AthEM Simulation

In this tutorial we will discuss the theory of the Athermal Electron Model (AthEM). To see more check out [1]. This model takes the computational efficiency of the TTM but adds a athermal electron subsystem to simulate electron-hole pairs. The source of the efficiency compared to the Boltzmann equation is the use of the relaxation-time approximation without the lack of energy and particle conservation. This gain in efficiency allows users to do more complex simulations in terms of dimensionality than the Boltzmann equation would allow, e.g. 1D. The discussion on the thermal componets of AthEM will be minimal here as it all follows the same as in the Two-Temperature Model (TTM) simulation. Here we shall first discuss a 0D AthEM simulation, to see about 1D, check out either the Surface DOS or Antenna-Reactor tutorials.

Theory of AthEM

Athermal Electrons

The athermal electron subsystem of AthEM, is described in a similar way to the Boltzmann equation for a distribution. In AthEM's case it is written as follows:

\[ \frac{\partial f^*(E, t)}{\partial t} = \left.\frac{\partial f^*}{\partial t}\right|_\text{absorb} + \left.\frac{\partial f^*}{\partial t}\right|_\text{el^*el} + \left.\frac{\partial f^*}{\partial t}\right|_\text{el^*ph}\]

Each of the partial differential's on the right-hand side denotes a scattering term with photons, thermal electrons, and phonons respectively. The athermal electron-athermal electron scattering is set to 0 as in most practical simulations the number of athermal electrons is so small that they are a minor component.

The scattering of the athermal electrons with photons is described by the sum of two Fermi's Golden Rule expressions. One for the hole generation and the other for the electron generation.

\[\begin{align} \left.\frac{\partial f^*}{\partial t}\right|_\text{absorb} &= \left.\frac{\partial f^*}{\partial t}\right|_\text{h^+} + \left.\frac{\partial f^*}{\partial t}\right|_\text{e^-} \\ \left.\frac{\partial f^*}{\partial t}\right|_\text{h^+} &= \frac{2\pi V}{\hbar}\text{DOS}(E+hv)|M_{E,E_+}|^2 f(E)[1-f(E+hv)] \\ \left.\frac{\partial f^*}{\partial t}\right|_\text{e^-} &= \frac{2\pi V}{\hbar}\text{DOS}(E-hv)|M_{E,E_-}|^2 f(E-hv)[1-f(E)] \end{align}\]

Here $V$ is the volume of the cell, everything in LightMatter.jl is per $nm^3$ so this is one in the code's implementation. $f(E)$ is the current total electronic distribution, $\text{DOS}$ is the electronic density-of-states (DOS), $hv$ is the laser photon frequency and $|M_{E,E'}|^2$ are matrix elements that are set such that the number of electrons and holes generated are equal as well as that the internal energy of the excitation matches the energy inparted by the laser. In the future there is plans to add further complexity to the matrix elements. The first thing to now notice, is that there is an explicit and unescapable dependence on the DOS. This is unlike the TTM which can use approximations that escape this. Currently there are no keyword arguments that can interact with this component.

\[\left.\frac{\partial f^*}{\partial t}\right|_\text{el^*el} = - \frac{f^*}{\tau_\text{ee}} + \frac{f^\text{eq} - f^\text{rlx}}{\tau_\text{ee}}\]

Here we have a relaxation time approximation (RTA) in two parts. Firstly, there is the component that sends the athermal system to 0 (the first component) and the second component drives the equilibrium distribution towards the state with the same internal energy as equilbrium and athermal combined ($f^\text{rlx}$). This additional effect could be considered as generating the generation of secondary athermal electrons.

The relaxation time itself has a couple of options for electrons. Either a constant (:constant) or Fermi- Liquid Theory (:FLT) relaxation time can be used. The Fermi-Liquid time is considered more acurrate due to the presence of the energy-dependence on the relaxation time. Both are given below
Athermal Electron Relaxation Time

In the FLT there is a new variable, FE, which isn't in the equation. This variable is the Fermi energy and is defined as the difference between the bottom and top of the valence band. This can be calculated in LightMatter via FE = FE_initialization(bulk_DOS) where bulk_DOS is the same string you provided to build_Structure assuming that the DOS Fermi energy is at 0.0 eV. This variable corrects for the the fact that in LightMatter.jl we treat the 0 K $μ$ at 0.0 eV rather than FE. Also, in the case of the FLT, τ now is a material- dependent parameter calculated from, $τ = 128 / \sqrt{3}\pi\omega_p$ where $\omega_p$ is the plasmon-frequency of the material. This you must provide yourselves.

For the interaction with the phonon system we have,

\[\left.\frac{\partial f^*}{\partial t}\right|_\text{el^*ph} = - \frac{f^*}{\tau_\text{ep}}\]

This is similar to the electron-electron RTA but the difference is there is no driving force on the phonons unlike the thermal electrons. The electron-phonon relaxation time can only be treated as (:constant). We typically use a relaxation-time derived from, $\tau_text{ep} = \tau_\text{fft}hv/k_B \theta_D$ where $\tau_\text{fft}$ is the free-flight time of electrons and $\theta_D$ is the Debye temperature.

Now that we have constructed the athermal-electron system let us define how it couples to both of the thermal baths.

Thermal Baths

The equations for how the internal energy of the electronic and phononic thermal systems are very similar between AthEM and the TTM. In 0D they are

\[ \frac{\partial u_\text{el}(t)}{\partial t} = - g(T_\text{el} + T_\text{ph}) + \left.\frac{\partial u_\text{el}}{\partial t}\right|_\text{el-el^*} \\ \frac{\partial u_\text{ph}(t)}{\partial t} = g(T_\text{el} + T_\text{ph}) + \left.\frac{\partial u_\text{ph}}{\partial t}\right|_\text{ph-el^*} \]

We have an extra term on the right-hand side which denotes the change in energy from the relaxation with the athermal electrons. The equation for these extra terms are,

\[ \left.\frac{\partial u_\text{x}}{\partial t}\right|_\text{x-el^*} = -\int \left.\frac{\partial f^*}{\partial t}\right|_\text{el^*-x} \text{DOS}(E) E dE\]