3465
J . Phys. Chem. 1992, 96, 3465-3474
Material Properties of Low-Dimensional Charge-Transfer Salts. The Temperature Dependence of the Direct Current Electrical Conductivity in Synthetic Metals of the Kohn Type Michael C. Bohm* and Arnulf Staib' Institut f u r Physikalische Chemie, Physikalische Chemie III, Technische Hochschule Darmstadt, 0-6100 Darmstadt, Germany (Received: July 22, 1991)
A Kubo-Mori projector formalism for one-phonon, one-electron (Iple) scattering processes has been employed to investigate the temperature dependence of the dc electrical conductivity, u( r), in low-dimensional synthetic metals where remarkable mode softening is possible. The single-particle theory is applied to a number of model systems with small anisotropies in the conductivity but where particle hole symmetry on the Fermi surface leads nevertheless to flat portions. The latter are the structural (topologic) prerequisites for the development of staggered Kohn mode softening. The T dependence of a( r ) in the considered materials follows essentially a T 2law of decay at ambient pressure. This behavior can be reproduced with remarkable accuracy in terms of the suggested single-particletheory restricted to (lple) scattering events. With decreasing anisotropy in the electrical conductivity, the T 2boundary law is gradually shifted to a T ilaw in all materials where the formation of effectively flat portions on the Fermi surface is not possible. The anisotropy ratio, ull/uL,is without decisive influence on a T"law of u ( T ) in 2:l materials with an integral charge transfer between donor and acceptor units maintaining dependence is also reproduced under particle hole symmetry on the Fermi surface. The classical Bloch formula with its T i high-pressure conditions, causing an attenuation of mode softening. The temperature dependence of a( r ) for temperatures, T , smaller than the Peierls transition temperature, Tp,can be described by a two-/three-parameter function defined by T i and T 2contributions of the metallic type and an exponential Boltzmann term measuring the charge density wave gap in the one-electron band spectrum.
1. Introduction
In the present contribution on material properties of low-dimensional organic and organometallic charge-transfer (CT) salts, we investigate the temperature dependence of the dc electrical conductivity u( 7') in synthetic metals. The rather unexpected analytic form of the u( r ) curves in C T compounds focused the interest of many experimentalists and theorists over the past 20 years. Any physically reliable transport theory has to take into account several experimental boundaries which will be shortly summarized. Already at the beginning of this article we want to stress, however, that it is beyond the accessible scope of the work to give a complete review covering previous theoretical approaches to explain the nonconventional u( T ) curves. Only those contributions are explicitly cited that are more or less tangled with the present analysis. In the past it has been an almost universal trend to describe the temperature dependence of u( T ) in low-dimensional organic solids in terms of a T 2law at ambient pressure. The latter T exponent corresponds to the stacking axis (11) in quasi-one-dimensional (1D) solids. Perpendicular (I) to the principal axis, T nlaws are realized with n values of ca. 0.5-0.7.1-3 A recent investigation has shown that n is sizeably reduced with increasing pres~ure.~In the kilobar range, n numbers approaching 1.O have been detected in TTF-TCNQ and TSF-TCNQ C T salts of 1:l stoichiometry. The abbreviations for the studied synthetic metals are collected in the Appendix. For the subsequent theoretical analysis, it is quite significant that the logarithmic slopes of u as a function of p are rather similar in the above C T salts. A microscopic explanation for the strong volume dependence of the a( 7') curves in synthetic metals has not been given in the above contribution, which is prevailingly restricted to the presentation of experimental data. Finally it should be mentioned that phonon frequencies increase drastically with increasing pressure; Le. Kohn mode softeningS in low-dimensional metals is thereby strongly attenuated. Kohn effects accessible in synthetic metals have been quantified in previous work of one of It turned out that it is convenient *Author
to
whom correspondence should be addressed.
'Present address: Department of Chemistry and Biochemistry, University
of Colorado, Boulder, CO 80309-0215.
0022-3654/92/2096-3465$03.00/0
to subdivide the synthetic metals susceptible to the formation of soft modes into two characteristic classes. The underlying ordering principles are shown schematically in Figure 1. In the diagram we have given an experimentally deduced interrelation between the measured temperature dependence of the electrical conductivity, u ( T ) T", and the associated anisotropy ratios, q / u , (central part). All important families of charge-transfer salts have been considered in the evaluation of the u,,!u, versus T"relation. The smaller vertical bars measure the width of the given correlation. The value of the exponent, n, in the T"law of u( T ) has been determined on the basis of measured u( T ) curves for temperatures larger than the metal-nonmetal transition temperature by means of a least-squares fit. At the right and left margins, the experimental T"numbers are related to Fermi surface (FS) properties realized in the organic CT salts. Materials with quasi-onedimensional ( 1D) surfaces have been called class I Kohn metals. In the figure this is symbolized at the right margin, where we have modeled a strict 1D example without any transverse dispersion, t,, and a strongly anisotropic one (two-dimensional projection). For convenience only, one lattice parameter, a, has been employed in the representation. kF labels the Fermi momentum, and kFo,used in Figure 1, the Fermi momentum in the absence of transverse hopping. For incommensurate CT salts of Dd+A" stoichiometry (D = donor, A = acceptor; 6 # 1: charge transfer), the anisotropy ratio measures directly the shape of the corresponding FS; incommensurability is here the necessary prerequisite. In the central hole part of the FS, reciprocal lattice vectors, = (0,0,2kF0),are displayed by broken lines which lead to a one-dimensional divergence of the electronic polarizability, II(4,w). The experimental outcome of this divergence is the above-mentioned mode softening. The limit w 0 corresponds to a metal-nonmetal transition of the Peierls type.* Mode sof-
-
-
( I ) Toombs, G . A. Phys. Rep. 1978, 40, 181. (2) Torrance, J . B. Ace. Chem. Res. 1979, 12, 79. (3) Jerome, D.; Schultz, H. J . Adu. Phys. 1982, 31, 399. (4) Cooper, J . R. Phys. Reo. B 1979, 19, 2404. (5) Kohn, W . Phys. Reu. Lett. 1959, 2, 393. (6) Bohm, M . C. Chem. Phys. 1991, I55, 49. (7) Bohm, M . C. J . Chem. Phys. 1991, 94, 5631. (8) Peierls, R. E. The quantum theory of solids; Oxford University Press: London, 1959.
0 1992 American Chemical Society
3466 The Journal of Physical Chemistry,Vol. 96,No.8,1992 class
U I T salts
class I CT salts n Tn low of a (TI In orgnnit CT snits .. ............... ...
tlossll
................... -kg
20/30 nesting vectors
---
a,,la,
0 +kg mh
10 nesting vectors - - -
Figure I . Schematic display of the variation of the exponent n in a T" law of decay of the electrical conductivity, u, as a function of the roomtemperature anisotropy, ul,/uL,in Kohn metals of the classes I and 11. Characteristic Fermi surfaces (FS) for the two subclasses are shown schematically on the right- and left-hand sides. On the corresponding surfaces, nesting vectors have been symbolized by broken lines. In class I1 materials, an exponent n of ca. 2 is found also in the limit of small anisotropies. Investigated model systems belonging to this class are located in the smaller u H / u Iinterval indicated by a full horizontal line. The label 'expected" indicates our expectations concerning the value of n in class I1 materials for increasing anisotropies. In type I systems, n is an increasing function of ull/uL. For further details see the text.
tening is only possible in the presence of a macroscopic number of identical wave vectors, 4'. As can be seen on the right-hand FS, transverse modulation is accompanied by a reduction in the number of 4' vectors (upper part on the FS). The strength of Kohn mode softening is attenuated proportional to the ratio (tl/tlI)l/*. A simple 1:1 correspondence between the measured anisotropy ratio, U , ~ / Uand ~ , the analytic structure of the FS is not conserved any longer in Kohn metals of class 11. A typical wraped FS, which shows particle hole symmetry, is shown schematically on the left-hand side of Figure 1. Fermi surface nesting allows for mode softening at wave vectors 4' = ( r / a , r / a , 2 k F )(3D solids) or 4' = (0,*/a,2kF) (2D systems with a dispersionless x axis). The underlying nesting vectors are again symbolized by broken lines. With respect to the latter vectors, the FS is effectively flat. In this context it is of fundamental importance to recognize that measured uli/uI ratios in the class I1 Kohn metals transmit no direct information on the symmetry of the corresponding FS. Even if the measured uI1/u1ratio changes remarkably as a function of T,it is nevertheless possible that the effective flatness of the FS in relation to characteristic nesting vectors is conserved. Boundary curves for the two types of lattice instabilities have been derived previously.6 For the class I1 Kohn metals, so-called metastable lattices have been defined. They label systems with strong mode softening, which does not propagate finally to an insulating low-temperature state; the Peierls transition with w = 0 is scarcely suppressed. In ref 7 it has been demonstrated that metastability in Kohn metals of class I1 is often realized in D2+Aand D+A< CT salts with smaller anisotropy ratios; see also below. In the present contribution we explain the microscopic origin leading to the experimental findings displayed in Figure 1. As a central topic we rationalize that the temperature exponent n of ~ ( 7 ' )in Figure 1 is determined by mode softening or more precisely by the electronic properties allowing for softening of lattice modes, in Kohn metals of the classes I and 11. To summarize, we find one region in Figure 1 where n increases linearly with ul,/ul. In strong quasi-1D metals, i.e. the i d e a l i i boundary of the class I Kohn metals, a T 2behavior of a( 7') is realized a t ambient pressure. Typical quasi-1D compounds with planar Fermi surfaces, for example, are the aforementioned TTF-TCNQ or TSF-TCNQ CT ~ a l t s . ul,/ul ~ ? ~ in the sulfur system reaches room temperature values >500; uI1/uI is reduced to ca. 300 in the selenium compound. The dc electrical conductivity in the former material and similar compounds follows throughout a T zlaw. The exponent is reduced in TSF-TCNQ, which is less one-dimensional. The 'T2 anomaly" symbolized on the extreme left of Figure 1 is a peculiarity observed in materials belonging to the class I1 Kohn metals. The T 2behavior is coupled to the existence of a stronger Kohn effect in metastable lattices. Experimental veri(9) Bohm, M . C.; Staib, A. Chem. Phys. 1991, 155, 27.
Bohm and Staib fications of mode softening in Kohn metals of class I1 are available for (TMTSF)2X10." and (BEDT-TTF)2X'2 compounds. The empirical correlation displayed in Figure 1 thus strongly supports the assumption that T 2laws of the dc electrical conductivity in synthetic metals are microscopically related to Kohn effects. At this place it is informative to mention that the observed T dependence of ull/ul fits well with the suggested subclassification of the Kohn metals. In the 1D compounds of class I, U , ~ / Uis~ strongly enhanced with decreasing T. This is caused by the T z modulation of u ( T ) in the stacking axis (11) direction and an exponent n = 0.5-0.7 in the perpendicular plane, I. In the class I1 Kohn metals, ull/ul is only weakly T dependent. But n is nevertheless of an order of magnitude of ca. 2.0.13 Previous to the above scenario, we decided to investigate the T dependence of u ( T ) by a Kubo-Mori projector scheme for one-phonon, one-electron ( l p l e ) scattering processes.14Js In a recent contribution we have studied quasi-1D metals with this approach.16 The corresponding single-particle method has been formulated for the first time in ref 17, where simplified analytic expressions for the respective relaxation times, T(T),for (lple) processes have been reported. In ref 16 we have reproduced the fundamental T 2 dependence of u( T ) at ambient pressure by taking into account the effect of Kohn mode softening. The present investigation follows several new directions. At first the transport theory is formulated in the absence of any remarkable frequency shift. This may be due to applied pressure (see above), FS properties preventing divergences of the electronic polarizability, or high-temperature conditions with negligible frequency shifts. Then we formulate the theoretical approach for Kohn metals of class I1 with remarkable mode softening on nonplanar Fermi surfaces. Metastability is here of particular importance. Finally we analyze the T variation of u ( T ) in the insulating low-temperature ground state of unstable quasi-1D metals. Previous theoretical models to account for the T"dependence of u( 7') in synthetic metals can be divided into at least two different branches. On one hand the enhanced T exponent has been traced back to strong electronic correlations.13J8 But the electronelectron coupling in materials with repulsive interactions is retarding: i.e. the T gradient of u( T ) is even flattened. In contrast to this many-particle ansatz, several singleparticle models for u( T ) have been published in the past years.1920 In some of these models, a superposition of (lple) and (2ple) scattering processes (libron contributions) has been suggested to explain the T 2behavior of u ( T ) in synthetic metals. But it seems that the experimental correlation in Figure 1 cannot be explained by the above schemes. Pure pressure/volume effects without consideration of microscopic restrictions, i.e. mode softening, fail also to reproduce the experimental correlation of Figure 1 . It is obvious that the characterized systems are much too complex to take into account all degrees of freedom in a suggested transport theory. Also the present adaptation is based on simplifying assumptions, abstractions, and idealizations. Nevertheless we believe that the physical key elements determining the material properties of the synthetic CT salts have been considered in our model.
(IO) Jacobsen, C. S.;Tanner, D. B.; Bechgaard, K. Mol. Crysr. Liq. Crysr.
1982, 79, 25. ( 1 1 ) Eldridge, J. E.; Bates, G.S. Mol. Crysr. Liq. Crysr. 1985, 119, 183. Ng, H. K.; Timusk, T. Mol. Crysr. Liq. Crysr. 1985, 119, 191. (12) Nowack, A.; Weger, M.; Schweitzer, D.; Keller, H. J. Solid Srore Commun. 1986, 69, 199. Nowack, A,; Poppe, U.;Weger, M.; Schweitzer, D.; Schwenk, H. 2.Phys. B 1981.86, 41. (13) Bulaevskii, L. N . Adu. Phys. 1988, 37, 433. (14) Kubo, R. J . Phys. SOC.Jpn. 1957, 12, 570. (15) Mori, H . Prog. Theor. Phys. 1965, 33, 423. (16) Staib, A. G.;Bohm, M . C. J . Chem. Phys. 1989, 91,4961. (17) Dieterich, K.; Wagner, M . Phys. Srorus Solidi B 1985, 127, 715. (18) Abrikosov, A. A.; Ryzhkin, 1. A. Adu. Phys. 1978, 27, 947. (19) Kaveh, M.; Gutfreund, H.; Weger, M. Phys. Reu. B 1978,18, 7171. Kaveh. M. Phvs. Reu. B 1981. 24. 619. (20) Conwell, E. M . Phys. keu: B 1980, 22, 1761. Hale, P. D.; Ratner. M. A. J . Chem. Phys. 1985,83, 5277. (21) Chu, C. W.; Harper, M. E.; Geballe. T. H.; Greene, R. L. Phys. Reu. Leu. 1973, 31, 1491.
Electrical Conductivity in Low-Dimensional Metals
Ni (PC)I Figure 2. Schematic representation of the Ni(PC)I structure along the (001) direction. The Ni site in the center of the 7r ligand is symbolized by a hatched area, the carbon and nitrogen atoms are symbolized by open and full circles. The location of the 13- acceptor unit is indicated on the upper right by a checked area.
We have selected (TMTSF)2PF6,22(BEDT-TTF)2C104,23and 8-(BEDT-M'F)21$4 in the present study as typical members of the class I1 Kohn metals of D2+A- stoichiometry. (2,5-CH3DCNQI)$u has been adopted as a D+A2- pendant.2s Additionally we have studied u(T) of the organometallic Ni(PC)I system, which is a member of the class I Kohn m e t a l ~ . ~ ~ JA' schematic representation of its structure is given in Figure 2. The organometallic C T salt belongs to a class of low-dimensional phthalocyanines and porphyrines, whose solid-state electronic structures have been studied previously by one of U S . ~ * - ~ I The theoretical background of the adopted single-particle theory for the T dependence of u ( n is summarized in section 2; a comparison of experimentally and theoretically calculated normalized conductivity curves, u(T)/u(298 K), is given in the next section. Final conclusions are then presented in section 4. In this context we formulate general ordering and building principles realized in low-dimensional CT salts. 2. Theoretical Background
To calculate u ( T ) in the employed Kubo-Mori scheme, we adopt a simple Frohlich H a m i l t ~ n i a nfor ~ ~a monatomic chain with an effectively flat FS. The effective flatness either can be the result of quasi- 1D interactions in an incommensurate lattice (class I Kohn metals) or can be caused by nesting on a nonplanar FS (class I1 systems). The associated reciprocal lattice vectors leading to the divergence of the electronic polarizability, II(&w), read 4' = (0,092kF) in the 1D case and 4' = (*/a,0,2kF), (?r/a, */b,2kF) for 2D/3D compounds. a and b are lattice constants, and 2kF labels throughout twice the Fermi momentum. The internal structure of the synthetic metals is not explicitly taken into account in our topological model. Whenever convenient in the subsequent formulations, we adopt q = d, which either is a consequence of the quasi-1D properties of the class I materials (22) Bechgaard, K.; Jacobsen, C. S.; Mortensen, K.; Petersen, H. J.; Thorup, N. Solid State Commun. 1980, 33, I 1 19. (23) Saito, G.; Enoki, T.; Toriumi, K.; Inokuchi, H. Solid State Commun. 1982, 42, 557. (24) Carlson, K. D.; Grabtree, G. W.; Hall, N.; Copps, P. T.; Wang, H. H.; Enge, T. J.; Beno, M. A.; Williams. J. M. Mol. Cryst. Liq. Cryst. 1985, 119, 357. (25) Aumiiller, A.; Erk, P.; Klebe, G.; Hiinig, S.; von Schiitz, J . U.; Werner, H.-P. Angew. Chem. 1986, 98, 759. (26) Schramm, C. J.; Scaringe, R. P.; Stojakovic, D. R.; Hoffman, B. M.; Ibers, J. A.; Marks, T. J . J . Am. Chem. SOC.1980, 102, 6702. (27) Martinsen, J.; Palmer, S. M.; Tanaka, J.; Greene, R. L.; Hoffman, B. M. Phys. Rev. B 1984, 30, 2612. (28) Bohm, M . C. Phys. Rev. B 1983, 28, 6914. (29) Bohm, M. C.; Ramirez, R.; Ole$ A. M. Ber. Bunsen-Ges. f h y s . Chem. 1987, 91, 717. Chem. Phys. 1987, 117, 405. (30) Ramirez, R.; Bohm, M. C. Int. J . Quantum Chem. 1988, 34, 73. (3 1 ) BBhm, M. C. One-dimensional Organometallic Marerials; Lecture Notes in Chemistry, 45, Springer: Heidelberg, 1987. (32) Frbhlich, H. froc. R . SOC.London, A 1952. 215, 291.
The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 3467
or is caused by the effective flatness of the FS in Kohn metals of class 11. In the metallic state of the low-dimensional solids, only ( l p l e ) processes are allowed in the calculation of the respective relaxation times. Note that the aforementioned D2+Aand D + A r compounds are still metallic although the corresponding Brillouin zone is folded. Folding leads here to a half-filled dispersion in the majority component. For temperatures smaller than the metal-insulator transition, Tp, we have taken into account electron-phonon scattering of the Umklapp type. Before we define the equations of the adopted Kubo-Mori projector scheme, two nontrivial problems should be mentioned: (i) One concerns the definition of the electron-phonon coupling constant X in metastable compounds. Here we adopt a simple mean-field interpolation relating X to T,,the temperature at the maximum of the a(T) curve. (ii) The second problem deals with the divergence of the thermal expectation values in the phonon part of the Hamiltonian in the neighborhood of T pas encountered in the mean-field approximation. By some kind of van Hove limit" in the evaluation of the relaxation time, r(r),for the considered (lple) processes, this nonanalyticity is suppressed. The symbol r stands for r = l / k ~ T with kBdenoting the Boltzmann constant. The underlying Fr6hlich Hamiltonian, H,for the coupled electron-phonon system is defined in eq 2.1 in terms of the unH = Ho + HO,ph+ HI = H,+ HI (2.1) and the unperturbed phonon perturbed one-electron operator, Ho, part, HO,ph, as well as the electron-phonon (ep) interaction, H I . For the present problem, it is convenient to combine the zerothorder elements Ho and Ho,* to a compound Hamiltonian, H,.The (ep) interaction in HI is described subsequently in a perturbational limit; this can be only an approximation in the studied synthetic metals. H,and H Iare defined by
Hz = 2 6 ( z ) C k + C k k
+ (l/v)1'3c(PqPq+ + WqZQqQq') B
(2.2)
and
HI =
gG(ki)Q&k+Ckl k.q'
(2.3)
V in (2.2) symbolizes the system volume. t(z) denotes the electronic dispersion. In the determination of the thermal expectation vaJues, we have used the simplest free-electron description, t(k) = h2k2/2m (m = electron mass). The c k + and c k stand for electron creation and annihilation operators for the kth Bloch state. Qq symbolizes a mass-reduced longitudinal normal coordinate, and Pqstands for_the crystal momentum of the phonon with wave vector, q. G(q',k) in (2.3) denotes an (ep) coupling element which depends in the most general case ,on both the electronic and phononic momentum coordinates, k and tj, respectively. The coupling element, G ( B , however, is only a function of the vibrational momentum quantum number, 4'. The phonon frequency, wq, is determined by the well-known Debye model
= vs*q (2.4) with us denoting the sound velocity. In the above relation and the subsequent ones, we have already assumed scalar definitions for the 4' vector. In (2.5) we have expressed the dc electrical conductivity, u(r), wq
= (r/vs)Wr(4 (2.5) as a function of r(r) for (lple) processes and us. us is alternatively defined for I D or staggered reciprocal lattice vectors in Kohn metals of the classes I and 11. The quantity C(r) in (2.5) symbolizes the scalar product (2.6), with Z(r) denoting the operator C(r) = (Z(O)IZ(r)) (2.6) of the electrical current. The current operator, I(O), reads
(2.7) (33) van Hove, L. Pfiysica 1955, 21, 517.
3468 The Journal of Physical Chemistry, Vol. 96, No. 8. 1992 with e standing for the electronic charge. The explicit form of a scalar product ( 4 Y ) for operators X and Y in Hilbert space is written as ( 4 Y ) = (l/r)J’db
( e x d b H W exp(-bH)Y)B
l/?(r) =
r2e2 2 h mrC(r)
4DN(-ko)&q)IG(q)l2 exP(ra[(y/2a
(2.19) DEN = (exp(ryqJ - 1)(1 + exp(ra[(y/2a (1
0
0
(2.11)
formula, which relates the dc electrical conductivity, u(O), to the current-current correlation function, (Z(ihz)IZ(t)).l4 Z ( t ) is the electrical current in Heisenberg representation, and z abbreviates z = i r / h with f denoting the time. The exponential term on the right-hand side of (2.11) follows from the underlying response theory in the projector approach and guarantees the systems memory loss with respect to the ”external perturbation” at t
-
--m *
+ 0.5q)2- 2kF])) X
+ exp(ra[(y/2a - 0.5q)2- ZkF]]) (2.20)
In the above equation we have used the following abbreviations. N(l1) is the electronic and 6(q)the phononic density of states, a, y, and ko are explained in the eqs 2.21-2.23. The Debye vector, a = h/2m (2.21)
use of the commutator relation [Z(O), H,] = 0 between the zeroth-order Hamiltonian, H,, and Z(0). In (2.1 1) we give the Kubo 1-0
+ 0.5q)’ - 2kF1k2 dq
DEN
(2.9)
where Tr denotes the trace over the canonical density matrix, p ; the latter is defined in (2.10). In the subsequent steps, we make p = expl-rHJ/(Tr expi-rH]) (2.10)
u(0) = (1/V) lim S s d r exp(-tt]lmdz (Z(ihz))Z(t))
X
(2.8)
For any operator X,the thermal average with respect to the system Hamiltonian, H, is given by
(X)!= Tr (P-W
Bohm and Staib
y
fko=
= ho,
(2.22)
aq2 f h w ,
(2.23) 2aq qD, in (2.19) has been estimated by the approximate identity = 2kP This interpolation has been used successfully in previous theoretical investigations of u(r) curves in terms of the described Kubo-Mori projector techniq~e.’~J’ To evaluate the electionic and phononic density of state distributions, N(-k,) and D(q), particle hole symmetry has been assumed. We have N(+ko) = N(-ko) and 6(q)= &-q)
(2.24)
The current-current correlation function in the Kubo equation (2.1 1) can be evaluated by an equation of motion method, which is supplemented by a projector technique in Hilbert space. The corresponding Mori equationI5 reads
The normalization factor, C(r),before the integrand of (2.19) is given by
d -e(t) = i u t ( t ) - C ’ ( r ) l ‘ d s ((1 - P)iLZ(O)I X dr 0 exp(s(1 - P)iL](1 - P)iLZ(o))fQt - s) (2.12)
with N(0) denoting the electronic density of states at the Fermi energy, eF. To determine the integrand in (2.19), we make use of some type of Sommerfeld an sat^^^ in a hypothetical high-temperature limit. Thereby we assume that the unequality E,(2k~) = ykF 5 2 k ~ T (2.26)
e(t)in the above expression is the normalized current-current correlation function
Qt) =
(z(o)lz(r))c?
(2.13)
with the “normalization constant”, C, given by
c = (W3lZ(O))
(2.14)
P in (2.12) symbolizes the projector onto the current at time = 0. We have p = V(O)) (z(0)lC-l
t
(2.15)
The Liouville operator, L,in Hilbert space can be defined formally by the conventional Heisenberg equation of motion (2.16). A iLZ(t) = (i/h)[H,Z(t)] (2.16) similar decomposition as for the system Hamiltonian, H (see eq 2.1), is convenient for the Liouville operator, L. In synthetic (2.17) L = L, L ,
C(r) = e2N(o)kF/ (3mr)
holds. E,(2kF) is the phonon energy at q = 2kF. The assumption of higher temperatures in the subsequent steps is more or less a synonym characterizing vanishing mode softening in the calculation of l/?(r). The given expression simulates therefore also the low-temperature behavior of u( T, in synthetic metals provided that ‘three-dimensionality” or applied pressure prevent sizeable frequency shifts. By means of a Sommerfeld ansatz, (2.19) can be simplified to l/?(r) =
1/ i ( r ) = C(r1-l Jmds (iLJ(0)I explisL,]iLII(0)) 0
(2.18)
time in the high-temperature regime and in synthetic metals where the FS prevents the softening of lattice modes. Additionally the expression is valid under high-pressure conditions where frequency shifts are more and more suppressed with increasing pressure, p . Evaluation of the integral on the right-hand side of (2.18) leads to (2.19).
3r2e2 IG(4)12k~N(o)6(2k~)(1 + (rYkk~)/2)~ 2h2rC(r) exp(2rykF) - 1 (2.27)
Insertion of (2.27) into expression 2.5 for the dc electrical conductivity gives (2.28) where we have summarized only the tem-
+
metals susceptible to Kohn mode softening, the latter processes are described by the L1term of the Liouville operator. A quite simple solution for the inverse relaxation time, ? ( r ) , in the above Kubo-Mori technique can be derived for conventional metals not susceptible to any shifts of the phonon frequencies, wq. The subsequent formula (2.18) describes the inverse relaxation
(2.25)
(2.28) perature-dependent terms entering a ( r ) . All constant material parameters have been collected into a normalization factor, R , not specified in the present context where we are only interested in the T variation of ?T(r). To simplify (2.28), we make use of the identity qD = 2kF and adopt the definitions us = wD/qD and wD = kBTD/h,introducing the Debye temperature into the expression for 5(r). Furthermore we neglect (qkF)/2 versus the additive constant 1 in the denominator of (2.28). This gives the simple proportionality (2.29) for C(r). The straightforward (34) Sommerfeld, A. Vorlesungen iiber Theorerische Physik. Geest & Portig: Leipzig, 1962; Vol. S. ( 3 5 ) Ziman, M . Electrons and Phonons; Oxford University Press: London, 1960.
The Journal of Physical Chemistry, Vol. 96, No. 8,1992 3469
Electrical Conductivity in Low-Dimensional Metals b(r)
-
exp(TD/q - 1
(2.29)
-
expansion of the exponential term on the right-hand side of (2.29) gives a remarkable result; 5(r) T D / Treproduces the classical Bloch formula for the dc electrical conductivity, which is consequently modulated by a T ilaw. The temperature gradient is enhanced with increasing Debye temperatures of the lattice. We have thus an intelligible explanation for the aforementioned pressure experiments on organic C T salts as well as the u(r) dependence in incommensurate solids with smaller u,,/uaLratios. Subsequently we discuss the results for u(r) in synthetic metals with Kohn mode softening, which is now explicitly taken into account in the evaluation of the relaxation time, d r ) , for (lple) processes. Only key steps are summarized in the present work; for further details see our previous investigation16and also ref 17. To make the discussion most transparent, we go back to eqs 2.1 2-2.16. The restriction of effectively flat Fermi surfaces in combination with w 0 for T Tp in systems with mode softening leads to remarkable modifications in the projector approach in comparison to the above formula for the dc electrical conductivity. The form of the FS in Kohn metals leads to the situation that only the autocorrelation of the electrical current is involved in the corresponding equation of motion entering the Kubo-Mori technique. The zero-frequency boundary condition has an interesting influence on the quantities liLZ(0)) and IZ(0)) defined in (2.12). w = 0 guarantees orthogonality between the state vectors liLZ(0)) and IZ(0)). This has the result that the projector (1 - P) acts on liLZ(0)) only as the unit operator with (1 - P)(iLZ(O)) = 0 (2.30)
-
-
T t e zero-frequency condition (2.31) leads then to (2.32) for d C ( r ) / d ~ By ~ ~taking into account the orthogonality condition d (2.31) iw = C1;i; (Z(0)~Z(t))~,=o =0 d dt
- e(t) = C ( r ) - I l ' d s (iLZ(0)) exp(s(1 - P)iLliLZ(O))e(t - s)
and phononic degrees of freedom. The corresponding solutions (2.37) and (2.38) are approximated by the mean-field description. Oq symbolizes the softened frequency of Kohn metals. Equation 2.39 relates the phonon frequency, oa,square to the softened
frequency square via the density response function. Ac(z)] stands for the Fermi distribution function. Equation 2.39 indicates that the softened frequency, 3,, is determined by the analytic shape of the density response function in the neighborhood of 4' = 2kF, which is here negative and increases in magnitude with decreasing T. In this context it is quite remarkable that the softened frequencies remain finite in metastable Kohn metals of class 11. This has its origin in the fact that only certain parts of the FS are truncated in 2D/3D materials. The simple mean-field solutions (2.37) and (2.38) for the electronic and phononic degrees of freedom are employed to calculate the respective inverse relaxation times, l/T(r), in the presence of mode softening on planar and nonplanar Fermi surfaces. The evaluation of the associated thermal expectation values has been explained in ref 16. The canonic density matrix associated with the soft mode would generate a nonanalytic value of (Q(O)*Q(O))F in the neighborhood of Tp. To suppress this nonanalyticy, we assume the validity of the harmonic approximation (2.40) for the phonon density, Tq. Combination of the above 7, = l/(exp(rhw,j - 1) (2.40) elements leads to (2.41) for the inverse relaxation time for (lple) l/T(r) = -lqDN(-ko)b(q)IG(q)(2 r2e2 exp(ra(+ko2 2hrC(r) o kF2)lq2K(q,r)/ii + exp{ra(+ko2 - kF2)l)(1 + exp(ra(-ko2 kF2)l)(ex~~rhw,l - 1) dq (2.41)
(2.30), the soluble standard integro-differential equation (2.33)
processes in the presence of mode softening. The parameter K(q,r) modulates the inverse relaxation time as a function of the frequency shifts accompanying mode softening. It is given by
l / ~ ( r )= C ( r ) - ' l m d s (iLZ(0)l exp(s(1 - P)iL)iLZ(O)) (2.33)
K(q,r) = (1 - :)2(exp(rh&q]
0
(2.32)
0
is derived for the inverse relaxation time l/T(r), Disappearance of the imaginary part, i&(t), indicates that pure relaxation processes are described by T ( r ) in the corresponding Kohn metals. Equation 2.33 has been derived from (2.32) by inserting some type of v,an Hove limit. In this step we have neglected relaxation effects in C(t - s), which is valid if the integralJerne1 in (2.32) decays faster than any accessible variation in C(t - s) can take place. Formally this corresponds to a subdivision of the respective time scale into one subscale for the fast motion and a second one for slow processes. Next we insert (2.33) into (2.12) and decompose the Liouville operator according to (2.17). This leads to l/T(r) = C ( r ) - l l m d s (iL,I(O)(exp{is(L, + LI - LIP))iLIZ(0)) (2.34) 0
To simplify the above equation, we make use of the orthogonality relation (2.30). As already outlined, the projector (1 - P) is approximated by the unit operator. For the solution of (2.34), we define equations of motion (2.35) and (2.36) for the electronic C d r ) = iwk'ck - (i/h)X:G(q)Q&v+q (2.35) 4
&q
= W q Z Q q - G*(q)&-&
(2.36)
k
ck(t)
ck(0)exp(it(k)t/h}
Qq(t) = Qq cos
&,t
+ (P,+/&J
sin 3,r
(36) Berne, B. J.; Harp, G.D. Ado. Chem. Phys. 1978, 1 7 , 63.
(2.37) (2.38)
+ exp(-rho,)) +
sinh (rh;,)
rho,
(2.42)
Insertion of (2.41) and (2.42) into definition 2.5 leads to the searched expression for the dc electrical conductivity, u( T), as a function of T i n synthetic metals with strong mode softening. The analytic structure of the latter equations is too complex to recognize by pure inspection the temperature exponent of u( T) in metals susceptible to strong Kohn effects. For quasi- 1D systems we have shown previously that u( 7') follows prevailingly a T2law of decay. With decreasing mode softening, &2kF wzI( the magnitude of n in a T"law is reduced. It is even possi6.ie to express n as a function of TDand the (ep) coupling constant, A, in quasi-1 D metals. The corresponding numerical interrelations read
-
n
= 1.82 + 3.83
X
10-3TD and n = 1.82A4,0715
(2.43)
The above theoretical elements suggest strongly that the sizeable Kohn effect in materials with effectively flat Fermi surfaces is responsible for the observed deviations in a( T ) from a T'law of decay. Any perturbation of this process due to either FS properties or external pressure will shift the T dependence of a( r ) back to the above T I law, which is the second marginal limit in the present transport approach. On the basis of eqs 2.29 and 2.43, one can understand that the divergence of the density response function in (2.39) renders
3470 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992
possible a continuous transition between the accessible marginal T”limits of a( 7‘) with n = 1 (3D metals, pressure) and those with n = 2 (Kohn metals). The above-mentioned n values
