Heat conduction and phonon localization in disordered harmonic lattices Anupam Kundu Abhishek Chaudhuri Dibyendu Roy Abhishek Dhar Joel Lebowitz Herbert Spohn Raman Research Institute NUS, Singapore February 2010 (RRI) July 2010 1 / 33
Outline Introduction Heat transport in disordered harmonic lattices Results from kinetic theory Results from localization theory A heuristic theory Numerical results Discussion (RRI) July 2010 2 / 33
Fourier s law For small T = T L T R and system size L : Fourier s law implies : J κ T L The thermal conductivity κ is an intrinsic material property. e.g at room temperature: κ = 0.025 W /mk air = 429 W /mk silver = 2000 W /mk Diamond (RRI) July 2010 3 / 33
Proof of Fourier s law Open problem: Microscopic derivation/verification of Fourier s law in a realistic model with Hamiltonian dynamics. Review article: Fourier s law: A challenge for theorists (Bonetto, Rey-Bellet, Lebowitz) (2000). (RRI) July 2010 4 / 33
Results so far Studies in one and two dimensional systems find that Fourier s law is not valid. The thermal conductivity is not an intrinsic material property. For anharmonic systems without disorder, κ diverges with system size L as: κ L α 1D L α, log L 2D L 0 3D Pinned anharmonic (momentum non-conserving) systems satisfy Fourier s law. A.D, Advances in Physics, vol. 57 (2008). (RRI) July 2010 5 / 33
Disordered Harmonic systems: Results in 1D Exact expression for nonequilibrium heat current. In classical case: J = k B T 2π Z dωt (ω), 0 where T (ω) is the phonon transmission function. Localization implies: T (ω) e N/l(ω) with l(ω) 1/ω 2 for ω 0. Hence frequencies ω < N 1/2 do not see the randomness and can carry current. These are ballistic modes. Hence J R N 1/2 0 T (ω)dω. (RRI) July 2010 6 / 33
Disordered Harmonic systems: 1D Form of T (ω) (at small ω) depends on boundary conditions. T(ω) 1 0.8 0.6 0.4 T(ω) 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 ω 0.2 0 0 0.5 1 1.5 2 ω Fixed BC: T (ω) ω 2 J 1/N 3/2 Free BC: T (ω) ω 0 J 1/N 1/2 If all sites are pinned then low frequency modes are cut off: hence J e N/l. [Matsuda,Ishii,Rubin,Greer,Casher,Lebowitz,Dhar] (RRI) July 2010 7 / 33
Higher dimensional systems What about systems in three dimensions? Is Fourier s law valid? Consider dielectric crystals. Heat conduction is through lattice vibrations. Harmonic approximation: solid can be thought of as a gas of phonons with frequencies corresponding to the vibrational spectrum of ordered harmonic crystal. Phonons get scattered by: (i) Impurities (ii) Other phonons (anharmonicitiy)...keiji s Talk. It is usually expected that these scattering mechanisms should lead to a finite conductivity. (RRI) July 2010 8 / 33
Three dimensional systems Heat conduction in a classical harmonic crystal with isotopic mass disorder. Look at L dependence of J in nonequilibrium steady state. (J 1/L? ) Look at phonon transmission: Effect of localization and boundary conditions. EPL 90, 40001 (2010), PRB 81, 064301 (2010). (RRI) July 2010 9 / 33
Heat conduction in mass disordered harmonic crystal H = X x m x 2 q2 x + X k 2 (qx q x+ê) 2 + X x,ê x q x: scalar displacement, masses m x random. k o = 0: Unpinned. k o > 0: Pinned. k o 2 q2 x (RRI) July 2010 10 / 33
Kinetic theory. In the harmonic approximation the solid can be thought of as a gas of phonons with frequencies corresponding to the vibrational spectrum. The phonons do a random walk. A phonon of frequency ω has a mean free path l K (ω). Thermal conductivity given by: Z ωd κ = 0 dω D(ω) ρ(ω) c(ω, T ) D(ω) = vl K (ω) = phonon diffusivity, v = phonon velocity, ρ = density of states, c = specific heat capacity, ω D = Debye frequency. (RRI) July 2010 11 / 33
Kinetic theory Rayleigh scattering of phonons gives a mean free path l K (ω) 1/ω d+1. For low frequency phonons l K > L, where L is the size of system (ballistic phonons). ρ(ω) ω d 1. Hence κ = Z ωd 1/N L 1/(d+1) dω ρ(ω) vl K (ω) 1D : κ L 1/2, 2D : κ L 1/3, 3D : κ L 1/4. Thus kinetic theory predicts a divergence of the conductivity in all dimensions. For a pinned crystal it predicts a finite conductivity. (RRI) July 2010 12 / 33
Problems with kinetic theory Is a perturbative result valid in the limit of weak disorder. Rayleigh scattering result for l K (ω) is obtained for single scattering of sound waves. Effect of Anderson localization is not accounted for since this requires one to consider multiple scattering of phonons. The kinetic theory result can also be derived directly from the Green-Kubo formula and l K (ω) can be computed using normal mode eigenvalues and eigenstates. Green-Kubo probably correct for normal transport. Probably WRONG for anomalous transport. (RRI) July 2010 13 / 33
Anderson localization Consider harmonic Hamiltonian: H = X x 1 2 mx q 2 x + X x,x q xφ xx q x. x denotes point on a d-dimensional hypercube of size N d. φ - Force-matrix. Equation for q th normnal mode, q = 1, 2,..., N d. [Closed system] m x Ω 2 q a q(x) = X x φ x,x a q(x ). Normal mode frequency: Ω q. Normal mode wave-function: a q(x). (RRI) July 2010 14 / 33
Character of normal modes 1 0.5 0-0.5-1 40 60 Extended periodic mode (Ballistic) 20 40 60 20 5 0-5 40 60 Extended random mode (Diffusive) 20 40 60 20 10 7.5 5 2.5 0 40 60 Localized mode (Non-conducting) 20 20 40 60 (RRI) July 2010 15 / 33
Predictions from Renormalization group theory S. John, H. Sompolinsky, and M.J. Stephen For unpinned system: In 1D all modes with ω > 1/N 1/2 are localized. In 2D all modes with ω > [log(n)] 1/2 are localized. In 3D there is a finite band of frequencies of non-localized states between 0 to ω L c, where ω L c is independent of system-size. System size dependence of current?? Nature of non-localized states?? Ballistic OR Diffusive OR... (RRI) July 2010 16 / 33
Nature of phonon modes Localization theory tells us that low frequency modes are extended. At low frequencies the effect of disorder is weak and so we expect kinetic theory to be accurate. Setting l K (ω) = N gives us the frequency scale ωc K l K (ω) = 1/ω d+1 this gives: 1D: ω K c N 1/2 2D: ω K c N 1/3 3D: ω K c N 1/4 below which modes are ballistic. Since (RRI) July 2010 17 / 33
Nature of phonon modes Kinetic theory and localization theory gives us two frequency cut-offs: ω K c and ω L c : ω Ballistic Diffusive Localized ω K c ω L c 1D : ωc K ωc L N 1/2 2D : ωc K N 1/3, ωc L [ln N] 1/2 3D : ωc K N 1/4, ωc L N 0 Try to estimate the N dependence of J. (RRI) July 2010 18 / 33
Heuristic theory Nonequilibrium heat current is given by the Landauer formula: J = T 2π Z dω T (ω), 0 where T (ω) is the transmission coefficient of phonons. Phonons can be classified as: Ballistic: Plane wave like states with T (ω) independent of system size. Diffusive: Extended disordered states with T (ω) decaying as 1/N. Localized states: T (ω) e N/l L. Find contribution to J of the different types of modes and estimate the asymptotic size dependence. Ballistic contribution to current depends on boundary conditions. (RRI) July 2010 19 / 33
Boundary conditions Free BC: T (ω) ω d 1 ω 0 Fixed BC: T (ω) ω d+1 ω 0 Pinned system: No low frequency ballistic modes (RRI) July 2010 20 / 33
Predictions of heuristic theory 1D unpinned fixed BC: J N 3/2 1D unpinned free BC : J N 1/2 1D pinned: J exp( bn) These have been proved rigorously. 3D unpinned fixed BC: J N 1 3D unpinned free BC : J N 3/4 3D pinned: J N 1 2D unpinned fixed BC: J N 1 (ln N) 1/2 2D unpinned free BC : J N 2/3 2D pinned: J exp( bn) We now check these predictions numerically. (RRI) July 2010 21 / 33
Disordered harmonic systems Equations of motion: m x q x = X^e (q x q x ^e) k oq x {+f x from heat baths } {m x} chosen from a random distribution. f x is force from heat baths and acts only on boundary particles. We choose white noise Langevin baths: f x = γ q x + (2γT ) 1/2 η x(t). η x(t)η x (t ) = δ(t t )δ x,x Periodic boundary conditions in transverse directions. (RRI) July 2010 22 / 33
Measured quantities In the nonequilibrium steady state we measure the energy current and temperature profile. These are given by the following time-averages: J = X x q 1,x q 2,x T x = X x m x,x q 2 x,x We use two approaches: 1. Numerical evaluation of phonon transmission and use the Landauer formula to find J. 2. Nonequilibrium Simulations. (RRI) July 2010 23 / 33
Current in terms of phonon transmission J = T 2π Z dωt (ω), 0 where T (ω) = 4γ 2 ω 2 G 1N 2 and G = [ ω 2 M + Φ Σ L Σ R ] 1 0 1 1 a 1 iγω 1 0... 0 1 a 2 I 0... 0 = B............... C @ 0... 0 1 a N 1 1 A 0... 0 1 a N iγω, a l = 2 + k o m l ω 2. (RRI) July 2010 24 / 33
Product of random matrices Transmission function T (ω) can be expressed in terms of a product of random matrices. ( G 1N = (1 iγ) ˆT 1 N ˆTN 1...ˆT 1 iγ ( ) 2 ml ω where ˆTl = 2 1 1 0 ) Generalization to higher dimensions (N d lattice): T (ω) can be expressed in terms of a product of 2N d 1 2N d 1 random matrices (Anupam Kundu). We implement this numerically for d = 2, 3. (RRI) July 2010 25 / 33
Numerical and simulation results Open system. We look at: Disorder averaged transmission: Current: = J R 0 dω T (ω). T (ω) = [T (ω)] N d 1 Closed system without heat baths. For the normal modes find: Density of states: ρ(ω) = 1 N d P q Inverse participation ratio: P 1 = Diffusivity D(ω) from Green-Kubo. δ(ω Ωq) P x a4 x ( P x a2 x )2. IPR is large ( 1) for localized states, small ( N d ) for extended states. Binary mass distribution: m 1 = 1 +, m 2 = 1 ( = 0.8. (RRI) July 2010 26 / 33
J-versus-N plots: 3D 0.01 Free BC Fixed BC Pinned J N -0.75 0.001 N -1 8 16 32 64 N (RRI) July 2010 27 / 33
J-versus-N plots: 2D 0.1 Free BC Fixed BC 0.01 N -0.6 J 0.001 0.001 0.0001 1e-05 1e-06 N -3.7 4 16 64 N -0.75 4 16 64 256 1024 N (RRI) July 2010 28 / 33
Comparision with numerical results 3D Heuristic prediction (Numerical result) Unpinned fixed BC: J N 1 [ N 0.75 ] Unpinned free BC: J N 3/4 [ N 0.75 ] Pinned: J N 1 [ N 1 ] 2D Heuristic prediction [Numerical result] Unpinned fixed BC: J N 1 (ln N) 1/2 [ N 0.75 ] Unpinned free BC: J N 2/3 [ N 0.6 ] Pinned: J exp( bn) [ exp( bn) ] Look at Phonon transmission functions. (RRI) July 2010 29 / 33
Phonon transmission: pinned 3D N T(ω) 0.32 (i) 0.24 0.16 0.08 N=8 N=16 N=32 ρ(ω) 1.5 0.5 0 2 2.4 2.8 3.2 1 (ii) 0 2 2.4 2.8 3.2 8 8.4 8.8 9.2 9.6 10 10.4 ω Disordered Ordered 8 8.4 8.8 9.2 9.6 10 10.4 ω (RRI) July 2010 30 / 33
Transmission: Unpinned 3D T(ω) 0.08 (i) 0.04 N=8 N=16 N=32 0 0 1 2 3 4 5 6 7 8 N T(ω) 0.8 0.4 (ii) N=8 N=16 N=32 ρ(ω) 0.4 0.2 0 0 1 2 1000 (iv) N=8 3 4 5 6 7 8 N=16 100 N 3 P -1 0 0 1 2 3 4 5 6 7 8 ω 1 0.8 0.6 10 (iii) Ordered Lattice Disordered Lattice 1 0 1 2 3 4 5 6 7 8 ω (RRI) July 2010 31 / 33
Effective phonon mean free path: 3D l eff = NT (ω)/ω d 1 10 ω 4 l eff (ω) 1 0.1 Fixed: N=16 N=32 Free: N=16 N=32 10 1 0.01 0.1 0.01 Fixed Free 0.1 1 0.1 1 ω (RRI) July 2010 32 / 33
Conclusions Unpinned case: Results sensitive to boundary conditions. Heuristic arguments suggest that Fourier s law valid for generic boundary conditions (fixed). Numerical verification presumably requires larger system sizes. Pinned system: First numerical verification of Fourier s law in a 3D system. Pinned system: Direct verification of a 2D heat insulator. Landauer and Green-Kubo formulas. N T (ω)/2π? =? D(ω) ρ(ω) (Proof) Our results suggest this is true for normal transport but not true for anomalous tranport. Future questions Exact results. Improving numerical methods and study larger system sizes. What happens in real systems? Effect of anharmonicity, other degrees of freedom. (RRI) July 2010 33 / 33