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I have been studying band theory and semiconductors in condensed matter physics and I am confused about the relation between mobility and velocity of electrons and holes in semiconductors.

My standard text book reference, Introduction to Solid State Physics, by Charles Kittel, says this:

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i.e., the velocities of electrons and holes are the same in a semiconductor.

However, I was also reading about the dependence of the Hall coefficient on temperature and found this:

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Now I can't understand how the mobilities of electrons and holes are different if their velocities are the same. What am I missing here?

Also, intuitively why should the mobilities be different for electrons and holes? Does it depend on doping too? Holes are just the gaps left behind by electrons and can practically be regarded as positive versions of electrons. Is it due to the mass factor coming into play due to electrons having some mass but holes being massless? Even then, holes should be more mobile than the electrons, right?

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The expression $$ \mathbf{v}(\mathbf{p}) = \nabla\epsilon(\mathbf{p}) $$ is the velocity of an electron with momentum $\mathbf{p}$. This velocity can be calculated for an electron anywhere in the band. On the other hand, the velocity associated with mobility is the drift velocity, $$v_d = \mu E,$$ which describes the velocity of carriers in a stationary current, which is obtained by solving the kinetic equation (or some equivalent equation) taking into account the accelerating electric field and the dissipation. The electrons participating in transport are usually the ones close to the Fermi surface. E.g., one could solve a Drude-like equation $$ \frac{d\mathbf{p}}{dt}=-e\mathbf{E} -\frac{1}{\tau}\mathbf{v}(\mathbf{p}) $$ and obtain the drift velocity or momentum.

Another important thing to keep in mind is that, while the hole velocity is the same as that of the missing electron, when discussing Hall effect we are talking not about missing electrons, but about electrons in the conduction band, and holes in the valence band, which have different dispersion relations and hence different velocity, different effective mass, and different mobility.

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  • $\begingroup$ Thanks! Can you elaborate more on the "different dispersion relations" part. Maybe a reference or something where it is elaborated more. $\endgroup$
    – Soham
    Nov 11, 2022 at 13:51
  • $\begingroup$ These wre two different energy bands, one has dispersion $\epsilon_c(\mathbf{k})$, while the other $\epsilon_v(\mathbf{k})$, see e.g., this figure Valence band is filled to the top, and a few electrons are removed - so we redefine the momentum and energy in the valence band to talk about the missing electrons as holes, but they are not the same electrons as those in the conduction band. In fact, in a typical semiconductor we have one conduction band and several overlapping valence bands. $\endgroup$
    – Roger Vadim
    Nov 11, 2022 at 13:59
  • $\begingroup$ What gives rise to the asymmetry of the valence and conduction band structures? Why are they not just mirror images of each other? Under what conditions would they be symmetric? $\endgroup$
    – Soham
    Nov 11, 2022 at 14:01
  • $\begingroup$ Why should they be symmetric? We solve the Schrödinger equation for periodic potential and obtain the eigenenergies $\epsilon_n(\mathbf{k})$, where $n$ is the band index. All these may be different, although we are usually interested only in the band (or bands) close to the Fermi level. $\endgroup$
    – Roger Vadim
    Nov 11, 2022 at 14:03
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    $\begingroup$ I attempted to improve the grammar of this answer. Hopefully, I didn't introduce any errors. $\endgroup$
    – PM 2Ring
    Nov 12, 2022 at 6:02
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The velocity of a hole is equal to the velocity of the missing electron

The missing context: in the same band.

Typically we talk about "electrons" in the conduction band and "holes" in the valence band. But the quote is talking about electrons filling or leaving a vacancy in the valence band.

The velocity of the hole is the same as the velocity of the electron whose absence means a hole exists.

Also, intuitively why should the mobilities be different for electrons and holes (does it depend on doping too?) ? Holes are just the gaps left behind by electrons and can practically be regarded as a positive version of an electron.

Charge carriers in a semiconductor behave differently than their free space counterparts. An electron in the conduction band may have a different effective mass than a hole in the valence band.

Holes are the missing electron, but in a different band with a different effective mass. So they act differently.

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