Thermal Transport in Crystals as a Kinetic Theory of Relaxons

Thermal conductivity in dielectric crystals is the result of the relaxation of lattice vibrations described by the phonon Boltzmann transport equation. Remarkably, an exact microscopic definition of the heat carriers and their relaxation times is still missing: Phonons, typically regarded as the relevant excitations for thermal transport, cannot be identified as the heat carriers when most scattering events conserve momentum and do not dissipate heat flux. This is the case for two-dimensional or layered materials at room temperature, or three-dimensional crystals at cryogenic temperatures. In this work, we show that the eigenvectors of the scattering matrix in the Boltzmann equation define collective phonon excitations, which are termed here "relaxons". These excitations have well-defined relaxation times, directly related to heat-flux dissipation, and they provide an exact description of thermal transport as a kinetic theory of the relaxon gas. We show why Matthiessen's rule is violated, and we construct a procedure for obtaining the mean free paths and relaxation times of the relaxons. These considerations are general and would also apply to other semiclassical transport models, such as the electronic Boltzmann equation. For heat transport, they remain relevant even in conventional crystals like silicon, but they are of the utmost importance in the case of two-dimensional materials, where they can revise, by several orders of magnitude, the relevant time and length scales for thermal transport in the hydrodynamic regime.

Thermal transport in low dimensions

Andrea Cepellotti

Lattice vibrations are the microscopic mechanism responsible for a large, if not dominant, contribution to heat transport in crystalline insulators. These vibrations are described in terms of phonons, collective excitations (or quasiparticles) in the form of waves of atomic displacements inside a crystal. Phonons are traditionally considered to be the quasiparticles responsible for carrying heat through the material. Heat transport is considered as a flux of a phonon gas, diffusing from hot areas (high phonon densities) to cold areas (low phonon densities) in an attempt to reestablish equilibrium, with phonon collisions being the source of heat flux dissipation. However, as dimensionality is reduced, the motion of phonons stimulated by temperature perturbations becomes correlated and this gas-like picture of thermal transport in terms of phonons becomes invalid. In this Thesis, we lay out an interpretation of thermal transport in 2D materials based on the Boltzmann transport equation in the form of collective excitations of phonons. These collective phonon excitations give raise to complex phenomena, such as high thermal conductivities, that are otherwise unexplained. As another example, collective excitations, at variance with conventional diffusive transport, can induce wave-like heat propagation, or second sound. This had been found only in a few exotic materials at cryogenic temperatures, but is present instead routinely in 2D materials at room temperature. The correlated-phonon description of heat transfer can be rationalised by introducing a new collective excitation, called 'relaxon', which is defined as the eigenstate of the collision operator. Whereas only oversimplifying assumptions endow phonons with well defined relaxation times (the average interval of time between collisions), relaxons have always well defined relaxation times and permit an exact description of thermal transport. The complex dynamic of heat transport in 2D is thus greatly simplified and a kinetic gas theory of thermal transport still applies, provided that the gas is not constituted by phonons, but by relaxons. Our work on thermal transport is part of a larger effort, aiming at the creation of a database of numerically computed properties of materials. The high-throughput production of simulated properties is a challenging task, since it necessitates the understanding of a physical model, but it also needs to face a myriad of technicalities and problems that hinder the execution of a large number of calculations. In order to allow the creation of computational materials databases, we developed AiiDA, an open-source automated interactive infrastructure and database for computational science. This platform tackles the problems of creation, management, analysis and sharing of data and simulations, summarized in the pillars of Automation, Data, Environment and Sharing. Automation is achieved by management of remote computational resources and the encoding of workflows, that allows the execution of complex sequences of calculations. The tight coupling between automation and data storage, handled by the platform, enables full reproducibility of the results and a suitable database design allows for efficient data analysis tools. Sharing of scientific knowledge is addressed by providing tools for distribution of data and of the underlying workflows that generated them, creating an ecosystem for computational materials science.

EPFL2016

Boltzmann Transport in Nanostructures as a Friction Effect

Andrea Cepellotti, Nicola Marzari

Surface scattering is the key limiting factor to thermal transport in dielectric crystals as the length scales are reduced or when temperature is lowered. To explain this phenomenon, it is commonly assumed that the mean free paths of heat carriers are bound by the crystal size and that thermal conductivity is reduced in a manner proportional to such mean free paths. We show here that these conclusions rely on simplifying assumptions and approximated transport models. Instead, starting from the linearized Boltzmann transport equation in the relaxon basis, we show how the problem can be reduced to a set of decoupled linear differential equations. Then, the heat flow can be interpreted as a hydrodynamic phenomenon with the relaxon gas being slowed down in proximity of a surface by friction effects, similar to the flux of a viscous fluid in a pipe. As an example, we study a ribbon and a trench of monolayer molybdenum disulfide, describing the procedure to reconstruct the temperature and thermal conductivity profile in the sample interior and showing how to estimate the effect of nanostructuring. The approach is general and could be extended to other transport carriers, such as electrons, or extended to materials of higher dimensionality and to different geometries, such as thin films.

Amer Chemical Soc2017

Cutting Materials in Half: A Graph Theory Approach for Generating Crystal Surfaces and Its Prediction of 2D Zeolites

Senja Dominique Barthel, Peter George Boyd, Berend Smit, Matthew David Witman

Scientific interest in two-dimensional (2D) materials, ranging from graphene and other single layer materials to atomically thin crystals, is quickly increasing for a large variety of technological applications. While in silico design approaches have made a large impact in the study of 3D crystals, algorithms designed to discover atomically thin 2D materials from their parent 3D materials are by comparison more sparse. We hypothesize that determining how to cut a 3D material in half (i.e., which Miller surface is formed) by severing a minimal number of bonds or a minimal amount of total bond energy per unit area can yield insight into preferred crystal faces. We answer this question by implementing a graph theory technique to mathematically formalize the enumeration of minimum cut surfaces of crystals. While the algorithm is generally applicable to different classes of materials, we focus on zeolitic materials due to their diverse structural topology and because 2D zeolites have promising catalytic and separation performance compared to their 3D counterparts. We report here a simple descriptor based only on structural information that predicts whether a zeolite is likely to be synthesizable in the 2D form and correctly identifies the expressed surface in known layered 2D zeolites. The discovery of this descriptor allows us to highlight other zeolites that may also be synthesized in the 2D form that have not been experimentally realized yet. Finally, our method is general since the mathematical formalism can be applied to find the minimum cut surfaces of other crystallographic materials such as metal–organic frameworks, covalent-organic frameworks, zeolitic-imidazolate frameworks, metal oxides, etc.

2018

Thermal transport in low dimensions

Andrea Cepellotti

EPFL2016