Quantum coherent transport in Meso- and Nanoscopic Systems

Transcription

1 Quantum coherent transport in Meso- and Nanoscopic Systems Philippe Jacquod U of Arizona

2 Quantum coherent transport Outline Quantum corrections to transport: Weak Localization Short tutorial on theoretical approach(es) Spin-rotational symmetric case Broken Spin-rotation symmetry Universal Conductance Fluctuations Symmetries and Reciprocity Relations in Multi-terminal transport

3 Coherent Mesoscopic Corrections to Transport I: Weak Localization QM destructive interferences induce negative corrections to Drude conductivity Expansion in λ F/l In 2D : Fig. From : van den Dries et al., PRL 81 Th. : Gor kov, Larkin, Khmel nitskii, JETP Lett. 79 Abrahams, Anderson, Ramakrishnan, PRL 79 Rev.: Bergmann, Phys. Rep. 83; Lee and Ramakrishnan, RMP 85

4 Coherent Mesoscopic Corrections to Transport I: Weak Localization - Quasiclassical Picture Break TRS - magnetic field φ A Probability to be at C: 2 2 P(C) ~ P+P =2 P (1+cos[φ]) C B QM interferences disappear = positive magnetoconductance

5 Weak Localization - Quasiclassical Picture Break TRS - magnetic field φ A B Probability to be at C: 2 P(C) ~ P+P =2 P (1+cos[2πφ/φ 0 ]) Averaging over loop length/flux distribution 2 C

6 Trademark of Weak Localization - Magnetoresistance (Chang et al., PRL 94) (Chan et al., PRL 95) Note: is the flux quantum ; depends on B-fied and time scales (stay tuned) shape differs for regular systems (diff. enclosed areas)

7 A theoretical approach to weak localization

8 Feynman path integrals Time evolution of wavefunction Evolution kernel Evolution kernel as sum over all possible paths among them classically allowed paths

9 Short wavelength limit : semiclassical theory Evolution kernel ~ sum over classically allowed paths only Classical action Sum over classical trajs. Trajectory s stability Keep only classical trajectories

10 Applications: Theory of Quantum Coherent transport Scattering approach Entrance / exit points Classical trajectories, stability and action R. Whitney and PJ, PRL/PRB 05/06; precursor theory: Richter and Sieber, PRL 02

11 Weak Localization in a nutshell Semiclassical expression x Semiclassical approximation (i) because N=W/λ F >> 1 (ii) Stationary phase approximation ~minimize variations of (energy averaging) S α -S β

12 Weak Localization in a nutshell Diagonal term : classical conductance diagonal approx weaklocalization Drude, i.e. classical contribution Note: multi-encounter expansion - Heusler, Muller, Braun and Haake PRL 06

13 Weak Localization in a nutshell Loop correction : weak localization diagonal approx Leading-order Quantum correction weaklocalization Note: multi-encounter expansion - Heusler, Muller, Braun and Haake PRL 06

14 Weak Localization in a nutshell (ii) Stationary phase approximation : diagonal approx weaklocalization *stability of paired orbits is the same, A α =A β *consider pairs with encounters (iii) Sum rule Classical probability To go from Y to Y 0 in t

15 Weak Localization in a nutshell (iv) Loop correction as a function of angle ε δs = 4 E F ε 2 /4λ SPA selects out Classical probability to go from Y to Y 0 in t with loop

16 Weak Localization in a nutshell Loop correction : weak localization diagonal approx Leading-order Quantum correction weaklocalization

17 Signature of Weak Localization : magnetoresistance External magnetic field B = curl A B<<mv/eL, with L: system size -> R=mv/eB >> L i.e. cyclotron radius much larger than system size -> neglect orbital effects -> only quantum phase effects at first Peierls substitution p -> p - ea/c Accumulated phase on given traj. Additional phase accumulated along a given loop with the flux enclosed by the loop

18 Signature of Weak Localization : magnetoresistance Additional phase accumulated along a given loop External B-field is fixed with the flux enclosed by the loop Directed area enclosed by classical scattering trajs. in chaotic system has Gaussian distribution P(A) with zero mean and a variance linear in the trajector duration* -> the flux distribution is the distribution P(AB) -> integrate phase times distribution of durations P(t)=exp[-t/t D ]/t D gives Lorentzian damping of weak localization *Baranger, Jalabert, Stone, Chaos 93.

19 Signature of Weak Localization : magnetoresistance Lorentzian damping of weak localization!!! Critical field much smaller than flux quantum!!! =O(1) (Chang et al., PRL 94)

20 Weak Antilocalization : spin-orbit interaction Landauer-Buttiker conductance for spin-dependent transport x x Mathur and Stone RL 92;Bolte and Keppeler PRL 98; Zaitsev, Frustaglia and Richter PRB 05; Bolte and Waltner PRB 07

21 Weak Antilocalization : spin-orbit interaction Landauer-Buttiker conductance for spin-dependent transport x x *Calculate average < > *Assume no correlation between orbital and spin d.o.f.

22 Weak Antilocalization : spin-orbit interaction diagonal approx weaklocalization

23 Weak Antilocalization : spin-orbit interaction As SOI increases, crossover of quantum corrections to conductance Theory vs. Experiments

24 Alternative approach: random matrix theory Scattering approach Chaotic cavity! S as a Random Matrix!!Distr. of T s!!conductance!!ucf symmetry index 1 with TRS 2 without TRS 4 without SRS, with TRS

25 Alternative approach: random matrix theory *S-matrix is unitary *If unitarity is the only constraint - broken TRS ->all elements have the same distribution -> conductance Beenakker, RMP 97; Baranger and Mello arxiv:cond-mat/

26 Alternative approach: random matrix theory *S-matrix is unitary *Systems with TRS and SRS -> S is symmetric, use representation blue pairing for i=j red pairing for all -> <U U U U> ~ <U U> <U U> (sum over all poss. pairings) ->average over randomly distributed variables

27 Alternative approach: random matrix theory ->Enhanced reflection probability ->Reduced conductance - weak localization

28 Alternative approach: random matrix theory *S-matrix is unitary *If SRS is broken but TRS is preserved ->S-matrix is antisymmetric ->diagonal elements are zero ->reduced reflection ->enhanced conductance, a.k.a. weak antilocalization Remarkable: semiclassics and RMT agree exactly!!!

29 Alternative approach: random matrix theory This is remarkable: semiclassics and RMT agree exactly!!! Two competing views: * semiclassics can only reproduce known results * semiclassics provides for a microscopic basis for RMT Beenakker, RMP 97; Baranger and Mello arxiv:cond-mat/