All semiconductor-based quantum devices rely on using electrodes to trap and manipulate some small number of charge carriers within the device. In Si/SiGe devices, electrons are confined to a thin layer of Si, creating an effective 2-dimensional electron gas (2DEG). The electron density within this 2DEG is manipulated via the voltages placed on the electrodes that sit above the 2DEG. It is useful to use simulations to estimate how the electrode voltages affect the electron density in the 2DEG.

COMSOL can be used to estimate the electron density within a simulated semiconductor device. While in principle one could use a self-consistent Schrodinger-Poisson solver to find the number of electrons in the 2DEG for a given combination of voltages, this method is in many cases prohibitively expensive. Instead, we make use of the Thomas-Fermi approximation. On this page, we explain the Thomas-Fermi approximation, and describe how it is implemented here.

The primary assumption of the Thomas-Fermi approximation is that charge is not quantized. As shown in Eq. 1.113 of The Physics of Low-Dimensional Semiconductors by John H. Davies, this leads to

where n_2D is the 2D density of the electrons within the 2DEG, n(E) is the 2D density of states, and f(E,E_F,T) is the Fermi-Dirac distribution. Assuming low temperature, the Fermi-Dirac distribution becomes a step function, leading to

where m^* is the transverse effective mass of an electron in Si (0.19 m_e) and E_C is the shift in the conduction band due to band bending. The two factors of 2 come from the valley degeneracy and the spin degeneracy.