Linearized quantum transport equations: ac conductance of a quantum wire with an electron-phonon interaction

Transcription

1 PHYSICAL REVIEW B VOLUME 53, NUMBER APRIL 1996-II Lineaized quantum tanspot equations: ac conductance of a quantum wie with an electon-phonon inteaction Pet Kál Institute of Physics, Academy of Sciences, Na Slovance, Paha 8, Czech Republic Received 6 Septembe 1995 A linea esponse fomalism is developed fo evaluation of tanspot coefficients in quantum many-paticle systems. The method is based on a systematic lineaization of nonequilibium Kadanoff-Baym tanspot equations in an integal fom in ac electic fields. Simple and consistent appoximations can be pefomed in the esulting set of tanspot equations, which eplace vetex equations in the Kubo fomalism. The method is illustated by the example of an ac and dc conductivity and conductance fo a quasi-one-dimensional electon system with an electon phonon inteaction fo thee-dimensional acoustical phonons. Afte appoximations standad in metals, the solved tanspot equations coincide with the Holstein equation. I. INTRODUCTION Tanspot phenomena in quantum many-paticle systems, in paticula with a esticted geomety, cannot be fully undestood without quantum tanspot theoy, which would be simple and eliable, but at the same time poweful enough to avoid inappopiate appoximations. The nonequilibium Geen s functions fomalism NGF of Kadanoff and Baym 1 and Keldysh can faithfully descibe quantum systems in nonequilibium. 3 Unfotunately, the complex stuctue of the theoy involving double time stuctue was slowing down an application of NGF. Theefoe, much effot has been devoted to educe NGF in stongly nonequilibium systems to a single time tanspot theoy. 4 Vey successful ae applications of the NGF to a deivation of linea esponse fomalisms of a geneal validity, whee usually the double time stuctue can be avoided. In NGF the Dyson equation in complex times and in the pesence of extenal fields is analytically continued to eal times. 5 This gives a set of tanspot equations fo nonequilibium coelation functions in eithe a diffeential 1 o an integal fom. 6 Lineaization of these tanspot equations in tems of the extenal fields poduces new equations, fom which the linea esponse coefficients can be found. In equilibium systems these theoies can be compaed with the complementay appoach of the Kubo fomula, 7 which expesses the linea esponse in tems of a special type of a two-paticle Geen s function. One way of its actual detemination is to lineaize the one-paticle Matsubaa Geen s functions 8 with espect to the extenal fields. This gives a Bethe-Salpete equation 9 fo the two-paticle Geen s function of complex times. Its solution can be, afte an analytic continuation to eal times, identified with the Kubo fomula with minimal vetex coections. 9 Seveal woks followed the way of lineaized NGF with the goal of developing a simple consistent scheme of incopoating appoximations fo caie scatteing. Lineaized Kadanoff-Baym tanspot equations in a diffeential fom fo dc electic fields have been used by Pange and Kadanoff 10 fo an electon-phonon inteaction. Ha nsch and Mahan 11 have elaboated and applied this appoach. Thei esults epoduced coectly the Holstein equation 1 fo the electonphonon scatteing in the Migdal appoximation. Chen and Su 13 have tied to augment this method to achieve a pope bookkeeping of the many tems appeaing as a esult of the appoximate lineaization of the diffeential fom. This extension is unavoidable fo moe complicated types of scatteing. Anothe genealization of the method 11 is to the case of the ac electic fields, by Wu and Mahan. 14 The last method does not explicitly coincide with the wok 11 in the dc limit 0, but the Holstein equation 1 the Boltzmann limit has been acquied too. A gauge invaiant fom of these diffeential equations has been ecently pesented by Levanda and Fleuov, 15 whee the ac equations coincide in the limit 0 with the dc equations. In this wok we lineaize quantum tanspot equations in the integal fom. 16 This way is pobably moe flexible than the diffeential methods, 11,15 because some intemediate steps can be avoided and the esulting system of equations appeas simple. The integal appoach is also inheent to the Kubo fomula, but in ou method one and two-paticle points of view ae moe clealy inteconnected. This opens a possibility to easily pefom consistent appoximations. When testing equilibium systems, the new method seems to agee with the Kubo fomula fo minimal vetex coections. 9 The appoach can be applied in stationay nonequilibium conditions, and its possible extension to tansient nonequilibium situations is ensued by its NGF oigin. We apply the new method to evaluate ac and dc conductivity and conductance fo a quasi-one-dimensional 1D electon system, embedded in a thee-dimensional 3D semiconducto and inteacting with bulk acoustical phonons. 17 Afte standad appoximations fo metals, the method gives the Holstein equation 1 fo this model. Its numeical solutions show that below T1 K scatteing of electons in a backwad diection feezes out and the homogeneous dc conductivity, given solely by scatteing in a fowad diection, goes like T 3. Simila phenomena appea also in the homogeneous o inhomogeneous ac conductivity. Fom the conductivity we evaluate an ac/dc conductance, defined by an absobed powe in a locally excited pat of the wie. 18 The one-channel dc conductance acquies the stan /96/5316/ /$ The Ameican Physical Society

2 53 LINEARIZED QUANTUM TRANSPORT EQUATIONS: ac dad value e /h as T 0 K. Above T1 K this conductance shaply falls down, due to the pesence of the backwad scatteing. Hee the ac conductance gows as a function of a fequency until the backwad scatteing becomes distubed. The pape is oganized as follows. In Sec. II we develop the new linea esponse method fom the quantum tanspot equations in the integal fom. We lineaize these equations and explain how tanspot coefficients can be evaluated fom the lineaized tanspot equations in a fequency-momentum epesentation. In Sec. IV we apply the new method to evaluation of an ac/dc conductivity and conductance fo a quasi-1d electon system with an electon-phonon inteaction. In this section some numeical esults ae pesented, which might be of impotance fo mesoscopics. In Appendix B we show how consevation laws esult fo the lineaized functions in the new method. The equivalence of the new lineaized tanspot equations with the Holstein equation is also pesented thee fo ou model. Finally elaxation times ae evaluated fom the solutions of the lineaized tanspot equations. II. LINEARIZED KADANOFF-BAYM EQUATIONS IN THE INTEGRAL FORM Conside an inteacting electon system excited by an electomagnetic field. This system can be descibed by the Hamiltonian HH s H ext, H ext e d 3,t e c d 3 A,t j e mc d 3 A,t, whee H s is the Hamiltonian fo the fee system and H ext epesents coupling to the fields, descibed by the scala (,t) o the vecto potential A(,t). In the following we will assume that the fields ae weak, so that without loss of geneality the potentials can be used in the foms,t 0 e itk t, A,tE 0 c eitk t 0. i The density () and the cuent density opeatos j() ae 9, j im. 1 3 U1,t 1 t e1 e cim A1 1 e mc A 1, whee the numbes j1, epesent a pai of coodinates ( j,t j ). The fields ae chaacteized by the opeato function U(1,), which is implicitly pesent also in the selfenegy (1,) some GF elationships ae summaized in Appendix A. A. Fomulation of the equations Linea esponse coefficients can be pecisely calculated fom the Bethe-Salpete equation 9 Kubo fomula o the diffeential fom of the Kadanoff-Baym equations. 1 Ou stating point in the lineaization scheme is the integal vesion of the Kadanoff-Baym tanspot equations A10. The NGF oigin of these equations allows thei application to stongly nonequilibium systems. Hee fo simplicity we conside that the studied system is in stationay nonequilibium conditions, maintained by the extenal fields and by a contact with esevoi. Then initial condition contained in the function G 0 dies out and the nonequilibium tanspot equations A10 become simplified as follows: G 1,G 1,3 3,4 G a 4,. The equation fo G esults by the exchange of by in 5, so that only the equation fo G (G /U) will be witten as a epesentative of the pai of equations fo G, G (G /U,G /U). The nonequilibium popagatos G,a (1,) can be found fom G, 1, by A3. In this wok we conside that the weak pobe fields test an equilibium system. Tem by tem vaiation of 5 ove these fields o A gives a new set of equations G 1,1 U pˆ UG 1,G,1G 1,G a,1 G 1,3 3,4 G 4,1 U G 1,3 a 3,4 G a 4,1 U G 1,3 3,4 G a 4,1. U In 6 the functional deivative of G (a) was substituted by the identity Fo ou puposes it is appopiate to descibe the system by Geen s functions, time odeed along a cuve in the complex plane 5 genealization of the Matsubaa GF. The electon Geen s function follows a Dyson equation in the integal o diffeential fom. Its inveted solution looks like G 1 1,G 0 1 1,U1,1,, 4 G 1,1 pˆ U UG 1,G,1 whee the fomulas G 1,3 3,4 G 4,1, U 7

3 P. KRÁL 53 G 1,3 U,4 G 1,5 G 1 5,6 G 6,3 8 U,4 and 4 wee used. The opeato is pˆ U1, (/im) ( ) fo Ue, (e/c)a, and the nonlinea tem in A is neglected due to weakness of the pobe fields. The single paticle functions G,,,a in 6, 7 ae taken in equilibium. The lineaized popagatos G,a /U could also be expessed fom the lineaized coelation functions G, /U in an equivalency analogical to that between the popagatos G,a and the coelation functions G, see Appendix A G 1,1 i U 11 G 1,1 G 1,1. U U 9 If the expessions fo the lineaized coelation functions fom 6 and its countepat G /U ae inseted into 9, then, afte some algeba with theta functions, the ight side of 7 esults. Theefoe 7 and 9 ae equivalent to each othe and ae consistent with the tanspot equations 6. Fom the identity 9 an impotant elation between the lineaized functions esults. If we take into account that G (1,1)1, then it follows that G 1,1 i U 1 1 A1,1 U 0, 10 G 1,1 G 1,1. U U In othe wods as 1 1 the two lineaized coelation functions degeneate into one. The linea esponse coefficients can be found fom the function G (1,1)/U(). To obtain this function of one agument (1) fo a stationay and homogeneous system, the moe geneal function G (1,3)/U() with two aguments (13),(1) must be found. Theefoe the equivalency 10 usually does not educe the numbe of solved equations, but in some systems it gives a hint to a possible simplification. The coupled system of tanspot equations 6 and its countepat fo G /U will be closed, if all functions ae expessed by G, /U. This can be achieved in two steps. Fist the functions,a /U should be epesented by, /U in a elation analogous to A6 and A7, holding between,a and,. Second the functions, /U must be expessed by G, /U in the same way as elate, and G, fo a given many-body appoximation. Let us see these elationships in moe details. Geneally the electon self-enegy is a sum of the singula Hatee- Fock o mean field pat HF and the egula o collisional pat 1 c. Both pats contibute to the self-enegy popagatos, but only the egula pat has nonzeo contibution to the coelation function of the self-enegy 1,3 HF 1,3 c 1,3, 1,3 c 1,3. 11 The lineaized self-enegy function fulfills the same elationships 1,3 U HF 1,3 c 1,3, U U 1,3 c 1,3. U U 1 The Hatee-Fock self-enegy, which is a mean field appoximation local in time of the electon-electon inteactions, has the etaded pat equal to HF 1,3t 1 t d 3 v 1 G,t 1 ;,t 1 t 1 t 3 v 1 3 G 1,t 1 ; 3,t 1. This popagato self-enegy can be lineaized as follows: HF 1,3 t U 1 t d 3 v 1 G,t 1 ;,t 1 t U 1 t 3 v G 1,t 1 ; 3,t U The collisional contibutions fulfill the same elationships as the elated self-enegy pats, namely A6 and A7 c 1,3 i U h t 1t 3 c 1,3 c 1,3. U U 15 Geneation of linea esponse functions by a functional deivative is known to lead in a safe way to many-body conseving appoximations 1 consevation laws ae fulfilled 9. To ensue this mandatoy consistency of the tanspot equations in the pesence of appoximations, the same appoximative steps must be pefomed in, and, /U. This holds not only fo the appoximations, which esult by selecting an appopiate class of Feynman diagams in the self-enegy, 19 but also fo auxiliay physically motivated appoximations. This makes it possible to keep unde contol a self-consistent stuctue of the one- and two-paticle levels of the solution. Some of these appoximations ae pesented in the next section. B. Fequency epesentation Weak pobe fields can be esolved into Fouie components, which can be consideed to act on the studied system independently. A linea esponse to composite weak fields can be calculated fom a linea supeposition of esponses to Fouie components of these fields. Since the weakly excited system emains space and time tanslationally invaiant, it is enough to know any lineaized function

4 53 LINEARIZED QUANTUM TRANSPORT EQUATIONS: ac F(1,3)/U(), fom the tanspot equations 6, fo the diffeences of aguments (13), (1). Ou esults can be moe simply compaed with the wok of Holstein, 1 if we define the Fouie tansfom in a little unsymmetic way F1,3 U d 1 d dk 1 dk F U 1,k 1 ;,k exp i 1 t 1 t 3 i t 1 t ik ik Intenal aguments of the functions ( 1,k 1 ) ae analogical as in G(k,). Extenal aguments (,k ) ae identical with those in the fields. Application of the tansfom 16 to Eqs. 6 gives G U k 1, 1 ;k, G k 1 k, 1 p U U k 1, 1 ;k, G k 1 k 1, G k 1 k, 1 p U a U k 1, 1 ;k, G ak 1 k, 1 G k 1 k, 1 U k 1, 1 ;k, G ak 1 k, 1, 17 whee the functional deivative of G was substituted by G U k 1, 1 ;k, G k 1 k, 1 p U U k 1, 1 ;k, G k 1 k, 1 18 and the momentum is p U 1, k 1 /m fo Ue,(e/c)A. Late we will also use the elationship 15 in a fequency epesentation c U 1 ; 1 d 1 i c U ; c U ;. Hee we have witten only the fequency vaiables. C. Tanspot coefficients 19 Tanspot coefficients can be easily calculated fom the solution of the tanspot equations 17. Fo example the total cuent density induced by weak pobe fields esults by aveaging of the velocity of electons ove the petubed state of the system 0 J,te j,t e mc A,t J co,tj diam,t d 3 dt,tt E,t, whee the definitions 3 have been applied. Aveaging of the cuent density opeato j gives the coelation cuent J co and aveaging of the poduct of the vecto potential A with the density opeato gives the diamagnetic cuent J diam. The coelation cuent can be expessed in a fequencymomentum epesentation as follows: J co k,e d3 k k d 3 m G k, e d3 k k d 3 m G 0 k, G e c k, ;k, A e c Ak,. 1 Combination of the fomulas 0 and 1 togethe with the elation between fequency components of the electic field and vecto potential E(i/c)A gives the components of the tenso fo ac conductivity 0

5 P. KRÁL 53, k, i n 0e m,, k,,, k, ie d3 k k 3 m d G e c k, ;k,. A Hee n 0 is the concentation of caies and the multiplication by esults fom the summation ove spin components. In many situations only the diagonal elements of the conductivity tenso ae nonzeo. Anothe physical popety which is often equied is the polaizability of a quantum many-body system. 0 It can be calculated fom the tanspot equations 17 as the linea esponse to the scala potential. The polaizability diffes fom the coelation pat in by deletion of the momentum pefacto and the substitution of G /A by G /. In the Appendix B we show how consevation laws ae fulfilled by the linea esponse functions G /A and G /. The question emains whethe the esulting linea esponse coefficients ae equivalent to those found by the Kubo fomula. To answe this question in full geneality would be quite had, because diffeent analytical stuctues in both methods lead to diffeent diagammatic expansions. Theefoe moe wok is necessay in this diection in the futue. III. ELECTRON-PHONON INTERACTION IN QUASI-1D In ode to see how the lineaized tanspot equations 17 can be applied, we find an ac/dc conductivity and conductance fo a quasi-1d electon system. The model has been intoduced and studied in Ref. 17, whee one-paticle popeties and the nonvetex conductivity and conductance have been investigated. Late a multiband extension of the model was also pesented. 1 In ou wok we evaluate vetex coections to the conductivity fom Ref. 17 by the new linea esponse method. The system is a single-quantum-well wie fomed in a bulk semiconducto. The wie is extended in the z diection and confined by a otationally symmetic paabolic potential in the x, y diections. The longitudinal motion along the wie can be descibed by plane waves with wave vectos fom a 1D band with a paabolic dispesion elation. The tansvese motion of electons is quantized into hamonic oscillato states. The electons in this quasi-1d system ae coupled to longitudinal 3D acoustic phonons. 0 Hee we discuss fo simplicity only tanspot in the lowest state of the well. This one-band model can be descibed by the modified Főhlich Hamiltonian 1,16 H k k m a k a k q k,ql a k a kql qt sqb q b q Fq t Mqb q b q, 3 Fq t exp 0 q x q y, 4 1 MqV 0 q 1/, cyst sq whee the gound state of the well is chaacteized by the wave function. 4 x,y 1 0 exp xy 0 In 3 s is the velocity of sound, 0 is the diamete of the wie, cyst is the density of matte, and V 0 is the stength of the electon-phonon inteaction. 0 The opeato a k ceates an electon with a 1D wave vecto k in the tansvesal state 4 and the opeato b q ceates an acoustical phonon with a 3D wave vecto q(q x,q y,q zl ), q t q x q y. In this wok the electon-phonon inteaction in 3 as well as the pobing longitudinal field fom ae not sceened. A. One-paticle popeties Electons and phonons in this system can be descibed by 1D-electon and 3D-phonon Geen s functions see the Appendix A. The influence of the quasi-1d electons on the 3D phonons is pobably small, so that the enomalization of phonons is neglected.,1 In expeiments the Femi level is usually high enough, which means that the electons on this level ae much faste than sound. Theefoe the Migdal appoximation of the electon self-enegy should be valid possible beakdown is mentioned late k;t,ti ql Fq t Mq Gkq l ;t,tdq;t,t.,q t 5 Its coelation function can be witten as k, 1 4 d dq l d q t Fq tmq G kq l, D q, d dq l Akq l, Aq l, n F n B, 6 whee we have used the expessions A5 and A9. The function A(q l,) can be found fom the -like spectal function fo 3D fee phonons A D q, q q s /s q q t l s q l. s q l 7 In the second expession of 7 the amplitude of the tansvesal momentum q t has been chosen as a new vaiable.

6 53 LINEARIZED QUANTUM TRANSPORT EQUATIONS: ac TABLE I. Paametes of the model. effective electon mass m m e density bulk GaAs cyst 4560 kg/m 3 velocity of a sound s 50 m/s inteaction constant V 0 7eV diamete of the wie 0 7nm Femi enegy E F 0 mev Substituting the function A D (k,) fom 7 into 6 and pefoming the integation ove q t, the function A(q l,) esults as follows: Aq l,c i exp 0 q s l. s q l 8 The coelations functions, diffe only by statistical factos and the etaded self-enegy can be calculated fom them as in the Appendix A. We have numeically tested the non-self-consistent fom of the Migdal self-enegy, called the fist-ode Tamm- Dankoff TD1 appoximation, which esults fom 6 by application of fee Geen s functions. These tests, with paametes in Table I, show that the magnitude of Im is about 10 mev, but the stuctue in Im has the width of the ode of mev. Theefoe the self-consistent electon spectal function should be shaply localized aound E F, so the enomalization constant is vey close to one, Z , and the Migdal self-enegy could be appaently substituted by the TD1 self-enegy. The tests also show that the TD1 selfenegy 6 depends vey weakly on the wave vecto k, because the momentum dependence fo the effective phonon spectal function A is much weake than that fo the spectal function fo electons A due to the small s/v F atio. Theefoe the TD1 self-enegy can be futhe appoximated by keeping constant the integation vaiable q l in A duing the q l integation. Since the electon dispesion law in 1D has two banches aound kk F ), the q l integation can be divided into two pats. They natually sepaate the scatteing in the fowad and backwad diections, fo which the constants in A can be chosen as q l0 and q lk F. By this appoximation the k vaiable can be integated out fom the electon spectal function A(kq l, ) in 6, sothe coelation function becomes k independent d m F f F b n F n B, 9 F f C i exp 0 s, FIG. 1. Schematic daft of the electon scatteing on acoustical phonons in the quasi-1d system. In the insets we can see contibutions to the imaginay pat of the self-enegy Im, as a function of enegy, fom scatteing in the fowad and backwad diections. A lineaized electon dispesion law has been used and the tempeatue is T1 K.AsT 0 K the Luttinge theoem is fulfilled, so that Im (E F ) 0. The dashed line epesents a deivative of a Femi- Diac distibution at the same tempeatue. F b C i exp 0 s s k F. k F Hee the functions F f () and F b () ae effective scatteing factos in the fowad and backwad diections. The above substitutions peseve inelastic chaacte of the scatteing in this model. They ae analogous to the momentum-indepedent appoximation of a self-enegy, 10 called also the fist quasiclassical appoximation. In Fig. 1 we demonstate scatteing of electons in the fowad and backwad diections. Contibutions fom these two scatteing channels to the self-enegy in 9 ae pesented in the uppe insets fo paametes in Table I. In the fowad scatteing phonons of vey low momenta q l0 and only these can be excited. Theefoe it is Im fowad () 0 even vey closely to the Femi enegy see the ight inset. In a stictly 1D system one-phonon scatteing of electons in the fowad diection is completely fobidden by consevation conditions, so that the fowad scatteing can eappea only in highe odes of a petubation theoy. 3 In the studied quasi-1d model, momentum consevation ules ae boken see 8, so that a weakly inelastic one-phonon fowad scatteing is pesent. In the elastic appoximation the fowad scatteing falls out completely. 4 In the backwad pat Im backwad a gap is pesent aound the Femi level, whee Im backwad 0 at low tempeatues. The gap has a half-width sk F see the left inset, given by scatteing on phonons with momenta q lk F. The distibution function dn F ()/d, fom a conductivity fomula, is localized aound E F and has the half-width k Bol T. Theefoe at low tempeatues this distibution pactically does not ovelap with the backwad self-enegy contibution

7 P. KRÁL 53 Im backwad, so that the backwad scatteing feezes out below the tempeatue T f sk F /k Bol (14 K hee. At lowe tempeatues, tails of the distibution dn F ()/d have only exponentially small ovelap with the backwad pat of Im and the fowad scatteing must dominate hee at T1 K. B. Two-paticle popeties We can find the lineaized tanspot equations 17 fo the pesent model, pobed by the weak longitudinal electic field A(,t)E 0 ce i(tk )t /i (k is paallel to A and to the wie. This vecto potential A(,t) can epesent one Fouie component of the longitudinal electic field, which appeas below a slit iadiated by a lase beam. 17 To keep the appoach self-consistent, the Migdal appoximation of the self-enegy 6 must be applied also in lineaized self-enegy functions fom Eqs. 17. In this many-body appoximation the function /A looks like we use a simplified notation (e/c)a A A k 1, 1 ;k, d dq l G A k 1q l, 1 ;k, Aq l, n B. 30 We have to pefom in /A all othe appoximations applied befoe to.in9 the integation vaiable q l in the function A is substituted by two igid values q l0 and q lk F, fo the fowad and backwad scatteing, and the q l integation is pefomed aound these values. If the same substitutions ae pefomed in 30, then the k 1 -independent function /A esults A 1 ;k, d n B F f G A 1 ;k, F b G A 1 ;k,, 31 whee the k 1 -independent but the sign of k 1 ) lineaized functions ae defined by G dk 1 G A 1 ;k, A k 1, 1 ;k,. 3 Hee the signs mean integation in the neighbohood of the values k 1k F, which coespond to the two banches of the electon dispesion law in 1D. Note that these signs ae evesed in the second tem of 31. The k 1 -independent lineaized popagatos fo a self-enegy,a /A elate to /A by 15. If Eqs. 17 fo the pesent model ae integated as in 3, then a new closed system of equations fo (G, /A )( 1 ;k, ) can be deived. Let us see this diectly. If the above deived k 1 -independent lineaized self-enegy functions in 31 ae substituted in 17, then on the ight side of this equation the vaiable k 1 appeas only in the pais of Geen s functions. Theefoe, when the k 1 integation fom 3 is applied to 17, these pais of Geen s functions can be integated sepaately fom othe tems. These integals of bilinea combinations of popagatos and coelation functions ae pefomed below. To simplify integations of these functions, the dispesion law fo electons E k can be lineaized in the neighbohood of the Femi wave vectos as follows: 0 E k kk f k f C m 0 kk f, 33 C 0 k F m v F. The same appoximation has been done in the self-enegy 9 fom Fig. 1. In fact at the Femi level, whee tanspot is impotant, this appoximation changes the self-enegy vey little. The appoximate popagato esults G 1 k, E k 1 1 C 0 k, k F, 34 C 0 and the functions G a (k,) and G, (k,) can be expessed fom 34. The integals fom bilinea combinations of the appoximate Geen s functions can be easily evaluated I 1 1 ;k, dk 1 G k 1k /, 1 G a k 1k /, 1 1 C 0 dk 1 1 k 1k / 1 1 k 1k /* 1 i C 0 1 C 0 k 1 a 1, I dk 1 1 ;k, G k 1k /, 1 G a k 1k /, 1 in F 1 I 1 1 ;k,, I 3 1 ;k, dk 1 G k 1k /, 1 G k 1k /, 1 in F 1 I 1 1 ;k,. 35

8 53 LINEARIZED QUANTUM TRANSPORT EQUATIONS: ac Afte the selection of one of the values k 1k F, the integations in 35 can be diectly pefomed. Extension of the integation inteval fom one of the banches of the electon dispesion law to the the whole eal axis of k 1 makes pobably only small eos. These integals can be used to define the new set of coupled equations analogously the equation fo (G /A )( 1 ;k, ) esults G A 1 ;k, k F m A 1 ;k, k F m a A 1 ;k, I 3 1 ;k, I 1 ;k, I 1 1 ;k, A 1 ;k,. 36 In 36 the momentum pefacto is appoximated by the Femi momentum and taken out of the integals. Application of the solution of 36 in the fomula, little modified fo the pesent case, gives the ac conductivity k, ie v F d G ;k, A G A ;k,. 37 Note that the coellation pat of the conductivity 37 does not cancel the diamagnetic tem in, due to the appoximations of the integals 35 see also Ref. 0. Fotunately the diamagnetic tem does not influence the expeimentally accessible eal pat of the conductivity. C. Reduction to one tanspot equation In 10 we have shown that G 1,3 U G 1,3 U as 3 1. Afte a Fouie tansfom to the momentum-fequency epesentation (k 1, 1 ;k, ) in 16, this equivalency is not fulfilled by sepaate fequency components, but only by thei integal ove the fequency 1. It is inteesting that the 1 integation could be effectively substituted in this ole by the k 1 integation in 3. If we apply in 36 and its countepat fo (G /A )( 1 ;k, ) the identity 19 and the elations 35 between I 1,,3, then afte some little algeba the mentioned equivalency esults G A 1 ;k, G A 1 ;k,, G,a A 1 ;k, Fulfillment of 38 is a consequence of the quasi-local chaacte of ou inteaction, which gives a vey weak momentum dependent electon self-enegy. This dependence can be neglected and the necessay 1 -enegy integation, leading to 10, can be shifted to the k 1 -momentum integation, due to the unambiguous pole elationship in 34. Relations 38 educe the system of lineaized tanspot equations to one equation, which is deived hee fo the electon-phonon inteaction. By the elationships 38 and the equivalency 19 the lineaized popagatos fo a selfenegy can be expessed fom 31 and its countepat fo /A as follows:,a A 1 ;k, 1 d 1 i d 1 F f 1 G A 1;k, F b 1 G A 1;k,. 39 We can substitute the expessions 39 into the tanspot equation 36, multiply both its sides by the inveted function I 1 ( 1 ;k, ), and eode tems. Then this single tanspot equation has the fom ic 0 C 0 k 1 a 1 G A 1 ;k, ik F m n F 1 n F 1 d 1 n F 1 n F 1 n B 1 F f 1 G A 1 1;k, F b 1 G A 1 1;k, in F 1 n F 1 d 1 1 d 1 F f 1 G A 1;k, F b 1 G A 1;k,. Equation 40 is simila to the Holstein equation, 1 deived fo the electon phonon inteaction with acoustical phonons, but in 40 no limitation to the stength of the inteaction has been applied. The validity of these equations is pobably the same as equations in Ref

9 11 04 P. KRÁL 53 Fo not vey stong inteactions a pole appoximation fo a self-enegy sometimes called the second quasiclassical appoximation, can be applied. Fo weak inteactions the quasipaticle spectum can be substituted by a fee paticle spectum. In 40 this can be pefomed as follows: 1 e k1, (E F ), 1e k1 e k1 q l, and d 1v k dq l C 0 dq l. This substitution diminishes the fequency vaiable 1 and einstalls the integated out k 1 vaiable. It would seem that the fequency vaiable 1 could be integated out fom (k 1, 1 ) at the beginning. Unfotunately, such an appoximation cannot be easonably pefomed, because the self-enegy (k 1, 1 ) is stongly dependent on 1, but weakly on k 1. Substitution of the fee pole values in the a Eq. 40 leads to the following Boltzmann-like equation: e k 1 e k 1 ic 0C 0 k G A e k 1 ;k, ik F m n Fe k1 n F e k1 C 0 dq l n Fe k1 n F e k1 F f e k 1 e k1 q l G A e k 1q l ;k, F b e k 1 e k1 q l in F e k1 n F e k1 d 1 e k1 C 0 dq l F f e k 1 e k1ql G e k1 e k1 q l A G A e k 1q l ;k, F b e k 1 e k1 q l ;k, G A n B e k1 e k1 q l e k1 e k1 q l ;k,. 41 The final fomula fo the conductivity 37 can be changed similaly. We should also pefom in 41 the lineaization of the electon dispesion law, to get an ageement with the appoximation applied at 33. The pefomed appoximations ae summaized in Table II the fist/second ow coesponds to the fowad/backwad scatteing. In the Appendix C a diect compaison shows that the Holstein equation in the pesent system is identical to 41. D. Numeical esults fo two-paticle popeties We have applied the tanspot equations 40 and 41 and the Holstein equation C3 to the pesent poblem. The numeical esults of these equations ae pactically the same, so we do not distinguish between them hee. The esults ae fist discussed in physical tems. To this goal elaxation times can be used see thei definition in the Appendix D. We discuss the aveage time between scatteing events s and the aveage momentum elaxation time p sepaately fo the fowad and backwad scatteing pocesses. Additional indices fowad, backwad ae used at these times, which chaacteize fictitious systems with only one type of a scatteing. In most situations one of the pocesses dominates, so that discussion of sepaate pocesses is easonable. Afte the intoductoy summay some numeical examples ae pesented. TABLE II. Scatteing channels appoximate egions of paametes. k q l e k e kql e kql k F 0 C 0 q l C 0 (kk F q l ) k F k F C 0 q l C 0 (kk F q l ) In Fig. 1 we have seen that magnitudes of the functions Im fowad/backwad (e k ) in the plateaus ae of the same ode. This means that the ates fo elaxing electons fom thei incoming states by fowad and backwad scatteing ae simila. Aveaging of 1/Im fowad/backwad (e k ) with the distibution function df 0 (k)/de k, as in D4, gives the aveage times between scatteing events s fowad/backwad. At high tempeatues the stuctue in Im fowad/backwad (e k ) disappeas and it appoximately holds s fowad s backwad. Below T1 K this appoximate equivalence fails, because the time s backwad diveges 4 like exp(ct 1 ), due to the fact that the distibution function df 0 (k)/de k does not ovelap the plateaus in Im backwad. On the othe hand, the electon momentum is elaxed by the fowad and backwad scatteing vey diffeently. Fowad scatteing does not change the sign of the electon momentum and pactically peseves its magnitude (kk), so the momentum is elaxed vey slowly. Backwad scatteing gives an opposite sign to the momentum and changes its magnitude slightly, so the momentum is elaxed quite fast. The diffeent chaacte of these scatteing pocesses becomes evident in the conductivity o equivalently in the magnitude of the momentum elaxation time p. Aleady at high tempeatues the two scatteing channels have vey diffeent momentum elaxation times p fowad p backwad. At low tempeatues it holds that p backwad exp(ct 1 ), similaly as befoe. These facts can be aanged as follows see also comment at Fig. 1 and Eq. D6. At tempeatues highe than T1 K backwad scatteing dominates, so the total time p is suppessed appoximately to the total time s. Below T1 K backwad scatteing feezes out and fowad scatteing dominates. Theefoe p, given hee pactically only by fowad

10 53 LINEARIZED QUANTUM TRANSPORT EQUATIONS: ac law of a momentum. Simila esults can be found fo the homogeneous ac conductivity (k0,0). We can also biefly comment on the numeical solution of the tanspot equations, solved hee by iteations. The iteations have stated fom the nonvetex solution and the vetex coections have been switched on vey slowly, because the convegence adius is closely appoached duing iteations. We have multiplied the vetex pat in 40 by a numbe, which was vaied in an inteval c ite duing the iteations. It is also suitable to stat the next solution fom the pevious one and not fom the new nonvetex solution. FIG.. The dc conductivity in the tempeatue inteval T0.110 K fo paametes in Table I. We elate the conductivity to a efeence value ef (e /h)l ef, whee the efeence length is l ef 1 m. At highe tempeatues vetex full line and nonvetex dotted line solutions coincide. In the intemediate tempeatues the backwad scatteing feezes out and the vetex conductivity apidly gows. Below T1 K only the fowad scatteing contibutes and both solutions follow a thid powe law T 3. scatteing, gows and becomes much lage than s. The times s and p ae appoximately in the same elationship like the nonvetex and vetex conductivity. If vetex coections to the conductivity ae included in ou model, they ae not vey impotant at high tempeatues, while they completely enomalize the conductivity at low tempeatues. 1. Homogeneous conductivity Let us fist discuss the homogeneous conductivity (k0,). At high tempeatues the fequency width of the distibution function dn F ()/d is boade than the gap in Im backwad (). Theefoe the backwad scatteing contibutes to the conductivity, whee it dominates ove the enomalized fowad scatteing. The nonvetex Kubo fomula gives hee appoximately the same dc conductivity like the tanspot equations. At lowe tempeatues the distibution function dn F ()/d ovelaps only little with the stuctues in Im backwad () and the nonvetex Kubo fomula fails. Below T1 K the distibution function is fully localized inside the gap in Im backwad (), so that the backwad scatteing completely feezes out and the fowad scatteing dominates in the conductivity. 16 In Fig. we pesent the homogeneous dc conductivity (k0,0) in the tempeatue inteval T0.110 K, calculated fom the tanspot equations 41 fo the paametes fom Table I. This figue clealy suppots conclusions fom the pevious paagaphs. At highe tempeatues vetex solution full line and nonvetex solution dotted line coincide. At lowe tempeatues backwad scatteing feezes out and below T1 K, whee the fowad scatteing dominates, the vetex solution is about two odes bigge than the nonvetex solution. Hee both solutions fo the dc conductivity give a low-tempeatue asymptotic behavio of the fom T 3. The pesence of the fowad scatteing at low tempeatues changes an exponential in T behavio of the conductivity exp(ct 1 ), fom feezing out backwad scatteing, 3,4 to a powe law in T. The thid powe is pobably a consequence of a confined geomety, which beaks down the consevation. Inhomogeneous conductivity The inhomogeneous conductivity (k0,) is moe complex. The eal pat of the conductivity Re (k,) isa symmetic function of the excitation wave vecto k, which, fo a nonzeo excitation fequency, is fomed by two peaks located at k es. Thei position is detemined by the Femi velocity v F as v F k es and thei width k can be appoximately elated with the dc tanspot time by p /kv F the width of the nonvetex solution non k fulfills the same elation fo the time s / non kv F ). As the fequency goes down 0, the two peaks join each othe and the total weight below the cuve Re (k,) can change in dependence on the type of a dominant scatteing pocess. If only the fowad scatteing is pesent in the system, then the weight is peseved. If also the backwad scatteing is pesent, then the weight goes down till 0 fo elastically appoximated backwad scatteing 16. Theefoe at high tempeatues backwad scatteing dominates the weight falls down geatly as 0 and deeply below the feezing tempeatue fowad scatteing dominates the weight is peseved. In Fig. 3 we pesent the behavio of the k dependent conductivity as 0, calculated fom the tanspot equations 40. The weight below the cuve Re (k,) is peseved fo the low tempeatue T1 K, while it goes down fo T K, whee the backwad scatteing is pesent. 3. Conductance These esults ae eflected in the ac conductance () do not confuse () with usual symbols fo an electon photon o electon phonon vetex. The conductance could be defined 18 by an absobed powe P() of a locally excited quantum wie in the electic field E(x,) P /, dx Ex,, 4 iad. eg. whee () is the change of the electic potential in the iadiated egion. In a k epesentation the absobed powe can be expessed by the conductivity and the acting field as P 1 dk Rek,Ek,. 43 If the adiation falls only in a space inteval of the wie with a finite length ExE 0 xl/xl/,

11 P. KRÁL 53 FIG. 4. The dc conductance fo the vetex full line and nonvetex solutions dotted line as a function of a tempeatue. As T 0 both solutions appoach the value e /h. Above T1 K the vetex solution shaply goes to a smalle value, detemined by the inelasticity of the backwad scatteing, while the nonvetex solution elaxes vey slowly. Fo a finite length L m a cut off is seen at some tempeatue in both solutions. It esults fom a competition of L with /k( non k), and coesponds to a tansfe in the Ohmic egime. FIG. 3. The k-dependent ac conductivity fo diffeent excitation fequencies and the tempeatue T1, K, calculated fom the solution of tanspot equations. As this fequency gows a esonant peak foms in Re (k,) and moves to highe esonant wave vectos k of the excitation field. As the fequency falls down 0, the suface below the cuve Re (k,) can be educed. In the uppe dawing (T1 Kbackwad scatteing is fozen up and the suface is peseved, while on the lowe dawing (T Kthe backwad scatteing is pesent and the suface is not conseved. then the ac conductance can be expessed as dk Rek, sinkl/ kl/. 44 The souce with the length L has a k spectum localized aound the oigin k0. Theefoe the conductance () in 44 becomes sensitive to the aea below the cuve Re (k,). Fo usual lengths L in ange of micons, the momentum width of the function sin(kl/)/(kl/ ) is lage than that of Re (k,). Theefoe the change of the suface below the cuve Re (k,) as 0see Fig. 3 is eflected in the conductance 44. In Fig. 4 the dc conductance fom 44 is evaluated fo the solution of the tanspot equations 41 o equivalently the Holstein equation C3. The excitation souce of the length Lm is consideed. We can see that the value e /h, chaacteistic fo a Femi liquid, 5 is pactically eached by both vetex and nonvetex solutions as T 0. Nevetheless, ecent theoies and expeiments suppot the Luttinge liquid behavio, which cannot be descibed by the Migdal self-enegy. The onset of a backwad scatteing above T1 K is accompanied by a shap falling down of the vetex solution. This can be undestood as a blocking of the wie by consequent scatteing of electons in backwad diections. The magnitude of the changed conductance is athe given by the degee of inelasticity of the backwad scatteing, than by the stength of this scatteing. At T5 K the nonvetex solution suddenly falls down, because / non k appoaches the excitation length L m. In the vetex solutions this falling is chaacteized by the length /k. Physically, this falling down coesponds to a tansfe fom the ballistic in the Ohmic egime of the conductance. We have also evaluated the ac conductance, in ode to see the influence of high-fequency electic fields on the blocking. The esults ae pesented in Fig. 5. Below T1 K the backwad scatteing is fozen up and the conductance is a flat function of, stating fom the asymptote (0)e /h in Fig. 4. Above T1 K this asymptote falls down, since the backwad scatteing suvives. This falling of () is distubed by the excitation field, if its fe- FIG. 5. The fequency-dependent ac conductance () fo T1, K. In the limit 0 these cuves appoach the dc conductance (0) fom the pevious figue. At low fequencies both cuves fo the ac conductance () ae flat. As the fequency gows, blocking at T K becomes distubed, so that both cuves satuate to the common plateau close to e /h. The nonvetex solution would be concave at all tempeatues as 0.

12 53 LINEARIZED QUANTUM TRANSPORT EQUATIONS: ac quency follows v F k/. Theefoe at T K the ac conductance () is a convex function of fequency as 0. The convexity of () has also been pedicted in capacitance effects, chaacteistic fo esonant tunneling devices. 6 The nonvetex solution, 17 which is a flat concave function of fequency as 0, is not pesented hee. At highe fequencies intefeence dips 17 fom the finite length of excitation L m might appea in all solutions. In expeiments usually a shot channel is connected to boad contacts. Then a conductance defined fom a tansmissivity of this shot channel 7 pobably does not fall down as much as in Fig. 4. This is because, in expeiments, backwad scatteing of electons in the ballistic egime usually takes pat outside of the shot channel. The scatteed electons do not etun back to this shot channel, but spead out in the contacts. As a esult the Femi level in the constiction is not well stabilized against shift by the electic field, and blocking is not possible. In moe ealistic calculations sceening should be also included. Ou nonsceened esults coespond only to the difted pat of the total conductivity. 8 IV. CONCLUSION We have developed 9 a linea esponse method fo quantum many-body systems by lineaization of the nonequilibium Geen s function equations 1 in the integal fom. The lineaized integal equations esult in a simple way than the lineaized diffeential equations, 11 because some tansfoms can be avoided. In the new method also the one- and twopaticle points of view ae vey closely inteconnected. Theefoe the method is moe diect than the Kubo fomula, whee vetex equations of diffeent analytical stuctues must be solved in the fist step. As a demonstation example, we have calculated a linea esponse of a quantum wie with an electon phonon inteaction to longitudinal electic fields. 9 Afte application of standad appoximations fom metals to the lineaized equations, we have obtained the Holstein equation, 1 being a weak scatteing limit of the Kubo fomula in the pesent poblem. The Holstein equation has been solved, and fom its solutions the ac/dc conductivity have been calculated. We have ealized that fo high tempeatues T1 K the solution of the nonvetex Kubo fomula 17 agees with the Holstein equation. At lowe tempeatues the scatteing of electons in a backwad diection feezes out and fo tempeatues T1 K the fowad scatteing completely dominates. Hee both the solution of the nonvetex Kubo fomula and the Holstein equation go like T 3, but the vetex solution of the Holstein equation gives a much lage conductivity. Simila effects have been found in the ac conductivity fo homogeneous o inhomogeneous pobe electic fields. These esults have been applied to evaluation of the ac/dc conductance, calculated fom a locally absobed powe by the system. 18 The one channel dc conductance appoaches its ballistic value e /h below T1 K. Above this tempeatue the dc conductance shaply deceases, since the backwad scatteing becomes active. This blocking of the tanspot by the backwad scatteing is distubed, if fequencies of the excitation field ae bigge than the backwad scatteing ate. We believe that the new method, intoduced and tested hee, can be applied in moe complicated linea esponse poblems. The method could be systematically genealized to pobe stationay nonequilibium 31 o nonstationay systems. 6 A nonlinea esponse of quantum systems could be defined by highe-ode functional deivatives ove extenal fields. ACKNOWLEDGMENTS The autho would like to thank B. Velický fo wide suppot of this poject and fo many stimulating discussions. He is also gateful to P. Lipavský, V.Špička, and J. Mašek fo discussions of the fomalism and many helpful comments. The numeical pat of this wok has been done on CRAY Y-MP EL at the Institute of Physics and on STARDENT in ASCOC Laboatoy, Pague. APPENDIX A: SUMMARY OF GREEN S FUNCTIONS In this appendix we define Geen s functions fo eal times analogously, the Matsubaa Geen s functions can be defined in complex times. Some standad elationships between functions with diffeent analytical stuctues ae also intoduced. The causal time-odeed femion (O) o boson (OA) Geen s functions 0 ae defined by G t 1, i TO1O, j j,t j j1,. A1 Coelation functions ae elated to the causal functions by ig t 1,G 1,O1O, t 1 t, ig t 1,G 1,O O1, t 1 t, A whee the uppe lowe sign applies fo femions bosons. The etaded and advanced Geen s functions ae defined by G 1, i 1G 1,G 1,, G a 1, i 1G 1,G 1,. A3 In space homogeneous systems the Geen s functions depend only on the diffeence 1, so that in the k epesentation they esult as G t k;t 1,t d n e i k G t k;t 1,t i n TOk,t 1 O k,t. A4 Hee the opeato is equal to O(k,t)a k (t) o O(k,t)b q (t)b q fo femions o bosons. In a themodynamic equilibium they depend also on the diffeence tt 1 t, and afte a Fouie tansfom ove times the femion and boson coelation functions ead 1

13 P. KRÁL 53 G k,n F,B A G k,, A5 G k,1n F,B A G k,, whee n F,n B denote the Femi-Diac and Bose-Einstein distibutions 1 n F,B e /kt 1 and the spectal function is defined by A G k, ImG k,g k,g k,. A6 Fom Kames-Konig ules it follows that the etaded femion o boson Geen s function can be calculated fom its spectal function A6 by G k, d A G k, i. A7 Most of the above fomulas hold not only fo the Geen s function but also fo the self-enegy. Fo example the spectal function fo the self-enegy can be epesented by coelation functions fo the self-enegy as in A5. Then the etaded self-enegy can be found fom its spectal function by A7. The nonequilibium Geen s functions ae deived by analytical continuation of the Matsubaa Geen s functions in complex times. 1 In NGF it is often necessay to find the etaded o small pat of a combination of functions in complex times. An example is the poduct A1,iB1,C1,, A8 whee A,B,C ae one-paticle causal Geen s functions o self-enegies and the numbes with bas mean that these pais of coodinates ae integated out. The equied functions can be found by analytical continuation along a complex cuve 5 and eduction to the eal axis A 1,B 1,C 1,, A 1,B 1,C 1,B 1,C a 1,. A9 Othe stuctues can be handled similaly. The integal vesion of the Dyson equation leads to the following equation fo the nonequilibium coelation function equation fo G esults by the exchange of signs and hee: G 1,G 1,3 G 0 1 3,4 G 0 4,5 G 0 1 a 5,6 G a 6, G 1,3 3,4 G a 4,. A10 The nonequilibium popagatos G,a can be found fom G, by A3. APPENDIX B: FULFILLMENT OF CONSERVATION LAWS Hee we pesent an analog of genealized Wad identities 9 fo the lineaized functions G /U, which can be applied to pove consevation laws 31 fo these functions. The deivation is the same as fo the linea esponse function L(1,;3,4), used in standad appoaches. 9,31 Conside that electic and magnetic fields acting on the studied quantum system ae zeo. The scala and vecto potentials, coupled by a gauge tansfom, can still be nonzeo. This equilibium system can be descibed by diffeential equations fo G, in the pesence of the potentials 11 A,t,t,,t 1 c t,t, B1 fo any easonable gauge function (,t). The solution of these equations can be expessed as 9 G, 1,;exp1G, 1,;0exp, B which can be checked by the substitution of B into the equations fo G,. The same solution should esult fom the integal fom of these equations A10. Vaiation of G in B ove the gauge function gives analogical expession esults fo G ) G 1,; 313G 3 0 1,;0. B3 Fom the othe side, fo an infinitesimal change of the gauge, the change of G in the potentials B1 is equal to G 1,; d3 G 1, 1 3 c t 3 3 G 1, A B4 Afte the pe-pats integation in B4, the vaiation esults as G 1,;0 1 G 1, 3 c t G 1,. A3 B5 The two vaiations B3, B5 must be equal. Theefoe we get the following analog of genealized Wad identities: 9 1 c G 1, t G 1, A3 313 G 1,, 0, B6 which can also be intepeted as a numbe chage consevation law, 31 obeyed by the lineaized functions G /U. Similaly, othe laws can be deived. APPENDIX C: COMPARISON WITH THE HOLSTEIN EQUATION In this appendix Eq. 41 is compaed with the Holstein equation, 1 which esults by application of the Kubo fomula to the electon phonon inteaction in metals. Holstein applies the Migdal appoximation of the electon self-enegy,

14 53 LINEARIZED QUANTUM TRANSPORT EQUATIONS: ac but does not exclude its momentum dependence. His equations esult fom pole appoximations in a vetex equation, which can be then inveted to a tanspot-like fom. This equation can be identified with a lineaized Boltzmann equation in the limit of small extenal fequencies. In the Holstein wok 1 the momentum k is bound to the pole enegy e k as (k,(e k )/). Then the tanspot equation can be witten as follows: i v k q k, e k a k, e k k v k i ql F s k,kq l kql, C1 whee the function F s in 61 esults fom the vetex equation. Fo ou poblem F s eads F s k,kq l qt n Fe kql n B q 1n F e kql n B q V q e k e kql q i e k e kql q i n F e kql n B q e k e kql q i 1n F e kql n B q e k e kql q i d Aq n F e kql n F e kql P l, e k e kql i e k e A kql q l, n B e k e kql n F e kql n F e kql. C In the second expession some algeba has been used, which togethe with summation of the tansvesal momentum q t in the poduct qt V(q) ( ) gives the effective spectal distibution 8. Theefoe the integation ove in C should be pefomed at the end. As a esult the above fomulas and the following equation C3 look athe diffeently than in Ref. 1. To explicitly compae the equations C1 and 41, we must pefom in C1 appoximations analogical those done in 41. When the pole substitution 1e k1 e k1 q l is pefomed in the exponent of the function A fom 8, then in this exponent the momentum q l appeas in two places (e k1 e k1 q l )/s q l. This function is appoximated by giving the momentum q l at the end of this expession (q l ) the value q l 0(q l k F ) fo the fowad backwad scatteing. The same substitutions can be pefomed in the exponents of the expession C. Afte these appoximations of A, the expession C can be substituted in the Holstein equation C1. We can also neglect the small diffeence between ou (e k )/ and k,(e k )/ in C1. Then the appoximated equation esults i e k 1 C 0 k a e k 1 e k 1 ;k, v k dq l F f e k 1 e k1 q l n F e k1 q l n F e k1 q l n B e k1 q l e k1 e k 1q l ;k, F b e k 1 e k1 q l i dq l n Fe k1 q l n F e k1 q l d 1 e k1 e k1 q l F f e k 1 q l e k 1q l ;k, ;k, F b e k 1 q l ;k,. C3

15 P. KRÁL 53 The elation between the distibution function in Ref. 1 and hee is k (e k )/, whee k esp. k coespond to k0 esp. k0. The conductivity can be found as in Ref. 1 e k ;q, q,e k dk v k n Fe k n F e k. C4 The lineaization of the electon dispesion law fom Table II must be also pefomed in C3 and C4. In ode to show that Eq. C3 is identical to the lineaized tanspot equation 41, it is necessay to compae all tems fom these equations. The following identification can be found eithe fom 41, C3 o fom the conductivity fomulas 37, C4: G C 0 A e k 1 ;k, i e k 1 ;k, n F e k1 n F e k1. C5 If this substitution C5 is pefomed in 41, and Eq. C3 is multiplied by in F (e k1 )n F (e k1 ), then both these equations can be diectly compaed. Since the left sides ae evidently the same, we have to compae the ight sides. Let us compae fist the leading fist tems of the ighthand sides in these equations. If the seemingly diffeent statistical factos ae equivalent, then these tems ae identical. These statistical factos in the fist tems of the ight-hand sides of 41 and C3 look as follows: n Fe k1 n F e k1 n B e k1 e k1 q l n F e k1 q l n F e k1 q l, n Fe k1 q l n F e k1 q l n B e k1 q l e k1 n F e k1 n F e k1. C6 The equivalency of these factos can be poven in seveal steps. The expessions in C6 can be multiplied by all diffeent denominatos with exponentials fom the distibutions n F,n B. Then all tems with diffeent powes of exp (e k1 e k1 q l )/ can be compaed. Afte some algeba the equivalency of factos in C6 esults. The last enomalization tems in Eqs. 41 and C3 have to be compaed in a diffeent way, because the distibution functions have diffeent aguments. Afte a detailed analysis, whee the same integation vaiables ae chosen, the two tems esult as equivalent, too. APPENDIX D: EVALUATION OF RELAXATION TIMES In pactice the elaxation time appoximation of the Boltzmann equation is boadly employed. 0,3,33 In this Appendix we would like to mention this appoximation in the context of linea esponse methods. The elaxation time appoximation can be intoduced in the Boltzmann equation as follows: 0 f k v f k k t t kf k f k t collis. f k f 0k. p k D1 In systems which ae close to equilibium the time p (k) can be identified with the equilibium tanspot time o equivalently the momentum elaxation time. 33 Then the two ight sides in D1, i.e., the exact scatteing integal and the elaxation time appoximation with p (k), give in most situations equivalent desciptions of the Boltzmann equation. Theefoe close to equilibium the time p (k) can be diectly evaluated fom the equation D1, as descibed below. In systems with an isotopic dispesion law fo electons the electon momenta change in a static homogeneous electic field as k/tee. Theefoe the solution of D1 is of the fom f k f 0 k e pke k f k f 0 k e pke k f 0 k. D The second expession in D is an Ansatz fo the distibution function f (k), which assumes that the electic field E is weak. 0,3 If this Ansatz function f (k) is substituted into the exact scatteing integal f (k)/t collis. in D1, then the time p (k) can be evaluated fom D1. Holstein has applied this appoach when compaed his fomulas 1 with the lineaized Boltzman equation. Thee the function k is consideed instead of p (k). The tanspot time p (k) is diffeent fom the time between scatteing events s (k), which is an invese imaginay pat of a self-enegy in a pole appoximation, multiplied by two. An expession fo s (k) can also esults fom Eq. D1, if outcoming tems ae peseved thee, but incoming tems ae neglected in the scatteing integal 3 ( the nonvetex Kubo fomula. Fo elastic scatteing the two times diffe by an impotant facto 1cos1k k/k, aveaged ove the outcoming momenta k with the squae of the matix element. 0 This diffeence expesses the fact that vetex coections fo a conductivity ae included in evaluation of p (k) but not in s (k). The lineaized Ansatz D can also be diectly used fo evaluation of an induced cuent in the weak field 0 E

16 53 LINEARIZED QUANTUM TRANSPORT EQUATIONS: ac Jen 0 en 0 d3 k 3 fk k m e n 0 d3 k 3 p kv k E v k df 0k de k. D3 Fom D3 a homogeneous dc conductivity in the elaxation time appoximation can be defined 0 e n 0 3 d3 k 3 v k p k df 0k de k. D4 In a fee electon metal at zeo tempeatue only the electons on the Femi suface contibute to the tanspot. Then the dc conductivity D4 gets a simple fom 0 e n 0 p k F. D5 m At nonzeo tempeatues p (k) is aveaged by the distibution function df 0 (k)/de k in D4, so that the time p (k F )in D5 should be substituted by the aveage tanspot time p. Only at vey high tempeatues might the aveaging again become tivial. Thee p (k) is usually weakly k dependent and it can be taken out of the integal D4. At high tempeatues also the vetices could be less impotant, since scatteing into all angles should become possible fo enegetical easons. Theefoe the substitution of p (k) by s (k) might be easonable. Fom the similaity of the fomula D4 with 37, it follows that an effective tanspot time p (k) can be defined diectly fom the Holstein equation C3 o fom ou tanspot equations 41 see C5. In homogeneous dc electic fields this time p (k) esults v k 0 1 p k k ic 0 lim v k n F e k n F e k A e k ;0,. G D6 If the nonvetex solution is used in the expessions D6, then the elaxation time between scatteing events s (k) esults instead of p (k). We can biefly discuss also the homogeneous ac conductivity, which is often descibed by a Dude fomula. This fomula applies the elaxation time appoximation in ac fields as follows: e n 0 3 d3 k 3 v k n 0e m p 1i p. p k 1i p k df 0 e k de k D7 The fist expession in D7 can be obtained fom the Boltzmann equation D1 in the same way as D4. The ac Ansatz diffes fom D by the multiplicative facto 1/1i p (k) in the ight side of D. In a moe exact appoach the second expession in D7 should be witten as p (k)/1i p (k)]. Numeical esults fo the ac conductivity seem to fulfill the Dude fomula D7 fo some time p (k). Theefoe we could ty to define the new ac elaxation time fom D7 similaly as in D6 the condition is kt) p k 1i p k k v k ic 0 v k G 1 n F e k n F e k A e k ;0,. D8 Hee the solutions of tanspot equations in homogeneous ac electic fields should be substituted. The elaxation time could depend also on the extenal fequency p (k,). 0 The definition D8 is not applicable, when p (k,) appeas to be complex o the imaginay pat is elatively big. Then the Dude-like fomula is not valid. 1 L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics Benjamin, New Yok, 196. L. V. Keldysh, Zh. Éksp. Teo. Fiz. 47, Sov. Phys. JETP 0, NGF was applied in many aeas of quantum tanspot; examples ae C. Caoli, R. Combescot, P. Nozieies, and D. Saint-James, J. Phys. C 5, 1197; D. C. Langeth and J. W. Wilkins, Phys. Rev. B 6, ; K. Hennebege and H. Haug, ibid. 38, ; H. Shao, D. C. Langeth, and P. Nodlande, ibid. 49, ; 49, P. Lipavský, V.Špička, and B. Velický, Phys. Rev. B 34, D. C. Langeth, in Linea and Nonlinea Electon Tanspot in Solids, edited by J. T. Devesee and E. van Boen Plenum, New Yok, A. P. Jauho, Phys. Rev. B 3, R. Kubo, J. Phys. Soc. Jpn. 1, T. Matsubaa, Pog. Theo. Phys. Kyoto 14, G. Stinati, Riv. Nuovo Cimento 11 1, R. E. Pange and L. P. Kadanoff, Phys. Rev. 134, A W. Ha nsch and G. D. Mahan, Phys. Rev. B 8, T. Holstein, Ann. Phys. 9, L. Y. Chen and Z. B. Su, Phys. Rev. B 40, J. W. Wu and G. D. Mahan, Phys. Rev. B 9, M. Levanda and V. Fleuov, J. Phys. Condens. Matte 6, The pesent pape foms a majo pat of the Ph.D. thesis of P. Kál, Institute of Physics, Pague, V. Špička, J. Mašek, and B. Velický, J. Phys. Condens. Matte, D. S. Fishe and P. A. Lee, Phys. Rev. B 3, ; B. Velický, J. Mašek, and B. Kame, Phys. Lett. A 140,

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