1. Introduction to Solid-state Physics Lecture 2
10/15/2012

2. Today’s Outline: Intrinsic and Extrinsic Semiconductors
Generation and Recombination
Carrier Injection and Minority Carrier Lifetimes

3. Summary of intrinsic semiconductor
Electrons
Holes
g(E)

4. Intrinsic carrier concentration and intrinsic Fermi level
ni is a constant at certain T
for Si at T = 300k is 1.45x1010cm-3
Intrinsic semiconductor : a pure, single crystal semiconductor in which all electrons in the conduction band are thermally excited from the valence band.
Note that np = NcNvexp(-Eg/kT) and np = ni is called the Law of Mass Action.
Since the thermal energy, kT, is << Eg
Example: for Si, kT at room temp is 0.026eV, i.e. 1/40 of the Si band gap.
Note that Ei is not exactly equal to (Ec + Ev)/2. Recall, Ei of Si at 300K is eV below the mid gap.
 The intrinsic Fermi level is very close to the mid point between the conduction band and the valence band. EV Ec Ei
(Ec+EV)/2

5. Extrinsic or doped semiconductors
n-type semiconductor
n-type with donor
Free electron (-) Ec Eo Ev Ei Ef Ed Ec Eo Ev Ei Ef Ea
Elemental impurities called dopants are added to form doped or extrinsic semiconductors. Impurity atoms must be in a substitutional site in Si in order to be electrical active.
Boron (acceptor atom) for p-type :
Accepting an electron from valence band that the Si contains excess holes. -vely charged when ionized.
Acceptors add allowed states just above the valence band edge (i.e. filled) .
Phosphorous and arsenic (donor atom) for n-type :
Donating an electron to conduction band i.e. excess electrons. +vely charged when ionized.
Donors add allowed electron states in the band gap close to the conduction band edge (i.e. empty).
Boron, P and As have ionized energy of 2kT.
Doping concentrations are generally quite low, e.g cm-3, compared to the number of host atoms, e.g. 5x1022 cm-3.
p-type semiconductor
p-type with acceptor
Free hole (+)

6. Extrinsic or doped semiconductors (cont.)
Non-degenerate semiconductor (i.e. Boltzmann approximations)
Degenerate semiconductor
Non-degenerate semiconductor :
If the Fermi level lies in the band gap more than a few kT (3kT) from either band edge.
The carrier concentration obeys Maxwell-Boltzmann statistics. The approximated expressions can be used.
Degenerate semiconductor (dopant conc. > 1x1018cm-3) :
If the Fermi level lies in the band gap less than a few kT (3kT) from either band edge. (i.e. the Fermi level moves into the conduction or valence band at very high doping concentration (about 1020 cm-3).)
The carrier concentration is to be calculated using Fermi-Dirac distribution function. Need to use the exact function.

7. Carrier concentration in doped semiconductors
Charge neutrality
Assuming complete ionization of dopants at elevated temperatures
Using
We get
Nd : conc. of donor atom (cm-3)
Na : conc. of acceptor atom (cm-3)
np = ni2 in equilibrium is constant, independent of the added impurities and Fermi level position.

8. Fermi level in doped semiconductors
At room temperature, the carrier conc. in degenerate Si
p-type semiconductor
n-type semiconductor
n-type : Nd -Na >> ni (or Nd >> ni since Nd >> Na) electrons are majority carriers and holes are minority carrier.
p-type : Na - Nd >> ni (or Na >> ni since Na >> Nd) holes are majority carriers and electrons are minority carrier.
The Fermi level in an extrinsic Si is not located at the midgap. The exact position of the Fermi level depends on both the ionization energy and the concentration of dopants.

9. Equilibrium carrier concentration (cont.)
Electron and hole concentration is large when (Ec - Ef) and (Ef - Ev) are small, respectively.
Effect of shifting Ef closer to the Ec

10. Fermi level in doped semiconductors (cont.)
(n-type)
(p-type)
Distance between Ef and Ei is a log fct of dopant conc. RT
degenerate
n-type
The distance between the Fermi level and the intrinsic Fermi level near the midgap is a logarithmic function of doping concentration.
Non-degenerate
p-type
degenerate

11. Equilibrium carrier distribution - intrinsic

12. Equilibrium carrier distribution - extrinsic

13. Generation
At T > O K, owing to thermal excitation, some electrons manage to ‘jump’ from the valence band (VB) to the conduction band (CB) and this leads to electron-hole pairs (EHPs) being generated.
In a generation process, an electron in CB and a hole (i.e., an empty state) in VB are created simultaneously, hence the term EHP.
Generation of EHPs can also be a result of irradiation (i.e., light) and in this case, it is known as photogeneration. This can only happen provided the photon energy is higher than the bandgap, Eg , of the semiconductor.
Generation of EHP is equivalent to the breaking of a covalent bond in the bond model : an electron in CB is a free electron and a hole in VB is a broken bond.
Direct generation
Generation of e-h pairs by absorbing energy > Eg
Thermal generation vs optical generation

14. Recombination
Recombination  a free electron in CB wandering around in a crystal, ‘meets’ a hole in VB and fills it since the latter is at a lower-energy state.
Direct band-to band recombination, or simply direct recombination.
In the energy band model, recombination of EHP is represented by returning an electron in CB into hole in VB, and this is accompanied by the release of energy, usually in the form of photon (light energy).
In an annihilation process (more commonly known as recombination), a free electron in CB wandering around in a crystal, ‘meets’ a hole in VB and fills it since the latter is at a lower-energy state. In the energy band model, recombination of EHP is represented by returning an electron in CB into hole in VB, and this is accompanied by the release of energy, usually in the form of photon (light energy).
The recombination process, just described is known as the direct band-to band recombination, or simply direct recombination.
Direct recombination
Recombination via band gap (band edge to band edge)
Energy is released : light (photons) or heat (phonons)

15. Recombination (cont.) Indirect recombination
Takes place with the help of a third entity – a recombination-generation (R-G) centre.
The R-G centres arise from ‘defects’ or impurity atoms present in a semiconductor crystal. In an indirect recombination, a free electron is captured by a R-G centre, become localised at this site, held there until a hole arrives to complete the process.
There is another type of recombination process that happens with the help of a third entity – a recombination-generation (R-G) centre. In this case, it is known as the indirect recombination.
The R-G centres have energy levels (ET) near the middle of the bandgap. This allow electrons (or holes) to make the transition between CB and VB in a number of small ‘hops’ instead of one big ‘jump’. Heat (or phonon) is absorbed or released in each ‘hop’.
Semiconductors, e.g., GaAs, where recombination of EHPs is more often via the direct band-to band process, are known as the direct bandgap materials. On the other hand, semiconductors such as Si or Ge are known as the indirect bandgap materials as the indirect recombination process usually dominates.
Indirect generation and recombination
via generation-recombination centers (R-G such as impurity atoms i.e. unintentional impurities or defects e.g. dislocation) having energy level Et. The conc. is much smaller than than that of donor or acceptor. Gold is a good recombination centers in Si.
Generation and recombination rate are faster.

16. Generation and Recombination
At T > O K, generation and recombination co-exist.
At equilibrium, since electron and hole concentrations are constant, the generation rate (G) and recombination rate ( R ) of EHPs must be equal
Thermal equilibrium
Non-equilibrium
R = G = dt dp dn
If R < G Þ
net generation
>
If R > G
net recombination
<
At equilibrium, since electron and hole concentrations are constant, the generation rate (G) and recombination rate ( R ) of EHPs must be equal, i.e.
G = R, and .   
Under non-equilibrium conditions,
 If G > R, which means ,
electron and hole concentrations will increase with time. We have a net generation and this usually happens when energy is provided to a semiconductor.
If G < R which means ,
electron and hole concentrations will decrease with time. We a net recombination and energy is released in this case.

17. Carrier injection
Carrier injection - a process of introducing excess carriers by optical, electrical mean etc.
excess electron concentration: n  n - no
excess hole concentration: p  p - po
Carrier injection refers to any increments of carriers due to nonthermal source, irrespective of the nature of the source.
It can be caused by illumination of light, forward biasing of a pn junction etc.
To achieve steady-state, there must be a net recombination of electrons and holes.
If energy is now provided e.g., by illuminating light with the appropriate photon energy (> Eg) uniformly onto the semiconductor, EHPs will be generated and the carrier concentrations will be increased to n > no and p > po. We now have a situation of carrier injection and can define the departure from the equilibrium by excess concentrations. In the above expressions, n and p denote respectively the electron and hole concentrations under arbitrary conditions. Take note that n = p and n p > ni2 (with carrier injection).
Carrier extraction
Can be caused by reversed biasing a pn junction.
To achieve steady-state, there must be a net generation of electrons and hole.

18. Carrier injection (cont.)
Two levels of carrier injection can be distinguished: (a) low level injection and (b) high level injection.
(a)  Low level injection
Excess carrier concentration is small (usually taken as less than 10%) compared with the equilibrium majority carrier concentration:
n-type semiconductor: nn & pn << nno
p-type semiconductor: np & pn << ppo
where nno is the equilibrium majority carrier (electron) concentration of the n-type semiconductor and ppo is the equilibrium majority carrier (hole) concentration of the p-type semiconductor. Take note that a second subscript ‘n’ or ‘p’ is introduced to signify the type of semiconductor.
(b)  High level injection
Excess carrier concentration is at least comparable to the equilibrium majority carrier concentration:
n-type semiconductor: nn & pn  nno  pn & nn  nno
p-type semiconductor:np & pp  ppo  np & pp  ppo

19. Carrier injection (cont.)
In semiconductor device operation, it is generally low level injection that is important.
nn & pn = excess carriers in n-type
np & pp = excess carriers in p-type
nno & ppo = thermal equilibrium value
Note that
1st subscript : type of semiconductor and
2nd subscript : thermal equilibrium value.
Low level injection in an n-type semiconductor does not change nn significantly but it affects pn drastically.

20. Minority carrier lifetimes
Under low level injection conditions, the net generation-recombination process is controlled by the availability of excess minority carriers. Majority carriers are not the limiting factor as they are present in large quantity.
The net generation-recombination rate is proportional to the excess minority carrier concentration.
p-type semiconductor:
n-type semiconductor:
Whenever there is a departure from equilibrium, e.g., a carrier injection condition, restoring force will also act so as to revert back to equilibrium. This means a net recombination will be present if there is carrier injection. On the other hand, if carriers are reduced below their equilibrium values (this corresponds to a carrier extraction condition), there is a net generation.
p and n are constants of proportionality. p is the minority carrier (hole) lifetime in an n-type semiconductor and n is the minority carrier (electron) lifetime in a p-type semiconductor. Minority carrier lifetime represents the average time an excess minority carrier will exist in the presence of majority carriers from its generation to recombination.
The minus sign in the above expressions indicates that when there are excess carriers, there is a net recombination and when there is a deficiency of carriers (i.e. below the thermal equilibrium values), there is a net generation.