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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.

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## Quantum transport and its classical limit

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**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 What to expect? G1+2 (e2/h) G(e2/h) dG B (mT) G1=G2=2e2/h B (10-4T) B R1+2=R1+R2 Magnetofingerprint Nonlocality 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**• More than one lead: • Nj is number of channels in lead j • Use amplitudes anj, bnj for incoming, • outgoing electrons, n = 1, …, Nj. sample 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, theny* 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 Ij,out mj, T In one dimension: = (nnkh)-1 Buettiker (1985)**Landauer-Buettiker formalism**Linear response mj = m – eVj sample Expand to first order in Vj: Ij,in Ij,out mj, T Zero temperature**Conductance coefficients**sample • Current conservation • and gauge invariance Ij mj=m-eVj • Time-reversal Note: only if B=0 or if there are only two leads. Otherwise 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 V I I V Benoit, 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 geometry**r’ t r, r’: “reflection matrices” t, t’: “transmission matrices” r t’ in e e f(e) f(e) eV out e e e (meV) f(e) Anthore, Pierre, Pothier, Devoret (2003) |t’|2 |r|2 |t|2 |r’|2**Quantum transport**Landauer formula r’ t 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 g 10 x 8 6 4 2 Vgate (V) 0 -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 Van Wees et al. (1988)**Quantum interference**a In general: dg small, random sign 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-1dek/dk r in lead j; r’ in lead k Substitute 1d 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**lF l L t tD tH h/eF terg Elastic mean free time Inverse level spacing: relevant for closed samples Ballistic quantum dot: t ~ terg ~ L/vF, l ~ L Diffusive conductor: terg ~ L2/D**Characteristic conductances**Conductances of the contacts: g1, g2 Conductance of sample without contacts:gsample if g >> 1 • ‘Bulk measurement’: g1,2 >> gsample • Quantum dot:g1,2 << gsample g dominated by sample g dominated by contacts general relationships:**Assumptions and restrictions**Always: lF << l. Well-defined momentum between scattering events Diagrammatic perturbation theory: g >> 1. This implies tD << tH Only ‘nonperturbative’ methods can describe the regime g ~ 1 or, equivalently, times up to tH. Examples are certain field theories, random matrix theory.**G**Quantum interference corrections Weak localization Small negative correction to the ensemble-averaged conductance at zero magnetic field Conductance fluctuations Reproducible fluctuations of the sample-specific 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 dg at zero magnetic field g dg B a = + b + permutations ‘Hikami box’ ‘Cooperon’ Interfering trajectories propagating in opposite directions**Weak localization (2)**Nonzero (negative) ensemble average dg at zero magnetic field g dg 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: (no derivation here) G(e2/h) Disordered quantum dot: B (10-4T) N1 channels N2 channels (derivation later) Mailly and Sanquer (1991) a b F**Weak localization (4)**dg G(e2/h) B (10-4T) a Magnetic field suppresses WL. 10-3 DR/R b 0 H(kOe) F -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 timetD: A(tD/terg)1/2. WL suppressed at F ~ (hc/e)(terg/tD)1/2 << hc/e.**Weak localization (6)**b 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 : dg oscillates with period hc/2e. a F ‘hc/2e Aharonov-Bohm effect’ Altshuler, Aronov, Spivak (1981) Note: phases picked up by individual trajectories are multiples of p, not 2p! Sharvin and Sharvin (1981)**Conductance fluctuations (1)**Fluctuations of dg with applied magnetic field dg Umbach, Washburn, Laibowitz, Webb (1984) a b a b “diffuson” interfering trajectories in the same direction b’ a’ a’ b’ a a “cooperon” interfering trajectories in the opposite direction b b a’ a’ b’ b’**Conductance fluctuations (2)**Fluctuations of dg with applied magnetic field dg Disordered wire: G(e2/h) Disordered quantum dot: B (mT) N1 channels N2 channels Jalabert, Pichard, Beenakker (1994) Baranger and Mello (1994) 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. N1 channels N2 channels Ansatz: S is as random as possible, with constraints of unitarity and time-reversal symmetry, “Dyson’s circular ensemble” Dimension of S is N1+N2. Assign channels m=1, …, N1 to lead 1, channels m=N1+1, …, N1+N2 to lead 2 Bluemel and Smilansky (1988)**RMT: Without time-reversal symmetry**Quantum dot 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 N1 channels N2 channels 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 N1 channels N2 channels Average conductance: Interference correction to average conductance**RMT: with time-reversal symmetry**Quantum dot Weak localization correction is difference with classical conductance N1 channels N2 channels For N1, N2 >> 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: N1 channels N2 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: lF << 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 ofa ma: Maslov index Aa: stability amplitude a q’ r r 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 a y**Semiclassical scattering matrix**Transmission matrix transverse momenta of a fixed at Legendre transformed action q a Stability amplitude y 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 time-reversed 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 b=a a One expects a corresponding reduction of the transmission. Where is it? The diagonal approximation gives 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 dg and varg 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 ~lF has reached dot size ~L: Aleiner and Larkin (1996) Richter and Sieber (2002) Interference corrections in ballistic quantum dot same as in disordered quantum dot if tE << tD L=lF exp(l t) t = “Ehrenfest time”

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