##### Description

The tutorial will give an overview of some unresolved theoretical and experimental problems in the field of nanoscale thermal transport. Particular attention will be devoted to the outstanding questions and techniques aimed at understanding non-diffusive transport regimes at the nanoscale where the Fourier law breaks down. The main goal of this tutorial is to present our current understanding of these issues, and give some ideas on how to move forward.

In the first part of the tutorial, Philip B. Allen will discuss when and why the Boltzmann transport theory for phonons fails, and present its possible extensions to the nanoscale. He will also discuss the outstanding issue of defining and measuring the local temperature.

In the second part, David G. Cahill will provide an overview of what is known and not known in the physics of thermal transport at the nanoscale with an emphasis on experimental studies of materials and their interfaces at temperatures near ambient. In particular, he will discuss the breakdown of the diffusion equation at small spatial and temporal scales.

**8:30 am—Heat Transport—Fundamentals and Theory for Nanoscale**

Philip B. Allen, Stony Brook University, The State University of New York

Crystals have quasiparticle excitations: electrons, holes, phonons, magnons, etc. These particles are “normal modes” of excitation. They have energy Ek, where k “labels” the mode (wave vector k, branch index n, and possibly other indices like spin). They also have velocities, vk=dEk/dk. In equilibrium, the number of particles in mode k is given by the Fermi–Dirac or Bose–Einstein distribution nk. If the system is out of equilibrium, the number of particles in mode k is Nk. These modes all transport heat if the system is out of equilibrium. The heat current is the sum of vkNk. Therefore, the fundamental object of study is the nonequilibrium distribution Nk-nk. The usual method for studying this is the Boltzmann transport equation (BTE). What are the important issues? (1) When and why can the BTE fail? (2) How should the BTE be extended to work for nanoscale heat sources? (3) How is the local temperature defined and measured? None of these questions is fully answered. Partial answers will be discussed. The discussion will focus on heat carried by vibrations in insulators. There are three reasons motivating this choice. (1) Phonons have very diverse mean free paths. Many insulators have lower frequency phonon modes with long mean free paths, which can exceed the dimensions of heating elements or measuring probes. These provide interesting challenges to theory and experiment. (2) Phonon quasiparticle modes are easier to deal with in one important sense. At higher temperatures, quantum aspects fade out, and classical ideas, including classical molecular dynamics simulation, become powerful tools. (3) Amorphous and other disordered insulators have vibrational normal modes of diverse character that can be modeled theoretically (e.g., in the “propagon/diffuson/locon” picture). Heat conduction in metals is also important, but the dominant electron and hole carriers of heat are less diverse. They are easy to treat in “conventional” crystalline metals, and hard to treat in “exotic” metals.

**10:00 am—BREAK**

**10:30 am—Current Understanding and Unsolved Problems in Thermal Transport at the Nanoscale**

David G. Cahill, University of Illinois at Urbana-Champaign

On length scales large compared to the mean free paths and equilibration lengths of the excitations that carrier heat, the diffusion equation is an accurate description of the relationship between temperature fields and heat fluxes. On small spatial and temporal scales, this simple description fails due to (i) scattering and finite transmission of excitations at boundaries; (ii) out-of-equilibrium distributions of heat carriers that are induced by heat flow across material interfaces; and (iii) nonequilibrium between phonons, electrons and magnons. In this tutorial lecture, I will provide an overview of what is known and not known in the physics of thermal transport at the nanoscale with an emphasis on experimental studies of materials and their interfaces at temperatures near ambient.