Lectures on the theory of Anderson localization

Transcription

1 Lectures on the theory of Anderson localization Enrico Fermi School on Nano optics and atomics: transport in light and matter waves June 3 to July 3, 009 Peter Wölfle Institut für Theorie der Kondensierten Materie (TKM), Universität Karlsruhe DFG-Center for Functional Nanostructures (CFN) Institut für Nanotechnologie (INT), Forschungszentrum Karlsruhe, Germany Karlsruhe Institute of Technology

2 Lecture I: Basic notions of Anderson localization Introduction Electrons and classical waves in disordered systems Strong localization and Anderson transition Weak localization One-dimensional systems Quasi-onedimensional systems

3 Introduction: Anderson localization Schroedinger s eq. for amplitudes a j on lattice sites j loc Probability distribution of self energy V c ext If V c 0 as s 0, with probability one: states at energy 0 are localized

4 Scattering of quantum particles in a random potential Multiple scattering of particle (wave packet) may lead to localization Envelope of wave function decays exponentially with distance

5 Electrons in disordered systems Schrödinger equation of an electron of mass m in a random potential V h r r r [ + V ( ) E] ψ ( ) = 0 m White noise random potential: r r r r V V V δ ( ) ( ') = ( ') Scattering rate (density of states N(E)): Mean free path (particle velocity v) : h ( ) τ = π N E V l = vτ

6 Classical waves in disordered systems Scalar wave equation for wave amplitude in a random medium r ω r ( ) 0 r ψ c ( ) + r = Wave velocity at position r: c(r) Realistic model: spheres at random positions R(i) and with velocity c 1 embedded in medium with velocity c 0 Comparison of wave equation and Schrödinger equation: ω r c ( ) m r ( E V ( )) h Main difference: randomness in wave equation gets weaker in limit ω 0, in contrast to Schrödinger equation in the limit E 0 It is much harder to localize classical waves

7 Strong localization and Anderson transition In the limit of strong disorder particles will get trapped at positions where the random potential forms deep wells Classical particles will be delocalized at energies E > E 0 Quantum particles may be localized at any energy, depending on spatial dimension, as a consequence of quantum interference The admixture of adjacent orbitals by hopping/tunneling is a minor perturbation, since nearby orbitals have very different energy ψ = ψ + ψ i i ψ 0 0 i 1 1 E 1 H ψ E i Orbitals with closeby energies are far apart in space

8 Strong localization and Anderson transition Localized orbitals may not coexist at the same energy with extended states. Any small hybridization will delocalize the localized orbital. Localized states are separated from delocalized states in energy The critical energy E c is termed mobility edge (N.F. Mott, 1968) Likewise, for fixed energy, as a function of disorder strength η, the nature of the states changes from localized to extended (if extended states are possible at all in the system considered) at a critical value η c Anderson transition

9 Phase diagram: electrons with on-site disorder Electrons on a 3-d cubic lattice with on-site disorder: uniform distribution of energies W/ < E < W/ Solid line: Self-consistent theory (J. Kroha, T. Kopp and P. Wölfle, 1990) Dashed lines: bounds on the band edge Dots: Numerical study (B. Kramer, A. MacKinnon, 1993)

10 Phase diagram: scalar waves with random scatterers Scalar waves of frequency ω in 3-d system of random point scatterers of average separation a and strength characterized by the average dielectric constant ε = 1/c Solid line: Self-consistent theory (J. Kroha, C. Soukoulis and P. Wölfle, 1993)

11 Weak localization Interference of quantum transition amplitudes for different paths. Transmission probability r r * T (, ') = A + A +.. = A + A Re{ A A +..} Interference terms fluctuate in sign, and will cancel on the average, except if particle returns to origin (r=r ), and amplitudes of time reversed paths A and A interfere: A A r r T (, ) = A + A' = T + T = T cl q cl A A A A provided time reversal symmetry is valid and therefore A=A. This constructive interference process leads to enhanced backscattering of the particle and hence reduces the mobility and therefore the conductivity. (Abrahams, Anderson, Licciardello and Ramakrishnan, 1979)

12 Weak localization The motion of the particle in the random potential is diffusive. The probability PdV for a particle starting at the origin at time t=0 to be in a volume element dv at point r after time t is a solution of the diffusion equation P 1 D P = D = v 0 0, 0 t d τ r P(, t) = (4 π D t) exp( r / 4 D t) d / 0 0 Interference will take place within a lateral width of the path of order wave length λ, and a length element vdt in the time interval dt, such that dv=λ d-1 vdt. The relative change of the conductivity in d dimensions caused by enhanced backscattering may then be estimated as (Gorkov, Larkin and Khmelnitskii, 1979) δσ σ τ φ d 1 vλ dt (4 π D t) 0 τ 0 d /

13 Weak localization Here the time integration extends from the shortest time τ possible for a diffusion process to the time τ φ at which phase coherence is lost. The weak localization (WL) quantum correction depends strongly on spatial dimension d: λ c3 (1 τ / τ ), d 3 φ = l δσ λ c ln( τ / τ ), d = φ σ 0 l c τφ τ d = ( 1 / 1), 1 In the limit of zero temperature, we expect complete quantum coherence and hence τ / τ φ Then, in dimension d=1 the WL correction is seen to diverge as wheras in d= the divergence is only marginal, ln( τ / τ ) In one and two dimensions all states are localized φ τφ / τ

14 Consequences of weak localization In any realistic system, the phase coherence time τ φ is finite. Processes limiting phase coherence are: (1) Inelastic scattering (e.g. by e-e interaction: lecture III): 1/ τ ~ p φ T () Spin flip scattering (magnetic impurities) (3) Spin-orbit scattering (4) Magnetic field H (Phase shift by the magnetic flux enclosed by a self-retracing path): 1/ τ = π DH / Φ φ 0 λ p / c3 (1 O( T )), d = 3 l δσ λ c ln( T0 / T ), d = σ 0 l c1 ( T0 / T 1), d = 1 λλ 1/ (1 O( H )), l δσ λ ln( H0 / H ), σ 0 l ( H0 / H 1), The negative magnetoresistance is widely observed

15 Systems of restricted dimensionality Characteristic length over which a wave packet retains phase coherence: L = Dτ φ φ Effective dimensionality of films of thickness a or wires of diameter depends on the ratio L / a φ a At L / a 1 the system is effectively threedimensional, φ while at L / a 1 the dimension is reduced to d= (film) or d=1 (wire) φ

16 One-dimensional systems Conductance of a one-dimensional wire is given by the transmission probability T(E) at the Fermi level (R. Landauer, 1957) G e = T ( EF ) πh The conductance depends exponentially on the length (Mott and Twose, 1961; Borland, 1963) G e α L where the coefficient α varies from sample to sample. The average value of α defines the inverse localization length α = 1/ ξ An exact result for weak disorder by Berezinskii, 1974 gives a localization length of the order of the mean free path ξ = 4l

17 One-dimensional systems The probability distribution of α in the case of true random disorder is Gaussian (Anderson, Thouless, Abrahams and Fisher, 1980). It follows that the conductance distribution is log-normal ξ g L P( g) exp ln + 4L 4 ξ The width of the distribution is seen to increase with length as L / ξ

18 Quasi one-dimensional systems Systems of one-dimensional character, but with N >> 1 transport channels per cross section, and at weak disorder, such that the mean free path is longer than the diameter of the wire, are called quasi onedimensional. Denoting the transmission probabilities of the eigenmodes by T i, the conductance is given by g = i T ( E) The distribution of eigenvalues may be calculated from a Fokker-Planck- type equation for the joint distribution function p({t}) i (Dorokhov, 198; Mello, Pereira and Kumar, 1988; Beenakker, 1997) termed DMPK equation: N p p 1 1 l = Tj Tj ( Tj 1) J ( Ti ), J = Ti Tj L N + 1 J ( Ti ) j Tj Tj i< j N N 1 p({ Ti }) = exp[ β H ], H = u( Ti, Tj ) + V ( Ti ) Z i i< j i The parameter β=1,,4 for orthogonal, unitary or symplectic symmetry

19 Quasi one-dimensional systems Since the localization length for quasi 1d systems is ξ ~ Nl, there is a metallic regime (ξ >> L) accessible, where the probability distribution of g is Gaussian 15 P( g) exp g ξ L Remarkably, the variance of this distribution does not depend on L or ξ and is therefore universal, of order unity. However, it depends on the symmetry class. universal conductance fluctuations (Altshuler, 1985; Lee and Stone, 1985) Deep in the insulating regime (ξ << L) the conductance is dominated by the single largest eigenvalue, such that the statistics is again log-normal, like in the exactly one-dimensional case

20 Quasi one-dimensional systems In the crossover regime (ξ ~ L) the distribution is very skewed, and can be characterized as a one-sided log-normal distribution (Muttalib and Wölfle, 1999) ξ P(ln g) L = 1 exp[ {cosh (1/ g )}], g < 1 8 / exp[ a( g 1) ], a (3/8) e L πξ, g 1 ξ / L = 0.7 ξ / L = 0.

21 Extension of DMPK-equation to 3d systems The DMPK-equation may be generalized to higher dimensions, by keeping the wave function correlations (matrix K ij ) over the cross section, instead of averaging over it: (Muttalib and Klauder, 00; Markos, Muttalib, Wölfle and Klauder, 004) N p p 1 1 l = Tj T j ( Tj 1) K jj J ( Ti ), J = Ti Tj L N + 1 J ( Ti ) j Tj Tj i< j K / K ij ii Approximating K ij in a simple way, the DMPK-equation may be solved. In the localized regime of 3d systems one finds deviations from the log-normal distribution: there are generically several correlated channels contributing to g Numerical data DMPK-result

22 Lecture I: Summary Wavepackets of quantum particles or classical waves may get localized in a random medium by scattering and interference At a given energy/frequency wavepackets are either localized or delocalized: Anderson transition at critical energy/disorder Coherent backscattering in the delocalized phase leads to weak localization corrections, indicating that all states are localized in d=1, dimensions The conductance of one-dimensional systems fluctuates wildly from sample to sample: log-normal distribution Quasi-onedimensional systems are in the localized or delocalized regime, depending on length. On the delocalized side the conductance fluctuations are universal

23 Lecture II: Theory of Anderson localization. Fundamental concepts Thouless conductance Scaling theory of conductance Renormalization group equation Critical exponents Dynamical scaling Symmetry classes Fractal structure of critical wave functions

24 Thouless conductance In pioneering work by Thouless (1974) the dependence of the conductance of a hypercubic sample of volume L d was analyzed, by first noticing that the conductance may be measured in units of the quantum e /( πh) A sample of size (L) d may be formed by assembling building blocks of size L d. Thouless conjectured that the properties of the larger sample should be entirely determined by the properties of a building block. Its eigenstates are linear combinations of those of the building blocks. The amount of admixture of states of neighboring blocks depends only on two characteristic quantities: The average level spacing: d δε = ( N L ) 0 1 The overlap of wave functions measured in terms of the energy shift induced by changing the boundary condition from periodic to antiperiodic: E

25 Thouless conductance Case A : E << δε = d ( N L ) 0 1 the wave functions are localized Case B : E >> δε = d ( N L ) 0 1 the wave functions are extended Thouless conductance: g = G e π = E /( / h) / δε

26 Scaling theory of localization According to Wegner (1976) the Anderson transition should be viewed as a continuous quantum phase transition exhibiting critical phenomena. There exists a correlation length ξ diverging as a function of disorder strength at the critical point ξ ( η) η η ν The conductivity of a hypercube in d dimensions has dimension (length) -d and should therefore scale as σ η ξ η η η η d s ( ) ( c ) ; < c, d > leading to the relation among exponents c s = ν ( d ) The conductance of a hypercube of length L should obey the scaling law g( L, η) = Φ( L / ξ )

27 Renormalization group equation The scaling hypothesis led Abrahams, Anderson, Licciardello and Ramakrishnan (1979) to propose that the dimensionless conductance g as a function of sample length L satisfies the Renormalization Group (RG) equation: d ln g = β ( g) d ln L The β-function may be calculated exactly in the limit of small and large g, and approximately for all g. At g << 1 all states are localized and we have g L At g >> 1 we have a good metal: β L / ξ ( ) e, ( g) ln( g / gc ) < 0 d g L, β ( g) = d

28 β-function The sign of β(g) governs whether g scales to infinity (positive β), or to zero (positive β). In d=1 dimension, presumably β < 0 for all g, and therefore we find that all states are localized. In d=3 dimensions β is positive at g >> 1 and negative at g << 1 : there exists a critical point at g=g c, where β(g c )=0: the Anderson transition. In d= dimensions the β-function tends to zero at g >> 1, so that a more accurate calculation is required in this limit. The leading L-dependent correction term to g may be determined from the weak localization correction, by replacing the dephasing length by L 1 L g( L) = σ 0 ln( ) π l 1 1 It follows that β ( g) =, d =, g 1 π g In d dimensions all states are localized

29 Importance of symmetry in d= dimensions This result is valid for orthogonal symmetry. For symplectic symmetry (case of spin-orbit scattering), β ( g) = +, d = g so that a transition exists in this case. For unitary symmetry (spin scattering), c s c u β ( g) =, d = g leading to localization. Unitary symmetry induced by a magnetic field: the Hall conductivity provides an additional scaling quantity. A delocalized state exists in the center of each Landau subband, leading to the Quantum Hall effect.

30 Approximate β-function for orthogonal symmetry Vollhardt, Wölfle, 198

31 Critical exponents in d=3 dimensions In the neighborhood of the critical point in d=3 dimensions the β-function may be expanded as 1 g gc β ( g) =, g gc gc y g c Integrating the RG equation for g>gc from g(l)=g0 to β 1 at large L one finds g(l)=σl, with 1 y 1 y σ [ g( l) g c ] [ η c η] l l We conclude that the inverse of the slope of the β-function is equal to the conductivity exponent : s=y On the localized side one finds y g( L) g exp[ c( g g( l)) L / l] g exp ~ [ L / ξ ] c c c and hence the localization length exponent is found as ν=y. Numerical studies give s=1.58 for orthogonal symmetry. (Kramer and MacKinnon, 1993; Slevin and Ohtsuki, 1999)

32 Numerical studies of disordered systems Numerical studies use the transfer matrix method, allowing to calculate the eigenstates of the Hamiltonian of rod-shaped systems of length L and width M. For any finite M the localization length ξ M is Found to obey the scaling relation ξm ξ Λ = = f ( d ) M M from which the correlation length of the infinite size sample may be obtained d= d=3 Numerical studies give s=1.58 for orthogonal symmetry. (Kramer and MacKinnon, 1993; Slevin and Ohtsuki, 1999)

33 One-parameter scaling in anisotropic systems Charge transport in anisotropic systems (anisotropic band structure, anisotropic scattering) is described by a conductivity tensor with (in d=3) three different eigenvalues σ ii. How do these eigenvalues scale? It can be shown that by scaling distances (momenta) in the x,y,z-direction according to the ratios of conductivity components σ ii, there remains only one scaling parameter, the geometric mean σ = ( σ σ σ ) xx yy zz 1/3 in excellent agreement with experiment. (Wölfle and Bhatt, 1984)

34 Scaling of frequency dependent conductivity Near the critical point spatial distances scale with the localization length ξ ~ t -ν, where t = η η c is distance of the disorder parameter from the critical value. Since the frequency of the critical modes is related to their momenta as ω ~ q z where z is the dynamical critical exponent, the conductivity σ(ω) in d dimensions takes the scaling form [Wegner, 1976; Shapiro and Abrahams, 1981] σ ω η ξ ξ ω ω d ( ; ) = Φ ( L ω / ), L ω = id ( ) / Here D(ω) is the diffusion coefficient, which is related to σ(ω) by the Einstein relation (N(E) is the density of states) σ ( ω) = N( E) D( ω) At the critical point in d=3 the scaling relation is a self-consistent equation 1 1/ 3 σ ( ω, η) ( iω), η = η, z = 3 L ω c

35 Accessing the scaling regime at finite temperature At finite temperature, the dephasing rate enters as the imaginary part of the frequency ω i / τφ Assuming that the dephasing rate obeys a temperature power law ~ T p, the T-dependent conductivity in the scaling regime takes in d=3 the form ω > (1/ τ )( ξ / l) z σ η ξ ξ 1 1 p / 3 ( T; ) = Φ( T ) Experimental data should collapse on to the scaling function, allowing to extract the critical exponent s. In electronic systems the Coulomb interaction between electrons is known to modify the critical behavior, so that the Anderson transition is not accessible in a clean way.

36 Conductivity data on Gd films at L φ < thickness Misra, Hebard, Muttalib and Wölfle, 009

37 Scaling of conductivity σ(t) of Gd films The conductivity data of Gd films at temperatures, where L φ < b (thickness) collapse onto scaling curves. The critical exponents obtained are z ~.4 and s ~ 1 (Misra, Hebard, Muttalib and Wölfle, 009)

38 Symmetry classes of Anderson localization The first symmetry classification of random systems has been given for Random Matrix Models and led to three classes. The scheme considers Time-reversal (T) and spin-rotation (S) symmetries. (Wigner, 1951; Dyson, 196) Gaussian orthogonal ensemble (GOE): T and S preserved Gaussian unitary ensemble (GUE): T violated, S preserved Gaussian symplectic ensemble (GSE): T preserved, S violated Later, additional 7 additional symmetry classes were discovered (Gade and Wegner, 1991; Altland and Zirnbauer, 1997; Zirnbauer, 1996) Three chiral symmetry classes (tight-binding models on bipartite lattices) Four Bogolyubov-de Gennes symmetry classes (Bogolyubov quasiparticles in unconventional superconductors) The ten symmetry classes correspond to the classical symmetric spaces

39 Nonlinear σ-model A field theoretic description of the impurity problem is complicated by the fact that there is only one particle in the system at any time: technically this means that there are no closed loop diagrams. In order to project out the closed loop diagrams Wegner (1979) employed the replica trick: considering N replicas of the system and taking the limit N 0 removes closed loop contributions. A mathematically better defined procedure is to introduce bosonic and fermionic fields, such that the two contributions cancel the closed loop contributions (Efetov, 1983). This requires a supersymmetric structure of the theory. The diffusion propagator Φ may then be expressed as Φ ( r, r '; ω) = G ( r, r ') G ( r ', r) = dqq Q e R A bb bb S[ Q] E+ ω / E ω / 1 1 where the σ-model action of 4x4 supermatrices Q is defined as π N( E) S Q d rstr D Q i Q 4 d [ ] = [ ( ) Λ ]

40 Fractal structure of critical wave functions The spatial structure of the wave functions at the critical point is of multifractal nature (Wegner, 1980; Evers and Mirlin, 008). The inverse participation ratios defined by q d r P d r ψ ( ), q real q = averaged over disorder, show anomalous scaling with the size L of the system d q ( ) ~ q Pq = L ψ r L τ One may define the fractal dimensions: D = ( q 1) / τ r The spectrum of fractal dimensions may be characterized by a function f(α): q f ( ) α + P ~ d L α α q In the limit L, τ is obtained from f(α): τ q = qα 0 f ( α0), q = f '( α0) q q

41 Spectral function and dimension of fractal structure (Mildenberger et al., 00; Evers and Mirlin, 008).

42 Lecture II: Summary Conductance and sensitivity to boundary conditions: a single parameter controlling transport Scaling theory of conductance: similarity to critical behavior near continuous phase transitions Renormalization group equation: β-function at small and large conductance Critical exponents near the Anderson transition Dynamical scaling and temperature dependence of conductivity Ten symmetry classes of disordered systems Multifractal structure of critical wave functions

43 Lecture III: Theory of Anderson localization. Diagrammatic approaches Density response in renormalized perturbation theory Identification of singular contributions to the inverse diffusion coefficient: self-consistent theory of diffusion Results of self-consistent theory Apparent Anderson transition in slab geometry Dephasing rate due to e-e interaction

44 Self-consistent theory of localization Building blocks of diagrammatic theory in momentum space: Disorder averaged retarded single particle Green s function R G ( E) = [ E k / m Σ ( E)] k R 1 k Random potential correlator: The self-energy Σ is noncritical and may be approximated by an imaginary constant i / τ in most cases Diffusion coefficient D and density response function at wave vector q and frequency ω V D( q, ω) q χ( q, ω) = χ 0 iω + D( q, ω) q In the absence of interaction: χ 0 = N( E F )

45 Particle-hole response function Density response in terms of particle-hole Green s function: ω ω χ( q, ω) = k, k '( q, ω) χ0 ( q, ω) χ0 π i Φ + = π i Φ + k, k ' Bethe-Salpeter equation involving irreducible vertex U Φ ( q, ω) = G G [ δ + U ( q, ω) Φ ( q, ω)], k = ( k ± q /, E ± ω / ) R A k, k ' k k k, k ' kk '' k '', k ' k '' + ± Reexpressed in the form of a kinetic equation: r r ( ω k q / m Σ ) Φ = G [ δ + U Φ ], Σ = Σ Σ + R A k k, k ' k k, k ' kk '' k '', k ' k k k k '' Summed over k,k : continuity equation ωφ( q, ω) qφ ( q, ω) = πin( E ), Φ ( q, ω) = Φ ( q, ω) j F j kk ' k, k ' Used Ward identity: Σ k = U kk ' Gk ' k ' Vollhardt a. Wölfle, 1980

46 General equation for the diffusion coefficient In the hydrodynamic limit ωτ 1 the current density is proportional to the gradient of the density Φ j ( q, ω) = iqd( q, ω) Φ( q, ω) Multiplying the Bethe-Salpeter equation by kq/m and summing over k and k one recovers the above relation where E D / D( q, ω) = 1 η ( k r qˆ ) G G U ( q, ω) G G ( k r ' qˆ )] + + R A R A 0 k k kk ' k ' k ' mn k, k ' And the disorder parameter is defined as η= π N( EF ) V = π E τ 1 F

47 Diffusion pole in the vertex function Symmetry of the full vertex function in the case of time reversal symmetry ( q, ) ( k r k r Γ rr ω = Γ r r r r r r + ', ω) kk ' ( k k ' + q)/,( k ' k + q)/ Leading singular contribution to Γ : diffusion pole (reducible diagrams) Γ ( q, ω) = D 1 1 π N( E ) τ iω + Dq F

48 Diffusion pole in U: self-consistent equation for D Leading singular contrib. to U : maximally crossed (irreducible) diagrams U ( q, ω) = 1 1 r r div kk ' π N( EF ) τ iω + D( k + k ') Substituted into equation for D (in limit q 0, assuming D>0) d 1/ l d 1 D0 kf Q = 1+ dq D( ω) π m iω + D( ω) Q 0

49 Results of self-consistent theory: conductivity Self-consistent equation for diffusion coefficient 1/ l d 1 d Q = 1 ηdkf dq 0 ω 0 D( ω) D i / D( ω) + Q (1) d.c. conductivity in <d<4 dimensions: η σ = σ 0(1 ), η < ηc = η c 1 3 π critical exponent s = 1 () a.c. conductivity in d=3 dimensions at the critical point: 1/3 ( ) = 0( ), = c σ ω σ ωτ η η critical exponent z = 3

50 Results of self-consist. theory: localization length (3) localization length ξ in <d<4 dimensions: = lim ( i / D( )), > c ξ ω 0 ω ω η η η ξ η η d = cdl 1, > ηc critical exponent ν=d- (4) localization length ξ in d dimensions: c 1 ξ = l = η 1/ exp 1, d ξ = cl, d = 1 coefficient c~.6, compared to exact result c=4

51 Results of self-consistent theory: β-function (6) RG β-function derived from L-dependent diffusion coefficient g gc β ( g) =, g > gc, d = 3 g 1 1+ x x x β ( g) = 1 e, g < g, 3 c d = π g 1+ x 1+ x where x=x(g) is given by 1 x 1 g = (1 + x) e (1 xarctan ), d = 3 π x Vollhardt, Wölfle, 198

52 Shortcomings of self-consistent theory Critical exponents s, ν, do not agree with numerical studies Width of critical region: validity of self-consistent theory Extensions of s.c. theory to unitary and symplectic cases not known q-dependence of diffusion coefficient not included Fractal structure of critical wave function not described

53 Numerical study of slab-shaped systems Numerical studies of finite-size quasi-twodimensional systems of dimension b << m << L indicate a localization transition as a function of thickness b [R.K.B. Singh and D. Kumar, Phys. Rev. B66, (00)] Model: Tight-binding model; site-diagonal disorder; box-shaped distribution of width W Method: Numerical calculation of transfer matrix; L<10 4, m<16, b<6 Criterion of localization: Localization length ξ(m) ξ, m [McKinnon, Kramer, 1981 ; Pichard, Sarma, 1981 ]

54 Scaling properties of conductance in disordered films Scale dependent correction to conductivity σ of a film of thickness b and area L in the limit of weak disorder: σ 1/ ( b, L) 3π 1 1 = 1 l r σ 0 ε Fτ kf bl q qx + qy + qz Momentum eigenvalues q=π(n x /L,n y /L,n z /b), n i integer ; lowest eigenvalue q=0 is excluded (particle number conservation); mean free path l; b << L integration over q x, q y σ sinh 1/ l 1/ l b ( b, L) l dq l l 3 = 1 λ 1 η ln, η b σ 0 b = = q 0 q q sinh ( ) z = + 1/ L z b L kfl Conductance in plane of film in regime l << b << L : kfl 1 L g( b, L) = kfb(1 η) ln 3π π b

55 Self-consistent equation: static limit I In the metallic phase (ξ > L) and iωξ D, and one finds a solution at weaker disorder η < η c than in 3d D l L = 1 λ(1 + ln ) = 1 λ(1 + x) D b b 0 For fixed η and L and tuning b the diffusion constant tends to zero at b=b c, and close to the transition one finds where b c and x c are a solution of D l bc b = (1 η + η ) = η( xc x) D b b 0 c c l b c L ln b c = x = c 1 η η The systems seems to approach a transition at a finite disorder value, but for any finite b in the limit L all states are localized.

56 Self-consistent theory: Phase diagram Phase boundary separating localized and delocalized phases in the thickness disorder plane at finite L Model: Tight-binding model; site-diagonal disorder; box-shaped distribution of width W Data points: numerical results by Singh and Kumar

57 Destruction of localization by dephasing processes At any finite temperature (or in any nonequilibrium state) inelastic processes, more precisely, dephasing processes limit the phase coherence of particles or wave packets to a finite time interval τ φ or equivalently a corresponding length L = Dτ An important mechanism for dephasing of electrons in disordered metals is provided by the Coulomb interaction between electrons. Its contribution to the dephasing rate may be estimated by interpreting the Coulomb potential experienced by a given electron on account of interaction with all other electrons as a fluctuating electric potential δv(t), leading to the phase shift (Altshuler, Aronov, Khmelnitskii, 198) = φ φ( t) dt eδv ( t ) 1 1 φ Assuming δv(t) to be Gaussian distributed, the phasefactor averages as i e φ 1 = φ exp[ ( ) ]

58 Dephasing by Nyquist noise By equating the exponent to t / τ, the following expression for the φ dephasing rate is obtained t φ t t 1 ( φ ) e dt1 dt δv ( t1 ) δv ( t ) 0 0 τ = = The thermal fluctuations of the voltage (Nyquist noise) are given by δ V ( t ) δ V ( t ) = TR δ ( t t ) 1 T 1 where T is the temperature and R is the resistance of a volume of the size explored by the particle in time h/t, which for a diffusing particle is LT = D / T so that R T = d 1 σ L T ( )

59 T-dependence of dephasing rates In d=3, approximating σ by the Drude conductivity, σ/e ~ E F τ, one finds T = τφ τ E E Fτ F 3/ In d= dimensions R=(σa) -1, where a ~ 1/k F we get 1 1 T ~ τ τφ E F is the thickness of the film, Where a more quantitative calculation leads to an additional factor of ln(e F τ) (Altshuler, Aronov, 198) d=1 dimension is special, since L < φ L T and RT = kf L φ / σ has to be used. The resulting self-consistent equation has the solution 1 1 ~ ( Tτ ) τ τ φ /3

60 W L correction in ferromagnetic Fe films Recent experiments on polycrystalline ferromagnetic Fe films on glass substrate show logarithmic temperature corrections in both, the longitudinal and the anomalous Hall conductivity R xx A ln( / 0 ) R R ( T ) L = A T T xx 0 00 Fe 14Å R=753Ω H=4T 1.1 T o =5.K σ xy /σo xy 1.0 R ( T ) R ( T ) L xy R xy 0 xx 0 00 = A AH ln( T / T ) R xx /R o xx L = e 00 π h T(K) Mitra and Hebard, 005 R xy /R o xy

61 Two mechanisms (dimension d=): Phase relaxation rate e-e-interaction (B. Altshuler, A. Aronov, 1981) τ ln( ε τ / ) T ε τ 1 F tr ee φ = F tr scattering off spin waves (G. Tatara et al., 004) 1 = τ sw φ J 4π ε F g T For lower resistance samples, estimate 1 1 sw τ τ φ ee φ In experiment, for T>5K : 1/ τ ( T ) >> 1/ τ,1/ τ, ω φ s so H ln( τ / τ ) = ln( T / T ) φ 0

62 Weak localization correction Skew scattering mechanism: (see also: Langenfeld a. Wölfle., 1991; Dugaev et al., 001) wl ( δσ xy / σ xy )( σ xx / L00 ) = ln( T / T0 ) AR AAH = 1 wl δσ xx = L00 ln( T / T0 ) A R = 1 Universal behavior Side-jump mechanism: WL-correction small in parameter 1/E F τ If both mechanisms are operative: σ = σ + σ ss sj xy xy xy A R 1 A = AH sj ss 1 + σ / σ Nonuniversal behavior xy xy

63 Disorder dependence of quantum corrections: Fe-films Two substrates: glass (triangles) sapphire (circles) Two regimes: R 0 < 3 kω : 1 A 1, A A 1 +σ / σ R R AH sj ss xy xy In agreement with theory for homogeneous system int (provided A 0 ) R 0 > 3 kω : Granular medium of weakly coupled crystallites R Mitra, Misra, Hebard, Muttalib and Wölfle, 007

64 Lecture III: Summary Density response: Renormalized perturbation theory in the disorder Diffusion coefficient expressed in terms of irreducible vertex function U Diffusion pole structure of U: Self-consistent theory (orthogonal symm.) Results of self-consistent theory: conductivity, localization length Destruction of localization by decoherence: Coulomb interaction