Inphysics,chemistry, andelectronic engineering, anelectron hole (often simply called ahole) is aquasiparticle denoting the lack of an electron at a position where one could exist in anatom oratomic lattice. Since in a normal atom or crystal lattice the negative charge of the electrons is balanced by the positive charge of theatomic nuclei, the absence of an electron leaves a net positive charge at the hole's location.
Holes in a metal[1] orsemiconductorcrystal lattice can move through the lattice as electrons can, and act similarly topositively-charged particles. They play an important role in the operation ofsemiconductor devices such astransistors,diodes (includinglight-emitting diodes) andintegrated circuits. If an electron is excited into a higher state it leaves a hole in its old state. This meaning is used inAuger electron spectroscopy (and otherx-ray techniques), incomputational chemistry, and to explain the low electron-electron scattering-rate in crystals (metals and semiconductors). Although they act like elementary particles, holes are ratherquasiparticles; they are different from thepositron, which is theantiparticle of the electron. (See alsoDirac sea.)
Incrystals,electronic band structure calculations show that electrons have a negativeeffective mass at the top of a band. Althoughnegative mass is unintuitive,[2] a more familiar and intuitive picture emerges by considering a hole, which has a positive charge and a positive mass, instead.
Insemiconductors, anelectron hole (usually referred to simply as ahole) is the absence of an electron from a fullvalence 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 semiconductorcrystal lattice is comparable to that of the bubble in a full bottle of water.[3]
More generally, a hole is defined as the absence of an electron relative to the system'sground state. This concept applies not only to semiconductors but also to metals with partially filled bands and other electronic systems. A hole with wavevector and spin is created by removing an electron with a wavevector and spin.[4][5]
The hole concept was pioneered in 1929 byRudolf Peierls, who analyzed theHall effect usingBloch's theorem, and demonstrated that a nearly full and a nearly empty Brillouin zones give the oppositeHall voltages.[6]
Hole conduction in avalence 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 theconduction 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 constituteconduction. 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 appliedelectric 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 positivecharge of +e, precisely the opposite of the electron charge.
In reality, due to theuncertainty principle ofquantum mechanics, combined with theenergy 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 ofunit cells. This is equivalent to being unable to tell which broken bond corresponds to the "missing" electron. Conduction band electrons are similarly delocalized.
The analogy above is quite simplified, and cannot explain why holes in semiconductors create an opposite effect to electrons in theHall effect andSeebeck effect. A more precise and detailed explanation follows.[7]
A dispersion relation is the relationship betweenwavevector (k-vector) and energy in a band, part of theelectronic band structure. In quantum mechanics, the electrons are waves, and energy is the wave frequency. A localized electron is awavepacket, and the motion of an electron is given by the formula for thegroup 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 relationE = ℏ2k2/(2m), wherem is the (real)electron mass and ℏ isreduced Planck constant. Near the bottom of theconduction band of a semiconductor, the dispersion relation is insteadE = ℏ2k2/(2m*) (m* is theeffective mass), so a conduction-band electron responds to forcesas if it had the massm*.
The dispersion relation near the top of the valence band isE = ℏ2k2/(2m*) withnegative effective mass. So electrons near the top of the valence band behave like they havenegative 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.
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 same band have positive effective mass, so the net motion is exactly zero. If an otherwise-almost-full valence band has a statewithout 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), andsubtract the current due to the electrons thatwould be in each hole state if it wasn't a hole. Sincesubtracting the current caused by anegative charge in motion is the same asadding the current caused by apositive 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.
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.
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 chargedquasiparticles.
In some semiconductors, such as silicon, the hole's effective mass is dependent on a direction (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 anelectron. This results in lowermobility for holes under the influence of anelectric 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 whyNMOS logic is faster thanPMOS 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 electronsand holes play an essential role. Examples includep–n diodes,bipolar transistors, andCMOS logic.
A hole in semiconductor physics, defined as the absence of an electron in a nearly full valence band, has a formal analogy to thepositron inPaul Dirac's relativistic theory of the electron (SeeDirac equation).[8] 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, theyannihilate each other and the energy is emitted as photons or other radiation. An analogous process,recombination, happens in semiconductors, and can be described as an electron falling to the empty hole state and filling it, emitting radiation.[9]
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.[10] The positron is a real particle with positiveinertial 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,excitons (electron–hole pairs) move rigidly with the lattice and carry no net momentum, unlikepositronium atoms (electron–positron pairs), which gain momentum and energy relative to an accelerating frame.[8]
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.[6]
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