27-30 June 2016 - Université Pierre et Marie Curie - Paris 6 (France)

Organizers: Lorenzo Paulatto (IMPMC, Université Paris 6/CNRS, FR) Giorgia Fugallo (ETSF/LSI, Ecole Polytechnique, Palaiseau, FR), Andrea Cepellotti (EPFL, CH) and Francesco Mauri (Università di Roma La Sapienza, IT).

This school will provide the conceptual and computational tools to predict the intrinsic phonon-driven thermal transport of materials, fully ab-initio. The school will be composed of two parts: (1) introduction to density functional theory, total energy calculations and phonons, and (2) anharmonicity and thermal transport, by means of the generalized density functional perturbation theory and the Boltzmann transport equation.

Confirmed speakers:

Nicola Bonini, King’s College London, UK

Andrea Cepellotti, École Polytechnique Fédérale, Lausanne, Switzerland

Giorgia Fugallo, École Polytechnique, Palaiseau, France

Francesco Mauri, Università di Roma La Sapienza, Rome, Italy

Lorenzo Paulatto, IMPMC/CNRS, Paris, France

Guilherme Ribeiro, IMPMC/UPMC, Paris France

Invited speakers to be confirmed:

Stefano de Gironcoli, SISSA, Trieste, Italy

Christian Carbogno, Fritz Haber Institut, Berlin, Germany

To register or to request any additional information please write to [email protected]. A limited number of travel and lodge allowances is available, if you wish to apply for one please join to your registration request a curriculum vitae and a short motivation letter and a recommendation letter from your supervisor. For more information a updates you can refer to the school website http://www.impmc.upmc.fr/~paulatto/qethermo2016/.

Ab-initio prediction of thermal properties for condensed matter has been a blossoming field in the last few years, thank to its numerous applications mainly in thermoelectric materials and heat dissipation technologies. This methodology however does require to master a certain amount of expertise, both technical and theoretical, in order to be executed rigorously and efficiently. In this school, we want to provide an in-depth view of the theoretical framework, without neglecting the importance of applying the theory on some practical examples. The school will be composed of theory sessions and hands-on tutorials suitable for anybody with graduate-level knowledge of condensed matter physics. This school will also serve as a launch event for the public release of open-source codes for thermal transport.

In the first part of the school we take every participant to a working-level knowledge of density functional theory. We want to treat the basic theory and the most common technical challenges of a plane-wave calculation, such as construction and optimization of simulation cells, choice of functional and pseudopotentials, convergence with basis set cutoff and k-points sampling. This introductory part should be covered in two days, comprising of lectures in the morning and hands-on sessions in the afternoon.

The core focus of the school will be on the advanced topics of the following two days. We will start from the harmonic model: ions are assumed to vibrate within a small radius around their equilibrium positions, under the effect of forces which can be accurately described as a superposition of 2-body harmonic potentials. Lattice vibrations can be decomposed as a set of monochromatic waves, called phonons, that can be easily characterized using plane waves calculations. Phonons of wavevector q can be obtained as eigenvectors of the dynamical matrix, i.e. the second derivative of the total energy with respect to a periodic atom displacement of periodicity dual to q.

Nevertheless, several interesting phenomena cannot be accounted in a purely harmonic model because of one key limitation: the phonons are stationary states of the harmonic hamiltonian that do not interact with each other in a harmonic crystal. Furthermore, they are bosonic quasiparticles, hence there is no limit to the amount of heat that can be carried by a phonon. In other words, the thermal conductivity of a perfect harmonic crystal is always infinite. In order to have a realistic description, anharmonicity has to be taken into account.

If the system is not close to melting point, anharmonicity can be treated perturbatively. The lowest term in perturbation theory requires the calculation of three-body force constants, i.e. the derivative of the total energy with respect to three harmonic perturbations. These derivatives can be computed from first principles using the “2n+1” theorem, which allows the calculation of both 2nd and 3rd order derivatives just by the knowledge of the first order wave function response. The phonon lifetime can be straightforwardly derived from their interaction and plugged into the phonon Boltzmann transport equation. Solving the Boltzmann equation is a difficult problem in itself which can be tackled in two ways. A simplified approach assumes thermalisation to be instantaneous, it is the so-called single-mode approximation (SMA), and it can be applied successfully to most materials. However, one side effect of the SMA is that it neglects the correlations that arise between phonons when a heat current is formed, that leads to the formation of collective phonon excitations and exotic second sound, or heat waves. To describe the effects beyond the SMA, one must employ more advanced techniques that take into account the complete phonon-phonon interaction matrix for an exact solution of the BTE.

During the last few years these methods have been developed, by the organisers, in two codes: (1) A code to efficiently compute the 3rd derivatives of DFT total energy, using the “2n+1” theorem on top of the open-source code distribution Quantum-ESPRESSO; these derivatives can then be used with any thermal transport code. (2)A suite of codes to compute phonon-phonon interactions and solve the Boltzmann transport equation, both in the SMA and the exact form; these codes can in principle be used with the 3-body force constants produced by any ab-initio code via “2n+1” DFPT or via finite differences, e.g. via Phonopy. Mastering all the necessary tools to perform the calculation in a rigorous and efficient way is still a non trivial task. An in-depth knowledge of the underlying physical process is required to understand the limits of the different methods and choose which is more suitable for each problem. The methods themselves have to be studied in order to understand the underlying technical parameters.

School Program

Every day will comprise a theory session of about 3 hours in the morning, followed by a hands-on session of about 4 hours in the afternoon. The meeting will be hosted at the Université Pierre et Marie Curie - Paris VI, from 27/6/2015 to 30/6/2015.

• Day 1 – Basics: Ground state DFT
  • Theory Session:
  • Ground state DFT (total energy, basis set, k-points, metals, magnetic systems)
  • Post processing (equation of state, charge density, energy bands, density of states)
  • Hand-on Session:
    • DFT calculations with Quantum-ESPRESSO
    • Post-processing
• Day 2 – Basics: Structural optimization and phonons
  • Theory Session:
  • Forces and intrinsic stress from the Hellmann-Feynman theorem
  • Density functional perturbation theory
  • Force constants and Fourier interpolation of phonon dispersion
  • Hand-on Session:
    • Structural optimization
    • Phonon calculation and dispersion interpolation
• Day 3 - Advanced: Phonon transport, anharmonic phonon-phonon interaction
  • Theory Session:
  • Phonon-phonon interactions, the 2n+1 theorem
  • Treatment of isotopes, defects and boundaries
  • Hand-on Session:
    • Computing the D3 matrices with Quantum-ESPRESSO
    • Phonon linewidths, lifetimes and mean-free-paths
• Day 4 - Advanced: solution of Boltzmann transport equation (BTE)
  • Theory Session:
  • Derivation and linearization of the BTE, detailed-balance theorem
  • Single-mode approximation solution
  • Exact variational solution of the BTE
  • Hand-on Session:
    • BTE calculation of thermal conductivity in the single-mode approximation
    • Exact solution of the BTE