Editing Electron hole (section)

== Detailed picture: A hole is the absence of a negative-mass electron ==
[[File:BandDiagram-Semiconductors-E.PNG|thumb|right|A semiconductor [[electronic band structure]] (right) includes the dispersion relation of each band, i.e. the energy of an electron ''E'' as a function of the electron's [[Wave vector|wavevector]] ''k''. The "unfilled band" is the semiconductor's [[conduction band]]; it curves upward indicating positive [[effective mass (solid-state physics)|effective mass]]. The "filled band" is the semiconductor's [[valence band]]; it curves downward indicating negative effective mass.]]
The analogy above is quite simplified, and cannot explain why holes in semiconductors create an opposite effect to electrons in the [[Hall effect]] and [[Thermoelectric effect#Seebeck effect|Seebeck effect]]. A more precise and detailed explanation follows.<ref name=Kittel>Kittel, ''[[Introduction to Solid State Physics]]'', 8th edition, pp. 194–196.</ref>

{{block indent | em = 1.5 | text = ''The [[dispersion relation]] determines how electrons respond to forces (via the concept of [[Effective mass (solid-state physics)|effective mass]]).''<ref name=Kittel />}}

A dispersion relation is the relationship between [[Wave vector|wavevector]] (k-vector) and energy in a band, part of the [[electronic band structure]]. In quantum mechanics, the electrons are waves, and energy is the wave frequency. A localized electron is a [[Wave packet|wavepacket]], and the motion of an electron is given by the formula for the [[group velocity|group velocity of a wave]]. An electric field affects an electron by gradually shifting all the wavevectors in the wavepacket, and the electron accelerates when its wave group velocity changes. Therefore, again, the way an electron responds to forces is entirely determined by its dispersion relation. An electron floating in space has the dispersion relation {{math|1=''E'' = ℏ<sup>2</sup>''k''<sup>2</sup>/(2''m'')}}, where ''m'' is the (real) [[Electron rest mass|electron mass]] and ℏ is [[Planck constant|reduced Planck constant]]. Near the bottom of the [[conduction band]] of a semiconductor, the dispersion relation is instead {{math|1=''E'' = ℏ<sup>2</sup>''k''<sup>2</sup>/(2''m''<sup>*</sup>)}} ({{math|''m''<sup>*</sup>}} is the ''[[Effective mass (solid-state physics)|effective mass]]''), so a conduction-band electron responds to forces ''as if'' it had the mass {{math|''m''<sup>*</sup>}}.

{{block indent | em = 1.5 | text = ''Electrons near the top of the [[valence band]] behave as if they have [[negative mass]].''<ref name=Kittel />}}

The dispersion relation near the top of the valence band is {{math|1=''E'' = ℏ<sup>2</sup>''k''<sup>2</sup>/(2''m''<sup>*</sup>)}} with ''negative'' effective mass. So electrons near the top of the valence band behave like they have [[negative mass]]. When a force pulls the electrons to the right, these electrons actually move left. This is solely due to the shape of the valence band and is unrelated to whether the band is full or empty. If you could somehow empty out the valence band and just put one electron near the valence band maximum (an unstable situation), this electron would move the "wrong way" in response to forces.

{{block indent | em = 1.5 | text = ''Positively-charged holes as a shortcut for calculating the total current of an almost-full band.''<ref name=Kittel />}}

A perfectly full band always has zero current. One way to think about this fact is that the electron states near the top of the band have negative effective mass, and those near the bottom of the band have positive effective mass, so the net motion is exactly zero. If an otherwise-almost-full valence band has a state ''without'' an electron in it, we say that this state is occupied by a hole. There is a mathematical shortcut for calculating the current due to every electron in the whole valence band: Start with zero current (the total if the band were full), and ''subtract'' the current due to the electrons that ''would'' be in each hole state if it wasn't a hole. Since ''subtracting'' the current caused by a ''negative'' charge in motion is the same as ''adding'' the current caused by a ''positive'' charge moving on the same path, the mathematical shortcut is to pretend that each hole state is carrying a positive charge, while ignoring every other electron state in the valence band.

{{block indent | em = 1.5 | text = ''A hole near the top of the valence band moves the same way as an electron near the top of the valence band '''would''' move''<ref name=Kittel /> (which is in the opposite direction compared to conduction-band electrons experiencing the same force.)}}

This fact follows from the discussion and definition above. This is an example where the auditorium analogy above is misleading. When a person moves left in a full auditorium, an empty seat moves right. But in this section we are imagining how electrons move through k-space, not real space, and the effect of a force is to move all the electrons through k-space in the same direction at the same time. In this context, a better analogy is a bubble underwater in a river: The bubble moves the same direction as the water, not the opposite.

Since force = mass × acceleration, a negative-effective-mass electron near the top of the valence band would move the opposite direction as a positive-effective-mass electron near the bottom of the conduction band, in response to a given electric or magnetic force. Therefore, a hole moves this way as well.

{{block indent | em = 1.5 | text = ''Conclusion: Hole is a positive-charge, positive-mass [[quasiparticle]]''.}}

From the above, a hole (1) carries a positive charge, and (2) responds to electric and magnetic fields as if it had a positive charge and positive mass. (The latter is because a particle with positive charge and positive mass respond to electric and magnetic fields in the same way as a particle with a negative charge and negative mass.) That explains why holes can be treated in all situations as ordinary positively charged [[quasiparticles]].