How to calculate the speed of electrons in a metal

Question

According to the Sommerfeld model, the electrons on the Fermi level has the relation

$$ \epsilon_F=\frac{\hbar^2k_F^2}{2m_e}=\frac{1}{2}m_ev_F^2 $$

i.e. $\hbar k_F=m_ev_F$ with $k_F=(3\pi^2n)^{1/3}$

But as it turns out, the calculated Fermi energy is higher than the experimental value for many metals. The energy band theory accounted for this by replacing $m_e$ with $[(m^*)^{-1}]=\nabla_\vec{k}\nabla_\vec{k}\epsilon/\hbar^2$. Now if I want to calculate the speed of electrons on the Fermi level, do I have to replace $m_e$ with $m^*$? It seems so, but some (not-so-reliable) source suggested otherwise. Is there some underlying relationship that canceled out and cause the electrons on Fermi level act like a naked electron?

Thanks!

Answer 1

Note that Sommerfeld's model simply generalizes Drude's theory of metals by taking into account the fact that electrons are fermions, so Pauli exclusion becomes a very important factor. In Sommerfeld's model, there's no effective mass to talk about, as one basically ignores the atoms(nuclei) in the system and considers free moving fermions. So there, your Fermi velocity is just given by: $$v_F=\frac{\hbar k_F}{m_e}.$$

With more advanced models, like the tight binding chain, one starts to take into account the periodic environment of the electron, namely the periodic Coulomb potential $V(\mathbf{r})=V(\mathbf{r}+\mathbf{R})$ (taken now in the $\mathcal{H}$ of the system) and with certain educated approximations a LCAO (linear combination of atomic orbitals) approach is used to solve the Schrödinger equation. As you already seem to know, this result is the famous band structure of electrons in solids, where an energetic gap between the valence and conduction bands appears (semiconductors, insulators). Whenever the bottom of the band (min of cond. band or max of valence band) can be approximated by a parabola, then the dispersion can be written as a constant part plus a quadratic term: $$E(k) \propto cte + \frac{\hbar^2 k^2}{2m^*}$$ with this approximation the electron in the tight binding chain of atoms, can be described as freely moving if the associated effective mass $m^*$ is used. Note that the term $cte$ is in fact $E_c$ or $E_v$ (just the band energies describing the gap).

To sum up, if you're talking about a free electron in the band structure model, then the effective mass is to be used. For more intuition, if you have seen some of the derivations, you may have noticed the hopping matrix element $t$ that pops out when solving for the energy eigenvalues, it's in fact the hopping strength (likelihood of the electron to jump between the atoms in the chain, in the tight binding model) that defines how different the effective mass $m^*$ of the electron is from its rest mass $m_e.$ In most cases they're simply inversely proportional $m^* \propto 1/t,$ the larger $t$, the lighter the electrons feel.