Dopant Models¶

This page describes the physics of the dopant models implemented in solvers.

Simple dopants¶

Donors¶

Donors can take on two charge states, ionized (+1) and neutral/filled (0), depending on the the Fermi level position relative to the donor energy level \(E\) (positive \(E\) indicating distance below the conduction band edge). The donor states may have degeneracy \(g\). The donor is specified in the material parameter database as follows:

material=GaN
    dopant=Si
        type: 'Donor'
        E: 29.7 meV
        g: 2

The grand canonical partition function for such a donor subsystem is

\[Z = \exp\left(\frac{E_F}{k_B T}\right) + g \exp\left(\frac{E_F-(E_C-E)}{k_B T}\right)\]

resulting in an expected charge of

\[\langle q \rangle = \frac{1}{1 + g \exp\left(\frac{E_F-(E_C-E)}{k_B T}\right)}\]

for each donor site. The density of donor sites is specified in the mesh definition.

Acceptors¶

Acceptors can take on two charge states, ionized (-1) and neutral/filled (0). The parameters are specified similarly, but the energy level \(E\) is measured upwards from the valence band edge.

The expected charge is given by

\[\langle q \rangle = \frac{-1}{1 + g \exp\left(\frac{E_V+E-E_F}{k_B T}\right)}\]

DX Centers¶

Experimental support for certain DX centers is enabled. DX centers can show a variety of behaviors with long lifetimes, but as PyNitride is a steady-state solver, we will take a simplified approach. These DX centers are taken to have three states: ionized donor (+1), singly-occupied (0) at energy \(E_0\) below the conduction band, and doubly-occupied (+1) at energy \(2E_1\) below the conduction band.

The grand canonical partition function is given by

\[Z = 1 + g_0 \exp\left(\frac{E_F-(E_C-E_0)}{k_B T}\right) + g_1 \exp\left(\frac{2E_F-2(E_C-E_1)}{k_B T}\right)\]

leading to a charge of

\[\langle q \rangle = \frac{1 - g_1 \exp\left(\frac{2E_F-2(E_C-E_1)}{k_B T}\right)} {Z}\]

In the cases considered, the singly-occupied state is significantly higher in energy than the double-occupied state, and thus its contribution to \(Z\) is dominated by either of the other terms depending on the position of the Fermi level. Taking this simplification, and assuming \(g_1=1\) (as per, e.g. a two-fold degenerate donor occupied by two electrons),

\[\langle q \rangle \approx -\tanh\left(\frac{E_F-(E_C-E_1)}{k_BT}\right)\]

PyNitride employs this form for now, thus only requiring a single energy specified:

material=AlN
    dopant=OxygenDX
        type: 'DX'
        E: 150 meV
        g: 1