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Electrons and holes behave differently in a silicon semiconductor (e.g. mobility of holes is one order of magnitude smaller than that of electrons, the collection time of holes at the same electric field is larger than for electrons... ). I was wondering, if holes are simply "a lack of electrons", they should behave in a mirrored way as electrons (if the latter move from $V_a$ to $V_b$ in a given time, the corresponding holes created when these electrons move should move in the opposite direction at the same speed). My question is: what is the origin of a different behavior between electrons and holes?

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One qualitative way to understand this concept overall is by thinking about the conduction band and valence band having orbital origin.

For example, let's assume the conduction band is primarily coming from hybridized s-orbitals, while the valence band comes from hybridized p-orbitals. As you might guess, particles in the s-orbital should move differently from the p-orbital, as they both have different spatial properties (s is more delocalized compared to p, and p also has nodes in the wavefunction).

Now it should be clear that a hole living in the valence band should behave differently compared to an electron in the conduction band, they aren't simple inverses of each other, but instead occupy different bands. In fact, for most semiconductors the electron and hole transport properties are quite different, with a common exception being Graphene.

Simplistic picture from phys.se 278844

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Check that electrons move in the conduction band, whereas holes "move" in the valence band. They have different energies, but they are also at a different border.

So the difference arises that the mass of electrons depends on the second derivative of the energy with respect to $k$. This derivative isn't equal in both bands.

Electrons are mostly near the minimum energy of the CB, called $E_C$. Their effective mass is possitive.

$$m_{e}^*|_{E_C}>0$$

Holes are mostly in the upper limit of the VB, at energies close to $E_V$. That's the maximum energy of the band, which leads to negative effective mass of electrons. If electrons have negative mass, holes have possitive mass.

$$m_{e}^*|_{E_V}<0 \Rightarrow \ \ m^*_h=-m_{e}^*|_{E_V}>0$$

So basically, holes are true "mirror" particles: opposite mass, quasimomentum, and velocities. But check that their mass is opposite only when compared at the same place, but they are at different places.

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For effective hole movement many valence electrons must move. For electron movement only a single conduction electron moves.

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  • $\begingroup$ Also the negative charge carriers in a semiconductor are "dressed" particles, not just a single naked electron. Difficult to predict relaxation times and effective mass. $\endgroup$ – Pieter May 8 '18 at 7:41
  • $\begingroup$ Of course, but the effects are much smaller for conduction electrons. Electrons have an effective mass of 1.04-1.09 m, holes have 0.59 m. en.wikipedia.org/wiki/… $\endgroup$ – my2cts May 8 '18 at 19:23
  • $\begingroup$ This is totally wrong, as it implies holes will always have larger effective masses than electrons, but that is false. See for example Germanium, or Silicon below 10K. $\endgroup$ – KF Gauss Jan 30 at 13:35
  • $\begingroup$ You should prove your assertion or withdraw your comment, @KF Gauss. $\endgroup$ – my2cts Jan 30 at 14:36
  • $\begingroup$ @my2cts I have proof right there, look at Germanium! The logic in your answer implies $m_h>>m_e$ which is clearly not true even for simple semiconductors. If it really was the case you would expect many, many orders of magnitude difference between the two effective masses, but that is experimentally false. $\endgroup$ – KF Gauss Jan 30 at 14:52

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