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Ludwig.jl Documentation

Overview

Ludwig provides a framework for generating the linearized Boltzmann collision operator for electron-electron scattering in two-dimensional materials and materials with a pseudo-two-dimensional band structure. This package also provides utilities for calculating conductivities and viscosities of the electron fluid from the generated collision matrix. For now, only square Brillouin Zones are supported.

Unicode

This package uses Unicode characters (primarily Greek letters) such as η, σ, and ε in both function names and for function arguments. Unicode symbols can be entered in the Julia REPL by typing, e.g., \eta followed by tab key. Read more about Unicode symbols in the Julia Documentation.

Units

For all calculations, $\hbar = k_B = 1.$ For converting output back to physical units, Ludwig includes the values of some important physical constants from the 2022 CODATA Recommended Values of the Fundamental Physical Constants.

Energy Scale

Since we take $k_B = 1$, temperatures must be expressed in the same energy scale used by the Hamiltonian. We recommend expressing all energies in units of eV for simplicity in multiband calculations where each band may have an independent natural energy scale. This is particularly important since many function involve the ratio of the energy to temperature; e.g. f0(E, T)

Ludwig.f0 — Function
    f0(E, T)

Return the value of the Fermi-Dirac distribution for energy E and temperature T.

\[ f^{(0)}(\varepsilon) = \frac{1}{1 + e^{\varepsilon/k_B T}}\]

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Moreover, all crystal momenta are normalized by $2\pi / a_i$ where $a_i$ denotes the the lattice spacing. This makes the computation of momentum integrals simplified:

\[\int \frac{d^2\mathbf{k}}{(2\pi)^2} \mapsto \frac{1}{a^2} \int d^2\mathbf{k}\]

Other Utilities

Ludwig.map_to_first_bz — Function
map_to_first_bz(k)

Map a vector k to the $d$-dimensional centered unit cube where $d$ is the dimension of k.

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References