Transport properties describe the flow of charge and/or heat in a solid. There are two common perspectives used to describe transport. Sometimes the electrons and phonons are considered to be particles that undergo collisions. This perspective is related to the classical theory of an ideal gas. The other perspective is based on the occupation of quantum wave functions. In thermodynamic equilibrium, the occupation of the electron states is described by the Fermi function, and the occupation of the phonon states is described by the Bose-Einstein function. Both the Fermi function and the Bose-Einstein function depend on the energy and the temperature. For every $\vec{k}$-state with energy $E$, there is a state $-\vec{k}$ with the same energy. In equilibrium, both of these states will have the same occupation, there are as many left-moving states as right-moving states, and the electrical and thermal currents are zero. To cause an electrical or thermal current to flow, an electric field or a thermal gradient must be applied to push the system out of equilibrium. The driving force for a current can be thought of as causing transitions between $\vec{k}$-states in such a way as to create an imbalance in the occupation of states $\vec{k}$ and $-\vec{k}$. The rate of these transitions can be calculated with Fermi's golden rule. Calculating all of the relevant transition rates is computationally intensive, and often an approximation is used. Here, only the simpler particle perspective for describing transport properties will be discussed.

Ballistic transport

We start considering the motion of electrons of charge $-e$ in vacuum where there are no collisions. This is called ballistic transport. Electrons in a constant electric field in vacuum follow parabolic trajectories like the motion of a ball in a constant gravitational field. If the electric field is in the positive $x-$direction, the force on the particle is, $$\vec{F} = -eE\hat{x} = m_e\vec{a},$$

where $E$ is the electric field, $m_e$ is the mass of the electron, and $\vec{a}$ is the acceleration of the electron. The acceleration can be integrated to determine the trajectory of the electron,

$$\vec{r}(t) = \vec{r}_0 +\vec{v}_0t -\frac{eE}{2m_e}t^2\hat{x},$$

where $\vec{r}_0$ is the position of the electron at time $t=0$ and $\vec{v}_0$ is the velocity of the electron at time $t=0$. The velocity of the electron is,

$$\vec{v}(t) = \vec{v}_0 -\frac{eE}{m_e}t\hat{x}.$$

Notice that the velocity increases linearly with time in ballistic transport. Electrons exhibit ballistic transport in the old vacuum tube technology.

Drift

Drift refers to the average motion of an electron in an electric field when collisions are present. In between collisions, the electrons follow the parabolic trajectories of ballistic transport. We assume that after a collision, the velocity is redirected in a random direction. The velocity after a collision at time $t_i$ is,

$$\vec{v}(t) = \vec{v}_i -\frac{e\vec{E}}{m^*}(t-t_i),$$

where $m^*$ is the effective mass. Averaging over many collisions gives us the average 'drift' velocity,

$$\vec{v}_d = \left< \vec{v}(t) \right> = \left< \vec{v}_i \right> -\frac{e\vec{E}}{m^*}\left<(t-t_i)\right>.$$

The average of the velocities just after the scattering events is $\left< \vec{v}_i \right>=0$ so the drift velocity is,

$$\vec{v}_d = -\frac{e\vec{E}}{m^*}\tau,$$

where $\tau = \left<(t-t_i)\right>$ is the scattering time. The drift velocity is constant in time and is in the opposite direction of the electric field because electrons have a negative charge. This is often rewritten as,

$$\vec{v}_d = -\mu_e\vec{E},$$

where $\mu_e$ is called the electron mobility. The drift regime is often encountered in metals, where the electrons scatter many times in between the electrodes, which are used to measure the resistance. The formula for the drift velocity is a statement of Ohm's law. To put it in a more conventional form, the current density can be written as $\vec{j} = -nev_d$, where $n$ is the electron density. The current density is in the opposite direction to the motion of the electrons. Substituting for the drift velocity yields,

$$\vec{j} = ne\mu_e\vec{E},$$

This can also be written in terms of the electrical conductivity $\sigma = ne\mu_e$,

$$\vec{j} = \sigma\vec{E}.$$

If the voltage across a conductor of with dimensions $L_x$, $L_y$, $L_z$, is measured in the $x-$direction, the voltage across the conductor will be $V = EL_x$ and the current is $I = jL_yL_z$. The relationship between the voltage and the current is then,

$$V = \frac{L_xI}{\sigma L_yL_z} = IR.$$

This is another form of Ohm's law where the resistance of the conductor is $R = \frac{L_x}{\sigma L_yL_z}$.

Diffusion

Electron will also diffuse from a high concentration to a low concentration. The electron current density due to diffusion is,

$$\vec{j} =eD\nabla n.$$

Here $D$ is the diffusion constant and $\nabla n$ is the gradient of the electron density. The gradient points from low concentration to high concentration, so the electrons flow in the direction of $-\nabla n$, but the current density is in the direction of $\nabla n$ because the electrons are negatively charged. Even when there is no electric field applied, electrons will diffuse because of the collisions. If the direction of the electron velocities is random after a collision, the electrons will perform a random walk.

Since electrons are conserved, there is a continuity equation for electrons,

$$e\frac{\partial n}{\partial t} = -\nabla\cdot \vec{j}.$$

Combining the last two equations results in the diffusion equation,

$$\frac{\partial n}{\partial t}= D\nabla^2n.$$

The diffusion equation is also called the heat equation. It is possible to consider the transport of heat as the diffusion of electrons and/or phonons in a solid.

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• Boltzmann transport theory