|𝔻⟩irac's Student: Drude Electrons

Most of us are familiar with the trope about physicists and their spherical cows. You'll certainly encounter this style of thinking when you take a solid-state physics course. But the reality is many of these toy model representations are actually not bad assumptions and they worked pretty well to describe discrepancies between different classes of materials like metals and insulators. I was organizing my bookself and cracked open [1] and was refereshed by one of my favorite of such model is the Drude theory of metals.

The discovery of the electron by J.J Thomson that described the corpuscles of charge in a metal got many at the turn of the 20th century thinking about how these charges moved inside a material. Paul Drude started to think about this problem and thought of conduction in the framework of Boltzmann gas kinetic theory. The main simplicity of the model is that electrons have some fundamental scattering interval (relaxation time or equally mean free path) represented by a collision rate $1/\tau$. Furthermore the scattering event randomizes the electron's momentum such that it loses all of its drift momentum after the collision. Finally, while an electron is moving freely (i.e. no scattering) it's moving under the Lorentz force from the applied electric and magnetic fields¹.

The primary equation that the Drude theory arrives at is momentum balance equation: $$ \begin{equation} \frac{d\langle \mathbf{p} \rangle}{dt} = -\frac{\langle \mathbf{p} \rangle}{\tau} + \mathbf{F} \label{eq:momentum_balance} \end{equation} $$

where $\langle p \rangle$ is the average momentum, $\tau$ is the relaxation time, and $F$ is the force. The derivation comes when you take the expectation value of the momentum balance equation over all possible states of the system.

The Drude model then approaches the problem by considering the behavior of electrons in an applied electric and magnetic field to determine the electric current density and then through relational observations of Ohm's law the conductivity of a material can be determined. A similar derivation is done for electrons in an electric and magnetic field to yield the Hall coefficient². For complete detail on the actual equations derived from $\eqref{eq:momentum_balance}$ take a look at [1].

The biggest win for this simple Drude model, where electrons are treated as a kinetic gas and then solved for in an applied electric and magnetic field, is that for idealized metals it worked pretty well as an analytical understanding of electrons in metals. It clearly falters because it's not a quantum theory and thus does not account for Fermi-Dirac statistics and electron correlations. But it was very easy to understand at the time as Maxwell's equations and Boltzmann's gas kinetic theory were well established frameworks.

Drude applied the theory to understand thermal conductivity of metals by assuming electron mobility was the primary carrier³. The application was straightforward because Drude just used the same kinetic theory to arrive at a thermal conductivity given as:

$$ \begin{equation} \kappa = \frac{4}{\pi}\frac{n\tau k_B^2 T}{m} \label{eq:thermal_conductivity} \end{equation} $$

where $n$ is the number of electrons per unit volume, $\tau$ is the relaxation time (i.e., scattering time), $k_B$ is the Boltzmann constant, $T$ is the temperature, and $m$ is the mass of the electron. This follows from the kinetic-theory form $\kappa = \frac{1}{3} c_v \bar{v} \ell$ with $c_v = \frac{3}{2} n k_B$, mean free path $\ell = \bar{v}\tau$, and Maxwell-Boltzmann average speed $\bar{v} = \sqrt{8 k_B T/(\pi m)}$. Combined with the Drude electrical conductivity, it yields a Lorenz number in rough agreement with experiment for many metals at room temperature. Below is shows a "Drude electron gas" animation,

I'm not going to go through numbers of different metals, again you can see all that in [2] or [1]. But the results from the Drude theory were used to compute the Lorenz number, Wiedemann-Franz law, and Seebeck coefficient for different metals. The Lorenz number and Hall coefficient were notable successes; specific heat (why Drude got this wrong) and thermopower were not. Generally speaking the theory was a good starting point with some success and failures. Once quantum theory became established Arnold Sommerfeld applied Fermi-Dirac statistics to the Drude model to arrive at the free electron model of metals that was more successful (Sommerfeld vs Drude).

Footnotes

In the Drude model, $\mathbf{F} = -e(\mathbf{E} + \mathbf{v} \times \mathbf{B})$ with externally applied $\mathbf{E}$ and $\mathbf{B}$; the fixed ion background provides charge neutrality. Electron-electron interactions enter through the scattering rate $\tau$, not as a separate mean-field force. ↩
The Hall coefficient $R_H$ relates the transverse Hall electric field to the longitudinal current and magnetic field, $E_y = R_H J_x B_z$. In the free-electron Drude model, $R_H = -1/(ne)$ (derivation and sign). ↩
In metals electrons are indeed the primary carriers of heat but for insulators and semiconductors lattice vibrations (i.e. phonons) are the primary carriers of heat. ↩