The Ehrenfest time dependence of the suppression of the weak localization
correction to the conductance of a {\em clean} chaotic cavity is calculated.
Unlike in earlier work, no impurity scattering is invoked to imitate
diffraction effects. The calculation extends the semiclassical theory of K.
Richter and M. Sieber [Phys. Rev. Lett. {\bf 89}, 206801 (2002)] to include the
effect of a finite Ehrenfest time.Comment: 3 Pages, 1 Figure, RevTe

A. Lodder

C.W.J. Beenakker

D. Taras-Semchuk

G. M. Zaslavsky

H. U. Baranger

H.U. Baranger

İ. Adagideli

I.L. Aleiner

K. Richter

M.G. Vavilov

O. Agam

P. G. Silvestrov

P.A. Lee

P.G. Silvestrov

Ph. Jacquod

R.A. Jalabert

S. Chakravarty

S. Oberholzer

English

arXiv

The Ehrenfest time dependence of the suppression of the weak localization correction to the conductance of a {\\em clean} chaotic cavity is calculated. Unlike in earlier work, no impurity scattering is invoked to imitate diffraction effects. The calculation extends the semiclassical theory of K. Richter and M. Sieber [Phys. Rev. Lett. {\\bf 89}, 206801 (2002)] to include the effect of a finite Ehrenfest time

Adagideli, Inanc

University of Regensburg Publication Server

arXiv:cond-mat/0306568v1  [cond-mat.mes-hall]  23 Jun 2003Ehrenfest time dependent suppression of weak localizationI˙. AdagideliInstituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands(Dated: November 16, 2018)The Ehrenfest time dependence of the suppression of the weak localization correction to theconductance of a clean chaotic cavity is calculated. Unlike in earlier work, no impurity scattering isinvoked to imitate diffraction effects. The calculation extends the semiclassical theory of K. Richterand M. Sieber [Phys. Rev. Lett. 89, 206801 (2002)] to include the effect of a finite Ehrenfest time.PACS numbers: 73.23.-b, 05.45.Mt, 73.20.FzThe average conductivity of a disordered metal is re-duced with respect to the classical value by quantuminterference. This phenomenon, known as weak local-ization, has been understood long ago [1, 2, 3] in termsof the constructive interference of time-reversed diffusivetrajectories. Weak localization exists also in quantumdots, which are so small and clean that impurity scatter-ing can be neglected [4]. In such ballistic cavities, quan-tum interference effects develop only after a time scaleon which a minimal wave packet has spread to cover theentire cavity. This time scale, known as the Ehrenfesttime [5], is of order τE = λ−1 ln kFL, with λ the Lya-punov exponent of the chaotic dynamics, kF the Fermiwavevector, and L the linear size of the cavity. The timescale τE becomes important if it is larger than the meandwell time τD of an electron in the quantum dot, coupledvia two point contacts to electron reservoirs.Suppression of weak localization in the Ehrenfestregime τD < τE was first proposed and studied by Aleinerand Larkin [6]. Their calculation played a seminal role inthe development of the subject, but it was unsatisfactoryin one key aspect: A small amount of impurity scatteringwas introduced by hand to imitate the effects of diffrac-tion in a ballistic system. The main aim of our workis to provide a derivation of the weak localization correc-tion in the Ehrenfest regime without recourse to impurityscattering. To our knowledge no such derivation exists.The theoretical framework that we shall adopt is thesemiclassical theory of Richter and Sieber [7], which isa well-understood and controlled approximation scheme.In Ref. [7] the effects of finite τE were not considered, sothere the weak localization correction was given by thevalue known from random matrix theory [8, 9]. We findthat the absence of interfering trajectories when τD < τEleads to the exponential suppression of the weak local-ization correction ∝ exp(−τE/τD), in agreement withRef. [6].Apart from the setting of weak localization, effects ofa finite Ehrenfest time have received much attention re-cently: The excitation gap in an Andreev billiard [10] aswell as the shot noise [11] of a ballistic cavity are pre-dicted to be suppressed when τE > τD. The latter effecthave received experimental support [12]. For these prob-lems there now exist semiclassical theories, which do notinvoke impurity scattering. However, all these theorieseyy'FIG. 1: The Richter-Sieber pair. The weak localization cor-rection to the transmission amplitudes comes from selfcross-ing angles ǫ .√λ~/EF . The characteristic time of suchorbits is the Ehrenfest time τE = (1/λ) ln(EF/λ~).deal only with leading order effects. Quantum correc-tions such as weak localization are beyond their reach.That is why in this work we follow an altogether differ-ent approach.Following Richter and Sieber, we consider a two-dimensional ballistic quantum dot to which two leadsof width w and w′ are attached. We assume that theclassical dynamics of this dot is chaotic, with Lyapunovexponent λ. The Landauer formula for the conductanceis given byG = 2e2hN∑n=1N ′∑m=1|tnm|2, (1)where tnm is the transmission amplitude between incom-ing and outgoing channels m and n at the Fermi energyEF and N(N′) is the number of channels of width w(w′).The semiclassical expression for tnm is given as a sumover classical trajectories γ joining two leads [7, 13]:tnm = −√π~2ww′∑γ(n¯,m¯)exp((i/~)Sγ)Φγ| cos θn¯ cos θm¯Mγ21|1/2. (2)2Here sin θn¯ = n¯π/kFw and sin θm¯ = m¯π/kFw′, n¯ = ±nand m¯ = ±m, and Φγ = sgn(m¯)sgn(n¯) exp(iπ(m¯y/w −n¯y′/w′ − µγ/2 + 1/4)). The term Sγ is the classical ac-tion, Mγ21 is an element of the monodromy matrix, andµγ is the Maslov index. The trajectory γ starts at trans-verse coordinate y in lead w with an angle θn¯ and endsat the transverse coordinate y′ in lead w′ with angle θm¯.When calculating |tnm|2 the double sum over trajec-tories γ and γ′ is approximated to leading order by thediagonal approximation γ = γ′ [13]. The first order quan-tum correction to the transmission amplitudes (responsi-ble for the weak localization effect [14]) is due to Richter-Sieber pairs [7]: γ is exponentially close to γ′ everywhereexcept in the vicinity of a crossing point of γ where γ′avoids that crossing. This is illustrated in Fig. 1. Theaction difference between γ and γ′ is: ∆S = EF ǫ2/λ,where ǫ is the angle at the crossing. In the diagonal ap-proximation, the sum over trajectories can be evaluatedvia the sum rule [7]∑γ(y′,θn;y,θm)δ(T − Tγ)|Mγ21|=cos θn cos θm2πmAdy dy′ ρ(T ), (3)where the sum is over all trajectories that begin in in-terval dy′ around y and end in interval dy around y,ρ(T ) ∝ exp(−T/τD) is the dwell time distribution andτD = mA/~(N +N′) (4)is the mean dwell time, we denote by m the effectiveelectron mass, by A the area of the cavity, and by N =kFw/π, N′ = kFw′/π the number of channels in the twoleads. The weak localization correction from Richter-Sieber pairs is given byδ|tnm|2 =2EF~πm2A2∫ π0dǫ∫∞TǫdT e−T/τD(T − Tǫ)2× cos(EF ǫ2/λ~) sin ǫ, (5)where Tǫ = −(2/λ) ln ǫ. The lower bound in the integralover T signifies that there are no orbits shorter than Tǫwith a selfcrossing angle ǫ.So far we have followed the calculation of Richter andSieber [7]. Now we depart from it. We first evaluate theT integral,δ|tnm|2 =4EF~τ3Dπm2A2∫ π0dǫ e−Tǫ/τD cos(EF ǫ2/λ~) sin ǫ.(6)In the semiclassical limit, the main contribution to thisintegral comes from ǫ .√λ~/EF ≪ 1. Thus we mayapproximate sin ǫ ≈ ǫ and extend the upper limit of theintegral to infinity. The result isδ|tnm|2 =4EF~πm2A2τ3D∫∞0dǫ ǫ1+2/λτD cos(EF ǫ2/λ~)= −(~τDmA)22λτDπsin(π2λτD)Γ(1 +1λτD)× exp(−τE/τD), (7)where τE = (1/λ) ln(EF /λ~) is the Ehrenfest time of thisproblem. In the relevant regime λτD ≫ 1 we haveδ|tnm|2 ≃(~τDmA)2e−τE/τD . (8)Finally, using Eq. (4) and the sum rule (3), we find theweak localization correction to the conductanceδG= −2e2hNN ′(N +N ′)2exp(−τE/τD), (9)in agreement with Ref. [6].Up to this point we have rederived a known result.Now we shall apply this technology to the magnetic fielddependence of the weak localization correction in theEhrenfest regime. This is done via the calculation of themagnetic field dependence of the density of self cross-ings [7]. Accordingly, Eq.(5) is modified as follows:δ|tnm|2 =4EF~τ2Bπm2A2∫ π0dǫ∫∞TǫdT cos(EF ǫ2/λ~) sin ǫ×e−T/τD(e(Tǫ−T )/τB − 1 +T − TǫτB), (10)where τB = φ20/(8π2βB2) is the magnetic time, φ0 isthe flux quantum, B is the magnetic field, and β isa system dependent parameter [7, 13]. As before, wefirst evaluate the T integral exactly and then evaluatethe ǫ integral in stationary phase approximation. Thisproduces the B dependent transmission matrix elementsδ|tnm(B)|2 = δ|tnm(0)|2(1 + τD/τB)−1. Finally, sum-ming over all channels we obtain the magnetic field de-pendence of the weak localization correction to the con-ductance,δG(B) = −2e2hNN ′(N +N ′)2e−τE/τD1 + τD/τB(11)We see that the Lorentzian lineshape of the weak local-ization peak is preserved in the Ehrenfest regime, whileits size is exponentially suppressed.In conclusion, we have presented a derivation of theEhrenfest time dependence of the weak localization cor-rection in a two dimensional chaotic billiard. All interfer-ence effects are fully accounted for within the frameworkof a controlled semiclassical approximation [7], withoutrequiring the artificial inclusion of impurity scattering [6].Interesting extensions include the appearance of a second3Lyapunov exponent in three dimensions, and the coex-istence of chaotic and mixed regions of phase space. Itwould also be of interest to extend the method to de-scribe universal conductance fluctuations in the Ehren-fest regime.This work was supported by the Dutch Science Foun-dation NWO/FOM. We thank C.W.J. Beenakker and J.Tworzydlo for helpful discussions.[1] Y. Imry, Introduction to Mesoscopic Physics (OxfordUniversity Press, Oxford, 1997).[2] P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57,287 (1985).[3] S. Chakravarty and A. Schmid, Phys. Rep 140, 193(1986).[4] C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 (1997).[5] G. M. Zaslavsky, Phys. Rep. 80, 157 (1981).[6] I.L. Aleiner and A.I. Larkin, Phys. Rev. B 54, 14423(1996).[7] K. Richter and M. Sieber, Phys. Rev. Lett. 89, 206801(2002).[8] R.A. Jalabert, J.L. Pichard, and C.W.J. Beenakker, Eu-rophys. Lett. 27, 255 (1994).[9] H.U. Baranger and P.A. Mello, Phys. Rev. Lett. 73, 142(1994).[10] A. Lodder and Yu. V. Nazarov, Phys. Rev. B 59, 5783(1998); D. Taras-Semchuk and A. Altland, Phys. Rev. B64, 014512 (2001); I˙. Adagideli and C.W.J. Beenakker,Phys. Rev. Lett. 89, 237002 (2002); P. G. Silvestrov, M.C. Goorden, and C. W. J. Beenakker, Phys. Rev. Lett.90, 116801 (2003); M.G. Vavilov and A.I. Larkin, Phys.Rev. B 67, 115335 (2003); Ph. Jacquod, H. Schome-rus, and C.W.J. Beenakker, Phys. Rev. Lett. 90, 207004(2003).[11] O. Agam, I. Aleiner and A. Larkin, Phys. Rev. Lett.89, 3153 (2000); P.G. Silvestrov, M.C. Goorden, andC.W.J. Beenakker, Phys. Rev. B 67, 241301(R) (2003);J. Tworzydlo, A. Tajic, H. Schomerus, and C.W.J.Beenakker, cond-mat/0304327.[12] S. Oberholzer, E. V. Sukhorukov, and C. Scho¨nenberger,Nature 415, 765 (2002).[13] H. U. Baranger, R.A. Jalabert, and A.D. Stone, Phys.Rev. Lett. 70, 3876 (1993).[14] Equivalently, one can calculate the quantum correctionto the reflection amplitudes, but this is more complicatedbecause then both diagonal terms and Richter-Sieberpairs contribute [7].

Ehrenfest time dependent suppression of weak localization

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Ehrenfest-time-dependent suppression of weak localization

Ehrenfest time dependent suppression of weak localization

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