Semiconductors Flashcards
Examples of semiconducting materials
Electronics Batteries Solar cells Self-cleaning glass LEDs
Traditional description of bonding in metals
Bonding in metals is described as a ‘sea of electrons’ (free electron model)
Electronic band structure of a solid
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
Band theory
Derives energy bands and band gaps by examining the quantum mechanical wave functions for an electron in a large lattice of atoms/molecules
What are bands composed of?
Closely spaced electron orbitals
Derivation of electronic band structure
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
Which electrons in atoms are involved in forming bands?
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
Band gap
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
Valence band
The highest range of electron energies in which electrons are normally present at T = absolute zero (0K)
Located below the Fermi level
Conduction band
The lowest range of vacant electron states
Located above the Fermi level
Fermi level (Ef)
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
Position of Fermi level in metals
Lies inside at least one band
Position of Fermi level in insulators and semiconductors
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
T = 0K
Probability of finding an electron below the Fermi level is 100%
Probability of finding an electron above the Fermi level is 0
T > 0K
Some electrons have E > Ef
The energy levels are filled following a Boltzmann distribution
What happens to an electron that has been promoted to the conduction band?
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
Drude model
Simple model for conduction
Treats the solid as a fixed array of ions with unbound electrons
Phonon
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)
Interactions between electrons and phonons
When electrons interact with phonons, they lose small amounts of energy but experience a large shift in momentum
Significance of electron-phonon interactions
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
Properties of the electrons in a band
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
Kinetic energy formula
KE = 1/2mv^2
p (momentum) = mv
Therefore KE = p^2/2m
(showing that energy and momentum are related)
In the absence of an electric field…
…there is no net movement of electrons
When an electric field is applied…
…the electrons experience electric force
The electrons gain momentum and are now flowing in the opposite direction to the electric field
Electric force equation
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)
Ways in which electrons can be excited
- By an electric field
- Thermally - absorption of heat by electrons is significant at low temperatures
- 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
What do electrons collide with in solids?
Valence electrons and ions
Phonons
Impurities
Crystal imperfections
How can we account for the very large range of electrical conductivity in solids?
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)
Fermi energy, Ef
= 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
Difference between Fermi level and Fermi energy
Fermi energy does not depend on temperature, whereas Fermi level does
Effect of temperature on conductivity of metals
Increasing temperature decreases metal conductivity due to increased thermal vibrations of the ions
Effect of temperature on conductivity of semi-conductors
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
Ebg < 3eV
Semi-conductor
Ebg > 3 eV
Insulator
Electron hole
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
Role of electron holes in conductivity
Electron holes can readily accept electrons
In semiconductors, current conduction by holes is as important as electron conduction in general
Doping a semi-conductor
Increases the number of conducting electrons by introducing suitable impurity atoms
Allows us to control the conductivity of the material
Doping Si with As
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
N-type semiconductors
Contain many mobile electrons and few holes
because electrons carry negative (N) charge
Doping Si with B
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
P-type semiconductors
Contain many holes and few mobile electrons
because holes carry positive (P) charge
Examples of donor impurities
As, P
Examples of acceptor impurities
B, Al, In
Amphoteric impurities
Can act as donors or acceptors
e.g. group IV Si/Ge in GaAs (group III and group V)
Which plane is Si cut along?
(1,0,0) plane
This is the best plane for uniformity and good device performance
Direct band gap
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
Indirect band gap
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
Radiative recombination
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)
Why can the valence band in semiconductors not carry an electric current?
The valence band is so nearly full of electrons that the electrons are not mobile, so cannot flow as electric current
Why is radiative recombination not possible in semiconductors with indirect band gaps?
Because photons cannot carry crystal momentum
Thus conservation of crystal momentum would be violated
How can radiative recombination occur in a material with an indirect band gap?
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
Why is radiative recombination far slower in materials with indirect band gaps, compared to those with direct band gaps?
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
Objects produced from direct band gap materials
Light-emitting and laser diodes
Recombination of electrons with electron holes within the device releases energy in the form of photons (heat/light energy)
Reverse of radiative recombination
Light absorption
Photovoltaic effect
The creation of voltage and electric current in a material upon exposure to light
Application = solar cells
What is the process that occurs in the photovoltaic effect?
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
Photodiode
A semiconductor device that converts light into an electrical current
This current is generated when photons are absorbed by the photodiode
Why is the material used to produce a photodiode so critical to defining its properties?
Because only photons with sufficient energy to excite electrons across the material’s band gap will produce significant photocurrents
Most common solar-cell material
Silicon
Crystal momentum
A momentum-like vector associated with electrons in a crystal lattice
Effect of different surface orientations
Different surface orientations have different properties such as rate of oxidation and electronic quality of the oxide/semiconductor interface
Brillouin zone
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
Bragg plane
A plane in reciprocal space that bisects a reciprocal lattice vector at a right angle
Critical points in a Brillouin zone
Points of high symmetry in the Brillouin zone are of special interest because they best capture the energy landscape
Different unit cells…
…have different Brillouin zones
How are band energies calculated?
In “k-space”/”momentum space”
K-space
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)
What type of material is best for LEDs?
Direct band-gap materials e.g. GaN
Can provide photoluminescence
N-type semiconductors
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
Majority and minority charge carriers in n-type semiconductors
Electrons = majority charge carriers and holes = minority charge carriers
Because there are more electrons than holes in n-type semiconductors
Effect of N-type dopants on the Fermi level
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
Where is the Fermi energy in semiconductors?
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
How does the presence of holes affect the conductivity of a material?
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
Number of holes in a pure sample of material
Equal to the number of electrons
What happens when a valence electron is excited into the conduction band?
It leaves behind a hole
Electron band structure of a conducting metal
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
Optical excitation of electrons
= 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
Main consideration for optical excitation of electrons
This process will only occur if the energy of the photon is at least as large as the band gap of the semiconductor
Energy band diagram for N-type semiconductors
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
Effect of temperature on the Fermi level in N-type semiconductors
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
Effect of dopant concentration on the Fermi energy in N-type semiconductors
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
Energy band diagram for P-type semiconductors
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
Effect of P-type dopants on the Fermi level
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
Effect of temperature on the Fermi level in P-type semiconductors
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
Rate of recombination in intrinsic semiconductors
Relatively low because the concentrations of conductions electrons and holes are very low
Rate of recombination in an n-type semiconductor
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
The conductivity of an intrinsic semiconductor…
…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
