In the past, I sticked to finite displacement method to calculate phonon dispersion thanks to the excellent implementation of ALAMODE. Also, direct calculations in real space are intuitive and straightforward. However, due to recent interest in the phonon-electron interaction which necessitates the calculations of deformation potential, DFPT is a must. In this post, I will try to learn the theory and practice of DFPT in phonon calculations.

The thermal conductivity tensor can be calculated by the autocorrelation function of the heat current, that is $\kappa_{\mu\nu}(t) = \frac{1}{k_{\text{B}}T^2 V}\int_0^t \Braket{J_{\mu}(0)J_{\nu}(t')}dt'$. It is a widely used method for simulating materials with complex structures (but only used for the bulk materials).

Sometimes you just want to see a quick preview of your data in the terminal, for example when you are working on a remote server. YouPlot is a simple tool that allows you to do just that.

Comparaisons of Python Numba, Fortran, Numba-CUDA, and Nvfortran in Monte Carlo simulations.

With jupyter notebook running on clusters like Hoffman2, maybe we will never need local environment again...

Despise vimer $\rightarrow$ understand vimer $\rightarrow$ become vimer.

In order to perform a fully ab initio calculation, for example, the VRMC simulation that we did in the last post, ab initio data, such as specific heat (spectral), phonon dispersion, and phonon lifetime, is needed. ALAMODE is a powerful tool to calculate these data, but the official documentation is not very clear about the formats...

The variance-reduced Monte Carlo simulation of phonon transport solves the following equation:

\[ \frac{\partial e^{\text{d}}}{\partial t} + \boldsymbol{v}_{\mathrm{g}} \cdot \nabla e^{\text{d}} = -\frac{e^{\mathrm{d}} - (e^{\mathrm{eq}}(T)-e^{\mathrm{eq}}(T_{\mathrm{eq}}))}{\tau} \]

Here, relaxation time approximation is used, and \(\tau\) takes ab initio calculated value, see the previous post. In this equation, \(e^{\mathrm{d}} = \hbar\omega (f(\omega)-f_{\mathrm{BE}}(\omega, T_{\mathrm{eq}}))\) for each phonon mode, where \(f\) is the distribution function (not necessarily equilibrium). While \(e^{\mathrm{eq}}(T)=\hbar\omega (f_{\mathrm{BE}}(\omega, T)-f_{\mathrm{BE}}(\omega, T_{\mathrm{eq}}))\) follows the Bose-Einstein distribution. Note that here \(T_{\mathrm{eq}}\) is a reference temperature, and \(T\) is the local temperature that deviates from \(T_{\mathrm{eq}}\).

Overview

The ab inito calculation of thermal conductivity is very complex and needs several softwares to work together. The basic idea is to calculate the interatomic force constants (IFCs) for both the harmonic and anharmonic terms, and then use the IFCs to calculate the phonon dispersion relation and phonon scattering time using knowledges that are summarized in the previous posts about anharmonicity and phonon dispersion.

Workflow

Overall, we use the following open-source softwares:

• phonopy that helps to generate the supercell we want.
• Quantum Espresso that carries out the ab initio calculation of electronic structures.
• alamode which fits the IFCs from the ab initio calculation and calculates the phonon dispersion relation and phonon scattering time and finally thermal conductivity.

We will discuss the Fermi's golden rule by perturbing a quantum system and calculate the first-order approximation of the transition rate. The golden rule is the basis of the derivation of the scattering rate, which is the key to understand the anharmonicity of the lattice vibration...