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. The electronic energy levels in the bands,
The following discussion assumes a basic knowledge of elementary statistical physics.
We know that
The electronic energy levels in the bands,
which are 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. Fundamental Properties:
Even though we normally treat these levels as if they were continuous, for 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
Results from quantum statistical physics:
Electrons have the following
Fundamental Properties:
They are indistinguishable particles
For statistical purposes, they are
Fermions, Spin s = ½

4. There can be at most, one e- occupying
So, electrons are indistinguishable Fermions, with Spin s = ½
This means that they 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)
Consider the band state labeled nk (energy Enk, & wavefunction nk). It can hold, at most, 2 e- :
1 e- with spin “up” (  ) e- with spin “down” (  ).
So, energy level Enk can have up to 2 e-:
 or 1 e- :  or 1 e- :  , or 0 e- : __

5. Fermi-Dirac Distribution Physical Interpretation
Statistical Mechanics Results for Electrons:
Consider a system of n e-, with
N Single e- Levels
(ε1, ε2, ε3, … εN-1, εN ) with degeneracies (g1, g2,…, gN)
at absolute temperature T:
In every statistical physics book, it is proven 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)

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

7. f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1
Now, lets look at the Fermi Function in more detail.
f(ε) ≡ (exp[(ε - εF)/kBT] +1)-1
Physical Interpretation of εF ≡ Fermi Energy:
εF ≡ The 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

8. 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

9. Fermi-Dirac 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

10. “Free Electrons” in Metals at 0 K
Simple calculations of the Fermi Energy EF & related properties for a Free Electron Gas.
Fermi Energy EF  Energy of the highest occupied state
Fermi Velocity vF  Velocity of an electron with energy EF
Fermi Temperature TF  Velocity of electron with energy EF
Fermi Number kF  Wave number of an electron with energy EF
Fermi Wavelength λF  Wavelength of an electron with energy EF
ηe  Electron Density in the material

11. 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

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

13. Effect of Temperature Fermi-Dirac Distribution 1 E Electron Energy,1 E F
Electron Energy,
Occupation Probability, f
Work Function,
Increasing T
= 0 K
k T B
Vacuum
Level

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

15. 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)
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

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

17. Consider an idealized carbon (diamond) lattice
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 : (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

18. But, it works both directions We’ve shown earlier how
Doped Materials
The 5th e- is really not free, but is loosely bound with energy
ΔED << Eg
The 5th e- moves when an E field is applied!
It becomes a conduction e-
Let: D  any donor, DX  neutral donor
D+ ionized donor (e- to the conduction band)
Consider the chemical “reaction”:
e- + D+  DX + ΔED
As T increases, this “reaction” goes to the left.
But, it works both directions
We’ve shown earlier how
to calculate this!

19.  the “Extrinsic” Conduction region.
Consider very high T  All donors are ionized
 n = ND  concentration of donor atoms
(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.

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

21. n vs. T

22.  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 : (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

23.  e+ moves when an E field is applied! 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!

24. “Compensated Material”
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)