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Holes and Electrons

Spin and Charge Transport

In a solid all positively-charged particles (the protons) are localized inside an atomic nuclear and they do not transport the charge and the spin. Only negatively-charged particles (the electrons) transport the charge and the spin. However, some electrons with energies lower than the Fermi energy behave like positively-charged particles. For example, in an electrical field they diffuse from a "+" to a "-" drain. Therefore, the collective movement of negatively-charged electrons are described as a movement of positively- charged particles called the holes.

 

The hole in a metal and a semiconductor is not real hole. It is not void or unfilled orbital. It is an electron, which has the properties similar to that of a positive particle in vacuum.


The same content can be found in this paper (V.Zayets JMMM 2018 ) .Chapter 6.

 

What do we know about holes in a solid? Is a hole a positive particle?

What do we know about holes in a solid? Is a hole a positive particle? Is there a real hole inside something?

The holes in a solid are "spin" states with energy lower than the Fermi energy. They are negative particles with spin, but their behavior and properties are very similar to behavior and properties of a positive particle.

There is no any real holes in the electron gas!!!!

What do we know for sure:

1. All positive particles are inside nuclear, they do not move inside a solid and their position in solid is fixed.

2. All charge and spin are transported by negatively-charged electrons.

3. Holes is a feature of electron gas, which makes electrons to behave like positive particles.

4. Holes are not unoccupied atomic orbitals. The transport of holes is not continuous jumping of electrons between neighbor atomic orbitals. A hole (electron) can not jump from an orbital to an orbital simply because the size of hole (electron) is substantially longer than the size of the orbital. For example in a semiconductor the size of a hole (the mean-free path) may exceed 1000-10 000 of atomic orbitals.

 

 

Q. Is there a real holes inside the electron gas?

No. There are only electrons in the electron gas. The electrons fills electrons states according to the Fermi-Dirac statistics. There are unfilled states, but they do not contribute to the transport.

There is a wrong view that a hole is unfilled atomic orbital and the electron from a neighbor orbital is scattered into this unfilled state. Therefore, the neighbor orbital becomes unfilled and the real hole literally moves from in the space. It makes the hole current. It is completely wrong view. If this were the case, the mobility of the hole would be extremely low. In contrast, the hole mobility is the same as the electron mobility in metals and it is only a little smaller than the electron mobility in the most of semiconductors. Even in some semiconductors (diamond, PbS, PbTe) the hole mobility is larger than the electron mobility .

The same as an electron current, the hole current is movement of electrons at a high speed in opposite directions. A little bit more electrons move in one direction than in another direction.

 

 

 



Classical model consider a hole literally as a hole or a void in the ocean of electrons

it is an incorrect view !!!!

click here or on picture to enlarge it

 

Q. Is any difference between holes in a semiconductors and holes in a metal? Are they of the same kind?

In a semiconductor, the electrons and the holes belong to different bands and have different symmetry. The electrons belongs to conduction band with s-symmetry and the holes belongs to the valence p-symmetry.

In a metal, there are near the same number of electrons and holes in each band. Often in a metal only one band crosses the Fermi energy and there are near-equal number of electrons and holes in this band.

 

In a semiconductor the hole and the electrons are easily distinguished, because they have different spacial symmetry:

The electrons have s-symmetry. They occupy the conduction band.

The hole have p-symmetry. They occupy the valence band.

 

In a metal in one band there are both the hole and the electrons. They have the same spacial symmetry, but they have different energy:

The energy of the electrons is above the Fermi energy.

The energy of the holes is below the Fermi energy.

 

 

What is the hole? Is it a void in the ocean of electrons? Or it is an electron with special features?

 

Fig.7 Number of quantum states occupied by one election (half-filled states) in a solid

click on image to enlarge it

Mainly the half-filled states participate in the charge, the spin and the heat transfer in a metal and a semiconductor.

Above the Fermi energy EF the slope is positive and the half-filled states are called the electrons

Below the Fermi energy EF the slope is negative and the half-filled states are called the holes

More details is here or V.Zayets, JMMM (2014)

.The holes are electrons, which energy is below the Fermi energy and which occupy quantum states, which is filled only by one electron.

Hole can be approximated as a void in the ocean of electron es in the case of electron energy substantially smaller than the Fermi energy. It is still relatively rough approximation.

In the vicinity of the Fermi energy there is no ocean of electrons and the void has no meaning at all.

 

Q. A negatively-charged particle is attracted to "+" potential and it is repelled from "-" potential. If there are no any positively-charged particles in electron gas, how it is possible that in hole-type semiconductor the charge flows from "+" to "-"? Therefore, the transport particles are repelled from "+" potential and they are attracted to "-" potential?

A .In electron gas the electrons do not simply follows the direction of an electrical field. The transport is more complex, which is explained as follows:

In a metal there are 1021 electrons per a cubic centimeter. It is a great number. It is about a billion times larger than the number of stars in our galaxy. In a semiconductor there are 1016 - 1019 electrons per a cubic centimeter.

All these electrons collide each other and move in all directions with a relatively high speed. Always the same amount of electrons move in any opposite directions.

An applied even-strong electric field does not change much. It does not reverse the movement direction of electrons. Still the electrons move in all directions. The speed of electrons does not change much as well. However, along the direction of electrical field there is a slight difference of number of electrons, which move in opposite directions. Even though this difference is a very tiny, it is sufficient to transfer the charge, the spin and the heat in the metal, because of the huge number of the conduction electrons.

The direction, in which a greater number of electrons move, depends on the energy distribution of electrons. The applied electrical field makes energy of electrons, which moves in the opposite directions, slightly different. A slope of energy distribution determines whether the number of electrons of a higher energy is larger or smaller. Therefore, it determines the direction of an average movement of electrons.

Figure 7 shows the energy distribution of quantum states in electron gas, which are occupied by only one electron (half-filled states). It has a maximum at the Fermi energy EF. It decreases for a higher energy, where all states are not occupied and it decreases for a lower energy, where all states are occupied by two electrons of opposite spins. Mainly the half-filled states participate in the charge, the spin and the heat transfer in a metal and a semiconductor.

As can be seen in Fig.7, in case of electron energy higher than the Fermi energy the slope of energy distribution is negative. It cause that the number of electrons, which move from "-" to "+", is greater than the number of electrons, which move from "+" to "-". Therefore, the average movement of the electrons is similar to a movent of negatively-charged particle, which all move in one direction along the electrical field from "-" to "+". Therefore, the electrons of energy higher than the Fermi energy are called the electrons.

In case of electron energy lower than the Fermi energy the slope of energy distribution is positive. It cause that the number of electrons, which move from "-" to "+", is smaller than the number of electrons, which move from "+" to "-". Therefore, the average movement of the electrons is similar to a movent of positively-charged particle, which all move in one direction along the electrical field from "+" to "-". Therefore, the electrons of energy lower than the Fermi energy are called the holes. (See Fig.8 below)

Classical model consider a hole literally as a hole or a void in the ocean of electrons

it is an incorrect view !!!!

click here or on picture to enlarge it

Q. From the energy conservation law, only two possible charge movement directions are possible. Ether negatively- charged particles move from "-" to "+" or positively-charged particles move from "+" to "-". How it is possible that a negatively-charged electron moves from "+" to "-" ?

In the case of holes the charge movement is in the correct direction. It is because along the movent of states, which are filled by one electron, there is a movement in opposite direction of states filled by two electrons. In total the negative charge of electrons moved from "-" to "+" as it should be.

However, most of the transport properties of the holes are determined by the half-filled states, but not full-filled states. For example, only the half-filled states may transport the spin. When a state is occupied by two electrons of opposite spins, the spin of the state is zero and such full-filled state can not transport the spin.

The contribution of the half-filled states to the ordinary Hall effect is substantially larger than the contribution of the full-filled states. It is because of a substantially different mean-free path for these two kinds of states.

 

Q. In the case of holes, is it possible to use a positively-charged void in the see of electrons instead of the negatively-charged electrons in order to describe the hole transport?

Yes, it is possible. However, it is only an approximation. In some cases this approximation can give an incorrect prediction.

It is better to verify the validity of this approximation for any particular cases.

The approximation of the positively- charged voids works well for electrons of an energy substantially lower than the Fermi energy.

 

Simple facts:

(1) Carries of the spin and the charge:

In a n-semiconductor the carries of the charge and the spin are electrons. In a p-semiconductor the carries of charge and spin are holes. In a metal the carries of charge and spin are both the electrons and holes.

(2) Direction of spin transfer by a drift current.

A drift current of the electrons transfers the spin a direction from "-" to "+". A drift current of the holes transforms the spin in the opposite direction from from "+" to "-". In a metal always there are two near-equal currents of the electrons and holes, which transform the charge in the same direction, but the spin in opposite directions. It is the reason why in a ferromagnetic metal the spin transfer by a drift current is not effective and the spin polarization of a drift current is much smaller than spin polarization of the electron gas in the metal.

Note: When a voltage is applied to a conductor, a current flows in the conductor. This current is called the drift current. Another possible current is diffusion current, which is a current between regions of a smaller and higher spin or charge accumulations.

(3) The Hall effect. The spin and charge accumulations

The Hall effect describes the fact that the spin and the charge are accumulated at sides of wire, when a magnetic field is applied perpendicularly to a current flowing in the wire. (details see here)

The charge accumulation, which is measured by a Hall voltage, is of opposite polarity of electron and hole currents. In contrast, the spin accumulation is of the same polarity for electron and hole currents. For this reason in a metal the Hall voltage is relatively small, but the spin accumulation due the Hall effect is large.

 

a quantum state filled by one electron.

Q? Is it half-full? (it is the electron)

or is it half-empty (it is the hole)

A. It is both. At a higher energy the "spin" states is the electron and at a lower energy the "spin" states is the hole!!!

Similar to a glass of wine, when a quantum state is filled only by one electron, it is called:

- half-filled or the electron

-half-empty or the hole.

 

The holes in a solid are "spin" states with energy lower than the Fermi energy. They are negative particles with spin, but their behavior and properties are very similar to behavior and properties of a positive particle. click here or on image to enlarge it

 

 

Q. Since a metal has both the electrons and holes, is it possible to make a transistor using only metals similar to a semiconductor transistor (a MOSFET transistor or a bipolar transistor?

Answer: No.

A semiconductor has two key properties, which a metal does not have.

Property 1: A semiconductor can be depleted. Therefore, a semiconductor can be switched from a conductor to an isolator.

Property 2: The electrons and the holes in a semiconductor belongs to different bands. It make the relaxation time between a hole and an electron to be relatively long about a few microseconds in a semiconductor. Since the relaxation between holes and electrons is slow, the electron and holes can coexist out from a common equilibrium for a long time. This property of a semiconductor makes possible the fabrication of a semiconductor laser and a bipolar transistor. Also, it makes possible existence of particles like an exciton.

In contrast, in a metal the relaxation time between a electron and a hole is very short about a few hundreds femtoseconds. The electrons and holes in a metal is always in a common equilibrium.

Note, the relaxation time is the average time during which a transition occurs of an electron from a quantum state in the conduction band into a quantum state in the valence band.


 

Each a quantum state for delocalized electrons (conduction electrons) may be filled maximum by two electrons of opposite spins. Main carriers of the charge, the spin and the heat are the quantum states, which are filled only by one electron. When the energy of these half-filled or half-empty quantum states is above the Fermi energy, these states are called the electrons. When their energy is below the Fermi energy, the quantum states of the conduction electrons are called the holes.

 

 


Does the hole has the spin? Is the spin of the hole spin different from the spin of the electron?

 

Does the hole have the spin? Is the spin of the hole different from the spin of the electron?

yes, the hole has the spin.

The spin of the hole is absolutely identical to the spin of the electron

click on image to enlarge it

 

Both the electron and the hole are a quantum state, which filled only by one electron (half-filled states). They are absolutely identical except their energy in the respect to the Fermi energy. Both the electron and the hole have the spin=1/2.

 

In a ferromagnetic metal, in which the electron gas is spin-polarized, the amount of the spin-polarized electrons and the spin-polarized electrons is nearly the same.

Under applied voltage the spin-polarized electrons and holes move in opposite directions. It explains the fact that the spin transfer by a drift current is very inefficient in bulk of a metal.

In the vicinity of interface the conductivities of holes and electrons become different, because of different mean-free paths. It makes the spin transport along an interface much more efficient.

In a metal with defects the mean-free path for the electron and the electron is different as well. Similar it makes the spin transport more efficient.

In a semiconductor the electrons and holes belong to different bands (the conduction band and the valence band). Therefore, they have different orbital moments. The electrons of the conduction band have none or very small orbital moment. The holes of valence band have a substantial orbital moment.

The spin may interact with orbital moment for both the conduction and localized electrons.

Depending whether the electron orbital moment is quenched or unquenched, the spin may interact or not with orbital moment.

The electrical field of a nuclear, the electrical field induced by charge accumulation or an external electrical field may induced substantial the spin-orbit interaction.

Both the hole and the electrons experience the spin-orbit interaction.

In a metal the electron and holes may have near the same magnitude, but opposite polarity of the spin-orbit interaction.

 

 

 

 


Are the holes really positively charged and the electrons negatively charged?

 

Fig. 33 Spacial charge distribution in a semiconductor and metals

Equilibrium

Thermo fluctuation

In the equilibrium the spacial charge distribution is very smooth and the total charge equals to zero everywhere. The spacial charge distribution the is the sum of charges of local and conduction electrons, holes, nuclears and dopants. Due to a thermo fluctuation some electrons may move slightly from right to left. At the left it will be more electrons. It is region of the charge accumulation, which is negatively charged. At the right, there is a region of the charge depletion, which is positively charged. Click on image to enlarge it.

A. No. All conduction in a solid are negatively charged and only nuclears are positively charged. The electrons and holes are very similar. Both the electrons and holes are half-filled states, in which place is filled by an electron and another place is not filled. Only the difference between holes and electrons is that the energy of electrons is above the Fermi energy and the energy of holes is below the Fermi energy.

However, there is a charge accumulation associated with the electron and the charge depletion associated with the hole. Therefore, the electron can be associated with a negative charge and the hole can be associated with a positive charge.

Charge accumulation regions:

In a solid there is an equal number of negatively-charged electrons and positively charged protons. Therefore, the solid is not charged. Even locally, the charge distribution in a solid is very smooth and equals to zero everywhere (Fig. 33 left). A thermo fluctuation (a scattering) can move electron out from the equilibrium and the regions of charge accumulation and charge depletion are formed.

At an energy lower than the Fermi energy, the most of states are filled by two electrons of opposite spins. At an energy lower than the Fermi energy, the most of states are empty (See Fig.35). It causes different sign of charge accumulation region of the hole and electrons. At an energy lower than the Fermi energy, there is an electron depletion at place of the half-filled state (the hole) and there is an electron accumulation at place of the full-filled state. Therefore, the hole is positively charged (See Fig. 34 right). At an energy higher than the Fermi energy, there is an electron accumulation at place of the half-filled state (the electron) and there is an electron depletion at place of the empty state. Therefore, the electron is negatively charged (See Fig. 34 left).

Short effective length (the mean-free path) of a half-filled state makes region of charge accumulation/ depletion at place of the electron and the hole to be narrow and with high magnitude (spark-like).

The effective length or mean-free path is short for half-filled state and it is long for a full-filled state at an energy lower than EF and for an empty state at an energy higher than EF (See here). That makes narrow spark-like charge accumulation/ depletion region at place of the half-filled state and very broad region at place of full-filled state and empty state.

Why does the mean-free path depend on electron energy and type of state?

It is because of a different scattering probability for different state. In order to be scattered from a state the electron free space where to be scattered. For example, at energy sufficiently below the Fermi energy near all states are filled by two electrons of opposite spins (Fig.35). Therefore, near there is no any unoccupied state where an electron could scattered into. Therefore, in this case the life time of full-filled state is very long and the mean-free path is long as well. In contrast, an electron from a half-filled state can be scattered into empty state and a higher energy and an electron from full-filled state can be scattered into the half-filled state. Therefore, it is always many states are available for a scattering from/into a half-filled state. It makes the life time of half-filled state is short and the mean-free path is short as well.

 

Fig. 34 Local charge of Electrons and Holes

Fig.35 The distribution of full-filled and empty states Fig.36. Mean-free path of states, which filled by two , one or none electrons

Electrons

Holes

The electron is the half-filled state with energy above the Fermi energy. When the half-filled state moves left, the right space is occupied by "empty" state. Length (mean-free path) of half-filled state is substantially shorter than the length of the "empty" state. As result, there is a sharp noticeable negatively-charged region of charge accumulation and there is a broad unnoticeable region of charge depletion. Therefore, the electron can be a negatively charged particle. The hole is the half-filled state with energy below the Fermi energy. When the half-filled state moves left, the right space is occupied by full-filled state. Length (mean-free path) of half-filled state is substantially shorter than the length of the full-filled state. As result, there is a sharp noticeable positively-charged region of charge depletion and there is a broad unnoticeable region of charge accumulation. Therefore, the hole can be considered a positively charged particle. Click on image to enlarge it.

The spin statistics is calculated here. Click on image to enlarge it.

Below the Fermi energy EF near-all states are occupied mainly by two electrons. Therefore, when the holes moves, its place is occupied by a full-filled state (See Fig. 34 right).

Above the Fermi energy EF near-all states are empty. Therefore, when the electron moves, its place remains empty(See Fig. 34 left).

The mean-free path or effective length of states, which filled by two, one or none electrons. Distance between defects is 1 µm. The mean-free path is calculated here. Click on image to enlarge it.

The holes and electrons (the half-filled states) always have a short effective length. Therefore, there is a sharp region of charge accumulation/ depletion at the place of an electron or hole (See Fig. 34).

In contrast, a full-filled state at energy lower than EF or a empty state at energy higher than EF have a very long effective length. Therefore, the region of the charge accumulation/ depletion is broad, with small magnitude and near unnoticeable.

 

 

 

Is there clear charge accumulation/ depletion region near each hole or electron?

A. No. Charge charge accumulation/ depletion region near each hole or electron can be distinguished only in a case when there are a few holes and electrons. For example in a semiconductor. When the density of holes and electrons become larger (for example, in a metal), the sum of all charged regions makes a very homogeneous and smooth charge distribution (Fig. 33 left) with any sparks.

Can the electron and hole attract each other?

A. Yes, in the case when there are positive and negative charge accumulation regions, they can attract each other. This can occur only in a semiconductor. See the question above.

Excitons

In a semiconductor the Fermi energy EF is inside band gap. Therefore, electrons belong to conduction band of the s-symmetry and the holes belong to the valence band with the p-symmetry. Because of the different symmetry, the scattering probability of the electron into the hole is very low. The scattering time is about a few microseconds. The electron and the hole attract each other and stay at the same place for rather long time of about a microsecond. The properties of this pair of electron and holes slightly different for properties of individual holes and electrons. Therefore, this pair is considered as a new particle named exciton. The exciton life time is about a microsecond. After that time the electron is scattered into the hole forming a empty state in the conduction band and a full-filled state in the valence band.

Can an exciton exist in a metal?

A. No. In a metal the holes and the electrons belong to the same energy band. Therefore, they have the same symmetry. It makes the scattering probability between them very high. The average scattering time between electron and hole is about 10-100 femtoseconds. This time is too short to form an exciton. Another reason is that there is no spark charge accumulation region at place of hole or electron in metal.

Dopants

The energy level of dopants is close to the the Fermi level. Therefore, a dopant atom can catch or give an electron and the region of the charge accumulation/ depletion is created at place of the donor. The reciprocal region of opposite charge depletion/ accumulation is created at place of the hole or electron.

 




Current of Holes

The Current of hole is the current of electron with energy lower than the Fermi energy.

The half-filled states, which are filled by one electron, and full-filled states, which are filled by two electrons of opposite spins, contribute to the hole current.

 

The electron current and the hole current, they are both the currents of negatively charged particles (electrons)!!!

 

The Hole current is very similar to the current of positively-charged particles. Why?

The facts, which makes the hole current similar to the current of positively-charged particles.:

(1) Hall coefficient (Hall effect) is positive, but not negative as it should be for a current of negatively-charged particles.

(2) The spin is transferred by a drift current in direction from "+" to "-", but not from "-" to "+" as it should be for a current of negatively-charged particles.

A: The half-filled states, which energy is below the Fermi energy, indeed move from "+" to "-" as a positively - charged particles. It explains all above features.


 

In an electron gas there two possible mechanisms of charge, spin and heat transfer:

(1) due to movement of an electron between scatterings.

It is most transport mechanism. It is major transport mechanism in bulk of metals and semiconductors.

I call this mechanism the current of the running-wave electrons.

(2) due to the scattering of electrons between quantum states.

It is substantially less efficient than previous mechanism.

It may become dominant mechanism for the transport in close vicinity of the interface, in a material with many defects or during a tunneling.

More details are here or V Zayets,, arXiv(2014) ;

 


 

Band Current

The band current is the major transport mechanism in the bulk of a conductor. It occurs due to movement of electrons between scatterings. Only the running-wave electrons contribute to this current.

 

 

Band Current

Simple example

electron current

hole current

There are 2 half-filled states moving from"+" to "-" against 3 half-filled states moving from"-" to "+". In total spin and "-" charge are moving from"-" to "+" There are 2 full-filled states moving from"+" to "-" against 3 full-filled states moving from"-" to "+". In opposite,, there are 2 half-filled" states moving from"+" to "-" against 3 half-filled states moving from"-" to "+". In total, the"-" charge is moving from"-" to "+", but the spin is moving in opposite direction from"+" to "-" ,

 

 

How the electrons transfer the charge and the spin in the band current?

In a conductor the electron always move between scatterings. In an equilibrium equal amount of electrons move in any two opposite direction. Therefore in average there is no current. When voltage is applied, there are a little bit more electrons flows along voltage than in the opposite direction. Therefore, in average there is a current along the applied voltage.

Why a negatively-charged electrons in the half-filled states move from "+" to "-" as positively -charged particles??

The conductivity is linearly proportional to the derivative of the the distribution function. The distribution for half-filled states is peak-like (See Fig.7). For electrons with energy > EF , the derivative is negative. It means that in this case the states move as negatively-charged particles. However, for electrons with energy < EF (holes), the derivative is positive. It means that in this case the states move as positively-charged particles. The derivative of distribution of full-filled states is always negative. It means that in this case the full-filled states always move as negatively-charged particles(For more details See here).

 

Why conductivity is proportional to the derivative of the the distribution function?

The band current occurs because there is a difference of electrons, which moves in the opposite directions. For example, if an electrical field is applied between left and right. Then, in two neighbor regions the electrons at left will have a little higher energy than electron in the right region. Because of the energy distribution (the dependence of the number of electron on their energy), the number of electrons in the left and the right regions will be slightly different. Therefore, it will difference of the number of electrons, which move from left to right, and the number of electrons, which move from right to left. The band current will flow. The difference is linearly proportional to the derivative of the the distribution function.

 

Conductivity of electrons and holes

 

Fig.7. Energy distribution of states, which filled by one and two electrons Fig.8 (a) Contribution from states filled by one electron Fig. 8 (b) Contribution from states filled by two electrons
The conductivity is linearly proportional to the derivative of the the distribution function. (For details See here) The conductivity is positive only for electrons with energy > EF (electrons). For electrons with energy < EF (holes), the conductivity is negative. It means that in this case the states move as positively-charged particles The conductivity is always positive. It means that in this case the states move as negatively-charged particles

More details are here or V Zayets,, arXiv(2014) ; Click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

It is important:

The electron and hole drift currents transfer the charge in the same direction, but the spin in the opposite directions (See here)

 

 

 

Electron Current

click here or on picture to enlarge it

Hole Current

click here or on picture to enlarge it

Running-wave electron current and scattering current of the running wave electrons.

When wave functions (dark yellow) of two states are overlap, an electron is scattered on a defect (blue ball) from a "spin" state into an "empty" state.

 

The conduction electrons contribute to the total conductivity unequally. The electrons of deep-energy levels nearly do not contribute at all. Only electrons with energy near the Fermi energy mainly conduct the spin and the charge in the electron gas.

Also, the conductivity is different for electrons occupying states, which filled by one electron and filled by two electrons of opposite spin.

Why?

Reason 1 (major) : The energy distribution of electrons occupying the half-filled and full-filled state are substantially different. (See above)

Reason 2 (minor): The mean-free path of electrons occupying the half-filled and full-filled state are substantially different. More details is here orV Zayets,, arXiv(2014)

 

 

 

 

 

 

 



 

Why is an electron is faster

and a hole is slower???

 

Why electron mobility is greater than hole mobility???

Why is the mobility of electron higher than that of hole?

Interesting question from ReaserchGate

A simple answer would be

a) It is because the effective mass of electron is much smaller than the effective mass of an hole

b) It is feature of the band diagram

It does not explain any Physics. Both the electrons and the holes experience the same scatterings from the same defects. What is the difference between them?

My answer is

It depends on the material. In a metal, the mobility of electrons and holes is almost the same.

In a semiconductor, the electron mobility is larger than the hole mobility (except for a few exceptions)

The major reason of different motilities in a semiconductor is different special symmetry for electrons and holes.

The special symmetry of electrons is s-orbital like (spherical). The special symmetry of holes is p-orbital like.

It takes a longer time for an electron to circle the p-orbital than the s-orbital. It makes a hole slower than an electron.

(See here)

 

click here for detailed explanation & answers

Abhishek Chauhan · Shri Mata Vaishno Devi University

Electrons travel in the conduction band between the two materials. Since the conduction band is formed by the uppermost shell, farthest from any of the nucleus, and since proton resides inside nucleus and hence more energy is required to move an proton than an electron


 

Joel T. Asubar · 20.37 · 25.34 · Hokkaido University

Holes are absence of electrons in covalent bond and hole movements are actually movement of electrons to fill up an adjacent hole in the covalent bond. Thus, the electrons moving this way are still under the influence of nuclear forces that scatter them. In contrast, free electrons (or just commonly referred to as electrons) move within the semiconductor material freely, without going from a covalent bond vacancy to another. Meaning, free electrons are less influenced by the scattering due to nuclear forces and hence the higher mobility.

my comment:

Explanation given by Joel T. Asubar is not correct.

The movement of holes is not movement of electrons to fill up an adjacent hole in the covalent bond. The movement of a hole is as "free" as the movement of an electron. In fact, both the electrons and the holes are the same delocalized electrons. For example, in a metal the electrons and holes can be only distinguished by their energy. The energy of an electron is above the Fermi energy and the energy of a hole is below the Fermi energy. Of cause, an electron and a hole interact differently with electrical field.

The mean-free path of a hole may reach several micrometers (typically in a semiconductor it is 50-200 nm at room temperature). If you see an electron as a wave, the mean-free path of an electron is just the size of the electron. Therefore, the size of a hole may be a micrometer. The distance between covalent bonds is about 0.1 nm. It means that simultaneously an electron or a hole is filling up or interacts with a million of covalent bonds. Surely, the movement of holes is not movement of an electron to fill up adjacent holes in the covalent bonds. Besides, in order to jump from one covalent bond to other covalent bond an electron needs to interact with a phonon. In the case if such jumping between the covalent bonds were the mechanism of the hole transport, the hole mobility would be very small. It would be even smaller than mobility in a material with hopping conduction.

In a metal the electron mobility and the hole mobility is the same. Only reason for a difference in the mobilities in a semiconductor is different spatial symmetry of the envelop function for an electron and a hole.

following question

Thank you Vadym Zayets for the detailed explanation considering spin-orbit interaction. However, I have a little doubt in my mind. As you pointed out, both are delocalized electrons having only difference in spatial symmetry,how would you explain the effective mass of electrons is smaller than hole in accordance with your explanation?

my answer

The mobility of electrons and holes is different, because the electrons and holes have different spatial symmetry. The electrons have s-orbital -like symmetry of the envelop wave function. The holes have p-orbital -like symmetry of the envelop wave function.

There are two reasons why the difference in the spatial symmetry causes the different mobility. They may sound different, but they explain the same fact.


The first reason, why the mobility is different, is that it takes a longer time for an electron to circle around the p-orbital than around the s-orbital.
A delocalized electron simultaneously moves along the crystal lattice and rotates around millions of atomic nuclei . It might sound strange that it is possible to rotate around several objects simultaneously. Since an electron is a wave, it is possible. For example, when light is diffracted by a diffraction grating , each photon is reflected from each of million steps of the grating) simultaneously. Any wave can interact with many objects simultaneously. It is not forbidden.
There are many experimental proofs that delocalized electrons have a non-zero orbital moment. For example, their g-factor is different from 2, they experience the center-symmetrical spin-orbit interaction and the existence of the split-off valence band is other proof. The non-zero orbital moment literally means that the delocalized electron is orbiting around atomic nuclei.

The second reason, why the mobilities (effective masses) are different for holes and electrons in a semiconductor, is that the magnitudes of the s-like and p-like envelope functions are different in the vicinity of the atomic nucleus. The electrons of different symmetry interact differently with the nucleus. Therefore, the amplitude of the lattice periodic potential is different for electrons and holes in a semiconductor. This causes the different effective masses.

 


Answer from Samares Kar

Mobility is proportional to the average time between two consecutive scattering events (collisions) and is inversely proportional to the effective mass. Electrons and holes are scattered by different types of crystal imperfections, which include phonons, impurities, defects, etc. In current MOSFET transistor, the channel electrons and holes are scattered mainly by impurity and interface charges (Coulomb scattering), interface roughness, and phonons. The average time between collisions (also known as the mean free time) is given by the Mathiessen's Rule. For most semiconductors (elemental, III-V compounds, II-VI compounds, etc.), the electron mobility is larger than the hole mobility, and in some cases much larger; but there are exceptions. Exceptions are: electron mobility/hole mobility in cm-2V-1s-1: C (Diamond) - 1800-2200/1600-1800; PbS - 600/700; PbTe - 2500-6000/1000-4000.

 

 


Mobility of s-, p-, d- electrons and holes.

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Question from Usman ul Muazzam

I have a question that mobility of holes in oxides having 2p orbital (e.g. gallium oxide) is less compared to mobility of holes in oxides having 3d orbital, e.g. copper oxide , why this is so? I have followed your answers on difference in mobility of electrons and holes but still didn't get answer to this question. Could you please explain it to me.

Answer:

Firstly, the mobility does not depend whether the electron is the “electron” or “hole”. For example, the mobility of the holes and electrons in a metal is absolutely identical. Whether it is a hole and an electron are only distinguished by the electron energy in respect to the Fermi energy. The energy of a hole is smaller than the Fermi energy, the energy of an electron is greater. Generally the energy of an electron of p-symmetry is smaller than an electron of s-symmetry and the energy of an electron of d-symmetry is smaller than an electron of p-symmetry as it follows the electron arrangement in a free atom (it is not always the case). Therefore, an electron of p- or d-symmetry is often the hole, but an electron of p- or d-symmetry is often the “electron”.
You can look in more details about this topic in the following my paper
V. Zayets, ", JMM 445, pp 53–65 (2018).
You can download it here

Secondly, the mobility of the electron depends more on crystal symmetry than atomic symmetry (s-,p-,d-). For example, it is believed that d-electrons in Fe metal are localized for e_1g symmetry and delocalized for t_2g symmetry. Therefore, the d-electrons in iron have zero mobility, when they have e_1g symmetry, and they have reasonably high mobility, when they have t_2g symmetry.
You can understand it as follows.
The mobility of an electron depends on the length of the electron. The longer length of the electron, the greater its mobility. (of course, the mobility is also depends on several other parameters)


The length of an electron depends how well the electron function of neighbor atoms are overlap. Within oversimplify view, more they overlap, longer the electron length is.

I do not now details about band structure of gallium oxide and copper oxide.
A possible explanations is as follows.
The oxide are mainly ionic crystals. The oxygen takes one or two s- or p- from Ga ore Cu. Therefore, s- or p- electrons becomes localized and they do not participate in the conductivity. In contrary, d- or p- electrons are still near its host atom, their neighbor-atom wave functions can overlap and they become delocalized conduction electrons. Of course, everything depends on crystal symmetry. Additionally, the wavefunction could be elongated due the Coulomb's repulsion from neighbor oxygen.
Everything depends on crystal symmetry and atom arrangement. An electron of d-symmetry may have greater mobility that an electron of p-symmetry.

 

 


Heavy hole & Light hole

What is difference between a heavy hole and a light hole??? Their Weight??

click here or on image to enlarge it

 

The heavy and light holes are defined only for a semiconductor, but not metal.

 

Even though the effective mass of heavy holes is usually larger than the mass of light holes, the heavy and light holes are distinguished by their spin and orbital moment mutual directions.

 

Heavy holes:

spin is parallel to the orbital moment

J=3/2 (L=1 S=1/2) and J=-3/2 (L=-1 S=-1/2)

 

 

Light holes:

spin is antiparallel to the orbital moment

J=1/2 (L=1 S=-1/2) and J=-1/2 (L=-1 S=1/2)

 

 

 

 

 

 

Orbitals of Heavy Holes

Shape of orbitals depends on direction of the orbital moment. If orbital moment and spin is unquenched, the orbital shape depends on spin direction as well.

Direction of Spin and orbital moment is perpendicular to the propagation direction

click here or on image to enlarge it

Direction of Spin and orbital moment is along to the propagation direction

click here or on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

Direct-band semiconductor

without strains compressively -strained tensile-strained

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Scattering current

3 scattering currents can be distinguished: electron current, hole current and full-empty currents

Both the standing-wave electrons and the running-wave electrons contribute to this current

The scattering current is the major transport mechanism in vicinity of a high-resistivity contact.

Electron scattering current

Hole scattering current.

Consequence scattering of an electron from a "spin" state into "empty" state corresponds to movement of "spin" states from right to left and to movement of "empty" from left to right . The spin and "-e" charge are transported in the same direction from the right to the left. Consequence scattering of an electron from a "full" state into a "spin" state corresponds to movement of "spin" states from right to left and to movement of "full" from left to right. The spin and "-e" charge are transported in opposite directions. The spin is transported from the right to the left. The "-e" charge is transported from the right to the left.

 

 

Scattering electron current

Scattering hole current

The scattering electron current in n-type semiconductor. The yellow cells indicates quantum states. In a n-type semiconductor the most states are "empty". There are a few half-filled states, which are filled only by one electron. Electrons are drifted through the sample by the scattering from a half-filled state into an "empty" state. At left side of the sample the directions of movement of half-filled and "empty" states are shown. At right side of the sample the directions of movement of a positive charge, a negative charge and spin are shown. Important: The spin moves from "-" to "+". It is opposite direction to the case of p-type semiconductor The scattering hole current in p-type semiconductor. Spin-up current. The yellow cells indicates quantum states. In a p-type semiconductor the most states are filled by two electrons of opposite spins. There are a few half-filled states. Electrons are drifted through the sample by the scattering from a full-filled state into a half-filled state. At left side of the sample the directions of movement of half-filled and full-filled states are shown. At right side of the sample the directions of movement of a positive charge, a negative charge and spin are shown. Important: The spin moves from "+" to "-". It is opposite direction to the case of n-type semiconductor

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

From Prameela: at some point in this discussion, it was asked about the higher values of electron mobilities. But there it was said that in metals, the mobilities of both electrons & holes is the same. Now my doubt is why will metals have holes at all. it would be only the electrons that are the charge carriers?

A. Dear Prameela. A metal has nearly equal amounts of electrons and holes. In a metal, an electron and a hole are almost the same thing. The difference between them is only their energy with respect to the Fermi energy. In the vicinity of the Fermi energy the difference between an electron and a hole is negligible. We have learned from a semiconductor. The case of a semiconductor is special. In a semiconductor the Fermi level is in gap between two bands of very different symmetry. As a result, all electrons are of s-like symmetry and they belong to the conduction band. All holes are of p-like symmetry and they belong to the valence band. Because of the different symmetry, the scatterings between electrons and holes are rare and it is is possible to assign different quasi-Fermi levels for electrons and holes. This feature is determines all unique properties of the semiconductor. Because of different symmetry, many properties (like the mobility) is different for electrons and holes. The case of a metal is very different. The holes and electrons have the same symmetry and belong to the same band. Therefore, many of their property (e.g. the mobility) are the same. Still it is important to distinguish between holes and electrons in a metal. For example, the polarity of Hall effect is different in an electron-dominated metal and in a hole-dominated metal (See here). Even though, both the electron and the hole are in fact the electrons, in a solid they behave very differently. The electron behaves as a negatively-charged particle. The hole behaves as a positively -charged particle.

 

 

I truly appreciate your comments, feedbacks and questions

I will try to answer your questions as soon as possible

 

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