Heat transport and phonon localization in mass-disordered harmonic crystals (original) (raw)

Heat conduction and phonon localization in disordered harmonic crystals

Europhysics Letters (epl), 2010

We investigate the steady state heat current in two and three dimensional isotopically disordered harmonic lattices. Using localization theory as well as kinetic theory we estimate the system size dependence of the current. These estimates are compared with numerical results obtained using an exact formula for the current given in terms of a phonon transmission function, as well as by direct nonequilibrium simulations. We find that heat conduction by high-frequency modes is suppressed by localization while low-frequency modes are strongly affected by boundary conditions. Our {\color{black}heuristic} arguments show that Fourier's law is valid in a three dimensional disordered solid except for special boundary conditions. We also study the pinned case relevant to localization in quantum systems and often used as a model system to study the validity of Fourier's law. Here we provide the first numerical verification of Fourier's law in three dimensions. In the two dimensional pinned case we find that localization of phonon modes leads to a heat insulator.

Thermal conductance of one-dimensional disordered harmonic chains

Physical review, 2020

We study heat conduction mediated by longitudinal phonons in one dimensional disordered harmonic chains. Using scaling properties of the phonon density of states and localization in disordered systems, we find nontrivial scaling of the thermal conductance with the system size. Our findings are corroborated by extensive numerical analysis. We show that a system with strong disorder, characterized by a 'heavy-tailed' probability distribution, and with large impedance mismatch between the bath and the system satisfies Fourier's law. We identify a dimensionless scaling parameter, related to the temperature scale and the localization length of the phonons, through which the thermal conductance for different models of disorder and different temperatures follows a universal behavior.

Role of pinning potentials in heat transport through disordered harmonic chains

Physical Review E, 2008

The role of quadratic onsite pinning potentials on determining the size (N) dependence of the disorder averaged steady state heat current , in a isotopically disordered harmonic chain connected to stochastic heat baths, is investigated. For two models of heat baths, namely white noise baths and Rubin's model of baths, we find that the N dependence of is the same and depends on the number of pinning centers present in the chain. In the absence of pinning, ~ 1/N^{1/2} while in presence of one or two pins ~ 1/N^{3/2}. For a finite (n) number of pinning centers with 2 <= n << N, we provide heuristic arguments and numerical evidence to show that ~ 1/N^{n-1/2}. We discuss the relevance of our results in the context of recent experiments.

Localization effects due to a random magnetic field on heat transport in a harmonic chain

Journal of Statistical Mechanics: Theory and Experiment, 2021

We consider a harmonic chain of N oscillators in the presence of a disordered magnetic field. The ends of the chain are connected to heat baths and we study the effects of the magnetic field randomness on heat transport. The disorder, in general, causes localization of the normal modes, due to which a system becomes insulating. However, for this system, the localization length diverges as the normal mode frequency approaches zero. Therefore, the low frequency modes contribute to the transmission, T N ( ω ) , and the heat current goes down as a power law with the system size, N. This power law is determined by the small frequency behaviour of some Lyapunov exponents, λ(ω), and the transmission in the thermodynamic limit, T ∞ ( ω ) . While it is known that in the presence of a constant magnetic field T ∞ ( ω ) ∼ ω 3 / 2 , ω 1 / 2 depending on the boundary conditions, we find that the Lyapunov exponent for the system behaves as λ(ω) ∼ ω for B ≠ 0 and λ(ω) ∼ ω 2/3 for B = 0 . Therefore,...

Phonon transport in a one-dimensional harmonic chain with long-range interaction and mass disorder

Physical Review E

Atomic mass and interatomic interaction are the two key quantities that significantly affect the heat conduction carried by phonons. Here, we study the effects of long-range (LR) interatomic interaction and mass-disorder on the phonon transport in a one-dimensional harmonic chain with up to 10 5 atoms. We find that while LR interaction reduces the transmission of low frequency phonons, it enhances the transmission of high frequency phonons by suppressing the localization effects caused by mass disorder. Therefore, long-range interaction is able to boost heat conductance in the high temperature regime or in the large size regime, where the high frequency modes are important.

Anderson Localization of Thermal Phonons Leads to a Thermal Conductivity Maximum

Nano Letters, 2016

Our elastic model of ErAs disordered GaAs/AlAs superlattices exhibits a local thermal conductivity maximum as a function of length due to exponentially suppressed Anderson-localized phonons. By analyzing the sample-to-sample fluctuations in the dimensionless conductance, !, the transition from diffusive to localized transport is identified as the crossover from the multi-channel to single-channel transport regime ! ≈ 1. Single parameter scaling is shown to hold in this crossover regime through the universality of the probability distribution of ! that is independent of system size and disorder strength.

Thermal Conductivity in Harmonic Lattices with Random Collisions

Springer eBooks, 2016

We review recent rigorous mathematical results about the macroscopic behaviour of harmonic chains with the dynamics perturbed by a random exchange of velocities between nearest neighbor particles. The random exchange models the effects of nonlinearities of anharmonic chains and the resulting dynamics have similar macroscopic behaviour. In particular there is a superdiffusion of energy for unpinned acoustic chains. The corresponding evolution of the temperature profile is governed by a fractional heat equation. In non-acoustic chains we have normal diffusivity, even if momentum is conserved.

Heat Transport in Harmonic Lattices

Journal of Statistical Physics, 2006

We work out the non-equilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain steady state properties such as currents and other second moments involving the position and momentum operators. The resulting expressions will be seen to be similar in form to results obtained for electronic transport using the non-equilibrium Green’s function formalism. As an application of the formalism we discuss heat conduction in a harmonic chain connected to self-consistent reservoirs. We obtain a temperature dependent thermal conductivity which, in the high-temperature classical limit, reproduces the exact result on this model obtained recently by Bonetto, Lebowitz and Lukkarinen.

Thermal conduction in classical low-dimensional lattices

2003

Deriving macroscopic phenomenological laws of irreversible thermodynamics from simple microscopic models is one of the tasks of non-equilibrium statistical mechanics. We consider stationary energy transport in crystals with reference to simple mathematical models consisting of coupled oscillators on a lattice. The role of lattice dimensionality on the breakdown of the Fourier's law is discussed and some universal quantitative aspects are emphasized: the divergence of the finite-size thermal conductivity is characterized by universal laws in one and two dimensions. Equilibrium and non-equilibrium molecular dynamics methods are presented along with a critical survey of previous numerical results. Analytical results for the non-equilibrium dynamics can be obtained in the harmonic chain where the role of disorder and localization can be also understood. The traditional kinetic approach, based on the Boltzmann-Peierls equation is also briefly sketched with reference to one-dimensional chains. Simple toy models can be defined in which the conductivity is finite. Anomalous transport in integrable nonlinear systems is briefly discussed. Finally, possible future research themes are outlined.