Introduction to Theory of Mesoscopic Systems

Transcription

1 Introduction to Theory of Mesoscopic Systems Boris Altshuler Princeton University, Columbia University & NEC Laboratories America Lecture 5

2 Beforehand Yesterday Today Anderson Localization, Mesoscopic Fluctuations, Random Matrices, and Quantum Chaos Interaction between electrons in mesoscopic systems Universal Hamiltonian, 0D Fermi liquid Continue. Inelastic relaxation rates. Dephasing, decoherence, What does it mean?

3 Quasiparticle decay rate at T = 0 in a clean Fermi Liquid. ω +ω Fermi Sea τ e e ( ) F ( 2 ) log( ) Conclusions: 1. For d=3,2 from << F it follows that τ >> e-e h, i.e., that the qusiparticles are well determined and the Fermi-liquid approach is applicable. 2. For d=1 τ e-e is of the order of h, i.e., that the Fermi-liquid approach is not valid for 1d systems of interacting fermions. Luttinger liquids 2 F F d d d = = 2 3 = 1

4 Quasiparticle relaxation rate in 0D case T=0 ω + ω Offdiagonal matrix element M δ ( ω,, ) 1 << g δ 1

5 Quasiparticle relaxation rate in 0D case T=0 Fermi Golden Rule h M τ δ e-e 0< ω< ω< < 0 1 ( ω,, ) 2 τ e-e h L 0D case: hd = δ1 ET 2 L < L, i.e., < E T M δ 1 (L)/g Each gives δ ( U.Sivan, Y.Imry & A.Aronov,1994 ) 1

6 Quasiparticle relaxation rate in disordered conductors T=0 Fermi Golden Rule h M τ δ e-e 0< ω< ω< < 0 1 0D case: L < L, i.e., < E T h hd τ L = d>0 case: L > L, i.e., > E T ( ω,, ) 2 e-e h 1 τe-e δ δ1 ET At L L the rate is of the order of the mean level spacing δ 1. It should not change, when we keep increasing the system size, i.e. decreasing the Thouless energy L 2

7 Quasiparticle relaxation rate in disordered conductors T=0 Fermi Golden Rule h M τ δ e-e 0< ω< ω< < 0 1 ( ω,, ) 2 0D case: L < L, i.e., < E T hd h L = τ e-e δ 1 ET 2 d>0 case: L > L, i.e., > E T h 1 τ δ e-e L h τ << A. Schmid 1973 e-e g L B.A. & A.Aronov 1979

8 Matrix elements at large,,ω >> E T, energies h M τ δ e-e 0< ω< ω< < 0 1 ( ω,, ) 2 h τe-e g L << M,, ( ω) 3 2+ d 2 1 L 1 Lω 2 d 2 2 δ δ ω ω D M ( ω,, ) δ ( L) 1 2 ω 0

9 Quasiparticle relaxation rate in disordered conductors T>0 h M τ δ e-e 0< ω< ω< < 0 1 ( ω,, ) 2 T=0 h M ( 1 n ) n ( 1 n ) τ δ (, T ) ω + ω + ω e-e ω 1 ( ω,, ) 2 T>0 n = exp 1 T 1 Fermi distribution function a) T=0 -no problems: ω and converges M 2 ω,, ω ω D b) T>0 -a problem: T and diverges! ω Abrahams, Anderson, Lee & Ramakrishnan d 2 d 2

10 T>0 -a problem: 1/τ e-e diverges h T τ ω e-e (, T) ω ( 1 n ) ω 2 d/2 d 2 D B.A., A.Aronov & D.E. Khmelnitskii (1983): Divergence of is not a catastrophe: 1/τ e-e has no physical meaning E.g., for energy relaxation of hot electrons processes with small energy transfer ω are irrelevant. h τ g L Q: Is it the energy relaxation rate? that determines the applicability of the Fermi liquid approach

11 T>0 -a problem: 1/τ e-e diverges h T τ ω e-e (, T) ω ( 1 n ) ω 2 d/2 d 2 D B.A., A.Aronov & D.E. Khmelnitskii (1983): Divergence of is not a catastrophe: 1/τ e-e has no physical meaning E.g., for energy relaxation of hot electrons processes with small energy transfer ω are irrelevant. h τ g L Phase relaxation: in a time t after a collision δ (2π ω t) / h processes with energy transfer ω smaller than 1/τ are irrelevant. h? τ

12 What is Dephasing? What is Dephasing? 1. Suppose that originally a system(an electron) was in a pure quantum state. It means that it could be described by a wave function with a given phase. 2. External perturbations can transfer the system to a different quantum state. Such a transition is characterized by its amplitude, which has a modulus and a phase. 3. The phase of the amplitude can be measured by comparing it with the phase of another amplitude of the same transition. Example: Fabri-Perrot interferometer beam splitter mirror

13 4. Usually we can not control all of the perturbations. As a result, even for fixed initial and final states, the phase of the transition amplitude has a random component. 5. We call this contribution to the phase, δ, random if it changes from measurement to measurement in an uncontrollable way. 6. It usually also depends on the duration of the experiment, t: δ = δ(t) 7. When the time t is large enough, δ exceeds 2π, and interference gets averaged out. 8. Definitions: δ( τ ) 2π τ phase coherence time; 1/τ dephasing rate

14 Why is Dephasing rate important? Imagine that we need to measure the energy of a quantum system, which interacts with an environment and can exchange energy with it. Let the typical energy transferred between our system an the environment in time t be δ(t). The total uncertainty of an ideal measurement is environment () t δ ; t t 0 t! δ } () t δ () t ( t ) δ ( t ) + t quantum uncertainty There should be an optimal measurement time t=t*, which minimizes (t) : (t*)= min t* τ! * * 1 t * min τ

15 Why is Dephasing rate important? δ t* * * 1 t * ( t ) δφ ( t ) min τ τ It is dephasing rate that determines the accuracy at which the energy of the quantum state can be measured in principle.

16 T>0 -a problem: 1/τ e-e diverges h T τ ω e-e (, T) ω ( 1 n ) ω 2 d/2 d 2 D B.A., A.Aronov & D.E. Khmelnitskii (1983): Divergence of is not a catastrophe: 1/τ e-e has no physical meaning E.g., for energy relaxation of hot electrons processes with small energy transfer ω are irrelevant. Phase relaxation: in a time t after a collision δ (2π ω t) / h processes with energy transfer ω smaller than 1/τ are irrelevant. (, T) h τ h T τ ω ω> h τ g L ( 1 n ) ω 2 d/2 d 2 D

17 e-e interaction Electric noise E α Fluctuation- dissipation theorem: Electric noise - randomly time and space - α dependent electric field α E r, t E ( k., ω ) Correlation function of this field is completely determined by the conductivity σ k,ω : E β = ω coth ω α β, ω k σ ω αβ αβ k (, k) 2T k (, k) 2 ω σ k T Noise intensity increases with the temperature, T, and with resistance g

18 E α E β = ω coth α β, ω k σ ω αβ αβ ω k (, k ) 2T k (, k ) 2 ω σ k T h ( L) - Thouless conductance def. e R( L) R( L) - resistance of the sample with length (1d) area (2d) g 2 { } L τ 1 T g L L D τ - dephasing length D - diffusion constant of the electrons

19 1 T τ This is an equation! g L

20 1 T τ This is an equation! g L ( L) L d 2 g L τ where is the number of dimensions: d=1 for wires; d=2 for films, L T ( ) 1 4 d τ T 2 4 ( ) 1 T d = 2 d T 23 d = 1

21 1 T L τ g L D τ Fermi liquid is valid (one particle excitations are well Tτ ( T) defined), provided that >

22 1 T L τ g L D τ Fermi liquid is valid (one particle excitations are well Tτ ( T) defined), provided that > 1. In a purely1d chain, g 1,and, therefore, Fermi liquid theory is never valid.

23 1 T L τ g L D τ Fermi liquid is valid (one particle excitations are well Tτ ( T) defined), provided that > 1. In a purely1d chain, g 1,and, therefore, Fermi liquid theory is never valid In a multichannel wire,provided that is smaller g L > L than the localization length, and Fermi liquid approach is justified

24 How to detect phase coherence? measure the dephasing/decoherence rate Quantum phenomena in electronic systems: Weak localization Mesoscopic fluctuations Persistent current yes yes no

25 Persistent current at zero temperature is a property of the ground state! J E Φ E is the ground state energy Φ J Interaction between electrons can change both E and J, but this does not mean that there is dephasing of the ground state wave function. Measurements of the persistent current as well as of other thermodynamic properties do not allow to extract the dephasing rate. Only transport measurements

26 Magnetoresistance Φ O No magnetic field 1 = 2 O With magnetic field H 1 2 = 2 2π Φ/Φ 0 Φ = HS - magnetic flux Φ = hc/e - through the loop 0 flux quantum

27 Weak Localization, Magnetoresistance in Metallic Wires I 3.1 K 1.17 K 210 mk B R/R L ~ 0.25 mm mk <L φ =(Dτ φ ) L = 1/2 D φ τ φ B (T) V 1d case; strong spin-orbital coupling 2 R h 1 1 A = R + R e 2 L L 2 3 L H L H = h eh A area of the wire cross-section

28 Can we always reliably extract the inelastic dephasing rate from the experiment Weak localization: NO - everything that violates T-invariance will destroy the constructive interference EXAMPLE: random quenched magnetic field B? Mesoscopic fluctuations: YES - Even strong magnetic field will not eliminate these fluctuations. It will only reduce their amplitude by factor 2. A But Slow diffusion of the impurities will look as dephasing in mesoscopic fluctuations measurements

29 Magnetic Impurities - before - after T-invariance is clearly violated, therefore we have dephasing Mesoscopic fluctuations Magnetic impurities cause dephasing only through effective interaction between the electrons. T 0 Either Kondo scattering or quenching due to the RKKY exchange. In both cases no elastic dephasing

30 Inelastic dephasing rate 1/τ can be separated at least in principle ω ω other electrons phonons magnons two level systems

31 THE EXPERIMENTAL CONTROVERSY Mohanty, Jariwala and Webb, PRL 78, 3366 (1997) T -3 T -2/3 Saturation of τ : Artifact of measurement? Real effect in samples?

32 e Zero-point Oscillations Collision between the quantum particle and a harmonic oscillator - energy counted from the Fermi level ( 1 E ) n = ω n T > ω 2. T << ω > ω; n > 0 The particle and the oscillator can exchange energy n=2 n=1 << ω; n = 0 No energy exchange between the oscillator and the particle n=0 Inelastic scattering dephasing Pure elastic scattering No dephasing

33 e Zero-point Oscillations Collision between the quantum particle and a harmonic oscillator - energy counted from the Fermi level ( 1 E ) n = ω n + 2 T << ω << ω; n = 0 No energy exchange between the oscillator and the particle Pure elastic scattering No dephasing 1 2 ω Zero-point oscillations

34 Chaos in Nuclei Delocalization? Without interactions between fermions energy of each of the particles is conserved, i.e., there are as many integrals of motion as there are excited particles. Fermi Sea For a finite Fermi Gas one should expect Poisson situation for the eigenstates of the whole system.

35 Chaos in Nuclei Delocalization? generations Fermi Sea.... Delocalization in Fock space Expansion of a typical eigenstate of the many-body system in the basis of states with given number of excitations involves a large number of terms

36 Quasiparticle relaxation rate in 0D case ω + ω τ e-e h δ1 ET ( U.Sivan, Y.Imry & A.Aronov,1994 ) 2 Becomes incorrect as < E δ soon as provided that T 1 = δ1 g δ1 ( B.A, Y.Gefen. A Kamenev & L.Levitov,1994 )

37 Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states cond-mat/ v1 23 Jun 2005 D. M. Basko Princeton University I.L. Aleiner Columbia University B. L. Altshuler Princeton University, Columbia University, NEC-Laboratories America

38 Phonon assisted hopping conductivity in the Anderson Insulator ph

39 Can hopping conductivity exist without phonons 1. Temperature is finite 2. All one-particle states are localized 3. Electrons interact with each other 4. They are isolated from the outside world Does DC conductivity vanish or it is finite.

40 Does DC conductivity vanish or it is finite σ ( T ) Model dependent? e 2 2 d loc T c h ζ σ = 0 T Insulator metal