Quantum transport and its classical limit Piet Brouwer

Quantum transport and its classical limit Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Lecture 1 Capri spring school on Transport in Nanostructures, March 25 -31, 2007

Quantum Transport About the manifestations of quantum mechanics on the electrical transport properties of conductors sample These lectures: signatures of quantum interference Quantum effects not covered here: • Interaction effects • Shot noise • Mesoscopic superconductivity

Quantum Transport These lectures: signatures of quantum interference d. G G (e 2/h) G 1+2 (e 2/h) What to expect? B (m. T) B (10 -4 T) Magnetofingerprint B Nonlocality G 1=G 2=2 e 2/h R 1+2=R 1+R 2 Figures adapted from: Mailly and Sanquer (1991) Webb, Washburn, Umbach, and Laibowitz (1985) Marcus (2005)

Landauer-Buettiker formalism Ideal leads sample y W x N: number of propagating transverse modes or “channels” N depends on energy e, width W an: electrons moving towards sample bn: electrons moving away from sample Note: |an|2 and |bn|2 determine flux in each channel, not density

Scattering Matrix: Definition sample 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” • |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.

Scattering matrix: Properties Linear relationship between anj, bnj: sample 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 formalism Reservoirs Each lead j is connected to an electron reservoir at temperature T and chemical potential mj. sample mj , T Distribution function for electrons originating from reservoir j is f(e-mj).

Landauer-Buettiker formalism Current in leads sample Ij, in In one dimension: Ij, out mj , T = (nnkh)-1 Buettiker (1985)

Landauer-Buettiker formalism Linear response mj = m – e. Vj sample Expand to first order in Vj: Ij, in Zero temperature Ij, out mj , T

Conductance coefficients sample • Current conservation and gauge invariance Ij mj=m-e. Vj • Time-reversal Note: Otherwise only if B=0 or if there are only two leads. and in general.

Multiterminal measurements In 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! V I I Benoit, Washburn, Umbach, Laibowitz, Webb (1986) V

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 2 e 2/h: “Dimensionless conductance”.

Two-terminal geometry r’ t r r, r’: “reflection matrices” t, t’: “transmission matrices” t’ e f(e) e in f(e) Anthore, Pierre, Pothier, Devoret (2003) out e e e (me. V) e. V |t’|2 |r|2 f(e) |t|2 |r’|2

Quantum transport Landauer formula t r r’ t’ sample What is the “sample”? • Point contact • Quantum dot • Disordered metal wire • Metal ring • Molecule • Graphene sheet

Example: adiabatic point contact N(x) Nmin 10 8 6 4 2 0 g x Vgate (V) -2. 0 -1. 8 -1. 6 -1. 4 -1. 2 -1. 0 Van Wees et al. (1988)

Quantum interference In general: dg small, random sign a b tnm, a , tnm, b : amplitude for transmission along paths a, b

Quantum interference Three prototypical examples: • Disordered wire • Disordered quantum dot • Ballistic quantum dot

Scattering matrix and Green function Recall: retarded Green function is solution of In one dimension: Green function in channel basis: ek = e and v = h-1 dek/dk r in lead j; r’ in lead k Substitute 1 d form of Green function If j = k:

Quantum transport and its classical limit Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Lecture 2 Capri spring school on Transport in Nanostructures, March 25 -31, 2007

Characteristic time scales L h/e. F t terg t. D Ballistic quantum dot: t ~ terg ~ L/v. F, l ~ L Diffusive conductor: terg ~ L 2/D t. H Inverse level spacing: relevant for closed samples l Elastic mean free time l. F

Characteristic conductances Conductances of the contacts: g 1, g 2 Conductance of sample without contacts: gsample if g >> 1 • ‘Bulk measurement’: g 1, 2 >> gsample • Quantum dot: g 1, 2 << gsample general relationships: g dominated by sample g dominated by contacts

Assumptions and restrictions Always: l. F << l. Well-defined momentum between scattering events Diagrammatic perturbation theory: g >> 1. This implies t. D << t. H Only ‘nonperturbative’ methods can describe the regime g ~ 1 or, equivalently, times up to t. H. Examples are certain field theories, random matrix theory.

Quantum interference corrections G Weak localization Small negative correction to the ensembleaveraged conductance at zero magnetic field Conductance fluctuations Reproducible fluctuations of the samplespecific conductance as a function of magnetic field or Fermi energy B G B Anderson, 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 field g dg a b B = + + permutations ‘Cooperon’ ‘Hikami box’ Interfering trajectories propagating in opposite directions

Weak localization (2) Nonzero (negative) ensemble average d g at zero magnetic field dg g B Sign of effect follows directly from quantum correction to reflection. a b 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: G (e 2/h) (no derivation here) Disordered quantum dot: (derivation later) a b F N 1 channels N 2 channels B (10 -4 T) Mailly and Sanquer (1991)

Weak localization (4) G (e 2/h) dg B (10 -4 T) a Magnetic field suppresses WL. b F 10 -3 0 DR/R H(k. Oe) -10 -3 Chentsov (1948)

Weak localization (5) a b F Typical dwell time for transmitted electrons: terg Typical 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 timet. D: A(t. D/terg)1/2. WL suppressed at F ~ (hc/e)(terg/t. D)1/2 << hc/e.

Weak localization (6) b F a In a ring, all trajectories enclose multiples of the same area. If F is a multiple of hc/2 e, all phase differences are multiples of 2 p : d g oscillates with period hc/2 e. ‘hc/2 e Aharonov-Bohm effect’ Altshuler, Aronov, Spivak (1981) Note: phases picked up by individual trajectories are multiples of p, not 2 p! Sharvin and Sharvin (1981)

Conductance fluctuations (1) Fluctuations of d g with applied magnetic field dg a b’ a b a’ a b b’ a’ a’ b b’ a’ Umbach, Washburn, Laibowitz, Webb (1984) “diffuson” interfering trajectories in the same direction “cooperon” interfering trajectories in the opposite direction

Conductance fluctuations (2) Fluctuations of d g with applied magnetic field dg G (e 2/h) Disordered wire: Disordered quantum dot: Jalabert, Pichard, Beenakker (1994) Baranger and Mello (1994) N 1 channels N 2 channels B (m. T) Marcus (2005)

Conductance fluctuations (3) dg b F a In a ring: sample-specific conductance g is periodic funtion of F with period hc/e. ‘hc/e Aharonov-Bohm effect’ Webb, Washburn, Umbach, and Laibowitz (1985)

Random Matrix Theory Quantum dot Ideal contacts: every electron that reaches the contact is transmitted. For ideal contacts: all elements of S have random phase. N 1 channels N 2 channels Ansatz: S is as random as possible, with constraints of unitarity and time-reversal symmetry, “Dyson’s circular ensemble” Dimension of S is N 1+N 2. Assign channels m=1, …, N 1 to lead 1, channels m=N 1+1, …, N 1+N 2 to lead 2 Bluemel and Smilansky (1988)

RMT: Without time-reversal symmetry Quantum dot N 1 channels N 2 channels Ansatz: S is as random as possible, with constraint of unitarity Probability to find certain S does not change if • We permute rows or columns • We multiply a row or column by eif Average conductance: No interference correction to average conductance

RMT: with time-reversal symmetry Quantum dot Additional 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 symmetric N 1 channels N 2 channels Average conductance: Interference correction to average conductance

RMT: with time-reversal symmetry Quantum dot Weak localization correction is difference with classical conductance N 1 channels N 2 channels For N 1, N 2 >> 1: Jalabert, Pichard, Beenakker (1994) Baranger and Mello (1994) Same as diagrammatic perturbation theory

RMT: conductance fluctuations Quantum dot Without time-reversal symmetry: With time-reversal symmetry: N 1 channels N 2 channels Jalabert, Pichard, Beenakker (1994) Baranger and Mello (1994) Same as diagrammatic perturbation theory There exist extensions of RMT to deal with contacts that contain tunnel barriers, magnetic-field dependence, etc.

Quantum transport and its classical limit Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Lecture 3 Capri spring school on Transport in Nanostructures, March 25 -31, 2007

Ballistic quantum dots Past 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 dots Assumptions and restrictions: l. F << l, g >> 1 Method: semiclassics, quantum properties are obtained from the classical dynamics

Semiclassical Green function Relation between transmission matrix and Green function Semiclassical Green function (two dimensions) a: classical trajectory connection r’ and r S: classical action of a ma: Maslov index Aa: stability amplitude a r’ r q’ r ’

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

Semiclassical scattering matrix Insert semiclassical Green function and Fourier transform to y, y’. This replaces y, y’ by the conjugate momenta py, py’ and fixes these to Result: Jalabert, Baranger, Stone (1990) Legendre transformed action q y a

Semiclassical scattering matrix Transmission matrix Legendre transformed action Stability amplitude transverse momenta of a fixed at q y a Reflection matrix

Diagonal approximation Reflection probability a b Dominant contribution from terms a = b. probability to return to contact 1

Enhanced diagonal reflection Reflection probability b=a a If m=n: also contribution if b = a timereversed of a: Without magnetic field: a and a have equal actions, hence Factor-two enhancement of diagonal reflection Doron, Smilansky, Frenkel (1991) Lewenkopf, Weidenmueller (1991)

diagonal approximation: limitations We found One expects a corresponding reduction of the transmission. Where is it? The diagonal approximation gives b=a a Note: Time-reversed of transmitting trajectories contribute to t’, not t. No interference! Compare to RMT: captured by diagonal approximation missed by diagonal approximation

Lesson from disordered metals a Weak localization correction to reflection: Do not need Hikami box. b Weak localization correction to transmission: Need Hikami box. a b = + + permutations ‘Hikami box’

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

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

Ballistic Hikami box? l: Lyapunov exponent Initial uncertainty is magnified by chaotic boundary scattering. Time until initial uncertainty ~l. F has reached dot size ~L: L=l. F exp(l t) t= “Ehrenfest time” Aleiner and Larkin (1996) Richter and Sieber (2002) Interference corrections in ballistic quantum dot same as in disordered quantum dot if t. E << t. D

Ballistic weak localization Probability to remain in dot: special for ballistic dot also for disordered quantum dot: included in RMT t. E t loop Aleiner and Larkin (1996) Adagideli (2003) Rahav and Brouwer (2005)

Semiclassical theory Landauer formula Jalabert, Baranger, Stone (1990) • Sa, Sb: classical action • angles of a, b consistent with transverse momentum in lead, • Aa, Ab: stability amplitudes

Semiclassical theory Landauer formula s, u: distances along stable, unstable phase space directions encounter region: |s|, |u| < c c: classical cut-off scale Action difference Sa-Sb = su tenc (0, 0) (s, u) s a e -l t u a e lt Richter and Sieber (2002) Spehner (2003) Turek and Richter (2003) Müller et al. (2004) Heusler et al. (2006)

Semiclassical theory Landauer formula s, u: distances along stable, unstable phase space directions tenc (0, 0) (s, u) t’ P 1, P 2: probabilities to enter, exit through contacts 1, 2 Aleiner and Larkin (1996) Adagideli (2003) Rahav and Brouwer (2005) c: classical cut-off scale

Classical Limit Take limit l. F/L 0 without changing the classical dynamics of the dot, including its contacts L diverges in this limit! 0 Aleiner and Larkin (1996) but… var g remains finite! Brouwer and Rahav (2006) THE END