Quasi-one-dimensional Peierls systems with quantum and thermal lattice
fluctuations can be modeled by a Dirac-type equation with a Gaussian-correlated
off-diagonal disorder. A powerful new method gives the exact disorder-averaged
Green function used to compute the optical conductivity. The strong subgap tail
of the conductivity has a universal scaling form. The frequency and temperature
dependence of the calculated spectrum agrees with experiments on KCP(Br) and
trans-polyacetylene.Comment: 11 pages (+ 3 figures), LATEX (REVTEX 3.0

A. A. Ovchinnikov

A. J. Heeger

C. Weisbuch

D. Berner

F. H. Long

G. Grüner

G. Leising

H. J. Fischbeck

H. Takayama

H. Wagner

I. M. Lifshits

J. M. Williams

J. Rammer

John W. Wilkins

K. Carneiro

Kihong Kim

L. Degiorgi

M. J. Rice

P. A. Lee

P. Brüesch

R. H. McKenzie

R. P. McCall

Ross H. McKenzie

S. A. Brazovskii

English

arXiv

Crossref

Physical Review Letters

Universal subgap optical conductivity in quasi-one-dimensional Peierls systems

Quasi-one-dimensional Peierls systems with quantum and thermal lattice fluctuations can be modeled by a Dirac-type equation with a Gaussian-correlated off-diagonal disorder. A powerful new method gives the exact disorder-averaged Green function used to compute the optical conductivity. The strong subgap tail of the conductivity has a universal scaling form. The frequency and temperature dependence of the calculated spectrum agrees with experiments on KCP(Br) and trans-polyacetylene

Kim, K

McKenzie, RH

Wilkins, JW

University of Queensland eSpace

VOLUME 71, NUMBER 24 PHYSICAL REVIEW LETTERS 13 DECEMBER 1993
Universal Subgap Optical Conductivity in Quasi-One-Dimensional Peierls Systems
Kihong Kim, Ross H. McKenzie, and John W. Wilkins
Department of Physics, The Ohio State University, Columbus, Ohio $M10
(Received 26 July 1993)
Quasi-one-dimensional Peierls systems with quantum and thermal lattice fiuctuations can be
modeled by a Dirac-type equation with a Gaussian-correlated ofF-diagonal disorder. A powerful
new method gives the exact disorder-averaged Green function used to compute the optical con-
ductivity. The strong subgap tail of the conductivity has a universal scaling form. The frequency
and temperature dependence of the calculated spectrum agrees with experiments on KCP(Br) and
trans-polyacetylene.
PACS numbers: 71.38.+i, 71.45.Lr
The interesting new physics exhibited by quasi-one-
dimensional electronic materials continues to expand as
advances in synthetic chemistry [1] and semiconductor
fabrication technology [2] make new materials available
for study, Many quasi-one-dimensional materials un-
dergo a structural instability known as the Peierls or
charge-density-wave instability [3—5]. Each chain has a
periodic lattice distortion with twice the Fermi wave vec-
tor, 2k~, resulting in a gap 2L in the electronic spec-
trum at the Fermi surface, suggesting that such a gap
should be clearly visible in the optical absorption spec-
trum. However, over the past twenty years, optical ab-
sorption measurements on a wide range of materials [6—8]
have consistently shown that there is a broad tail be-
low the gap. Moreover, as the temperature increases the
spectrum broadens considerably. There is no accepted
quantitative theory of the shape of the spectrum or its
temperature dependence. While considerable efFort has
been made to understand excitation energies of solitons
and polarons [3] in these compounds, there is also no
accepted theory for their absorption spectrum.
The zero-point and thermal lattice motions have a sig-
ni6cant efFect on the electronic properties of the Peierls
state and are, we believe, key to understanding the sub-
gap absorption [9]. A recent luminescence and Raman
study of a metal halogen mixed-valence (MX chain) com-
pound produced results consistent with this view [10].
If the relevant phonon frequency at q = 2k~, w2k~, is
much smaller than the optical frequency, as is the case
in the optical absorption in most Peierls materials, the
quantum and thermal lattice fluctuations can be mod-
eled [9,11] by a static, Gaussian-random, backscatter-
ing potential, ((x), with zero mean. Further, if the
phonon dispersion near q = 2k~ is ignored, ((x) is
fully characterized by the disorder-averaged correlator
(((x)(*(y)) = pb(x —y). In general, the dimensionless
disorder parameter rt = p/hv~A, where v~ is the Fermi
velocity, has contributions from extrinsic disorder such
as due to impurities, g„as well as from the intrinsic dis-
order due to lattice fiuctuations, g, . The latter is related
to the dimensionless electron-phonon coupling constant
A, the phonon frequency at 2k~, and temperature [9]:
1.0
0.8
7l
K
"0.6b
0.4
b
0.2
Experim
0.0
0.0
Theory
1
0.5 1.0
G3 6)peak
FIG. 1. Comparison of the real part of the fre-
quency-dependent conductivity, o(w), obtained from our cal-
culations for a disorder parameter g = 0.43 with experiment
on KCP(Br) at T = 40 K [6] and the conductivity obtained
from a Lorentzian convolution of the rigid-lattice conductiv-
ity for a dimensionless damping parameter ( = 0.42 [12]. The
frequency and the conductivity are scaled by the frequency at
the peak, ur ~~k, and by op@&k —o (ll)pe&k) respectively.
rl(T) =rl, +rh(T) = rl, +A '" coth
~
'" [. (1)2A i, 2k~T )
In this Letter, we develop a novel and powerful
method for computing the electronic properties of one-
dimensional Peierls semiconductors with static disorder.
We use this method to calculate the real part of the op-
tical conductivity. We find (1) a strong subgap tail in
the optical absorption; (2) the remarkable fact that the
subgap conductivity, when properly scaled, follows a uni-
versal scaling curve independent of the disorder param-
eter rl; (3) good agreement between the computed con-
ductivity and the experimental conductivity for KCP(Br)
[6] and trans-polyacetylene [7]; and (4) a universal scal-
ing curve for the experimental conductivity for KCP(Br)
at difFerent temperatures with g having the temperature
dependence given by Eq. (1). Figure 1 contrasts the rel-
atively good agreement with experiment achieved by our
calculation against that for a Lorentzian convolution of
0031-9007/93/71(24)/4015(4) $06.00
1993 The American Physical Society
4015
VOLUME 71, NUMBER 24 P H YS ICAL R EVI E% LETTERS 13 DECEMBER 1993
the rigid-lattice conductivity [12].
We consider the standard continuum model [11,13] of
noninteracting electrons in one dimension with a Peierls
ground state, which, without the random term, has been
studied extensively (see p. 795 in [3]) to understand the
soliton and polaron excitations in conducting polymers.
Our system is described by a Dirac-type equation [14) for
the wave functions gi(x) and gq(x) for electrons moving
with v~ to the right and to the left, respectively, in the
interval —L ( x ( L with a complex, Gaussian-random,
backscattering potential ((x):
t ihv~— 6+((x) ) p
~
( & + q.(.) ,&. ' ) i ~.(*) (&(*))= ((( )&( )) = o ((( )&*( )) = ( — ). (2)
The above equations correspond to the case where the Fermi wavelength of the electron is incommensurate with the
lattice period [15]. Previous work has succeeded in computing the density of states and localization length for this
model [16]. Here we report the first exact calculation of the Green function [17].
We use G+ (G ), the retarded (advanced) 2 x 2 matrix Green function to compute the real part of the frequency-
dependent conductivity for hu » kIiT [14,18):
6 2
Ah
dx Re j+ (E+ ~, E]x —y) —j++(E+ Ku, E x —y)
where A is the cross-sectional area of a chain and
&"(E',E]x y) = (n-v~)'(T ~sG'(y, x]E')~,G'(x, y~E) ). (4)
mrs is a Pauli matrix. Here we report the computation of the conductivity using the approximation (GG) = (G)(G),
which we expect to be valid for hu » p/hv~, i.e. , for optical but not dc conductivities. However, our method can be
used to compute (GG) directly.
Exact calculation of the Green function. —Two linearly independent wave functions g and g satisfy the boundary
conditions Q( L) = 1 an—d Q(L) = 1. These @ and @ will be statistically independent in the limit L + oo [14]. For
x&y,
G(x, y]E) = & &i(x)&2(y) 4i(x)&i(y) &
vhi('tp Qi2—$2fpi) ( 0 ( 2)0x2(y) 42(x)gi(y) ) (5)
where (@i@2—$2gi) is a constant independent of x. We obtain G+ (G ) by solving for wave functions with ImE a
small positive (negative) number. Then the denominator can be Taylor expanded and
where
)C()()D(*)()
C ( y)y+ ( ) D-( y)y-+ ( )
42(x) C „&i(x) "&~(y) D „4i(x) "4'i(y)
.4.(). M.()' " ' .M.(). 4.()
(6)
Next, we take the L ~ oo limit and utilize the statisti-
cal independence of Q and @ to factor the average of the
Green function into products of averages of C (D) and
y. The quantities C„—:(C„(x,y)) and D„—:(D„(x,y))
are functions of (x —y) and y„—:(y„(x)) is indepen-
dent of x. Thus the averaged Eq. (6) can be written in
terms of y„, C„(x —y), and D„(x —y). The following
equations for y„and C„can be derived from Eq. (2)
using standard Fokker-Planck methods [19], subject to
the conditions yo = 1 and C~(0) = y„, which follow
from Eq. (7) for yo, and from Eq. (7) and the equality
((Q2/Qi)") = ((gi/i/'q)") [14] fo C„(0):
E
l ~~
+ '////) // —// +/ —// —/ = 0,
hvF dC„. E
= i(2n+ 1)—C„—i(n+ 1)C„+idx —y
—inC„, —q (n'+ n+ —,') C„. (9)
D„satisfies the same equation as C„with the initial con-
dition D„(0) = y„+i.
The key to our method is the fact that y„, C„, and
D„decay very rapidly and exponentially as a function of
n.—The decay rate is determined by the dimensionless
disorder parameter rj = p/hv~A: the larger q, the more
rapid the decay. To solve Eqs. (8) and (9) numerically,
we truncate the number of equations by setting y = 0
for n & ¹ For example, at r) = 0.1, the Green function
4016
VOLUME 71, NUMBER 24 PH YSICAL REVIEW LETTERS 13 DECEMBER 1993
1.5
1.0
0.5
(a)
1.0
0.8
I—
o 0.6
0.43
b
0.2
0.00.0
1.0
0.5 1.0 1.5 0.0 I I I I I
—1 0 1
[ro ropeak(T)]/I (T)
0.8
a 0.6
R.
~0.4
b
0.2
0.0
—2
I I I I I I
—1 0 1
L~-~~ k(n)]/~(n)
FIG. 2. Universal form of the computed subgap optical
conductivity. (a) Conductivity spectrum for different val-
ues of the disorder parameter q [Eq. (1)]. The spectrum
broadens as g increases. The conductivity scale is set by
oo —e vs /AE, where A is the cross-sectional area of a chain.
(b) Scaling plot of o (w), where the frequency is scaled by I'(q),
the half-width for the low frequency side of the peak. Note we
claim only that the subgaP (iv ( imp, k) conductivity scales.
Inset: Dependence on the disorder parameter g of o.p k r,
and bu)(q):—2A —~vp~~k(q) .
is converged to 1 ppm for N = 30. The method can be
generalized to the calculation of (GG). The number of
resulting equations scales as N2.
We calculate the conductivity using the exact Green
function obtained by solving Eqs. (8) and (9) and the
approximation (GG) = (G)(G). In Fig. 2(a) we show
the result for several difI'erent q values. The g = 0 curve
shows no absorption below the energy gap and a diver-
gent behavior just above the gap. For finite g values, the
singularity is absent, and the spectrum broadens rapidly
as q increases. In the inset of Fig. 2(b) we plot the depen-
dence on g of three quantities characterizing the shape of
the conductivity curve: I', which is the half-width for the
low frequency side of the peak, the inverse of the conduc-
tivity at the peak, and b~, the shift of the peak frequency
from 2A. Quite remarkably, when we scale the conduc-
tivity by the peak value and the frequency by I", we find
that the scaled curves for all g values have a universal
form independent of g below the peak frequency [20,21].
Experimental data on the platinum, linear-chain com-
pound K2Pt(CN)4Bros 3HqO or KCP(Br) [6] show a
similar scaling behavior. In Fig. 3 we plot the data for five
FIG. 3. Scaling plot of experimental data for KCP(Br) [6]
and trans-(CH) [7] at different temperatures. For each tem-
perature, the corresponding disorder parameter value is ob-
tained by choosing q such that the theoretical I'(q)/cup k('l7)
value is equal to I'(T)/wp, k(T). Inset: Comparison of the
temperature dependence of q for KCP(Br) with Eq. (1). We
fit the data with a curve in which the first number (0.3)
characterizes extrinsic disorder eA'ects and the second number
[q, (0) = 0.07] depends sensitively on the value of iv2k~ (twice
the third number in K) which we took from experiment [5].
The possibility of the scaling curve extending to temperatures
above the three-dimensional ordering transition at Tp —120
K [5] is based on the persistence of a "pseudogap" above Tr
[25].
temperatures. The sealing curve below the conductivity
peak has approximately the same form as the theoretical
curve in Fig. 2(b). We deduce the effective disorder pa-
rameter for each temperature by choosing g such that the
computed I (rl)/urp, k(rj) [22] is equal to I'(T)/imp, k(T).
This criterion was used to choose the value g = 0.43 in
Fig. 1. In the inset of Fig. 3 we plot the g values thus de-
terrnined versus temperature. The temperature depen-
dence of rI is compared with the postulated form (1):
an extrinsic temperature-independent g, plus an intrin-
sic temperature-dependent g, . The fit value of extrinsic
disorder is large. This is not surprising since x-ray and
neutron scattering studies have shown that the bromine
ions randomly occupy two difFerent sites [23]. This dis-
order could cause significant scattering of the electrons
along the platinum chains [24].
We have analyzed the optical conductivity data for
trans-polyacetylene (CH) at room temperature [7].
Since the phonon frequency at 2k~ for this material is at
a much larger energy scale than the room temperature,
the optical conductivity does not depend sensitively on
temperature. Consequently, unlike in KCP(Br), we can-
not determine the separate contributions from extrinsic
and intrinsic disorder. In Fig. 3, we show that the scaled
(CH) subgap conductivity curve agrees well with those
of KCP(Br).
The peak of the conductivity is the product of ao =
e v~ /AE and the dimensionless ratio crp, k(q) joo deter-
mined from fitting the experimental curves. Accordingly
the peak conductivity yields o.o and hence vp since the
4017
VOLUME 71, NUMBER 24 PHYSICAL REVIEW LETTERS 13 DECEMBER 1993
TABLE I. Fermi velocity v~ and disorder parameter g determined from fits of experimental
optical conductivity to our calculations. The peak height determines vz which is compared to
the results of band structure calculations [3,5]. The il values obtained from the fits are compared
to the intrinsic rl, (T = 0) = (bu /up) (Aa/hvF), calculated using the band structure vF, and the
zero-point motion b'u and lattice distortion up in Table I of Ref. [9]. The electron-phonon coupling
A is calculated from the estimated value of ri, and Eq. (1).
KCP(Br)
(CH)
vF (10 cm /sec)
Fit Band structure
5.2 11.0
7.1 9.3
Fit
i7, =0.3, rl, (0)=0.07
rl, +rl, (0)=0.15
Estimate of g,
0.07
0.18
A
0.96
0.48
gap 2L and cross-sectional area A are known. Table I
shows that the vF's obtained this way are smaller than
estimates based on band structure calculations. The val-
ues of rl, (T = 0) for KCP(Br) and rl(T = 0) for (CH)
obtained from the fitting are compared with estimates
from the lattice zero-point motion.
In conclusion, we have developed a new method
for computing the electronic properties of quasi-one-
dimensional Peierls systems including disorder and lat-
tice Huctuation eKects and used it to obtain the absorp-
tive conductivity at all temperatures. Our results com-
pare well with data on KCP(Br) and trans-polyacetylene.
The Subgap conductivity curve has a universal scaling
form.
We thank Leo Degiorgi for helpful discussions and for
showing us his results before publication. This research
has been supported in part by the U.S. DOE, Basic En-
ergy Sciences, Division of Materials Sciences, and the
OSU Center for Materials Research.
[1] Proceedings of the International Conference on Science
and Technology of Synthetic Metals, Goteberg, Sipeden,
IP. I8 August —I992 [Synth. Met. 55—57 (1993)].
[2] For a discussion of quantum wires see, e.g. , C. Weisbuch
and B.Vinter, Quantum Semiconductor Structures (Aca-
demic, Boston, 1991), Chap. 6.
[3] A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su,
Rev. Mod. Phys. 60, 781 (1988).For (CH), vF = 2tpa/h
where 4to —10 eV is the tight-binding bandwidth and
a = 1.24 A is the intrachain atomic separation. The cross-
sectional area per chain A = 1.55 x 10 cm .
[4] G. Griiner, Rev. Mod. Phys. 60, 1129 (1988).
[5] K. Carneiro, in Electronic Properties of Inorganic Quasi
One-Dimensional Compounds, Part II, edited by P. Mon-
ceau (Reidel, Dordrecht, 1985), p, 1. KCP(Br) has a
free-electron-like band, vF = hkF/m. kF = 0.94 A
g = 4.90 )& 10
[6] P. Briiesch, S. Strassler, and H. R. Zeller, Phys. Rev. B
12, 219 (1975) [KCP(Br)]. KCP(Br) is an acronym for
KqPt(CN)4Brp, s 3(HzO).
[7] G. Leising, Phys. Rev. B 38, 10313 (1988) [(CH) ].
[8] R. P. McCall et al. , Phys. Rev. B 39, 7760 (1989) (TCNQ
compounds); D. Berner et al. , J. Phys. (France) IV C2 3,
255 (1993) [(TaSe4)2I]; L. Degiorgi, J. Phys. IV (France),
Colloq. 3, C2-103 (1993) (Kp, sMoOs).
[9] R. H. McKenzie and J. W. Wilkins, Phys. Rev. Lett. 69,
1085 (1992); Synth. Met. 55—57, 4296 (1993).
[10] F. H. Long, S. P. Love, B. I. Swanson, and R. H. McKen-
zie, Phys. Rev. Lett. 71, 762 (1993).
[11] S. A. Brazovskii and I. E. Dzyaloshinskii, Zh. Eksp. Teor.
Fiz. 71, 2338 (1976) [Sov. Phys. JETP 44, 1233 (1976)].
[12] If we convolute the rigid-lattice conductivity with a
Lorentzian, we obtain
1dx Ql+ x [1+x —(u)/6+ i() /4]
where (4 corresponds to an energy-independent damp-
ing of the transition. Z.-b. Su and L. Yu [Commun. Theor.
Phys. (Beijing) 2, 1341 (1983)] previously pointed out
that this form does not agree quantitatively with the ob-
served subgap absorption.
[13] H. Takayama, Y. R. Lin-Liu, and K. Maki, Phys. Rev. B
21, 2388 (1980).
[14] A. A. Ovchinnikov and N. S. Erikhman, Zh. Eksp. Teor.
Fiz. 78, 1448 (1980) [Sov. Phys. JETP 51, 728 (1980)].
[15] In particular, the condition ($(x)((y)) = 0 implies that
there is no umklapp scattering. A nonzero value occurs
for a real random backscattering potential which corre-
sponds to the half-filled band (commensurate) case, an
example of which is polyacetylene (CH)
[16] H. J. Fischbeck and R. Hayn, Phys. Status Solidi (b)
158, 565 (1990), and references therein.
[17] Our method can be further generalized to the case with
forward, backward, and umklapp scattering processes. In
fact, we can reproduce the results of previous density of
states calculations [16) with great efficiency.
[18] J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323
(1986).
[19] I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Intro
duction to the Theory of Disordered Systems (Wiley, New
York, 1988), p. 146.
[20] This universal behavior is quite distinct from that obeyed
by the Urbach tails seen in three-dimensional crystalline
and amorphous semiconductors.
[21] The scaling curve in Fig. 2(b) below the absorp-
tion peak is fitted with the function o(w)/o(uri, „q)
exp[ —0.49[(~ —~...,)/r I ' —0.20 I (~ —~o"-k) /I'I ')
[22] The dependence of I'/~~, q on the disorder parameter il
can be fitted by I'(rl)/ui„k(rt) = g'"(0.414+ 0.077').
[23] J. M. Williams et al. , Phys. Rev. Lett. 33, 1079 (1974).
24] H. Wagner et al. , Solid State Commun. 13, 659 (1973).
[25 P. A. Lee, T. M. Rice, and P. W. Anderson, Phys. Rev.
Lett. 31, 462 (1973); M. J. Rice and S. Strassler, Solid
State Commun. 13, 1389 (1973).
4018


Universal Subgap Optical Conductivity in Quasi-One-Dimensional Peierls Systems

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.305.8582

Universal subgap optical conductivity in quasi-one-dimensional Peierls
  systems

Universal subgap optical conductivity in quasi-one-dimensional Peierls systems

Abstract

Similar works

Full text

Available Versions

Crossref

University of Queensland eSpace