tc which is about
94
2∆=15cm−1 =1.86meV in these samples. The form of the thin film interference
exhibited by the substrate is responsible for the slight disagreement between
the fits to the data displayed in Fig. 7.6. However the agreement between both
spectra is evident thus suggesting that the particular manner in which the normal
state is obtained is not relevant.
Chapter 8
Conclusions
We have demonstrated both by numerical simulation and experiment that an
effective medium of only conducting elements responds predominantly to the
magnetic field of incident electromagnetic fields. Remarkably, there is a band of
frequencies for which µef f (ω) can be negative, here manifested as a region of attenuation in scattering from a finite section of material. We have combined this
material with a negative ef f (ω) material to form a “left-handed” medium, forming a propagation band with negative group velocity where previously there was
only attenuation. We are now in a position to further investigate the fascinating
electrodynamic effects anticipated for such composite metamaterials.
By tracking the evolution of the multi-channel features through the underdoped region of the phase diagram we have been able to gain valuable insights
into the nature of the Mott transition in undoped counterparts as well as effects
underlying the ”non-Fermi liquid” transport at moderate and high doping levels.
95
96
Several approaches to extract m∗ produce consistent results and indicate that
the transition to the Mott insulating state occurs with constant effective mass
and vanishing carrier density. Surprisingly, despite a drastic change in the carrier
density, the masses appear to be fairly light even at dopings in immediate proximity to the Mott insulating state. This suggests that the transport is governed
by excitations topologically compatible with an antiferromagnetic background.
Specific schemes permitting a constant effective mass Mott insulator transition
include spin-charge separation[31, 32] and electronic phase separation.[33, 34]
These scenarios propose explanations of high-Tc superconductivity which departs from standard BCS theory. The dynamical hole behavior discussed here
is common to both LSCO and YBCO thus unveiling the generic properties of
doped MH systems independent of the details of a particular host material.
A strong magnetic response at THz frequencies from a non-magnetic, conducting microfabricated metamaterial has been demonstrated. The THz magnetic effect is broad band and tunable by scaling the SRR dimensions. The
scalability of these magnetic metamaterials throughout the THz range and potentially into optical frequencies promises many exciting applications such as
biological[51] and security imaging, biomolecular fingerprinting, remote sensing
and guidance in zero visibility weather conditions. Additionally the effect is
nearly an order of magnitude larger than that obtained from natural magnetic
materials.[52] Structures with a negative magnetic response, when combined with
97
plasmonic wires that exhibit negative electrical permittivity[53, 54, 3, 4], have
the potential to produce a negative refractive index material at these very high
frequencies, enabling the realization of devices that are needed to fill the gap of
the terahertz regime. These new materials extend the frequency range of metamaterials by over two orders of magnitude from previous microwave frequency
results. This novel result sets a new upper frequency bound to the extension of
artificial magnetic and natural magnetic materials alike.
Given the experimental evidence discussed above, the Slater mechanism emerges
as a viable model of the MIT in Cd2 Os2 O7 . Therefore, this compound may be
the first well documented case of a three-dimensional SDW material[77], subject
to further direct verification of the spin structure. An unexpected feature of
the antiferromagnetically-driven MIT is a mismatch between the TM IT 200K
and the frequency range involved in the redistribution of the spectral weight
in the insulating state Ω 20000K. Similar mismatch is commonly found
throughout the spectroscopic studies of the so-called pseudogap state in highTc superconductors[78], in which antiferromagnetic fluctuations are perceived as
a likely cause of the pseudogap state. Finally, it is worth mentioning that other
pyrochlore compounds reveal very different properties at the verge of the metalinsulator transition. For instance, the transition to the insulating regime in the
closely related Tl2 Ru2 O7 system appears to be of first order and additionally is
accompanied by charge ordering effects judging from the transformations of the
98
phonon spectra[79]. It is yet to be determined what microscopic factors define
the peculiar character of the MIT in Cd2 Os2 O7 .
We have observed all expected infrared active modes in a series of strontium
doped La2−x Srx CuO4 from undoped crystals across the phase diagram into the
superconducting regime. All of the aforementioned dependence of phonons happens smoothly and monotonically as a function of temperature and doping. This
suggests that the phase transitions from the antiferromagnetic insulator to the
metallic regime and also into the superconducting regime are of second order.
The observation of electronic anisotropy for all crystal dopings in the regime
0% ≤ x ≤ 4% support the interpretation that spin stripes are intrinsically conducting and confirm the observation of the spin stripe re-orientation across the
normal-superconducting transition.
We have presented a magneto-optical system capable of broad band spectroscopy from the THz to optical range at field up to 9 Tesla. As a measure
of the system performance we have characterized several canonical samples including: high-Tc superconductors, cyclotron resonant materials, and classical superconductors. The versatile design permits measurements in both Faraday and
Voight geometry and simultaneous transmission and reflection characterization.
By utilizing an intermediate focus the system can characterize the off-diagonal
magneto-optical permittivities for crossed polarized measurements. Yet another
benefit of the optical setup is the ability to perform referenced measurements
99
with dual channel acquisition. This compensates for long acquisition times, system noise and drift, and allows one to obtain absolute values.
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