# To which real densities do carrier densities in the semi-classical model of a crystal correspond?

In the semi-classical model of a crystal in solid state physics, electrons and holes are assigned effective masses that account for their different mobilities. E.g. in silicon, holes have a bigger mass than electrons. This results in different electron/hole densities as the temperature increases.

Why do those different densities not violate conservation of charge? In what way do these imaginary electrons/holes correspond to real particles, in what way do they not?

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I do not understand your question - how can changing the effective mass affect conservation of charge. –  BebopButUnsteady Apr 16 '13 at 17:11
The densities of electrons and holes depend on the corresponding effective masses. Thus when the effective masses are different, the electron and hole densities are different, as e.g. in silicon. But in general, silicon is not positively charged! Maybe the point is that different energy bands are considered when calculating electron/hole densities…? –  Deniz Apr 16 '13 at 17:18
Why do the density of electrons and holes depend on the corresponding effective masses? Are you assuming that the total mass of all electrons is equal to the total mass of all holes? Because that's not true. –  Steve B Apr 16 '13 at 17:56
The particle density is an integral over the (Fermi-Dirac) distribution function and the density of states. The density of states depends upon the effective mass. Doesn't there need to be a hole for every electron and vice versa? This is what I mean when I see a problem with charge conservation. –  Deniz Apr 16 '13 at 18:13
Obviously, I must misunderstand something. But where is the error? –  Deniz Apr 16 '13 at 18:16

The density of electrons / holes depends on BOTH the effective mass of the electron / holes (which, as you correctly realize, affects the density of states) AND the fermi level.

If the fermi level moves towards the conduction band, you get more electrons and fewer holes; if the fermi level moves towards the valence band, you get fewer electrons and more holes.

Therefore, there is some possible location (in the band gap) of the fermi level where the number of electrons equals the number of holes. For a big piece of undoped semiconductor, that is exactly where you'll find the fermi level.

If the electron and hole effective masses are different, then the intrinsic (# holes = # electrons) fermi-level depends on temperature. Therefore, as you change the temperature of the semiconductor, the fermi level will shift up or down a bit. It will not be exactly halfway between the conduction and valence bands; it will have a temperature-dependent offset.

There is no law that says that the fermi level has to be independent of temperature.

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