2D electron systems beyond the diffusive regime

Transcription

1 2D electron systems beyond the diffusive regime Peter Markoš FEI STU Bratislava 9. June 211 Collaboration with: K. Muttalib, L. Schweitzer Typeset by FoilTEX

2 Introduction: Spatial distribution of the electron propagating through a disordered sample Weak disorder: homogeneous: electron is everywhere Strong disorder: selected path through the sample Is it possible to describe both limits analytically? Typeset by FoilTEX 1

3 Scattering experiment: A B Conductance g. C D Transfer matrix: ( ) C = T D ( A B ) T = ( t 1 rt 1 r t 1 t r t 1 r ) (1) Conductance g ( Economou and Soukoulis) g = e2 π h Tr t t. Typeset by FoilTEX 2

4 T = ( u u u,v are N N matrices Parametrization of the Transfer matrix )( )( ) 1 + λ λ v λ 1 + λ v. (2) Orthogonal symmetry: u = u and v = v. Symplectic symmetry: u = ku k T, v = kv k T, k = Conductance (e 2 /π h = 1) ( 1 1 ). (3) g = Tr t t = Tr v 1 (1 + λ) 1 v = N a= λ a = N a=1 4 cosh 2 x a /2 (4) g does not depend on matrices u and v. Typeset by FoilTEX 3

5 Generalized DMPK Equation p L z(λ) (L z /l) = 1 J N a λ a [ λ a (1 + λ a )K aa J p ], (5) λ a l is the mean free path J N a<b λ a λ b γ ab, γ ab 2K ab K aa. (6) Original DMPK Equation: K aa 1/L, 1 orthogonal β = 2 unitary 4 symplectic γ ab β (the symmetry parameter) (7) Universal properties of weakly disordered Q1D systems. Typeset by FoilTEX 4

6 General form of the matrix K K ab k ab... represents the ensemble average. Symmetry k ab K ab in diffusive limit Orthogonal Unitary k O ab = L α=1 v αa 2 v αb 2 K O ab = 1 + δ ab L + 1 L k U ab = v αa 2 v αb 2 K U ab = 1 L Symplectic α=1 L k S ab = v αa v αb v αbv α a α=1 K S ab = 2 δ ab 2L 1 v = ( v 11 v 21 v 12 v 22 ) v = ( v 11 v 12 v 21 v 22 ) v = ( v22 v 12 v 21 v 11 ) (8) Typeset by FoilTEX 5

7 Approximation Consider only two parameters: K aa K 11 γ ab γ for all a, b (a b). K O ab = K U ab = b b b K S ab = 1. (9) Typeset by FoilTEX 6

8 Physical interpretation (Muttalib) In the localized regime, K 11 L and lim L K 12 = Therefore γ = 2K 12 K 11 in the insulating regime. The localized regime differs from the metallic regime because the wave function is not homogeneous. For the strong disorder, there is only a few possible paths through the sample: u(x) u(x) γ = β γ K aa... inverse participation ratio for the electron on the back side of the sample Typeset by FoilTEX 7

9 Motivation: Distribution of the conductance in the localized regime.1 L = 1 <ln g> = -16 var ln g = 2.6 L = 2 <ln g> = var ln g = 33.4 L = 4 <ln g> = var ln g = 1,2,15 3D L=26, W=16 ξ=5.41 <ln g>=-1.66 Eq. (83) Γ=.63, γ=γ/2, <ln g>= D L=1, W=26 ξ=1.8 <ln g>=-1.94 P (ln g) P(ln g),1.5, ln g ln g The distribution P(ln g) is not Gaussian (2D symplectic and 3D orthogonal systems) Typeset by FoilTEX 8

10 2D disordered systems Numerical estimation of the matrix K Identification of the critical point from the L- dependence of K and γ What is the role of the symmetry parameter β in the localized regime? (γ ) Typeset by FoilTEX 9

11 Diffusive regime K 11 S.55 K 11 O W = 2 K 12 S 1.7 K 12 O 1.73 W = 2 K 11 U 1.6 K 11 U 1.1 K 11 K /L /L K O ab = 1 + δ ab L + 1. KU ab = 1 L. KS ab = 2 δ ab 2L 1 (1) Typeset by FoilTEX 1

12 Critical regime - symplectic symmetry 1.5 Metal: W < W c γ S 2. W = 2 W = 3 W = 4 W = 5 W = 5.25 W = 5.5 W = W = 5.75 W = 6 W = <g> Insulator: W > W c L γ S = 2K 12 K L Metal: Critical Insulator γ S 4 γ S 2.6 γ S Typeset by FoilTEX 11

13 Absence of critical regime - orthogonal symmetry 2D orthogonal system box disorder γ O γ U W 1 1 L W = 1 W = 2 W = 3 W = 4 <g> 1 W = 1 W = 2 W = 3 W = L Typeset by FoilTEX 12

14 Random matrix theory 1 p( ).5 W = 2 W = 5.84 L = 3 W = 5.84 W = 2 W = 5 What is the form of the probability distribution of = x 2 x 1 In particular, we expect that P( ) β In the limit of strong disorder, γ What is the true - dependence pf P( )? Typeset by FoilTEX 13

15 Probability distribution P(ln ) (β = 4) p(ln ) W <ln g> P P P ln lnp(ln ) (β+1)ln Symplectic models: Numerical data for L = 14, N stat 1 8. Slope: 4.8, 3.78, 3., 3.3 Typeset by FoilTEX 14

16 Probability distribution P(ln ) (β = 1) 1 P P P 1 p(ln ) 1-6 W L <ln g> slope ln Typeset by FoilTEX 15

17 Correlation of x 1 and x 2 (symplectic model) 2 W = 3 L = 2 3 W = 8 L = x x 2.5 x 2 -x x 2 -x x x x x 1 g = λ a = a 4 cosh 2 (x a /2), λ a = [coshx a 1]/2 x 1 < x 2 <... The conductance is large when x 1 is small. However, small value of x 2 x 1 occurs only in samples with large value of x 1. The samples with small have small g and do not affect the statistical distribution of the conductance. Typeset by FoilTEX 16

18 Conclusions Generalized DMPK is suitable for the description of transport in strongly disordered systems New parameters: K 11 and γ can be used to detect critical points γ β in the diffusive regime γ = const < β at the critical point γ in the localized regime Symmetry parameter β determines the distribution P( ) β The statistical properties of the conductance can be estimated from the model with β PRB 65, 14297, PRB 72, , PRB 82, 9423, PRB 82, Typeset by FoilTEX 17