Ann. Physik 3 (1994) 28-39 Annalen der Physik 0 Johann Ambrosius Barth 1994 Electronic transport time through mesoscopic devices with contact barriers Alexander Altland Institut fiir Theoretische Physik, ZUlpicher StraRe 77, D-50937KC)ln, Germany Received 15 October 1993, accepted 2 November 1993 Abstract. Information about the transport time of electrons through a quasi one-dimensional sample is obtained by calculating the energy auto-correlation function of the conductance. Depending on the length of the sample and its coupling to the external device (here modelled by perfectly conducting leads), the transport time undergoes a smooth crossover between two different limiting regimes. In the case of long samples and good coupling it coincides with the diffusion time. In the opposite limit of short and weakly coupled systems, however, the transport time is given by the reciprocal of the quantum mechanical decay width into the leads. The transition between both regimes is discussed in terms of a few model independent concepts. Keywords: Mesoscopic fluctuations; Scattering theory; Nonlinear a-model. 1 Introduction Significant information about the dynamical aspects of charge transport can be obtained by studying the typical transport time ttr of charge carriers through metallic systems. In the case of a mesoscopic sample which is ideally coupled with some measuring device, for instance, ftr coincides with the classical diffusion time revealing the diffusive nature of the transport process. A few years ago, as a result of revolutionary advances in the field of GaAstechnologies, it became possible to create mesoscopic samples which may be gradually decoupled from the external measuring device by means of contact gates. Nowadays such samples are used in numerous experimental applications and a natural question to ask is how the introduction of a decoupling mechanism affects the electronic transport. Progress in this direction has been made by Iida et al. who introduced a simple microscopic model system describing such an experimental setup and studied its transport properties [2]. It is the purpose of the present article to investigate the role of the gates within the same model by analyzing their influence on the transport time. I will show that depending on the ratio between extension of the mesoscopic sample and permeability of the decoupling barriers, ttr undergoes a transition between two qualitatively different regimes. By interpreting the analytic expression for t , in the limiting cases of perfectly coupled and nearly decoupled samples, I will show that the electronic transport is governed by an interplay between two absolutely different mechanisms. A. Altland, Electronic transport time through mesoscopic devices with contact barriers 29 The dependence of the transport time on the connection between sample and device has been investigated earlier by Serota et al. [l]. Considering a mesoscopic sample which is coupled with the outside world by leads of variable width they found that tt, scales with the ratio between cross section of sample and leads. This turned out to be a consequence of the lengthening of a typical diffusive path through the system due to the relatively enlarged width of the bulk sample. In the present work, however (which is based on an alternative modeling of the coupling mechanism), I obtain qualitatively different results which can no longer be interpreted within the standard diffusive picture of the electronic transport alone. Information about ttr can be obtained by calculating the dependence of the energy auto-correlation function (ECF) on the energy difference o.Here g(E)is the energy dependent conductance of a system with Fermi energy E, and (. . .) denotes the configurational averaging. As a function of o,the ECF decays monotonously. This fact can easily be understood as an effect which is caused by a loss of quantum interference [3]: Consider a typical Feynman path contributing to a quantum mechanical description of the conductance. As long w = 0 the quantum mechanical phase corresponding to that path will be the same for both values of the conductance. For this reason, their product contains a phase independent interference term which survives the configurational averaging and gives rise to the phenomenon of conductance fluctuations. For o >0, however, the phases differ by an amount t d h , where t is the time needed to traverse the path, thereby leading to an interference reduction. As soon as t t r o / h s1, the suppression of the interference term is nearly complete whence o h/t,, should be the characteristic energy scale governing the decay of the ECF. Calculations performed on diffusive and perfectly coupled samples [3] indeed showed that the half width of the energy correlation function coincides with the diffusion time 1, = E;', where E, = D K2/ L2 , D is the diffusion constant, L the length of the mesoscopic device and A has been set to unity. Relying on this knowledge and the semiclassical argument given above, one may interpret the half width AE,,, of F as a qualitative measure for the inverse transport time through the system. Iida et al. originally applied their model, which will henceforth be denoted as IWZmodel, to a calculation of the disorder averaged conductance and its fluctuations, i.e. F(0). Finite energy differences o as they appear in 1, however, break a symmetry which is essential for the applicability of the formalism developed in Ref. [2] and render it useless for an analysis of the ECF. A reformulation of the model which may straightforwardly be generalized to incorporate finite energy differences has been introduced in Ref. [4]and will be used in this paper. In order to achieve a self contained presentation, I will begin with a definition of the model and a concise re derivation of the formalism. - 2 Definition of the model The IWZ-model in its simplest form describes a quasi one-dimensional weakly disordered (kfIP 1) sample of length L and constant cross section L t (1;I 1-4L 4 1,) which is at both ends connected with infinite, perfectly conducting leads. Here, kr is the Fermi-momentum, 1 the elastic mean free path and 1, the localization length. The connection between sample and leads is established by potential barriers of variable - 30 Ann. Physik 3 (1994) permeability. What follows is a concise description of the model introducing the notation required in subsequent sections. A detailed introduction to the model can be found in the original Ref. [2]. The Hilbert space of the complete system consists of two parts, a "scattering space" representing the electrons in the leads and a "bound space" corresponding to the disordered region. Conduction electrons in the leads are represented by states x:, where a is an abbreviation for the quantum numbers (typically transverse momentum, spin and direction of the longitudinal momentum component) characterizing a state at the Fermi energy Efand C = L (R)for states populating the left (right) lead. To specify the bound space, the disordered region is cut into K = L/l% I ficticious slices of thickness I and cross section L", Each sub-space belonging to a certain slice i is formed by N - /$A,: 1 local states IF'>. According to the definition of the Hilbert space, the Hamiltonian can be represented by a block structure H= (" ") Hbs , Hbb where indices s(b) correspond to states in the leads (disordered region). Without loss of generality, H, may be taken to be diagonal in the states xf. To model the disorder, Hbb is represented by a Gaussian distributed, real symmetric random matrix Hiv := + ' I & , Iv j ) defined by the first two moments ' where the correlation matrix controls the strength of the hopping between states of identical ( A ) and neighboring (0) slices, resp. Typically, u2/A2 - (kfl)-'[2]. The transition between sample and leads is governed by the matrix elements of Hsb = ff&: It may be assumed [2] that the matrix elements W s fulfil an orthogonality relation c wg w; = x,6,:' , pi where the real quantity xu controls the coupling strength between disordered region and lead channel a. The permeability of the barriers separating leads and disordered region is conveniently characterized in terms of the so-called transmission coefficients T,, i.e. the probabilities for an electron coming in channel a to enter the sample. In terms of the mean level spacing A and xu, T, = 4x,A/(x, + A ) 2 [2]. Only the case of time reversal invariant systems will be considered in this paper whence the entire Hamiltonian may be taken to be real symmetric. A. Altland, Electronic transport time through mesoscopk devices with contact barriers 31 To calculate the conductance the IWZ model employs the Landauer-Buttiker formula where is the on-shell scattering matrix element between an incoming state xi and an outgoing state xf and @'[ WI d ) : = c~ W$ W z . c 3 Calculation of the ECF According to Eqs. (I), (3) and (4), the ECF is essentially determined by the expression ((D (E+ w / 2 ) ) j $ ,(D - I t @ + w/2))pk$, ' x (D - (E- w/2))& (D- I t (E- ~ / 2 ) )P2~6 22)' z, (5) where I have introduced the symmetrized set (E+012, E- 012) of energy arguments (instead of (E,E+ 0))for the sake of notational convenience2. To evaluate expressions of this type, the IWZ-model employs the graded nonlinear a-model, i.e. a formalism relying on functional integration over both commuting and anti-commuting degrees of freedom [6].A detailed discussion of the method in its present context and the supersymmetric generating functional corresponding to the case o = 0 can be found in Refs. [2, 4, 51. The generalization of its derivation to the case finite energy differences w does not impose any conceptual difficulties and results in the functional Z[J^]= j d [ Q l e x p [ . . A ^ ^ where the quantities M = Q,J, W;S are 16-N.K-dimensional matrices with an index structure A?=@$::~), i,j= I , . ..,& ~ , p =1,...,4; p , v = 1,...,N; a , p = I ,...,4, g = C-' and the abbreviation "trg" denotes the generalization of the matrix trace to graded spaces (cf. Ref. [7]). While the indices i, j and p, v correspond to the physical sites of the model and have already been introduced in Section 2, A, p and a , p account for the internal structure of the natrix-fields: The indices A, p label the four propagators I , p = 1,2 (3,4) corresponding to the first, third, (second, appearing in expression (3, fourth) D - ' (i.e. index values 1,2 (3,4) are assigned to retarded (advanced) propagators). A fourth set of indices a, p is necessiated by the grading (that is the subdivision of the superspace in commuting and anti-commuting degrees of freedom) and the time reversal invariance o f the model under consideration (cf. Ref. [4]). Introducing * Due to the independence of all final resuits on E, this is an inessential manipulation. 32 Ann. Physik 3 (1994) l,, 14, lK, l N as unit matrices acting in the spaces of a , I , i, windices, resp., L = diag (1,1, - 1, - 1) @ 1, and I = -)3 the matrices appearing in the generating functional are given by W= W Q L Q = (Q$,,&l71@ IN , where JA are real symmetric but otherwise arbitrary matrices serving as source fields, i.e. the disorder averaged product of propagator matrix elements ( 5 ) is obtained by a fourfold differentiation with respect to suitably chosen matrix elements of JAyA = 1 , . . .4(cf. Eq. (28) of Ref. [5]). The matrices Qi, i = 1, . . .,K are the degrees of freedom of the nonlinear a-model. A detailed description of the Q-matrices, their algebraic properties and the integration measure d[Q] can be found in Refs. [6,7] and will not be repeated here as it is not essential for an understanding of the present application. Shifting the Q-matrices according to transforms Eq. (6) into Z[J^]= j d [ Q ] e x p [ -i 1 giitrg(QiQ'-oQiS)--trgIn(iW+J^-&) 417 2 1 , (8) i.e. in a functional with an action depending only linearly on o.In what follows this representation turns out to be more convenient than the r.h.s. of Eq. (6). Computational details of the evaluation of a nearly identical (up to the o-term) functional have been presented in Refs. [2, 51. To avoid repetitions, I will restrict myself to a brief outline of the calculation here, emphasizing only a few changes necessiated by the presence of the o-term. Functionals like Z ( 3 ) are generally evaluated by means of a saddle point approximation, i.e. the Q-integration is restricted to the subset of Q-matrices fulfilling the stationarity condition -_ 6Q' - o+ d((kfl)-'), (9) where S [Q] denotes the action of the functional. In the limit kfl+ 00 the saddle point approximation becomes exact. Among various other possibilities, the solutions of Eq. (9) can be parameterized according to A. Altland, Electronic transport time through mesoscopic devices with contact barriers 33 A where A = -and the matrix structure corresponds to a (12, 34)-grouping of the KNn A-indices. The 8-dimensional matrices t f 2= {t!za4) and ti1 are defined by symmetry relations I.PP resulting mediately from the fundamental symmetries of the model Hamiltonian, namely Hermiticity and time reversal invariance [4]: til = - I t f 2 t i l = -Mt12MTI , where M= acts in the space of a-indices. According to the definition of the I-indices and the matrix structure Eq. (lo), the indices p , p ' = i(2) are related to conductance l(2). This statement will become more transparent below. Inserting the parameterization Eq. (10) into the functional Eq. (8) and carrying out the differentiation with respect to the sources J leads to ~ ( w=)j d[t121exp (~[t~,,t,~1)0[t2~,t,21 , (13) where t12,t21is a shorthand notation for the set of all t{2,til, i = 1 , . . .,K,S[t12,t21] denotes the action S[Q(t12,t2J]expressed in terms of the t-matrices and the pre-exponential factor 0 [t12,t2J contains contributions from the J-differentiation as well as the Jacobian associated with the transformation d[Q] +d[t12] (note that the matrix elements of t12and tzl are related via the symmetry relations Eq. (11) whence only t12 appears as an independent integration variable). As a consequence of the models nonlinearity (cf. Eq. (lo)), both S and 0 contain infinite powers of t-matrices. Due to the structure of the functional Eq. (8) and the choice of parameterization Eq. (lo), however, the energy difference w affects the action S only to 2ndorder in t. Defining where S'2"'[t] = O(t2"), the 2ndorder term can be written as where Z 7 - l is a matrix defined by 34 Ann. Physik 3 (1994) is a discretized Laplace operator, Cij = 6@(6"+ S i K ) accounts for the coupling to the leads and y = @ A ) - ' T,.The physical meaning of this expression will be discussed a few paragraphs further down. Not only the Gaussian part S(*) but also all higher contributions S(2n),n > 1 (whose explicit form can be found in Refs. [2, 51) scale with the channel number A % 1, whence A may serve as an expansion parameter for a perturbative evaluation of the functional. To calculate the ECF, it suffices to keep the leading order contribution to the A-expansion which turns out to be independent of the channel number, thereby reflecting the universality of the conductance fluctuations. As a first step of the perturbational analysis exp ( S [ t ] )as well as O [ t ]have to be expanded around the Gaussian part ) powers of t-matrices. Individual terms of the thus resulting series can exp ( S ( 2 ) [ t ]in be written symbolically as ~ i ~ jaij++'-2aij+ai+lj where A,B, . . . are abbreviations for matrix products involving matrices t12,tZland I. Expressions of this type can be evaluated by means of a generalization of Wick's theorem to matrix valued fields which states that Eq. (17) equals the sum of all possible contractions, contractions being defined by rules like with The complete list of all contraction rules can be found in Ref. [S]. As a consequence after the contracof @"" = @(A a l), each vertex S ( 2 " ) [ t ]leads to a factor A tion IS]. This means that only a finite number of terms in the series contributes to the result to @(A?. More precisely speaking one may identify 17 different leading order terns whose contraction yields a page filling polynomial in the matrix elements L"jP.' which can no longer be handled analytically. Instead, one has to employ a computer algebraic system to substitute concrete expressions for the matrix elements l 7 l J P p ' and to sum over all indices. B e h e turning to a discussion of the thus obtained results, it is instructive to consider the case of asymptoticaliy long samples in which all calculations can be performed fully analytically. In comparison with the general case, the analysis of long samples is much simpler for (i) only contributions of leading order in L/141 need to be considered (as has been discussed in Ref. [51, only three of the @(A@)terms meet this additional condition) and (ii) the slice index i may be regarded as a quasi continuous variable, which means that summations over i turn into more handy integrations. To leading order in L/I, the free part of the action can be written as -' '-' 35 A. Altland, Electronic transport time through mesoscopic devices with contact barriers where (17Pp‘)-’(x)is the continuum version of the matrix (17-’)PP‘iu and acts as a local differential operator to be specified below. The continuum form of the higher order terms S ( 2 n ) [ t ]n, > 1 and all relevant pre-exponential terms can be found in Ref. [5]. The contraction of these terms leads to three different contributions of leading order in L/I [5]. In the case of a finite energy difference, the sum of these, that is the leading order contribution to F(o),reads a s d x d x ’ ( 4 ~ , F ( x , x ) ~ ~ F ( x ‘ I, x=xa313=xt ’)~, F ( w )= I -a + 1712 (1 ,x’)172, (2,x? F(x,3) F(x,4) + n’2(x,3)172, (x, 4)F( 1 , x’)F(2,x‘) +4F(x,xI2a,&I,=2=,1712(1,2)+(1712t.1721)1+ W L ) 9 (21) where 17:=17*’= 1722 and F(x,x’):=1 7 ( 0 , x ) ~ ( x ’ , L ) + 1 7 ( 0 , ~ ‘ ) 1 7 ( xTo , L )evaluate . this expression further one needs to specify ’the “matrix elements” IIPp’(x,y).Eq. (16) implies that they fulfil the differential equation - 2 71 v P(Da; + (-)P (1 - P’) W ) n q X , y )= 6 ( x - y ) (22) subject to the boundary conditions (r ra, + y )17PP‘ (x,y ) I ,= =0 . (23) Here I have used the relations [2] D = 4A A l 2 / N and v = N / ( n A f i 3 )between the difthe density of states v at the Fermi surface and the IWZ-parameters fusion constant 0, A, N , A . According to Eq. (22), the continuum 17 is just the diffusion propagator of a quasi one-dimensional disordered conductor. The energy difference w renders the propagators nPp‘, p # p’ “massive”: Regarded as a function of the coordinate difference Ix-yl, they decay exponentially rather than linearly as does the “massless” 17 (see below). Technically speaking, this exponential decay is responsible for the destruction of the conductance fluctuations by finite energy differences. Eqs. (22) and (23) define a Sturm-Liouville boundary value problem which can be solved by a standard procedure. For the massless propagators one obtains 17(X,X’) = ( 1 + y ( a + x , ) / l ) ( l +y ( a - x , ) / l ) , 8A y(2+ y L / l ) where x , ( x , ) denotes the smaller (larger) of the arguments x and x‘ and I have in- 36 Ann. Physik 3 (1994) troduced coordinates such that the disordered region extends form - a to a = L/2. The solution of the differential equation for the massive propagators simplifies drastically if Dirichlet boundary conditions rather than the more complex Eq. (23) are imposed. This is a legitimate simplification since the propagators nPP’, p f p ‘ are mostly evaluated in the bulk sample (cf. Eq. (21)) where n P p ’ ( x , x )= @(L/l).Boundary variations of 8(1) are therefore inessential to leading order in L/I and the almost totaI decay of the diffusion propagator at the boundaries may be replaced by strict Dirichlet conditions. The solution of the thus simplified differential equation reads i a where a , = n/L. Insertion of Eqs. (24) and (25) in Eq. (21) yields after a somewhat cumbersome calculation 4 F( w, L )= -- P+ coth(P+) coth2@+) + +(P+ - P J + @(I/L) 9 P: P$ where p , := La,. For small values of the energy shift, o = b(E,), F decays Lorentzian like, 0-E, F(o,L) 8 - 1 with a half width AEIR = V W E , = E,, According to the physical interpretation of AElI2 given above, this means that the transport time of charge carriers through an asymptotically long sample coincides with the classical diffusion time E , * as expected. For large values of the energy shift o%E,,F behaves as Both results, Eq. (27) and Eq. (28) have been obtained earlier in models relying on the impurity diagram technique 131. For a physical interpretation of the power law decay in the regime of large energy shifts, the reader is referred to Ref. [31. A. Altland, Electronic transport time through mesoscopic devices with contact barriers 37 I now turn to the discussion of samples of arbitrary (within the limitations of the model) length and barrier permeability. In this case all contributions of O ( A 4 to the functional, most of them describing the influence of the coupling between sample and leads, have to be taken into account. Although the result of the total contraction of all leading order terms can no longer be handled analytically, it is still possible to process it with the aid of a computer algebraic system. As a result of inserting the (analytically calculated) matrix elements and performing all index summations one obtains a still rather complex expression involving the parameters A , N,K, o and y. Instead of displaying it explicitly, I will restrict myself to a purely graphical presentation of the results. Fig. 1 shows two representative curves F(w,K = const.). The dashed lines are Lorentzian fits to the small o regime. Qualitatively all curves F ( o , K = const.) show the same behaviour as F(o,L-* m) discussed above. There is, however, an important quantitative parameter describing deviations between different curves F(.,K), namely the K-dependent half width A E I l 2 ( K )plotted in Fig. 2. As was discussed above, AE1,2(K)-*Ec for asymptotically large values of K. More interesting is the regime of small values of K where AE,,2 approaches an entirely different energy scale, viz. the so called decay width 1 I-=-A T, This behaviour can easily be understood within the interpretation of AE,,, as the inverse transport time: An electron, initially moving in the disordered region, escapes with probability T, to a certain lead channel a upon hitting the interface between sample and leads. Quantum mechanically, the electron is to be represented by a wave package which reproduces itself every Poincare-recurrence time 2 7cd The escape rate into channel a is therefore given by T0A/(27r).In order to obtain the total escape rate into -'. .5 k3 .3 W .4 .2 .o - solid lines: ECF dashed lines: Lorentzion fits tti\ \ K=20 ' ,000 .005 .O10 I I I I .Ol5 ,020 ,025 .030 energy shift in arbitrary units Fig. 1 Solid lines: ECF versus w for fixed length L = 201 and L = 601, resp. Dashed lines: Lorentzian fits. 38 Ann. Phvsik 3 (1994) h .-2 lo-' C 3 1 loo I I J 10' lo2 sample length in units of l. Fig. 2 Half width AE,,* versus length of the disordered region. the leads one finally has to sum over all channels thereby arriving at Eq. (29). (The extra factor of two before the channel summation comes from the fact that the decay into both leads is equalIy probable in the case of asymptotically short samples.) The location of the crossover region between the two asymptotic regimes is determined by the condition where p = (AK)-' 0: (rnkfL: l)-' is the density of states in each individual site, i.e. the smaller the c o d i n g between sample and leads (7'') the larger is the characteristic sample length at which the transition takes place. 4 Summary and conclusion Subject of this paper was the analysis of the transport time ttr of charge carriers through a quasi one-dimensional disordered sample which is gradually decoupled from the external device by potential barriers. Depending on both, sample length and coupling strength, tt, may exhibit entirely different behaviour. Owing to the diffusive nature of the transport in the interior of the disordered sample, the transport time through asymptotically long samples coincides with the classical diffusion time E; thereby increasing quadratically in the sample length. In the opposite limit of a short and weakly coupled system, however, the time necessary to traverse the disordered region eventually becomes negligible in comparison with the penetration time through the decoupling barriers. In this case tt, is given by the inverse decay width f which increases only linearly in the same length. Remaining always larger than both, r-' and E;', tt, undergoes a smooth crossover between these two limiting scales as the sample length increases. The location of the transition region is determined by the condition E;' = T-'*K= 7r3Dp(12C, I"')-' (kff)2( ,T,)-', i.e. it decreases with both, the coupling strength to the leads and the disorder. ' - A. Altland, Electronic transport time through mesoscopic devices with contact barriers 39 These results have been obtained for the simplest possible model of a non-isolated mesoscopic device, i.e. a quasi one-dimensional disordered region coupled with two ideally conducting leads. On the other hand, the qualitative behaviour of the transport time could be interpreted in terms of a few elementary concepts of quantum mechanics and did not depend on any model specific features whence it seems to be likely that the above discussed picture of an interplay between diffusion and decay into the attached device also applies to more complex systems, e.g. multi-lead devices or hetero-structures with decoupling tunnel contacts. I thank H. A. Weidenmiiller for many helpful discussions. I am also indebted to S. Iida and A. MiillerGrCjling for several remarks and suggestions. References [l] S. Iida, H.A. Weidenmiiller, J.A. Zuk, Ann. Phys. 200 (1990) 219 [2] R.A. Serota, S. Feng, C. Kane, P.A. Lee, Phys. Rev. B36 (1987) 5031 [3] P.A. Lee, A.D. Stone, H. Fukuyama, Phys. Rev. F35 (1987) 1039 [4] A. Altland, Z. Phys. B82 (1991) 105 [5] A. Altland, Z. Phys. B86 (1992) 101 [6] K.B. Efetov, Adv. Phys. 32 (1983) 53 [7] J. J.M. Verbaarschot, H.A. Weidenmiiller, M.R. Zirnbauer, Phys. Rep. 129 (1985) 367

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