Condensed Matter – Free Electron Theory in Metals Explained

Question

We study "Free electron theory in metals" in Condensed Matter Physics. While discussing this theory (both classically and quantum mechanically), we assume the electrons moving inside the metal to be a "free electron gas". But in Condensed Matter Physics, we mainly discuss about the "condensed" phases of matter, then why do we consider a gas in this model? (I know this gas is not same as any ordinary gas and it's concentration is much more than that of an ordinary gas, also by assuming this, the calculation gets simplified by many folds, my question is on what physical grounds do we consider the freely moving electrons in the metal to be a gas?)

Answer 1

That is simplification based on the Wannier theorem, later generalized by Luttinger and Kohn in their 1955 article "Motion of Electrons and Holes in Perturbed Periodic Fields". Basically, if you have energy band depending on the wave vector as $\varepsilon_{0} (\mathbf{k})$, and apply external potential $V(\mathbf{r})$, weak compared to crystal potential and slowly varying compared to unit cell size, then Schrödinger's equation can be written like that:

$$ \left(\varepsilon_{0} \left(- i \mathbf{\nabla}\right) + V(\mathbf{r})\right) \Psi(\mathbf{r}) = \varepsilon \Psi(\mathbf{r}) $$

Here $\nabla = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right)$. For details and proof go to the 1955 paper mentioned above or to Anselm's "Semiconductor theory", Ch. IV paragraph 3. The latter is simpler.

In many cases, especially in semiconductors, at low values of $k$ energy is parabolic: $\varepsilon_{0}(\mathbf{k}) \approx C * \left| \mathbf{k} \right| ^2$. Renaming $C = \frac{\hbar^2}{2 m^{\ast}}$, where $m^{\ast}$ is called effective mass, we have

$$ \left(- \frac{\hbar^2}{2 m^{\ast}} \nabla^{2} + V(\mathbf{r})\right) \Psi(r) = \varepsilon \Psi(\mathbf{r}), $$

which is very similar to the equation of actual free electron gas.

Actually, the question you asked puzzled scientists for a while, but they asked "why electrons in materials behave like free electron gas".