### 0.1 The Hall Effect


**Definition 0.1.1 **  The **cyclotron frequency** denoted $\omega_c$ is the frequency at which an electron would spin in a magnetic field of strength $B$, with elemental charge $e$ and electron mass $m_e$.
\[\omega_c = \frac{eB}{m_e}\]

**Definition 0.1.2 **  The **hall effect** is the production of an electric field $\vec{E}_{\text{hall}}$ (called the **hall field**) across a material in the direction of the cross product between the external electric field $\vec{E}_{\text{ext}}$ and the magnetic field $\vec{B}$.

**Result 0.1.3 **  The **classical hall effect** is the hall effect as predicted by solving the equilibrium condition $\frac{\partial \vec{p}}{\partial t} = 0$ for the [Drude Model](https://kaedon.net/l/73pc) in 2 dimensions with a magnetic field $\vec{B}=B\hat{z}$ perpendicular to the plane and an in-plane electric field $\vec{E} = E_x\hat{x} + E_y\hat{y}$.
\[0=-eE_x - \omega_c p_y - \frac{p_x}{\tau}, \quad 0=-eE_y + \omega_c p_x - \frac{p_y}{\tau}\]
\[\vec{p} = \frac{-e\tau}{1+(\omega_c\tau)^2}\begin{pmatrix}1 & -\omega_c\tau\\ \omega_c\tau & 1\end{pmatrix}\vec{E}\]

**Figure 0.1.4 Hall Effect Diagram **  

<img style="width:calc(min(100%,300px)); filter:invert(var(--invert-images));" src="https://kaedon.net/l/a520"></img>
Hall Effect Measurement Setup for Electrons. An external field $E_x$ is applied in the x direction and a magnetic field $B_z$ is applied in the z direction, resulting in a hall field $E_y$ in the y direction. Public Domain, <a href="https://commons.wikimedia.org/wiki/File:Hall_Effect_Measurement_Setup_for_Electrons.png">Link</a>

**File 0.1.4 **&nbsp; Hall_Effect_Measurement_Setup_for_Electrons.svg

**Result 0.1.5 **&nbsp; The **classical hall field**  $\vec{E}_{\text{hall}}$ for external electric field $\vec{E}_{\text{ext}}$ and magnetic field $\vec{B}$ can be written in terms of the [cyclotron frequency](https://kaedon.net/l/k3cc) $\omega_c$ and [mean scattering time](https://kaedon.net/l/73pc) $\tau$
\[\vec{E}_{\text{hall}} = \omega_c\tau(\vec{B}\times\vec{E}_{\text{ext}})\]

**Definition 0.1.6 **&nbsp; The **hall coefficient** denoted $R_H$ is the measurable ratio between the hall field $E_y$ and the product of the current $J_x$ and $B_z$ applied to drive that hall field. The Drude model predicts that this quantity is related to the charge carrier density $n_e$.
\[R_H = \frac{E_y}{J_xB_z} = \frac{-\omega_c\tau E_x}{J_x B_z} = \frac{-1}{en_e}\]