Electron & Hole Statistics in Semiconductors
A “Short Course”. BW, Ch. 6 & S. Ch 3
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.
• 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
• 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” (  ) + 1 e- with spin “down” (  ).
So, energy level Enk can have up to 2 e-:
 or 1 e- :  or 1 e- :  , or 0 e- : __
Fermi-Dirac Distribution
• 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)
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)
• 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
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  0.025 eV
So, obviously we always have εF >> kBT
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
“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

 2 k F2  2
2
EF 

3  e
2m
2m
1

2
vF  3  e 3
m


3
2
EF
TF 
kB
• 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 
EF 
F
EF  Fermi Energy
F  Work Function
Energy

 2 k F2  2
EF 

3 2 e
2m
2m
1

2
vF  3  e 3
m

3
2

TF 
EF
kB
• Using typical numbers in the formulas for several
metals & calculating gives the table below:
Element Electron
Density, e
[1028 m-3]
Cu
8.47
Au
5.90
Fe
17.0
Al
18.1
Fermi
Energy
EF [eV]
7.00
5.53
11.1
11.7
Fermi
Temperature
TF [104 K]
8.16
6.42
13.0
13.6
Fermi
Wavelength
F [Å]
4.65
5.22
2.67
3.59
Fermi
Velocity
vF [106 m/s]
1.57
1.40
1.98
2.03
Work
Function
 [eV]
4.44
4.3
4.31
4.25
Effect of Temperature
f
Occupation Probability,
Fermi-Dirac
Distribution
1
f E  
 E  EF 
1  exp 

 k BT 
kBT
1
T=0K
Vacuum
Level
IncreasingT
0
Electron Energy,E
EF
Work Function,F
Number and Energy Densities
N 
Number density: e    f  E De  E dE;
V 0

E
Energy density: e  e   Ef  E De  E dE
V 0
Density of States  Number of electron states
available between energies E & E+dE.
For 3D spherical bands only, it’s easily shown that:
De  E  
m
 
2 2
2mE
2

T Dependences of e- & e+ Concentrations
n  concentration (cm-3) of e-, p  concentration (cm-3) of e+
• Using earlier results & making the MaxwellBoltzmann 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
Intrinsic Concentration vs. T
Measurements/Predictions
Note the different scales on the right & left figures!
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
Doped Materials
• The 5th e- is really not free, but is loosely bound with energy
We’ve shown earlier how
to calculate this!
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
• 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.
n vs. 1/T
Almost no ionized
donors & no
intrinsic carriers
lllll
  High T
Low T  
n vs. T
• 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
• 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!
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)