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DEE4521
Semiconductor Device Physics
Lecture 3A:
Density-of-States (DOS), Fermi-Dirac Statistics, and Fermi Level
Prof. Ming-Jer Chen
Department of Electronics Engineering
National Chiao-Tung University
October 1, 2013
1
What are States?
• Pauli exclusion principle:
No two electrons in a system can have the same set
of quantum numbers.
• Here, Quantum Numbers represent States.
2
DOS
• We have defined the effective masses (ml*
and mt*) in a valley minimum in Brillouin zone.
• We now want to define another type of
effective mass in the whole Brillouin zone to
account for all valley minima: DOS Effective
Mass m*ds
• Here DOS denotes Density of States.
• States (defined by Pauli exclusion principle) can be
thought of as available seats for electrons in
conduction band as well as for holes in valence band.
3
DOS
Ways to derive DOS and hence its DOS effective mass:
•Solve Schrodinger equation in x-y-z space to find
corresponding k solutions
•Again apply the Pauli exclusion principle to these k
solutions – spin up and spin down
•Mathematically Transform an ellipsoidal energy
surface to a sphere energy surface, particularly for Si
and Ge
Regarding this point, textbooks would be helpful.
4
3-D Carriers
S(E): DOS function, the number of states per unit energy
per unit volume.
mdse*: electron DOS effective mass, which carries the information
about the DOS in conduction band
mdsh*: hole DOS effective mass, which carries the information
about the DOS in valence band
*
dse 3 / 2
2
1 2m
S(E)  2 (
)
2 
*
dsh 3 / 2
2
1 2m
S(E)  2 (
)
2 
E  EC
EV  E
5
3-D Case
1. Conduction Band
• GaAs:
mdse* = me*
•
Silicon and Germanium:
mdse* = g2/3(ml*mt*2)1/3
where the degeneracy factor g is the number of ellipsoidal
constant-energy surfaces lying within the Brillouin zone.
For Si, g = 6;
For Ge, g = 8/2 = 4.
2. Valence Band – Ge, Si, GaAs
mdsh* = ((mhh*)3/2 + (mlh*)3/2)2/3
(Here for simplicity, we do not consider the Split-off band)
6
Fermi-Dirac Statistics
Fermi-Dirac distribution function gives the probability of
occupancy of an energy state E if the state exists.
f (E) 
1
1 e
( E  E f )/ kBT
1 - f(E): the probability of unfilled state E
Ef: Fermi Level
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Fermi level is related to one of laws of Nature:
Conservation of Charge
Extrinsic case
8
2-13
Case of EV < Ef < EC (Non-degenerate)
C = (Ef – EC)/kBT
Electron concentration

n   S ( E ) f ( E )dE
EC
n  NC exp(C)
EV
p   S ( E )(1  f ( E ))dE

Hole concentration
Effective density of states
in the conduction band
p  NV exp(V)
V= (EV – Ef)/kBT
NC = 2(mdse*kBT/2ħ2)3/2
NV = 2(mdsh*kBT/2ħ2)3/2
Effective density of states
in the valence band
Note: for EV < Ef < EC, Fermi-Dirac distribution reduces
to Boltzmann distribution.
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Case of EV < Ef < EC (Non-degenerate)
n  NC exp(C)
C = (Ef – EC)/kBT
p  NV exp(V)
V= (EV – Ef)/kBT
NC = 2(mdse*kBT/2ħ2)3/2
NV = 2(mdsh*kBT/2ħ2)3/2
For intrinsic case where n = p, at least four statements can be drawn:
•Ef is the intrinsic Fermi level Efi
•Efi is a function of the temperature T and the ratio of mdse* to mdsh*
•Corresponding ni (= n = p) is the intrinsic concentration
•ni is a function of the band gap (= Ec- Ev)
10
11
12
(Continued from Lecture 2)
Conduction-Band Electrons and
Valence-Band Holes and Electrons
Hole: Vacancy of Valence-Band Electron
13
No Electrons in Conduction Bands
All Valence Bands are filled up.
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