What actually are electron holes?

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Electron holes are defined as the absence of electrons in a crystal lattice, creating a space that allows for the movement of adjacent electrons into these vacancies. This movement results in the apparent flow of holes in one direction while electrons move in the opposite direction. The Hall Effect presents a conundrum, as positive Hall Voltage in certain metals suggests the presence of positive charge carriers, which are attributed to holes, despite their movement being a result of electron dynamics. The concept of effective mass explains that electrons near the top of a band can exhibit negative mass, leading to unexpected behavior in response to external forces. Thus, holes are more than just missing electrons; they represent the absence of electrons with negative effective mass, a nuance that is often highlighted in solid-state physics literature.
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What actually are electron holes?

I have come to read that electron holes are nothing but the blank space that is left behind when an electron is missing in a crystal where it ought to have been present. I have also learned that the movement of holes are due to adjacent electrons jumping into the holes (hence creating a hole in its former position). So a hole movement in one direction is created by electron movement in another direction.

But what bothers me is the Hall Effect. Hall Voltage in some metals like Zn are positive which is only possible if the charge carriers are positive.
Saying that holes are a kind of positive charge carrier doesn't answer the riddle because the holes movement are actually the electron movement in the other direction! so the magnetic force acts on these electron and hence negative Hall voltage should have been established?!.

Whats the point I am missing here?
 
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The "effective mass" of an electron is given by the curvature of the band. Hence electrons near the top of a band have a negative mass and also the velocity is anti-parallel to the driving force. So you are right, the electrons really move in the other direction than what you would expect if they were free. The hole picture fits in here nicely. A missing electron of negative mass will behave like a particle with positive mass. The reason for the negative effective mass is that an increase in the crystal momentum of the electrons will lead to an increased Bragg scattering of the electrons. Near the top of the band, this increase in the backscattered wave outweights the increase in crystal momentum.
 


So, you mean that, when measuring Hall voltage, the magnetic force causes the electron drift towards negative direction due to its effective negative mass so that the hall voltage is +ve. Thank you, I understand that.
But, shouldn't it be emphasized that holes are not merely absent of electrons but absent of electrons with negative effective mass?
 


Yes, and I think any good book on solid state theory does that (try e.g. Ashcroft/ Mermin).
 

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