PHYS 260 Lecture Notes - Lecture 3: Number Density, Drift Current, Gallium Phosphide
25 May 2018
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Department
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Professor
SEMI CONDUCTORS
At the end of this chapter you are able to:
• List the properties of Fermi Function
• Compute carrier densities in Bands
• Derive Hall Effect
• Contrast Drift & Diffusion Currents.
The Fermi-Dirac probability distribution of energies is given by P
e
(E) =
KTEFE /)(
exp1
1
−
+
Properties of the Fermi Dirac statistics:-
1. It is derived on the condition that equilibrium exists and is therefore strictly valid only in
equilibrium.
2. Ef denotes an energy level and is called the Fermi level. From the above it also follows
that the Fermi level concept is valid only. Strictly speaking at equilibrium.
3. The Fermi-Dirac distribution applies to all particles obeying Pauli’s exclusion principle
and is equally applicable regardless of the type of solid (Insulator, Semiconductor,
metal) doping of semiconductor etc.
4. The Fermi-Dirac distribution considers statistically the entire collection of fermions in
the volume. Thus it considers all electrons in the semi conducting solid and not merely
electrons in a band.
5. Whenever an electron state is empty it can be characterized as a hole. The Fermi-Dirac
distribution function for holes in the solid would correspond to the statistical distribution
of vacant states. Denoting the hole distribution function as P
h
(E). We obtain P
h
(E)=1-
P
FD
(E).
KTEEF
h
EP
/)(
exp1
1
)(
−
+
=
6.
At an energy E = E
F
the probability of occupancy of the electron state by an electron or
hole is ½. This provides us with a definition of the Fermi level.
7.
At OK P
e
(E) = 1 for E <E
F
and P
e
(E) = 0 for E>E
F
This implies that at absolute zero
temperature all states up to the Fermi level are full i.e., occupied by electrons and all
states above the Fermi level are empty.
8.
The distribution function is a strong function of temperatures only at energies close to E
F
.
plots of P
e
(E) at different temperatures are as follows.
9.
The Fermi-Dirac distribution gives the probability of an electron state of energy “E”
being occupied and thus represents the fraction of available states being occupied. The
actual number of occupied states would be given by the number of states available times
the proportion of some occupied. The number of available states per unit volume are
given by g
e
(E), g
h
(E) for electrons and holes respectively. The number density n of
electrons and P of holes at energy E would be given by
)()()(
EPEgEn
ee
=
)()()(
EPEgEP
hh
=
10.
An extension of the principle can be used to obtain carrier densities n
o
and P
o
in a range
or band of energies from E
1
to E
2
in equilibrium.
∫
=2
1)()(
E
Eeeo
EdEpEgn
∫
=2
1)()(
0
E
Ehh
EdEpEgP
Electron and Hole densities:-
The most important application of Fermi-Dirac statistics in semiconductors is for computing
free electron and hole densities.
Consider an extrinsic n-type semiconductor. The impurity of donor level Ed lies very close to
the conduction band I the energy gap. Let N
D
be the number of impurity atoms per unit
volume. Each of the impurity atoms gives rise to a single electron state at Ed . If E
f
is the
Fermi energy level at a temperature T. The number of electrons at the donor level Ed is given
as nd = g
e
(Ed) P
e
(Ed) where g
e
(Ed) is the number of available states per unit volume P
e
(E)
is the Fermi function, Hence
KT
EFE
N
nd
D
−
+
=exp1 -------------------------------(1)
Where nd represents the un-ionized donor atoms out of the available total N
D
states per unit
volume. Thus the remaining vacant states are to be treated as nd
+
ionized donors per unit
volume is given by
KT
EEF
N
ndNnd
D
D
−
+
=−=
+
exp1 ------------------------(2)
Where each of the nd
+
levels being empty is associated with a +ive charge that corresponds to
the absence of an electron.
n
d
= electrons n
d
+
= Hole
P
P
P
P
P
Valence Band
Conduction Band
E
D
n
a
= Holes
Valence Band
Conduction Band
n
a
-
= electrons
A
A
A
A
A
Conduction Band
E
A
for semiconductors doped with N
A
acceptor atoms per unit volume that give rise to acceptor
level E
a
that is a little above the valancy energy E
V
. If some of the N
A
atoms get ionized by
accepting an electron then the number of acceptor atoms n
a
occupying E
a
level is given by
KT
EFE
N
an
A
−
+
=exp1 -----------------(3)
Then the number of unionized acceptor atoms n
a
will be =
KT
EE
N
F
A
−
+exp1 Hence the
acceptor levels also can be either filled or empty.
Carrier density in Energy bands:-
The two types of free carriers of interest in a semiconductor are the electrons in the
conduction band and holes in the valance band.
Let n be the number of electrons per unit volume of a homogeneously doped semiconductor
crystal in equilibrium. If the conduction band extends from Ec to Ect then the number of
electrons n per unit volume in the conduction band is
∫
=Ect
Ec ee EdEPEgn )()(
∫∫ ∞∞
−=−= Ect e
Ec ee IIEdEPEgeEdEPEgn 21
)()()()(
The Fermi level lies always below Ec and for reasonable widths of conduction band
Ect-EF >> Kt. In I
2
, all energies E being considered are greater that Ect and can be written as
Ect + ∆E where ∆E is positive.
E
F
E
C
E
CT
∞ ∞
Document Summary
At the end of this chapter you are able to: list the properties of fermi function, compute carrier densities in bands, derive hall effect, contrast drift & diffusion currents. The fermi-dirac probability distribution of energies is given by pe (e) = Properties of the fermi dirac statistics:: it is derived on the condition that equilibrium exists and is therefore strictly valid only in equilibrium, ef denotes an energy level and is called the fermi level. From the above it also follows that the fermi level concept is valid only. Thus it considers all electrons in the semi conducting solid and not merely electrons in a band: whenever an electron state is empty it can be characterized as a hole. The fermi-dirac distribution function for holes in the solid would correspond to the statistical distribution of vacant states. Denoting the hole distribution function as ph (e).
