Polaron transport in quasi-1d organic conductors and in some narrow band systems

PHYSICS REPORTS (Review Section of Physics Letters) 157, No. 6 (1988) 347—391. North-Holland, Amsterdam

POLARON TRANSPORT IN QUASI-iD ORGANIC CONDUCTORS AND IN SOME NARROW BAND SYSTEMS A.A. GOGOLIN L.D. Landau Institute for Theoretical Physics, Moscow, USSR Received March 1987 Contents: 1. Introduction 2. Strong-coupling large polaron mobility 2.1. Interaction with optical phonons 2.2. Polaron scattering 2.3. Polaron mobility in id conductors 2.4. Polarization interaction 2.5. Polaron mobility in the 3d ionic crystals 2.6. Interaction with acoustic phonons 2.7. Drift velocity for an acoustic polaron in strong dcctron fields

349 356 356 359 360 363 364 365 367

3. Small polaron transport 3.1. Effective Hamiltonian 3.2. Anisotropy of polaron bands 3.3. Small polaron mobility in a strong electric field 4. Gunn effect in narrow band conductors 4.1. Main equations of Gunn effect 4.2. Moving domain structure 4.3. Stability of inhomogeneous solutions 5. Conclusions References

371 371 373 376 380 380 382 385 387 388

Abstract: A detailed review of large and small polaron transport theory in quasi-id organic conductors and conducting polymers. and in some narrow band systems including superlattices, superionic conductors and superconductors with heavy fermions is presented. The strong-coupling large polaron mobility is evaluated in Id and in 3d crystals taking account of optical phonon dispersion. The current-voltage characteristic (CVC) for a Id acoustic polaron with saturation of the drift velocity t’~fE)in strong fields E—~°near the sound velocity S is found. This effect has been recently observed by Donovan and Wilson in polydiacetylene PDA TS [8]. The small polaron (SP) spectrum in narrow band conductors is investigated and its strong anisotropy in quasi-Id organic conductors is proved. The CVC for SP is calculated and the characteristic maximum with the negative differential conductivity is found. An exact theory of the Gunn effect in these id systems is developed and the explicit analytic expressions for a domain structure are obtained. The domain stability is studied and the possibility of their experimental observing is discussed.

Single orders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 157, No. 6 (1988) 347—391. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 34.00, postage included.

0 370-1573/88/$15.75

© Elsevier

Science Publishers B.V. (North—Holland Physics Publishing Division)

POLARON TRANSPORT IN QUASI-iD ORGANIC CONDUCTORS AND IN SOME NARROW BAND SYSTEMS

A.A. GOGOLIN L.D. Landau Institute for Theoretical Physics, Moscow, USSR

(~1 NORTH-HOLLAND

-

AMSTERDAM

A. A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

349

1. Introduction A great amount of various experimental data on electrical, optical, thermodynamic and magnetic properties of quasi-id organic conductors and conducting polymers have recently been obtained [1—9]. A great interest in these substances is connected with their unusual electrical properties, including superconductivity [3]. First, it is necessary to mention a very high anisotropy of conductivity o~1o1 5 in PDATS [10], which is about 106 in polyacetylene (CH)~[ii]. A mobility ~ in these substances is i0 also rather large. In TTF-TCNQ this value is about i_102 cm2/V~sec [12], in (CH)~it is about i02_103 cm2/V sec [13,14] and in PDATS even i04—iOt cm2/V- sec [10]. A detailed analysis of experimental data on optIcal spectra in TCNQ salts [15—18] and in (CH) 1 [19—23]demonstrates strong electron—phonon interaction in these substances and in other quasi-id organic conductors [1—9].As a result, the polaron states arise in id systems [20—38].The theory of these states has been developed first by Holstein [24] and by Rashba [25]. The characteristic size of a polaron X0 in this case in dimensionless units determined by the lattice constant a0 = 1 is of the order of the ratio M°I~’Fc where M°is the initial bandwidth and ~FC is the Franck—Condon energy. The value of2w ~FC is determined by the dimensionless coupling constant g and by the phonon frequency ~ ~FC = g 0 [24—26,39—42]. It is necessary to mention 2here, that in the is usually introtheory [27,31, of large34—36]. polarons dimensionless coupling constant a = isg( ~‘FC’ M°)’’ duced In another conducting TCNQ salts the value of M° about 0.1 eV [1—8]and in conducting polymers it is about 1 eV [5—9].So at g ~ 1 and w 0 0.01 —0.1 eV [15—23]the value of -~

~FC ~ M°in TCNQ salts and small polaron (SP) states arise in the system. In conducting polymers the value of ~FC ~ M°,so large polaron states can be observed in this case. There are many substances with ~FC M°,so intermediate polaron states with X0 1 can be found in this case also. The theory of large polarons has been developed in the works of Landau [43], Pekar [44], Bogolyubov and Tyablikov [45],Feynman [46], Lee, Low, Pines [47] and Fröhlich [48]. In refs. [43—45] the adiabatic perturbation theory has been developed. Using this theory the ground state energy H0 and the renormalized mass M* for a strong coupling polaron in 3d ionic crystal with a polarization interaction have been found [43—47](see also the review [26]). The values of H0 and M* can be expressed through the dimensionless constant for polarization electron—phonon interaction [48]: (1.1) -~

=

2

h

H112 (4_4) ~

according to the formulas [26,43—46]:

H

2hw

4m, a~ 1 (1.2) 0, M* 0.023a in the lowest order in a 1. Here m is the effective electron mass and the numerical coefficients in the formulas (1.2) are found’ with the help of the variational procedure [26, 43—47]. At weak coupling a ~ 1 the values of H 0 and M* can be calculated quite easily using the well-known perturbation theory [48], so in the lowest order in a they are described by the following expressions [26]:

0 = —0.109a

-

H0

=

—a!1a0,

-~

M* = m(1 + a16).

(1.3)

Comparing these results with the formulas (1.2) we can easily see, that the strong coupling approximation is valid for 3d polaron at a >6 [26].In real 3d crystals the value of a is usually about 4—~

350

A.A. Gogolin, Polaron transport in qua.si-Id organic conductors and in some narrow band systems

[26]. Only in some ionic crystals of NaCI type this value is about 8—9 [26]. Thus, the range of validity for the strong coupling approximation in real 3d crystals is rather narrow. It is necessary to mention also, that in 3d crystals formation of polaron states is possible only for polarization electron—phonon interaction [26,43—48]. The characteristic polaron size r0 = (0.66a) (2mhw~) [26] in this case is less than the lattic constant a0 at a > 10. Thus, SP states arise in the system at a > 10 and the usual large polaron theory [43—48]is invalid in this case. These restrictions are not so important in id systems because formation of polaron states in this case is possible for all types of electron—phonon interaction [24—27].For example, the strong coupling approximation in the id case is valid at a >2 [27, 34—36]. An effective coupling constant a for short-ranged electron—phonon interactions is about 4—5 in (CH)5 and is more than 10 in PDATS [34—36].A characteristic longitudinal polaron size X() ~ I in this case is not so small and the large polaron theory [24—27]can be used in this case. Evaluation of strong-coupling polaron mobility is one of the most difficult problems in the general polaron theory [26]. The first attempts of its solution were done by Pekar [441 et al. (see the review of Appel [26]). The correct method of a strong-coupling polaron mobility evaluation in 3d systems was developed by Volovik, Mel’nikov and Edel’stein in refs. [49—51].In their works an effective separation of the center-of-mass motion was done using the Bogolyubov—Tyablikov [45]and Lee, Low, Pines [47] transformations. As a result an effective Hamiltonian of a polaron scattering by phonons was obtained. They have proved exactly, that the main contribution to the polaron mobility p. at low temperatures T ~ COo arises from the two-phonon processes of the Compton type [49—5 1]. In these works the general integral equation for a full scattering amplitude WKK. was obtained and its special solutions were studied [51]. As a result the correct dependence p. over T and a was obtained [51]:

p.(T) = y~p.~n0(T),

7~

~,

p.

=

e

mw10 a

2

n0(T)

exp(w0IT).

=

(1.4)

COo

The numerical coefficient y0 1 was determined by the system of two non-linear differential equations of the fifth order, so it was not found in ref. [511. It is necessary to mention, that Volovik, Mel’nikov and Edel’stein [49—5 1] have considered the usual polaron model [26,43—51] with dispersionless optical phonons only. This model is quite reasonable for a polaron energy calculation [43—48]because for large polarons with r0 ~ a0 the characteristic phonon momenta K h are small enough in comparison with the Brillouin momentum KB, determining an essential optical phonon dispersion. This model, however, is not reasonable for evaluation of a scattering amplitude WKK connected with the corresponding Born amplitude VKK. by the equation [49—51]: —

-~

WKK=VKK—~VKK.. ~2

~

WK.K~.

(1.5)

K’

Here ~ = Ky describes the phonon frequencies with account of the Doppler shift and V— \/T/M* is the polaron velocity. In the absence of a phonon dispersion the small denominator of the order of KV arises in eq. (1.5). As a result its solution is rather complicated and is not obtained yet [51].These problems, however, are absent in the more realistic model taking account of a phonon dispersion [35,36]. The dispersion of —

A A. Gogolin, Polaron transport in quasi-i d organic conductors and in some narrow band systems

351

optical phonons in this case can be described at low temperatures by the corresponding quadratic terms of the Taylor expansion for wK near its minimum w0 at K = K0: + (K

= W1~

21M

—

K0 )

0.

(1.6)

The corresponding phonon dispersion (1.6) is rather essential at M* M~[35,36]. This condition is possible at strong coupling a > 6 (for example in NaC1 according to some estimations [26]the value of a 9, so M* 300 m and M0 30 m [52]). In these crystals with a simple cubic structure the optical phonon modes with K0 KB and K0 = 0 are present [53],so we shall consider both these situations. In the first case (K0 KB) the account of a phonon dispersion diminishes strongly the corresponding integral term in eq. (1.5) at low temperatures [35, 36], so the scattering amplitude WKK. = VKK. is well described by the Born approximation. As a result, the simple expression for a polaron mobility p. [35,36]: ~-

-~

-~

‘~

p.

=

‘y p.0 n(T),

y=

/

YITf

K~ \4 ~

~~ )

-s--

/

—1,

n(T)

IWO1 = ~

exp(w0IT)

(1.7)

iri~W0

~.

differs from that of Volovik, Mel’nikov and Edel’stein (1.4) [51] only by the exact expression for the numerical coefficient ‘y 1 and by the pre-exponential factor in the temperature dependence of n(T). It is necessary to mention here, that the numerical coefficient y 1 in formula (1.7) now depends on the dimensionless phonon dispersion factor K~/M0COo 1. It is interesting, that the dependence of p. on a in this case is just the same and is determined by the value of p.0 (1.4). At K0 = 0 eq. (1.5) can be solved exactly [35,36] and the resulting expressions for the values of y and of n(T) have the following form: —~

—

—

y=

--~

(~~)4,

C2

~

1,

n(T) =

(~)4

exp(w0/T).

(1.8)

The numerical coefficient C2 1 is determined by the exact electron wave functions in a polarization well [35,36], so it can be calculated by numerical is interesting, that the 6 is quite analogousmethods. to that ofItthe Pekar [44]. Thus, thecorresponding characteristic dependence of p. on a: /.L(a) a dependence of p. on a for the realistic models of 3d crystals is not the universal one and depends on a concrete structure of a phonon spectrum. The estimations of p. at a = 8, m = m 0 (m0 is the 2/V~sec for all the cases (1.4,free 1.7,electron 1.8) in mass), w0 = 0.01 eV give at T—~ w~ the value p. 10~ cm agreement with the corresponding experimental data for alkali-halides [26]. The value of M 0 in this case is about 100 m in agreement with the corresponding calculations of a phonon spectra [52, 531. The role of a phonon dispersion in id systems is even more important. Indeed, the mobility p. for a heavy polaron at a 1 is determined by the two-phonon scattering with a small enough momentum transfer [35,36], so it is described by the Fokker—Planck equation. In id systems the corresponding forward scattering is an inelastic one and is absent in the case of dispersionless optical phonons. In the present review a mobility of large and small polarons in id conductors is investigated in detail. The mobility p. for large polarons at strong coupling a 1 is calculated using the general method of Volovik, Mel’nikov and Edel’stein [49—51].The corresponding expressions for p. are quite analogous to those in the formulas (1.4, 1.7, 1.8), so at the same parameters the typical value of p. is of the order of —~

—~

—

-~

~‘

~-

352

A. A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

2!V~sec in agreement with the corresponding experimental data for (CH) 10~cm 5 [13, 14] and for other conducting polymers [5—9]. A great progress of the large polaron theory in id conductors makes it possible to solve exactly a lot of interesting theoretical problems. For example, it is necessary to mention first of all the polaron theory in Peierls dielectrics [20—23],the theory of polaron bound states with phonons [27],the theory of acoustic polarons [30—381,the theory of bipolarons [21,29] and the polaron transport theory for id conductors [28,31—36]. As a result, a great amount of different experimental data concerning magnetic and optical properties of (CH)~[191 and other conducting polymers [5—9] were interpreted using the corresponding polaron and soliton models [20—23].It is necessary to mention first of all a characteristic saturation of the drift velocity 170(E) in PDATS [10] near the sound velocity S in strong electric fields E (1—i0~)V/cm. These effects were observed in some conducting polymers [54],molecular crystals and in other organic compounds [55—59]. A characteristic saturation of V~(E)in strong fields is a typical one for acoustic polarons [34—36].It is provided by a corresponding saturation of the polaron velocity V( p) at large momenta p —* near the sound velocity S [30—38].In the case of 3d piezoelectric polarons this effect was studied first by Rona and Whitfield [60]and by Volovik and Edel’stein [61]. asymptotic behaviorand of the polaron[61]: spectrum 2mS wasThe obtained by Volovik Edel’stein E(p) in this case at the large momenta p cr p.0

—

~‘

E(p)=Sp(1_ ~ (ams)ln(f 5)),

V(p)=~=S(1_ ~

ln

~.

(1.9)

2mS is reached rather slowly, according to the As a result, laws the asymptotic V(P)a polaron S at p ~spectrum a logarithmic (1.9). In idvalue systems for the deformation interaction with Id phonons was evaluated by Whitfield and Shaw [30]. The corresponding asymptotic behavior of E(p) and V( p) at p a 2mS described in this case by the power laws [30—361: ~‘

E(p)=Sp(1_ ~ (2mS)),

V(p)=S(1_ ~ (2~mS)).

A characteristic saturation of V(p) at p—*

(1.10)

gives rise to the corresponding saturation of V

0(E) in strong fields. This effect was observed by Donovan, Wilson et al. [10] in PDA TS. Wilson [321has attempted to calculate the function V0(E). It was, however, not found in his work [32]because of some arbitrary assumptions and certain errors. This problem was solved by Gogolin [34—36].As it was shown in his works [34—36]the velocity V0(E) saturates in rather weak field E 1 V/cm in2/Vagreement with sec arises in the the experimental data [10]. As a result, the extremely large mobility p. i0~—i0~ cm system [34—36]. In the case of acoustic polarons all their parameters depend essentially on a polaron velocity V. The corresponding renormalization of electron—phonon coupling constant c~= a(1 (V/S)2)’ diverges at V—* S. So in the narrow interval near the sound velocity S a strong decrease of the polaron size r ‘-~

—

—

0j

arises. As a result the value of r0 tends to a0 and a large polaron transforms to a small one [26]. The general theory of small polarons (SP) was developed by Holstein [24], Firsov [39], Klinger [40,41] et al. [26]. The2) most result[62] of this is an exponential narrowing of polaron foundinteresting by Tyablikov and theory Holstein [24]. In anisotropic quasi-id organic bands M = M° exp(— g conductors this narrowing is strongly anisotropic [63], so the additional anisotropy of electron bands

A A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

353

arises in the system. This effect makes it possible to explain an essential difference by the order of magnitude [63] between the experimentally observed, width of electron bands in quasi-id organic superconductors (TMTSF)2X (X = Cl04, PF6, AsF6 and others) [3] and its theoretical value [64], found numerically without any account of the polaron effects. This effect makes it possible to interpret an extremely high anisotropy of electron bands of the order of 105_106 in (CH)~[ii]. It is quite clear also, that an extremely small electron bandwidth M i0~eV in UBe13 et al. [65,66] can be explained using these polaron effects. The effective diagonalization of the corresponding Hamiltonian for linear electron—phonon interaction is performed usually with the help of a special canonical transformation [39—42]developed in the SP theory. The main idea of this transformation was suggested by Tyablikov [62] and then was developed by many authors [39—42].The corresponding transformation describes a shift of equilibrium positions for ions in a lattice in the presence of an electron in the site. So this transformation provides an approximate diagonalization of an effective Hamiltonian only. The accuracy of this diagonalization is determined by a small parameter X0 ~ 1 [42]. As a result, in the case of the intermediate polaron size X0 1 the summation of the whole diagram row over X0 [42]is necessary for evaluation of the polaron spectrum. The attempt of this summation was done by Gogolin [42] using the Elyutin diagram technique [67], which is quite useful in some other problems of a solid state theory [68]. As it was shown by Gogolin [42] the spectrum of an intermediate polaron at strong coupling g 1 contains a great amount of corresponding to theanalogous bound states of a polaron a large number 2 ~bands, 1. These states are quite to those for SP, with studied in detail by of phonon modes: N g Bryksin and Firsov [69]. The corresponding band structure makes it possible to interpret the thermodynamic, electric and magnetic properties of some narrow band metals like superconductors with the A-15 structure [70]. At small X 0 ~ 1 the binding energy for these states tends to zero and the polaron spectrum transforms to separate polaron and phonon states. This fact was pointed out by Tyablikov and Moskalenko [71], Gogolin [42] and Tkach [72]. In the meson theory the analogous results were obtained by Schweber [73] and by Gribov, Levin and Migdal [74]. The theory of SP mobility was developed by many authors [26, 39—41]. In these works the hopping mechanism of SP motion was studied first of all. This mechanism arises at very high temperatures T w0 due to the strong temperature dependent narrowing of electron bands [26, 39—41] providing their destruction by the strong electron—phonon scattering. These effects give rise to a strong localization of SP in a lattice site and provides a hopping mechanism of its mobility. At lower temperatures T ~ w0 the usual band mechanism of SP motion describes a polaron transport [26,40, 41, 75—78]. At M < to0 one-phonon scattering processes are forbidden by the energy and momentum conservation laws. So, as in the case of a large polaron [26,49—51] the main contribution to SP scattering arises from the two-phonon processes of the Compton type [41,75, 77, 78]. Thus a polaron transport in this case is well described by the Fokker—Planck equation [77, 78] for a heavy particle [79]. The corresponding expression for SP mobility p. at low temperatures T ~ to0 was obtained by Kagan, Klinger [75]and by Gogolin [77, 78]. In this temperature region the main contribution to the mobility arises from acoustic phonons. Their spectrum in most part of quasi-id organic conductors is a 3d one [1—8],so the temperature dependence of p.(T) T~[75,77, 78] with n = 8 at T M [78]and n = 10 at M -~The T to0 [75,77, 78] is afor power one. TTheto value of to0 in temperatures this case corresponds to the Debye frequency t0D~ analogous result p.( T) at very low was obtained by Andreev and -~

—

~‘

—

~‘

—

-~

—~

-

-~

354

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

Lifshits [80] in the case of the vacancy motion in quantum crystals. The analogous expressions describe the low temperature mobility for all heavy quasi-particles in crystals: light ions, heavy fermions etc. [81,82]. The general theory of their quantum motion in crystals were developed by Kagan and Maximov [76]. Gogolin [77, 78] has evaluated the current-voltage characteristics (CVC) j(E) for SP in strong electric fields. As a result, the characteristic maximum of j(E) at E = E10 = M/el was observed with a value of E0 determined by a mean free path 1. At E> E0 there is a decrease of j(E), corresponding to the negative differential conductivity UD = djldE <0. This effect is a typical one for all narrow band conductors like UBe13 [65, 66], superionic conductors [81, 82] and superlattices [83]. All these systems at M < to0 are described by the same model with a two-phonon scattering mechanism of a heavy particle in the corresponding Boltzmann equation. The general structure of this equation with account of a strong electric field was studied in detail by Seminozhenko [84]. At high temperatures T M the dependence of j(E) for narrow band conductors has the simple form [77, 78]: ~‘

2) s = EIE~ /3 = MIT. (1.11) j(E) = ~/3j0e/(1 + e The value of j 0 = Mp0a0 is determined by the average charge density p0. This result was obtained from the exact analytical solution of the Boltzmann equation [77, 78], which is transformed to the Fokker—Planck one for narrow band conductors. The corresponding phenomenological expressions of the type (1.11) were obtained for superlattices in many works (see the review [83]). They describe quite perfectly the experimental data [85]. The formula (1.11) well describes the initial part of CVC for anthracene [58, 59]. The strong temperature dependence of E0~(T)makes it possible to explain the strengthening of non-linearity in CVC with the relatively small decrease of T from 290°Kto 140°K[58]. At weak electric fields E E0 the dependence of j(E) is a linear one, so the mobility p. = f3Ma0/2E0e describes a polaron motion. strong< electric s~E0 the function j(E) (E0/E), so 0D =AtdjldE 0 arises fields in the Esystem. The main physical reason for the negative differential conductivity this effect is a restricted character of electron motion in narrow bands at strong fields. This fact gives rise to a localization of electrons. Thus, the hopping mechanism of electron motion arises in the system due to inelastic phonon scattering processes. This effect is analogous to that in disordered systems [86] with localized electron states. A breakdown of a band mechanism for electron motion in narrow band systems takes place at a strong electric field. At E ~ E 0 a characteristic size of localized electron states 10 = MIeE is rather large, so it is much more than the mean free path 1. Thus, an electron motion looks like a usual diffusion process and the localization effects give small corrections to the low field mobility p.. At E E0 this size is much less than 1, so an electron motion is a hopping one like in disordered systems [86]. The negative differential conductivity °b<0 for narrow band systems in strong electric fields was pointed out first by Keldysh [87] and then by Bychkov and Dykhne [88]. A characteristic electric field, corresponding to this effect, E0 = M/el is not so large. Thus, it can be taken into account in the frameworks of the Boltzmann equation, because all the quantum effects studied in detail by Berezhkovsky and Ovchinnikov [90], are inessential at E E0 At these electric fields the special resonant effects studied by Suns [91]are inessential also because they arise at the larger field E = MIea0 E0 due to the usual condition 1 a0. As it was shown by Suns [91] these effects give rise to the increase of j(E) at E E~~ E0. As a result, the N-shaped CVC arises in narrow band systems in very strong electric ,

,

-~

—

~‘

-~

-

—~

~-

~

~-

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

fields E

355

E1. The analogous effect arises in the system with few electron bands due to effective electron tunneling through an energy gap Eg at a strong electric field E E2 = Egi ea0 according to Franz—Keldysh effect [92]. The negative differential conductivity UD <0 at strong electric fields [77, 78] gives rise to the domain formation and the Gunn effect [93—96].The general phenomenological theory of this effect was developed in the works [97—103]The corresponding domain instability in the systems with cr~<0 gives rise to a periodic modulation of the charge density p, the electric field E and the current density j. These periodic structures move in the system with some velocity VD, determined by an external current J. The exact analytic theory of this effect was developed by Gogolin [93—96]for narrow band conductors. This theory was developed using the exact solution of the Fokker—Plancl~equation, describing a motion of heavy particles in strong electric fields [77, 78]. It is the first exact solution for the Gunn instability, which makes it possible to carry out a comparative analysis of the various phenomenological approaches developed now [99—103]and to test the validity of the corresponding physical assumptions. In the frameworks of the phenomenological approach, suggested by Knight and Peterson [99], the non-linear diffusion equation for the space and time dependence of the electric field E(X, t) was obtained for id Gunn systems: —

-~

-

E5—D(E)E~~+V(E)(E~+~p0)= ~ J.

(1.12)

This equation corresponds to the usual balance between the field current and the diffusion current with account of the displacement current, described by the corresponding terms in the left-hand side of eq. (1.12). The full current J in this case is equal to the sum of these currents as usual [99—103]. In the frameworks of the phenomenological theory this equation was not proved exactly. As a result, the functions D(E) and V(E) were not evaluated and a great amount of different phenomenological approaches with the various relations between them was developed [99—103]. This problem, however, was studied qualitatively by Knight and Peterson [99] and by BonchBruevich, Mironov and Zvyagin [101] using the general theory of differential equations [104—106]. As a result, some properties of eq. (1.12) were studied and the phenomenological theory of the Gunn effect was developed. The existence of periodic solutions for eq. (1.12) was proved in these works [99—103] and the exact expression for their velocity VD = f/p0 independent on a domain shape was obtained. Gogolin [93—96]has proved, that eq. (1.12) is always valid for narrow band systems large 112 1. with This the result is characteristic domain size X.of the order of the Debye radius TD = (KT/4lTep0) valid in the lowest order in small /3 -~1 and in small space and time gradients of the order of liX~~ l/r~ 1. It is interesting, that in the lowest order over /3 1 the Einstein relation D(E) = T p.(E) = T V(E) IE [79] is valid in the system [93—96].Here T is the phonon temperature. This fact demonstrates some advantages of Knight—Peterson phenomenological approach [99], based on this relation, in comparison with the other approaches [100—103],where the dependence of D(E) was not taken into account. It is necessary to mention here, that for narrow band systems the Einstein relation is valid in the lowest order in /3 4 1 only. Just the same limitations are valid for eq. (1.12) also, so all the phenomenological approaches cannot describe in detail the theory of the Gunn effect. The detailed review of these approaches was performed in the books [101—103]. It is necessary to mention, that the exact solution [93—96]for the Gunn effect makes it possible to solve a lot of different problems for such systems. For example, in these works [93—96]the effects of ~-

-~

-.~

356

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

thermodiffusion were studied for narrow band systems and the dependence of VD on a domain shape was investigated. It is interesting, that this effect is rather small [94, 96] and we can neglect it in the general theory of the Gunn effect for narrow band conductors [93—96]. In conclusion it is necessary to point out, that the polaron theory for quasi-id organic conductors and for some other narrow band systems is rather successful now. So it is possible to interpret a wide range of experimental data and to make some interesting predictions for further experimental and theoretical investigations.

2. Strong-coupling large polaron mobility 2.1. Interaction with optical phonons There are many quasi-id organic conductors and conducting polymers with not so narrow electron bands, described by a bandwidth M~ 1 eV [1—9](for example (CH)5 [19] and PDA TS [10]). In these —

systems the formation of large id polarons with r0 ~ a0 is possible according to the general polaron theory [26]. The id structure of these states is connected a high spectrum 5 with in PDA TS anisotropy [10] and isofofelectron the order of 106 in in these substances. This anisotropy is of the order of i0 (CH) 5 [ii]. Evaluations of the electron—phonon coupling constant a in these substances [20—23,32—36] give a 10, so the formation of strong coupling polarons is possible there. The general theory of these states was developed in the works [24—38]. The Hamiltonian of the strong electron—phonon interaction in the units h = m = 1 has the form [25]: —

H=

—!

~

+

~ K

VK

e’~~QK+

2K

~

(2.1)

+ ~K~K)’

(COKWKQKQK

Here QK are the phonon coordinates and ~K = —i aIdQK are their effective momenta. The values QK and ~K can be expressed through the phonon operators bK, b [107]: QK =

(2wK)

—0/2

+

.

(bK + b_K),

‘3k =

i(wK/2)

t/2

+

(bK

—

b_K).

2.2

The analysis of experimental data [1—9]for quasi-id organic conductors and conducting polymers demonstrates the existence of id and 3d phonons in these substances. So the sums over K in the Hamiltonian (2.1) are the 3d ones. In the present part we consider the interaction of electrons with optical phonons in the quasi-id organic compounds [1—9].There are two types of optical phonons in these substances. First of all there are intramolecular vibrations with a very small dispersion ~1(K)4 con. There are also id intermolecular vibrations with a large longitudinal dispersion w 0(K1). In conducting polymers like (CH)5 [19—23]the electron bandwidth M~ 1 eV, so the characteristic polaron size r0 a0 and we can use the continuous approximation [20—38].All the characteristic momentum changes I~K~ 4 KB in this case at low temperatures, so we can consider the quadratic terms in the expansion of w0(K5) near the corresponding minimum value w0(K0) quite analogous to eq. (1.6): —

w0(K~)=

2/M

to0

+

(K~ K0) —

0.

~-

(2.3)

A A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

357

In the case of intramolecular modes with a very small dispersion 4(K) the value of K~4 KB and at T z.l we neglect the longitudinal dispersion z.~(K1),so the transverse momentum dependence of 4 on K1 only is to be taken into account. This dependence is usually well described in organic compounds [108]by the tight-binding approximation [53], so it has the simple form: wo(K±)=wo+z.11(cosKr+cosKz),

414w0.

(2.4)

Here the transverse lattice constants a,, = a~= 1 and the value of 4~has an arbitrary sign. The effective interaction of an electron with optical phonons in quasi-id organic compounds and conducting polymers is usually a short-range one [15—18],so the effective coupling constant VK depends weakly on K. In dimensionless units to0 = 1 it can be expressed through a in the following form: 2. (2.5) VK = (2a)~” In some ionic compounds with a charge transfer like TCNQ salts [1—8]the electron—phonon polarization interaction can take place. In this case the effective coupling constant VK is described by the expression [26, 43—51]: VK

K~’(2V~1Ta)~2.

(2.6)

The effective dimensionless coupling constant a in this case is described by eq. (1.1) and the crystal volume 12 = 1. It is interesting that the polaron formation in id case is possible at an arbitrary structure of effective electron—phonon interaction [24—38],as it was shown first by Rashba [25]. In the 3d case large polarons arise only for polarization interaction [26, 43—51]. The evaluation of the polaron spectrum at strong coupling a ~- 1 in id conductors can be performed using the Pekar method [44].As it was shown in his work [44]the characteristic Franck—Condon energy shift ~FC a2 ~ 1 at a 1, so the effective number of phonons taking part in the polanon formation, N 2 ~ 1 is large enough. As a result in the lowest order in a 4 1 lattice vibrations are described by0 thea classical model [44].Thus the polaron spectrum in this limiting case is determined by the classical minimum of the Hamiltonian H (2.1) in phonon coordinates QK. The corresponding lattice kinetic energy is described by the terms ~ in eq. (2.1) and can be neglected in the lowest order in a~’41 ~‘

[44]. The minimization of the Hamiltonian H (2.1) in

QK

=

Q~ v

00(K) (wKwK)~,

Vnm(K)

=

QK

VK

f

gives the following expression for these values:

dx ~(x) e~~Im(x).

(2.7)

Here ~ (x) are the electron eigenfunctions in a polaron well. These functions are determined by the minimum the Hamiltonian H (2.1) at QK = Q~(2.7) and correspond to the large eigenvalues 2 ~- 1of[24—38]. a The effective polaron mass M* is determined by the formula of Landau and Pekar [43, 44]:

M* = ~

K~Q~Q~K.

(2.8)

358

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

The value of M* /2 is defined in this case as the coefficient in the quadratic term in the low velocity expansion for the polaron energy H in the moving system [43, 441. It is necessary to mention that at large polaron size r00 a0 all the characteristic phonon momenta ~-

K,, 4 KB, so we can neglect the optical phonon dispersion for the evaluation of the polaron spectrum. This dispersion, however, is rather essential for the evaluations of the polaron mobility p., determined by two-phonon scattering processes [35, 36]. In the case of the short-range interaction (2.5)~the ground state energy H)) of a polaron and the 2~’~ are determined in the units = ax by the simple equations [24—38]: eigenvalues e~= a H

0=

~a2

J

2/d~2 d~[(8~~/0~)2

-

4~]=

-

-

~

(2.9)

=.

[-d

~a2

These equations can be obtained by the minimization of the Hamiltonian H (2.1) in QK (2.7). The ground state wave function ~/i 0( ~) for a polaron in a potential well and the wave functions i~i~( ~)of the continuous spectrum have the form [109]: 1,

~=-1,

~(=e~th

~0(~)=(~chfl

1~.~,

(2.10)

~

There is only one level i~in a polaron well and the wave functions i/i~(~) describe the absence of scattering waves.2~This fact isproperty a consequence of one non-scattering character of the effective [109].This is a typical of all the exactly solvable models for id potential polarons 4[37]and ~/i~(~) = 2 ch~ solitons [110]. The analogous equations arise also in the theory of Lifshits tails [111—1151 for id disordered systems. This fact demonstrates some analogy between the localization [86, 112] and autolocalization [20—38]of electrons in id conductors. The values Voq(K) (2.7) in this case have the simple form:

v (K) Oq

i’TTKVK =

-

x

,

\/~(i—iq)ch[~ir(K,,+q)]

7TKVK

v

00(K) = 2sh(~irK~) -

,

K

X

=

aK

,

q

=

aq (2.11)

These values4/15 determine the coordinates (2.7)inand the case scattering amplitude VKK,numerical [34—36].The values (2.8) and H 216Q~ (2.9) this contain the large factors in of M* = 8a with the 3d case 0[26, = —a comparison 43—51], where M* = 0.023a4 and H 2. We are to take 0 = for —0.i09a account of the factor \/~also introduced to the constant a (2.5) in id case simplicity. With account of this factor the numerical coefficients for H 0 and M* are equal to 1/3 and 32/15 correspondingly. The comparison of these formulas for H0 and M* with the corresponding perturbation theory results at weak coupling a 4 1: H0=—aiV~,

M*1+a

(2.12)

demonstrates a wide range of validity for the strong coupling approximation in id case. It begins at a >2, which is much less than the corresponding condition a >6 for 3d case [261.As it follows from these formulas the id systems are to be well described by the special adiabatic perturbation theory of Bogolyubov et al. [45, 49—511, corresponding to the expansion in a -‘ 4 1.

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

359

2.2. Polaron scattering The evaluation of the mobility p. for strong coupling polarons at a 1 is one of the most difficult and interesting problems of the general polaron theory [26]. The main difficulties of this problem are connected with the polaron scattering evaluation in the polaron Hamiltonian using its expansion in a_t 4 1. The first attempts of the mobility evaluation has been performed by Pekar [44] (see also [26]). The exact method for this problem has been developed by Volovik, Mel’nikov and Edel’stein [49—51]. In the work of Volovik and Edel’stein [49] an effective separation of the center-of-mass coordinates has been performed in the Hamiltonian (2.1) using Bogolyubov—Tyablikov [45]and Lee—Low—Pines [47] transformations. As a result, the effective Hamiltonian of two-phonon scattering processes for a polaron has been obtained in the lowest order in a~14 1 [49—51]: ~‘

H= H

0+

t0K’ = t0K

-~-

—

Here by the relation:

Ky,

+~

(O1ç~(b;.bj~. +

~)+

~ V~(b + b_K)(b, + bK).

(2.13)

V= PIM* and the Born amplitude VKK is connected with the values Vnm(K) (2.7)

(~Kto_K,)tl2 VKK =

~ v

0~(K)v~0(—K’)(~— s~)~

(2.14)

.

Another method for deriving eqs. (2.13, 2.14) using the Feynmann integrals [46] was proposed by Mel’nikov and Volovik [50]. 01K WK is connected with the center-of-mass motion A Doppler of theinto phonon frequencies [49—51] It can shift be taken account in the Hamiltonian (2.1) by the transformation to the system of coordinates moving with the polaron velocity V [34—36].At V— a_t 4 1 this effect gives rise to a dependence of s~and M* on V. This dependence in 3d ionic crystals was studied in detail by Davydov and Enol’sky [116] using the computational methods. It is necessary to mention here that only the usual polaron model with dispersionless optical phonons —~

-

=

to

0 has been considered in refs. [49—51]. But the general method for deriving eqs. (2.13, 2.14) in the work of Volovik and Edel’stein [49] is not connected with this assumption. This statement follows immediately from the analogous result for a piezoelectric polaron [49]. The one-phonon scattering correspond processes are included to the transfer Hamiltonian low tempera2 these processes to anot large momentum K~ (2.13). a (K~ At (M*)~2 a2 at tures T 4 a T ~ 1 and K~ (M*IT)1’2 a at 1 4 T 4 a2). As a result, an exponentially small contribution arises in the corresponding scattering amplitude [49, 50]. ~‘

—~

~-

W°(P,K~)= vQO(K)/~K—exp(—ClKXIa) where C

1 1. This fact is a general one due to well-known analytic properties of the ground state wave function ~i0(x)[49, 50], which has no singularities at real x. In our consideration the value ~,r/ 2 contains 2 of C1a=small according to eq. (2.11) for v00(K). So the one-phonon scattering probability W°~ factor i0~even at a = 2. Thus the quadratic terms in the phonon operators bK, b, describing the two-phonon “Compton” scattering, are only to be taken into account in the Hamiltonian (2.13). The corresponding equation for the full scattering amplitude WKK for a heavy polaron in the id case has the usual structure (1.5) with the substitution of K, K’, K” by K~,K~,K~[34—36].It is just the same

360

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

at an arbitrary value of the polaron velocity V [36], as it can be seen easily from the general method [49]. In the framework of this method it is necessary first of all to consider a moving coordinate system and then to make use of Bogolyubov—Tyablikov transformation [45]. The physical sense of this transformation corresponds to an expansion of the phonon coordinates QK in a 4 1 near Q~ [49].The corresponding linear terms in the phonon operators give the expression (2.14) for the Born scattering amplitude VK K, in the second order of the perturbation theory [49—51]. At low temperatures T < 1 the polaron mobility p. is determined by the usual kinetic equation [49—51].Due to a large polaron mass M* a~ 1 the corresponding momentum transfer AK,, T 4 P (M* T)”2 is rather small for two-phonon processes [49—51],so we can use the Fokker—Planck expansion of the Boltzmann equation [79]: —

~‘

—

A=BVIT.

(2.15)

The coefficient B in this equation is connected with the scattering amplitude WK phonon filling numbers NK = (exp(WK/T) — 1)_I by the formula [49—51]: B=

~

NK(NK +

i)~ WK

KJ

(K~ K’,,)2

~(wK

—

—

K’

and the Planck

(2.16)

WK).

The polaron mobility p. in weak electric fields E 4 E 3B is determined by the coefficient B according to the simple relation: p. = eTIB. This value is 00the pmain characteristic of electric properties for quasi-id organic conductors and conducting polymers [1—9]. -~-

2.3. Polaron mobility in ld conductors The main mechanism of electron coupling with optical phonons in conducting polymers is a short-range contact interaction (2.5) [6—9]due to a covalent character of chemical bonds in these compounds [9]. Thus we can use the formulas (2.9, 2.10) for the polaron spectrum at strong coupling a >2. As a result, the simple expression for VK K can be obtained quite easily and eq. (1.5) for WK K can be studied in detail. In the case of intermolecular vibrations (2.3) with a large dispersion [6—9]the corresponding effective mass M 2m at a > 3. 0 iOm [108] much less than theofpolaron effective mass M* Thus at low temperatures T 4 1can thebethermal fluctuations a phonon momentum ~KT iO(TM 2 are much less than the polaron momentum p (TM*)t/2. This fact provides the validity of the 0)U Fokker— Planck approximation (2.15) at low temperatures and eliminates the corresponding limitations for it, arising in the usual polaron model with dispersionless optical phonons [35, 36]. The dispersion laws for optical phonons in Organic compounds contain both the modes with K)) = 0 and K 0 KB, so we shall consider both of these two situations. At K0 KB we can use an asymptotic expansion of VKK~at large K1, K~ K0 a [35, 36]: —

—

—

—~

~), ~-

5K VKK.=~(~/O)K~K~(12

K 5,K~a,

(K5—K~)—a

(2.17)

12 are small enough in comparison with because the thermal fluctuations of (K1 K 1, K,~ K0 a. ~‘

—

K)

—

z~KT (TM00)t -‘--

A - A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

361

Neglecting the small Doppler terms V i~K14 (I~K~)2IMin 0 ~0(K1) we can transform the id eq. (1.5) to the usual equation for the elastic scattering amplitude [109]of a quantum particle with an effective mass M012 in the potential U( ~)= (2 ~( ~)iK0)~.The corresponding Schrödinger equation with the function i/i~(~) (2.10) can be solved exactly [109]. As a result, we obtain for WK K~ at K~= — K1 + M*V~~ — ~ the following expression: 2lTK 2K)’ K = ~ir(—i + i6M 1’2—1. (2.18) WK_K = (1 + sh 1/ch 0iK~) The mobility p. = p. 2/M 0 y n(T) in this case at T 4 T0 = a 0 41 is determined in the usual units by the formula: T. (2.19) 7 2i~(amIM0)~, n(T) = e0)0/ There is a small factor y 4 1 in this formula in comparison with the formulas (1.4, 1.7) for the 3d case and there are no preexponential factors in n(T). The condition T 0 4 1 is not a rigorous one, because it is necessary for the evaluation of numerical coefficients in the formula (2.19) only. So in the real polymers like (CH)1 [23] at a = 5 and M0 = 50m the value of T0 = ~is not very small and we may consider the case T0 1. 2/VIn this y = ~andwith the seccase in agreement estimation of p. at T T0 to0 = 0.01 eV, m = 0. im0 gives p. —~0.ip.0 10 cm the experimental data [6—9,13, 14]. At large M 0 the value of T0 41, so at T0 4 T 4 1 the mobility p. is described by the formula (1.4) with the numerical coefficient ‘y0(K):

,

—

f

3/4ft(K),

At

4 KIKdK(ch2K+ch2K)2.

(2.20)

1(K)=ch

70(K)=7r

0 the function f 1(0) = 0.073 and y~(0) ~ 1 sec theatfunction = K2i8 and 31K. The corresponding estimations of p.= 106 p. and at K2iVthe samef1(K) parameters are 70(K) = 27r 0 102 cm also in agreement with the experimental data for (CH) 1 [6—9,13, 14]. At K0 = 0 the values K1, K~~ a at T ~ T0 and we cannot use the asymptotical expansion (2.17) for VK K’~ Hence the scattering problem for the non-local potential U(~,i’), corresponding to the Fourier components of the function VK K’ (2.14), is rather complicated. So the evaluation of p. in this case can be performed at low temperatures T 4 T0 only. In this limit case we can use a more simple method of the id eq. (1.5) solution, quite analogous to that of Volovik and Edel’stein [49] for a piezoelectric polaron. At T 4 T0 the thermal phonon momenta K1, K 4 K’~ a so in the lowest order in this parameter = 1 in the id eq. (1.5). Using the corresponding expansion of VKK at K1, K~ 4 1: K=

V~=~ K1KC0,

Co4Jd~(1+~2)2(1+ch1r~)~i=0.888

and making use of the general relations [45] for VKK and V 2(toK to_K.) 1/2 VKK.]KXQK. , 0 =0 —

—

~K [toK CO_K~KK~+

—

—

(2.21)

Q~: (2.22)

362

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

we obtain in the lowest order in TIT0 4 1 the following solution for WKK~[35, 36]: WKK

=

2K,,K(Q~/Q~) (K~/M~) (7rC00/a).

(2.23)

Substitution of this solution to the id eq. (1.5) turns to zero its right-hand side in the lowest order in T/T0 41. Its left-hand side in this case is also small enough due to the condition K~IM() T ~ T0~4 i. Using the formula (2.16) we get the following expression for the mobility p.: 2mIM 4, n(T) = (w 4 ewh/T. (2.24) y = (6~rC0)t(a 0) 00/T) The value of y in this case is also rather small, but it is not so small as in eq. (2.19). This factor, however, is compensated partially by the small numerical coefficient (6 11.C0)_t 0.05.estimation, So at the same 2/V~ sec. =This howparameters we obtain the following estimation for p. 0.01 p.00 I cm ever, increases rapidly with a temperature decrease at T < to 0, so at T = w0/2 the value of p. 0.ip.0 2/V- sec in agreement with the experimental data for (CH) 10 cm 1. Our exact solution is valid only for the case of a 4K large enough optical phonon dispersion. In the case ~ to of intramolecular phonons with a small dispersion 0 the evaluation of p. can be performed using the general method of Volovik, Mel’nikov and Edel’stein [51]for dispersionless optical phonons. In this case the large K1, K~ a with a small difference (K1 K~) a are essential in the integrals (2.16) [35, 36], so we can use an asymptotic expansion (2.17) for VKK. Performing the Fourier transformation on (K1 K~)in the id eq. (1.5) and introducing a new variable ~ and a new function ~(i, K,,): —

—

—

~‘

—

—

—

-iJ d~’W(~’,K~)

=

we get in the units a

=

Vj[1

-

~,

K~)]

(2.25)

1 the following equation for ,~: (2.26)

The value of B (2.16) is connected with the function B=

J

e~w0/T

j by the

relation:

2 1-

~

Introducing the amplitude

ii

j(_oc,

(2.27)

ki)~

and the phase ~ of the function ~:

K~)= ~i, K~)exp{i~K~~ ~ K~)}

~

(2.28)

and using the expansion (2.17) for V(x, K~ ia /ax) we obtain the following equations for —

~4(a2~V~~3)—1 (1

—

ij)

~

a

In

-

a =

~ ~

2-

‘~‘-~,

~

~ = —4 ~i~(x) ~i~i —

a’ and ,~: (2.29)

~-~) 8~~’ .

(2.30)

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

As it follows from eq. (2.30) the value of expressed through the function ~ only: (a~m\( ~ =

~-M’~)~

K) =

tT(—c~s,K~)=

1, so the mobility p. can be

T

\2(3\

~,-~--),

cT(+cn,

363

n(T) =

— n0(T)

(2.31)

where C1=J ~

2(~),

The dependence of ~

~

~)=~(-co,

j

fl,

j(+cc,

~)=0.

(2.32)

is determined by eq. (2.29):

fdi[Z~13+Z~h13_2],

Z

1’2, 1=i+Z0+(Z0(Z0+2))

Z

2i. 0=54~/ch (2.33)

At ~41 the function

~(~)

~ and at ~ ~ 1 this function ~

~

The value of C

1 21.5 can be found numerically and determines the value of y (2.31). This coefficient y contains also the large parameter (w0ia41 )2 ~ 1. This factor is provided by the id character of a polaron forward scattering, which is a purely elastic one at 4~ 0 and does not give any contribution to the mobility p. in this limit case. Thus the value of p. in this limit case is determined by an exponentially small backward scattering amplitude. These processes, studied in detail by Holstein [28], are essential only at a very small ~ a’~exp(— iral2). This inequality gives at a >3 the estimation 41/to0 ~ i0~, which is in contradiction with the corresponding experimental data [1—9]:41ito0 —0.1—0.01. Thus the mobility p. is determined by eq. (2.31)2IV~ andsec theinvalue of y at a = 5 and T to0 is of the order agreement with the experimental data for of 1. As a result, the mobility p. 102 cm (CH) 1 [6—9,13, 14]. It is necessary to mention here, that the range of validity of expression (2.31) is rather narrow. Indeed, in this case the characteristic momenta K~ Due aT -1/6 aresmall smallvalues enough 2T1’2 phonon only at a3’2 4 T 41. to the of in T comparison with the polaron momentum p a powers in these formulas the range of validity for the corresponding results is rather narrow. It is necessary to strengthen here the wide range of validity for the expressions (2.19, 2.24), describing the polaron mobility at a large dispersion of optical phonon modes. At low temperatures T < to 0 the phonon dispersion 4~is supposed to be large in comparison with the 1 T2IBV2 4 4~.As a result, all the localization effects due to a polaron scattering probability r~ quasi-elastic scattering [86, 117, 118] are eliminated in the system. —*

—

—

—

2.4. Polarization interaction The polarization interaction (2.6) is absent usually in conducting polymers, but it is essential in ionic compounds like TCNQ salts [1—7].The evaluation of VKK in this case according to the formulas (2.7, 2.14) gives VKKI S a 4 1, so the integral term in the id eq. (1.5) is small enough at least for the temperatures T ~ a -2 [35, 36]. Thus the Born approximation WKK. = VKK. is valid for WKK.. At a sufficiently large longitudinal dispersion (2.3) the condition T ~ a -2 is eliminated [35,36] and the Born ‘~

364

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

approximation is valid at T < 1. In this case at K0

KB

~‘

a we can use the corresponding expansion

(2.17) for VKK. at K, K’ K0 4(M~Im)2 a and at T 4n(T) T10 we get: , = e’~’ (2.34) y = (i6~)°(K~/M0w0) Thus at the same parameters a = 5, = 0.01 eV, m = 0.1m0, M0 = 50m the coefficient y so 2/V- sec is a little bit more, than in the case of a short-range interaction (2.6).10,This p. provides lOp.0 ~ a cm fact low intensity of the corresponding scattering in quasi-id organic conductors. At K 12 a 0 = 0 and T 4 T0 we can use the corresponding expansion of VKK. at small K, K’ (TM)) quite analogous to eq. (2.21). The exact expressions for the wave functions ~t~(x)in this case are absent, so the unknown numerical coefficient of the order of 1 arises in this expansion. Thus a qualitative estimation of p.: ~-

.

to

0

-‘--

-~

—

y—(a2mIM 4 , n(T) = (w 2 1n2(w/T) e’°~’ (2.35) 0) 0IT) is possible in this case. At the same parameters it gives p. p.)) 102 cm2/V~sec. At T 0 4 T < 1 the 2m/M, n(T) = n estimation of p. is more simple, so y a 0(T) and p. p.0 also. At a small enough phonon dispersion (2.4) the mobility is described by the formula [35, 36]: ‘—

-y---a

—i

2

(tooia4±)

4

(a mIM

*

)

3/2

,

n(T)=(T/w0)

5/2

wIT

e 0

(2.35a)

So at ~ -‘-‘O.lto0 and the same other parameters the value of y—’ also.than As athe result, all 2iV- sec. These values are 1aand littlep. bitp.00more typical the estimations of p. give p. p.0 10~ cm experimental meaning: p. 1—10 cm2/V- sec [1—8]in TCNQ salts at T = 300°K.They are in agreement with the maximum of p. for ITF-TCNQ [119]only. In some cases these large values are diminished by the corresponding numerical coefficients [35, 36] to some extent. —

2.5. Polaron mobility

in

3d

ionic

crystals

As it was shown in the previous sections the phonon dispersion is rather essential for the calculations of polaron mobility in id conductors. These effects are of great importance also for the 3d crystals, because the dispersion of optical phonons in these substances is not small enough [53]. The account of these effects changes drastically the expression for p. and simplifies essentially the polaron scattering problem [35, 36]. As a result, the exact analytical solution for the polaron mobility can be obtained including the numerical coefficients. These results are not obtained now in the framework of the usual dispersionless model [51]. In the works of Volovik, Mel’nikov and Edel’stein [49—51]the effective 3d Hamiltonian of a polaron scattering quite analogous to eqs. (2.13, 2.14) was obtained. The mobility p. in this case is determined by the 3d Fokker—Planck equation of the type (2.15) and the scattering amplitude WKK. is determined by eq. (1.5). The general structure of these equations is just the same in the case of optical phonons with the usual dispersion law (1.6) [35, 36]. In the present part we shall calculate p. at M 4m (1.2). At strong coupling a 10 this 0 4 M* a condition is usually fulfilled, because M 0 lOm [53] and M* ~ lOOm [26]. For example, in NaCI according to some estimations [26] the values of a = 9, M* = 300m0 and Mo 30m0 [52]. In these —

A A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

365

crystals with a simple cubic lattice both the optical phonons with K0 = 0 and with K0 KB are present [53], so we shall consider these two cases. 2/M 1124a are small, so At K0=O and T4 T0—a 041p.the thermal phonon momenta K—(TM0) K <

—

VKK,

=

(C

3) (KK’)/(KK’),

C 2 —1.

2/a

(2.36)

We can neglect also the small momenta K’ 4 K” a in the denominator of the second term in eq. (1.5) and we can use the 3d eq. (2.22) [35, 36]. As a re~ult,using the method (2.21—2.23) we obtain in the lowest order in TIT0 41 and T0 4 i the following expression for WKK., quite analogous to eq. (2.23): —

2/Mo)VKK.. (2.37) WKK. = (2Q~/VK.)(K The phonon coordinates Q~in this case are determined by the 3d eq. (2.7). Substituting these expressions to the 3d eq. (2.16) we get for p. eq. (1.8) [35, 36]. This expression (1.8) differs from the formula (i.4) of Volovik, Mel’nikov and Edel’stein [5i] by the factor (T 4 4 1 and by the dependence of y(a) and n(T). In real crystals, however, at a = 9 and T = to 01T) 0 this factor is not so large. It is interesting also, that our expression (1.8) for p. is valid at low 2 4 T 4 to temperatures and its validity is not restricted by the condition w0/a 0 [51].This condition gives rise to a 6rather narrow range of validity for eq. (1.4) [51]. It is interesting also, that the dependence of in eq. (1.8) is quite analogous to the corresponding Pekar result [44]. y(a) a At K 0 KB we can use the asymptotical 3d expansion (2.17) for VKK. at K, K’ K0 ~- a [35, 36]. The evaluations of the integral term in eq. (1.5) demonstrate that it is small enough due to the conditions 2/M a 0 41 and a/K0 41. Thus, the Born approximation for WKK. VKK. = 4iraV~/K~ [35, 36] is valid in this case. As a result, the corresponding expression for p. has the form (1.7) with ‘y 1 due to the condition K~IM0 to0. The expression (1.7) differs from formula (1.4) the temperature 3 only. It is necessary to point out,[Si] thatbya small value of the pre-exponential factor n(T)1n0(T) = (w0IT) Born scattering amplitude WKK. = VKK a IK~4 a ~ does not effect practically the mobility p. p. 0 at T to0. This fact is connected with the diminishing of the full scattering amplitude W~.I 4 VKK. I by the integral term in eq. (1.5). This effect is quite obvious because only a small enough value of WKK. can compensate a small value of the corresponding denominator in this term. A small value of WKK. is compensated partially by a large phase volume in the integrals (2.16) over K, K’, so the main estimation of p. p.0 at T— to0 is just the same for both formulas (1.4, 1.7). The situation with K0 KB is a usual one for ionic crystals [53], so formula (1.7) gives the most useful 2IV resultsecfor = 10, m (1.4, m0 1.7, and1.8) to0 —0.01 eV the with valuethe of corresponding p. p.0 at T—experimental to0 is about forp..allAt theaformulas in agreement 102 cm data [26] for alkali-halides. —

—

—

—

—

—

I

—

2.6. Interaction with acoustic phonons The important mechanism of polaron formation in quasi-id organic conductors and conducting polymers [5—9]is a deformation interaction of electrons with acoustic phonons [20—38].As it was shown by Rashba [25]the formation of these states in quasi-id conductors is possible in all the cases, because

366

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

the corresponding bound states arise in id systems even at weak attractive interaction. In 3d systems there are high potential barriers for these states [120,121], so their formation is rather difficult. In quasi-id organic conductors and conducting polymers there are id and 3d acoustic modes [1—9]. The electron interaction with id acoustic phonons is usually rather strong [10,13—23], so we shall consider it first of all. It is interesting, that the corresponding id acoustic modes were observed in the experiments [122]. In these experiments the low temperature lattice heat capacity was investigated in (CH)1 [122] and the corresponding linear term was observed. The Hamiltonian of the deformation interaction of with phonons has the form 1/K electrons = K 2w id2. acoustic The dimensionless coupling (2.1) with the substitution K—* K1, WK = ~ 11 (K5 1)t/ constant a is connected with the deformational potential ED, the mass of a unitary cell MA and the lattice constant a 0 by the relations [20—38]: 2Wt,

to

a = E~/2MAS

1

=

2S/a0.

(2.37a)

Substituting into (2.37a) the typical values ED = 3 eV, S = 106 cm/sec, a0 = 1.4 A, MA = 13 for (CH)1 [23]we get a = 4. In the case of PDATS [10]at ED = 3.7 eV, S = 3.6 X iO~cm/sec, a0 = 4.9 A, MA = 420 the value of a = 12. So we shall consider the strong coupling approximation [24—38]with a ~ 1. For the evaluation of the spectrum E( p) for the acoustic polaron at an arbitrary velocity v ~ S we can use the general method [34—36].According to this method it is necessary to consider the coordinate system, moving with the velocity v. This transformation of the Hamiltonian (2.1) leads to the Doppler shift of the phonon frequencies toK~~*WK [34—36].The polaron spectrum E(p) at a 1 is determined by the classical minimum of the Hamiltonian (2.1) over QK. As a result, just the same expressions (2.7) for Q~arise in the system with account of the corresponding Doppler shift WK ~ WK only. Taking account of the corresponding Doppler shift pu for the polaron energy E( p) in the moving coordinate system we get the following expression for E( p) in the units /l. = m = S = 1: 2 ~ VV i roo K,,, K~ 2 2 E=J+pv, J= 1r dxy~—) ‘ä’~ (tIJo)K (~/o)_K (2.38) uX K~toK ~‘

~

—

-

-

.

‘-

1to_K,

The functional J is determined by the ground state wave function ~/i0(x)and the polaron momentum p = M*v by the effective polaron mass M* (2.8), depending on the velocity v due to the Doppler shift of the phonon frequencies [34—36]: 2VV K M* = ~ (~ ~ (2.39) K ~toK to_K)

Equations (2.38, 2.39) determine the spectrum E( p) and give just the same result as the usual Lagrangian equations [61]. The minimum of the functional J over ~/i~ is determined by eqs. (2.9, 2.10) with the account of a corresponding coupling constant renormalization: ~ = a/(1 v2). As a result, the well-known expressions for M* and J, obtained first by Whitfield and Shaw [30] with the help of the Lagrangian method [61], arise: —

~

6(1—v~ ,

M*= 3(1—v2)3

-

(2.40)

A A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

367

The corresponding expression for the functional .1(2.40) was obtained also by Davydov [37, 38]. At v—>i the value of pv—a2(i—v)3 is large enough in comparison withf—a2(i—v)2, so the polaron dispersion law E( p) = p is a linear one at p a2 The asymptotic behavior of E( p) and V(p) = aEIt3p is described in this case by eq. (1.10) and is rather slow in comparison with the 3d piezoelectric polaron (1.9) [61]. The jhysical reason for a saturation of V( p) at p —~~ is connected with the limited velocity of a deformation potential well motion. This potential well cannot move with a velocity v > 1, SO at large enough v an electron is free. As a result, the bound polaron states are absent at v > 1, as it follows from eq. (2.38) for J. In this equation the sign of the second term changes at v > 1, so the potential’ well transforms to the potential barrier. At v > 1, however, the free electron states with the spectrum Ee ( p) = p2i2 exist. The energy of these states at fixed momentum p is much more than the polaron energy E( p) p 4 p2i2 at p —~oo. Thus, at low temperatures T 4 a2 the free electrons go down to a polaron well with a characteristic time of the order of reciprocal phonon frequencies. At high temperatures T ~ a2 the polaron states can decay thermally, but due to a large energy difference (p2/2 E(p)) ~ T the most part of the time is spent by the electron in a polaron well [34—36].As a result, a great amount of interesting effects arise in the system even at high temperatures. For example, the magnetic susceptibility and the electron heat capacity are determined by the electron density of states, which increases rapidly with an effective mass, so they may be described by the polaron model. It is necessary to mention that all these effects do not depend on the concrete mechanism of the acoustic polaron formation. So our qualitative analysis is valid for all the acoustic systems. As a result, a great amount of experimental data about the saturation of a drift velocity at strong electric fields in many organic compounds [54—59]can be explained [34—36]. ~‘

—

2.7. Drift velocity for

an acoustic

polaron in strong electric fields

The interesting experimental data about a characteristic saturation of a drift velocity V 0(E) near S at large E for charge carriers in PDATS have been obtained by Donovan and Wilson [10]. Thus the interesting theoretical problem of an acoustic polaron mobility evaluation is to be considered in the present review. Some attempts of its solution were done by Wilson [32, 33], but the correct expression for p. was not found in these works, because the general method [49—51]was not considered and some errors in the Fokker—Planck equation (2.15) were done [34, 36]. The correct solution of this problem was obtained by Gogolin [34, 36] using the general method [49—51]. As it was shown in the previous section at a ~ 1 the effective polaron mass M* is large enough, thus the Fokker—Planck2 (2.15) equation describes the1/polaron mobility. This equation (2.15) at low determines the drift velocity temperatures T 4 a 0(E) an acoustic polaron a strong 2e1 is strong enough, so for the characteristic electroninenergy eElelectric at the field. The electric field E E0 a mean free path 1— T21BV is compared with the binding energy ~FC a2 and we can see the saturation of V 0(E) —* 1 at E ~ E0. This field, however, is not so strong due to a small enough polaron size a 4 1,2 so the polaron not destructed by phonon this fieldthermal due tomomenta the condition eEa 2 The condition T 4 a provides a smallisenough value of the KT T in 4 comparison with the polaron ones p a2 so the Fokker—Planck equation (2.15) is valid in this case. For the evaluation of the function V 0(E) it is necessary to find the dependence of the coefficient B (2.16) on the velocity v < 1. This dependence is determined by the scattering amplitude WK K’~ This amplitude is described by the id eq. (1.5), which is just valid at v i with account of the Doppl~rshift —

—

—

—

—

—

368

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

wK_~*i~K [34,

36]. This is the consequence of the general procedure [49—51]with account of the coordinate system motion [34, 36]. At very low temperatures T4 a the id eq. (1.5) can be solved easily by the general method of Volovik and Edel’stein [49] for a piezoelectric polaron. Using the corresponding expansion [49] of eq. (1.5) at small K1, K— T4K~—a and eq. (2.22) we get the following expression for WKK [34, 36]: WKK.

=

1 VKVKKXK~ -~-~-~~._

\3/2

-

(2.41)

-

11

This solution corresponds to a second order perturbation theory in the Hamiltonian (2.1) with account of a polaron mass renormalization, so it describes the two-phonon “Compton” scattering processes. Thus the perturbation theory result (2.41) is valid at a 1 and at a 41, so it is valid at a i also. Substituting this expression into eq. (2.16) we obtain the following expression for B [34, 36]: 5/a2 , A)) = ‘4(2i~)’ (2.42) B = A0 B0(T) çli0(v), B0(T) = T where the function ~‘ 0(v) has the form: p0(v) = ~p1(v) + ~-

15

~

f

4dKexp{K(~ K

0—1)} 5hK5hKX0

1—v

x0=

(2.43)

.

The function ~~(O)= ~( 1) = 1 and differs from 1 only by a few percent in the interval 0 ~ v ~ 1. At v 4 1 this function determines the polaron mobility p. in weak electric fields. For the evaluation of the function V0(E) it is necessary to solve the Fokker—Planck equation (2.15). This solution at an arbitrary dependence of B(p) has the following form [34, 36]: f(p)

=

C exp(_ ~ E(p) + eEf dqiB(q)),

Vo=JdpV(p)f(p),

C~tJdpf(p).

(2.44)

2 in the Here p and v are connected by eq. (2.40). As it follows from these formulas at T 4 E(p) a strong electric field E E 0 = A0 B0(T) leT the drift velocity V0(E) is determined by the extremum of In f(p). So the corresponding equations for V0(E) have the simple form [34, 36]: —

—

~ ~(V0) E,

E = EIE0,

E0

=

A0 B0(T)IeT.

(2.45)

This equation is a general one and does not depend on the structure of ~‘0(V0)and the coefficients A0, B0. It determines the functions E(V0) and V0(E), so the current-voltage characteristics (CVC) are described by this equation straightforwardly. The corresponding graph of V0(E) at T 4 a is shown in fig. 1. At E < 1 the function V0(E) is a quasi-linear one and its slope determines the mobility p. = StE0. The value of p. in this case at T— a, -

-

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

369

Fig. 1. Current—voltage characteristic for an acoustic polaron at low temperatures.

a 10, 5— ~ cm/sec is of the order of i02_103 cm2tV~sec. This estimation agrees with the corresponding experimental data for (CH) 5 [5—9]but it is much less than the corresponding valuefor in B. PDATS 2 in eq. (2.42) This [10]. This fact is connected with a large numerical coefficient A0 i0 coefficient diminishes strongly at high temperatures, as it will be shown further. At E 1 the value of V 0(E) i and does not depend on E. As a result, a characteristic saturation of V0(E) at finite E arises, which was observed by Donovan and Wilson [10]in PDATS. This fact follows from eq. (2.44) for V0, where the corresponding divergency in the integrals over p arises at p ~ in the case_of E> 1. In the real systems this divergency is cut off at the Brillouin momentum KB a, 50 T/0(E) = V(KB) at E > 1, where V(KB) < 1 is determined by eq. (i.iO). As a result, the upper limit for V0(E) is to be less than 1 in agreement with the corresponding experimental data of Donovan and Wilson [iO], where 1/0(E) 0.7 in a 2wide range of electric corresponding to the fields. experiments of Donovan and Wilson [iO], higher term temperatures a 4eq.T 4(1.5) a is small enough at (1 V) 4 1. So at V—s’ 1, describing the theAtintegral in the id experiment [10] in PDATS, the amplitude WKK. = VKK. is described by the Born approximation. The main contribution to the V 0(E) dependence in tI~iscase arises from the narrow interval of V near 1, so we can use the asymptotic expression for A0 and co0(V) in the coefficient B = B0(T) A0 tp0(V): —

—~ ~‘

—

2a, B0(T)

=

T

çc~ 0(V)= 1 — V,

A0

~

J

dK £6 ~22(~)

=

1.14

(2.46)

where

~)=

f

(2.47)

2]-2[h(~)+h(1~)]-1

d~[1+(~- 1j)

The formulas (2.46) are valid at (1 — V) ~ alT, so the function co

0(V)

not tend to zero in this limiting case.

—

alT 4 1 at V—+ 1 and does

370

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

As it follows from eq. (2.44) for 17~the integrals over p at p ~ a2 diverge in this case even at weak electric fields E ~ E 3 at large 1 aE0tT 4 E0 the momenta decrease pof the B(p) momenta. This divergency is to be cut due off attolarge Ta coefficient a2, so at very weakp~ electric fields E4 E 1 the dependence of 1/0(E) is a linear one.2(amS2)2 As a result, p. in ofthis case is andtheis mobility of the order i05limiting cm2/V~ sec at determined by the simple relation p. S/E1 e5 a 10, S i05 cm/sec, m 0.1m 0. This estimation agrees with the corresponding data of 2)2teSexperimental at T— 100°Kgives Donovan and Wilson [10] for PDATS. The estimation of the field E1 (amS at the same parameters the quite reasonable result: E 1 1 V/cm in agreement with the corresponding experimental data [10].These estimations of the mobility p. and the electric field E1 make it possible to explain this experimental data. The experimental results [10] are also in agreement with the value of V0(E) < 1 at E> E1. Indeed, the maximum value of V0(E) in this case is determined by the formula (1.10) at p KB, so we get the quite reasonable result for V0(E) 0.7 at K8 = In conclusion one should mention that some attempts of the drift velocity V0(E) evaluation for PDATS were done by Wilson [32] in the frameworks of the same model. In his work, however, the correct expressions for the effective Hamiltonian of a polaron scattering (2.13, 2.14) was not obtained and some mistakes in the Fokker—Planck expansion (2.15) and the calculations of the scattering amplitude WK K’ were made. As a result, the2tVcorrect foragreement V0(E) was with not obtained by Wilson sec expression were not in the corresponding [32] and his estimations of p. 106_i07 cm experimental data [10], where p. i0~—i05cm2/V- sec. It is necessary to mention also that the value of p. 102 cm2/V- sec in (CH) andis p.also 2/V~sec in PDATS is rather large. The corresponding mean free path I in these 1cases ~ cm rather large, so in (CH) 2a 5a 1 1’— i0 0 and in PDATS 1— 10 0. It is interesting to understand in this case the physical reasons for the low effectiveness of impurity scattering in these substances. The impurity concentration in doped (CH)1 is about 1O_2 [5—9],but it is possible to show, however, that the corresponding scattering at room temperatures is rather low. As it was shown by the X-ray data [5—9]the charged impurities in (CH)1 are situated between the polymer chains and are described by the usual Coulomb potential. Thus, the backward scattering 2 at amplitude in 300°K[35, the Born approximation is quite small for a large polaron momentum p (TM*)u/ M* iO2m,W,,,T= 36]: —

—

—

—

~‘

—

—

—~

—

—

—

= ~-~2

~

s(2~ =

~-~2

K

2(1T)l/2

0(2pa1)

e2~

at pa 1 >1.

2e

(2.48)

The value of a1 = a112 is determined 4m/15 by the~500m interchain 4.4 A [23] and the value of (a =distance 6) and ma1= =O.2m p = 0.6 X 108 cm’ at T= 300°K,M* = 8a 0 [23]. So, the2e2 parameter 4 e2 at pa1 101.4 and the McDonald function K0 (2.8) 0.03 [123]. As a result, the value of W~ i0 [5—9]is rather small. Thus, the impurity mean free path 1~with respect to a backward scattering contains the large factor i04 together with a large reciprocal concentration C~1 i0~[5—9].So, the value of 1~is rather large and does not effect the polaron mobility. The large l~gives rise also to the absence of all the localization effects in these substances, because these effects at high temperatures are strongly suppressed by inelastic phonon scattering processes [86]. In the case of PDATS the impurity concentration C 1 i0~[10]is rather small, because this compound is polymerized in the solid phase and the high quality of the corresponding crystals is achieved easily [10]. —

A. A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

371

It is necessary to mention in conclusion, that theory of large polarons at strong coupling describes quite perfectly the transport phenomena in conducting polymers and other quasi-id organic conductors, where the electron bands are not rather narrow. Some difficulties, however, arise in this theory at a small enough polaron size r0 a0 at a > 10. As a result, the theory of small polarons (SP) is necessary —

for description of polaron states in narrow band organic conductors. This theory was developed in the works [77, 78] and is to be analyzed in detail in the following chapters.

3. Small polaron transport 3.1. Effective Hamiltonian The characteristic size of polaron states in quasi-id~organic conductors with narrow electron bands M~~ 0.1 eV like TCNQ salts [1—8] at g 1 is of the order of the lattice constant2toa0 [42]. This fact is connected with the large enough value of the Franck—Condon energy shift = g 0 M0°in this case. As a result the intermediate and the small polaron states arise in the system. The theory of these states was developed by Holstein [24], Firsov [39], Klinger [40, 41] et al. (see the reviews [26, 39—41]). The most useful method for this problem is the method of the special canonical transformations [39], which was developed first by Tyablikov [62]. This method gives an effective expansion for SP energy spectrum in the powers of the dimensionless polaron size x0 = ~‘Fc/Mo.This theory was developed further by Gogolin [42,78]. For these purposes we shall consider the usual Hamiltonian of electron—phonon interaction in molecular crystals [24—26]: —

~

H—110+EHmn,

=

Here i/i,

~

~

&n=M~nnhI1~Pn,

M~~= M~m 1~(bqj+ b~iqj).

+ ~ to~1b~b~1 + ~

(3.1) (3.2)

~Fqj~IJI1J~ e

are the electron operators, b~,bqj are the phonon ones, toqj are the phonon frequencies, are the overlap integrals. The index j denotes different phonon modes and the vectors m, n describe positions of lattice sites, formed by the lattice vectors a 0, b0, c0. The full number of different phonon modes in quasi-id organic conductors is rather large and is about 102 [15—18].The main contribution to the electron—phonon coupling arises usually in these substances from the interaction with acoustic and intramolecular optical phonons. The dispersion of these optical vibrations is rather small and the dependence of on q is rather weak, so in this case we can neglect these dependences on q: toqj=to/’ I’~=I. (3.3) i/is

1,, are the electron—phonon coupling constants and ~

The frequencies of intramolecular vibrations vary usually from iOO°Kto 5000°K[15—18].Their interaction with electrons is usually rather large, so the dimensionless coupling constantsg. = 1/w 1 —0. i [15—18]and in some cases g1 1 [15, 16]. —

372

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

The interaction with acoustic phonons is described by the deformation potential E,(e). This potential in anisotropic crystals depends on the direction of the phonon momentum q = qe, el = 1. The analogous dependence of the sound velocity S.(e) gives rise to the corresponding dependence of tog, and Fqj: 51(e) q,

to~=

2.

(3.4)

L~= E1(e) (q/2M~SJ(e))”

Here MA is the mass of a unitary cell. The diagonalization of the Hamiltonian (3.1) can be done with the help of the special canonical transformation [391: ~

=

=

~ SJ~,

s~ =

~

~i~n

— bqj).

(3.5)

(b~qj

This transformation describes the displacement of ions in the presence of an electron in a site and corresponds to the polaron coupling of the electron and phonon operators in the Hamiltonian (3.1). The effective Hamiltonian H 0 in this case is diagonalized exactly and the effective Hamiltonian Hmn~ describing a polaron translation motion, has the form [39]: JTImn =

~

exp[~ ~

—~

b~(e~—

m— e~t~m)]exp[-~

e~)].

bqj(&~

~

(3.6) Here the values fmn

=

~

(Fqj/toqj)2[l

-

cos(q(m

-

(3.7)

n))J

determine the polaron renormalization of the effective transfer integrals Mmn~describing the electron bandwidth M. This renormalization is quite analogous to the usual Debye—Waller factor [124]and leads to the exponential narrowing of electron bands at g, ~ 1. This effect makes it possible to interpret an existence of extremely narrow electron bands with M iO~eV in UBe 13, U2Zn17 and others [65, 66]. At high temperatures the additional factor —

2Nqj

+

1

=

cth(toqj/2T),

determined by the Planck filling numbers Nqj~arise in eq. (3.7), so the polaron narrowing is even much stronger. As a result the effective breakdown of the polaron bands can take place at high temperatures due to the strong polaron scattering by phonons. Thus the hopping mechanism is to describe a polaron motion at high temperatures. This mechanism has been studied in detail by Firsov [39] and the usual band one was investigated by Gogolin [77, 78]. In conclusion it is necessary to mention, that the usual condition ~FC M~is necessary for the validity of this canonical transformation [24, 26, 39—41]. Here ~‘

~FC =

~ F~jttoqj

(3.8)

A. A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

373

and this condition makes it possible to neglect the off-diagonal terms (3.6) in the polaron Hamiltonian [42]. 3.2. Anisotropy of polaron bands The effective Hamiltonian (3.6) describes a polaron translational motion accompanied by a phonon scattering. The corresponding renormalization of the bandwidth M is described by the zero term of the expansion of the values Hm~(3.6) in b~,bqj and is determined by the exponential factor exp(—fmn). In the nearest neighbor approximation the values (m — n)

=

a0, b0, c0

correspondingly, so these factors describe a polaron renormalization of electron transfer integrals along these crystal axes. As a result the additional anisotropy of electron bands arises in the system due to a strong dependence of the values fmn on (m — n) (3.7) [63].As it was shown by Gogolin [63]these effects are strong enough in quasi-id organic conductors and give rise to an essential enlargement of the corresponding anisotropy of electron spectra. All the phonon modes are included to the formulas (3.7, 3.8) for the values fmn and ~FC in an additional manner, so we may consider them independently. The corresponding contributions to the renormalization factors exp(—fmn) are multiplied by each other, so they are also independent. For example, the main contribution to the effective anisotropy of polaron bands arises usually from the strong coupling with low-frequency acoustic phonons [63]due to a strong dependence of the values I~ and on q (3.4). At the same time the main contribution to the Franck—Condon energy shift ~‘FC may arise from the strong coupling with high-frequency intramolecular optical phonons. The great amount of these modes and a strong enough electron—phonon coupling for them in quasi-id organic conductors [15—18]makes it possible to fulfill the condition ~ even at M°—0.ieV. In quasi-id organic conductors an essential anisotropy of acoustic phonon spectra sometimes takes place. In strongly arüsotropic quasi-id hexagonal crystals there are usually two anisotropic acoustic modes [125]with the corresponding angular dependence of S1(e) and E1(e): S1(e) = \IS~11e~ + S~1e~ ,

E1(e)

=

\/~11e~+

E~1e~ .

(3.9)

The additional anisotropy of polaron bands may arise even in the absence of strong anisotropy of acoustic phonon spectra [63], but we shall consider first of all the situation with the large enough ratios E111/E1~~ S111/S11 8~1 for simplicity. Substituting the expressions (3.4, 3.9) into eq. (3.7) we get at (m — n) = a0 in the lowest order in the small parameter S111S111 4 1 the following result for the renonnalization of the longitudinal bandwidth M1~: M11 where

=

/ M~exp~,,,—A1~ g~),

E~11a0 g~11= 2MASJII

(3.10)

/

Ai=!f~(i_cosx)=O.47.

(3.11)

374

A.A. Gogolin. Polaron transport in quasi-id organic conductors and in some narrow band systems

It is necessary to emphasize that there are no singularities in the integrals over q11 in eq. (3.7) in this case, so these integrals determine the numerical coefficient A1 in eq. (3.10). In the case of transverse electron bands (m

— n)

=

b00, c0

cos(q(m

,

—

n))

=

cos(q1(m

—

(3.12)

n))

so the corresponding singularities in the integrals over q11 at q11 —*0 arise in eq. (3.7). In the case of E111/E11 4 S1~lS,1these singularities are the logarithmic ones and in the case of E1111E11 ~ S1~/S~1 they are even the power ones. These singularities are to be cut off at small q11 q1(S11/S,11) 4 q1 b~so in the first case we get in the main logarithmic approximation: —

M1

=

[ 1 expL— —~

M~,

‘77~ j

g~ln~~— —a)] /1

0

—

(3.13)

.

We have omitted here the trivial terms of the order of g~11 in the exponential factor (3.13), corresponding to the usual isotropic narrowing of polaron bands. Thus the additional anisotropy due to SP effects is determined in this case by the factor M1IM~in eq. (3.13). For the5 evaluation cmtsec, E of this factor we shall use the typical values MA = 500, a0~= 3.6 A, b00 = 7.2 A, S = 2 x i0 111 = 0.5 eV for (TMTSF)2X(X = C104, PF6, A3F6) [3]. Substituting these values into eq. (3.10) we get g~11= 2, so at S111 /S~~ = 3 (according to some indirect experimental data [3]) the additional anisotropy is about 10. Thus the SP effects eliminate the discrepancies between the experimental data [3, 126] and the corresponding numerical calculations of Grant [64] for this anisotropy, done without any account of the polaron effects. In conclusion it is necessary to mention that the polaron bandwidth renormalizations (3.13) in the case of id phonon modes are the typical ones of quantum tunneling of heavy particles in crystals and in other periodical structures. This fact was pointed out in the works of Kagan, Prokofiev [127], Bulgadaev [128] and Schmidt [129]. In the case of weakly anisotropic deformation potential EJ~1/EJ1,the corresponding singularities in the integrals (3.7) over q11 are the power ones, so the polaron renormalization of transverse bandwidths is even stronger [63]:

M1

=

[ —1 ~ g~1(S111/S11)12], M°~ exp[_

2

g~ 1=

E 1a00 2MASJII

.

(3.14)

IT

Thus the corresponding renormalization factor for effective anisotropy is of the order of i02_103 and at larger values of E. and smaller values of MA it is about i~~—iü~ and even more. The strong narrowing of the transverse electron bands makes it possible to explain an extremely high degree of microscopic anisotropy of the order of 106 for electron spectra in (CH) 1, observed with the help of NMR technique [ii]. This anisotropy is by 4—5 orders of magnitude larger than that of the numerical calculations [130], so it is also connected with the polaron effects. It is necessary to emphasize here, that the main contribution to the effective polaron anisotropy arises from the strong narrowing of the transverse electron bands. Their small initial bandwidth M°14 ~FC provides the small transverse radius of the corresponding polaron states. As a result all these effects are well described by the usual SP theory (3.6, 3.7, 3.13, 3.14).

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

375

The longitudinal polaron size in quasi-id organic compounds is usually of the order of the lattice constant a0 and even more. For example, this effect takes place in (CH)1 [5—9],so the corresponding longitudinal renormalization of the effective polaron mass may be described by the formulas (2.8, 2.39, 2.40) for large polarons. In the case of the intermediate polaron size r0 a0 these evaluations of the polaron effective mass are in qualitative agreement with the corresponding results (3.6, 3.7) for SP at <10. That is why all the qualitative effects for the polaron anisotropy described by eqs. 6 cm/sec,may MAbe = 13, a (3.i3, 3.14). Making use of the typical values E111 = 3eV, ~ = i0 0 = 1.4 A for (CH)1 [23]4we = 8. and So even at b0 i05 = 4.4 and(3.14). S111/S/2It =is 2rather we obtain the additional anisotropy of the forget eq. g~11 (3.13) forAeq. interesting that the real anisotropy of order of i0 the sound velocity in (CH) 1 may be even more, because the linear phonon terms were observed in the temperature dependence of the heat capacity in this substance [122]. These terms demonstrate the existence of quasi-id acoustic modes in (CH)1. In our previous consideration we supposed the strong anisotropy of the sound velocity. In the most part of quasi-id organic compounds this anisotropy is not more than 3—4 [1—9]and in many cases it is absent at all, but due to the well-pronounced anisotropy of the crystal lattices in these substances the strong anisotropy of electron bands arise even for isotropic sound velocity [63]. For example, we shall consider the usual isotropic Debye model for acoustic phonons in the anisotropic hexagonal lattice with the 3 isotropic acoustic modes and the usual deformation potential ED: 2, =

toq =

Sq,

q~= (6ir2/a

E~(ql2M~S)”

113.

(3.15)

0b0c0)

Using these expressions we easily obtain from eq. (3.7) the following results for fmn:

21’

f2sin(~q~~m—n~)\2]

q~lmnI )

fmn3g [1_~~

j,

2

g

~

E2D

~

MAS2toD

-

(3.16)

It can be easily seen from this formula that the value of fmn increases rapidly with the increase of rn — nI. As a result the transverse electron bands, corresponding to the large value of rn — = b 0,

a0, are narrowing much than theeV, longitudinal the typical parameters 2 =stronger, 1 for ED = 0.5 MA = 500,ones. S = 2Thus, X i05using cm/sec, toD O.Oi eV and of at (TMTSF)2X [3] we get g a 0 = 3.6 A, b0 = 7.2 A, c0 = 14 A we obtain the additional anisotropy of about 5. Thus the essential additional anisotropy of electron bands due to SP effects may arise in all quasi-id organic conductors even in the case of the isotropic sound velocity. In the case of (CH)1 at a0 = 1.4 A, b0 = c0 = 4.4 A [131] using eq. (3.16) we get the following expressions for the corresponding values fmn at rn — n = a0, b0, C0: c0>

2, f~ 2. (3.17) fa00.5g 0=f~0=2.9g At ED = 2eV, S = i06 cm/see, MA = i3, toD = 0.06 eV [23] the value of g2 = 4.5, so the additional polaron anisotropy is determined by the expression: M

M°

~=exp(f

M° 5~

6_~fa)zi0

.

(3.18)

376

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

This result agrees also with the corresponding experimental data [11]. In conclusion it is necessary to emphasize that all the acoustic modes in quasi-id organic compounds give the essential contribution to the effective anisotropy of the polaron bands. 3.3. Small polaron mobility in a strong electric field As it was shown in the previous section the effective polaron renormalization of electron spectra in quasi-id organic conductors leads to the essential increase of its anisotropy. As a result the quite reasonable explanation for the quasi-id character of electron spectra in these substances [1—9]can be suggested in this case. The high anisotropy of electron spectra in quasi-id organic compounds makes it possible to use the purely id models for the corresponding transport theory in these materials. In many substances with quasi-id electron spectra the effective anisotropy of phonon spectra is not so high and thus the usual 3d phonon models can be considered for them in this case. The most interesting transport phenomena arise in the narrow-band conductors in strong electric fields [77, 78]. In strong electric fields an inelastic electron scattering provides the main mechanism of the energy relaxation processes, so it determines the characteristic features of a charge transport in these systems. In the narrow-band conductors the effective energy transfer in the inelastic scattering processes is rather small due to a large electron effective mass, so we can use the corresponding Fokker—Planck approximation (2.15) for the transport theory [77,78]. The corresponding coefficient B in this equation for the narrow-band conductors is also determined by the two-phonon scattering processes of the “Compton” type [77, 78]. This fact for the systems with the small enough bandwidth M < toph is quite obvious because all the one-phonon scattering processes are forbidden in this case by the energy and momentum conservation laws. The value of B is connected with the corresponding scattering amplitude WKK by eq. (2.16) and the value of WKK is determined by the effective Hamiltonian H2 for the two-phonon scattering processes [77, 78]: 112

=

N ~

q1.q2.n

~n~nYq~q~

exp{i(q1

—

q2)n}(bq1 +

(3.19)

btqj)(b~q,+ b~).

For acoustic phonons the Born scattering amplitude yqq is determined by the corresponding deformation potential E2D according to the usual formula [77, 78]: 2l2M~S E2~(q1q2)~ In the case of intramolecular optical phonons the value of 7qq,

(3.20)

.

=

=

g

2~w0does not depend on the momenta q1, q2 as usual [77, 78]. The effective two-phonon Hañiiltonian H2 (3.19) describes the quadratic terms in the ion displacements, so they are small enough in comparison with the linear ones (3.2). As a result the corresponding coupling constant is small enough, so we can use the Born approximation for WKK = YKK’~ It is necessary to emphasize the Hamiltonian H2 (3.19) does not change its structure after the polaron transformation (3.5), so the corresponding scattering amplitude ‘Yqq ~5 7q also just the same. Thus the effective value of 1q2 is not renormalized by this transformation and is to be much more than the corresponding amplitudes of two-phonon polaron scattering processes arising in the Hamiltonian (3.6). The corresponding terms in the effective Hamiltonian (3.6) are renormalized by the small enough exponential factor e~ and do not give any contribution to the full scattering amplitude WKK.

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

377

As a result the value of B is determined by the Born amplitude YKK’ and does not depend on the polaron momentum p. In the case of the deformation interaction with 3d acoustic phonons (3.20) the corresponding expression for B has the form [77, 78]: B=

(E2D/MAS2)2(TttoD)9.

toD ~D ~2

(3.21)

Here 11 is the volume of a unitary cell. The temperature dependence of B(T) is rather strong in this case, as it was pointed out in refs. [41, 75—78, 80], and is typical of all the two-phonon scattering processes for a heavy particle [132]. In a more general case the value of B depends on the polaron momentum p due to the corresponding two-phonon terms in the effective Hamiltonian (3.6). The corresponding solution of eq. (2.15) with account of the periodic boundary conditions for the distribution function f( p) = f( p + 2 IT) has the form [96]:

~ 0(IT)

f(p) Here a0

=

R0

J

dq~0(q)+%(_IT)Jdq~o(q)

B(p) ~‘0(p)(~~(IT)—

1 and the function 1

p0(p)

.

(3.22)

is determined by the relation:

=B(p)exp[_ i~E(p)+eEfdq/B(q)].

(3.23)

0

The constant R0 in eq. (3.22) is determined by the usual normalization condition:

fdpf(p)=p.

(3.24)

Here p is the charge density. The electric current density j is defined in a usual manner:

1J

dpV(p) f(p).

(3.25)

The corresponding decrease of the current-voltage characteristic (CVC) in strong electric fields follows immediately from eqs. (3.22—3.25) in the most general case. This property is typical of all narrow-band conductors and is connected with a general structure of the corresponding Boltzmann equation (2.15) with a restricted electron spectrum. Indeed, at large E —* this equation is to be expanded in the powers of E 1, so in the lowest order the corresponding trivial solution f( p) = B = const. does not give any contribution to the current density j(E). As a result, the value of j(E) is determined at E—* by the next term of the corresponding expansion and is proportional to E’. Thus, the simple asymptotic behavior of j(E) E~at E—* ~ is valid for all the narrow-band systems [77, 78, 96]. The main physical reason for —

378

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

this effect is connected with a constant character of the corresponding heating Ej in strong electric fields due to a restricted character of electron spectra and the fixed energy flow to the phonon bath, provided by inelastic phonon scattering processes. This fact was pointed out first by Keldysh [87] and by Bychkov and Dykhne [88]. In our case (3.22) the most simple expression for j(E) arises at B(p) = B = const. In this case the general expression (3.22) has a rather simple form, so it is convenient to introduce the new function x( p): f(p)=x(p)exp(—E(p)/T).

(3.26)

After the Fourier transformation in p we get easily the following simple expression for

x( p):

F e’m’~

imB—eE~

(3.27)

Here a

0 = 1 and the values Fm describe the Fourier components of the function exp(E(p)/T). Substituting this expression (3.27) into eq. (3.25) we get the following result for the corresponding drift velocity V(E) j(E)lp: *

V’E~— ~

~

~1mm

328

m

x~(E)_~imB — eE

‘

where imB-eE

=

(3.29)

and the values Gm describe the Fourier components of the function exp(—E(p)/T). At E(p) = — M cos( p) the values Fm and Gm can be expressed easily through the modified Bessel functions Im (/3): mGm=(~i)mIm(/3), /3=M/T. (3.30) Fm=(~i)

In weak electric fields E 4 E 10

=

B/e the function j(E) is a linear one and determines in a usual

manner the mobility p. [96]: p.(T)

=

GF /3 ~ 0

(~

FmG~ FOGO), —

p.1

=

M/eE00.

(3.31)

This expression can be simplified at E(p) = —M cos(p), so in this case [77, 781: p.(T)

=

-~-(I~(/3)- i)I~2(/3).

(3.32)

The temperature dependence of the polaron mobility p.(T) in this situation is determined by the function B(T). So in the case of the two-phonon scattering processes for 3d acoustic phonons (3.21) this dependence is a power one: p.(T) T” with the large enough value of n = 10 at T ~ M and of n = 8 at T 4 M [77, 78]. Thus, an extremely rapid increase of the SP mobility p.(T) with the decrease of —

A. A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

379

temperature takes place in narrow-band conductors. This fact was pointed out in the works of Kagan, Maximov and Klinger [40, 41, 75, 76] and of Andreev and Lifshits [80].The analogous rapid increase of p.(T) takes place at low temperatures in all the systems with the deformation mechanism of electron—phonon interaction [49,132] and is typical of all acoustic phonon scattering processes. In strong electric fields E a~E0 the essential decrease of j(E) takes place in the narrow-band systems. It is described by the simple asymptotic law [77, 78, 96]:

j(E)

i =

Thus, at E

L~

E 1~ 2FmG,~~] 1 1~FmG;~] Jo ~ m

L~

(3.33)

.

E

0 there is a characteristic maximum of j(E) with the negative differential conductivity o~<0 in strong fields. The graph of this dependence forj(E) at E(p) = —M cos(p) and /3 = 0.5; 1; 2 is shown in fig. 2. It is interesting that the characteristic maximum of j(E) increases in the units of Jo = 1 with an increase of T and the corresponding electric field Emax increases also in this case. The general expression (3.28) for j(E) is to be simplified at high temperatures T a~M. Intothis m/m! [123]leads eq.limiting (1.11) case the corresponding expansion of the Bessel functions ‘m(13) = (,812) for j(E) in the lowest order in /3 4 1 [77, 78, 93—96]. This dependence contains the maximum of j(E) at = E/E 0 = 1 with the height Jmax = ~ 4j0. The value of Jmax <
~‘

—

i/jo

0.5

0~

E/Eo

Fig. 2. Current—voltage characteristic for a small polaron in a narrow-band conductor.

380

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

4. Gunn effect in narrow band conductors 4.1. Main equations of Gunn effect

The negative value of 0~D<0 gives rise to a domain formation in narrow-band conductors in strong electric fields [93—96]. In this case the exact analytic theory for the corresponding Gunn effect has been developed by Gogolin [93—96] using the explicit analytic solutions of the kinetic equation (2.15) for inhomogeneous electric fields E(X, t). With account of the time and space dependences for the distribution function f(p, X, t) this equation has the usual form [79]: (4.1) The explicit analytic solutions of this equation makes it possible to study in detail a shape of electric domains and to find their velocity [93—96].As a result, the detailed comparative analysis of the various phenomenological approaches in the theory of the Gunn effect [99—103]is possible. We shall consider the case of a very smooth space and time dependences with very small corresponding derivatives, so the value of E(X, t) changes slowly at X 1 4 rD and at t = liv 4 rD/v [93—96].It is assumed here, that the mean free path 14 rD due to a small electron density in the system. In this limiting case we can neglect also the effects of electron—electron scattering in eq. (4.1). As a result we can expand eq. (4.1) in small space and time gradients of the order of x 1 and t In the lowest order in these parameters we can use the general solution (3.22) for f(p, X, t) with smooth time and space dependences for E and p. Substituting this solution into the terms with time and space gradients on the left-hand side of eq. (4.1), we get easily the effective non-linear diffusion equation (1.12) for E(X, t) [96].This general procedure can be simplified essentially in the case of B(p) = const., describing quite perfectly the charge transport in narrow-band systems [93—96]. For the evaluation of the function D(E) in this case we shall consider in detail eq. (4.1) with E( p) = M cos p. Performing the Fourier transformation in p, we find the following equations for the corresponding Fourier components fm ——

.

—

~

(4.2)

We see from the general structure of this equation that at /3 4 1 and X 1 = MIB it can be expanded in these parameters, so we find a closed system of equations for the three lowest harmonics f ~

0, f1, f_1.

describing the values p, p =~,

1= ~

j and the

average electron kinetic energy W respectively:

(f1 -p).

W=

-

~ (f1 +f).

In these notations, the corresponding system of equations for p, —Ej + (B

(~

+

+

f3Bp = 0

2EW+ ~M2OplOX=0.

+B) j+e

(4.3)

j, E,

W has the form [93—96]:

(4.4) (4.5)

A. A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

381

The equations (4.4, 4.5) describe correspondingly the energy and current relaxation processes with account of heating and diffusion. These equations together with the Poisson equation =

41T(p — p

0) IK

(4.6)

and the usual relation —OJ/OX

Op/8t

(4.6a)

form the closed system of equations for the Gunn effect [93—96].For example, the well-known

relationship between [99—103]:

j

and the given external current J follows immediately from eqs. (4.6, 4.6a)

(4.7) The conditions t 2 =atB’ X ~ 1 densities. hold clearly for in thetheGunn domains with a characteristic low and electron Thus, lowest order in time gradients we size can X r~ (rt Tl4irep0)” omit the= terms containing time derivatives from eqs. (4.4, 4.5), so they have the following form: ~‘

—

BW= Ej —

—Bj = ~M2 ~

/3Bp,

+

In terms of dimensionless variables x = X/l 0, r = 10 = V(E0)r0

=

KE0/41rp0,

J0

=

(4.8)

e2EW.

tlT0,

f0

=

JIJ0 with the usual values l~,J0 [99—i03]:

p0 V(E0)

(4.9)

these equations for dimensionless quantities 1p r=EIE0,

p—p

0,

Jj1J0,

w—4irW/rtE~

(4.10)

2

(4.11)

have the following form:

i=f0—

~,

j5= ~ + 1,

bw — ej= —215,

bJ+ Ewb2

b =(2rD/l) =

— -~15~,.

(4.12)

The subscripts x and r mean differentiation. Eliminating the variables J, 15, w we find the following equation for the electric field e(x, r): —d(s) e~+ u(E)(e~+ 1)

+ s~

f

0,

e~ —i

(4.13)

where 2. v(e) =

2~/(i+

2)

d(e) = d0/(i +

2)

d0

=

2(11/3rD)

(4.14)

382

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

As a result the explicit analytic expressions for the corresponding coefficients in the usual gradient expansion (1.12) are obtained for narrow-band conductors and the exact theory of the Gunn effect in these systems is developed [93—96].It is necessary to mention here that the condition /3 4 1 is necessary only for the evaluation of the functions d(r), v(e) and does not effect the general structure of eq. (4.13). The explicit analytic expressions for d(r), v(r) describe the decrease of these functions in strong electric fields. The value of the diffusion coefficient d(r) in this case is determined by the parameter d0, 2 4 1. As a result, both which is awith ratiod of the two independent small parameters /324 1 and (1/rD) situations 0 ~ 1 and d0 4 1 are possible in the system as far as the intermediate value of d0 1. Thus, the various interesting situations arise in the system, depending on the value of d0 [93—96]. It is necessary to emphasize that in our consideration the Einstein relation D(E) = T p.(E) between D(E) and p.(E) = V(E)/eE is valid for narrow-band systems [93—96]. This fact demonstrates the general character of the Einstein relation, which is valid not only for equilibrium systems, and was assumed first by Night and Peterson [99] in the frameworks of their phenomenological theory of the Gunn effect. In their work the general structure of eq. (4.13) was investigated in detail at arbitrary dependence of d(E), v(E) and some exact analytic results for the domain velocity VD were obtained. Night and Peterson have assumed the N-shape structure of v(r), so our situation differs first of all by the decrease of v(e) at large r. It is necessary to emphasize the advantages of their approach in comparison with the other ones [100—103],where only the case d(r) = const. was considered. In the frameworks of these phenomenological theories, however, some interesting qualitative results were obtained and all the main features of real Gunn systems were described [100—103]. 4.2. Moving domain structure The equation (4.13) makes it possible to study in detail the structure of moving electric domains in the Gunn systems. It is determined by the corresponding travelling-wave solutions r(x, ‘r) = C0T) = e( ~) with a periodic ~ dependence corresponding to the condition of overall electrical neutrality of the system [99—103].Substituting this solution into eq. (4.13) for a constant external current f0 we find the following ordinary differential equation for r( ~): —

+ (v(r)

—

C0) e~=ft

—

v(E),

r~

—1.

(4.15)

This equation can be rewritten for an arbitrary domain velocity C00 as [99]: —(e~+1)d(r)~C—r~(f0—C0)

(4.16)

where _C=ln(e~+i)_e~+Jd~d~(E)(v(e)_C0).

(4.17)

It follows from (4.16, 4.17) that eq. (4.15) has closed integral curves and periodic solutions e( fl only with C0 = f0, so that the domain velocity at /3 4 1 is determined exclusively by the external current. Taking into account the explicit functions d(r), v(r) in (4.14), we can put the general expression for the integral of motion (4.17) in the following form:

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

d0(s~—ln(e~+i))=C+s2—sf0(i+~e2)ns~(s), C=Cd0.

383

(4.18)

The existence of the periodic solutions e( fl is provided by the existence of three real roots ~ s~for the equation ~ = 0, which is equivalent to the condition ~‘(s) = 0. So the constant C must satisfy the conditions: C~~ C~C,

C~= 1—

312].

(4.19)

[1±(i—f~)

The quantities ~ ~ determine the maximum and minimum values of the electric field in a domain. In the general case d 0 1, we would need to solve the transcendental equation (4.18) for e~.Figure 3 shows a plot of the corresponding function e( ~) for the values d0 = 1, f0 = ~and C = — i. We note that the shape of the domain is very asymmetric in this case; we also note that the maximum field in the domain is quite high, and with C <0 it goes off to infinity in acccordance with Ei = 3/f0 in the limit f0 —*0. This situation does not arise in the case of an ordinary N-shaped CVC [99—103],in which case we have f0 f1 and f1 is the local minimum of v(e). As a result, some interesting effects, connected with a strong field, can take place in narrow-band systems, differing them from the ordinary Gunn ones [97—103]. The asymmetry of the domain shape depends strongly on d0 and disappears entirely at d0 ~ 1, as obviously follows from eq. (4.15), where terms of even parity in ~ are predominant at large values of d0. In the case of d0 4 1, the asymmetry becomes very pronounced, and the domain structure converts into a saw tooth curve with characteristic values e~ 1 and e~= —1 [93—96]: -~

~‘

for

e~>0 and

r~+i=exp(—~’(e)/d0)4i for e~<0.

(4.20)

The period (V0) of the domain structure is e~— 62 in this case. If d0 ~21, leftasside of transcendental equation (4.18) can be expanded in the small quantity 41,theand a result we find: d~ ~(s)=2(

d

1/2

o)

F(~,K),

K2= ~

~,

K2sin2 ~ =

(4.21)

where F(~,K) is an elliptic integral of the first kind. In this case the function s( ~) can be expressed in

Fig. 3. Field distribution in a Gunn domain for the case d 0

=

1,

f0 ~, C =

=

—1.

384

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

terms of the elliptic sine sn(~’,K) [133]: =

dn2(~’,K) [62— 63K2 sn2(~’,K)]

— 1 ~

—

2~ d 0

,

dn2(~’,K)

=

1— K2 sn2(~’,K) (4.22)

E3~h/2

!

2 for f Figure 4 shows a corresponding curve of e versus ~ = ~/d0~ 0~= ~and C of the superstructure is determined in this case by eq. (4.21) with ~ = =

2 K(k) (d0/(61 —

63))

=

1/2

—1. The period (Q00)

(4.23)

where K(k) is the complete elliptic integral of the first kind. In ordinary units, we would have ~ rD at d0 ~ 1 and (1~ 1~ (I/f3r~)~ ~ rD at d0 4 1. In the2.case f0-.-* 1, the domain structure is smoothed we have 63 1, and f2~ Analogously, this structure disappears as C—* over, C_ forand arbitrary d (1 —f0)” 0>1, with 61 60, where -=

—

~—

62

(4.24) In this case, expressions (4.21—4.23) become valid for arbitrary d0 ~ 1 and simplify considerably [133]: 2 (4.25) r(fl= 62 +(r~ sin In a similar way we find the functional dependence s( ~) as C—* C~with 62 = 63 [133]: ~‘.

—62)

(4.26) In this case,

(10

tends toward infinity logarithmically.

Fig. 4. Field distribution in a Gunn domain for the case d 0

8’ 1,

f0

=

~, C

=

—1.

A. A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

385

The structure of the moving domains with d0 ~ 1 is analogous to that of the ordinary soliton solutions of the Korteweg—de Vries equation [79, iio]. This comment applies, however, only to travelling-wave solutions. It does not mean that the original equation, (4.13), is integrable by the method of the inverse scattering problem [110].Furthermore, analysis of eq. (4.13) with the help of the general criteria for the integrability of non-linear equations suggested by Sokolov and Shabat [134] proves that it is not integrable by the method of the inverse scattering problem or by other methods of non-linear wave theory [134, 135]. 4.3. Stability of inhomogeneous solutions

Knight and Peterson [99] have developed a general theory for the stability of the solutions of eq. (4.13) for arbitrary functions d(s), v(e). They proved that the domain structures are unstable in the case of a fixed external current f0, while they are stable in the case of a fixed voltage with a negative static impedance Z(0) = dEldf0 (which describes the behavior of the current f0 through a sample with a domain as a function of the average field E) [99]. Making use of the explicit functional dependence E(~) at d0 ~ 1, in (4.21—4.26), we can put these general results in a more concrete form and estimate the instability increment A0 at a fixed current. We can also calculate Z(0) and determine the region in which domain structure exists. The quantity A~ characterizes the region of frequency dispersion, Z(to), at small to ~ A0/r0 and determines the extent to which weak signals are amplified at low frequencies imposed on the strong static field. To evaluate A0 we need to linearize eq. (4.13) near the solution ~( ~), determined at d0 ~ 1 by eqs. (4.21—4.26), and analyze a small increment i1(x, r) = e~1(~) = ~(~)— i~(fl.As a result we find an eigenvalue equation for the instability increment A [93—96]: (4.27) This equation is equivalent to a Schrödinger equation with an effective potential U(~)=A~(~)+2(i—f0e~(~))

(4.28)

which depends on A. We are interested in the maximum positive values of A0, so we should find the ground state in the potential U(~)(4.28) with the negative energy (—A0). The existence of this state follows from the fact, that the eigenvalue A = 0 in (4.27) corresponds to the solution ~ ( fl = which has zeros at the points = 0 (4.18), so it is not the ground state. The inequality A0
386

A.A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

semiclassical, and A0 is determined by the position of its bottom at ~0( ~) C—* C, we thus find:

~0

in (4.24). In the limit

A0(f0)=f~-i+~i-f~.

(4.29)

2 and the maximum value A Atf0 41 we have A0(f0)=f~/2;asf0—* 1 we have A0~2(i f)l/ 0 = ~ is reached at f0 = v~/2.The small value of A0 at f0 4 1 and at f0—* 1 demonstrates an essential dispersion of the impedance Z(to) at low frequencies to 4 r~. The periodic structure of ~( ~) gives rise to an entire narrow band of A, so that the results above refer to the maximum value of A0, corresponding to the top of the band. The criterion of the stability of the domain solutions of eq. (4.13) at fixed external voltage reduces to the condition Z(0) = d~/df0<0 [99—103].This condition follows from the general structure of eq. (4.13), since as s(~)is varied with unfixedf0 in (4.27) there is a shift of A by the factor Z’(O) = ~f0/~e. As a result the negative value of A0 may arise in the system, which would mean the stability of the domain solutions. To calculate Z(0) at d0> 1 we use the explicit expression for r( fl in (4.22). Averaging r over the period (1~in (4.23), and using the corresponding relations for the elliptic functions [133] we find [93—96]:

E

E~+

~ —

63)

E(k)/K(k)

(4.30)

where E(k) is the complete elliptic integral of the second kind. Figure 5 shows CVC f0(~) (4.30) for the case C = —1. At C <0, the CVC begins2); at the point ~ = ~, ft = f5, determined from the conditions it then decreases to zero as ~—~x in accordance with a law C_(f~) = C and ~ =f~’(i + (1 —f~)~ determined by the asymptotic relation ~(f 0)= —6/f0 ln f0 asf0—* 0. At ~> ~ we have Z(0) <0, and the domain structure is stable. At the point = t~ there is an abrupt change in f0, corresponding to a metastability of the lower branch of the CVC at ~ < ~, which is typical of Gunn systems of all types [99—103]. In the case of 0 < C < ~, at large values of ~ there is also6 ashows criticala value 2). Figure plot of~c1’f determined from the conditions C = C+(f~1), ~ =f~(i + (1 —f~1)~ 0(~)for C = 0.05. As C—* 0 we have ~c1~ ~ as C—* ~we have ~cI ~ ~ and the region of the domain stability contracts to zero: the change in the 2.current at the point ~c1 also occurs abruptly; the current vanishes as f0—* 1 in accordance with (1 — f0)

I 0

-—

e

Fig. 5. Current—voltage characteristic of a sample with a domain at C = —1 in the case d 8’ 1.

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

~_~1

0

387

~

Fig. 6. Current—voltage characteristic of a sample with a domain at C = 0.05 in the case d

0 8’!.

With a large number of domains the condition d~/df0<0 corresponds to stability with respect to periodic perturbations with a period 1l~[99—103].The system, however, may be unstable with respect to distortions of the periodic structure; this instability is described by the same increment A0 [99—103]as in the case of a fixed value of f0. This instability, however, develops relatively slowly, at least at low currents, f ~ 0.1. Indeed, in this case the value of A0 ~ ~ according to eq. (4.29), so a rather large number of domains, of the order of A~,can be observed in the system. In conclusion it is necessary to emphasize, that all the results obtained are the exact ones, so the various phenomenological theories [99—103]are to be analyzed now using these methods.

5. Conclusions A polaron transport in quasi-id organic conductors and in some narrow band systems is studied in detail in the present review. One of the most interesting problems of this theory is connected with intermediate polarons. Some attempts of their energy spectra calculations were done in the work of Gogolin [42]. In this work the special diagram technique, suggested by Elyutin [67],has been used and some partial summation of the corresponding diagrams has been performed. The exact analytic solution of this problem, however, has not been obtained, so it has to be solved in the future. Some numerical results for intermediate polarons in id systems were obtained by Venzl and Fischer [136]. The most interesting effect, studied in the present review, is connected with non-linear CVC in strong electric fields for small polarons (SP) [77,78] and for large acoustic ones [34—36].This non-linearity in narrow-band conductors is so strong, that even the negative differential conductivity and the corresponding Gunn domain instability arise in these systems [93—96].The exact analytic theory of these effects was developed in refs. [93—96].The interesting results connected with thermodiffusion effects were obtained in refs. [94, 96]. These effects give rise to the corrections for the domain velocity C0, so this value differs a little from f0 [94,96]. These corrections, however, are rather small, so they can be neglected in many cases [94, 96]. This fact agrees with the results of the corresponding computer simulations [ioi—i03]. It is necessary to mention, that the typical values of electric fields E0 Miel, 4—i05 characterizing V/cm. Asthea non-linear effects, are not so large as at M 0.01 eV, 1 100 A they are about i0 result, the direct experimental observation of these effects in real quasi-id organic conductors and in some other narrow-band systems is possible. The most favorable conditions for observing this are in quasi-id organic semiconductors such as Cs 2 (TCNQ)3, (TEA) (TCNQ)2 etc. [1]. The effective charge carrier concentrations n in these substances —~

—

-=

388

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

are rather low: n 1014_lOb cm~3and the room temperature conductivity 6 is rather small: 6 i02_iO_5 f1~cm’ [1]. As a result, the maximum of current density j~ 6E() in this case at M 0.01 eV and 1 (iO_i02) A is also not so large and is about (0.1—1) A/cm2. Thus the total current J in this case for thin samples with diameter D 3 A, so their heating will not 1 -=0.1 cm will be about i0lead to a large enough value of destruct them. The low charge carrier concentrations in these substances the Debye radius rD (102—i03)A. This value determines the characteristic domain width and is to be much larger, than the mean free path 1. The analogous situation takes place also in semiconductors with superlattices [83,85]. The observation of the Gunn effect in high conducting compounds with a large electron concentration n—(i018—i019)cm3 and with o.~(102_103)1l~ cm~ at T=300°K [1—8]is rather difficult. Indeed, the effective Debye radius rD in these substances is rather small: rD (1—10) A and the effective current density J~ i05 A/cm2 at E i03 V/cm, so their heating will be very strong and the main criteria of our theory are not fulfilled. Just the same situation takes place, for example, in the superconductors with heavy fermions, such as U [65,66], though M the value of E0 in 3 V/cm) 2Zn17, due to UBe13 a smalletc. enough bandwidth i0~eV. these not so large (E0 i0 transport theory was developed using the Fokker—Planck In substances the presentis review the polaron approximation. The range of validity of this approach in quasi-id organic conductors is rather wide even at M —0.1 eV [1—9].The reason is that the main mechanism for the inelastic electron scattering in these substances is connected with high-frequency intermolecular vibrations with a very small dispersion iO~eV 4 to 0 0.1 eV. The one-phonon processes in this case are forbidden at M < to0, so only the two-phonon ones of the “Compton” type are essential in the system. The analogous situation also takes place in the case of low-frequency acoustic phonons, due to a small energy transfer L~E toD i0~eV 4 M. Thus the results obtained above are applicable to a great amount of organic substances and molecular crystals. —

-=

—

-=

-=

‘-=

—

-=

—

-=

In conclusion I wish to thank A.S. Alexandrov, V.L. Bonch-Bruevich, S.A. Brazovsky, Yu.A. Firsov, I.B. Levinson, A.A. Ovchinnikov, E.I. Rashba, R.A. Suns and G.E. Volovik for useful discussions of some problems, considered in the present review.

References [1] IF. Schegolev, Phys. Stat. Sol. A12 (1972) 9. [2] GA. Tombs, Phys. Rep. C40 (1978) 181. [31D. Jerome and H.J. Schulz, Adv. Phys. 31(1982) 299. [4] Yu.A. Firsov, V.N. Prigodin and K. Seidel, Phys. Rep. 126 (1985) 246. [5] Physics in One Dimension, eds. J. Bernasconi and T. Schneider, Springer Series “Solid State Phys. “ (Springer Verlag, 1981). [61Proc. Intern. Conf. on Low-Dimensional Conductors, Boulder, Colorado 1981, eds. A.J. Epstein and E.M. Conwell, Molec. Cryst. Liq. Cryst. 77 (1981) N 1—4. [7] Proc. Intern. Conf. on Low-Dimensional Conductors (Les Arc Conference) 1982, J. Physique (Paris) Colloque C3, supplement No 6 (1983) 44. [8] Proc. Intern. Conf. on Physics and Chemistry of Low-Dimensional Synthetic Metals (Albano-Terme, Italy, 1984) Molec. Cryst. Liq. Cryst. (1985). [9] Proc. Intern. Conf. on Conducting Polymers (Vienna 1985), Springer Series Solid State Phys.” (Springer Verlag 1985). 1101 K.J. Donovan and E.G. Wilson, Phil. Mag. B44 (1981) 9,31; J. Phys. C18 (1985) 151; B. Movaghar, D.W. Murray, K.J. Donovan and E.G. Wilson, J. Phys. C17 (1984) 1247. [11]M. Nechtschein, F. Devreux, R.L. Greene, T.C. Clarke and GB. Street, Phys. Rev. Lett. 44 (1980) 356. [12]MI. Cohen, LB. Coleman, A.F. Garito and A.J. Heeger, Phys. Rev. BlO (1974) 1298.

A. A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems

389

[13] C.K. Chiang, C.R. Fincher, Y.Q. Park, A.J. Heeger, H. Shirakawa, E.J. Louis, S.C. Gau and AG. Mac Diarmid, Phys. Rev. Lett. 39(1977) 1089; K. Ehinger, W. Banhoffer, K. Menke and S. Roth, J. Physique 44, C3 (1983) 115. [14] K. Seeger, ‘ND. Gill, T.C. Clarke and GB. Street, Solid State Commun. 28 (1978) 873; W. Ross, A. Phillip and K. Seeger, J. Physique 44, C3, (1983) 127. [15] MI. Rice, Phys. Rev. Lett. 37 (1976) 36; MI. Rice, C.B. Duke and NO. Lipari, Solid State Commun. 17 (1975) 1089; M.J. Rice and NO. Lipari, Phys. Rev. Lett. 38 (1977) 437. [16]M.J. Rice, NO. Lipari and S. Strässler, Phys. Rev. Lett. 39 (1977) 1359; Mi. Rice, L. Pietronero and P. Brüesch, Solid State Commun. 21(1977) 757. [17]MG. Kaplunov, T.P. Panova, E.B. Yagubsky and Yu.G. Borod’ko, Zh. Strukt. Khim. 13 (1972) 440. [18]L.P. Gor’kov and El. Rashba, Solid State Commun. 27 (1978) 1211. [19]B.R. Weinberger, J. Kaufer, A.J. Heeger, A. Pron and A.G. Mac Diarmid, Phys. Rev. B20 (1979) 2231; CR. Fincher, M. Ozaki, M. Tanaka, D. Peebles, L. Lauchlan and Al. Heeger, Phys. Rev. B20 (1979) 1589; N. Suzuki, M. Ozaki, S. Etemad, A.J. Heeger and A.G. Mac Diannid, Phys. Rev. Lett. 45(1980) 1209, 1463; C.R. Fincher, M. Ozaki, A.J. Heeger and AG. Mac Diarmid, Phys. Rev. B19 (1979) 4140. [20]W.P. Su, J.R. Schrieffer and A.J. Heeger, Phys. Rev. Lett. 42 (1979) 1698; Phys. Rev. B22 (1980) 2099. [21] S.A. Brazovsky, Zh. Eksp. Teor. Fiz. 78 (1980) 677 [Soy. Phys. JETP 51(1980)3421; S.A. Brazovsky and N.N. Kirova, Pis’ma Zh. Eksp. Teor. Fiz. 33 (1981) 6 [JETP Lett. 33 (1981) 4]; S.A. Brazovsky and S.I. Matveenko, Zh. Eksp. Teor. Fiz. 81(1981)1542. [22] M.J. Rice, Phys. Lett. A71 (1979) 152; Mi. Rice and El. Mele, Solid State Commun. 35(1980)487. [23] E.J. Mele and M.J. Rice, Phys. Rev. Lett. 45 (1980) 926. [24]T. Holstein, Ann. Phys. 8 (1959) 325, 343. [25] El. Rashba, Opt. Spektrosk. 2 (1957) 75, 88. [26]J. Appel, Solid State Phys. 21 (1968) 193. [27] VI. Mel’nikov, Zh. Eksp. Teor. Fiz. 72 (1977) 2345 [Soy. Phys. JETP 45(1977)1233]. [28] T. Holstein, Mol. Cryst. Liq. Cryst. 77 (1981) 235. [29] EM. Conwell and IA. Howard, Phys. Rev. B31 (1985) 7835. [30] G. Whitfield and P.B. Shaw, Phys. Rev. B14 (1976) 3346. [31] H.B. Schüttler and T. Holstein, Phys. Rev. Lett. 51(1983) 2337; Ann. Phys. 166 (1986) 93. [32] E.G. Wilson, J. Phys. C16 (1983) 6739. [33] E.G. Wilson, J. Phys. C16 (1983) 1039; C15 (1982) 3733. [34] A.A. Gogolin, Pis’ma Zh. Eksp. Teor. Fiz. 43 (1986) 395 [JETP Lett. 43 (1986) 514]. [35] A.A. Gogolin, Zh. Eksp. Teor. Fiz. 92 (1987) 152. [36] A.A. Gogolin, Preprint of L.D. Landau Institute N 1986-12 (Chernogolovka, 1986). [37] A.S. Davydov, Usp. Fiz. Nauk 138 (1982) 603 [Soy. Phys. Usp. 25 (1982) 898]. [38] AS. Davydov, Zh. Eksp. Teor. Fiz. 78 (1980) 789; AS. Davydov and NI. Kislukha, Zh. Eksp. Teor. Fiz. 71(1976) 1090. [39] Yu. A. Firsov, Polarons (Nauka, Moscow, 1975). [40] P.S. Zyryanov and MI. Klinger, Quantum Theory of Electron Transport in Crystal Semiconductors (Nauka, Moscow, 1976). [41] MI. Klinger, Phys. Rep. 94 (1983) 184; Usp. Fiz. Nauk 146 (1985) 105. [42] A.A. Gogolin, Phys. Stat. Sol. b109 (1982) 95; Preprint series in theoretical physics HU-TFT-81-26 (University of Helsinki, Helsinki, 1981). [43] L.D. Landau, Collections of Scientific Papers (Nauka, Moscow, 1969). [44] SI. Pekar, Investigations in Electron Theory of Crystals (GTI’I, Moscow 1951). [45] N.N. Bogolyubov, Ukr. Matem. Zh. 2 (1950) 3; S.V. Tyablikov, Zh. Eksp. Teor. Fiz. 21(1951) 377. [46] R.P. Feynman, Statistical Mechanics (Mir, Moscow, 1978). [47] T.D. Lee, FE. Low and D. Pines, Phys. Rev. 90 (1953) 297. [48] H. Frbhlich, Adv. Phys. 3 (1954) 325. [49] G.E. Volovik and V.M. Edel’stein, Zh. Eksp. Teor. Fiz. 65(1973)1947 [Soy. Phys. JE1’P 38 (1974) 972]. [50]VI. Mel’nikov and G.E. Volovik, Th. Eksp. Teor. Fiz. 65 (1973) 1637 [Soy. Phys. JETP 38 (1974) 819. [51] G.E. Volovik, VI. Mel’nikov and V.M. Edel’stein, Pis’ma Zh. Eksp. Teor. Fiz. 18 (1973) 138 [JETP Lett. 18 (1973) 81]. [52]A. Maradudin, E. Montroll and J. Weiss, Dynamic Theory of a Crystal Lattice in Harmonic Approximation (Mir, Moscow, 1965). [53] M. Bom and Khuan Kun, Dynamic Theory of a Crystal Lattice (IL, Moscow, 1958). [54] H. Hansel, J. Hanspach, H. Knoblich and ‘N. Neumann, in: Proc. Second Intern. Conf. “Electrical and Related Properties of Organic Solids”, Karpacz 1978, Poland, Sci. Pap. Techn. Univ. of Wroclaw, N 16 (Wroclaw, 1978). [55]G.W. Bak, Polymer Bulletin 8 (1982) 95; 11(1984) 443; Thin Solid Films 105 (1983) 177, 183.

390

A.A. Gogolin, Polaron transport in quasi-id organic conductors and in some narrow band systems

[56] J.J. Halady, J. Michalski and A. Szymanski, in: Proc. Second Intern. Conf. “Molecular Crystals”, Wdzydze 1981, Poland, Sci. Pap. Techn. Univ. of Gdansk (Gdansk, 1981). [57] W. Mycielski and A. Lipinski, Thin Solid Films 48(1978)133. [58] S. Nakano and Y. Maruyama, Solid State Commun. 35(1980) 671. [59] W. Warta and N. Karl, Phys. Rev. B32 (1985) 1172. [60] M. Rona and G. Whitfield, Phys. Rev. B7 (1973) 2727. [61] G.E. Volovik and V.M. Edel’stein, Zh. Eksp. Teor. Fiz. 67(1974)273 [Soy. Phys. JETP 40 (1975) 137]. [62] S.V. Tyablikov, Zh. Eksp. Teor. Fiz. 23(1952)381. [63] A.A. Gogolin, Solid State Commun. 50 (1984) 265. [64] P.M. Grant, J. Physique (Paris) 44 C3 (1983) 847, 1121. [65] G.R. Stewart, Rev. Mod. Phys. 56 (1984) 755. [66] NB. Brandt and VV. Moschalkov, Adv. Phys. 34 (1985) 1. [67] P.V. Elyutin, Opt. Spektrosk. 43 (1977) 542. [68] MV. Sadovsky, Zh. Eksp. Teor. Fiz. 77 (1979) 2070. [69] VV. Bryksin and Yu. A. Firsov, Fiz. Tverd. Tela 10 (1968) 1364. [70] AS. Alexandrov and V.F. Elesin, Fiz. Tverd. Tela 25 (1983) 456; A.S. Alexandrov, V.F. Elesin and V.V. Kabanov, Zh. Eksp. Teor. Fiz. 86 (1984) 1937. [71] CV. Tyablikoy and V.A. Moskalenko, Uch. Zapiski Kishiney Univ. 55(1960)113, 129, 143. [72] NV. Tkuch, Teor. Mat. Fiz. 61(1984) 400. [73] S.S. Schweber, An Introduction to Relativistic Quantum Field Theory (Row, Peterson and Co., NY, 1961). [74] V.N. Gribov, EM. Levin and A.A. Migdal, Zh. Eksp. Teor. Fiz. 59 (1970) 2140. [75] Yu.M. Kagan and MI. Klinger, I. Phys. C7 (1974) 2791. [76] Yu.M. Kagan and L.A. Maximov, Zh. Eksp. Teor. Fiz. 65(1973) 622; 84 (1983) 792; 87(1984)348 [Soy. Phys. JET? 38 (1974) 307; 57 (1983) 459; 60 (1984) 201]. [77] A.A. Gogolin, Pis’ma Zh. Eksp. Teor. Fiz. 32 (1980) 30 [JETP Lett. 32 (1980) 28]. [78] A.A. Gogolin, Phys. Stat. SoI. b103 (1981) 397; Preprint L.D. Landau Institute N 1980-14 (Chernogolovka, 1980). [79] E.M. Lifshits and L.P. Pitaevsky, Physical Kinetics (Nauka, Moscow, 1979). [80] A.F. Andreev, Usp. Fiz. Nauk 118 (1976) 251 [Soy. Phys. Usp. 19 (1976) 137]; A.F. Andreev and I.M. Lifshits, Zh. Eksp. Teor. Fiz. 56 (1969) 2057 [Soy. Phys. JETP 29 (1969) 1107]. [81] K. Funke, Progr. in Solid State Chem. 11(1976)345. [82] Yu.A. Gureyich and Yu.I. Kharkats, Phys. Rep. 139 (1986) 203. [83] F.G. Bass and A.P. Tetervov, Phys. Rep. 140 (1986) 237. [84] V.P. Seminozhenko, Phys. Rep. 91(1982)103. [85] R.A. Hopfel, I. Shah and A.C. Gossard, Phys. Rev. Lett. 56 (1986) 765; R.A. Hbpfel, I. Shah, AC. Gossard and W. Wiegmann, Physica 134B (1985) 509. [86] A.A. Gogolin, Phys. Rep. 86 (1982) 1. [87] LV. Keldysh, Zh. Eksp. Teor. Fiz. 43 (1962) 661. [88] Yu.A. Bychkov and AM. Dykhne, Zh. Eksp. Teor. Fiz. 48 (1965) 1168 [Soy. Phys. JETP 21(1965) 779]. [89] G.H. Wannier, Phys. Rev. 117 (1960) 432. [90] AM. Berezhkovsky and A.A. Ovchinnikov, Phys. Stat. Sol. b107 (1981) 157, 299. [91] R.A. Suns, Fiz. Tekn. Poluprov. 7 (1973) 154. [92] LV. Keldysh, Zh. Eksp. Teor. Fiz. 33 (1957) 994. [93] A.A. Gogolin, Solid State Commun. 54 (1985) 409. [94] A.A. Gogolin, Zh. Eksp. Teor. Fiz. 88 (1985) 2063 [Soy. Phys. JETP 61(1985)1220]. [95] A.A. Gogolin, Preprint L.D. Landau Institute No 1985-1 (Chernogolovka, 1985). [96] A.A. Gogolin, Preprint L.D. Landau Institute N 1985-5 (Chemogolovka, 1985). [97] lB. Gunn, Solid State Commun. 1(1963) 88; IEEE Trans. Electron Devices 10 (1966) 30; 14 (1967) 720. [98] BK. Ridley and TB. Watkins, Proc. Phys. Soc. 78(1961) 293; BK. Ridley, Proc. Phys. Soc. 82 (1963) 954; BK. Ridley and R.G. Pratt, Phys. Lett. 4 (1963) 300; 1. Phys. Chem. Solids 26 (1965) 21; C. Hilsum, Proc. IRE 50 (1962) 185. [99] B.W. Knight and GA. Peterson, Phys. Rev. 155 (1967) 393. [100]SI. Anisimov, VI. Mel’nikov and El. Rashba, Pis’ma Zh. Eksp. Teor. Fiz. 7 (1968) 253 [JETP Lett. 7 (1968) 196]. [101]V.L. Bonch-Bruevich, I.?. Zvyagin and AG. Mironoy, Domain Electrical Instability in Semiconductors (Nauka, Moscow, 1972). [102]ME. Levinshtein, Yu.K. Pozhela and MS. Schur, The Gunn Effect (Soy. Radio, Moscow, 1975). [103]V.V. Vladimirov, A.F. Volkov and E.Z. Meilikhov, Semiconductor Plasmas (Atomizdat, Moscow, 1979). [104]V.V. Stepanov, Course of Differential Equations (Nauka, Moscow, 1966). [105]N.N. Bogolyubov and Yu.A. Mitropol’sky, Asymptotic Methods in the Theory of Nonlinear Oscillations (Nauka, Moscow, 1974).

A. A. Gogolin, Polaron transport in quasi-Id organic conductors and in some narrow band systems [106]A.A. Andronov, A.A. Vitt and S.E. Khaikin, Theory of Oscillations (Nauka, Moscow, 1981). [107]AS. Davydov, Solid State Theory (Nauka, Moscow, 1976). [108]Al. Kitaigorodsky, Molecular Crystals (Nauka, Moscow, 1971). [109]L.D. Landau and E.M. Lifshits, Quantum Mechanics (Nauka, Moscow, 1974). [110]V.E. Zakharov, SY. Manakov, S.P. Noyikov and L.P. Pitaevsky, Theory of Solitons (Nauka, Moscow, 1980). [111]J. Zittartz and IS. Langer, Phys. Rev. 148 (1966) 741; B.I. Halperin and M. Lax, Phys. Rev. 148 (1966) 722; I.M. Lifshits, Zh. Eksp. Teor. Fiz. 53 (1967) 743. [112] I.M. Lifshits, S.A. Gredeskul and L.A. Pastur, Introduction to the Statistical Theory of Disordered Systems (Nauka, Moscow, 1981). [113]B.I. Halperin, Phys. Rev. A139 (1965)104; Ady. Chem. Phys. 13 (1967) 123. [114]A.A. Gogolin, Zh. Eksp. Teor. Fiz. 83 (1982) 2260. [115] A.A. Gogolin, Z. Phys. B52 (1983) 19; B58 (1985) 205. [116]A.S. Davydov and V.Z. Enol’sky, Zh. Eksp. Teor. Fiz. 81(1981)1088. [117]A.A. Gogolin, Zh. Eksp. Teor. Fiz. 85 (1983) 1131. [118]A.A. Gogolin, Solid State Commun. 47 (1983) 753. [119]MI. Cohen, L.B. Coleman, A.F. Garito and A.J. Heeger, Phys. Rev. BlO (1974) 1298. [120]M.F. Deigen and SI. Pekar, Zh. Eksp. Teor. Fiz. 21(1951) 803. [121]El. Rashba, Izv. Soy. Acad. Nauk Ser. Fiz. 40 (1976) 1793. [122]D. Moses, A. Denenstein, A. Pron, Al. Heeger and A.G. Mac Dianmid, Solid State Commun. 36 (1980) 219. [123]E. Yanke, F. Emde and F. Lösch, Special Functions (Nauka, Moscow, 1977). [124]A.S. Davydov, Quantum Mechanics (Nauka, Moscow, 1969). [125]L.D. Landau and EM. Lifshits, Theory of Elasticity (Nauka, Moscow, 1965). [126] D. Jerome, J. Physique (Paris) 44 C3 (1983) 775. [127] Yu.M. Kagan and NV. Prokofley, Zh. Eksp. Teor. Fiz. 9 (1986) 2176. [128] S.A. Bulgadaev, Zh. Eksp. Teor. Fiz. 90 (1986) 634. [129] A. Schmid, Phys. Rev. Lett. 51(1983)1506. [130] P.M. Grant and I.P. Batra, Solid State Commun. 29 (1979) 225. [131] K. Holczer, J.P. Boucher, F. Devreux and M. Nechtschein, Phys. Rev. B23 (1981) 1051. [132] G.I. Thorbergsson and J. Sak, Phys. Lett. A116 (1986) 325; J. Phys. C19 (1986) 1.615. [133]IS. Gradshtein and I.M. Ryzhik, Tables of Integrals (Nauka, Moscow, 1971). [134]V.V. Sokolov and A.B. Shabat, Soy. Sci. Rev. C4 (1984) 221. [135]E.A. Kuznetsov, AM. Rubenchik and yE. Zakharov, Phys. Rep. 142 (1986) 104. [136]G. Venzl and S.F. Fischer, Phys. Rev. B32 (1985) 6437. [137]A.A. Gogolin, VI. Mel’nikov and El. Rashba, Zh. Eksp. Teor. Fiz. 69 (1975) 327.

391