## Ohm's law and Hall effect ### Ohm's law in semiconductor devices For semiconductor devices, we typically use the local form of Ohm's law, $\mathbf{j}(\mathbf{r}) = \boldsymbol{\sigma} \, \mathbf{E}(\mathbf{r})$ $\mathbf{j}(\mathbf{r})$ is the current density, $\mathbf{E}(\mathbf{r})$ is the electric field. $\boldsymbol{\sigma}$ is electrical conductivity; for 2DEG, it's a $2 \times 2$ tensor. $\boldsymbol{\sigma}=\begin{pmatrix} \sigma_{xx} & -\sigma_{xy} \\ \sigma_{xy} & \sigma_{xx} \end{pmatrix}$ $\sigma$ has the unit $\Omega^{-1}$, the specific resistance, $\boldsymbol{\rho}=\boldsymbol{\sigma}^{-1}$ has the unit $\Omega$. >[!Note] >Due to the tensor nature, $\boldsymbol{\rho}=\boldsymbol{\sigma}^{-1}$, we have >$\rho_{xx} = \frac{\sigma_{xx}}{\sqrt{\sigma_{xx}^2 + \sigma_{xy}^2}}$ >$\rho_{xy} = \frac{\sigma_{xy}}{\sqrt{\sigma_{xx}^2 + \sigma_{xy}^2}}$ >$\sigma_{xx},\ \rho_{xy}$ can be 0 at the same time (like in integer QHE) Different materials can have the same specific resistance, $\boldsymbol{\rho}$ is related to density of charge carriers participating in electron transport and scattering time (or relaxation time) $\tau$ (scattering rate is $\frac{1}{\tau}$). In the diffusive transport regime, the scattering of electrons takes place on length scales that are small compared to the size of the sample. ### Hall effect When current $I$, magnetic field $B$ applied to the material, a Hall voltage exists as: $U_{H}=R_{H}BI$$R_H$ is the Hall coefficient, $\frac{U_H}{I}$ is called Hall resistance. This Hall coefficient is related to the electron density and thickness of the material in the $B$ direction, i.e., $R_H^{3D} = -\frac{1}{n \lvert e \rvert d}$ But for 2D case, it is $R_{H}^{2D}=-\frac{1}{n_{s}|e|}$ >[!Info] >See [[Drude model with magnetic field]] for how Hall effect is like udner Drude model (and the very important Hall bar).