elphem 0.1.0

Elphem: Calculating electron-phonon interactions with the empty lattice.

Elphem

Python Library for Calculations of Electron-Phonon Interactions with Empty Lattice

Support

• Python 3.10 and 3.11

Installation

git clone [e-mail]:cohsh/elphem.git
cd elphem
pip install .

Features

Currently, Elphem allows calculations of

• (reciprocal) lattice vectors from lattice constants.
• electronic structures with empty lattice approximation.
• phonon dispersion relations with Debye model.
• first-order electron-phonon couplings.
• one-electron self-energies.

Examples

Calculation of the electronic band structure

import numpy as np
import matplotlib.pyplot as plt

from elphem import SpecialPoints, Energy, Length, LatticeConstant, EmptyLattice, FreeElectron

def main():
    # Example: Li (BCC)
    a = 2.98 * Length.ANGSTROM["->"]
    alpha = 109.47

    lattice_constant = LatticeConstant(a,a,a,alpha,alpha,alpha)
    lattice = EmptyLattice(lattice_constant)
    
    n_cut = np.array([2] * 3)
    electron = FreeElectron(lattice, 1)
        
    k_names = ["G", "H", "N", "G", "P", "H"]
    k_via = [SpecialPoints.BCC[name] for name in k_names]

    x, eig, x_special = electron.get_band_structure(n_cut, *k_via)

    fig, ax = plt.subplots()
    for band in eig:
        ax.plot(x, band * Energy.EV["<-"], color="tab:blue")
    
    ax.vlines(x_special, ymin=-10, ymax=50, color="black", linewidth=0.3)
    ax.set_xticks(x_special)
    ax.set_xticklabels(k_names)
    ax.set_ylabel("Energy ($\mathrm{eV}$)")
    ax.set_ylim([-10,50])

    fig.savefig("test_band_structure.png")

if __name__ == "__main__":
    main()

Calculation of the phonon dispersion

import os
import matplotlib.pyplot as plt

from elphem import LatticeConstant, EmptyLattice, DebyeModel, AtomicWeight, Energy, Mass, Length, SpecialPoints, Prefix

def main():
    # Example: \gamma-Fe (FCC)
    a = 2.58 * Length.ANGSTROM["->"]
    lattice_constant = LatticeConstant(a, a, a, 60, 60, 60)
    lattice = EmptyLattice(lattice_constant)
    
    debye_temperature = 470.0

    phonon = DebyeModel(lattice, debye_temperature, 1, AtomicWeight.table["Fe"] * Mass.DALTON["->"])

    q_names = ["G", "X", "G", "L"]
    q_via = [SpecialPoints.FCC[name] for name in q_names]
    
    x, omega, x_special = phonon.get_dispersion(*q_via)
    
    fig, ax = plt.subplots()

    ax.plot(x, omega * Energy.EV["<-"] / Prefix.MILLI, color="tab:blue")
    
    for x0 in x_special:
        ax.axvline(x=x0, color="black", linewidth=0.3)
    
    ax.set_xticks(x_special)
    ax.set_xticklabels(q_names)
    ax.set_ylabel("Energy ($\mathrm{meV}$)")

    fig.savefig("test_phonon_dispersion.png")

if __name__ == "__main__":
    main()

Calculation of the electron-phonon renormalization (EPR)

import numpy as np
import matplotlib.pyplot as plt

from elphem import LatticeConstant, EmptyLattice, FreeElectron, DebyeModel, SelfEnergy2nd
from elphem.const import Mass, Energy, Length, AtomicWeight, SpecialPoints

def main():
    # Example: Li (BCC)
    a = 2.98 * Length.ANGSTROM["->"]
    alpha = 109.47
    lattice_constant = LatticeConstant(a,a,a,alpha,alpha,alpha)
    lattice = EmptyLattice(lattice_constant)

    electron = FreeElectron(lattice, 1)
    
    mass = AtomicWeight.table["Li"] * Mass.DALTON["->"]
    
    debye_temperature = 344.0

    phonon = DebyeModel(lattice, debye_temperature, 1, mass)

    self_energy = SelfEnergy2nd(lattice, electron, phonon)

    n_g = np.array([2]*3)
    
    g = lattice.grid(n_g).reshape(-1, 3)
    
    n_g_inter = np.array([1]*3)
    n_q = np.array([10]*3)

    k_names = ["G", "H", "N", "G", "P", "H"]

    k_via = [SpecialPoints.BCC[name] for name in k_names]

    x, k, special_x = lattice.reciprocal_cell.path(20, *k_via)
    
    selfen = np.empty((len(k)), dtype=complex)

    fig = plt.figure()
    ax = fig.add_subplot(1, 1, 1)

    temperature = 2 * debye_temperature

    for n in range(len(g)):
        eig = electron.eigenenergy(k + g[n])
        for i in range(len(k)):
            selfen[i] = self_energy.calculate(temperature, g[n], k[i], n_g_inter, n_q)

        epr = selfen.real

        ax.plot(x, eig * Energy.EV["<-"], color="tab:blue")
        ax.plot(x, (eig + epr) * Energy.EV["<-"], color="tab:orange")
    
    for x0 in special_x:
        ax.axvline(x=x0, color="black", linewidth=0.3)
    
    ax.set_xticks(special_x)
    ax.set_xticklabels(k_names)
    ax.set_ylabel("Energy ($\mathrm{eV}$)")
    ax.set_ylim([-7,Energy.EV["<-"]])
    ax.set_title("Example: Band structure of bcc-Li")

    fig.savefig("test_epr.png")

if __name__ == "__main__":
    main()

Calculation of scattering rate

import numpy as np
import matplotlib.pyplot as plt

from elphem import LatticeConstant, EmptyLattice, FreeElectron, DebyeModel, SelfEnergy2nd
from elphem.const import Mass, Energy, Prefix, Time, Length, AtomicWeight

def main():
    # Example: Li (BCC)
    a = 2.98 * Length.ANGSTROM["->"]
    alpha = 109.47
    lattice_constant = LatticeConstant(a,a,a,alpha,alpha,alpha)
    lattice = EmptyLattice(lattice_constant)

    electron = FreeElectron(lattice, 1)
    
    mass = AtomicWeight.table["Li"] * Mass.DALTON["->"]
    
    debye_temperature = 344.0

    phonon = DebyeModel(lattice, debye_temperature, 1, mass)

    temperature = debye_temperature

    self_energy = SelfEnergy2nd(lattice, electron, phonon)

    n_g = np.array([1]*3)
    n_k = np.array([6]*3)
    g, k = electron.grid(n_g, n_k)
    
    n_g_inter = np.array([1]*3)
    n_q = np.array([10]*3)
    selfen = self_energy.calculate(temperature, g, k, n_g_inter, n_q)

    epsilon_nk = electron.eigenenergy(k + g)

    fig = plt.figure()

    ax1 = fig.add_subplot(2, 1, 1)
    ax2 = fig.add_subplot(2, 1, 2)

    for ax in [ax1, ax2]:
        ax.scatter(epsilon_nk * Energy.EV["<-"], selfen.imag / (Time.SI["<-"] / Prefix.PICO), label="$\mathrm{Im}\Sigma^\mathrm{Fan}$")

        ax.set_ylabel("Scattering rate ($\mathrm{ps}^{-1}$)")
        ax.legend()

    ax2.set_xlabel("Electron energy ($\mathrm{eV}$)")
    ax2.set_yscale("log")
    ax1.set_title("Example: Scattering rate of bcc-Li")
    
    file_name = "test_scattering_rate.png"
    fig.savefig(file_name)

if __name__ == "__main__":
    main()

License

MIT