Transcript F - TTU Physics

Electron & Hole Statistics
in Semiconductors
A “Short Course”. BW, Ch. 6 & S. Ch 3
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.
• 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
• Results from quantum
statistical physics:
Electrons have the following
Fundamental Properties:
• They are indistinguishable
• For statistical purposes, they are
Fermions, Spin s = ½
• 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
occupying a given quantum
state (including spin)
e
• Electrons obey the
Pauli Exclusion Principle:
• So, when doing statistics for the occupied states:
There can be at most, one eoccupying 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_
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)
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)
• Look at the Fermi Function in more detail.
f(ε) ≡ (exp[(ε - εF)/kBT]
Physical Interpretation:
-1
+1)
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
The Fermi Function:
f(ε) ≡ (exp[(ε - εF)/kBT]
• Limit T  0:
for all T:
-1
+1)
f(ε)  1, ε < εF
f(ε)  0, ε > εF
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
• 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
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
2 2
2
2

 kF 
EF 

3 2 e
2m
2m
1

2
vF  3  e 3
m


3
EF
TF 
kB
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]
• Sketch of a typical experiment. A sample of
metal is “sandwiched” between two larger sized
samples of an insulator or semiconductor.
Vacuum Level 
Band Edge 
EF 
Metal
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

EF
TF 
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
Fermi-Dirac Distribution
Occupation
Probability
1
f E  
 E  EF 
1  exp 

 k BT 
kBT
1
T=0K
Increasing T
0
Electron Energy
EF Work Function F
Number and Energy Densities
N 
Number Density: e  V   f E De  E dE;
0

E
Energy Density: e  e   Ef E De E dE
V 0
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:
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
Maxwell-Boltzmann approximation to
the Fermi Function for energies near
EF, it can be shown that
np =
3
CT exp[-
Eg /(kBT)]
(C = material dependent constant)
• 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
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.
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
th
• The 5 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 eIf there are enough of these, a
current is created
Doped Materials
• Let: D  any donor, DX  neutral donor
• D+ ionized donor (e- to conduction band)
• Consider the chemical “reaction”:
e
+
+
D

X
D +
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
(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.
n vs. 1/T
Almost no ionized donors
& no intrinsic carriers
lllll
  High T
Low T  
n vs. T
  Low T
High 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.
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 = N A
“n-Type Material”  ND > NA
(n dominates p: n > p)
“p-Type Material”  NA > ND
(p dominates n: p > n)