## Characteristic quantities of 2DEG ### Electron density $n_s$ depends on voltage, typical value is $10^{11} \sim 10^{12} \, \text{cm}^{-2}$. For 2DEG, this could be measured by [[Drude model with magnetic field|Drude model]] (Hall bar), by [[Magnetotransport in 2D and Shubnikov–de Haas effect|Shubnikov–de Haas effect]] ($\cos$ behavior), or by [[Integer quantum Hall effect|quantum hall effect]] (filling factor $\nu$). ### Dispersion relation For GaAs, it's parabolic for the motion in-plane: $E_{nk} = E_n + \frac{\hbar^2 \mathbf{k}^2}{2m^*}$ $\mathbf{k} = (k_x, k_y)$ an in-plane vector. ### Wavefunction State $(n, \mathbf{k})$ comes with an envelope wave function: $\psi_{nk}(x, y, z) = \chi_n(z) e^{i (k_x x + k_y y)}$ $\chi_n(z)$ is quantization in the $z$ direction. ### DOS $D_{2D}(E) = \frac{g_s g_v m^*}{2 \pi \hbar^2}$ This can be measured from quantum capacitance $C_{q}$, as shown in [[Electrostatics of GaAs-AlGaAs heterostructures]], but it's better to calculate it directly. $g_s$ is the spin degeneracy (2 for GaAs and Si). $g_v$ is the number of degenerate conduction band minima, or say valley degeneracy (1 for GaAs, 2 for Si). ### Quantum limit For 2DEG, only 1 sub-band gets occupied. Typically, when we say 2DEG, we mean 2DEG in the quantum limit. ### Fermi energy and electron density For 2DEG, $n_s = D_{2D} \cdot (\mu - E_0) = D_{2D} E_F$ $\mu$ is the electrochemical potential; $E_0$ is the quantization energy of the ground state sub-band; $E_F = \mu - E_0$, the Fermi energy, whose definition here is greatly differ from bulk 3D semiconductors. This $E_{F}$ is not a quantity that could be obtained easily. It could be obtained if $n_{s}$ is known. ### Fermi wave vector, Fermi energy, electron density $k_F = \sqrt{\frac{2m^* E_F}{\hbar^2}} = \sqrt{\frac{4 \pi n_s}{g_s g_v}}$ $\lambda_F = \frac{2 \pi}{k_F} = \sqrt{\frac{g_s g_v \pi}{n_s}}$ $\lambda_F$ is on the order of mean electron separation, $\frac{1}{\sqrt{n_s}}$. ### Fermi velocity Defined as the (phase) velocity of an electron at the Fermi energy, $v_{F}=\frac{{\hbar k}}{m^{*}}$ One have to know the effective mass to compute this term. ### Effective Bohr radius and Rydberg energy $a_B^* = \frac{4 \pi \varepsilon_0 \hbar^2}{m^* e^2} = \frac{\varepsilon m_e}{m^* a_B}$ $E_{Ry}^* = \frac{e^4 m^*}{2(4 \pi\varepsilon \varepsilon_0)^2 \hbar^2} = \frac{m^*}{m_e \varepsilon^2} E_{Ry}$