Editing Electron hole

{{short description|Conceptual opposite of an electron}}
{{distinguish|Holon (physics)}}

[[Image:Electron-hole.svg|thumb|When an electron leaves a [[helium]] atom, it leaves an electron hole
in its place. This causes the helium atom to become positively charged.]]

In [[physics]], [[chemistry]], and [[electronic engineering]], an '''electron hole''' (often simply called a '''hole''') is a [[quasiparticle]] denoting the lack of an electron at a position where one could exist in an [[atom]] or [[crystal structure|atomic lattice]].  Since in a normal atom or crystal lattice the negative charge of the electrons is balanced by the positive charge of the [[atomic nucleus|atomic nuclei]], the absence of an electron leaves a net positive charge at the hole's location.

Holes in a metal<ref name="ashcroftandmermin">{{cite book|last1=Ashcroft and Mermin|title=Solid State Physics|date=1976|publisher=Holt, Rinehart, and Winston | isbn=978-0-03-083993-1|pages=[https://archive.org/details/solidstatephysic00ashc/page/299 299–302]|edition=1st|url-access=registration |url=https://archive.org/details/solidstatephysic00ashc/page/299}}</ref> or [[semiconductor]] [[crystal lattice]] can move through the lattice as electrons can, and act similarly to [[electric charge|positively-charged]] particles. They play an important role in the operation of [[semiconductor device]]s such as [[transistor]]s, [[diode]]s (including [[light-emitting diodes]]) and [[integrated circuit]]s. If an electron is excited into a higher state it leaves a hole in its old state. This meaning is used in [[Auger electron spectroscopy]] (and other [[x-ray]] techniques), in [[computational chemistry]], and to explain the low electron-electron scattering-rate in crystals (metals and semiconductors).  Although they act like elementary particles, holes are rather [[quasiparticle]]s; they are different from the [[positron]], which is the [[antiparticle]] of the electron. (See also [[Dirac sea]].)

In [[crystal]]s, [[electronic band structure]] calculations show that electrons have a negative [[effective mass (solid-state physics)|effective mass]] at the top of a band. Although [[negative mass]] is unintuitive,<ref>For these negative mass electrons, [[crystal momentum|momentum]] is opposite to [[group velocity|velocity]], so forces acting on these electrons cause their velocity to change in the 'wrong' direction. As these electrons gain energy (moving towards the top of the band), they slow down.{{cn|date=May 2025}}</ref> a more familiar and intuitive picture emerges by considering a hole, which has a positive charge and a positive mass, instead.

== Definition ==

In [[Semiconductor|semiconductors]], an ''electron hole'' (usually referred to simply as a ''hole'') is the absence of an electron from a full [[valence band]]. A hole is essentially a way to conceptualize the interactions of the electrons within a nearly full valence band of a crystal lattice, which is missing a small fraction of its electrons. In some ways, the behavior of a hole within a semiconductor [[Crystal structure|crystal lattice]] is comparable to that of the bubble in a full bottle of water.<ref>{{cite journal|last=Weller|first=Paul F. |date=1967 |title=An analogy for elementary band theory concepts in solids|journal=J. Chem. Educ. |volume=44 |issue=7 |pages=391 |doi=10.1021/ed044p391 |bibcode = 1967JChEd..44..391W}}</ref>

More generally, a hole is defined as the absence of an electron relative to the system's [[ground state]]. This concept applies not only to semiconductors but also to metals with partially filled bands and other electronic systems. A hole with wavevector ''<math>k</math>'' and spin <math>\uparrow</math> is created by removing an electron with a wavevector ''<math>-k</math>'' and spin <math>\downarrow</math>.<ref name="i776">{{cite book |last=Coleman |first=Piers |title=Introduction to many-body physics |date=2015 |publisher=Cambridge University Press |isbn=978-1-139-02091-6 |publication-place=Cambridge |pages=82-83}}</ref><ref>{{cite book |last=Nazarov |first=Julij V. |title=Advanced quantum mechanics: a practical guide |last2=Danon |first2=Jeroen |date=2013 |publisher=Cambridge University Press |isbn=978-0-511-98042-8 |publication-place=Cambridge |pages=99-100}}</ref>

The hole concept was pioneered in 1929 by [[Rudolf Peierls]], who analyzed the [[Hall effect]] using [[Bloch's theorem]], and demonstrated that a nearly full and a nearly empty Brillouin zones give the opposite [[Hall voltage]]s.<ref name=":0" />

== Simplified analogy: Empty seat in an auditorium ==
[[File:15-puzzle-02.jpg|thumb|A children's puzzle which illustrates the mobility of holes in an atomic lattice. The tiles are analogous to electrons, while the missing tile ''(lower right corner)'' is analogous to a hole.  Just as the position of the missing tile can be moved to different locations by moving the tiles, a hole in a crystal lattice can move to different positions in the lattice by the motion of the surrounding electrons.]]

Hole conduction in a [[valence band]] can be explained by the following analogy:

Imagine a row of people seated in an auditorium, where there are no spare chairs. Someone in the middle of the row wants to leave, so he jumps over the back of the seat into another row, and walks out. The empty row is analogous to the [[conduction band]], and the person walking out is analogous to a conduction electron.

Now imagine someone else comes along and wants to sit down. The empty row has a poor view; so he does not want to sit there. Instead, a person in the crowded row moves into the empty seat the first person left behind. The empty seat moves one spot closer to the edge and the person waiting to sit down. The next person follows, and the next, et cetera. One could say that the empty seat moves towards the edge of the row. Once the empty seat reaches the edge, the new person can sit down.

In the process everyone in the row has moved along. If those people were negatively charged (like electrons), this movement would constitute [[Electrical resistivity and conductivity|conduction]]. If the seats themselves were positively charged, then only the vacant seat would be positive. This is a very simple model of how hole conduction works.

Instead of analyzing the movement of an empty state in the valence band as the movement of many separate electrons, a single equivalent imaginary particle called a "hole" is considered. In an applied [[electric field]], the electrons move in one direction, corresponding to the hole moving in the other. If a hole associates itself with a neutral atom, that atom loses an electron and becomes positive. Therefore, the hole is taken to have positive [[electric charge|charge]] of +''e'', precisely the opposite of the electron charge.

In reality, due to the [[uncertainty principle]] of [[quantum mechanics]], combined with the [[Bloch's theorem|energy levels available in the crystal]], the hole is not localizable to a single position as described in the previous example. Rather, the positive charge which represents the hole spans an area in the crystal lattice covering many hundreds of [[crystal structure|unit cells]]. This is equivalent to being unable to tell which broken bond corresponds to the "missing" electron.  Conduction band electrons are similarly delocalized.

== 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]].

== Role in semiconductor technology ==
[[File:Silicon doping - Type P.svg|thumb|An array of [[Silicon]] atoms [[Doping (semiconductor)|doped]] with [[Boron]] creates holes. This type of [[Extrinsic semiconductor|extrinsic semiconducting material]] is dubbed ''Type P''.]]
In some semiconductors, such as silicon, the hole's effective mass is dependent on a direction ([[Anisotropy|anisotropic]]), however a value averaged over all directions can be used for some macroscopic calculations.

In most semiconductors, the effective mass of a hole is much larger than that of an [[electron]]. This results in lower [[Electron mobility|mobility]] for holes under the influence of an [[electric field]] and this may slow down the speed of the electronic device made of that semiconductor. This is one major reason for adopting electrons as the primary charge carriers, whenever possible in semiconductor devices, rather than holes. This is also why [[NMOS logic]] is faster than [[PMOS logic]].
[[OLED]] screens have been modified to reduce imbalance resulting in non radiative recombination by adding extra layers and/or decreasing electron density on one plastic layer so electrons and holes precisely balance within the emission zone.  
However, in many semiconductor devices, both electrons ''and'' holes play an essential role. Examples include [[p–n diode]]s, [[Bipolar junction transistor|bipolar transistors]], and [[CMOS logic]].

== Comparison to positron ==
A hole in semiconductor physics, defined as the absence of an electron in a nearly full valence band, has a formal analogy to the [[positron]] in [[Paul Dirac]]'s relativistic theory of the electron (See [[Dirac equation]]).<ref name=":1">{{Cite journal |last=Kohn |first=W. |date=1970 |title=Electrons, positrons and holes |url=https://linkinghub.elsevier.com/retrieve/pii/0025540870901054 |journal=Materials Research Bulletin |language=en |volume=5 |issue=8 |pages=641–654 |doi=10.1016/0025-5408(70)90105-4}}</ref> In both cases, the system is described as a filled sea of negative-energy or valence states, and the removal of an electron leads to a positively charged entity that can carry current. The analogy extends to their electromagnetic behavior: both holes and positrons have a charge that is equal and opposite to that of an electron. When an electron and positron collide, they [[Annihilation|annihilate]] each other and the energy is emitted as photons or other radiation. An analogous process, [[Carrier generation and recombination|recombination]], happens in semiconductors, and can be described as an electron falling to the empty hole state and filling it, emitting radiation.<ref name="d593">{{cite book |last=Simon |first=Steven H. |url=https://www.worldcat.org/title/853504907 |title=The Oxford Solid State Basics |date=2013-06-20 |publisher=OUP Oxford |isbn=978-0-19-968077-1 |publication-place=Oxford |page=183 |oclc=853504907 |access-date=2025-05-18}}</ref> 

However, there are also limitations to this analogy. Due to the symmetries of Dirac's theory, positron and electron have exactly the same mass, while holes and electrons in crystals generally have different masses.<ref>{{cite encyclopedia |title=Semiconductors, History of |encyclopedia=Encyclopedia of Condensed Matter Physics |publisher=Elsevier |last=Bassani |first=G. F. |date=2005 |editor-last=Bassani |editor-first=G. F. |publication-place=Amsterdam Boston |pages=325 |isbn=978-0-12-369401-0 |last2=La Rocca |first2=G. C.}}</ref> The positron is a real particle with positive [[inertial mass]] and rest energy, while the hole is a quasiparticle whose inertial mass is negative. For this reason the responses differ in non-inertial frames: in an accelerating crystal lattice, a positron lags behind, whereas a hole moves forward with the lattice. These differences also appear in composite systems; for example, [[Exciton|excitons]] (electron–hole pairs) move rigidly with the lattice and carry no net momentum, unlike [[positronium]] atoms (electron–positron pairs), which gain momentum and energy relative to an accelerating frame.<ref name=":1" />

The concept of an electron hole in solid-state physics predates the concept of a hole in Dirac equation, but there is no evidence that it would have influenced Dirac's thinking.<ref name=":0">{{Cite book |last=Pippard |first=Brian |title=Twentieth century physics |publisher=American Institute of Physics Press |year=1995 |isbn=978-0-7503-0310-1 |volume=III |pages=1296–1298 |chapter=Electrons in solids}}</ref>

== See also ==
* [[Band gap]]
* [[Effective mass (solid-state physics)]]
* [[Electrical resistivity and conductivity]]

== References ==
{{Reflist}}

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[[Category:Electronics concepts]]
[[Category:Quasiparticles]]
[[Category:Quantum chemistry]]
[[Category:Charge carriers]]
[[Category:Holes]]