Heat conduction in harmonic crystals

Transcription

1 Heat conduction in harmonic crystals Abhishek Dhar International centre for theoretical sciences TIFR, Bangalore FPSP-XIII, Leuven June 16-29, 2013 (FPSP-XIII) June / 67

2 Outline Introduction - why is heat conduction an interesting problem? Heat conduction in harmonic crystals Derivation of a general formula for heat current in harmonic systems using the quantum Langevin equations approach. Applications of the heat current formula towards understanding heat conduction in disordered harmonic crystals. Current fluctuations and Large deviation functions for heat conduction. Discussion (FPSP-XIII) June / 67

3 References One dimensional harmonic chains Rieder, E. Lieb and J. L. Lebowitz (JMP, 1967) Casher and Lebowitz (JMP, 1971), O. Ajanki and F. Huveneers (CMP, 2011) Rubin and Greer (JMP, 1971), T. Verheggen (JMP, 1979) A. Dhar (PRL, 2001) S. Das and A. Dhar (EPJB, 2012) Higher dimensions A. Dhar and D. Roy (JSP, 2006) A. Kundu, D. Chaudhuri, D. Roy, A. Dhar, J.L. Lebowitz and H. Spohn (EPL, 2010) A. Kundu, D. Chaudhuri, D. Roy, A. Dhar, J.L. Lebowitz and H. Spohn (PRB, 2011) A. Dhar, K. Saito and P. Hanggi (PRE, 2012) Current fluctuations and large deviations P. Visco (JSM, 2006) K. Saito and A.Dhar (PRL, 2007) A. Kundu, A. Dhar and S. Sabhapandit (JSM, 2011) A. Dhar and K. Saito (PRL, 2011) H. Fogedby and A. Imparato (JSM, 2012) J.S. Wang, B. K. Agarwalla, and H. Li (2012) (FPSP-XIII) June / 67

4 Heat CONDUCTION Heat transfer through a body from HOT to COLD region. T L J T R T L T(x) x T R (FPSP-XIII) June / 67

5 Fourier s law Fourier s law of heat conduction J = κ T (x) κ thermal conductivity of the material. Using Fourier s law and the energy conservation equation ɛ t + J = 0 and, writing ɛ/ t = c T / t where c = ɛ/ T is the specific heat capacity, gives the heat DIFFUSION equation: T t = κ c 2 T. Thus Fourier s law implies diffusive heat transfer. (FPSP-XIII) June / 67

6 Proving Fourier s law Proving Fourier s law from first principles (Newton s equations of motion) is a difficult open problem in theoretical physics. Review article: Fourier s law: A challenge for theorists Bonetto, Rey-Bellet, Lebowitz (2000). It seems there is no problem in modern physics for which there are on record as many false starts, and as many theories which overlook some essential feature, as in the problem of the thermal conductivity of nonconducting crystals. R. Peierls (1961) (FPSP-XIII) June / 67

7 Theory of heat conduction Equilibrium Systems As far as equilibrium properties of a system are concerned there is a well defined prescription to obtain macroscopic properties starting from the microscopic Hamiltonian Statistical mechanics of Boltzmann and Gibbs. For a system in equilibrium at temperature T : F = k B T ln Z Z = Tr [ e βh ] Phase space density : ρ(r 1,..., r N, p 1,..., p N ) = e βh Z Nonequilibrium Systems Heat conduction is a nonequilibrium process and there is no equivalent statistical mechanical theory for systems out of nonequilibrium. We do not have a nonequilibrium free energy. For a system in contact with two heat baths we not know what the appropriate phase space distribution is. (FPSP-XIII) June / 67

8 Theories of heat transport Kinetic theory Peierl s-boltzmann theory Green-Kubo linear response theory Landauer theory This is an open systems approach especially useful for non-interacting systems. Nonequilibrium Green s function formalism Generalized Langevin equations formalism (FPSP-XIII) June / 67

9 Kinetic theory Heat conduction in a dilectric crystalline solid: Basic microscopic picture Ion Electron Heat is carried by lattice vibrations i.e phonons. (FPSP-XIII) June / 67

10 Kinetic theory for phonon gas x Simplest derivation of Fourier s law. Heat carried by phonons which undergo collisions at time intervals τ - hence do random walks. Let ɛ(x) = be energy density at x and v = average velocity of particles. Then: J = 1 2 v [ɛ(x vτ) ɛ(x + vτ)] = v 2 τ ɛ T T x = vl K c T x Therefore J = κ T x with κ = vl K c where c = specific heat, l K = mean free path. Can calculate thermal conductivity κ if we know average velocity, mean free path and specific heat of the phonons. Note that this is not a proof since it assumes diffusive motion! But the heat carriers could be Levy walkers!! (FPSP-XIII) June / 67

11 The set-up for a direct verification of Fourier s law. TL J L TR Dynamics in bulk described by a Hamiltonian: H = N l=1 p 2 l 2m l + U(x l x l 1 ) + V (x l ) where U(x) is the inter-particle interaction potential and V (x) an external pinning potential. Boundary particles interact with heat baths. Example: Langevin baths. m 1 ẍ 1 = H/ x 1 γẋ 1 + (2γk B T L ) 1/2 η L m l ẍ l = H/ x l l = 2,..., N 1 m N ẍ N = H/ x N γẋ N + (2γk B T R ) 1/2 η R In nonequilibrium steady state compute temperature profile and current for different system sizes. k B T l = m l ẋ 2 l and J l = v l f l 1,l, where f l,l 1 is the force on l th particle from (l 1) th particle. (FPSP-XIII) June / 67

12 Heat current and heat conductivity J TL L TR Connect system of length L to heat baths with small temperature difference T = T L T R. We can measure the heat current J and this is always finite and it can be proven that J = G T, i.e that is the current response is linear. Fourier s law J = κ T implies J = κ T L. The size-dependence is difficult to prove. We define a size-dependent conductivity κ L = J L T. We can directly measure this and ask whether lim L κ L exists and agrees with what is given by the Green-Kubo formula. (FPSP-XIII) June / 67

13 Results so far Direct studies (simulations and some exact results) in one and two dimensional systems find that Fourier s law is in fact 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 1/3 1D L α, log L 2D L 0 3D Lepri, Livi, Politi, Phys. Rep. 377 (2003), A.D, Advances in Physics, vol. 57 (2008). Beijeren (PRL, 2012), Spohn (2013) - analytic understanding, connection to KPZ. A.D, K. Saito, B. Derrida (PRE, 2013) - Levy walk picture. (FPSP-XIII) June / 67

14 Experiments. PRL 101, (2008) P H Y S I C A L R E V I E W L E T T E R S week ending 15 AUGUST 2008 Breakdown of Fourier s Law in Nanotube Thermal Conductors C. W. Chang, 1,2, * D. Okawa, 1 H. Garcia, 1 A. Majumdar, 2,3,4 and A. Zettl 1,2,4,+ 1 Department of Physics, University of California at Berkeley, California 94720, USA 2 Center of Integrated Nanomechanical Systems, University of California at Berkeley, California 94720, USA 3 Departments of Mechanical Engineering and Materials Science and Engineering, University of California at Berkeley, California 94720, USA 4 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Received 11 March 2008; revised manuscript received 9 July 2008; published 15 August 2008) We present experimental evidence that the room temperature thermal conductivity () of individual multiwalled carbon and boron-nitride nanotubes does not obey Fourier s empirical law of thermal conduction. Because of isotopic disorder, s of carbon nanotubes and boron-nitride nanotubes show different length dependence behavior. Moreover, for these systems we find that Fourier s law is violated even when the phonon mean free path is much shorter than the sample length. DOI: /PhysRevLett PACS numbers: n, Gh, Fg, Kc FIG. 3 (color online). (a) Normalized thermal resistance vs normalized sample length for CNT sample 4 (solid black circles), best fit assuming ¼ 0:6 (open blue stars), and best fit assuming Fourier s law (open red circles). (b) Normalized thermal resistance vs normalized sample length for BNNT sample 2 (solid black diamonds), best fit assuming ¼ 0:4 (open blue stars), and best fit assuming Fourier s law (open red circles). (FPSP-XIII) June / 67

15 Experiments: graphene. Very large thermal conductivity. κ log(l) has been reported. [Balandin etal, Nat. Phys. (2011), Li etal (2012)] (FPSP-XIII) June / 67

16 Experiments. Science 10 August 2007: Vol. 317 no pp DOI: /science Prev Table of Contents Next REPORT Ultrafast Flash Thermal Conductance of Molecular Chains Zhaohui Wang 1,*, Jeffrey A. Carter 1,*, Alexei Lagutchev 1,*, Yee Kan Koh 2, Nak-Hyun Seong 1,*, David G. Cahill 2,3 and Dana D. Dlott 1,3, + Author Affiliations To whom correspondence should be addressed. dlott@scs.uiuc.edu * These authors contributed equally to this work. ABST RACT At the level of individual molecules, familiar concepts of heat transport no longer apply. When large amounts of heat are transported through a molecule, a crucial process in molecular electronic devices, energy is carried by discrete molecular vibrational excitations. We studied heat transport through self-assembled monolayers of long-chain hydrocarbon molecules anchored to a gold substrate by ultrafast heating of the gold with a femtosecond laser pulse. When the heat reached the methyl groups at the chain ends, a nonlinear coherent vibrational spectroscopy technique detected the resulting thermally induced disorder. The flow of heat into the chains was limited by the interface conductance. The leading edge of the heat burst traveled ballistically along the chains at a velocity of 1 kilometer per second. The molecular conductance per chain was 50 picowatts per kelvin. (FPSP-XIII) June / 67

17 Experiments. (FPSP-XIII) June / 67

18 The Simplest models Ordered harmonic Chain [Rieder, Lebowitz, Lieb (1967)]. T L T R This case can be solved exactly and the nonequilibrium steady state distribution function computed. One finds J (T L T R ) unlike expected (T L T R )/N from Fourier law. Flat Temperature profiles. No local thermal equilibrium. Fourier s law is not valid. This can be understood since there are no mechanism for scattering of phonons and heat transport is purely ballistic. In real systems we expect scattering in various ways: Phonon-phonon interactions: In oscillator chain take anharmonic springs ( e.g Fermi-Pasta-Ulam chain ). Disorder: In harmonic systems make masses random. WE WILL DISCUSS THIS IN DETAIL. (FPSP-XIII) June / 67

19 Heat conduction in disordered harmonic crystals This is best understood using the Landauer approach. Open systems approach to transport. Write coupled Hamiltonian for system and two baths. Evolve initial state ρ = ρ L (T L ) ρ R (T R ) ρ S and obtain the nonequilibrium steady state density matrix. Exact results for non-interacting systems ( both phononic and electronic systems with quadratic Hamiltonians). Current in terms of transmission coefficients. Results can be derived using: Langevin equations approach Keldysh Green functions techniques Path integral approach Simple derivation is the Langevin equations approach which I will derive in this lecture. (FPSP-XIII) June / 67

20 Landauer formula for heat current T L T R J = k B(T L T R ) 4π dωt (ω), 0 where T (ω) = 4γ 2 ω 2 G 1N 2, (Phonon transmission) with the matrix G = [ ω 2 M + Φ Σ] 1 M = mass matrix, Φ = force matrix Σ is a self-energy correction from baths. Quantum mechanical case: J = 1 dω ω T (ω) [ f (T L ) f (T R ) ] 2π 0 where f (ω) = (e β ω 1) 1. (FPSP-XIII) June / 67

21 Harmonic chains The Hamiltonian is ( N p 2 H = l + kox ) 2 N l k(x l+1 x l ) m l=1 l 2 2 l=0 Boundary conditions Fixed BC: x 0 = x N+1 = 0. To drive a heat current in this system, one needs to connect it to heat reservoirs. Various models of baths exist and here we discuss Langevin type baths. (FPSP-XIII) June / 67

22 The classical Langevin equation Langevin baths: Add extra terms in the equation of motion of the particles in contact with baths. In the simplest form, the additional forces from the heat bath consist of a dissipative term, and a stochastic term. For a single oscillator connected to a heat bath, the Langevin equations of motion are ẋ = p m, ṗ = kx γ m p + η(t) where the noise term given by η is taken to be Gaussian, with zero mean, and related to the dissipation coefficient γ by the fluctuation dissipation relation η(t)η(t ) = 2k B T γ(. t t ). The Langevin dynamics ensures that the system reaches thermal equilibrium in the long time limit. More general Langevin baths where the noise is correlated can also be used an example being the Rubin model. Quantum description also possible. (FPSP-XIII) June / 67

23 Quantum Langevin baths- Rubin model System ko mo m k Heat Bath The bath is itself a semi-infinite harmonic oscillator chain with Hamiltonian H b = l=1 P2 l /2m + l=0 k(x l X l+1 ) 2 /2 The bath is initially assumed to be in thermal equilibrium at temperature T and then coupled, at time t =, to the system. System-bath coupling is of the form k x X 1. Starting with Heisenberg equations of motion for x, p it can be shown that the effective equation of motion of the particle is a generalized Langevin equation of the form t ẋ = p/m o, ṗ = k ox + dt Σ(t t )x(t ) + η(t), where the Kernel Σ(t) is given by: Σ(ω) = dtσ(t)e iωt = k{1 mω 2 /2k + iω(m/k) 1/2 [1 mω 2 /(4k)] 1/2 } = ke iq, 0 and q is defined through cos(q) = 1 mω 2 /2k. (FPSP-XIII) June / 67

24 Quantum baths The noise is Gaussian and the correlations are now given by: η(ω) η(ω ) = Γ(ω) π [1 + f (ω, T )] δ(ω + ω ), where η(ω) = (1/2π) dtη L(t)e iωt, Γ(ω) = Im[ Σ(ω)], f (ω, T ) = Alternative form: 1 2 η(ω) η(ω ) + η(ω ) η(ω) = Γ(ω) 2π coth ω 2k B T δ(ω + ω ). Ohmic bath Σ(ω) = iγω. High temperature classical limit gives white noise. 1 e ω/k B T 1. Note Γ(ω) is non-zero only within bath bandwidth infinite for Ohmic baths. (FPSP-XIII) June / 67

25 Definition of Current To define the local energy current inside the wire we first define the local energy density associated with the l th particle (or energy at the lattice site l) as follows: ɛ l = p2 l 2m l + V (x l ) [ U(x l 1 x l ) + U(x l x l+1 ) ]. Taking a time derivative of these equations, and after some straightforward manipulations, we get the discrete continuity equations given by: ɛ l = j l+1,l + j l,l 1 where j l,l 1 = 1 2 (ẋ l 1 + ẋ l )f l,l 1, and f l,l 1 = xl U(x l x l 1 ) is the force that the (l 1) th particle exerts on the l th particle. Hence j l+1,l is the energy current from site l to l + 1. In the steady state using the fact that du(x l 1 x l )/dt = 0 it follows that ẋ l 1 f l,l 1 = ẋ l f l,l 1 and hence: j l,l 1 = (ẋ l + ẋ l 1 )f l,l 1 /2 = ẋ l f l,l 1, and this has the simple interpretation as the average rate at which the (l 1) th particle does work on the l th particle. Similarly it can be shown that the current flowing from reservoirs into the system is given by j L = ẋ 1 f LB, j R = ẋ N f RB, for the two baths respectively. (FPSP-XIII) June / 67

26 Solving the linear Langevin equations for steady state properties-simple example Consider single harmonic oscillator coupled to a single Langevin-type heat bath. ẋ = p/m, ṗ = kx γ p m + η(t), η(t)η(t ) = 2γk B T δ(t t ). Equivalently the system is described by the Fokker-Planck equation for the phase-space probability distribution P(x, p, t) [ ] P(x, p, t) = p t x m P + p kxp + γp p m P + γk BT 2 P p 2. First two terms on r.h.s Hamiltonian evolutionsecond two terms heat bath dynamics. In steady state P SS / t = 0. Here steady state is the equilibrium distribution: P(x, p) = e β [ p 2 ] 2m + kx2 2 Z, where Z = dx [ dp e β p 2 ] 2m + kx2 2. From P SS (x, p) we can then obtain steady state properties, for example x 2 = k B T /k, p 2 = mk B T. (FPSP-XIII) June / 67

27 Method 2: Fourier transforms Define the Fourier transforms x(t) = dωe iωt x(ω), We then get x(ω) = G + (ω) η(ω), where G + (ω) = Noise correlations Hence Similarly η(ω) η(ω ) = γk BT π (. ω + ω ). x 2 (t) = = dω dω = γk BT π u 2 (t) = γk BT π 1 mω 2 + k iωγ. dω e i(ω+ω )t x(ω) x(ω ) dω e i(ω+ω )t G + (ω)g + (ω ) η(ω) η(ω ) dω G + (ω) G + ( ω) = k BT k dω ω 2 G + (ω) G + ( ω) = k BT m. η(t) = dωe iωt η(ω).. (FPSP-XIII) June / 67

28 Steady state current Classical problem with white noise. Gaussian steady state measure. Let q = (x 1,..., x N, p 1,..., p N ) T. Then P SS (q) e qc 1q/2, where C is the correlation matrix with elements x i x j, x i p j and p i p j. Two ways to obtain the correlation matrix C. Method 1 Write the Fokker-Planck equation: P(q, t) = LP(q, t), t Solve LP SS = 0. Linear equation for C which can be explicitly solved for ordered chain Note that current and temperature profile are both given by two-point correlations. (Rieder,Lebowitz, Lieb, 1967). (FPSP-XIII) June / 67

29 Simpler and more general approach- Langevin equations m 1 ẍ 1 = k 1 x 1 k(x 1 x 2 ) + dt Σ + 1 (t t )x 1 (t ) + η 1 (t) m l ẍ l = k ox l k(2x l x l 1 x l+1 ) 1 < l < N m N ẍ N = k N x 1 k(x N x N 1 ) + dt Σ + N (t t )x N (t ) + η N (t). Noise correlations are η L (ω) η L T (ω ) = Γ L(ω) π η R (ω) η R T (ω ) = Γ R(ω) π [ 1 + f (ω, T L ) ] δ(ω + ω ), [ 1 + f (ω, T R ) ] δ(ω + ω ). (FPSP-XIII) June / 67

30 Solution by Fourier transforms As for the single particle case, the cheapest way to obtain steady state properties is by Fourier transforming the equations of motion. We get m 1 ω 2 x 1 (ω) = k 0 x 1 k( x 1 x 2 ) + Σ + 1 (ω) x 1(ω) + η 1 (ω) m l ω 2 x l (ω) = k 0 x l k(2 x l x l+1 x l+1 ) 1 < l < N m N ω 2 x N (ω) = k 0 x N k( x N x N 1 ) + Σ + N (ω) x N(ω) + η N (ω) Useful to write in matrix form [ Mω 2 + Φ Σ + L (ω) Σ+ R (ω) ] X(ω) = ηl (ω) + η R (ω), where X T = { x 1, x 2,... x N }, η L T = { η 1, 0, 0,...}, η R T = {0, 0, 0,... η N}. Σ + L, Σ+ R are N N matrices whose only non-vanishing elements are [Σ + L ] 11 = Σ+ 1 (ω), [Σ+ R ] NN = Σ+ N (ω). (FPSP-XIII) June / 67

31 Thus we get the following solution of the equations of motion: X(ω) = G + (ω) [ η L (ω) + η R (ω) ], where G + (ω) = 1 M ω 2 + Φ Σ + L (ω) Σ+ R (ω). Recall single particle solution: x(ω) = G + (ω) η(ω), where G + (ω) = Noise correlations are now η L (ω) η L T (ω ) = Γ L(ω) π η R (ω) η R T (ω ) = Γ R(ω) π 1 m ω 2 + k iγω. [ 1 + f (ω, T L ) ] δ(ω + ω ), [ 1 + f (ω, T R ) ] δ(ω + ω ). where Γ(ω) = Im[Σ + (ω)]. Solution in time is X(t) = dω X(ω)e iωt. (FPSP-XIII) June / 67

32 Computing steady state current We now calculate the heat current and the local kinetic energies (temperature) in the steady state. These are two point correlation functions. The steady state current can be found by evaluating the current on any bond. To get Landauer form evaluate the boundary current which is [ ] J = ẋ 1 (t) η 1 (t) + dt Σ + 1 (t t ) x 1 (t ). Using matrix notation thi is J = = i [ Ẋ T (t) η L (t) + dω dω e i(ω+ω )t ω = i dω dω e i(ω+ω )t ω ] dt Ẋ T (t) Σ + L (t t ) X(t ) Now use the solution X = G + (ω) [ η L + η R ]. We get [ X T (ω) η L (ω ) + X T (ω) Σ + L (ω ) X(ω ] ) [ Tr η L (ω ) X ] [ T (ω) + Tr Σ + L (ω ) X(ω ) X ] T (ω). (FPSP-XIII) June / 67

33 Computing the current J = i + Tr where we used G +T = G +. ( [ ] dω dω e i(ω+ω )t ω Tr η L (ω ) ( η L T (ω) + ηt R (ω) ) G + (ω) [ Σ + L (ω ) G + (ω ) ] ) ( η L (ω ) + η R (ω ) ) ( η L T (ω) + ηt R (ω) ) G + (ω), Now collect terms in J which depend only on T R - [ J R = i dω dω e i(ω+ω )t ω Tr Σ + L (ω ) G + (ω ) η R (ω ) η R T (ω) ] G + (ω). Using the noise correlations we then get: J R = i dω ω Tr[ Σ + L π ( ω) G (ω) Γ R (ω) G + (ω) ] f (ω, T R ), where we have defined G + (ω) = G ( ω). We also used 1 + f ( ω) = f (ω) and Γ R (ω) = Γ(ω). Now take the complex conjugate of this. (FPSP-XIII) June / 67

34 Computing the current J R = i dω ω Tr[ Σ + L π ( ω) G (ω) Γ R (ω) G + (ω) ] f (ω, T R ), Using (Tr[A]) = Tr[A ]) JR = i dω ω Tr[ G (ω) Γ R (ω) G + (ω) Σ L π ( ω) ] f (ω, T R). Using the cyclic property of trace we get JR = i dω ω Tr[ Σ L π ( ω) G (ω) Γ R (ω) G + (ω) ] f (ω, T R ). Hence taking the real part (J R + J R )/2 and noting that Γ L( ω) = Im[Σ + L ( ω)] = Γ L(ω) gives J R = 1 π dω ω Tr [ G + (ω) Γ L (ω) G (ω) Γ R (ω) ] f (ω, T R ). (FPSP-XIII) June / 67

35 Computing the current Contribution of the left reservoir to the current must have a similar form (since for T L = T R we must have J = 0 hence J = 1 dω ω Tr [ G + (ω) Γ L (ω) G (ω) Γ R (ω) ] [ f (ω, T L ) f (ω, T R ) ] π 1 = dω ω T (ω) [f (ω, T L ) f (ω, T R )] 4π where T (ω) = 4Tr[G + (ω) Γ L (ω) G (ω) Γ R (ω) ] is the phonon-transmission coefficient. Linear response T = T L T R << T, where T = (T L + T R )/2 J = T f (ω, T ) dω T (ω) ω. 4π T Classical limit - ω/k B T 0. J = k B T dω T (ω). 4π (FPSP-XIII) June / 67

36 Local kinetic energy The average kinetic energy at a site is given by k i = [MK ] ii, where the correlation matrix K is given by K = Ẋ Ẋ T = dω ω π [ G+ (ω)γ L (ω)g (ω) ω(1 + f (ω, T L )) + G + (ω)γ R (ω)g (ω) ω(1 + f (ω, T R )) ] = dω ω π [ G+ (ω)γ L (ω)g (ω) ω 2 coth( ω ) + G + (ω)γ R (ω)g (ω) ω 2k B T L 2 coth( ω ) ] 2k B T R where the last line is easily obtained after writing K = (K + K )/2 since K is a real matrix. Classical limit - K = ẊẊT = k BT L dω ω G + (ω)γ L (ω)g (ω) + k BT R π π dω ω G + (ω)γ R (ω)g (ω). (FPSP-XIII) June / 67

37 Note: The expressions for current and kinetic energy density have been written using a somewhat formal matrix notation. The reason for doing so is that in this form they generalize immediately to the case when the wire is not one dimensional but any harmonic system. In the next section we again go back to our one-dimensional chain and will write more explicit expressions. (FPSP-XIII) June / 67

38 Comments on the Landauer formula T is the transmission probability of phonons from the left to right reservoir (Or the transmission probability of phonons from right to left). Same derivation works for arbitrary harmonic networks in any dimension (assuming a unique steady state exists). Derivation can also be done for electronic systems described by tight-binding type Hamiltonian where one can obtain Landauer-type expressions for both the heat and particle current. Results for electron -Landauer conductance ADD Interacting problems (FPSP-XIII) June / 67

39 Current in terms of phonon transmission Consider classical case with Σ 1 (ω) = Σ N (ω) = Σ(ω). J = T 2π dωt (ω), 0 where T (ω) = 4Γ 2 (ω) G 1N 2 and kg = k[ ω 2 ˆM + ˆΦ ˆΣL ˆΣ R ] 1 = Z 1 a 1 Σ (ω) a where Z = a N 1 1, a N Σ (ω) a l = (2k m l ω 2 )/k, Σ = Σ/k. (FPSP-XIII) June / 67

40 For the two heat bath models (a) Casher-Lebowitz and (b) Rubin-Greer, Σ(ω) = iγω model(a) Σ(ω) = k{1 mω 2 /2k iω(m/k) 1/2 [1 mω 2 /(4k)] 1/2 } = k e iq model(b), G + = 1 k Z 1 G + 1N = 1 k Z 1 1N = 1 k 1 N where N = Det[Z ]. Let us define l,m to be the determinant of the submatrix of Z beginning with the l th row and column and ending with the m th row and column. Also let D l,m be defined to be the determinant of the submatrix of [ ω 2 M + Φ]/k beginning with the l th row and column and ending with the m th row and column. Using recursion relations such as 1,N = (a 1 Σ ) 2,N 3,N it is easy to see that N (ω) = D 1,N Σ (D 2,N + D 1,N 1 ) + Σ 2 D 2,N 1 ( ) ( ) = (1 Σ D1,N D ) 1,N 1 1 D 2,N D 2,N 1 Σ (FPSP-XIII) June / 67

41 Product of random matrices Again using relations such as D 1,N = a 1 D 2,N D 3,N we can write ( ) D1,N D ˆD = 1,N 1 = D 2,N D ˆT 1 ˆT2...ˆT N 2,N 1 ( 2 ml ω where ˆTl = 2 ) /k Thus finally we find that G 1N and thus the transmission T (ω) can be expressed in terms of a product of random matrices. N = [kg 1N ] 1 = (1 Σ ) ˆT N ˆTN 1...ˆT 1 ( 1 Σ ) where ˆTl = ( 2 ml ω ) 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. (FPSP-XIII) June / 67

42 Ordered case In this case it is easy to show that D 1,N = sin(n + 1)q/ sin q, ω 2 = 2 (k/m) [1 cos(q)]. Rubin-Greer model: Σ = ke iq, therefore Γ = k sin(q) and is non-zero only for ω within the bath spectrum. It then follows that T = 1 as expected. Hence J = k B T 2π ωm 0 1 = k B T ω m 2π. Casher-Lebowitz model: In this case T 1 but for large N, only phonons within the spectral band of system transmit, and the integral over ω can be converted to one over q. We get J = 2γ2 k B T k 2 π π 0 dq dω dq ωq 2 N 2. The integrand has the form g 1 (q)/(1 + g 2 (q) sin Nq) and we use the result π g 1 (q) π lim dq N g 2 (q) sin Nq = g 1 (q) dq 0 [1 g2 2, (q)]1/2 to get in the limit N the Rieder-Lebowitz-Lieb result J = kk [ B T 1 + ν 2γ 2 ν ] where ν = mk 2 ν γ 2. (FPSP-XIII) June / 67

43 Disordered Harmonic systems: Results in 1D In this case we use the following reult: Furstenberg s theorem for product of random matrices - D 1,N e N/l(ω) (This is infact a proof of Anderson localization). This then 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 which give a current J N 1/2 0 T (ω)dω. Form of T (ω) (at small ω) depends on boundary conditions and from our earlier result for ordered case we get: Casher-Lebowitz model: T (ω) ω 2 J 1/L 3/2 Rubin-Greer model: T (ω) ω 0 J 1/L 1/2 If all sites are pinned then low frequency modes are cut off. Hence we get: J e L/l. [Ford and Allen (1966), Matsuda, Ishi (1968), Rubin, Greer, Casher, Lebowitz (1972), Verheggen(1979), Dhar (2001), Ojanki and Huveneers (2012)] (FPSP-XIII) June / 67

44 Higher dimensional systems What about systems in two three dimensions? Is Fourier s law valid? 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. A. Kundu, D.Chaudhuri, A. Dhar, D. Roy, J.L. Lebowitz and H. Spohn, EPL 90, (2010), PRB 81, (2010). (FPSP-XIII) June / 67

45 Heat conduction in mass disordered harmonic crystal H = x m x 2 q2 x + k 2 (qx q x+ê) 2 + x,ê x q x: scalar displacement, masses m x random. k o = 0: Unpinned. k o > 0: Pinned. k o 2 q2 x (FPSP-XIII) June / 67

46 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: 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. (FPSP-XIII) June / 67

47 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 = ωd 1/L 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. (FPSP-XIII) June / 67

48 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. (FPSP-XIII) June / 67

49 Anderson localization Consider harmonic Hamiltonian: H = x 1 2 mx q 2 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 φ x,x a q(x ). Normal mode frequency: Ω q. Normal mode wave-function: a q(x). (FPSP-XIII) June / 67

50 Character of normal modes Extended periodic mode (Ballistic) Extended random mode (Diffusive) Localized mode (Non-conducting) (FPSP-XIII) June / 67

51 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... (FPSP-XIII) June / 67

52 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 (FPSP-XIII) June / 67

53 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. (FPSP-XIII) June / 67

54 Heuristic theory Nonequilibrium heat current is given by the Landauer formula: J = T 2π 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. (FPSP-XIII) June / 67

55 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 (FPSP-XIII) June / 67

56 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. (FPSP-XIII) June / 67

57 Disordered harmonic systems Equations of motion: m x q 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. (FPSP-XIII) June / 67

58 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 q 1,x q 2,x T 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. (FPSP-XIII) June / 67

59 Current in terms of phonon transmission J = T 2π dωt (ω), 0 where T (ω) = 4γ 2 ω 2 G 1N 2 and G = [ ω 2 M + Φ Σ L Σ R ] 1 1 a 1 iγω a 2 I = a N a N iγω, a l = 2 + k o m l ω 2. (FPSP-XIII) June / 67

60 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 = ) 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. (FPSP-XIII) June / 67

61 Numerical and simulation results Open system. We look at: Disorder averaged transmission: Current: = J 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 q Inverse participation ratio: P 1 = Diffusivity D(ω) from Green-Kubo. δ(ω Ωq) x a4 x ( 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. (FPSP-XIII) June / 67

62 J-versus-N plots: 3D 0.01 Free BC Fixed BC Pinned (b) J N N N (FPSP-XIII) June / 67

63 J-versus-N plots: 2D 0.1 Free BC Fixed BC 0.01 N -0.6 J e-05 1e-06 N N N (FPSP-XIII) June / 67

64 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. (FPSP-XIII) June / 67

65 Phonon transmission: pinned 3D N T(ω) 0.32 (i) N=8 N=16 N=32 ρ(ω) (ii) ω Disordered Ordered ω (FPSP-XIII) June / 67

66 Transmission: Unpinned 3D T(ω) 0.08 (i) 0.04 N=8 N=16 N= N T(ω) (ii) N=8 N=16 N=32 ρ(ω) (iv) N= N= N 3 P ω (iii) Ordered Lattice Disordered Lattice ω (FPSP-XIII) June / 67

67 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. (FPSP-XIII) June / 67