mirror of https://gitlab.com/QEF/q-e.git
365 lines
12 KiB
TeX
365 lines
12 KiB
TeX
\documentclass[12pt]{article}
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%\usepackage{fancyhdr}
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\usepackage{amsmath}
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\usepackage{graphicx}
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%\pagestyle{fancy}
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%
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\topmargin=-20mm
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\textheight=25cm
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\textwidth=17cm
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\oddsidemargin=-0.24 cm
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%\evensidemargin=-2.04cm
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%1inchi = 2.54cm
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%A4 = 21.0 * 29.7
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\begin{document}
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\title{Note for the tetrahedron method in DFPT and electron-phonon calculations}
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\author{Mitsuaki Kawamura, ISSP}
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\date{\today}
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\maketitle
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%
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\section{Definitions}
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\begin{itemize}
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\item $\varepsilon_{k n \sigma}, \varepsilon_{k+q n' \sigma}, \cdots$ : Kohn-Sham eigenvalues
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\item $\varepsilon_{\rm F}$ : Fermi energy
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\item $\omega_{q \nu}$ : Phonon frequency
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\item $g^{q \nu}_{n k n' k+q}$ : electron-phonon vertex
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\item $N(\varepsilon_{\rm F})$ : Density of states (for both spin) at the Fermi energy
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\end{itemize}
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%
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\section{Equations in DFPT and Electron-Phonon}
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We employ the tetrahedron method in the following equations in the DFPT with
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the ultrasoft pseudopotential \cite{DFPT-US}:
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\begin{itemize}
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\item $\delta(\varepsilon_{\rm F} - \varepsilon_{i \sigma})$ in Eqn. (25).
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It is stored in a variable \verb|dfpt_tetra_delta(1:nbnd,1:nks)|
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computed in subroutines \verb|dfpt_tetra_main| and \verb|dfpt_tetra_calc_delta|.
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\item $\theta(\varepsilon_{\rm F} - \varepsilon_{k v \sigma})$ in Eqn. (B17).
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It is \verb|wg(1:nbnd,ik)/wk(ik)|.
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\item Eqn. (B19),
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\begin{align}
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w_{k v \sigma, k+q v' \sigma} =
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\theta(\varepsilon_{\rm F} - \varepsilon_{k v \sigma})
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\theta(\varepsilon_{k v \sigma} - \varepsilon_{k+q v' \sigma})
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+
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\theta(\varepsilon_{\rm F} - \varepsilon_{k+q v' \sigma})
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\theta(\varepsilon_{k+q v' \sigma} - \varepsilon_{k v \sigma}).
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\end{align}
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It is stored in a variable \verb|dfpt_tetra_ttheta(1:nbnd,1:nbnd,1:nks)|
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computed in subroutines \verb|dfpt_tetra_main|, \verb|dfpt_tetra_calc_beta1|,
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\verb|dfpt_tetra_calc_beta2|, \\
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and \verb|dfpt_tetra_average_beta|.
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\item Eqn. (B28),
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\begin{align}
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\beta_{k v \sigma, k+q v' \sigma} &=
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\theta(\varepsilon_{\rm F} - \varepsilon_{k v \sigma})
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\theta(\varepsilon_{k v \sigma} - \varepsilon_{k+q v' \sigma})
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+
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\theta(\varepsilon_{\rm F} - \varepsilon_{k+q v' \sigma})
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\theta(\varepsilon_{k+q v' \sigma} - \varepsilon_{k v \sigma})
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\nonumber \\
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&+ \alpha_{k+q v' \sigma}
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\frac{\theta(\varepsilon_{\rm F} - \varepsilon_{k v \sigma})
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- \theta(\varepsilon_{\rm F} - \varepsilon_{k+q v' \sigma})}
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{\varepsilon_{k v \sigma} - \varepsilon_{k+q v' \sigma}}
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\theta(\varepsilon_{k+q v' \sigma} - \varepsilon_{k v \sigma}).
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\end{align}
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It is stored in a variable \verb|dfpt_tetra_beta(1:nbnd,1:nbnd,1:nks)|
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computed in subroutines \verb|dfpt_tetra_main|, \verb|dfpt_tetra_calc_beta1|,
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\verb|dfpt_tetra_calc_beta2|, \\ \verb|dfpt_tetra_calc_beta3|,
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and \verb|dfpt_tetra_average_beta|.
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\end{itemize}
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We also employ the tetrahedron method in
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calculations of the Fl\"ohlich parameter
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\begin{align}
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\label{fml_lambdaq}
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\lambda_{q \nu} = \frac{2}{N(\varepsilon_{\rm F}) \omega_{q \nu}} \sum_{k n n'}
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|g_{n' k + q n k}^{\nu}|^2
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\delta(\varepsilon_{n k} - \varepsilon_{\rm F})
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\delta(\varepsilon_{n' k+q} - \varepsilon_{\rm F})
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\end{align}
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(when \verb|electron_phonon="lambda_tetra"|),
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and
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\begin{align}
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\lambda_{q \nu} = \frac{2}{N(\varepsilon_{\rm F}) \omega_{q \nu}^2} \sum_{k n n'}
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[\theta(\varepsilon_{\rm F}-\varepsilon_{n k})-\theta(\varepsilon_{\rm F} - \varepsilon_{n' k+q})]
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\delta(\varepsilon_{n' k+q} - \varepsilon_{n k} - \omega_{q \nu}) |g^{q \nu}_{n k n' k+q}|^2
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\label{fml_goldenrule}
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\end{align}
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(when \verb|electron_phonon="gamma_tetra"|).
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\subsection{Tetrahedron method for DFPT}
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First, we cut out one or three tetrahedra where
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$\theta(\varepsilon_{\rm F} - \varepsilon_{n k})=1$ from tetrahedron $T$
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and evaluate $\varepsilon_{n k},\varepsilon_{n' k+q}$ at the corners of $T''$
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(See Appendix A and B of the previous study\cite{opt_tetra}).
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Second, we perform the following integration in each tetrahedra [Eqn. (C3) in that paper] :
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\begin{align}
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W_{i} &= 6 V'' \int_0^1 dx_1 \int^1_0 dx_2 \int_0^1 dx_3 \int_0^1 dx_4
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\frac{x_i \delta(x_1+x_2+x_3+x_4-1)}{d_1 x_1 + d_2 x_2 + d_3 x_3 + d_4 x_4}
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\nonumber \\
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&= -V'' \sum_{j=1,j \ne i}^4
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\frac{d_j^2
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\left( \frac{\ln d_j - \ln d_i}{d_j - d_i} d_j - 1 \right)}
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{\prod_{k=1,k \ne j}^4 (d_j - d_k)},
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\end{align}
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where $d_i$ is $\varepsilon_{n' k+q} - \varepsilon_{n k}$ at the corner of the trimmed tetrahedron.
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For avoiding a numerical error, we should not use the above formula as is.
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Practically, we use the following formulae according to the degeneracy of $d_1, \cdots d_4$.
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\begin{itemize}
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\item When $d_1, \cdots d_4$ are different each other,
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\begin{align}
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A_2 &= \left(\frac{\ln(d_2) - \ln(d_1)}{d_2 - d_1} d_2 - 1 \right) \frac{d_2}{d_2 - d_1},
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\quad
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A_3 = \left(\frac{\ln(d_3) - \ln(d_1)}{d_3 - d_1} d_3 - 1 \right) \frac{d_3}{d_3 - d_1},
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\nonumber \\
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A_4 &= \left(\frac{\ln(d_4) - \ln(d_1)}{d_4 - d_1} d_4 - 1 \right) \frac{d_4}{d_4 - d_1},
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\quad
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B_2 = \frac{A_2 - A_3}{d_2 - d_3} d_2,
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\quad
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B_4 = \frac{A_4 - A_3}{d_4 - d_3} d_4,
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\nonumber \\
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W_1 &= \frac{B_4 - B_2}{d_4 - d_2}.
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\end{align}
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\item When $d_1 = d_4$,
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\begin{align}
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A_2 &= \left(\frac{\ln(d_2) - \ln(d_1)}{d_2 - d_1} d_2 - 1 \right) \frac{d_2^2}{d_2 - d_1} - \frac{d_1}{2},
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\quad
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B_2 = \frac{A_2}{d_2 - d_1},
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\nonumber \\
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A_3 &= \left(\frac{\ln(d_3) - \ln(d_1)}{d_3 - d_1} d_3 - 1 \right) \frac{d_3^2}{d_3 - d_1} - \frac{d_1}{2},
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\quad
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B_3 = \frac{A_3}{d_3 - d_1},
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\nonumber \\
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W_1 &= \frac{B_3 - B_2}{d_3 - d_2}.
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\end{align}
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\item When $d_3 = d_4$,
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\begin{align}
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A_2 &= \frac{\ln(d_2) - \ln(d_1)}{d_2 - d_1} d_2 - 1 ,
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\quad
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B_2 = \frac{d_2 A_2}{d_2 - d_1},
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\quad
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A_3 = \frac{\ln(d_3) - \ln(d_1)}{d_3 - d_1} d_3 - 1 ,
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\quad
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B_3 = \frac{d_3 A_3}{d_3 - d_1},
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\nonumber \\
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C_2 &= \frac{B_3 - B_2}{d_3 - d_2},
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\quad
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C_3 = \frac{\ln(d_3) - \ln(d_1)}{d_3 - d_1} d_3 - 1 ,
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\quad
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D_3 = 1 - \frac{2 C_3 d_1}{d_3 - d_1},
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\quad
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E_3 = \frac{D_3}{d_3 - d_1},
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\nonumber \\
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W_1 &= \frac{d_3 E_3 - d_2 C_2}{d_3 - d_2}.
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\end{align}
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\item When $d_4 = d_1$ and $d_3 = d_2$,
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\begin{align}
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A_1 &= 1 - \frac{\ln(d_2) - \ln(d_1)}{d_2 - d_1} d_1,
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\quad
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B_1 = -1 + \frac{2 d_2 A_1}{d_2 - d_1},
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\quad
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C_1 = -1 + \frac{3 d_2 B_1}{d_2 - d_1},
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\nonumber \\
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W_1 &= \frac{C_1}{2 (d_2 - d_1)}.
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\end{align}
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\item When $d_4 = d_3 = d_2$,
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\begin{align}
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A_1 &= 1 - \frac{\ln(d_2) - \ln(d_1)}{d_2 - d_1} d_1,
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\quad
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B_1 = -1 + \frac{2 d_2 A_1}{d_2 - d_1},
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\quad
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C_1 = -1 + \frac{3 d_2 B_1}{d_2 - d_1},
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\nonumber \\
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W_1 &= \frac{C_1}{2 (d_2 - d_1)}.
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\end{align}
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\item When $d_4 = d_3 = d_1$,
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\begin{align}
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A_1 &= -1 + \frac{\ln(d_2) - \ln(d_1)}{d_2 - d_1} d_2,
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\quad
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B_1 = -1 + \frac{2 d_2 A_1}{d_2 - d_1},
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\quad
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C_1 = -1 + \frac{3 d_2 B_1}{2 (d_2 - d_1)},
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\nonumber \\
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W_1 &= \frac{C_1}{3 (d_2 - d_1)}.
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\end{align}
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\item When $d_4 = d_3 = d_2 = d_1$,
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\begin{align}
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W_1 = \frac{1}{4 d_1}.
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\end{align}
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\item Other weights are calculated by using the permutation.
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\end{itemize}
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\section{Tetrahedron method for electron-phonon}
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\subsection{Eqn. (\ref{fml_lambdaq})}
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First, we cut out one or two triangles where
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$\varepsilon_{n k} = \varepsilon_{\rm F}$ from a tetrahedron
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and evaluate $\varepsilon_{n' k+q}$ at the corners of each triangles as
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\begin{align}
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\varepsilon'^{k+q}_{i} = \sum_{j=1}^4 F_{i j}(
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\varepsilon_1^{k}, \cdots, \varepsilon_{4}^{k}, \varepsilon_{\rm F})
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\epsilon_{j}^{k+q}.
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\end{align}
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Then we calculate $\delta(\varepsilon_{n' k+q} - \varepsilon{\rm F})$
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in each triangles and obtain weights of corners.
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This weights of corners are mapped into those of corners of the original tetrahedron as
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\begin{align}
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W_{i} = \sum_{j=1}^3 \frac{S}{\nabla_k \varepsilon_k}F_{j i}(
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\varepsilon_{1}^k, \cdots, \varepsilon_{4}^k, \varepsilon_{\rm F})
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W'_{j}.
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\end{align}
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$F_{i j}$ and $\frac{S}{\nabla_k \varepsilon_k}$ are calculated as follows
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($a_{i j} \equiv (\varepsilon_i - \varepsilon_j)/(\varepsilon_{\rm F} - \varepsilon_j)$):
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\begin{figure}[!tb]
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\includegraphics[width=16cm]{pic/elph_tetra.pdf}
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\caption{\label{fig_elph_tetra}
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How to divide a tetrahedron
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in the case of $\epsilon_1 \leq \varepsilon_{\rm F} \leq \varepsilon_2$ (a),
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$\varepsilon_2 \leq \varepsilon_{\rm F} \leq \varepsilon_3$ (b), and
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$\varepsilon_3 \leq \varepsilon_{\rm F} \leq \varepsilon_4$ (c).
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}
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\end{figure}
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\begin{itemize}
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\item When $\varepsilon_1 \leq \varepsilon_{\rm F} \leq \varepsilon_2 \leq \varepsilon_3 \leq\varepsilon_4$
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[Fig. \ref{fig_elph_tetra}(a)],
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\begin{align}
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F &=
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\begin{pmatrix}
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a_{1 2} & a_{2 1} & 0 & 0 \\
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a_{1 3} & 0 & a_{3 1} & 0 \\
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a_{1 4} & 0 & 0 & a_{4 1}
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\end{pmatrix},
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\qquad
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\frac{S}{\nabla_k \varepsilon_k} = \frac{3 a_{2 1} a_{3 1} a_{4 1}}{\varepsilon_{\rm F} - \varepsilon_1}
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\end{align}
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\item When $\varepsilon_1 \leq \varepsilon_2 \leq \varepsilon_{\rm F} \leq \varepsilon_3 \leq\varepsilon_4$
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[Fig. \ref{fig_elph_tetra}(b)],
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%
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\begin{align}
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F &=
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\begin{pmatrix}
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a_{1 3} & 0 & a_{3 1} & 0 \\
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a_{1 4} & 0 & 0 & a_{4 1} \\
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0 & a_{2 4} & 0 & a_{4 2}
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\end{pmatrix},
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\qquad
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\frac{S}{\nabla_k \varepsilon_k} = \frac{3 a_{3 1} a_{4 1} a_{2 4}}{\varepsilon_{\rm F} - \varepsilon_1}
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\end{align}
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\begin{align}
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F &=
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\begin{pmatrix}
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a_{1 3} & 0 & a_{3 1} & 0 \\
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0 & a_{2 3} & a_{3 2} & 0 \\
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0 & a_{2 4} & 0 & a_{4 2}
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\end{pmatrix},
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\qquad
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\frac{S}{\nabla_k \varepsilon_k} = \frac{3 a_{2 3} a_{3 1} a_{4 2}}{\varepsilon_{\rm F} - \varepsilon_1}
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\end{align}
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\item When $\varepsilon_1 \leq \varepsilon_2 \leq \varepsilon_3 \leq \varepsilon_{\rm F} \leq \varepsilon_4$
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[Fig. \ref{fig_elph_tetra}(c)],
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\begin{align}
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F &=
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\begin{pmatrix}
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a_{1 4} & 0 & 0 & a_{4 1} \\
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a_{1 3} & a_{2 4} & 0 & a_{4 2} \\
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a_{1 2} & 0 & a_{3 4} & a_{4 3}
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\end{pmatrix},
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\qquad
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\frac{S}{\nabla_k \varepsilon_k} = \frac{3 a_{1 4} a_{2 4} a_{3 4}}{\varepsilon_1 - \varepsilon_{\rm F}}
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\end{align}
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\end{itemize}
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Weights on each corners of the triangle are computed as follows
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[($a'_{i j} \equiv (\varepsilon'_i - \varepsilon'_j)/(\varepsilon_{\rm F} - \varepsilon'_j)$)]:
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\begin{itemize}
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\item When $\varepsilon'_1 \leq \varepsilon_{\rm F} \leq \varepsilon'_2 \leq \varepsilon'_3$
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[Fig. \ref{fig_elph_tetra}(d)],
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\begin{align}
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W'_1 = L (a'_{1 2} + a'_{1 3}), \qquad
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W'_2 = L a'_{2 1}, \qquad
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W'_3 = L a'_{3 1}, \qquad
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L \equiv \frac{a'_{2 1} a'_{3 1}}{\varepsilon_{\rm F} - \varepsilon'_{1}}
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\end{align}
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\item When $\varepsilon'_1 \leq \varepsilon'_2 \leq \varepsilon_{\rm F} \leq \varepsilon'_3$
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[Fig. \ref{fig_elph_tetra}(e)],
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\begin{align}
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W'_1 = L a'_{1 3}, \qquad
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W'_2 = L a'_{2 3}, \qquad
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W'_3 = L (a'_{3 1} + a'_{3 2}), \qquad
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L \equiv \frac{a'_{1 3} a'_{2 3}}{\varepsilon'_{3} - \varepsilon_{\rm F}}
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\end{align}
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\subsection{Eqn. (\ref{fml_goldenrule})}
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In this case, we cut tetrahedra in the same manner to the case of
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\begin{align}
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\frac{\theta(\varepsilon_{\rm F} - \varepsilon_{k v \sigma})
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- \theta(\varepsilon_{\rm F} - \varepsilon_{k+q v' \sigma})}
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{\varepsilon_{k v \sigma} - \varepsilon_{k+q v' \sigma}}
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\theta(\varepsilon_{k+q v' \sigma} - \varepsilon_{k v \sigma})
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\end{align}
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in the DFPT calculation.
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Then we evaluate $\delta(\varepsilon_{n' k+q} - \varepsilon_{n k} - \omega_{q \nu})$
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in the trimmed tetrahedra.
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\end{itemize}
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\begin{thebibliography}{99}
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\bibitem{DFPT-US}
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A. Dal Corso, Phys. Rev. B {\bf 64}, 235118 (2001).
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\bibitem{opt_tetra}
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M. Kawamura, Phys. Rev. B {\bf 89}, 094515 (2014).
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\end{thebibliography}
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\end{document}
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