Semiconductors

Question 1

Examples of semiconducting materials

Answer 1

Electronics
Batteries
Solar cells
Self-cleaning glass
LEDs

Question 2

Traditional description of bonding in metals

Answer 2

Bonding in metals is described as a ‘sea of electrons’ (free electron model)

Question 3

Electronic band structure of a solid

Answer 3

Describes the range of energies an electron within a solid may have (energy bands) and the range of energies it may not have (band gap)
i.e. electrons are not allowed in the band gap - this is a forbidden energy level

Question 4

Band theory

Answer 4

Derives energy bands and band gaps by examining the quantum mechanical wave functions for an electron in a large lattice of atoms/molecules

Question 5

What are bands composed of?

Answer 5

Closely spaced electron orbitals

Question 6

Derivation of electronic band structure

Answer 6

When atoms are far apart, the electrons of the atom occupy atomic orbitals of discrete energy
When N atoms come closer together to form a solid/crystal, their atomic orbitals overlap
The Pauli exclusion principle dictates that each atomic orbital must split into N discrete molecular orbitals, each with a different energy
Because the number of atoms in a solid is very large (N~10^22), the number of orbitals is also very large, and thus they are very closely spaced in energy (~10^-22 eV)
The energy spacing between adjacent orbitals is so small that there is effectively a continuous band of energies

Question 7

Which electrons in atoms are involved in forming bands?

Answer 7

Valence electrons (i.e. the outermost electrons)
Inner electron orbitals do not overlap to a significant degree - they do form bands but they are very narrow

Question 8

Band gap

Answer 8

An energy range in a solid where no electron states can exist
The energy difference (in eV) between the top of the valence band and the bottom of the conduction band

Question 9

Valence band

Answer 9

The highest range of electron energies in which electrons are normally present at T = absolute zero (0K)
Located below the Fermi level

Question 10

Conduction band

Answer 10

The lowest range of vacant electron states

Located above the Fermi level

Question 11

Fermi level (Ef)

Answer 11

The energy level with a 50% probability of being occupied for all finite temperatures
Expresses the work required to add an electron to the body, or the work obtained by removing an electron

Question 12

Position of Fermi level in metals

Answer 12

Lies inside at least one band

Question 13

Position of Fermi level in insulators and semiconductors

Answer 13

Lies inside a band gap

But in semiconductors the bands are close enough to the Fermi level to be thermally populated with electrons or holes

Question 14

T = 0K

Answer 14

Probability of finding an electron below the Fermi level is 100%
Probability of finding an electron above the Fermi level is 0

Question 15

T > 0K

Answer 15

Some electrons have E > Ef

The energy levels are filled following a Boltzmann distribution

Question 16

What happens to an electron that has been promoted to the conduction band?

Answer 16

It is accelerated
(but cannot do so forever due to collisions with ions/phonons)
Follows the Drude model - electrons experience collisions as they move through the crystal and are scattered by phonons

Question 17

Drude model

Answer 17

Simple model for conduction

Treats the solid as a fixed array of ions with unbound electrons

Question 18

Phonon

Answer 18

Quantised lattice vibration
Small energy, large momentum

At T = 0K, a crystal lattice is in its ground state and contains no phonons (so electrons primarily scatter off impurities/other defects in the crystal)
At non-zero T, a lattice has an energy that fluctuates randomly about a mean value (i.e. is not constant)
This energy fluctuation is caused by random lattice vibrations, which can be viewed as a gas of phonons (i.e. phonons provide the primary scattering mechanism for electrons)

Question 19

Interactions between electrons and phonons

Answer 19

When electrons interact with phonons, they lose small amounts of energy but experience a large shift in momentum

Question 20

Significance of electron-phonon interactions

Answer 20

Under normal conditions (i.e. at room temp.), the interaction between phonons and electrons is basically insignificant (i.e. electrons are basically unaffected by phonon energy)
When temperature is increased, phonon vibration increases so they interact more with electrons (hence, metal conductivity decreases with increasing temperature)
At low temperatures, phonons vibrate less, leading to fewer collisions and increased conductivity

Question 21

Properties of the electrons in a band

Answer 21

All states up to the Fermi level are full
Difference in energy between states is very small (~10^-22 eV)
Zero potential energy (because electrons can move), therefore the total energy must be equal to the kinetic energy

Question 22

Kinetic energy formula

Answer 22

KE = 1/2mv^2
p (momentum) = mv

Therefore KE = p^2/2m
(showing that energy and momentum are related)

Question 23

In the absence of an electric field…

Answer 23

…there is no net movement of electrons

Question 24

When an electric field is applied…

Answer 24

…the electrons experience electric force

The electrons gain momentum and are now flowing in the opposite direction to the electric field

Question 25

Electric force equation

Answer 25

F=qE Where q = charge (so for an electron this is negative, which means the force is in the opposite direction to the electric field, E) (current flows in the direction of the field, opposite to the electrons)

Question 26

Ways in which electrons can be excited

Answer 26

1. By an electric field 2. Thermally - absorption of heat by electrons is significant at low temperatures 3. By EM radiation of an appropriate wavelength E = hc/lambda, so lambda = hc/E, where E(cutoff) is the minimum energy required to promote an electron

Question 27

What do electrons collide with in solids?

Answer 27

Valence electrons and ions Phonons Impurities Crystal imperfections

Question 28

How can we account for the very large range of electrical conductivity in solids?

Answer 28

Metallic solids e.g. Ag, Cu have high conductivity because their electrons are delocalised, so can move in response to an electric field Covalent solids e.g. S have very low conductivity because the electrons are strongly bound to particular atoms (i.e. such materials are insulators)

Question 29

Fermi energy, Ef

Answer 29

= the 'cut off' point between filled and vacant energy states At T = absolute 0, the probability that a state below the Fermi energy is occupied is 100 % and the probability a state higher than the Fermi energy is occupied is 0 % i.e. Ef = the highest occupied energy level at absolute zero i.e. the Fermi level at absolute zero, where the Fermi level is the energy level with a 50% chance of being occupied at finite temperature

Question 30

Difference between Fermi level and Fermi energy

Answer 30

Fermi energy does not depend on temperature, whereas Fermi level does

Question 31

Effect of temperature on conductivity of metals

Answer 31

Increasing temperature decreases metal conductivity due to increased thermal vibrations of the ions

Question 32

Effect of temperature on conductivity of semi-conductors

Answer 32

Increasing temperature increases conductivity Conc. of electrons in the conduction band increases with temperature At high temperatures, Si has approximately the same conductivity as a metal

Question 33

Ebg < 3eV

Answer 33

Semi-conductor

Question 34

Ebg > 3 eV

Answer 34

Insulator

Question 35

Electron hole

Answer 35

The lack of an electron at a position where one could exist in an atom/atomic lattice (Basically what the electron leaves behind after being excited into a higher state) Holes in a metal/semi-conductor crystal lattice can move through the lattice like electrons can, behaving like positively-charged particles - i.e. holes can contribute to conductivity

Question 36

Role of electron holes in conductivity

Answer 36

Electron holes can readily accept electrons | In semiconductors, current conduction by holes is as important as electron conduction in general

Question 37

Doping a semi-conductor

Answer 37

Increases the number of conducting electrons by introducing suitable impurity atoms Allows us to control the conductivity of the material

Question 38

Doping Si with As

Answer 38

As = group V element so has 5 valence electrons - 4 of these electrons are shared with neighbouring Si atoms, leaving one electron free to become mobile As = 'donor' impurity, because it donates electrons No hole is created in conjunction with the creation of a conduction electron

Question 39

N-type semiconductors

Answer 39

Contain many mobile electrons and few holes | because electrons carry negative (N) charge

Question 40

Doping Si with B

Answer 40

``` B = group III element so has 3 valence electrons - each B atom can therefore accept electrons to satisfy the covalent bonds, thus creating a hole B = 'acceptor' impurity, because it accepts electrons ```

Question 41

P-type semiconductors

Answer 41

Contain many holes and few mobile electrons | because holes carry positive (P) charge

Question 42

Examples of donor impurities

Answer 42

As, P

Question 43

Examples of acceptor impurities

Answer 43

B, Al, In

Question 44

Amphoteric impurities

Answer 44

Can act as donors or acceptors | e.g. group IV Si/Ge in GaAs (group III and group V)

Question 45

Which plane is Si cut along?

Answer 45

(1,0,0) plane | This is the best plane for uniformity and good device performance

Question 46

Direct band gap

Answer 46

When the minimal-energy state in the conduction band and maximal-energy state in the valence band have the same k-vector (crystal momentum) An electron can be directly excited from the valence band to the conduction band without a change in crystal momentum

Question 47

Indirect band gap

Answer 47

When the minimal-energy state in the conduction band and maximal-energy state in the valence band have different k-vectors (crystal momentum) An electron cannot shift from the highest-energy state in the valence band to the lowest-energy state in the conduction band without a change in momentum This means a photon can't be emitted because photons cannot carry crystal momentum

Question 48

Radiative recombination

Answer 48

The process by which an electron that has been promoted to the conduction band loses energy and re-occupies the energy state of an electron hole in the valence band, releasing the excess energy as a photon This is possible in a direct band gap semiconductor if the electron has a k-vector near the conduction band minima (the hole also shares the same k-vector)

Question 49

Why can the valence band in semiconductors not carry an electric current?

Answer 49

The valence band is so nearly full of electrons that the electrons are not mobile, so cannot flow as electric current

Question 50

Why is radiative recombination not possible in semiconductors with indirect band gaps?

Answer 50

Because photons cannot carry crystal momentum | Thus conservation of crystal momentum would be violated

Question 51

How can radiative recombination occur in a material with an indirect band gap?

Answer 51

Radiative recombination in indirect band gap materials must involve the absorption or emission of a phonon, where the phonon momentum equals the difference between the electron and hole momentum Crystallographic defects can also perform essentially the same role

Question 52

Why is radiative recombination far slower in materials with indirect band gaps, compared to those with direct band gaps?

Answer 52

The involvement of a phonon in radiative recombination in indirect band gap materials makes the process much less likely to occur within a given span of time

Question 53

Objects produced from direct band gap materials

Answer 53

Light-emitting and laser diodes Recombination of electrons with electron holes within the device releases energy in the form of photons (heat/light energy)

Question 54

Reverse of radiative recombination

Answer 54

Light absorption

Question 55

Photovoltaic effect

Answer 55

The creation of voltage and electric current in a material upon exposure to light Application = solar cells

Question 56

What is the process that occurs in the photovoltaic effect?

Answer 56

When sunlight or other sufficiently energetic light is incident upon the photodiode, electrons in the valence band absorb the energy and are excited to the conduction band, where they are free This creates an electron-hole pair The excited electrons diffuse and some will reach the p-n junction, where they are accelerated into a different material, generating an electromotive force that is converted into electric energy

Question 57

Photodiode

Answer 57

A semiconductor device that converts light into an electrical current This current is generated when photons are absorbed by the photodiode

Question 58

Why is the material used to produce a photodiode so critical to defining its properties?

Answer 58

Because only photons with sufficient energy to excite electrons across the material's band gap will produce significant photocurrents

Question 59

Most common solar-cell material

Answer 59

Silicon

Question 60

Crystal momentum

Answer 60

A momentum-like vector associated with electrons in a crystal lattice

Question 61

Effect of different surface orientations

Answer 61

Different surface orientations have different properties such as rate of oxidation and electronic quality of the oxide/semiconductor interface

Question 62

Brillouin zone

Answer 62

A unique primitive unit cell in reciprocal (k-)space that is directly related to the unit cell of the real crystal i.e. the Brillouin zone of a crystal is the set of all unique k-vectors It is the set of points in k-space that can be reached from the origin without crossing any Bragg plane Reciprocal lattices are broken up into Brillouin zones The boundaries of this unit cell are given by planes related to points on the reciprocal lattice

Question 63

Bragg plane

Answer 63

A plane in reciprocal space that bisects a reciprocal lattice vector at a right angle

Question 64

Critical points in a Brillouin zone

Answer 64

Points of high symmetry in the Brillouin zone are of special interest because they best capture the energy landscape

Question 65

Different unit cells...

Answer 65

...have different Brillouin zones

Question 66

How are band energies calculated?

Answer 66

In "k-space"/"momentum space"

Question 67

K-space

Answer 67

An abstract space intimately related to real space The k wavevector is a convenient way to calculate and present the energies of orbital interactions in solids (Wavevector rather than wavenumber because we are in 3D)

Question 68

What type of material is best for LEDs?

Answer 68

Direct band-gap materials e.g. GaN | Can provide photoluminescence

Question 69

N-type semiconductors

Answer 69

Created by doping an intrinsic semiconductor with an electron-donor impurity e.g. doping Si with P This increases the number of charge carriers available in the material for conduction

Question 70

Majority and minority charge carriers in n-type semiconductors

Answer 70

Electrons = majority charge carriers and holes = minority charge carriers Because there are more electrons than holes in n-type semiconductors

Question 71

Effect of N-type dopants on the Fermi level

Answer 71

N-type dopants raise the Fermi level At T = 0K, all electron states up to and including the donor state are occupied and the lowest vacant state is in the conduction band Using the same argument as previously, the Fermi level is therefore halfway between the donor state and the conduction band edge

Question 72

Where is the Fermi energy in semiconductors?

Answer 72

In the middle of the band gap | At T = 0K, the probability a valence state is occupied = 1 and the probability a conduction state is occupied = 0

Question 73

How does the presence of holes affect the conductivity of a material?

Answer 73

When an electric field is applied to a semiconductor, the conduction electrons move in one direction (towards the positive end of the crystal) and the holes move in the opposite direction (towards the negative end of the crystal) Because these 2 sets of particles have charges of opposite polarity, the total current is equal to the sum of the electron current and the hole current

Question 74

Number of holes in a pure sample of material

Answer 74

Equal to the number of electrons

Question 75

What happens when a valence electron is excited into the conduction band?

Answer 75

It leaves behind a hole

Question 76

Electron band structure of a conducting metal

Answer 76

The highest occupied orbital in the energy band diagram for a metal is only partially filled The electrons in this band can gain an arbitrarily small amount of energy by moving into a nearby unoccupied state Therefore, when an electric field is applied to the material, the electrons can increase their kinetic energy by being accelerated in the direction of the electric field

Question 77

Optical excitation of electrons

Answer 77

= photoconductivity As an alternative to thermal excitation The energy of the photon is transferred completely to an electron, giving the electron enough energy to enter the conduction band The photon has been absorbed by the material

Question 78

Main consideration for optical excitation of electrons

Answer 78

This process will only occur if the energy of the photon is at least as large as the band gap of the semiconductor

Question 79

Energy band diagram for N-type semiconductors

Answer 79

The donor electron does not take part in bonding so there is no room for it in the valence band It also cannot belong in the conduction band because it is still weakly attracted to the donor atom e.g. P Supplying a small amount of energy liberates the electron from the donor atom into the conduction band, therefore this electron must occupy an energy state just below the conduction band (i.e. it is in the band gap) The crystal is no longer perfect so this is now allowed

Question 80

Effect of temperature on the Fermi level in N-type semiconductors

Answer 80

At high temperatures, the number of intrinsic carriers (i.e. those already in the material) becomes comparable with the number of donor electrons The material behaves like an intrinsic semiconductor and the Fermi energy is close to the centre of the band gap i.e. the Fermi level moves down in energy as the temperature is increased

Question 81

Effect of dopant concentration on the Fermi energy in N-type semiconductors

Answer 81

When the dopant concentration is low, the Fermi energy is near the centre of the band gap at moderate temperatures Because under these conditions the number of intrinsic carriers is comparable to the number of donor electrons When the doping concentration is high, the number of intrinsic carriers is negligible even at very high temperatures, so the Fermi energy never moves far below the conduction band edge

Question 82

Energy band diagram for P-type semiconductors

Answer 82

Because an acceptor impurity has one fewer electron than the atom it replaces, there is an incomplete bond, resulting in a hole in the valence band that is 'bound' to/associated with its atom So that the hole can move through the valence band, a valence electron needs to be excited into the incomplete bond and the acceptor impurity ionised Therefore, the acceptor state is just above the valence band edge

Question 83

Effect of P-type dopants on the Fermi level

Answer 83

P-type dopants lower the Fermi level At T = 0K, the highest occupied state is the top of the valence band and the lowest vacant state is the acceptor state, so the Fermi energy is halfway between these

Question 84

Effect of temperature on the Fermi level in P-type semiconductors

Answer 84

At high temperatures, the intrinsic carriers 'push' the Fermi energy up towards the middle of the band gap At room temperature, the Fermi energy is much closer to the valence band than the conduction band

Question 85

Rate of recombination in intrinsic semiconductors

Answer 85

Relatively low because the concentrations of conductions electrons and holes are very low

Question 86

Rate of recombination in an n-type semiconductor

Answer 86

Higher than intrinsic semiconductors The higher concentration of conduction electrons means there are far more opportunities for conduction electron to recombine with a hole This means the concentration of holes is lower than for an intrinsic semiconductor

Question 87

The conductivity of an intrinsic semiconductor...

Answer 87

...varies exponentially with the magnitude of the band gap | This leads to a huge difference in conductivity between materials that only have a small difference in band gap