Quantum transport and its classical limitPiet BrouwerLaboratory of Atomic and Solid State PhysicsCornell UniversityLecture 1Capri spring school on Transport in Nanostructures, March 25-31, 2007

Quantum TransportAbout the manifestations of quantum mechanics on the electrical transport properties of conductorsThese lectures: signatures of quantum interferenceQuantum effects not covered here:Interaction effectsShot noiseMesoscopic superconductivitysample

Quantum TransportThese lectures: signatures of quantum interferenceWhat to expect?MagnetofingerprintNonlocalityR1+2=R1+R2B (mT)G1+2 (e2/h)G1=G2=2e2/hBdGB (10-4T)G (e2/h)Figures adapted from: Mailly and Sanquer (1991)Webb, Washburn, Umbach, and Laibowitz (1985)Marcus (2005)

Landauer-Buettiker formalismsamplexyWN: number of propagating transverse modes or channelsan: electrons moving towards samplebn: electrons moving away from sampleNote: |an|2 and |bn|2 determine flux in each channel, not densityN depends on energy e, width WIdeal leads

Scattering Matrix: Definitionsample |Smj;nk|2 describes what fraction of the flux of electrons entering in lead k, channel n, leaves sample through lead j, channel m. Probability that an electron entering in lead k, channel n, leaves sample through lead j, channel m is |Smj;nk|2 vnk/vmj.More than one lead: Nj is number of channels in lead j Use amplitudes anj, bnj for incoming, outgoing electrons, n = 1, , Nj.Linear relationship between anj, bnj:S: scattering matrix

Scattering matrix: PropertiessampleLinear relationship between anj, bnj:S: scattering matrix Current conservation: S is unitary Time-reversal symmetry:If y is a solution of the Schroedinger equation at magnetic field B, then y* is a solution at magnetic field B.

Landauer-Buettiker formalismReservoirssampleEach lead j is connected to an electron reservoir at temperature T and chemical potential mj.mj, TDistribution function for electrons originating from reservoir j is f(e-mj).

Landauer-Buettiker formalismCurrent in leadssamplemj, TIj,inIj,outIn one dimension:= (nnkh)-1Buettiker (1985)

Landauer-Buettiker formalismLinear responsesamplemj, TIj,inIj,outmj = m eVjExpand to first order in Vj:Zero temperature

Conductance coefficientssamplemj=m-eVjIj Current conservation and gauge invariance Time-reversalNote: only if B=0 or if there are only two leads.Otherwise andin general.

Multiterminal measurementsIn four-terminal measurement, one measures a combination of the 16 coefficients Gjk. Different ways to perform the measurement correspond to different combinations of the Gjk, so they give different results!IVVVIIBenoit, Washburn, Umbach, Laibowitz, Webb (1986)

Landauer formula: spin Without spin-dependent scattering: Factor two for spin degeneracy

With spin-dependent scattering: Use separate sets of channels for each spin direction. Dimension of scattering matrix is doubled.Conductance measured in units of 2e2/h: Dimensionless conductance.

Two-terminal geometryr, r: reflection matricest, t: transmission matricestrinouteef(e)f(e)ee|t|2|r|2tr|t|2|r|2eVe (meV)f(e)Anthore, Pierre, Pothier, Devoret (2003)

Quantum transportLandauer formulatrsampletrWhat is the sample? Point contact Quantum dot Disordered metal wire Metal ring Molecule Graphene sheet

Example: adiabatic point contactN(x)Nminxg1062480Vgate (V)-2.0-1.0-1.8-1.6-1.4-1.2Van Wees et al. (1988)

Quantum interferenceIn general: dg small, random signtnm,a , tnm,b : amplitude for transmission along paths a, bab

Quantum interferenceThree prototypical examples: Disordered wire Disordered quantum dot Ballistic quantum dot

Scattering matrix and Green functionRecall: retarded Green function is solution ofIn one dimension:ek = e and v = h-1dek/dkGreen function in channel basis:r in lead j; r in lead kSubstitute 1d form of Green functionIf j = k:

Quantum transport and its classical limitPiet BrouwerLaboratory of Atomic and Solid State PhysicsCornell UniversityLecture 2Capri spring school on Transport in Nanostructures, March 25-31, 2007

Characteristic time scalesh/eFttergtDtHlFlLBallistic quantum dot: t ~ terg ~ L/vF, l ~ LDiffusive conductor: terg ~ L2/DInverse level spacing: relevant for closed samplesElastic mean free time

Characteristic conductancesConductances of the contacts: g1, g2Conductance of sample without contacts: gsampleif g >> 1 Bulk measurement: g1,2 >> gsample Quantum dot: g1,2

Assumptions and restrictionsAlways: lF > 1.This implies tD

Quantum interference correctionsWeak localizationSmall negative correction to the ensemble-averaged conductance at zero magnetic field

Conductance fluctuationsReproducible fluctuations of the sample-specific conductance as a function of magnetic field or Fermi energyGBBAnderson, Abrahams, Ramakrishnan (1979)Gorkov, Larkin, Khmelnitskii (1979)Altshuler (1985)Lee and Stone (1985)

Weak localization (1)Nonzero (negative) ensemble average d g at zero magnetic fieldbagB=+Hikami box+ permutationsd gCooperonInterfering trajectories propagating in opposite directions

Weak localization (2)Nonzero (negative) ensemble average d g at zero magnetic fieldgBd gbaSign of effect follows directly from quantum correction to reflection.Trajectories propagating at the same angle in the leads contribute to the same element of the reflection matrix r. Such trajectories can interfere.

Weak localization (3)Disordered wire:Disordered quantum dot:N1 channels N2 channelsB (10-4T)G (e2/h)Mailly and Sanquer (1991)(no derivation here)(derivation later)baF

Weak localization (4)baB (10-4T)G (e2/h)FMagnetic field suppresses WL.d gChentsov (1948)0-10-3DR/R 10-3H(kOe)

Weak localization (5)Typical dwell time for transmitted electrons: tergTypical area enclosed in that time: sample area A.WL suppressed at flux F ~ hc/e through sample.Typical area enclosed in time terg: sample area A.Typical area enclosed in timetD: A(tD/terg)1/2.WL suppressed at F ~ (hc/e)(terg/tD)1/2

Weak localization (6)In a ring, all trajectories enclose multiples of the same area. If F is a multiple of hc/2e, all phase differences are multiples of 2p : d g oscillates with period hc/2e.hc/2e Aharonov-Bohm effect

abFNote: phases picked up by individual trajectories are multiples of p, not 2p!Altshuler, Aronov, Spivak (1981)Sharvin and Sharvin (1981)

Conductance fluctuations (1)Fluctuations of d g with applied magnetic fieldd gdiffuson interfering trajectories in the same directioncooperon interfering trajectories in the opposite directionbaabbaabbaabbaabUmbach, Washburn, Laibowitz, Webb (1984)

Conductance fluctuations (2)Fluctuations of d g with applied magnetic fieldd gDisordered wire:Disordered quantum dot:N1 channels N2 channelsB (mT)G (e2/h)Jalabert, Pichard, Beenakker (1994)Baranger and Mello (1994)Marcus (2005)

Conductance fluctuations (3)d gabFIn a ring: sample-specific conductance g is periodic funtion of F with period hc/e.hc/e Aharonov-Bohm effectWebb, Washburn, Umbach, and Laibowitz (1985)

Random Matrix TheoryQuantum dotN1 channels N2 channelsIdeal contacts: every electron that reaches the contact is transmitted.For ideal contacts: all elements of S have random phase.Ansatz: S is as random as possible,with constraints of unitarity and time-reversal symmetry,Dimension of S is N1+N2. Assign channels m=1, , N1 to lead 1, channels m=N1+1, , N1+N2 to lead 2Dysons circular ensembleBluemel and Smilansky (1988)

RMT: Without time-reversal symmetryQuantum dotN1 channels N2 channelsAnsatz: S is as random as possible,with constraint of unitarityProbability to find certain S does not change if We permute rows or columns We multiply a row or column by eifAverage conductance:No interference correction to average conductance

RMT: with time-reversal symmetryQuantum dotN1 channels N2 channelsAdditional constraint: Probability to find certain S does not change if We permute rows and columns, We multiply a row and columns by eif,while keeping S symmetricInterference correction to average conductanceAverage conductance:

RMT: with time-reversal symmetryQuantum dotN1 channels N2 channelsWeak localization correction is difference with classical conductanceFor N1, N2 >> 1:Same as diagrammatic perturbation theoryJalabert, Pichard, Beenakker (1994)Baranger and Mello (1994)

RMT: conductance fluctuationsQuantum dotN1 channels N2 channelsWithout time-reversal symmetry:With time-reversal symmetry:Same as diagrammatic perturbation theoryThere exist extensions of RMT to deal with contacts that contain tunnel barriers, magnetic-field dependence, etc.Jalabert, Pichard, Beenakker (1994)Baranger and Mello (1994)

Quantum transport and its classical limitPiet BrouwerLaboratory of Atomic and Solid State PhysicsCornell UniversityLecture 3Capri spring school on Transport in Nanostructures, March 25-31, 2007

Ballistic quantum dotsPast lectures: Qualitative microscopic picture of interference corrections in disordered conductors; Quantitative calculations can be done using diagrammatic perturbation theory Quantitative non-microscopic theory of interference corrections in quantum dots (RMT).This lecture: Microscopic theory of interference corrections in ballistic quantum dotsAssumptions and restrictions:lF > 1Method: semiclassics, quantum properties are obtained from the classical dynamics

Semiclassical Green functionRelation between transmission matrix and Green functionSemiclassical Green function (two dimensions)a: classical trajectory connection r and rS: classical action of ama: Maslov indexAa: stability amplitudearrrq

Comparison to exact Green functionSemiclassical Green function (two dimensions)Exact Green function (two dimensions)Asymptotic behavior for k|r-r| >> 1equals semiclassical Green function

Semiclassical scattering matrixInsert semiclassical Green functionand Fourier transform to y, y. This replaces y, y by the conjugate momenta py, py and fixes these toResult:Legendre transformed actionayqJalabert, Baranger, Stone (1990)

Semiclassical scattering matrixLegendre transformed actionStability amplitudetransverse momenta of a fixed atayqTransmission matrixReflection matrix

Diagonal approximationReflection probabilityDominant contribution from terms a = b.probability to return to contact 1ab

Enhanced diagonal reflectionReflection probabilityIf m=n: also contribution if b = a time-reversed of a:ab=aWithout magnetic field: a and a have equal actions, henceFactor-two enhancement of diagonal reflectionDoron, Smilansky, Frenkel (1991)Lewenkopf, Weidenmueller (1991)

diagonal approximation: limitationsab=aOne expects a corresponding reduction of the transmission. Where is it?Note: Time-reversed of transmitting trajectories contribute to t, not t. No interference!Compare to RMT:captured by diagonal approximationmissed by diagonal approximationWe foundThe diagonal approximation gives

Lesson from disordered metalsba=+Hikami box+ permutationsWeak localization correction to transmission: Need Hikami box.baWeak localization correction to reflection: Do not need Hikami box.

Ballistic Hikami box?In a quantum dot with smooth boundaries: Wavepackets follow classical trajectories.

Ballistic Hikami box?Marcus groupBut quantum interference corrections d g and var g exist in ballistic quantum dots!

Ballistic Hikami box?Initial uncertainty is magnified by chaotic boundary scattering.l: Lyapunov exponentAleiner and Larkin (1996)Richter and Sieber (2002)Time until initial uncertainty ~lF has reached dot size ~L:L=lF exp(l t)t =Ehrenfest timeInterference corrections in ballistic quantum dot same as in disordered quantum dot if tE

Ballistic weak localizationProbability to remain in dot:tEt loopalso for disordered quantum dot: included in RMTspecial for ballistic dotAleiner and Larkin (1996)Adagideli (2003)Rahav and Brouwer (2005)

Semiclassical theoryLandauer formula Sa, Sb: classical action angles of a, b consistent with transverse momentum in lead,

Aa, Ab: stability amplitudesJalabert, Baranger, Stone (1990)

Semiclassical theoryLandauer formula(0,0)(s,u)s, u: distances along stable, unstable phase space directionsAction difference Sa-Sb = suRichter and Sieber (2002)Spehner (2003)Turek and Richter (2003)Mller et al. (2004)Heusler et al. (2006)tencc: classical cut-off scales a e-ltu a eltencounter region: |s|,|u| < c

Semiclassical theory(0,0)(s,u)t tencc: classical cut-off scaleP1, P2: probabilities to enter, exit through contacts 1,2Landauer formulas, u: distances along stable, unstable phase space directionsAleiner and Larkin (1996)Adagideli (2003)Rahav and Brouwer (2005)

Classical Limit Lbut var g remains finite!Brouwer and Rahav (2006)Take limit lF/L 0 without changing the classical dynamics of the dot, including its contactsdiverges in this limit!0THE ENDAleiner and Larkin (1996)