1. Electron & Hole Statistics in Semiconductors A “Short Course”. BW, Ch
Electron & Hole Statistics in Semiconductors A “Short Course”. BW, Ch. 6 & S. Ch 3

2. Electronic Energy Levels in the Bands
The following assumes basic knowledge of elementary statistical physics.
We know that
Electronic Energy Levels in the Bands
(Solutions to the Schrödinger Equation in the periodic crystal)
are actually NOT continuous, but are really discrete. We have always treated them as continuous, because there are so many levels & they are so very closely spaced.

3. treat them as discrete for a while
Though we normally treat these levels as if they were continuous, in the next discussion, lets
treat them as discrete for a while
Assume that there are N energy levels (N >>>1):
ε1, ε2, ε3, … εN-1, εN
with degeneracies: g1, g2,…,gN

4. Fermions, Spin s = ½ Electrons have the following
Results from quantum statistical physics:
Electrons have the following
Fundamental Properties:
They are indistinguishable
For statistical purposes, they are
Fermions, Spin s = ½

5. Pauli Exclusion Principle:
Indistinguishable Fermions, with
Spin s = ½.
This means that they must obey the
Pauli Exclusion Principle:
That is, when doing statistics (counting) for the occupied states:
There can be, at most, one e- occupying a given quantum state (including spin)

6. Energy level Enk can have
Electrons obey the
Pauli Exclusion Principle:
So, when doing statistics for the occupied states:
There can be at most, one e- occupying a given quantum state (including spin)
Consider the band state (Bloch Function) labeled nk (energy Enk, & wavefunction nk):
Energy level Enk can have
2 e- , or 1 e- , or 1 e- , or 0 e- _

7. Fermi-Dirac Distribution
Statistical Mechanics Results for Electrons:
Consider a system of n e-, with N Single e- energy Levels (ε1, ε2, ε3, … εN-1, εN ) with degeneracies (g1, g2,…, gN) at absolute temperature T:
See any statistical physics book for the proof that the probability that energy level εj (with degeneracy gj) is occupied is:
(<nj/gj ) ≡ (exp[(εj - εF)/kBT] +1)-1
(<  ≡ ensemble average, kB ≡ Boltzmann’s constant)
Physical Interpretation: <nj ≡ average number of e- in energy level εj at temperature T
εF ≡ Fermi Energy (or Fermi Level, discussed next)

8. f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1 Physical Interpretation:
Define:
The Fermi-Dirac Distribution Function
(or Fermi distribution)
f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1
Physical Interpretation:
The occupation probability for level j is
(<nj/gj ) ≡ f(ε)

9. Physical Interpretation:
Look at the Fermi Function in more detail.
f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1
Physical Interpretation:
εF ≡ Fermi Energy ≡ Energy of the highest occupied level at T = 0.
Consider the limit T  0. It’s easily shown that:
f(ε)  1, ε < εF
f(ε)  0, ε > εF
and, for all T
f(ε) = ½, ε = εF

10. f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1
The Fermi Function:
f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1
Limit T  0: f(ε)  1, ε < εF
f(ε)  0, ε > εF
for all T: f(ε) = ½, ε = εF
What is the order of magnitude of εF? Any solid state physics text discusses a simple calculation of εF.
Typically, it is found, (in temperature units) that
εF  104 K.
Compare with room temperature (T  300K):
kBT  (1/40) eV  eV
So, obviously we always have εF >> kBT

11. Fermi-Dirac Distribution ≡ Maxwell-Boltzmann Distribution
NOTE! Levels within ~  kBT of εF (in the “tail”,
where it differs from a step function) are the ONLY ones which enter conduction (transport) processes!
Within that tail, f(ε) ≡ exp[-(ε - ε F)/kBT]
≡ Maxwell-Boltzmann Distribution

12. “Free Electrons” in Metals at 0 K
Properties of the Free Electron Gas:
The Fermi Energy EF & related properties
Fermi Energy EF  Energy of the highest
occupied state.
Related Properties
Fermi Velocity vF  Velocity of an electron with
energy EF
Fermi Temperature TF  Effective temperature of an electron with energy EF
Fermi Wavenumber kF  Wave number of an electron with energy EF
Fermi Wavelength λF  de Broglie wavelength of an electron with energy EF

13. ηe  Electron Density in the material
Free Electron Gas:
Fermi Energy EF  Energy of highest occupied state.
Fermi Velocity vF  Velocity of electron with energy EF
Fermi Temperature TF  Effective temperature of an
electron with energy EF
ηe  Electron Density in the material
Fermi Wavenumber kF  Wave number of an electron
with energy EF: EF = [ħ2(kF)2]/(2m)
 kF  (3π2ηe)⅓
Fermi Wavelength λF  Wavelength of an electron with
energy EF : λF  (2π/kF)  λF  [2π/(3π2ηe)⅓]

14. Sketch of a typical experiment
Sketch of a typical experiment. A sample of metal is “sandwiched” between two larger sized samples of an insulator or semiconductor.
Vacuum Level 
Metal
Band Edge  F
EF 
EF  Fermi Energy
F  Work Function
Energy

15. Using typical numbers in the formulas for several metals & calculating gives the table below:

16. Fermi-Dirac Distribution1 EF
Electron Energy
Occupation
Probability
Work Function F
Increasing T
T = 0 K
kBT

17. Number and Energy Densities
Number Density:
Energy Density:
Density of States De(E)  Number of electron
states available between energies E & E+dE.
For 3D spherical bands only, it’s easily shown that:

18. T Dependences of e- & e+ Concentrations
n  concentration (cm-3) of e-
p  concentration (cm-3) of e+
Using earlier results & making the Maxwell-Boltzmann approximation to the Fermi Function for energies near EF, it can be shown that
np = CT3 exp[- Eg /(kBT)]
(C = material dependent constant)

19. ni = C1/2T3/2exp[- Eg /(2kBT)]
For all temperatures, it is always true that
np = CT3 exp[- Eg /(kBT)]
(C = material dependent constant)
In a pure material: n = p  ni (np = ni2)
ni  “Intrinsic carrier concentration”. So,
ni = C1/2T3/2exp[- Eg /(2kBT)]
At T = 300K
Si : Eg= 1.2 eV, ni =~ 1.5 x 1010 cm-3
Ge : Eg = 0.67 eV, ni =~ 3.0 x 1013 cm-3

20. Intrinsic Concentration vs. T Measurements/Predictions
Note the different scales on the right & left figures!

21.  P is a DONOR (D) impurity
Doped Materials: Materials with Impurities! As already discussed, these are more interesting & useful!
Consider an idealized carbon (diamond) lattice
(we could do the following for any Group IV material).
C : (Group IV) valence = 4
Replace one C with a phosphorous P.
P : (Group V) valence = 5
4 e-  go to the 4 bonds
5th e- ~ is “almost free” to move in the lattice
(goes to the conduction band; is weakly bound).
P donates 1 e- to the material
 P is a DONOR (D) impurity

22. It becomes a conduction e-
Doped Materials
The 5th e- isn’t really free, but is loosely bound with energy ΔED << Eg
(Earlier, we outlined how to calculate ΔED!)
The 5th e- moves when an E field is applied!
It becomes a conduction e-
If there are enough of these, a current is created

23. But, it works both directions
Doped Materials
Let: D  any donor, DX  neutral donor
D+ ionized donor (e- to conduction band)
Consider the chemical “reaction”:
e- + D+  DX + ΔED
As T increases, this “reaction” goes to the left.
But, it works both directions

24.  The “Extrinsic” Conduction Region.
Consider very high T  All donors are ionized
 n = ND  concentration of donor atoms
(a constant, independent of T)
It is still true that
np = ni2 = CT3 exp[- Eg /(kBT)]
 p = (CT3/ND)exp[- Eg /(kBT)]
 “Minority Carrier Concentration”
All donors are ionized
 The minority carrier concentration is T dependent.
At still higher T, n >>> ND, n ~ ni
The range of T where n = ND
 The “Extrinsic” Conduction Region.

25. Almost no ionized donors & no intrinsic carriers
n vs. 1/T
Almost no ionized donors
& no intrinsic carriers
lllll
  High T Low T  

26. n vs. T
  Low T High T  

27.  B is an ACCEPTOR (A) impurity
Again, consider an idealized C (diamond) lattice.
(or any Group IV material).
C : (Group IV) valence = 4
Replace one C with a boron B.
B : (Group III) valence = 3
B needs one e- to bond to 4 neighbors.
B can capture e- from a C
 e+ moves to C
(a mobile hole is created)
B accepts 1 e- from the material
 B is an ACCEPTOR (A) impurity

28. e++A-  AX + ΔEA NA  Acceptor Concentration
The hole e+ is really not free. It is loosely bound by energy ΔEA << Eg
Δ EA = Energy released when B captures e-
 e+ moves when an E field is applied!
NA  Acceptor Concentration
Let A  any acceptor, AX  neutral acceptor
A-  ionized acceptor (e+ in the valence band)
Chemical “reaction”:
e++A-  AX + ΔEA
As T increases, this “reaction” goes to the left.
But, it works both directions
Just switch n & p in the previous discussion!

29. “n-Type Material”  ND > NA
Terminology
“Compensated Material”
 ND = NA
“n-Type Material”  ND > NA
(n dominates p: n > p)
“p-Type Material”  NA > ND
(p dominates n: p > n)