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Microwave properties of potassium caused by the charge-density-wave broken symmetry

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PURDUE UNIVERSITY
GRADUATE SCHOOL
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This is to certify that the thesis prepared
gy
M i-R e P ark
Entitled
Microwave P r o p e rtie s o f Potassium Caused bp the C harge-D ensitu Wave Broken Symmetry.
Complies with University regulations and m eets the standards of the Graduate School for
originality and quality
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MICROWAVE PROPERTIES OF POTASSIUM CAUSED BY THE
CHARGE-DENSITY-WAVE BROKEN SYMMETRY
A Thesis
Submitted to the Faculty
of
Purdue University
by
Mi-Ae Park
In Partial Fulfillment of the
Requirements for the Degree
of
Doctor of Philosophy
August 1996
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UMI Number: 9713578
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To the memory of my mother
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ACKNOWLEDGMENTS
I would like to thank Professor Albert W. Overhauser for his assistance in
the preparation of this thesis and for his patience, encouragement, and guidance
throughout the course of this work. His endless power of thinking on one subject
made the thesis more perfect than I planned.
I thank Professor Ronald G. Reifenberger, Professor Sherwin T. Love, and
Professor Stephen M. Durbin for being on my committee.
I am particularly indebted to professor James G. Mullen. His encouragement,
counseling, and friendship have truly been invaluable in making the past few years
at Purdue pleasant for me.
I am proud to acknowledge the loving support of my family whose encour­
agement contributed significantly to the completion of this work. Cheers and
encouragement from my father were essential. My sister and her husband were
most helpful financially and mentally.
Especial appreciation goes to Dr. Yong-Jihn Kim. As a colleague and husband,
he has been very supportive.
The one person I should thank most is my late mother. She was modest, but
she had vision. Her wisdom gave me strength to aspire for a Ph. D. Without her
confidence in my ability, I would not have pursued graduate study.
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iv
TABLE OF CONTENTS
Page
LIST OF TABLES....................................................................................
v
LIST OF FIG U R E S.................................................................................
vi
A BSTRA CT.............................................................................................
ix
1. INTRODUCTION...............................................................................
1
2. SURFACE RESISTANCE IN A PERPENDICULARMAGNETIC FIELD
2.1
2.2
2.3
2.4
2.5
7
Introduction.....................................................................................
7
Effect of the heterodyne g a p s ......................................................... 10
Resonance from the Fermi-surface cylinder.................................... 16
Conclusions..................................................................................... 23
Reconciliation of the surface resistance and microwavetransmission 25
3. MICROWAVE TRANSMISSION IN A PERPENDICULAR MAG­
NETIC FIELD....................................................................................... 33
3.1
3.2
Gantmakher-Kaner oscillations ......................................................
High-frequency oscillations..............................................................
4. THEORY OF THE CYCLOTRON RESONANCETRANSMISSION
4.1
4.2
4.3
4.4
4.5
34
35
. . 42
Introduction.....................................................................................
Microwave transmission in an anisotropic, non-localmedium . . . .
Effect of minigaps on microwave transm ission..............................
Conductivity tensor from a tilted Fermi-surfacecylinder.................
Conclusions.....................................................................................
BIBLIOGRAPHY.....................................................................................
42
46
52
57
61
66
APPENDICES
Appendix A:
Appendix B:
Calculation of the conductivity .....................................
Polarization of the field inside an anisotropic metal . . .
V I T A .........................................................................................................
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71
74
80
LIST OF TABLES
Page
Table
1.1
2.1
Unexpected phenomena for a nearly-free electron model of potas­
sium : all of them require the existence of a CDW broken symmetry.
5
Calculated values, from Ref. 8, of the first five minigaps and het­
erodyne gaps for K. The main CDW gap was taken to be 0.62 eV
and the zone-boundary energy gap was 0.40 eV................................
32
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vi
LIST OF FIGURES
Figure
Page
1.1 The Brillouin zone of potassium on an (001) plane in A-space. The
angular tilt, relative to [110], of the CDW wave vector, Q, has been
exaggerated for clarity. The minigaps and heterodyne gaps are asso­
ciated with the periodicities of Eqs. (1.3) and (1.4). The horizontal
line is a [110] direction which is perpendicular to a smooth potas­
sium surface. The Fermi sphere for a free-electron model is also
shown..................................................................................................
6
2.1 Surface resistance of potassium versus magnetic field (u/c =
eH/m*c). The data, due to C. C. Grimes (1969), is for T=2.5K, and
circularly polarized radiation at u/2-ir = 23.9GHz. The dips near
±0.77 are due to particles of CiiSO^-hH^O, embedded in the cavity
walls during fabrication. The cyclotron resonance, at uic/u> = —1,
occurs when H = 1.03T. The small resonance, at ujc/ u) = 1 is caused
by a small admixture of the opposite polarization. The theoretical
curve is for a purely spherical Fermi surface, which potassium would
have in the absence of a CDW broken symmetry. ..........................
26
2.2 The Brillouin zone of potassium on an (001) plane in fc-space. The
angular tilt, relative to [110], of the CDW wave vector, Q, has been
exaggerated for clarity. The minigaps and heterodyne gaps are as­
sociated with the periodicities of Eqs. (2.1) and (2.2). The shaded
areas are the two halves of the Fermi-surface cylinder, which form
between the CDW gap and the first minigap. The axis of the cylin­
der is Guo —Q, which is also the direction of the heterodyne-gap
vectors. The dc magnetic field, H, is applied parallel to [110], which
is the habitual texture direction, perpendicular to smooth potassium
surfaces. The (ideal) Fermi sphere is also shown...............................
27
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vii
Figure
Page
2.3 Theoretical surface resistance for a Fermi sphere having only
heterodyne-gap intersections. The parameters of Eq. (2.6), which
quantify the loss in effective cyclotron motion on equatorial orbits,
are / = 0.8 and /? = 20. The drop in R for \H\ > Hc increases with
decreasing / . The steepness of the decline increases with increas­
ing /?. The magnetic-breakdown field is Ho = 4T. The electron
scattering time corresponds to wr = 30.............................................
28
2.4 Theoretical surface resistance of conduction electrons having ut =
30 on the Fermi sphere and u/rc = 150 on the Fermi-surface cylinder
(containing ~ 4 x 10-4 electrons per atom). The axis of the cylinder
is, here, parallel to H, and the heterodyne gaps (intersecting the
sphere) are ignored. A 4 : 1 ratio of left to right circular polarization
is assumed..........................................................................................
29
2.5 Theoretical R(H) for potassium based on the heterodyne-gap pa­
rameters of Fig. 2.3 and the Fermi-surface cylinder model of Fig.
2.4, except that the cylinder’s axis is tilted 45° from [110]. (The tilt
is required to m inim ize the elastic stress of the periodic lattice dis­
tortion needed to neutralize the electronic CDW.) This calculated
behavior should be compared with C. C. Grimes’ data in Fig. 2.1. . 30
2.6 Theoretical R(H) for potassium based on the Fermi surface model
of Fig. 3.4. The Q-domain size assumed for the specimen mea­
sured by C. C. Grimes is D=1.5/im. The other parameters used are
enumerated in Sec. 2.5.......................................................................
31
3.1 Rectangular microwave cavities used for the observations. Small
fused quartz windows are placed on both sides of the sample...........
38
3.2 Microwave transmission signal vs. H through a potassium slab in
a perpendicular magnetic field. (H = 3.42T at u>c/ uj = 1) The
microwave frequency is 79.18 GHz and the temperature is 1.3K. The
field at ujJ oj = 1 is 3.42T. The phase of the microwave reference
was adjusted so that the cyclotron resonance is symmetric. The slab
thickness is L = 85fim. The data, provided by G. L. Dunifer, were
obtained from sample K4, one of fifteen samples listed in Ref. 1. . . 39
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viii
Figure
3.3 Landau-level oscillations near uc/u = 1.2 and 0.6. The periodicity
in (a) requires a cylinder radius, kc ~ kpf 8, corresponding to the
middle cylinder of Fig. 3.4. The periodicity in (b) requires a cylinder
radius, kc ~ fcp/16, appropriate to the smallest cylinder in Fig. 3.4.
Page
40
3.4 Fermi surface cylinders of potassium. The horizontal axis is parallel
to [110] and to the dc magnetic field. The shaded cylinders are cre­
ated by the CDW gap and the first three minigaps. The thicknesses
of the cylinders have been exaggerated by a factor of ten. Each of
the half-cylinders shown is joined to a partner on the opposite side
of the Fermi surface by Bragg reflection at the energy-gap planes.
The complete Brillouin zone is shown in the inset.................................41
4.1 Theoretical microwave transmission signal versus uic/ uj for potas­
sium if the Fermi surface is spherical. Only Gantmakher-Kaner
oscillations appear (since the electron-spin magnetic moment is ne­
glected). The sample parameters are ujt0 = 150 and L = 85/zm. . . 63
4.2 Theoretical transmission signal when interruption of cyclotron mo­
tion by the CDW minigaps is modeled by the vz dependent relax­
ation time, Eq. (4.4), and with magnetic breakdown of the minigaps
described by Eq. (4.5). The parameters, y0 = 21 and H0 = 5.8T,
were adjusted so that the GK amplitude at uc/u = 1.47 agrees with
the (calibrated) data from Fig. 3.2 and with its observed growth by
a factor of five from low to high fields................................................... 64
4.3 Theoretical transmission signal when the non-local conductivity of
the largest Fermi surface cylinder, shown in Fig. 3.4, is added to the
main Fermi surface conductivity (employed in Fig. 4.2). The tilt,
6 — 50°, of the cylinder’s axis was adjusted so that ratio of the main
CR to the first subharmonic (at u)c/u = 0.5) is ~ 10, consistent with
Fig. 3.2. The cylinder radius, ke = 3kp/S, in the (110) plane was
adjusted so that the ratio of the CR to the high-field GK amplitude
agrees with that observed in Fig. 3.2. ujtq = 150 and L = 85(im. . . 65
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ABSTRACT
Park, Mi-Ae. Ph.D., Purdue University, August 1996. Microwave Properties of
Potassium Caused By the Charge-Density-Wave Broken Symmetry. Major Profes­
sor: Albert W. Overhauser.
The microwave surface resistance of potassium in a perpendicular magnetic
field, measured by C. C. Grimes in 1969, has never been completely explained.
The sharp cyclotron resonance peak (at a magnetic field Hc) is caused by a small
cylindrical section of Fermi surface created by the charge-density wave (CDW)
m in igaps,
having periodicities Kn = (n + 1)Q —nG110. The shape of the observed
resonance requires a tilt of the CDW vector Q away from [110], predicted by
Giuliani and Overhauser in 1979. An abrupt drop of the surface resistance for
\H\ > |Bc| is caused by the heterodyne gaps, which have periodicities Kn =
n(Guo —Q). These very small gaps, which begin to undergo magnetic breakdown
for fields H > IT, interrupt the cyclotron motion of equatorial orbits. The abrupt
drop in surface resistance for |B| > |BC| is caused by a partial loss of carrier
effectiveness for electrons having velocities nearly parallel to the surface.
Microwave transmission through potassium by Dunifer et al. shows five sig­
nals. They are Gantmakher-Kaner (GK) oscillations, conduction-electron spin res­
onance, high-frequency oscillations, cyclotron resonance, and cyclotron-resonance
subharmonics. Only the spin resonance has been successfully explained using a
free electron model. However, such a model predicts GK oscillations which are
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too large by several orders of magnitude. Lacueva and Overhauser have shown
that CDW energy gaps which cut through the Fermi surface reduce the GK signal.
The high-frequency oscillations were shown to result from Landau-level quantiza­
tion in a Fermi-surface cylinder created by the CDW. In this study we show that
oscillatory motion, parallel to the field, of electrons in a tilted cylinder cause the
cyclotron-resonance tr ansm ission. This signal and its subharmonics would be com­
pletely absent without the tilt. Consequently, four of the five transmission signals
require a CDW broken symmetry.
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1
1. INTRODUCTION
The crystalline potential of potassium acting on the conduction electrons is
so small that the degenerate Fermi sea is free-electron-like. Therefore the Fermi
surface of a potassium should be almost spherical, and simply connected. De Haasvan Alphen measurements were thought to show that the Fermi surface deviates
from a perfect sphere by only one or two parts per thousand.1
During the last thirty years a variety of magnetoconductivity anomalies have
been discovered in potassium, the simplest monovalent metal.2,3 Unlike Li and Na,
which undergo a crystallographic transformation to the 9R structure4 when cooled
to low temperature, a single crystal of K is not destroyed by cooling. Without a
charge-density wave (CDW) broken symmetry, the bcc lattice of K would support a
spherical conduction-electron Fermi surface. Low-temperature transport anomalies
could not then arise. Nevertheless, in dc experiments extraordinary phenomena
occur which require the Fermi surface to be multiply connected.3 Unexpected
phenomena also appear in the microwave properties of potassium.5, 6 Thirty two
anomalous examples are listed in Table 1.1. Most of these data require that the
translation symmetry of the b.c.c. lattice be broken by an incommensurate CDW
structure.
The Brillouin zone of an alkali metal has twelve congruent faces, each perpendicular to a [110] reciprocal lattice vector. The distance of each face from k = 0
is 14% larger than kp- The energy gaps at the zone faces, caused by the ionic
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2
potential, are about OAeV in potassium and leave the Fermi surface undistorted
from its spherical shape. However, the discovery by Mayer and El Naby of an
intense optical absorption with a threshold of Q.6eV created a theoretical crisis.7
This optical anomaly indicates that there is another periodic potential with a wave
vector Q nearly parallel to one of the [110] axes, for which the reciprocal lattice
vector is G. The angle between Q and G is expected to be only about one degree.
The modulated electron charge density is partially neutralized by a sinusoidal de­
formation of the positive ion background. This lattice deformation contributes
extra diffraction peaks in x-ray or neutron-scattering experiments. CDW satellite
reflections have been observed by Giebultowicz et al.8, 9, and the CDW wave vector
was determined accurately:
O7r
Q = (0.995,0.975,0.015)— ,
a
(1.1)
where a is a lattice constant. |Q| differs by only 1.5% from |G|
To find the energy spectrum for conduction electrons in a CDW, one has to
solve a Schrodinger equation having potential terms:
V{r) = 0.4cos(G • f) -1- 0.6cos(Q • f).
(1.2)
The first term on the right arises from the lattice and the second from the CDW
instability. The solution of Schrodinger’s equation with this potential shows that
the Fermi surface of potassium is very anisotropic and multiply connected,10 i.e.,
the Fermi surface suffers a distortion, and is sliced into several pieces by extra
energy gaps. An approximate shape of the Fermi surface is shown in Fig. 1.1. The
horizontal axis is parallel to [110] and perpendicular to a smooth potassium surface.
The Fermi surface is pierced by two families of extra energy gaps: minigaps and
heterodyne gaps. Minigaps, shown by the solid lines, are caused by higher order
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3
perturbations having periodicities:
Kn = (n + 1)Q —riGno>
n = 1,2,...
(1.3)
A periodicity with n = 0 in this equation gives rise to the main CDW gaps.
The influence of the minigaps has been noticed in a study of the GantmakherKaner oscillations.11 Several small cylindrical sheets of Fermi surface are created
by the minigaps. One such cylinder explains both the cyclotron resonance in the
perpendicular-field surface resistance12 and the Landau-level oscillations in the
microwave transmission.13
Heterodyne gaps, shown by the dashed lines in Fig. 1.1, are caused by pertur­
bations with periodicities:
Kn = n(Guo - Q ) ,
n = 1,2,...
(1.4)
The occurrence of open orbits caused by the heterodyne gaps14 and minigaps causes
the open-orbit magnetoresistance peaks observed by Coulter and Datars.15,16
The purpose of this study is to investigate the microwave properties that result
from the CDW in potassium. In Chapter 2 we study the surface resistance of
potassium when a dc magnetic field is applied perpendicular to the surface. Grimes
found a sharp resonance in the surface resistance at |/T| = Hc followed by a drop
for \H\ > Hc.5 (Hc = m’u/c/e.) Both of these anomalies have been unexplained for
twenty five years. In the experiment, circularly polarized microwaves are incident
on the potassium surface. Electrons having velocities nearly parallel to the surface
dominate the surface resistance. Because these electrons encounter the heterodyne
gaps, their cyclotron motion is interrupted. However, when H becomes large, the
electrons will undergo magnetic breakdown and act as if they were free. We model
the consequences of the heterodyne gaps, and include a reduction of their influence
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4
at high fields from magnetic breakdown. The sharp resonance is shown to arise
from the minigaps, which create a tilted Fermi-surface cylinder. The tilt of the
cylinder is essential for understanding the shape of the resonance.
Microwave transmission through a potassium slab will be studied in Chapters 3
and 4. The data reported by Dunifer et. al. show signals that cannot be explained
using a free-electron model.6 Linearly polarized microwaves are incident on a
potassium slab with a dc magnetic field perpendicular to the slab. For microwave
transmission, electrons having large velocity components parallel to the field are
most important. Such electrons encounter the minigaps. We introduce magnetic
breakdown of the minigaps to explain the field-dependence of the GantmakherKaner signal. A Fermi-surface cylinder created by the CDW plays a dominant
role in the transmission. Because the cylinder is tilted with respect to the dc
magnetic field, electrons in the cylinder have a velocity component parallel to the
field. In chapter 4, we show that the cyclotron resonance transmission together
with its subharmonics are caused by this longitudinal oscillation of the electrons
in the tilted cylinder. Therefore, in this thesis, we explain four of the anomalous
phenomena from Table 1.1, 7, 12, 28, and 31, by taking account of the CDW
broken symmetry.
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5
Table 1.1
Unexpected phenomena for a nearly-free electron model of potassium : all of them
require the existence of a CDW broken symmetry.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
1963
1963
1966
1968
1968
1968
1969
1969
1971
1971
1973
1974
1975
1976
1977
1978
1980
1980
1981
1982
1982
1983
1983
1985
1986
1987
1988
1989
1990
1992
1993
1996
Mayer-El Naby optical anomaly.
Optical anisotropy.
Conduction-electron spin-resonance splitting.
Non-saturating transverse magnetoresistance.
Kohler-slope variability.
Doppler-shifted cyclotron-resonance discrepancy.
Perpendicular-field cyclotron resonance.
Longitudinal magnetoresistance.
Four-peaked induced-torque anisotropy.
Residual-resistance variability.
Hall-coefficient discrepancy.
Cyclotron-resonance transmission.
Oil drop effect.
Deviations from Matthiessen’s rule.
Residual-resistivity anisotropy.
Low-temperature phason resistivity.
Open-orbit magnetoresistance resonances.
Phason peak in point-contact spectroscopy.
Variability of electron-electron scattering resistivity.
Phason heat capacity peak.
Temperature-dependence of the surface impedance.
Field-dependence of the residual-resistance anisotropy.
Four-peaked phase anomalies.
Fermi-energy photoemission peak.
Diffraction satellites.
Magneto-serpentine effect.
Splitting of paramagnetic-spin-wave sidebands.
Subharmonic cyclotron-resonance transmission.
Infrared inverse-photoemission (across minigaps).
Landau-level oscillations from the cylindrical Fermi surface.
Gantmakher-Kaner oscillations (too small by 104).
Anomalous CESR of K-Fe bilayers.
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6
CDW GAP
F.S. SPHERE
[110]
HETERODYNE GAPS
GAPS
BZ
Figure 1.1
The Brillouin zone of potassium on an (001) plane in A:-space. The angular tilt,
relative to [110], of the CDW wave vector, Q, has been exaggerated for clarity.
The m inigaps and heterodyne gaps are associated with the periodicities of Eqs.
(1.3) and (1.4). The horizontal line is a [110] direction which is perpendicular to
a smooth potassium surface. The Fermi sphere for a free-electron model is also
shown.
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7
2. SURFACE RESISTANCE IN A PERPENDICULAR MAGNETIC FIELD
2.1 Introduction
Cyclotron resonance of the conduction electrons in potassium was first observed
by Grimes and Kip1 using the Azbel-Kaner configuration,2 for which the dc mag­
netic field H is parallel to the metal’s surface. The effective mass was found to
be m* = 1.21m. Resonant peaks in the (microwave) surface resistance also occur
at subharmonic values,3 He/ n , n = 2,3,4,..., in addition to the fundamental res­
onance which occurs at Hc = m'uc/e. For conduction electrons having an energy
spectrum E(k) that is spherically symmetric, a resonance in the surface resistance
should never occur if H is perpendicular to the surface.4
Nevertheless Grimes, using a perpendicular-field configuration, found a sharp
fundamental resonance in the surface resistance of potassium.5 His data are shown
in Fig. 2.1 together with the theoretical R(H), which has no resonant structure at
all. The magnetic-field sweep, expressed as cjc/ uj (where u//27r is the microwave fre­
quency, 23.9GHz, and uc = eH/m’c), includes both positive and negative values
because the microwave field was circularly polarized. The sharp cyclotron reso­
nance, at u!c/u> = —1, corresponds to Hc = 1.03T. Not only was the existence of
the resonance unexpected, but the sharp drop of R(H) for \H| > Hc has remained
unexplained for twenty five years.
The reason why a resonance is not expected in a perpendicular field is easily
understood. The skin depth is ~ 10~4cm and the Fermi velocity is ~ 108cm/s.
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8
Accordingly, the time an electron (having the Fermi velocity) remains in the mi­
crowave field (~ 10"12sec) is an order of magnitude shorter than the microwave
period. (Electrons do not return periodically to the skin depth in a perpendicular
field, as occurs if the parallel-field configuration is employed.)
The resonance cannot be an Azbel-Kaner signal from an oblique surface patch
(at the sample’s edge) since there are no subharmonics. Neither can the resonance
be attributed to electrons in a (110) surface-state band, since the bottom of such
a band lies ~ 0.45eV above the Fermi level.
The only satisfactory explanation of the resonance in a perpendicular field is
based on the charge density wave (CDW) broken symmetry of potassium.6 Many
anomalous properties (now numbering more than thirty) require the presence of
a CDW,7 which causes two sequences of small energy gaps to cut the Fermi sur­
face,8 as illustrated in Fig. 2.2. The “minigaps” are higher-order gaps created by
periodicities:
Kn = (n + 1)Q —tiGno,
(n = l,2 ,...),
(2.1)
where Q is the CDW wave vector and Guo is the (110) reciprocal lattice vector
parallel to H. (It is known from optical properties that Q and one of the {110}
reciprocal lattice vectors are nearly perpendicular to a smooth potassium surface.9)
The calculated values8 of the first five minigaps are given in Table 2.1.
In Fig. 2.2 the black regions outline a small cylindrical section of Fermi surface
formed by the CDW energy gap and the first minigap. Only a small fraction, tj ~
4 x 10-4, of the conduction electrons are enclosed by this Fermi-surface cylinder.
Nevertheless, these electrons are responsible for the cyclotron-resonance structure
in the surface resistance.10 Landau-level oscillations caused by the cylinder have
been observed in microwave transmission.11 The periodicity of the oscillations
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9
(versus 1/H) indicates that the cylinder radius is fcf/8.12 The small velocities of
the cylinder electrons enable them to remain in the microwave skin region and to
exhibit a sharp resonance absorption.
The prior treatment of this resonance succeeded in identifying the cylindrical
Fermi-surface component as its cause.10 However two puzzles remained: The cal­
culated shape of the resonance was antisymmetric rather than (nearly) symmetric.
It was possible to “fix” this problem by mixing almost equal amounts of surface
reactance and surface resistance. A small amount of such mixing could be toler­
ated experimentally,13 but the required mixing angle of ~ 47° seems excessive. In
Sec. 2.3 we will show that this problem disappears when one recognizes that the
cylinder’s axis is ~ 45° from the [110] (and H) direction. This axis tilt is required
theoretically,14 and has been verified experimentally by the location of the CDW
diffraction satellites.15 (The cylinder’s axis is parallel to Guo —Q, which is tilted
~ 45° when Q is only ~ 1° away from [HO].8) The experimental resonance shape
can then be ascribed to the surface resistance alone.
The second puzzle is the sharp drop in R{H) for \H\ > Hc, mentioned above.
In the following section, we will show that this effect arises from the “heterodyne”
gaps, created by the periodicities,
Kn = n(G110 - <?),
(n = 1,2,...).
(2.2)
The energy-gap planes of this family are shown by the dashed lines in Fig. 2.2,
which cut at an angle, ~ 45°, through the central region of the Fermi “sphere”.
The calculated values8 of the first five heterodyne gaps are given in Table 2.1.
Cyclotron orbits for which kz is near zero can be “Bragg” reflected by the periodic
potentials associated with {Kn}i E<1- (2.2). When such reflections occur, the elec­
trons become “ineffective” with regard to their cyclotron rotation. A quantitative
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10
model for this phenomenon is presented in Sec. 2.2; and the observed behavior of
R{H) when \H\ > |# c| is explained.
2.2 Effect of the heterodyne gaps
In this section we will develop a model to account for the disruption of cyclotron
motion caused by the heterodyne gaps, which cut through the central section of the
Fermi sphere, as shown by the dashed lines in Fig. 2.2. (The dc magnetic field H is
parallel to the horizontal, z axis.) The main contribution to the surface resistance
R(H) arises from electrons having velocities nearly parallel to the surface;
so
these electrons (with kz ~ 0) necessarily encounter the heterodyne gaps.
An electron which meets a heterodyne gap during its cyclotron motion can
suffer a momentum transfer ±hK n, given by Eq. (2.2). The result is a disruption
of its cyclotron motion (in the xy plane); and the change in z component of its
velocity can cause it to rapidly leave the microwave skin depth, so its cyclotron
motion is no longer fully effective. We introduce a factor, / < 1, which describes
the probability that the electron behaves “effectively”, i.e., as if there were no gaps.
An electron encountering a small energy gap can also continue on its path
in ib-space, as it would if the gap were not present. This phenomenon is called
“magnetic breakdown”. The breakdown probability P depends exponentially on
H:ls
(2.3)
The parameter Hq depends critically on the energy gap Eg and the orbit geometry:
(2.4)
where H is a unit vector parallel to H, and v is the electron’s velocity at the
energy-gap plane (if Eg were zero). This invariant form17 for Ho is equivalent to
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11
the result derived by Blount.16 It is clear from Fig. 2.2 that an electron with
kz ~ 0 will encounter several heterodyne gaps. For simplicity, we will still employ
Eq. (2.3) to describe the net result of all such encounters. The effective fraction,
on taking into account magnetic breakdown, is then
= 0) = / + (1 - / )e -3 “.
(2.5)
At very high fields, when magnetic breakdown is complete, / e// = 1, i.e., the
electrons behave as they would without a CDW.
For small H, / ./ / = / , the
parameter we introduced above. /, a constant, will be adjusted to fit the data, (f
is not zero because electrons with kz ~ 0 sustain part of their cyclotron motion.)
On account of the complexity, the breakdown parameter
reliably; but we have estimated it to be:
H q~
H q cannot
be calculated
4T .
Equation (2.5) applies only to orbits for which fc* ~ 0; so we must generalize
the effective fraction for all kz. Electrons having a rapid speed along z don’t remain
in the skin layer very long anyway, so the interruption of their xy motion by the
heterodyne gaps is of little consequence. Thus, their effectiveness will approach
unity as |fcz| increases. This behavior can be described heuristically by,
—
fra g i
—
•
( 2 -6 )
The constant 0 will be adjusted to fit the surface-resistance data. The fitted values
are / = 0.8 and 0 = 20. It is clear that f ejj approaches unity rapidly as kz becomes
appreciable; and (of course) / e// equals Eq. (2.5) when kz = 0.
The foregoing ideas are needed to correct the theoretical electron-gas conduc­
tivity, aap(q,uj), which is obtained by solving the Boltzmann transport equation.
For an isotropic, free-electron metal the solution is standard. However, we display
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12
axx and c^y, the components derived from Eqs. (12) and (13) of Ref. 10:
+ ixt + 1 —
ixtt]'
axs = ~ i L * (1 ~ e)[l - i aia+
++
- i ta_
a. +
+ ix
(?xy —
3lCT° f 1 dt( 1 - t2)[------- ------------------- -------- ],
8 J-1 v
1 —ta+ + ixt 1 -ia_.
— +
+ ix
ixtth
(2.7)
K 1
where
a+
=
ne2r
m*
(a; + a;c)r,
a_
=
(a; —a>c)r,
x =ql = qvpT,
t
t is the scattering time,
s
kF
(2.8)
and the magnetic field, H (parallel to £), appears linearly in
a;c, the cyclotron frequency, eH/m*c. The cartesian components of a are displayed
here, instead of the circularly polarized ones, to anticipate the requirements of Sec.
2.3.
Notice that the factor (1 —£2) in the integrand of Eq. (2.7) is proportional
to the cross-sectional area of the Fermi surface for t = kz/k F, i.e., to the number
of electrons in the slice of width dt. However, as argued above, the heterodyne
gaps reduce the effective number by the factor, Eq. (2.6). Consequently, we must
replace:
(1 - i2)- (1 - i2) /« //( U
(2.9)
when the integrals are evaluated. Fortunately these integrals can be found analyti­
cally because, as is evident in what follows, the surface resistance involves a further
integration over the wave vector q, which can only be carried out numerically. The
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13
a.na.lyt.ir expressions for (Txx and <r*y which incorporate the substitution, Eq. (2.9),
are given in Appendix A.
Now, the surface impedance Z for an isotropic metal, having an xy surface at
z = 0, is defined by
2 = kT3x{z)dz
3 v °k -
(210)
With the use of Stoke’s theorem for a circuit in the yz plane and the two Maxwell
curl equations,
£ (0 ) = ^ f i ° M z ) d z .
The
p rim p
(2.11)
indicates d/dz, and the time dependence of the fields is taken as
exp(—iu/t). It follows that,
_ 4ttzu; £*(0)
c2 £'x(0)'
,
}
{' }
Solution of Maxwell’s equations in the metal with specular boundary conditions
at z = 0 can be found in Kittel,18 whom we follow. For the a = x,y components
of polarization,
<p£a(z)
u)2 . . .
47riu/ fcW
... ,
/n *o\
(2.13)
Solution of this equation may be obtained by Fourier transform. It has been
shown experimentally19 that conduction electrons are specularly reflected from
shiny potassium surfaces. Under these conditions, one can treat the metal as
infinite, instead of semi-infinite, provided £(z) is extended symmetrically to the
region z < 0.This means that at z = 0, S' must undergo a jump from- £ '( 0) to
S'{0). Accordingly integration by parts gives
f ° S'e-iqzdz = ( f~° + [°°)£'e~iqzdz = -2£'(0) - q2E(q).
7 —oo
7+o
7 —oo
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(2.14)
14
The Fourier transform of Eq. (2.13) is then
( V + ~ ) E a{q) =
<2-l5>
where for each component, a = x,y,
E (q)
=
-
£ / > * * * ■
S(z) = -i= r E{q)&**dq,
V i'K J —oo
j(z) = - L P
J(q)c*dq.
(2.16)
VZ7T •/—oo
Equation (2.15) is actually a pair of coupled equations because the conductivity
tensor, (2.7), has off-diagonal components. On using <Xy to eliminate Ja{q), Eq.
(2.15) becomes,
!><,(»,«)£,(«) = - ^ 3 ( 0 ) ,
(2.17)
where
( q2 _ ^ _ * ™ axx
Dij(q, uj) =
I
V
~ ^ a xy
2
“ ^ o -y x
q2 - ^ r ~
\
.
j •
“p ^ y y J
(2 -18)
For a spherical Fermi surface, axx = <Tyy and ayx = —<Jxy Eq.(2.17) can then be
solved:
_ , ,
[2
- g - * g V „)3 (0 ) +
(2 , 9)
We now introduce circularly polarized waves accordingly to the convention:
4 (z) = (x ± iy)f±(0)e‘(9z- w<).
(2.20)
3 ( 0) = ± 13 (0).
( 2.21)
It follows that
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15
This relation allows one to solve Eq. (2.19) for Ex(q)/S'X(Q). Subsequently, the
third relation of Eq. (2.16), with z=0, can be used to find £s(0)/£'x(0), which is
all one needs to evaluate the surface impedance (2.12). The final result is, after
restricting the integration to positive q,
Z±{H) = - 8t u f o ^ q2 _ u 2 _ A^ iu ^ xx±iaxyy
(2.22)
(That the integrand is even in q follows from the symmetry of £(z) mentioned
above.) The integration in dq must be carried out numerically with the expressions
for oxx and
from Appendix A. It was found sufficient to sum from q = 0 to
500,000 in 50,000 steps. (Doubling the range or reducing the step size by 10 did
not alter the output noticeably.)
Inspection of the experimental data of Fig. 2.1 reveals that the cavity was not
driven in a pure
” mode. Accordingly, we have calculated the surface resistance
given by,
R(H) = Real[0 .8 Z.(H ) + 0.2Z+(H)].
(2.23)
The residual-resistance ratio of potassium, p(Z00K)/p(4K), is typically ~ 5000.
This value implies a scattering time, r ~ 2 x 10- 10sec. For 23.9GHz, ojt = 30.
R(H) calculated from Eq. (2.23) is shown in Fig. 2.3. The heterodyne gaps cause
the surface resistance to decrease when \H\ > Hc and to level off near \ucfu\ ~ 2.
Not shown is the eventual recovery of R(H) to the ideal Fermi-sphere result for
\u ju \ > 3. The rate of this high-field approach to the ideal R(H) depends on
the magnetic-breakdown parameter, Ho’, so Hq can in principle be estimated by
studying R(H) in the high-field regime. Baraff has reported20 that unpublished
data of Grimes does indeed show the recovery of R(H) just described. (We have
not seen this particular data.)
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16
Interruption of the cyclotron motion for electrons having kz ~ 0, caused by the
heterodyne gaps, reproduces the observed behavior of R(H) when |f/| > He. The
sharp peaks at cyclotron resonance, however, are caused by the cylindrical section
of Fermi surface shown in Fig. 2.2, and will be explained below. The observed
resonance dips near uijw — ±0.77 have nothing to do with the potassium sample.
They are caused by embedded particles of Cu^SO^ • 5 H2 O in the cavity walls
created during fabrication.13
2.3 Resonance from the Fermi-surface cylinder
The minigaps, shown by the short, solid lines in Fig. 2.2, correspond to the
periodicities of Eq. (2.1). The sizes of the first few minigaps,8 tabulated in Table
2.1, are substantial. The two black patches in Fig. 2.2 represent a small Fermisurface cylinder which forms between the first minigap and the main CDW gap
(having periodicity Q). It has already been shown10 that such a cylinder can
explain the occurrence of the sharp cyclotron resonance observed by Grimes, and
reproduced in Fig. 2.1.
The size of the resonance requires the volume of the cylinder (pieced together
from the two halves) to be a very small fraction, 77 ~ 4 x 10~4, of the Fermisphere-volume. It is noteworthy that this volume fraction agrees with the value
calculated from the product of the cylinder’s length and its cross-sectional area.
The former is obtained from the neutron-diffiraction measurement of Q,15 and the
latter from the periodicity of the Landau-level oscillations,12 observed in microwave
transmission.11 The cylinder’s radius is kc = kp/8, and its length (projected along
[110]) is 0.015(?no. Although Q is tilted from [110] by about 1°, the cylinder’s
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17
axis, Q' = Guo —Q, is tilted about 45° from [110].
O' « (0.025,0.015,0.005)— .
a
(2.24)
Its cross section is approximately circular in a plane perpendicular to [110].
It is of interest to calculate first the surface resistance, R(H), caused by a
cylinder having its axis parallel to the magnetic field, H. On account of the
cylinder’s small size, electron velocities on the Fermi surface of the cylinder are
also small. Accordingly we will use a local conductivity tensor for the cylinder.
The dc conductivity in the xy plane is t/<toc and crzz = 0, where <jQc is ne2rc/m m.
The sharpness of the observed resonance corresponds to
cjtc
~
150. That rc
(on the cylinder) should be ~ 5 times larger than r on the main Fermi surface is
reasonable because of the smaller velocities of the cylinder electrons. The cylinder’s
conductivity tensor is then:
( I - i0JTc
a cyl _
_________ ^ O c
(1 - iurc)2 + (ujctc)2
LJCTC
0
—coct c
0\
1 —iUTc 0
0
(2.25)
0J
For this exercise we will neglect the effect of the heterodyne gaps. Consequently
ocvt, Eq. (2.25), is added to the conductivity, Eq. (2.7), for an ideal Fermi sphere.
The surface impedance is still given by Eq. (2.22), and R(H) for 80% circular
polarization is obtained from (2.23). The result is shown in Fig. 2.4 with utc = 150.
A sharp cyclotron resonance is obtained but, unlike the data of Fig. 2.1, the shape
is asymmetric.
The sharp, asymmetric resonance shown in Fig. 2.4 was obtained previously,10
but the remedy attempted then involved introduction of a more than 50-50 admix­
ture of surface reactance and surface resistance. However, a remedy not involving
such an admixture is possible. Since the cylinder’s axis must, theoretically, be
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18
tilted ~ 45° from [110],M, an angle confirmed by neutron diffraction,15 we now
study the effect of such a tilt on the resonance shape.
The equation for a cylindrical surface of constant energy, e = Ep, having an
axis at an angle 6 relative to the direction of H, and with k relative to the cylinder’s
center, is
€ —€0 =
tan e)2 +
(2-26)
This cylinder has a circular cross section in the xy plane. Consequently the cy­
clotron frequency, with H along z, is unchanged. (For the cylinder of interest here,
€ —eo = Ep/64.) On account of its small size, as already discussed, the electron
velocities on this surface are ~
Vp / S .
We will therefore employ local equations of
motion to find the tilted cylinder’s conductivity tensor, cr0*1. The Lorentz equation
for motion in the electric and magnetic fields is
'v = —eM~l (k)[E + - v x H] —
C
Tc
(2.27)
where hv = Vfce(^), and the effective mass tensor is,
(2-28>
Then
Vx — ■■(^>x fcjtan^),
m
Vy
*
—
+ky,
m*
vz — — ~ (k x — kz tan 0) tan 6.
m*
(2.29)
Equations (2.26)-(2.29) can now be used to find the conductivity of the cylinder.
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19
W dc
tf* =
(1
/
+ (wcrc) 2
1—
iu)Te
iuTc ) 2
ucTe
—<
jJctc
—tan 0(1 —iu>Tc) ^
1 —iuTc
- tan 9(ojcrc)
v—tan 0(1 —iurc) tan 6((Jctc)
(2.30)
2
tan2
0(1 —ia/rc) /
After compaLiring this tensor with Eq. (2.25), for which 0 = 0, it is clear that the
electric field may now have a longitudinal, z component. Jx and Jy of Eq. (2.15)
now involve Ez because crxz and ayz are no longer zero. However, we can express Ez
in terms of Ex and £y by using the requirement that the total longitudinal current
Jz be zero everywhere. Accordingly,
(2.31)
The longitudinal conductivity of the spherical portion of the Fermi surface must be
calculated nonlocally using the Boltzmann transport equation. For a longitudinal
electric field, proportional to
<zzl h
=
e
t
d
t
\
—
r
+
2 J —i
1 — iujr
v
ixt ’
3<70
[2x —r + 2urp + uj2t 2t + i(p —2u)tx + 2utt —u»2r 2p)], (2.32)
2x*
with
P
2 1 + (x + a;r)2
r = tan-1 (x —ur) + tan-1 (x + vr).
(2.33)
Equation (2.31) together with Eq. (2.15) changes Eq. (2.17) as follows:
r2
u2
Airiuj ,
I? ~
^2
4iriu ,
^
2
.2
~
+ [? ~~
471i u ,
l x
u2
g
xy
V
= - ^ ( 0),
Amu ,
g
a y y \E y
2
—
y 7r y
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(2.34)
20
where ovy = a i f 1+ a^j1. These two equations can be expressed compactly:
Ai(?, «)£<(») =
(2-35)
with
£>«(,,*) =
■*.
4*«>/ _ /
1
,
_2
?~°V*
T
.
or
4
4tiw _ /
-^ -°v y .
1
•
(2-36)
Here,
4 = ** -
(2-37)
O-zz
i and j denote x or y. The difference between Eq. (2.36) and Eq. (2.18) of Sec.
2.2 is that all transverse conductivities, cr^, in Eq. (2.18) are replaced by
in Eq. (2.36). For example,
a'xx includes axz, azx1 and azz as well as
The
longitudinal motion of electrons in the cylinder leads to creation of an electric field
in the z direction. The Sz which arises (to preserve charge neutrality) plays a role
in producing the transverse currents j x and j y due to the nonzero values of axz
and
ayz. (The tilted Fermi-surface cylinder mixes the transverse and longitudinal
motions.) Even though the number of electrons in the cylinder is small, this mixing
causes a large change in the surface impedance. The total conductivity tensor has
the following properties:
(Tyx = ~ 0 ’x yi
&zy = ~ & y zi
Accordingly, from Eq. (2.37), <7^ = —
*^
E ( )
v^
a zx ~ &xz-
(2.38)
Equation (2.35) may now be solved:
- 4.niu}<T,yy)£x(0 ) +
V 7r {q2c? —ui2 —A.'Kiuia'xx){q2(? — a/2 — Akujcx^) + {kxiwo'^ ) 2 ’
_ f2
P
(q2 c? - J
2
A‘KiuioJXySx(0) + (gV - u 2 - 4iriuja,xx)£y(0)
V 7T(q2c* —u}2—47riwa'I )(g 2c 2 — w2 — iiriujcr^) -f (Airiua!^ )2
(2.39)
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21
Using Eq. (2.11), we express £•(0) in terms of the total current density J,-,
Ji=
f
Jo
ji(z)dz.
(2.40)
The third equation of (2.16), together with (2.38)-(2.40), give the electric field at
z=0:
m
r00, ________ (gV - u? - 4xiu}<T,yy)Jx -f 4'Kiua/:cyJy________
X h
^(g2^ — oj2 —A^iuia'xx){q2(^ —u 2 — 47riu;o^y) + (47rio;cT^y)2 ’
£*«>) = - f
/•“
Jo
, _______ -iwiu)<j,xyJx + ( g V — u ) 2 - 4itiua/xx)Jv_______
^(q2c? —uj2 —47riu/<r'3.)(g2c2 —to2 —47riu;a^y) 4 - {i.iriua,xy)2'
(2.41)
These expressions can be written compactly:
£r(0) = ZXXJX + ZxyJy,
£y(0) =
—ZxyJx + Zyyjyi
(2-42)
which by inspection of (2.41) defines the four components of ZQ$, the surface
impedance tensor.
It is clear fromEqs.
(2.40) and (2.41) that JQ (a = x,y) depend intricately
on the bulkelectricfields. Anisotropy caused by thecylinder’s
tilt causes Ja
to be a complicated function of the conductivity components. This asymmetry
also prevents the field from having perfect circular polarization. This behavior is
studied in Appendix B. Nevertheless, on account of the small size of the cylinder,
the electric field polarization is almost circular. Accordingly,
Hy(Q) « ±iHx(0),
Jy
« ±iJx.
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(2.43)
22
The electric field at the surface will be Sx = Soe- *"* and £v = i£x, which corre­
sponds to right circular polarization. (So is real.) Then from Eq.(B.25),
J
'
— c^ Dr ~ iwt
_
2ir
’
J» =
(2,44)
The power absorbed per unit area per unit time is
S, = ^-{& [£(0)) x Jie[£(0)]}„
47T
= ^ { B e [^ (0)]Be[H,(0)] - J8e[^(0)]He[«,(0)]},
= Be[£«(0)]fle[JJ + Re[£,(0)\Re{J,\.
(2.45)
We now separate Zap [defined by (2.41) and (2.42)] into their real and imaginary
parts, i.e.,
Zap = Ra0 + iIQ0,
(2.46)
where R a 0 is the real part of Z a 0 and I a 0 its imaginary part. It follows that,
/»
Re(Jx) = —- cos(a;t),
27T
Re(Jy) = ~siii(w i),
Re(Sx) = -r-^[i2xx cos(wt) + Ixx sin(ut) 4- iZ™sin(ut) - 1 ^ cos(u>£)],
2/ir
cSn
Re(Sy) = ~ [ R n sin(a;t) —Im cos(a;i) —i?™cos(ut) —Ixy sin(cjt)]. (2.47)
27T
By using these expressions in Eq. (2.45) and averaging over time, we find the
absorbed power.
Sz = Re[Sx(Q)]Re[Jx] + iZe[£v(0)]/2e[Jv],
i
C2^ 2
87T2' ( R x x ■+" R y y ~ 2/ry)»
z"2/*2 1
^ . * e [ i ( Z xx + Z„) + iZxy\.
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(2.48)
23
The effective surface resistance is therefore,
R = Re[-(Zxx + Zn ) + iZxy].
(2-49)
From Eqs. (2.41) and (2.42), and £y = i£z for right circular polarization, the
surface impedance is,
ZR = -(■£«: + Zyy) + iZxy,
dq{[q2<? - u P - 2 'Kiu(o'xx + a'm)\ - 4?r
=
X{q2<P—up —4iriuj<T/xx)(q2<P—up —47rio;<7£v) -+• {Aitiuia'^)2'
For a left circularly polarized wave on the front surface, i.e., £y = —i£x, the surface
impedance is,
Z
l
=
—( Z x x + Z y y ) — i Z x y ,
=
JQ dq{[q c ? -U
2
2
-
2 niuj(</xx + <
T'yy)] +
4Ku><T,xy}
(q2P —up- — 4iziuja'xx){q2c2 — up — Aniua'yy) 4- (Airtua'^)2 ^ ^
Eqs. (2.50) and (2.51) must be evaluated numerically, as in Sec. 2.2. The
effective surface resistance applicable to the experiment, for which the polarization
was about a 4:1 admixture of L and R, is now
R(H) = Real[0.8ZL{H) + 0.2ZR(H)].
(2.52)
The theoretical R{H), which includes effects from both the tilted cylinder and the
heterodyne gaps, is shown in Fig. 2.5. The agreement with the experimental data
of Fig. 2.1 is remarkable.
2.4 Conclusions
Inspection of Figs. 2.1-2.5 allows one to recognize that the CDW in potas­
sium6, 7>8 has profound consequences in studies of the perpendicular-field cyclotron
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24
resonance. The fact that cyclotron resonance even exists (in the surface resistance,
R vs. H) attests to the presence of the small Fermi-surface cylinder (the dark areas
of Fig. 2.2), created by the CDW gap and the first minigap. A theory based on
only a spherical Fermi surface does not allow any structure near a/c = a/, as shown
by the top curve of Fig. 2.1.
The shape of the R{H) resonance (compare Figs. 2.4 and 2.5) reveals that the
cylinder’s axis is tilted away from [110] (the field direction) by ~ 45°, as was found
theoretically.15, 8 (The reason for the tilt is to minimize the elastic-stress energy
involved in creating the periodic lattice distortion, of wave vector Guo —Q, needed
to screen the electronic CDW.14)
The drop in R for \H\ > |# c|, see Fig. 2.3 and the experimental data of Fig.
2.1, arises from the heterodyne gaps (Fig. 2.2), which interrupt the cyclotron
motion of equatorial orbits, and cause a partial loss in carrier effectiveness.
The volume of the Fermi-surface cylinder (corresponding to i) = 4 x 10~4
electrons/atom) was determined from the size of the resonance relative to R(HC) —
R{O).10 The fact that this volume equals the product of the cylinder’s length (along
[110]), determined from Q (observed in neutron diffraction15) and the cylinder’s
cross section (perpendicular to [110]), defined by the periodicity of Landau-level os­
cillations observed in microwave transmission,12 indicates a compelling consistency
among relevant phenomena.
Fracture of potassium’s Fermi surface by CDW minigaps and heterodyne gaps,
Fig. 2.2, is not only evident in the surface resistance anomalies studied here,
but is the cause of many other magnetotransport effects, the most spectacular of
which are the multitudinous open-orbit resonances21 created by the minigaps and
heterodyne gaps. These open-orbit spectra have been explained within the same
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permission o fth e copyright owner. Further reproduction prohibited without permission.
25
framework employed here.17 Without a broken symmetry, potassium would be the
simplest metal of all since, unlike Li22 or Na, it would retain its cubic symmetry
to helium temperature. However, as a consequence of its CDW, potassium has
provided (during the last thirty three years) a veritable universe of unanticipated
behavior - a challenge to all who seek to understand electrons in metals.
2.5 Reconciliation of the surface resistance and microwave transmission
In Chap. 4 it will be found that a theory of the CR transmission and the
CR subharmonics requires the presence of a cylindrical Fermi surface having a
radius in the xy plane, k& ~ 3fcf/8. The question arises, of course, how this
additional cylinder affects the surface resistance behavior. This complication has
been resolved by taking into account the size of Q-domains, which are likely to be
small in Grimes’s specimens, since they were rolled (in oil) between mylar films,
and so would be expected to have small-scale undulations.
Only cyclotron orbits which do not intersect Q-domain boundaries would con­
tribute to the surface-resistance cyclotron resonance. The effective fraction of cylin­
der electrons is then, 77^ [l —(2RifD)], where Ri is the cyclotron radius (i=l,2,3),
hckd/eH, and D is the width of the (laminar) Q-domains. (If the factor in square
brackets is negative, it must be set equal to zero.) A modified theory of the surface
resistance which includes all three cylinders in Fig. 3.4 is shown in Fig. 2.6. The
Q-domain size is D=1.5/xm,
6
— 50°, yo = 3, u/rc = 150, Hq = 4, f=0.78, and
(3 = 20. In this fashion both the CR transmission and the surface resistance can
be understood quantitatively utilizing the same Fermi surface model.
Reproduced with
permission of the copyright owner. Further reproduction prohibited without permission.
26
CO
c
Theory
3
CO
DPPH
Marker
o
S:
X
DC
Experiment
-
1.5
1
-
0.5
o
0.5
1
1.5
ooc / c o
Figure 2.1
Surface resistance of potassium versus magnetic field (loc = eH/m*c). The data,
due to C. C. Grimes (1969), is for T=2.5K, and circularly polarized radiation at
v/2 ir = 23.9GHz. The dips near ±0.77 are due to particles of C 11SO 4 • 5H2 0,
embedded in the cavity walls during fabrication. The cyclotron resonance, at
u)c/u> = —1, occurs when H = 1.03T. The small resonance, at (jjJoj — 1 is caused
by a small admixture of the opposite polarization. The theoretical curve is for a
purely spherical Fermi surface, which potassium would have in the absence of a
CDW broken symmetry.
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27
CDW GAP
r* F.S. SPHERE
[1
1 0
]
HETERODYNE GAPS
GAPS
BZ
Figure 2.2
The Brillouin zone of potassium on an (001) plane in fc-space. The angular tilt,
relative to [110], of the CDW wave vector, Q, has been exaggerated for clarity. The
minigaps and heterodyne gaps are associated with the periodicities of Eqs. (2.1)
and (2.2). The shaded areas are the two halves of the Fermi-surface cylinder, which
form between the CDW gap and the first minigap. The axis of the cylinder is Guo—
Q, which is also the direction ofthe heterodyne-gap vectors. The dc magnetic field,
H, is applied parallel to [110], which is the habitual texture direction, perpendicular
to smooth potassium surfaces. The (ideal) Fermi sphere is also shown.
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28
1.04
o
£
i
tt
0.96
1
0
1
C0 c / CO
Figure 2.3
Theoretical surface resistance for a Fermi sphere having only heterodyne-gap in­
tersections. The parameters of Eq. (2.6), which quantify the loss in effective
cyclotron motion on equatorial orbits, are / = 0.8 and 0 = 20. The drop in R for
|F | > Hc increases with decreasing /. The steepness of the decline increases with
increasing 0. The magnetic-breakdown field is Hq = 4T. The electron scattering
time corresponds to u t = 30.
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29
O
cc
x
oc
0.9
1
0
1
G)c / CO
Figure 2.4
Theoretical surface resistance of conduction electrons having u;r = 30 on the Fermi
sphere and ujtc = 150 on the Fermi-surface cylinder (containing ~ 4x 10-4 electrons
per atom). The axis of the cylinder is, here, parallel to H, and the heterodyne
gaps (intersecting the sphere) are ignored. A 4 : 1 ratio of left to right circular
polarization is assumed.
R eproduced will, perm ission o fth e copyright owher. Further reproduction prohibited without perm ission.
30
1.04
©
CC
I
CC
0.96
1
0
1
(0 C / G)
Figure 2.5
Theoretical R (ff) for potassium based on the heterodyne-gap parameters of Fig.
2.3 and the Fermi-surface cylinder model of Fig. 2.4, except that the cylinder’s
axis is tilted 45° from [110]. (The tilt is required to minimize the elastic stress
of the periodic lattice distortion needed to neutralize the electronic CDW.) This
calculated behavior should be compared with C. C. Grimes’ data in Fig. 2.1.
Reproduced with permission o fth e copyright owner. Further reproduction prohibited without permission
31
1.04
©
£
x
£
0.96
1
0
1
(Dc / 0)
Figure 2.6
Theoretical R(H) for potassium based on the Fermi surface model of Fig. 3.4. The
Q-domain size assumed for the specimen measured by C. C. Grimes is D=1.5/zm.
The other parameters used are enumerated in Sec. 2.5.
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32
Table 2.1
Calculated values, from Ref. 8, of the first five minigaps and heterodyne gaps for
K. The main CDW gap was taken to be 0.62 eV and the zone-boundary energy
gap was 0.40 eV.
n Minigap
Heterodyne gap
1 90 meV
16 meV
2 67
14
3 51
12
4 34
8
5 15
3
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3. MICROWAVE TRANSMISSION IN A PERPENDICULAR MAGNETIC
FIELD.
The cyclotron resonance is measured using a microwave transmission technique.
The metallic sample forms a common wall between the transmit and receive mi­
crowave cavities. The microwave cavities used are rectangular and are excited in
the TE101 mode. As a consequence the driving fields at the surface of the sample
are linearly polarized, and the receiving cavity accepts only the linearly polarized
field having the same orientation as the driving field. As shown in Fig. 3.1 the
potassium sample is compressed between two quartz windows. To improve the
surface quality of the samples, the compression was carried out in vacuum. The
small area of the quartz window relative to the sample area provides microwave
isolation between the send and receive cavities. A dc magnetic field is oriented
perpendicular to the plane of the sample. The transmitted signal is detected and
displayed by conventional techniques.
Dunifer, Sambles, and Mace reported transmission data on 15 potassium sam­
ples.1 Figure 3.2 shows the transmission signal in sample K-4 as the magnetic field
is swept from 0 to 5.5kG at T = 1.3K. The field is expressed in terms of the
dimensionless quantity uc/u> where o/c = eH/m*c is the cyclotron frequency. The
effective mass m* is 1.21m.2 The microwave frequency is 79.18GHz. Therefore the
field at u)c/u> = 1 is 3.42T. The conduction-electron spin resonance (CESR) and
associated spin wave modes lie in the region 0.8 < ujc/ oj < 0.9. For display pur-
with permission o fth e copyright owner. Further reproduction prohibited without permission.
34
poses, the gain in this region has been reduced by a factor of five on account of the
strength of the spin resonance. Close to uc/ uj = 1 there is a strong transmission
signal and is, of course, the cyclotron resonance (CR). The general oscillatory sig­
nal seen over most of the field sweep is the Gantmakher-Kaner (GK) oscillations.
There are high-frequency (HF) oscillations near u/c/u/ = 1.2 and 0.6. Finally, super­
imposed on the GK oscillations can be seen subharmonics of the CR at u c = u/2
and w/3. Therefore there are five signals which appear in the microwave transmis­
sion of potassium, (a) CESR, (b) CR, (c) GK oscillations, (d) HF oscillations, and
(e) CR subharmonics. The study of the CESR and the spin wave side bands is well
understood.3 This is the only signal that can be explained using a free-electron
model. Even for this signal, there are splittings of some spin-wave side bands which
can’t occur if the Fermi surface is spherical. The splittings were explained by in­
volving CDW domains.4 Among the four non-spin related signals only two have
been successfully explained.They are the GK oscillations and the HF oscillations.
In the following sections we briefly review these two signals.In Chap. IV, we will
study the (until now) unexplained signals, CR and CR subharmonics.
3.1 Gantmakher-Kaner oscillations
The oscillatory signal seen throughout the field sweep are the GantmakherKaner (GK) oscillations.5 We can clearly see them near o/c/o/ = 1.3 in Fig. 3.2.
When a magnetic field is applied perpendicular to the surface, the electrons with
the fastest velocity parallel to the field cause the GK oscillations.
Suppose that an electron travels for a time equal to an integer number of
cyclotron periods :
_
27tm*c
t = n T = ti— — .
eH
,.
(3.1)
We suppose that during this time the electron travels from one surface of the slab
R eproduced with perm ission o fth e copyright owner. Further reproduction prohibited without permission
35
to the other. This condition requires,
2ttm*c
vFt = n— tz—vf = L,
eJi
, .
(3.2)
n = n eL- - H.
2ftcm*vp
(3.3)
whereupon
The oscillations are therefore periodic in the magnetic field. If we rewrite this
equation in terms of a/c and a/,
uc/u> = —— n.
Loj
(3.4)
For An = 1,
A(wc/w) =
(3.5)
The periodicity of the GK oscillations is determined by the Fermi velocity of the
conduction electrons and the sample thickness only. Therefore, with L =
vp = 7.36 x 107cm/sec, and u = 79GHz, we obtain A(a/c/u;) for each GK oscilla­
tion to be ~ 0.1. This agrees with the data in Fig. 3.2. Note that the amplitude
of the GK oscillations increases as the field increases in Fig. 3.2. In Chapter IV,
we will explain this behavior using a magnetic breakdown effect at the minigaps.
3.2 High-frequency oscillations
There are two sets of high-frequency oscillations in Fig. 3.2, one near uc/u =
0.6 and the other near u/c/u/ = 1.2. We show enlarged views of those regions in
Fig. 3.3. Lacueva and Overhauser studied the data in Fig. 3.3(a) to discern the
origin of the oscillations.6 They found that the periodicity of the oscillations is
linear in 1/H. This is the signature of Landau-level oscillations. Landau-level
quantization causes a periodic variation of any physical property that depends
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36
on conduction-electron response.7 A property A will then acquire an oscillatory
component,
n = integer,
A ~ cos(2irn),
(3.6)
where
» - (*£>/<*■>•
chfcj?
~2eH'
F
H'
_
“
(3.7)
F is the de Haas-van Alphen frequency,
=
F
27re
(3-8)
A is the extremal area in k space (perpendicular to H ) of the Fermi surface in­
volved. For a free-electron Fermi sphere A = nkp = 1.74 x 1016cm-2 for potassium
with a lattice constant, a = 5.2295A. Then the de Haas-van Alphen frequency for
a Fermi sphere is
F0 = 1.828 x 104T.
(3.9)
The de Haas-van Alphen frequency from the data is
Fi = 266T.
(3.10)
The ratio between F0 and Fi is,
Cl
= 69.
(3.11)
Because the frequency is proportional to the extremal area as in Eq. (3.8), we cam
estimated the cross-sectional area which causes the signal.
£ = |
= 69.
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(3.12)
37
Therefore the equivalent radius of the new Fermi surface is,
ki
kp/%,
(3.13)
The only possible Fermi surface (in addition to a sphere) is a cylinder created by
the CDW as shown in Fig. 3.4.
As one can see in Fig. 3.3(b), there are also high-frequency oscillations near
ujJ oj
= 0.6. From Eq. (3.7), one can find the periodicity of these oscillations
versus field.
|A"I =
P -14)
The interval between oscillations is then,
* * « IS -
(3i5)
If the oscillations near ojc/ u> = 0.6 were caused by the same cylinder as for uc/u =
1.2, the spacing between them would be reduced by a factor of four. However, the
periods of both oscillations are about the same, so the Fermi surface area causing
the oscillations in Fig. 3.3(b) is four times smaller than the Fermi surface area
causing the oscillations in Fig. 3.3(a). Accordingly,
These Fermi-surface cylinders are the two smallest ones shown in Fig. 3.4.
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38
From Klystron
(79 GHz)
To Detector
(0 - 5.5 T)
Potassium slab
(L = 85 urn)
Figure 3.1
Rectangular microwave cavities used for the observations. Small fused quartz
windows are placed on both sides of the sample.
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39
Cyclotron Resonance
CESR
Landau-Level
Oscillations
from FS cylinder
\ /
Cyclotron Resonance
Subharmonics
0
Gantmakher-Kaner
Oscillations
1
0 .5
1.5
G)c / CO
Figure 3.2
Microwave transmission signal vs. H through a potassium slab in a perpendicular
magnetic field. (H = 3.42T at cjJoj = 1) The microwave frequency is 79.18 GHz
and the temperature is 1.3K. The field at u)c/u) = 1 is 3.42T. The phase of the
microwave reference was adjusted so that the cyclotron resonance is symmetric.
The slab thickness is L = 85pm. The data, provided by G. L. Dunifer, were
obtained from sample K4, one of fifteen samples listed in Ref. 1.
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40
W
c
3
■
JD
a.
<0
-I
<
z
o
CO
1.1
1.15
1.2
(Dc t
1.25
1.3
0.65
0.7
CO
w
c
3
n>_
<0
-i
<
z
C5
CO
0.5
0.55
0.6
COc /( D
Figure 3.3
Landau-level oscillations near a/c/cu = 1.2 and 0.6. The periodicity in (a) requires
a cylinder radius, kc ~ kp/8, corresponding to the middle cylinder of Fig. 3.4.
The periodicity in (b) requires a cylinder radius, ke ~ k p /16, appropriate to the
smallest cylinder in Fig. 3.4.
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41
Figure 3.4
Fermi surface cylinders of potassium. The horizontal axis is parallel to [110] and
to the dc magnetic field. The shaded cylinders are created by the CDW gap and
the first three minigaps. The thicknesses of the cylinders have been exaggerated
by a factor of ten. Each of the half-cylinders shown is joined to a partner on the
opposite side of the Fermi surface by Bragg reflection at the energy-gap planes.
The complete Brillouin zone is shown in the inset.
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42
4. THEORY OF THE CYCLOTRON RESONANCE TRANSMISSION
4.1 Introduction
During the last thirty years a variety of magnetoconductivity anomalies have
been discovered in potassium, the simplest monovalent metal.1, 2 Unlike Li and Na,
which undergo a crystallographic transformation to the 9R structure3 when cooled
to low temperature, a single crystal of K is not destroyed by cooling. Without a
charge-density wave (CDW) broken symmetry, the bcc lattice of K would support a
spherical conduction-electron Fermi surface. Low-temperature transport anomalies
could not then arise. Nevertheless, in dc experiments extraordinary phenomena
occur which require the Fermi surface to be multiply connected. Examples are
the four-peaked induced-torque patterns of single-crystal spheres,4i 5 the many
open-orbit resonances,6, 7 and the magnetoserpentine effect8.
It is not surprising, therefore, that unexpected phenomena also appear in the
microwave properties of K. Fig. 3.2 shows the microwave transmission signal (at
a frequency, u;/27r = 79.18GHz) through a K slab in a perpendicular magnetic
field. Dunifer et al.9 studied this phenomenon in fifteen samples at T = 1.3K.
The data shown (from sample K-4) were kindly selected by G. L. Dunifer, since
it revealed clearly all five transmission signals. The horizontal axis of Fig. 3.2 is
ujc/ u , which
is proportional to the external magnetic field, H, since ojc = eH/m*c.
The cyclotron mass is m* = 1.21m;10 so the field for cyclotron resonance, ujc/ oj = 1,
is Hc = 3.42T.
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43
The signal at and near the conduction-electron spin resonance (CESR) has been
explored extensively.11 It is the only feature in Fig. 3.2 that can be explained by
a free-electron model. Gantmakher-Kaner (GK) oscillations should also appear,
but their amplitude should be a hundred times larger.12 The cyclotron resonance
(CR), its subharmonics, and the rapid oscillations shown near u c/ui = 1.2 should
not even exist (without a CDW).
The purpose of this study is to show that the CDW broken symmetry of potas­
sium explains CR transmission and CR subharmonics. One must, of course, solve
self-consistently a Boltzmann transport equation (for conduction electrons) to­
gether with Maxwell’s equations. Of crucial importance is the influence of CDW
energy gaps on the Fermi-surface topology. A schematic illustration of the com­
plexity introduced by the CDW is shown in Fig. 3.4.
It is known from observed optical anisotropy13 that K has a single CDW. The
CDW wave vector Q is tilted about a degree from a [110] direction. From detection
of neutron diffraction satellites,14,15 it was found that,
-•
27T
Q = (0.995,0.975,0.015)— .
a
(4.1)
The magnitude of Q is 1.5% smaller than that of the smallest reciprocal lattice
vector, Guo. The phonon mode which screens the electronic CDW has wave vector,
Q ' = G u o - <?•
(4.2)
Minimization of the elastic energy required to neutralize the electronic CDW leads
to a tilt of Q' about 45° away from [110].16 The “heterodyne gaps”, shown by
the dashed lines of the inset in Fig. 3.4, are created by the periodicities, nQ\
n = 1,2,... They are important in explaining the open-orbit spectra7 and in un­
derstanding the shape of the CR signal observed in perpendicular-field microwave
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44
surface resistance.17 The heterodyne gaps do not significantly affect the microwave
transmission, so we shall ignore them in what follows.
The horizontal axis of Fig. 3.4 is parallel to [110], parallel to the magnetic
field, and perpendicular to the surface of the K slab. Although such samples are
polycrystalline, they have recrystallized in contact with the smooth, amorphousquartz plates used to hold the K slab in the window between the transmit and
receive cavities.9 It is known from low-energy electron diffraction that thin alkalimetal samples, so deposited (or recrystallized), are epitaxially oriented with closepacked (110) planes parallel to the surface. Furthermore, such surfaces are smooth
enough for conduction electrons to be specularly reflected.18 Interfacial energy will
be optimized when Q is also perpendicular to the surface and, therefore, nearly
parallel to the [110] surface normal.
The important energy gaps for the present study are the main CDW gap and
the sequence of “minigaps”, shown in Fig. 3.4. The wave vectors that describe
these small gaps are:
K = (n + l)Q —tzGho-
(4-3)
The sizes of these gaps have been estimated theoretically,19 and are listed in Table
2.1. The minigaps create several cylindrical sheets of Fermi surface. Each of the
three shaded surfaces shown in Fig. 3.4, formed by the CDW gap and the first three
minigaps are joined (by Bragg reflection) to equivalent surfaces on the opposite
side of the Brillouin zone. Accordingly, each cylinder has twice the length that
appears in the figure. The cross section of each cylinder in a plane perpendicular
to [110], i.e., not perpendicular to the cylinder’s axis, is circular.
The rapid oscillations near uic/u = 1.2, shown in Fig. 3.2, have been found
to be periodic in l/H to very high precision.20 The periodicity corresponds to a
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
cross-sectional area 69 times smaller than irkp, the extremal area of the ideal Fermi
sphere. Consequently this Landau-level oscillation pattern arises from a cylinder
with a radius (in a plane perpendicular to [110]) equal to kp/8. We believe this
cylinder is the one formed by the first and second minigaps. The reason for this
assignment is that the analogous, rapid oscillations near v c/u = 0.6 corresponds
to a cylinder with a radius, ~ kpj 16. This cylinder is likely the one formed by the
CDW gap and the first minigap. The third cylindrical surface, shown in Fig. 3.4, is
the one to which we will attribute the CR transmission and the CR subharmonics.
Its radius is estimated in what follows to be ~ Zkp/8.
The influence of the minigaps has also been noticed in a study of the GK
oscillations.12 If a free-electron Fermi-surface sphere is employed, the transmitted
power ratio will be 10-18 instead of the value, 10-22, observed in sample K-4. Since
the microwave transmission signal is carried primarily by electrons having a rapid
velocity, u2, parallel to [110], they are the ones most affected as they encounter
the minigaps while undergoing cyclotron rotation in the xy plane. The n’th GK
oscillation occurs when the time, L/vp, to traverse the slab (of thickness L) equals
n cyclotron periods. Accordingly, the GK oscillations are periodic in H, as shown
by the transmitted signal amplitude in Fig. 4.1. This signal was calculated from a
free electron model (without spin). Not only are the features (described above) of
the observed signal absent, but the gradual growth from small to large uc of the
GK oscillations (a factor of five in Fig. 3.2) does not appear.
The effect of the minigaps on the cyclotron motion of electrons with large |uz|
can be modeled by a v2-dependent scattering time.12
r(vz) = -----—r,
1 + y\v»/vF\
(4.4)
where r0 is the scattering time attributable to impurities, and y is an adjustable
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46
parameter intended to account for the interruption of cyclotron motion by mini­
gaps. Magnetic breakdown of electron trajectories at the small minigaps implies
that y will be a function of H. The simplest way to model this effect is to let,
y(H) = y0[l - exp(-H 0/H)],
(4.5)
where y0 is a constant and Ho is a magnetic-breakdown field. Incorporation of
(4.4) and (4.5) in the Boltzmann equation allows one to reduce the GK signal by
the required two orders of magnitude and to fit its observed field dependence. This
consequence of the minigaps is treated in Sec. 4.3.
The major challenge of this study, however, is to include the influence of a tilted
Fermi-surface cylinder in the Boltzmann transport theory, solved self-consistently
with Maxwell’s equations. Complexity arises from the lack of axial symmetry
about the magnetic field direction. The general theory is developed in Sec. 4.2
and is applied to the microwave transmission of K in Sec. 4.4.
4.2 Microwave transmission in an anisotropic, non-local medium
Microwave propagation (which we take to be along z, perpendicular to the
metal surface) is governed of course by Maxwell’s equations :
*7
ld S
c
V x £
-
_
_
T-i
4x-? 1 d£
V xH = —j +
.
c
c at
(4.6)
These six equations can be reduced to three by taking the fields proportional to
e-iwt and eliminating H :
dP6a{z) 4niuj
- gzT ~ +
.
47riu;
d2
.
u)2 „
- g - h + ~^;£z -
a = *,y>
„
0.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.,
(4.7)
47
The current density j and the electric field £ depend only on z. However, since the
conduction-electron mean free path, I = vpr, in potassium near T=0 is typically
~ 10_2cm, whereas the microwave skin depth is approximately 2 x 10~5cm, the
relation between j and £ is nonlocal:
M Z) = J 2 [ Klm(z,z',u)£m(z')dz'.
m=lJo
(4.8)
Kim{z,z’1u) is the (non-local) conductivity tensor, and L is the thickness of the
metal slab.
Specular reflection of conduction electrons at smooth K surfaces18 allows a
simplification of Eq. (4.8) if the Fermi surface has axial symmetry about the
surface normal (and the dc field H). The integration limits can be extended to
infinity provided the microwave field is described by a Fourier cosine series (so that
£ and j are symmetric about z=0, L).21 Accordingly,
jiiz) = Y l [
Ktm{z - z\u})£m{z')dz'.
(4.9)
m = l J ~ °°
The Fourier expansions for £ and j are :
£*{z) = £ £ £ c o s ( g nz),
n=0
Jaiz) = 53
n=0
C0S(?n2),
(4-10)
n = integer.
(4-11)
where
T17T
qn = -p ;
.
The fundamentalreason Eq. (4.9) is allowed arises from the fact that on specular
reflection vz changes sign but not its magnitude. The reflection symmetry of £a(z),
Eq. (4.10), implies that the past history of an electron approaching 2 = 0from the
right is the sameas that imputed for an electron approaching from theleft along
a path which joins that of the original reflection.
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48
In general Eq. (4.9) would not be valid for a Fermi surface lacking axial sym­
metry, e.g., an ellipsoid having its axis tilted from the surface normal. However, for
the special case of an anisotropic conductivity caused by a tilted, cylindrical Fermi
surface, Eq. (4.9) is still valid because vz retains its magnitude upon reflection.
The formalism developed here anticipates that our application in Sec. 4.4 will be
for a Fermi surface having both a spherical and a (tilted) cylindrical piece, created
by the CDW broken symmetry. Nevertheless, there is a hidden approximation
which is discussed at the end of Sec. 4.3.
The Fourier coefficients in Eq. (4.10) are obtained by multiplying each equation
by cos qmz and integrating over the interval (0, L).
E% =
- - -/ ° m [
L
£a(z)cos(qmz)dz,
JO
^°™
JQ Ja(z) cos(qmz)dz.
(4.12)
Furthermore, from Eqs. (4.9) and (4.10), the Fourier components of the conduc­
tivity tensor, crjm(<7„,o/), can be related to the coefficients in (4.12).
J? = ffim(9n,w)££,
(4.13)
All nine components of <Jim will be non-zero when, in Sec. 4.4, we include the
effects of a tilted cylinder.
The electric-field components E£ are obtained by taking the Fourier cosine
transform of Eq. (4.7),
(«; -
+
=
! ^ S [ £ J ( £ ) ( - 1) * - S ( 0)],
(4.14)
=
^ K ( £ ) ( - 1 )* -5 (0 )],
(4.15)
=
0.
(4.16)
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49
Since S'a(L) is 10 orders of magnitude smaller than £4(0) in a slab 10/xm thick,
we can neglect S’JJL). We next use Eq. (4.13) to express
in terms of Bp. The
longitudinal field can then be related to the transverse field from Eqs. (4.16) and
(4.13),
+ CTzVB^
(4.17)
1+
With the help of these substitutions Eqs. (4.14) and (4.15) may be written in a
-?~&xX
4-ru.j _ /
—3~ °»x
„2
9
1
xy
cl
u>2
'E £
to
1
ixiuJ_/
4jrtri; _ /
1
fci2
4»i«j _./
w -l
A
i
r -2
9
r
coupled format:
L
(4.18)
. ^ ( 0).
where
&CLZ&Z&
a a0
~
~
(4.19)
The solution for the x component of the electric field is, for each Fourier component,
E* =
<?(2 - <Son)
L
(c2^
-up- - £TTuoolyy)£,x{Q) 4- 47Tia;g4y£ '( 0 ) _________
{<?q2 - u P - 4Trioja'xx)((?q2 - up- - 4x*w<r^) - (4iriu)2cr'iy<7'yx ’
(4.20)
At this point we must specify the boundary conditions at the front surface of
the sample, so that ££(0) and £'(0) in Eq. (4.20) can be determined. Rectangular
microwave cavities, excited in a T E m mode, were used.9 Consequently, from Eq.
(4.6),
uo
(4.21)
The magnetic field Hx{0) is zero, so that
£1(0) = ~ f l , ( o ) = o.
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(4.22)
50
Accordingly the solution, given by (4.20) is completely specified. We need only to
evaluate SX{L) to find the transmitted signal.
From an analysis given by Cochran22 and by Lacueva and Overhauser12, the
transmitted electric field just outside the rear surface is twice the incident field at
L in the infinite medium. From Eqs. (4.10) and (4.20), the electric field at z = L
is therefore,
_ ....
2cu/t__
£X(L) — — — Hy(0)
^QnXc2^ - u 2- foiujcr'yy) cos(nTr)____________
X b o (c2^ - W2 - 4xiua'xx) (c2?2 - w 2- 47riuja^) - (±iriujya'xya'yx'
(4.23)
y
____________(2 -
We now replace {gn} with their values from Eq. (4.11), so that the integers {n}
appear.
C7T
£ 5 1"2 - ( £ ) 2 - (“ g V J l " * - (“§)! (4.24)
Since the sum in Eq. (4.24) converges very slowly, one needs to include terms
to n = 107 to obtain reliable values for SX{L). Now, for large n, the a[m terms are
much smaller than n2, so we can take advantage of this disparity as follows.21
2w£* rr /rt\ TO ^
/yjr2 *' '
‘
n=A f+1
("I)"
«2
U
f __________(2 - & .) ( - l) V - (ffi2 - d * P K .)__________
I"2- O 2- (^KJ[»a- (#)2- (*gV„] - (^ir)2^
(4.25)
The break point, M, is large. The final sum on the right hand side can be reex­
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perm ission o f the copyright owner. Further reproduction prohibited without permission.
51
pressed by using the exact value,
£
(-1 )” _ V ( - 1)” , V (_1)“
^
~ - £ “
+ J +1~
— s-
M-fil
(4-26)
Therefore the transmitted field is
^ ) = M 1 f v(ok£ + 2 E ^
_ f __________ (2 - f t.) ( - l) V
h [n2 -
- (at)* - ( » g V j ___________
c-^-yjin2 ~(t )2 ~( tS tK J -
(*£)2 -
'
(4.27)
On account of the large reflectivity, the field at the front surface, Hy{0), is
twice the incident field H*.12 In the vacuum, just outside of the metal, £* — Hy.
Therefore the signal, defined to be the ratio of the transmitted electric field (just
beyond z = L) to the incident field at z=0, is
„ 5,(5) 5,(1)
25,(L)
s ~
- ~ H f ~ -W M '
(4'28)
Prom Eqs. (4.27) and (4.28), the signal is,
C7T2
_ f
6
“
n2
__________ (2 - f t .) ( - i ) V - ( g )2 - ( ^ V j ___________ ,
„.o [n2 - ( f )2 - ( “ ^ f e K
- ( ^ ) 2 - ( * $ V J - ( ^ ) 2< t ^
(4.29)
We have taken the upper limit of n to be M=50 000, since doubling M doesn’t
change S by more than one part in 105.
Eq. (4.29) can be evaluated once the conductivity tensor cr/m(gn,a»), defined by
Eqs. (4.13) and (4.19), is specified. In the following section we shall determine a[m
for the main (spherical) part of potassium’s Fermi surface, including the influence
of the minigaps on the scattering time r(vz), discussed in the introduction. In
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Sec. 4.4 the contribution of the tilted cylindrical part of the Fermi surface will be
incorporated.
4.3 Effect of minigaps on microwave transmission
The conductivity tensor derived in this section will be that for a spherical Fermi
surface. However, the effect of the minigaps will be modeled by a
-dependent
relaxation time described by Eqs. (4.4) and (4.5). The Boltzmann transport
equation for the electron distribution function f(k, r, t) is,
(4.30)
where fo(k) is the equilibrium distribution. For linear response one takes,
/(£, r, £) = /o(fc) + fi{k, f, £),
(4.31)
with the understanding that fi is first order in the microwave field. Accordingly
for each Fourier component,
£(r, t) = E(q,uj) exp{iq- f - iut),
f i ( k ,f ,t ) = fq(k) exp(iq- r —iujt).
(4.32)
The linearized transport equation is then
(4.33)
It is convenient to change coordinates from k to e, kz, <f>,where
<j> = tan l {kv/k x),
(4.34)
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53
<f> is
the azimuthal angle of an electron in its cyclotron orbit. On introduction
of these variables, the last term on the left-hand side of Eq. (4.33) reduces to
uc(.dfq/d<i>), so
[1 + i(q ■v - w)r]/f +
= erE • v ^ - .
(4.35)
The velocity in Eq. (4.35) is
= m
(4 3 6 )
Accordingly,
U
h 2m*e
vx = — kx = —7J — 2 - - k 2z cos4>,
771*
771* V
h
n
h / 2m*
.
v, = A * ,.
m*
(4.37)
Only propagation vectors, q = qz, are of interest. Since we shall assume here
that vz is independent of <f>, Eq. (4.35) is easily solved:
U = Uc OS rJ—oo « S ■" « p { t l -+-iY(gl,;-~
‘j)1( ^ - « } .
UJCT
(4.38)
The Fourier component of the current density is, naturally,
A M = -J ffi f
(4.39)
and the relation between current density and the electric field is
3
J{ = £ crimEm.
(4.40)
m =l
The tensor components of conductivity are then
O’t'm —
^2
x I* # W ( ^ M e x p {(1 + <T^
J —oo
c
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MD
54
Here the volume element, dzk, in Eq. (4.39) has been transformed to
(m* ffi2)dedkzd<p, appropriate for the cylindrical coordinates of Eq. (4.34). Three
of the four integrations in (4.41) can be evaluated after use of Eqs. (4.37) and
(4.4). All nine components of the conductivity tensor, <rim(g,u;), can be identified
1
3a° f dt(
l~e
l~*
1
8 7-i 1 + y|£| —ia+ + ixt
1 + y|£| —ia_ + ixt
3ia0 r'
1 - t 2_____________ l - t 2
8 7-i
1 + y|£| —ia+ + ixt 1 + y|£| —ia_ + ixt
CTxy
2 7-1
Cyx —
1 —iwTQ
v, + ixt ’
&xyi
®yy — &xxi
CTxz
=
CTzi = (Tyx = <Tzy = 0 .
( 4 -4 2 )
where
t
a±
= it
kF'
= (u/± (jc)tq,
x
= qvFTQ,
a0
ne2r° .
= ——
m
a(4.43)
(The lower limit of the integration in d$, which is the first to be executed, con­
tributes nil on account of the exponential factor.) Notice that azz depends only on
To, the impurity-scattering relaxation time, since we have allowed the minigaps to
interrupt only the cyclotron motion, as given by Eq. (4.4). The major influence of
the minigaps on azz and the nil components of a in (4.42) will be treated in Sec.
IV, where the existence and tilt of a Fermi surface cylinder will be incorporated.
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55
It is fortunate that the remaining integrals in (4.42) can be evaluated analyt­
ically since the transmitted signal, Eq. (4.29), requires a further sum over the
allowed values of q. The integrations are rather tedious; the final expressions are:
= x l ^ ? ^ - 7 T ? +/+p+" J+r+ + /;i/+ " s'+r; +
/ - p-
- g - r - + fjp L - 9-7- + i[f+T+ + 9+P+ + /+*+ + 9+P+ + f - T-
= ^ { 2(^ (v g; %
—
+ ' - r - + *-p- +
+
u - -
-9+P+ - f+r+ ~ 9+P+ + *[f+P+ - 9 +r+ + /+p'+ - g+r'+ - /_p_
+9 - r - - fLpL + plr'_]},
(4.44)
where
a:3) 4- 2(a± + yx){3y2x —x3)]/(x2 4- p2)3,
f ± = [(1 —a | —p2 + x2)(3y2x —
g± = [2(a± + px)(p3 - 3yx2) + (1 - a | - p2 + x2)(3p2x - x3)]/(x2 + y2)3,
1 .(a± —x)2 + (p + l)2,
p± = 2 -------(T T 4 )------- ]’
r± = tan"1!
-Q^ ± ------- ]•
p + 1 + a± —xa±
(4.45)
Furthermore, f ±(x) = /± (-x ), g'±(x) = g±(-x), p'±(x) = p±{-x), and r'±(x) =
r±(—x), and
3cr0
azz
2p)],
zz =
~ ^ [ 2 x - r + 2wT0p + u 2TqT + i(p - 2u;r0x + 2u/r0r - o;2r0
2x?
(4.46)
where
n - 1 inr1 + (J - u;ro)2l
P ~ 2 1 + (x + a;ro)2
r = tan-1(x —wr0) + tan-1(x + cdr0).
(4-47)
The results must now be incorporated in Eq. (4.29) to obtain S. The trans­
mitted signal shown in Fig. 4.1 is obtained by setting p = 0, in Eqs. (4.4), (4.44),
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56
and (4.45), i.e., by neglecting all interruptions of the cyclotron motion by the minigaps. The impurity scattering rate corresponds to
ujt0
= 150. The observed GK
oscillations in Fig. 3.2 are smaller in amplitude by two to three orders of mag­
nitude.12 What is perhaps more puzzling is the growth of the GK amplitude by
about a factor of five between uc/w = 0.7 and 1.5. We have adjusted y0 = 21 and
Ho = 5.8T in Eq. (4.5) to model the effect of magnetic breakdown of the minigaps.
The revised transmission signal is shown in Fig. 4.2. The gradual growth of S with
increasing uc/v , as shown in the figure, is typical for samples having a thickness
L = 85(j.m (or larger). At very high fields the GK oscillations may decrease,9 as
might be anticipated if there are slight variations in L across the sample surface.
We have explored the consequences of adding to the conductivity a contribution
from a small Fermi-surface cylinder having its axis parallel to the applied field H.
The calculated signal was indistinguishable from the curve shown in Fig. 4.2.
This result justifies an approximation which we have not yet discussed. When an
electron in the cylinder is reflected from the surface, vx changes sign as well as vz
(but vy does not). Consequently the xy cyclotron motion is interrupted, and Eq.
(4.9) does not strictly apply. However, since the current in the xy plane arising
from the cylinder is too small to affect the transmission signal, as we have just
mentioned, the error which has been tolerated is minimal. The CR transmission
signal is caused by the z component of the cylinder’s current, as shown in the
following section. Since the sign reversal of vz on reflection is treated correctly
by Eq. (4.9), the non-local theory for the cylinder’s conductivity, given below,
should be adequate for microwave transmission (but not for a theory of the surface
impedance).
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57
4.4 Conductivity tensor from a tilted Fermi-surface cylinder
A striking consequence of the minigaps, which correspond to the periodicities
of Eq. (4.3), is the creation of small Fermi-surface cylinders, shown in Fig. 3.4.
Giuliani and Overhauser showed that the CDW wave vector, Q, is rotated from
the [110] direction by a small angle, ~ 1°. The cause of this rotation is the need to
minimize the elastic stress energy of the positive-ion lattice distortion which arises
to screen the charge of the CDW. The wave vector, Q' = Guo ~ Q> °f the phonon
mode involved is ~ 45° from the [110]. The cylindrical Fermi surfaces created by
the minigaps have axes parallel to O'.19 We will find below that a quantitative fit
for the cyclotron resonance signals in Fig. 3.2 requires the tilt angle of Q' to be
9 ~ 50°.
It is necessary to obtain the (non-local) conductivity of a tilted cylinder by
solving once again the Boltzmann transport equation (4.30). The cross section of
a tilted cylinder in a plane perpendicular to [110] is circular.19 Accordingly the
conduction electron energy spectrum for the cylinder is,
£ = 2m*
~ kz tai10)2 + *9'
(4-48)
It is again appropriate to change notation to cylindrical coordinates : e, fcz, and (f>.
(h = tan-1(
—------ ).
,
[2mfe
i
.
»x = y —^5“ cos^ + «2ta n 0,
K
= y "T7” Sin<£.
(4.49)
The velocity components required for the transport equation are :
h ,,
=
^
h I2m*e
x~
z ta n
i
f
cos 01
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v
m*V
*v h
m*
t/* =
— (—tan0)(kx —kz tan0) = — (—tanfl)</- ^ - cos<ft.
m*
to*
Since vz is here a function of
V *
(4.50)
solution of the differential equation (4.35) for this
case is, instead of (4.38),
fs = — ^
Wc
f
d<f>'E • v exp[ia(<£' —<f>)+ i/5(sin <£' - sin <f>)],
J—o o
C7£
(4.51)
where,
a
=
.1 - iwTo
VcTq
—t -------------- ,
(3 = (—qh tan 6/ m*u;c) ^
^ 2
~•
(4-52)
The relations, (4.39) and (4.40), between current density and electric field can
be used to identify the tensor components of the cylinder’s conductivity.
^
=
“4 ? /
x/
r<t>
J~ OO
r
d(f>'vm(e, fc2,0;) exp[ia(<£' —<p) + i/5(sin<£' —sin 0)]- (4.53)
The integration in dfi is enabled by expanding exp (i/3 sin 4>) in a Bessel function
series.23
etfsi»*= £ Jm(/3)eif^ .
m=—
oo
(4.54)
We provide details only for crzx.
Ott. =
2e2 m*
— /
(2tr)3fi2,a;, j
/
* ,/« ,
de J
where,
/«
f 2Td<f>cos0e-,a^
=
m
f) Jn(-0)eT+
n= —00
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(4.55)
59
x [* i # c x f e * *
-
i
y
j
Z
00
m= —00
W )J
: 3) <'(m + a)T
JM e™
(4.56)
Using the relations,23
■W-0) = (-1 )m^n(/3),
J_m(/3) = ( - l ) roJm(^),
andletting 6 bethe length of the cylinder along kz (i.e.,twice the length
in Fig. 3.4),
(4.57)
shown
and withkc the radius of the circular cross section (in the plane
perpendicular to z), we may evaluate Eq. (4.55).
where,
/3p = (—qh tan. 6/m*u}c)kc.
(4.59)
Let 77 be the fraction of the conduction electrons contained in the cylinder, i.e.,
’ - f t -
(4-60)
Eqs. (4.52), (4.58), and (4.60) then determine the component, <rXI, of the cylinder’s
conductivity tensor :
_
axx
ne2r0 ^
V rn*
2 ,
1 - i{u - m uc)T0
"* F (cjcT0)2 + [ l - t ’(a;-m a;c)ro]2'
In a similar fashion the eight remaining components can be found :
_
CXV
ne2r0 ^
2 . _________ - uctq________
V m’ mioo m F (wcr0)2 + [1 - i(w - mcjc)r0]5
&XZ =
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(4.61)
60
CXyx — &xv>
°yy = &xxi
O’yz
— (Tjytill 0 ,
axx = —axxtan.0,
o’zy =
o'Xy tan 0,
(TZz — &xx tan20.
(4.62)
The transmission signal, S, including the effect of the tilted cylinder, can be
calculated from Eq. (4.29) by adding the conductivity tensor for the main Fermisurface, Eqs. (4.42)-(4.47), to the tensor just derived for the tilted cylinder, Eqs.
(4.61) and (4.62). The microwave detector is sensitive to the phase of a reference
signal from the klystron; and this phase can be adjusted (as was done for the data
in Fig. 3.2) so that the main cyclotron resonance appears symmetric.9 Accordingly,
the calculated electric field, ET, of the transmitted signal will depend on a phase
X :
E t = Re(S) cos x + Im(S) sin x-
(4.63)
The calculated cyclotron resonance signal, shown in Fig. 4.3, is symmetric with
X
= 280°. The experimental9 value for r0 corresponds to wr0 = 150.
A crucial feature of Eqs. (4.61) and (4.62) is the resonance denominator which
becomes small whenever ujc = u}/{m +1), m = 0,1,2... This feature is responsible
for the occurrence of the cyclotron-resonance subharmonics (as well as the main
resonance for m=0), and is similar to the Azbel-Kaner oscillations24 which occur
in parallel-field surface resistance studies.25
The parameters associated with the cylinder’s geometry are 6, kc, and 6.
The cylinder’s length is determined by the CDW diffraction satellites,14, 15 and
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61
is 0.015 [Guo | - The ratio of the main cyclotron resonance to the first subharmonic
(at ucf<j) = 0.5) is sensitive to the tilt angle 9. The observed ratio, ~ 10, in Fig.
3.2 is reproduced by the calculated ratio in Fig. 4.3 with 6 ~ 50°. The absolute
size of the main resonance is sensitive to fcc, which we find to be ~ Zkp/8. This
value, together with 6, indicates that the conduction-electron fraction of the third
and largest cylinder shown in Fig. 3.4 is tj ~ 0.004. (The absolute size of the
main CR is based on the amplitude of the GK oscillations near uc/ui = 1.5. The
GK amplitude was determined in Ref. 12 from the original data and instrumental
calibrations kindly provided by G. L. Dunifer.) Only the largest cylinder, shown
in Fig. 3.4, was included in the foregoing calculation. We have found that the
influence of the smaller cylinders on the height of the main CR is ~ 7%. How­
ever, their presence is necessary to account for the Landau-level oscillations near
£jc/u;=0.6 and 1.2.
4.5 Conclusions
The complex Fermi surface of potassium, illustrated schematically in Fig. 3.4,
is based on energy-band calculations19 that incorporate the periodic potential of
an incommensurate CDW having the wave vector Q, Eq. (4.1). The presence of
the three Fermi-surface cylinders, shown shaded (and with thickness exaggerated
by a factor of ten), are manifested by the transmission data of Fig. 3.2. The rapid
oscillations near o;c/o; = 0.6, arise from Landau-level quantization of the smallest
cylinder (having a radius ~ k p /16). The Landau-level oscillations near ujc/ ui = 1.2
arise from the next smallest cylinder, which has a radius ~ kp/8.20 Fig. 3.3 is an
enlarged view of the data in Fig. 3.2 showing the Landau-level oscillations near
ujc/ ui=Q.Q and
1.2. The largest cylinder shown in Fig. 3.4 has a radius ~ 3kp/8, a
value estimated from the main CR amplitude.
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62
The shape of the main CR varies from sample to sample, as shown in Fig. 3 of
Ref. 9. Resonances from thick samples, L > lOOfim, often exhibit structure. This
feature is to be expected as a consequence of CDW domains.7 Four possible CDW
Qs are nearly parallel to a [110] direction. They can be obtained by interchanging
the first two components of Eq. (4.1) and by reversing the sign of the third
component. It is likely that all four domains occur across the sample area. If the
magnetic field is exactly parallel to [110], the cylinders of all four domains will
have the same cyclotron frequency. However, if H deviates from [110] by even a
fraction of a degree, each cylinder will have its own u>c:
UJrcos 9:
^ =
i = 1'2'3’4'
'464>
where 0, is the angle between the cylinder’s axis, Q'0 given by Eq. (4.2), and
the projection of H on the plane containing [110] and Q'{. (9 is the angle each
axis would have if H were exactly parallel to [110].) The CR line can be merely
broadened if the deviations of 9{ from 9 are very small. Splitting of spin-wave
sidebands of the spin resonance signal has also been attributed to CDW domain
structure.26
The major conclusion of this study is, of course, that transmission CR and
the associated CR subharmonics arise in potassium from a cylindrical section of
Fermi surface created by CDW minigaps. Without the CDW broken
sy m m e try
the transmission signal would be the one shown in Fig. 4.1.
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63
L
U-40
CM
-80
0
1
0.5
1.5
coc / c o
Figure 4.1
Theoretical microwave transmission signal versus loJ uj for potassium if the
Fermi surface is spherical. Only Gantmakher-Kaner oscillations appear (since
the electron-spin magnetic moment is neglected). The sample parameters are
u/7o = 150 and L = 85/zm.
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64
0.2
0.1
-
0.1
-
0.2
-
0.3
0
1
0.5
1.5
C0C / CO
Figure 4.2
Theoretical transmission signal when interruption of cyclotron motion by the CDW
minigaps is modeled by the vz dependent relaxation time, Eq. (4.4), and with
magnetic breakdown of the minigaps described by Eq. (4.5). The parameters,
yQ= 21 and Hq = 5.8T, were adjusted so that the GK amplitude at ojJ oj = 1.47
agrees with the (calibrated) data from Fig. 3.2 and with its observed growth by a
factor of five from low to high fields.
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65
0.5
0
1
0.5
1.5
G)c / CO
Figure 4.3
Theoretical transmission signal when the non-local conductivity of the largest
Fermi surface cylinder, shown in Fig. 3.4, is added to the main Fermi surface
conductivity (employed in Fig. 4.2). The tilt, 6 = 50°, of the cylinder’s axis was
adjusted so that ratio of the main CR to the first subharmonic (at u/c/u; = 0.5) is
~ 10, consistent with Fig. 3.2. The cylinder radius, kc = 3kp/8, in the (110) plane
was adjusted so that the ratio of the CR to the high-field GK amplitude agrees
with that observed in Fig. 3.2. a;r0 = 150 and L = 85/zm.
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BIBLIOGRAPHY
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66
BIBLIOGRAPHY
C hapter 1
1. D. Shoenberg and P. J. Stiles, Proc. Roy. Soc. A281, 62 (1964).
2. A. W. Overhauser, in Highlights of Condensed-Matter Theory, Proceedings
of the International School of Physics “Enrico Fermi”, Course LXXXDC,
Varenna on Lake Como, 1983, edited by F. Bassani, F. Fumi, and M. P. Tosi
(North Holland, Amsterdam, 1985), p. 194.
3. A. W. Overhauser, Adv. Phys. 27, 343 (1978).
4. A. W. Overhauser, Phys. Rev. Lett. 53, 64 (1984).
5. G. A. Baraff, C. C. Grimes, and P. M. Platzman, Phys. Rev. Lett. 22, 590
(1969).
6. G. L. Dunifer J. F. Sambles and D. A. H. Mace, J. Phys. Condens. Matter
1, 875 (1989).
7. H. Mayer and M. H. El Naby, Z. Phys. 174, 269 (1963).
8. T. M. Giebultowicz, A. W. Overhauser, and S. A. Werner, Phys. Rev. Lett.
56, 2228 (1986).
9. S. A. Werner, A. W. Overhauser, and T. M. Giebultowicz, Phys. Rev. B41,
12536 (1990).
10. Yong Gyoo Hwang and A. W. Overhauser. Phys. Rev. B39, 3037 (1989).
11. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B48, 16935 (1994).
12. G. Lacueva and A. W. Overhauser, Phys. Rev. B33, 3765 (1986).
13. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B46, 1273 (1992).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
67
14. F. W. Holroyd and W. R. Datars, Can. J. Phys. 53, 2517 (1975).
15. P. G. Coulter and W. R. Datars. Can. J. Phys. 63, 159 (1985).
16. M. Hubennan and A. W. Overhauser. Phys. Rev. B25, 2211 (1982).
C hapter 2 and Appendices
1. C. C. Grimes and A. F. Kip, Phys. Rev. 132, 1991 (1963).
2. M.Ya. Azbel and E. A. Kaner, Zh. Eksp. Teor. Fiz. 32, 896(1957)
[Sov.Phys.-JETP 5, 730 (1957)].
3. D. C. Mattis and G. Dresselhaus, Phys. Rev. I l l , 403 (1958).
4. R.
G. Chambers, Philos. Mag. 1, 459 (1965).
5. G.A. Baraff, C. C. Grimes, and P. M. Platzman, Phys. Rev. Lett. 22, 590
(1969).
6. A. W. Overhauser, Phys. Rev. 167, 691 (1968).
7. A. W. Overhauser, Adv. Phys. 27, 343 (1978); in Highlights of CondensedMatter Theory, Proceedings of the International School of Physics “Enrico
Fermi”, Course LXXXDC, Varenna on Lake Como, 1983, edited by F. Bassani, F. Fumi, and M. P. Tosi (North Holland, Amsterdam, 1985), p. 194.
8. Yong Gyoo Hwang and A. W. Overhauser, Phys. Rev. B39, 3037 (1989).
9. A. W. Overhauser and N. R. Butler, Phys. Rev. B14, 3371 (1976).
10. G. Lacueva and A. W. Overhauser, Phys. Rev. B33, 3765 (1986).
11. G. L. Dunifer, J. F. Sambles, and D. A. H. Mace, J. Phys. Condens. Matter
1, 875 (1989).
12. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B46, 1273 (1992).
13. C. C. Grimes, private communication.
14. G. F. Giuliani and A. W. Overhauser, Phys. Rev. B20, 1328 (1979).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
15. T. M. Giebultowicz, A. W. Overhauser, and S. A. Werner, Phys. Rev. Lett.
56, 2228 (1986); Phys. Rev. B41, 12536 (1990).
16. E. I. Blount, Phys. Rev. 126, 1636 (1962).
17. M. Huberman and A. W. Overhauser, Phys. Rev. B25, 2211 (1982).
18. C. Kittel, Quantum Theory of Sohds, (John Wiley and Sons, New York
1963), p. 313.
19. P. A. Penz and T. Kushida, Phys. Rev. 176, 804 (1968).
20. G. A. Baraff, Phys. Rev. 187, 851 (1969), first paragraph.
21. P. G. Coulter and W. R. Datars, Can. J. Phys. 63, 159 (1985).
22. A. W. Overhauser, Phys. Rev. Lett. 53, 64 (1984).
Chapter 3
1. G. L. Dunifer, J. R. Sambles, and D. A. H. Mace, J. Phys. Condens. Matter
1, 875 (1989).
2. C. C. Grimes and A. F. Kip, Phys. Rev. 132, 1991 (1963).
3. D. A. H. Mace, G. L. Dunifer, and J. R. Sambles, J. Phys. F; Met. Phys. 14,
2105 (1984).
4. Yong Gyoo Hwang and A. W. Overhauser, Phys. Rev. B38, 9011 (1988).
5. F. Gantmakher and E. A. Kaner, Soviet Phys. JETP 21, 1053 (1965).
6. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B46, 1273 (1992).
7. D. Shoenberg, Magnetic Oscillations in Metals (Cambridge University Press,
London, 1984), Chap. 4.
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69
C hapter 4
1. A. W. Overhauser, in Highlights of Condensed-Matter Theory, Proceedings
of the International School of Physics “Enrico Fermi”, Course LXXXIX,
Varenna on Lake Como, 1983, edited by F. Bassani, F. Fumi, and M. P. Tosi
(North Holland, Amsterdam, 1985), p. 194.
2. A. W. Overhauser, Adv. Phys. 27, 343 (1978).
3. A. W. Overhauser, Phys. Rev. Lett. 53, 64 (1984).
4. J. A. Schaefer and J. A, Marcus, Phys. Rev. Lett. 27, 935 (1971).
5. F. W. Holroyd and W. R. Datars, Can. J. Phys. 53, 2517 (1975).
6. P. G. Coulter and W. R. Datars, Can. J. Phys. 63, 159 (1985).
7. M. Huberman and A. W. Overhauser, Phys. Rev. B25, 2211 (1982).
8. A. W. Overhauser, Phys. Rev. Lett. 59, 1966 (1987).
9. G. L. Dunifer, J. R. Sambles, and D. A. H. Mace, J. Phys. Condens. Matter
1, 875 (1989).
10. C. C. Grimes and A. F. Kip, Phys. Rev. 132, 1991 (1963).
11. D. A. H. Mace, G. L. Dunifer, and J. R. Sambles, J. Phys. F: Met. Phys. 14,
2105 (1984).
12. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B48, 16935 (1993).
13. A. W. Overhauser and N. R. Butler, Phys. Rev. B14, 3371 (1976).
14. T. M. Giebultowicz, A. W. Overhauser, and S. A. Werner, Phys. Rev. Lett.
56, 2228 (1986).
15. S. A. Werner, A. W. Overhauser, and T. M. Giebultowicz, Phys. Rev. B41,
12536 (1990).
16. G. F. Giuliani and A. W. Overhauser, Phys. Rev. B20, 1328 (1979).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
17. Mi-Ae Park and A. W. Overhauser, Phys. Rev. B54, (1996).
18. P. A. Penz and T. Kushida, Phys. Rev. 176, 804 (1968).
19. Yong Gyoo Hwang and A. W. Overhauser, Phys. Rev. B39, 3037 (1989).
20. Graciela Lacueva and A. W. Overhauser, Phys. Rev. B46, 1273 (1992).
21. B. Urquhart and J. F. Cochran, Can. J. Phys. 64, 796 (1986).
22. J. F. Cochran, Can. J. Phys. 48, 370 (1970).
23. N. W. McLachlan, Bessel Functions for Engineers, (Oxford University Press),
London 1941), p. 158.
24. M. Ya. Azbel and E. A. Kaner, Zh. Eksp. Teor. Fiz. 32, 896 (1957)
[Sov.Phys.-JETP 5, 730 (1957)].
25. D. C. Mattis and G. Dresselhaus, Phys. Rev. I l l , 403 (1958).
26. Yong Gyoo Hwang and A. W. Overhauser, Phys. Rev. B38, 9011 (1988).
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APPENDICES
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71
Appendix A: Calculation of the conductivity
Equation (2.6) can be rearranged as follows:
i+m
_n
(1 - / ) ( « - » - ! )
i + m
'
Prom Eqs. (2.7) and (2.9) we have expressions for
*xx =
(A1)
1
j
and
1 + P\t\
* L1 —ia+ -+- ixt
1 —ia_ + ixt
=
+ [(1 - f ) ^ ~ 1 )K ,,
_ 3«ff0
.
. .
/ + (l-/)e -> -l.
- — y-l ‘“ ( 1 _ ? ) [ 1 + —
rT 3 tj—
1
r
1
1
1 —ia+ + ixt
1 —ia_ + ixt
i
X [ -----------------------------------------------------------------------J ,
= < 3 + [ / + ( l - / ) < r # - l K >,
(A.2)
where
/
**
,
ffxy
= ^8 7-if1
i f
1 ~
(2
|
1
i
1
1 + P\t\ 1 —ia+ + ixt 1 —ia_ + ixt
3iao yi 1 - 12 .
1__________1
8 7-i l + B\t\ 1 —ia+ + ixt 1 —za_ + ixt
and a™9 and a™9 are the same as
and
1
in Eq. (2.7). These expressions
were evaluated previously.10.
a™9 = ^ -{ 2 a +p+ + 2a_p_ - 2 + r+(x2 + 1 —a+) + r_x2 + 1 -
a2)
+t[a+ + a_ + p+(x2 + 1 —a2) —p~(x2 + 1 —a2) —2a+r+ —2a_r_},
axy9 =
~ a + + P - f c 2 + 1 — <*1) - P + ( z 2 + 1 — a2 ) +
2a+r+
— 2a_r_
+i[2a+p+ —2a_p_ + r+(x2 + 1 - a+) —r_x2 + 1 —ai)]},
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(A.4)
72
where
a± =
( u ± o/c)r,
x = gZ,
P±
= J _ inr l ± j £ ± ^ ) ! 1
Ax 1 + (x —a±)2
r± = -^[tan_1(x + a±) + tan~l (x —a±)].
2x
(A.5)
Integration of Eq. (A.2) is tedious but straight forward. The final forms are:
<TXX =
j^ { 2 a +p+ + 2a_p_ - 2 + r+(x2 + 1 - a2) + r_(x2 + 1 - a2_)
+[(i - /)(e-^ -
i)][x(u
n
- p a + ( u + r +)
—0°—U - + /- ) ~ 0(9+ + 9+ + 9- + 9-) + (s+ + s+ + s- + O ^ 2]
+i[a+ + a_ + p+(x2 + 1 —a+) + p~(x2 + 1 —a?_) —2a+r+ —2a_r_
+*'[(1 - /)(«“^ " 1)][*(^+ - 9 + + 9 - - 9 - ) - fa+(9+ + 9+)
—0a-(9- + 9~) + 0(f+ + /+ + / - + /I ) + (t+ +
CTxy
=
{a_ - a+ + p_(x2 + 1 -
a 2_ ) —
+ t'_)x2]},
p+(x2 + 1 — a2 ) + 2a+r+ - 2a_r_
+[(1 ~ /) ( e " ^ “ !)][*(»- ~ 9 - ~ 9 + + 9+) + 0a+(9+ + 9+)
-/3a-(g. + g'_) + /?(/_ + f'_ - f + - f'+) + (t_ + f_ - t + - t'+)x2}
+i[2a+p+ —2a_p_ —r_x2 + 1 —a2_) + r+(x2 + 1 —a2)]
+*■[/+(i - /)<=-% - 1]w /+
1
'-) - /5 M /+ + /;>
+ 0a- ( f - + / I ) ~ 0(9 + + 0+ ~ 0- “ O + («+ + s+ - s_ - O x 2]},
(A.6)
where
l, t
u±
“
2
l+ « ±
i
1 + ~( ^ '+ q ± )'2
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73
v± = tan 1(a: + a ± )—tan x(a±),
f±
=
I jS*+ (X-
~ (4 ~**~1)~ x1'
x(x — 2 a ± ) .
2 ^ + ( x - ^ 1~ 2a±1,± + '1±(a±~ 3! _ 1 ) ---------2------11
S± “
[~ 2 a± U ±
1
r ~
s± = 2 j8> + ( l - 0a±)2
_ 1 -P(x - 0a±) u l
*
2 Z?2 + (x —/3a±)2 0
/
2
2
-x
^ 111(1 + ^ + 0* (1 ~ 2
lv ,
, !/,
P\\
^
pp (
2
(K 7\
( ^
Purthennore, /±(x) = /± (—x), $±(x) = 3±(-x), s'^x) = s±(—x), and tf±(x) =
t±(—x). The foregoing results are to be used in the integrand of Eq. (2.22), which
must then be evaluated numerically.
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74
Appendix B: Polarization of the field inside an anisotropic metal
Consider a metal in a high-frequency electromagnetic field. To learn how the
wave is polarized we shall treat the normal skin effect for which Ohm’s law, J = aE,
is valid and the conductivity is local. The relevant Maxwell equations are
IdH
„
Vx£ V x H
= — J.
(B.l)
c
We neglect the displacement current. Let us assume that the metal fills the z >
0 half-space, and that the wave is incident normal to the surface. For a wave
propagating in the z-direction we shall seek a solution proportional to exp(iqz —
iut). Eliminating the magnetic field H from Eq. (B.l), we can easily find:
A'jr
_
-V2£ + V (V - £) + ~
j
= 0,
(B.2)
which reduces to
d2
47riaj .
a + ~ Ja~ '
(B.3)
j z = o,
(B.4)
where a = x,y. The conductivity of a nearly-free electron system in the local
approximation is,
a8 =
ne2r
{ 1 —iu)T
—U)CT
lOcT
1 —iuJT
0
0
(B.5)
(l—iUT)2+(wcT)2
V o
o
l—tWT
The conductivity of the Fermi-surface cylinder, as calculated in Sec. 2.3, is
771* (1 — IU>t ) 2 + (u /c T )5
ac =
7ine^Tc
771*
/
1
(1 -
iiJ T c ) 2
+ (o>crc)2
1 — iu r c
—U c Tc
VCTC
1 —iuJTc
^ —tan 0(1 —iu>Tc) tan 6(u>crc)
—tan 0(1 —iurc) \
—tan 0(u)ctc)
tan20(1 —iojrc) j
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(B-6)
75
The total conductivity is a9+crc. The usual expression for the conductivity tensor
is:
( <*xx
G xy
& xz\
0yz
&yy
& yz
\ & zx
& zy
&zz)
a=
(B.7)
Because the number of electrons enclosed by the cylindrical Fermi surface is only
a fraction, i/ = 4 x 10-4, of the total, the following inequalities prevail.
0~xzi &yzt & zx j &z y
^ x x i & yyi & zzi &xi11 Gyx~
(B.8)
Using Ohm’s law to express Eqs. (B.3) and (B.4), we find a set of homogeneous
equations:
.2
V?
47ria; 4xiu>
^ *Txy£y
^ ^xz^z —0>
47tzu; .
^2 &xx)^x
47ria;
_ , 2 47rzu;
x „ 47ria;
^2 Gyx^x + (?
^ ayy)^y
^ Gyz^z ~ 0)
&ZX^X “i" &z y £ y " f GZZ^Z = 0 .
(B.9)
We next eliminate £z in favor of £x and £y, using the third equation of (B.9). This
allows us to express (B.9) with £x and £y only.
, ,
(?
47rio; . . _
~g~°xx)^x
47rio> . _
. ,
47rzu; . _
^ axy^v — 0)
47rio; . . „
+ (« -
= °.
(B.10)
where
&O0 —
&az&zf}
(B.ll)
Here a, 0 indicate x or y components only. This change of crQ/3 to a'a0 is the
main contribution of the electrons in the tilted cylinder. Transverse conductivities
are mixed with longitudinal conductivity on account of the longitudinal motion
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76
of electrons
in the cylinder. The determinant of (B.10) must vanish; andthis
condition leads to the allowed propagation vectors:
= Y ^rW xx + ff'n + >/(*£*-^ y P + ^V^v*]= y ^ r W z x + <r'vv- 'J(a'z x - ° ,n )2 + *°,xy°,vzl
(B-12)
Therefore the two electric-field modes are
Ainoj(T'xy
rCgjC2 - /k'Kiu)a'xx)a zy + A'Kiua'xv(Tzx
Airicjcr'zyazz
s, =
Anuoo'^
_r «£I—~ Aziuja'^a^ + Arriucr^ajx
Aniuja'^a^
(b .13)
The amplitudes of the transmitted wave, £\a and £2o, can be obtained in terms of
the amplitudes of the incident wave £Tby requiring the tangential field components
to be continuous at the boundary. There are incident, reflected, and transmitted
electric fields on the surface, z=0:
E1 = (S'x + Sfye***-^,
£* = (SZx + e f y + S^e-***-**,
f? = £l0(x + aiy + piz)eullt~,ut + £2O(x + a2y + lhz)e%q2Z~%ut, (B.14)
where T indicates the incident wave propagating along z withwave vector 50 =
uj/c, ‘R’ indicates the reflected wave traveling along —z with wave vector —qa,
and ‘T ’ indicates the transmitted wave. Equation (B.l) requires the microwave
magnetic field to have x and y components only.
H1 = (£j* + £/y)e<fl0*-<wt,
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77
=
^ . ( _ ai£ + y)ewi* - ^ + ^ 2 ( _ a2i + y)ewJ^
|
(B.15)
where
Pi
=
g^c2 - 47Tta;q^z
47ria;a^y
(g^c2 - 4'KiuoJxx)aty + $xiwo,xvozx
4iKVj}0,Xy0’zz
(B.16)
For the purposes of this appendix we treat potassium as a nearly-free electron gas
characterized by the following parameters: effective mass, m* = 1.21m, electron
density, n = 1.4 x 1022cm-3, Fermi radius, kp = 0.75 x 108cm-1, and electron
scattering time, r = 2.0 x 10-10sec (which is appropriate at T = 2.5°K). The
frequency of the applied microwave field is 23.9GHz. Accordingly, u j t = 30 is used
for electrons on the spherical Fermi surface. On account of the small velocity for
electrons in the Fermi surface cylinder urc = 150. (This value is required to fit the
observed width of the cyclotron resonance peak in Fig. 2.1.)
The inequalities of Eq. (B.8) are so extreme that ax and a 2 differ from i and
—i by ~ 10~6. Specifically,
ai « i,
q2
(B.17)
~ —t.
The ratio of the x or y components to the z component is about 100, so
Calculation of £\0 and £2o is straightforward by using the continuity of the tan­
gential field at z=0. The final results are
; ----------------
-A a iE l - £ / )
(ai —a 2)(go + ?i)
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78
* • =
(B19)
The amplitude of the transmitted wave may be found by specifying the incident
wave. For right-circular polarization,
e ‘ = is ’.
(B.20)
On account of extreme inequality, (B.8), one mode dominates the other by a factor
of at least
107 for all magnetic fields, i.e.,
€lo » €20 .
(B.21)
£ ' = - iS i,
(B.22)
For left-circular polarization,
and £2o is much larger than £\0.
The magnetic field at z=0 can be found from Eq. (B.15).
H
=
2aig1(a2gr ~ gy) _ 2a2g2(ai£7 - £j)
(ax — a 2)(g0 + qi) ( — a 2)(?o + ) ’
0:1
fl-,(0) =
- W i - £ i )
(ai —a 2)(9o + 9i)
From Eq. (B.17) and the fact that qi, q2
0) e T2i£j,
ff„(0) » 2£j,
92
W i - £ ‘y)
(ai —<*2)(?o + ?2)
(B23)
q0, the magnetic field at the surface is,
/o r
£[ =
/o r
S i = ±iS’.
Therefore the total current defined by Eq. (2.40) is
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(B.24)
79
The foregoing results are incorporated in the calculations of Sec. 2.3 at Eqs.
(2.43) and (2.44). It must be appreciated that the Fermi-sphere electrons are
treated non-locally in Sec. 2.3. The purpose of this appendix is to show that
the microwave modes in the metal are essentially circularly polarized, (despite the
broken axial symmetry caused by the tilt of the Fermi-surface cylinder) on account
of the small value (4 x 10-4) of 77.
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VITA
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80
VITA
Mi-Ae Park was bom on February 3, 1964, in Seoul, Korea. She attended
elementary and secondary school in Seoul, graduating from Jung-Eui High School
in February, 1982. In March, 1982, she entered Ewha Womans University, where
she received her Bachelor and Master of Science degree in physics in February,
1986 and February, 1988.
In August, 1990, she entered Purdue University to study for a her Ph. D. Her
doctoral research has been in the area of theoretical solid state physics. She was
a Graduate Teaching Assistant from 1991 to 1996. In 1996 she was awarded the
Edward Akeley Prize for Theoretical Physics. She is expected to complete her Ph.
D. degree in August, 1996.
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