Theory of Disordered Condensed-Matter Systems

Transcription

1 Electronic hopping transport in disordered semiconductors Theory of Disordered Condensed-Matter Systems Walter Schirmacher University of Mainz, Germany Summer School on Soft Matters and Biophysics, SJTU Shanghai July 3-6, 2017 July 5, 2017 Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 1 / ,

2 Electronic hopping transport in disordered semiconductors Outline 1 Disorder and the Glass transition 2 Diffusion in quenched-disordered systems Electronic hopping transport in disordered semiconductors Phonon-assisted tunneling Mott optimization Master equation and Percolation construction Random walk on a lattice and regular diffusion diffusion in a disordered system: spatially fluctuating diffusivity Perturbation theory Analogy with a mass-spring vibrational problem disordered diffusivity and AC conductivity Coherent-potential approximation (CPA) Comparison of CPA calculations with measured AC conductivity data Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 2 / ,

3 Electronic hopping transport in disordered semiconductors Electrons in doped/disordered semiconductors Phonon-assisted hopping transport between localized states Lightly doped crystalline semiconductors Structurally disordered semiconductors (OLED) phonon E i hω = E E j i e electron ψ i ψ j E j localized states Transition probability/time for an electron from ψ i to ψ j W ij = ν 0 e β(e j E i )/k B T e 2α r i r j β = 1/k B T, α = inverse localization length Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 3 / ,

4 Electronic hopping transport in disordered semiconductors Mott optimization and T 1/4 law Mott s argumentation: The electron will try to use a hopping path, in which neither the first nor the second part of the exponential becomes too small. The optimal hop will maximize W ij, i.e. minimize the exponent. The hopping distance will obey on the average ( r) 3 = [N F E] 1 N F = DOS at the Fermi level so that the exponent becomes ξ ij = β E +2α r = β N F ( r) 3 +2α r which is minimized for r opt = ( 3β 2αN F ) 1/4 W opt e (T0/T)1/4 T 0 α 3 /N F Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 4 / ,

5 Electronic hopping transport in disordered semiconductors Master equation and equivalent circuit The random walk of a hopping electron is described by the following Markovian Master equation d dt P i(t) = ) W ij (P i (t) P j (t) P i = Probability to be at i j In the steady state d dt P i(t) = 0 this becomes equivalent to Kirchhoff s node equation with P i being local voltages and W ij conductances between nodes i and j. If the conductances vary exponentially one can use the following percolation construction to estimate the conductance of the network A network of sites, connected with bonds i j is considered Each bond is assigned a conductance Wij = W 0 e ξij Sites with ξij < ξ threshold are connected, where E threshold is a variable theshold, which is increased from a low value Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 5 / ,

6 Electronic hopping transport in disordered semiconductors Increasing the threshold ξ Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 6 / ,

7 Electronic hopping transport in disordered semiconductors Increasing the threshold ξ Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 7 / ,

8 Electronic hopping transport in disordered semiconductors Increasing the threshold ξ Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 8 / ,

9 Electronic hopping transport in disordered semiconductors Increasing the threshold ξ Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU5, Shanghai 2017 July 9 / ,

10 Electronic hopping transport in disordered semiconductors Increasing the threshold ξ Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 10 July / ,

11 Electronic hopping transport in disordered semiconductors Percolation construction Mathematically this construction can be described in terms of the percolation threshold p c ( ) p c = θ ξ c ξ( r, E) ( ) = N( E ) 4π( r) 2 d r d Eθ ξ }{{} c ξ( r, E) N F = πn F 2α 3 β ξc 4 N F T α 3 ( x 2 dxdyθ ξ c x y) which gives the same result as Mott s optimization. Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 11 July / ,

12 Random walk on a lattice and regular diffusion Random walk on a lattice Master equation for a random Walk on a 3-dimensional lattice with lattice constant a and nearest-neighbor hopping W 0 d dt P ( ) i(t) = W 0 P i (t) P j (t) Initial condition: Pi (0) = δ j i,i0 Laplace transform P i (s) = 0 e st P i (t) s = iω +ǫ ( ) sp i (s) P i (0) = W 0 P i (s) P j (s) Bloch s theorem: j which leads to ( P i (s) = e ikr i u(k,s) ) su(k,s) 1 = 2W cos(k x ax)+cos(k y ay)+cos(k z az) 3 u(k, s) Solution u(k,s) = 1 s +2W[3 cos(k x ax)+cos(k y ay)+cos(k z az)] Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 12 July / ,

13 Random walk on a lattice and regular diffusion Random Walk on a lattice Bloch s theorem: which leads to P i (s) = e ikr i u(k,s) ( ) su(k,s) 1 = 2W cos(k x ax)+cos(k y ay)+cos(k z az) 3 u(k, s) Solution u(k,s) = 1 s +2W[3 cos(k x ax)+cos(k y ay)+cos(k z az)] G 0(k,s) This is the Green s function for the regular random walk. For k 0 it becomes 1 G(k,s) s +Wa 2 k 2 This is the solution in k,s space for the diffusion equation d dt P(r,t) = D 2 P(r,t) D = Wa 2 Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 13 July / ,

14 Random walk on a lattice and regular diffusion Ordinary diffusion equation Diffusion equation d dt P(r,t) = D 2 P(r,t) Spatial Fourier transform Laplace transform d dt P(k,t) = Dk2 P(k,t) sp(k,s) P(k,t = 0) = Dk 2 P(k,s) P(k,s) = P(k,t = 0) s +Dk 2 G(k,t) = e Dk2 t [ ] 3 1 G(r,t) = e r2 /4Dt 4πDt = P(k,t = 0)G(k,s) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 14 July / ,

15 diffusion in a disordered system: spatially fluctuating diffusivity diffusion in a disordered system: spatially fluctuating diffusivity We want to explore (as before) the diffusive motion of particles in a disordered system, which move between sites i and j d dt P i(t) = ) W ij (P i (t) P j (t) j We first try to relate this equation to a kind of diffusion equation Then we solve this diffusion equation By perturbation theory By the coherent-potential approximation (CPA) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 15 July / ,

16 diffusion in a disordered system: spatially fluctuating diffusivity Conduction and diffusion in a disordered system Let us consider a d-dimensional hypercube with side length L and volume V = L d. Face-to-face conductance g = σl d 2 σ = N e 2 V k B T D }{{} Einstein relation Let us now tile the volume V into a mesh of smaller hypercubes (voxels) with volume V c. For each of the voxels we can calculate the conductance and apply Einstein s relation to arrive at diffusivities D i = D(r i ), where r i is the center of a certain voxel. In a continuum description we arrive at a diffusion equation d P(r,t) = D(r) P(r,t) dt Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 16 July / ,

17 diffusion in a disordered system: spatially fluctuating diffusivity Conduction and diffusion in a disordered system Let us now tile the volume V into a mesh of smaller hypercubes (voxels) with volume V c. For each of the voxels we can calculate the conductance and apply Einstein s relation to arrive at diffusivities D i = D(r i ), where r i is the center of a certain voxel. In a continuum description we arrive at a diffusion equation and for the Green s function d P(r,t) = D(r) P(r,t) dt d dt G(r r,t t )+ D(r) G(r r,t t ) = δ(t t,r r ) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 17 July / ,

18 Perturbation theory Perturbation theory d dt G(r r,t t )+ D(r) G(r r,t t ) = δ(t t,r r ) We assume that the fluctuations of D(r) are small: D(r) = D 0 + D(r) D 0 = D The unperturbed Green s function is G 0 (k,s) = 1 s +D 0 k 2 Defining G(k,k,s) = 1 V dr dr e ikr e ik r G(r,r,s) we can re-write the diffusion equation for the full problem in k,s space G(k,k,s) = G 0 (k,s)δ k,k ( ) dq d k q D Vc 2π (k q)g(q,k,s) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 18 July / ,

19 Perturbation theory Perturbation theory G(k,k,s) = G 0 (k,s)δ k,k ( ) dq d k q D Vc 2π (k q)g(q,k,s) (1) We represent now the averaged Green s function in terms of a self-energy function Σ(k, s) as G(k,k 1,s) = δ k,k G0 1 (k,s) Σ(k,s) 1 Σ(k,s) = G 0 (k,s) 1 G(k,k,s) k=k Iterating (1) twice and expanding the fraction in (2) to second order in the fluctuations we obtain (using D Vc = 0) Σ(k,s) = 1 (2π) d d d pk(k p)(k p) 2 G 0 (p,s) where K(k) = V c d d re ikr δd(r 0 +r)δd(r 0 ) is the Walter Schirmacher Fourier-transformed (University of Mainz, Germany[.5cm] Theory correlation of Disordered Summer function Condensed-M. School Soft Systems Matters the diffusivity and Biophysics, July SJTU fluctuations. 5, 2017 Shanghai 19 July / , (2)

20 Perturbation theory Perturbation theory Iterating (1) twice and expanding the fraction in (2) to second order in the fluctuations we obtain (using D Vc = 0) Σ(k,s) = 1 (2π) d d d pk(k p)(k p) 2 G 0 (p,s) where K(k) = V c d d re ikr δd(r 0 +r)δd(r 0 ) is the Fourier-transformed correlation function of the diffusivity fluctuations. For short-range correlated fluctuation K(κ) is constant and D 2 and we obtain G 1 0 (k,s) = s +D 0k 2 k 2 D 1 s d/2 s +D(s) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 20 July / ,

21 Perturbation theory Perturbation theory For short-range correlated fluctuation K(κ) is constant and D 2 and we obtain with D 1 D 2. G 1 0 (k,s) = s +D 0k 2 k 2 D 1 s d/2 s +D(s) It can be shown that D(ω) is the Laplace transform of the velocity autocorrelation function D(ω) = 0 e st v x (t)v x (0) }{{} Z(t) This implies Z(t) t (d+2)/2 non-exponential long-time tail Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 21 July / ,

22 Analogy with a mass-spring vibrational problem Analogy with a mass-spring vibrational problem Consider a system of point masses connected with springs with equation of motion for scalar elongations ρ m d 2 dt 2u i = j W ij (u i u j ) By the replacement d 2 dt 2 d dt or ω 2 iω one can transfer conclusions from the disordered diffusion problem to the disordered vibrational problem. Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 22 July / ,

23 Analogy with a mass-spring vibrational problem Analogy with a mass-spring vibrational problem Unperturbed Green s function Unperturbed Green s function G 0 = 1 iω+ǫ+k 2 D 0 G 0 = 1 ω 2 +iǫ+k 2 c 2 0 Green s function with disorder G = 1 iω +ǫ+k 2 D(ω) Frequency-dependent diffusivity D(ω) Nonanalytic property D(ω) ω d/2 Green s function with disorder G = 1 ω 2 +iǫ+k 2 c 2 (ω) Frequency-dependent sound velocity c 2 (ω) Nonanalytic property Im{c(ω)} Γ(ω)/ω ω d Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 23 July / ,

24 Analogy with a mass-spring vibrational problem Analogy with a mass-spring vibrational problem Green s function with disorder G = 1 iω +ǫ+k 2 D(ω) Frequency-dependent diffusivity D(ω) Nonanalytic property Longtime tail D(ω) ω d/2 Green s function with disorder G = 1 ω 2 +iǫ+k 2 c 2 (ω) Frequency-dependent sound velocity c 2 (ω) Nonanalytic property Im{c(ω)} Γ(ω)/ω ω d Rayleigh scattering Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 24 July / ,

25 disordered diffusivity and AC conductivity Diffusivity in disordered materials and AC conductivity Nernst-Einstein relation σ(ω) = N V e 2 k B T D(ω) Loss function ǫ (ω) σ (ω) σ(0) ω ω 1/2 Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 25 July / ,

26 disordered diffusivity and AC conductivity Typical AC hopping conductivity data in amorphous insulators/semiconductors log σ (ω) Temperature ω s ~ s < 1 regime of strong frequency dependence anomalous low frequency regime (only visible in σ(ω) = σ(ω) σ(0) ) log ω Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 26 July / ,

27 Coherent-potential approximation (CPA) Coherent-potential approximation (CPA) on a lattice We consider again the Master equation d dt P i(t) = ) W ij (P i (t) P j (t) j where we now assume that the sites are on a simple-cubic lattice and the transition rates W ij are random numbers with some distribution density P(W ij ). The Green s function of this equation G ij (t,t ) obeys the equation d dt G ij(t,t )+ l Defining a Hamiltonian Matrix { j H ij = W ij W il (G ij G lj ) = δ ij δ(t t ) W ij i = j i j and Laplace transforming we obtain the following matrix equation for the Green matrix < i G(ω) j >= G ij (ω) ( s H ) G = ( iω 2 +ǫ H ) G = 1 Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 27 July / ,

28 Coherent-potential approximation (CPA) Coherent-potential approximation (CPA) on a lattice The CPA is now constructed as follows We invent an effective medium, which is not disordered (i. e. it has the cubic symmetry), but the force constants are frequency dependent: W eff ij (s) = Γ(s) Let Z = 6 be the coordination number of the sites. Then effective-medium Hamiltonian is j Weff ij (s) ZΓ( z) i = j H ij = Wij eff (s) = Γ(s) i j The Green s function of the effective medium obeys the equation of motion sg ij δ ij = ZΓ(s)(G lj G ij ) We have solved this equation already with the result G(k,s) = ij e ikri rj G ij (s) = l arbitrary neighboring site 1 s +Γ(s)[6 2cos(k x a)+2cos(k y a)+2cos(k z a)] Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 28 July / ,

29 Coherent-potential approximation (CPA) Coherent-potential approximation (CPA) on a lattice We now dig a hole into the effective medium. As we deal with pairs of sites this hole must contain a pair (i 0,j 0 ) Inside the hole we replace the effective-medium force constant Γ( z) by the actual one W i0j 0 so that we obtain a perturbation v i0j 0 (s) = W i0j 0 Γ(s) The corresponding perturbing Hamiltonian matrix V has four non-zero entries, namely v i0 i 0, v j0 j 0, v i0 j 0, and v j0 i 0 In the i 0 j 0 subspace we have ( vi0j0 (s) v i0j 0 (s) V = v i0j 0 (s) v i0j 0 (s) ) Effective Medium special pair W ij i 0 j 0 Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 29 July / ,

30 Coherent-potential approximation (CPA) Coherent-potential approximation (CPA) on a lattice We now demand that introducing this perturbation should have on the average no influence on the effective medium which is equivalent to demanding that the Green s function of the effective medium should be equal to the configuratonally averaged Green s function of the disordered system. This means that the T matrix corresponding to V should vanish on the average V T = 1 VG This is a 2 2 matrix equation, with 4 identical matrix elements W Γ( z) 1+(W Γ( z)2[g ij (s) G ii (s)] = 0 and, taking the medium equation of motion into account W Γ( z) 2 1+(W Γ( z) ZΓ( z) (1 zg = 0 ii( z) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 30 July / ,

31 Coherent-potential approximation (CPA) Coherent-potential approximation (CPA) on a lattice T = V 1 VG This is a 2 2 matrix equation, with 4 identical matrix elements W Γ( z) = 0 1+(W Γ( z)2[g ij (s) G ii (s)] and, taking the medium equation of motion into account W Γ( z) 2 1+(W Γ( z) ZΓ( z) (1 zg = 0 ii( z) This is the self-consistent CPA equation for the frequency-dependent rate Γ(s), which is proportional to D(s) on the lattice with fluctuating W ij Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 31 July / ,

32 Coherent-potential approximation (CPA) Off-lattice CPA for a spatially fluctuating diffusivity S. Köhler, G. Ruocco, WS, PRB 2013 continuous effective medium D i local diffusivity in a volume element V i obeying the coarse-grained equation ( iω +D(ω)k 2 )ρ(k,ω) = 0 Self-consistent CPA condition: D(ω) D i (r) 1 ν 3 [D(ω) D = 0 i(r)]λ(ω) E Λ(ω) = 1 N k2 iω+d(ω)k 2 1 D(0) for ω 0 k <k ξ derived as saddle-point of replica field theory can also be obtained as the continuum limit of the lattice CPA. Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 32 July / ,

33 Coherent-potential approximation (CPA) The dc diffusivity in CPA and percolation CPA equation in the dc (ω 0) limit: ν 1 D(0) = D(0) D(0) D i ν 3 = D(0) D i if D(0) 0 Percolation model: P(D i) = pδ(d i D)+(1 p)δ(d i) ν 3 = p D(0) D The parameter ν 3 D(0) = 3 2ν D(p p c) with p c = ν 3 D(0) = 0 for p < p c is the percolation threshold pc. Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 33 July / ,

34 Coherent-potential approximation (CPA) Activated diffusivity and percolation Activated diffusion: D i = D 0e E i/k B T with a given P(E i) CPA equation for ω = 0: ν 3 = pc = D(0) D i Parametrizing D(0) = D 0 2 e E D/k B T we obtain 1 p c = dep(e) e (E E D)/k BT +1 For low enough temperatures this gives p c = ED 0 dep(e) D(0) obeys an Arrhenius law irrespective of the shape of P(E)! Equation (*) corresponds to the percolation construction of hopping conductivity (Ambegaokar, Halperin, Langer, 1971, Shklovski, Efros 1984) ( ) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 34 July / ,

35 Coherent-potential approximation (CPA) Activated diffusivity and percolation The summed energies up to the percolation threshold E D must be equal to the bond-percolation concentration: p c = ED 0 dep(e) D(0) = 1 2 D 0e E D/k B T This equation is identical to the low-temperature limit of the dc CPA equation. Therefore the CPA in this limit (which is equal to Bruggeman s (1935) EMA) is equivalent to the percolation construction. (This result is already known for some time See e.g. Dyre, Schrøder, Rev. Mod. Phys. 72, 873, 2000) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 35 July / ,

36 Coherent-potential approximation (CPA) CPA ac diffusivity calculation for a constant barrier distribution 10 0 σ (ω) / Ω cm K 1273 K 756 K 456 K (ω/2π) / Hz The data points are ac ionic conductivity data compiled by Wong and Angell in their book Glass: Structure by spectroscopy Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 36 July / ,

37 Coherent-potential approximation (CPA) Loss function for the same data Loss function = ( ) σ (ω) σ(0) /ω ω 0.5 ω 0.33 [σ(ω) - σ(0)] /ω ω/σ(0) For small frequency D(ω) exhibits a non-analyticity of the form D(ω) = D(0)+Aω 3/2 Because D(ω) is just the Laplace transform of the velocity autocorrelation function Z(t) this is equivalent to a long-time behaviour Z(t) t 5/2 ( long-time tail ) Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 37 July / ,

38 Comparison of CPA calculations with measured AC conductivity da Measured ac conductivity data in amorphous silicon Loss function [ σ (ω) σ (0) ] /ω -2.4 (A. R. Long et al. Philos. Mag. B 58, 153 (1988) ) log 10 {[σ(ω) σ(0)]/ω} ω 1/2 evaporated a-si at different T (Variable-range hopping) ω -(1-s) log 10 ω Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 38 July / ,

39 Comparison of CPA calculations with measured AC conductivity da Take-home messages In electronic and ionic hopping transport with exponentially varying hopping rates there exists an underlying percolation aspect: The carriers choose the way of the least resistance. Diffusive transport can be well described with a spatially fluctuating diffusivity. Lowest-Order perturbation theory provides an explanation for the long-time tail of the velocity-autocorrelation function. In an analogy between the heterogeneous diffusion equation and heterogeneous sound propagation the diffusional anomaly corresponds to Rayleigh scattering. For disordered diffusion there exists an extensive frequency regime with a strong frequency dependence of the conductivity. The coherent-potential approximation (CPA) describes well the percolative aspect and the frequency-dependence. Walter Schirmacher (University of Mainz, Germany[.5cm] Theory of Disordered Summer Condensed-M. School Soft Systems Matters and Biophysics, July SJTU 5, 2017 Shanghai 39 July / ,