Ballistic transport: if there is no scattering, electrons follow parabolic trajectories in an electric field and prolate cycloid trajectories in a crossed electric and magnetic fields.
The total current density is a sum of drift and diffusion $\vec{j} = ne\mu_e\vec{E} + eD\nabla n$.
The electron mobility $\mu_e=e\tau/m^*$ depends on the scattering time $\tau$. Electrons can scatter from defects, phonons, or other electrons. The scattering contributions add in a reciprocal fashion following Matthiessen's rule, so that more scattering results in a shorter scattering time and a lower mobility.
In the Hall effect, a magnetic field is applied in a direction perpendicular to the current flow, and the Hall voltage is measured perpendicular to both the magnetic field and the current. The current flows at the Hall angle $\theta_H = \text{atan}(-\mu B_z)$ to the electric field.
The Hall coefficient $R_H=\frac{E_y}{j_xB_z} = −\frac{1}{ne}$ can be used to measure the free electron density $n$.
Hall sensors are used to measure magnetic fields.
The resistivity of metals increases as the temperature increases because of the increase in electron-phonon scattering. For metals that can be described by the free electron model, the resistivity is proportional to temperature near room temperature.
The resistivity of intrinsic semiconductors decreases with increasing temperature because the concentration of electrons and holes increase dramatically with increasing temperature.
Einstein argued that under steady-state conditions, the drift current must cancel the diffusion current.
The Einstein relation states that the electron mobility is proportional to the electron diffusion constant, $\mu_e = \frac{eD}{k_BT}$.
A consequence of the Einstein relation is that good electrical conductors are good thermal conductors. This is quantified in the Wiedemann-Franz law, $\frac{K_e}{\sigma} = LT,\qquad L = \frac{\pi^2 k_B^2}{3e^2} = 2.44\times 10^{-8}\,\text{W}\,\Omega\,\text{K}^{-2}.$
The thermal conductivity goes to zero as $T\rightarrow 0$. There is a peak in thermal conductivity at low temperature, and the thermal conductivity decreases as $1/T$ near room temperature.
Heat is carried by just the phonons in insulators and semiconductors. In metals, both the electrons and phonons contribute to the thermal conductivity.
