W.K. Chen Electrophysics, NCTU 1
Chapter 4
The semiconductor in
equilibrium
W.K. Chen
Electrophysics, NCTU
2
Equilibrium (Thermal equilibrium)
No external forces such as voltages, electric fields, magnetic fields, or
temperature gradients are acting on the semiconductor
All properties of the semiconductor will be independent of time at
equilibrium
Equilibrium is our starting point for developing the physics of the
semiconductor. We will then be able to determine the characteristics that
result when deviations from equilibrium occur
W.K. Chen
Electrophysics, NCTU
3
Outline
Charge carriers in semiconductor
Dopant atoms and energy levels
The extrinsic semiconductor
Statistics of donars and acceptors
Charge neutrality
Position Fermi energy level
W.K. Chen
Electrophysics, NCTU
4
4.1 Charge carriers in semiconductors
Current is the rate at which charge flow
Two types of carriers can contribute the current flow
Electrons in conduction band
Holes in valence band
The density of electrons and holes is related to the density of state function
and FermiDirac distribution function
dEEfEgdEEn
Fc
)()()( =
dEEfEgdEEp
F
)](1)[()( −=
υ
n(E)dE: density of electrons in CB at energy levels between E and E+dE
p(E)dE: density of holes in VB at energy levels between E and E+dE
W.K. Chen
Electrophysics, NCTU
5
Carriers concentration in intrinsic semiconductor
at equilibrium
dEEfEgdEEn
Fc
)()()( =
dEEfEgdEEp
F
)](1)[()( −=
υ
W.K. Chen
Electrophysics, NCTU
6
n
o
equation
∫∫
==
top
Ec
Fc
top
Ec
o
dEEfEgdEEnn )()()(
The thermal equilibrium concentration of electrons in CB
k
T
EE
Ef
f
F
)(
exp1
1
)(
−
+
=Q
For electrons in CB,
⇒
>>−⇒> kTEEEE
fC
)(
on)aproximati (Boltzmann
)(
exp
)(
exp1
1
)(
kT
EE
k
T
EE
Ef
f
f
F
−
−
≈
−
+
=Q
c
n
c
EE
h
m
Eg −=
3
2/3*
)2(4
)(
π
W.K. Chen
Electrophysics, NCTU
7
∫
−−
⋅−=
top
Ec
f
c
n
o
dE
kT
EE
EE
h
m
n
)(
exp
)2(4
3
2/3*
π
kT
EE
h
kTm
n
fC
n
o
)(
exp
2
2
2/3
2
*
−−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
π
We define effective density of states in CB,
The thermalequilibrium electron concentration in CB
2/3
2
*
2
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
h
kTm
N
n
C
π
kT
EE
Nn
fC
Co
)(
exp
−−
=
W.K. Chen
Electrophysics, NCTU
8
Example 4.1 electron concentration
31519
cm108.1)
0259.0
25.0
exp()108.2(
)(
exp
−
×=
−
×=
−−
≈
kT
EE
Nn
fc
co
300
K
at CBin ion concentratelectron theFin
d
below eV 0.25 is
cm108.2)300(
319
⇒
×=
−
cf
c
EE
KN
Solution
W.K. Chen
Electrophysics, NCTU
9
p
o
equation
∫∫
−==
top
Ec
F
top
Ec
o
dEEfEgdEEpp ))(1)(()(
υ
The thermal equilibrium concentration of holes in VB
k
T
EE
k
T
EE
Ef
ff
F
)(
exp1
1
)(
exp1
1
1)(1
−
+
=
−
+
−=−
Q
For holes in VB,
⇒
>>
−
⇒
<
kTEEEE
f
)(
υ
on)aproximati (Boltzmann
)(
exp
)(
exp1
1
)(1
kT
EE
k
T
EE
Ef
f
f
F
−
−
≈
−
+
=−
Q
EE
h
m
Eg
p
−=
υυ
π
3
2/3*
)2(4
)(
W.K. Chen
Electrophysics, NCTU
10
∫
−−
⋅−=
υ
υ
π
E
fp
o
dE
kT
EE
EE
h
m
p
0
3
2/3*
)(
exp
)2(4
kT
EE
h
kTm
p
fp
o
)(
exp
2
2
2/3
2
*
υ
π −−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
We define effective density of states in CB,
The thermalequilibrium electron concentration in CB
2/3
2
*
2
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
h
kTm
N
p
π
υ
kT
EE
Np
f
o
)(
exp
υ
υ
−−
=
W.K. Chen
Electrophysics, NCTU
11
2/3
2
*
2
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
h
kTm
N
n
C
π
N
c
& N
υ
values
N
c
& N
υ
are determined by the parameters of effective masses and
temperature. The effective mass is nearly a constant value ( a slight function
of temperature) for a semiconductor, which implies the N
c
& N
υ
will increase
in values with power of 3/2 with increasing temperature
Assume m
n
*=m
o
, then the value of the density of the effective density of
state at 300K is
which is lower the density of atoms in semiconductor
The effective mass of the electron in semiconductor is larger or smaller than
mo, but still of the same order of magnitude
, if cm105.2
*319
onC
mmN =×=
−
2/3
2
*
2
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
h
kTm
N
p
π
υ
W.K. Chen
Electrophysics, NCTU
12
Example 4.2 hole concentration
31519
cm1043.6)
03453.0
27.0
exp()1060.1(
)(
exp
−
×=
−
×=
−−
=
kT
EE
Np
f
o
υ
υ
319
2/3
19
2/3
cm1060.1
300
400
)1004.1()400(
300
400
)300(
)400(
−
×=
⎟
⎠
⎞
⎜
⎝
⎛
×=⇒
⎟
⎠
⎞
⎜
⎝
⎛
= KN
KN
KN
υ
υ
υ
Solution:
400Kat VBin ion concentrat hole theFin
d
above eV 0.27 is
cm1004.1)300(
319
⇒
×=
−
υ
υ
EE
KN
f
W.K. Chen
Electrophysics, NCTU
13
4.1.3 Intrinsic carrier concentration
For intrinsic semiconductor
The concentration of electrons in CB is equal to the concentration of holes in VB
W.K. Chen
Electrophysics, NCTU
14
kT
EE
Nnn
i
fC
Cio
)(
exp
−−
==
kT
EE
Npp
i
f
io
)(
exp
υ
υ
−−
==
torsemiconduc intrinsicfor
ii
pn =
kT
EE
NN
kT
EE
N
kT
EE
Nn
C
C
ffC
Ci
ii
)(
exp
)(
exp
)(
exp
2
υ
υ
υ
υ
−−
=
−−
⋅
−−
=
kT
E
NNn
g
Ci
−
= exp
2
υ
The intrinsic carrier concentration is a function of bandgap, independent
of Fermi level
W.K. Chen
Electrophysics, NCTU
15
Example 4.3 intrinsic carrier concentration
36
122
cm1026.2)300(
1009.5)
0259.0
42.1
exp()300()300()300(
−
×=
×=
−
=
Kn
KNKNKn
i
ci υ
Solution:
400K and300K at ion concentratcarrier intrinsic theFind
eV 42.1
cm100.7)300(
cm107.4)300( GaAs,For
318
317
⇒
=
×=
×=
−
−
g
c
E
KN
KN
υ
kT
E
NNn
g
Ci
−
= exp
2
υ
310
212
cm1085.3)400(
1048.1)
03885.0
42.1
exp()400()400()400(
−
×=
×=
−
=
Kn
KNKNKn
i
ci υ
W.K. Chen
Electrophysics, NCTU
16
kT
E
NNn
g
Ci
−
= exp
2
υ
The intrinsic carrier concentration is
nearly increase exponentially with
temperature
W.K. Chen
Electrophysics, NCTU
17
4.1.4 The intrinsic Fermilevel position
torsemiconduc intrinsicfor
ii
pn =
2/3
*
*
ln
2
1
)(
2
1
ln
2
1
)(
2
1
)(
exp
)(
exp
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
++=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
++=
−−
=
−−
p
p
cf
c
cf
ffC
C
m
m
kTEEE
N
N
kTEEE
kT
EE
N
kT
EE
N
i
i
ii
υ
υ
υ
υ
υ
2/3
*
*
ln
4
3
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+=
p
p
midgapf
m
m
kTEE
i
midgap
E
c
E
υ
E
fi
E
)(
2
1
υ
EEE
cmidgap
+=
W.K. Chen
Electrophysics, NCTU
18
Example 4.3 Intrinsic Fermi level
Solution:
level Fermi intrinsic theFin
d
56.0
08.1
*
*
⇒
=
=
op
on
mm
mm
2/3
*
*
ln
4
3
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+=
p
p
midgapf
m
m
kTEE
i
meV8.12eV0128.0
08.1
56.0
ln)0259.0(
4
3
2/3
−=−=
⎟
⎠
⎞
⎜
⎝
⎛
+=
midgapf
EE
i
W.K. Chen
Electrophysics, NCTU
19
4.2 Dopant atoms and energy levels
The intrinsic semiconductor may be an interesting material, but the real
power of semiconductor is extrinsic semiconductor, realized by adding small,
controlled amounts of specific dopant, or impurity atom.
ntype semiconductor
A group V element, such as P atom is added into Si. 5 valence electrons, 4
of them contribute to the covalent bonding, leaving the 5th electron loosely
bound to P atom, referred as a donor electron
W.K. Chen
Electrophysics, NCTU
20
ntype semiconductor
The energy level Ed is the energy state of donor impurity
If a small amount of energy, such as thermal energy, is added to the donor
electron, it can be elevated into CB, leaving behind a positively charged P
ion.
This type of impurity donates an electron to CB and so is called donor
impurity atoms, which add electrons to contribute the CB current, without
creating holes in VB
W.K. Chen
Electrophysics, NCTU
21
ptype semiconductor
For silicon, a group III element, such as B atom is added. 3 valence
electrons are all taken up in covalent bonding, one covalent bonding position
appears to be empty. (Fig. a)
The valence electrons in the VB (Fig. b) may gain a small amount of thermal
energy and move to the empty state of group III element, forming empty
state in VB
The energy level E
a
is the energy state of group III element in Si..
VB
Acceptor energy level
W.K. Chen
Electrophysics, NCTU
22
ptype semiconductor
The empty positions in the VB are thought of as holes.
The group III atom accepts an electron from the VB and so is referred to as
an acceptor
The holes in the VB, formed due to the adding of acceptor impurity atoms
without creating electrons in the CB, can move through the crystal
generating a current
hole
acceptor
B
W.K. Chen
Electrophysics, NCTU
23
4.2.2 Ionization energy
Ionization energy
The energy required to elevate the donor electron into the conduction band
nn
c
r
m
r
e
F
2*
2
2
4
υ
πε
==
The coulomb force of attraction between the electron and ion equal to the
centripetal force of the orbiting electron
+
Due to the quantization of angular momentum
)(
*
υ
mrn
n
⋅== hl
o
o
rn
a
m
m
n
em
n
r
⎟
⎠
⎞
⎜
⎝
⎛
==
*
2
2*
22
4
ε
πε h
a
o
: Bohr radius
o
o
o
em
a
A53.0
4
2
2
==
hπε
2
nr
n
∝
2
2
*
υ=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅ mr
n
n
h
*3
22
2
*
*
2
2
4 mr
n
mr
n
r
m
r
e
nnnn
⋅
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
=
hh
πε
W.K. Chen
Electrophysics, NCTU
24
Consider the lowest energy state of donor impurity in silicon material, in
which n=1
o
o
rn
a
m
m
n
em
n
r
⎟
⎠
⎞
⎜
⎝
⎛
==
*
2
2*
22
4
ε
πε h
o
o
o
o
r
aar
m
m
n
>>==⇒
===
A9.2345
26.0 ,7.11)Si( ,1
1
*
ε
.
Because of the difference in the dielectric constant and effective electron
mass, the electron orbit in a semiconductor is much larger than that in a free
atom and the ionization energy of a donor state is considerably smaller than
that of hydrogen atom
The radius of lowest energy state in Si is about 4 lattice constant of Si, so
the radius of the orbiting donor electron encompasses many silicon atoms
The energy required to elevate the donor electron into the conduction band
W.K. Chen
Electrophysics, NCTU
25
VTE +=
22
4*
2*
)4()(22
1
πε
υ
hn
em
mT ==
22
4*2
)4()(4 πεπε
h
n
em
r
e
V
n
−
=
−
=
22
4*
)4()(2
Energy Ionization
πεhn
em
VTE
−
=+=
eV m8.25)Si( :silicon ofenergy Ionization
eV 6.13)H( :hydrigen ofenergy Ionization
−=
−
=
E
E
W.K. Chen
Electrophysics, NCTU
26
22
4*
)4()(2
πεh
n
em
VTE
−
=+=
Ordinary impurity
For ordinary impurities, such as P, As, and Sb in Si and Ge, the hydrogen
model works quite well.
Amphoteric impurity
For IIIV semiconductor, the group IV such as Si and Ge is amphoteric
impurity. If a silicon atom replace a Ga atom, The Si impurity will act as a
donar, but if the Si atom replaces an As atom, then the Si impurity will act as
an acceptor
W.K. Chen
Electrophysics, NCTU
27
4.3 Extrinsic semiconductor
Intrinsic semiconductor
A semiconductor with no impurity atoms present in the crystal
Extrinsic semiconductor
A semiconductor in which controlled amounts of specific dopant or impurity
atoms have been added so that the thermalequilibrium electron and hole
concentrations are different the intrinsic carrier concentrations.
W.K. Chen
Electrophysics, NCTU
28
4.3.1 Equilibrium distribution of electrons & holes
Adding donor or acceptor will change the distribution of electrons and holes in
semiconductor
Since the Fermi level is related to the distribution function, the Fermi level will change
as dopant atoms are added
kT
EE
Nn
fC
Co
)(
exp
−−
=
kT
EE
Np
f
o
)(
exp
υ
υ
−−
=
W.K. Chen
Electrophysics, NCTU
29
ntype semiconductor
When the density of electrons in CB is greater than the density of holes in VB
ptype semiconductor
When the density of holes in VB is greater than the density of electrons in CB
kT
EE
Nn
fC
Co
)(
exp
−−
=
kT
EE
Np
f
o
)(
exp
υ
υ
−−
=
fifoo
fifoo
EEnp
EEpn
<⇒>
>⇒>
typep
typen
W.K. Chen
Electrophysics, NCTU
30
Example 4.5 thermal equilibrium concentrations
3419
31519
cm107.2
0259.0
87.0
exp)1004.1(
cm108.1
0259.0
25.0
exp)108.2(
×=
−
×=
×=
−
×=
o
o
p
n
kT
EE
Np
f
o
)(
exp
υ
υ
−−
=
ionsconcentrat hole &electron equlibrium thermal theFind
cm1004.1 cm108.2
band conduction below eV 0.25 isenergy Fermi
eV 1.12 is bandgap Si,For
319319 −−
×=×=
υ
NN
C
Solution:
kT
EE
Nn
fC
Co
)(
exp
−−
=
The electron and hole concentrations change by order of magnitude as the
Fermi energy changes by a few tenths of an electronvolt
W.K. Chen
Electrophysics, NCTU
31
Another form of equations for n
o
and p
o
⎥
⎦
⎤
⎢
⎣
⎡
−
⎥
⎦
⎤
⎢
⎣
⎡
−−
=
⎥
⎦
⎤
⎢
⎣
⎡
−+−−
=⇒
−−
==
−−
=
kT
EE
kT
EE
Nn
kT
EEEE
Nn
kT
EE
Nnn
kT
EE
Nn
fiFfiC
Co
fiFfiC
Co
fC
Cio
fC
Co
i
)(
exp
)(
exp
)()(
exp
)(
exp intrinsic
)(
exp typen
)(
exp
⎥
⎦
⎤
⎢
⎣
⎡
−
=
kT
EE
nn
fiF
io
) , type(n
fifio
EEnn >>
)(
exp
⎥
⎦
⎤
⎢
⎣
⎡
−−
=
kT
EE
np
fiF
io
) , type(p
fifio
EEnn >>
W.K. Chen
Electrophysics, NCTU
32
4.3.2 The n
o
p
o
product
kT
EE
Nn
fC
Co
)(
exp
−−
=
kT
EE
Np
f
o
)(
exp
υ
υ
−−
=
kT
EE
N
kT
EE
Npn
ffC
Coo
)(
exp
)(
exp
υ
υ
−−
⋅
−−
=
2
exp
i
g
Coo
n
kT
E
NNpn =
−
⋅=
υ
The product of n
o
and p
o
is always a constant for a given semiconductor material at a
given temperature, no matter it is intrinsic or extrinsic semiconductors,
The above equation is valid only when Boltzmann approximation is valid
The constant n
o
p
o
product is one of the fundamental principles of the semiconductor
in thermal equilibrium
W.K. Chen
Electrophysics, NCTU
33
4.3.3 The FermiDirac Integral
If the Boltzmann approximation does not hold, the thermal equilibrium carrier
concentrations is expressed as FermiDirac distribution function
on distributi DiracFermi elsewhere
holdsion approximatBoltzmann
3 typep
3 typen
⇒
⇒
⎪
⎭
⎪
⎬
⎫
>−−
>−−
kTEE
kTEE
f
fC
υ
c
E
υ
E
f
E
kT3
kT3
Boltzmann
approximation
FermiDirac
FermiDirac
W.K. Chen
Electrophysics, NCTU
34
∫
−
+
−
=
c
E
f
c
no
dE
kT
EE
EE
m
h
n
)exp(1
)(
)2(
4
2/1
2/3*
3
π
2/3
2
*
2
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
h
kTm
N
n
C
π
kT
EE
kT
EE
cF
F
c
−
=
−
= ηη and let
∫
∞
−+
=
0
2/1
2/3
2
*
)exp(1
)
2
(4 η
ηη
η
π d
h
kTm
n
F
n
o
∫
∞
−+
=
0
2/1
2/1
)exp(1
)( η
ηη
η
η dF
F
F
)(
2
2/1 Fco
FNn
η
π
=
)(
2
2/1 Fo
FNp
η
π
υ
=
2/3
2
*
2
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
h
kTm
N
p
π
υ
W.K. Chen
Electrophysics, NCTU
35
4.3.4 Degenerate and nondegenerate
semiconductors
Nondegenerate semiconductor
If the doping is low, the impurity atoms are spread far enough apart so that
there is no interaction between donor (acceptor) electrons (holes), the
impurities introduced discrete, noninteracting energy states
W.K. Chen
Electrophysics, NCTU
36
Degenerate semiconductor
If the doping is high, the impurity atoms are close enough so that
donor/acceptor electrons/holes will begin to interact with each other. When
this occur, the single discrete energy state will split into a band of energy.
When the concentration of donor (acceptor) electrons (holes) exceeds the
effective density of states (N
c
or N
υ
), the Fermi level lies with the band. This
type of semiconductors are called degenerate semiconductors.
f
cfc
EENp
EENn
>>
>>
υυ
,tor semiconduc degenerate typep
,tor semiconduc degenerate typen
o
o
W.K. Chen
Electrophysics, NCTU
37
4.4 Statistics of donors and acceptors
Suppose we have Ni electrons and gi quantum states at ith impurity energy
level
Each donor level at least has two possible spin orientations for donor
electron, thus each donor has at least two quantum states
The distribution function of donor electrons in the donor energy states is
slightly different than the FermiDirac function
bands allowedfor
)exp(1
1
)(
k
T
EE
Ef
f
F
−
+
=
statesdonor for
)exp(
2
1
1
1
)(
k
T
EE
N
n
Ef
f
d
d
F
−
+
==
n
d
: the density of electrons occupying the donor level
N
d
: the concentration of donor N
d
+
: the concentration of ionized donor
E
d
: the energy level of the donor
+
−=
ddd
NNn
W.K. Chen
Electrophysics, NCTU
38
statesacceptor for
)exp(
1
1
1
)(
kT
EE
g
N
p
Ef
af
a
a
F
−
+
==
+
−=
aaa
NNp
g is normally taken as four for acceptor level in Si and Ge because of the
detailed band structure
W.K. Chen
Electrophysics, NCTU
39
4.4.2 Complete ionization and freeout
⎥
⎦
⎤
⎢
⎣
⎡
−−
=
−
≈
−
+
==
>>−
kT
EE
N
kT
EE
N
kT
EE
N
EfNn
kTEE
fd
d
fd
d
f
d
Fdd
fd
)(
exp2
)exp(
2
1
)exp(
2
1
1
)(
If
For nondegenerate semiconductor,
kT
EE
Nn
fC
Co
)(
exp
−−
=
The total number of electrons in and near the conduction band is
do
nn +
The electron concentration in the conduction band is
typenfor )(
ddio
nNnn −+≈
W.K. Chen
Electrophysics, NCTU
40
The ratio of electrons in the donor states to the total number of electrons
in and near the conduction band is
kT
EE
N
kT
EE
N
kT
EE
N
nn
n
fC
C
fd
d
fd
d
od
d
)(
exp
)(
exp2
)(
exp2
−−
+
⎥
⎦
⎤
⎢
⎣
⎡
−−
⎥
⎦
⎤
⎢
⎣
⎡
−−
=
+
kT
EE
N
N
nn
n
dC
d
C
od
d
)(
exp
2
1
1
−−
+
=
+
c
E
υ
E
f
E
kT3
d
E
+
+
+
+
+
W.K. Chen
Electrophysics, NCTU
41
Example 4.7
%41.00041.0
0259.0
)045.0(
exp
)10(2
108.2
1
1
16
19
==
−×
+
=
+
od
d
nn
n
316
cm10
doping sphosphorou 300K,T tor,semiconduc Si
=
=
d
N
Solutions:
Nondegenerate semiconductor
For shallow doped semiconductor, only a few electrons are still in the donor
states, essentially all of the electrons from the donor states are in the
conduction band. In this case, the donor states are said to be completely
ionized
W.K. Chen
Electrophysics, NCTU
42
kT
EE
N
N
nn
n
dC
d
C
od
d
)(
exp
2
1
1
−−
+
=
+
kT
EE
N
N
pp
p
a
a
oa
a
)(
exp
4
1
1
υυ
−−
+
=
+
At T=300K
W.K. Chen
Electrophysics, NCTU
43
At T=0 K
All electrons are in their lowest possible energy states
df
fd
fd
d
d
F
EE
kT
EE
kT
EE
N
n
Ef >⇒−∞==
−
⇒=
−
+
== )exp(0)exp( 1
)exp(
2
1
1
1
)(
At T= 0 K, all the energy states below the Fermi level are full,and all the
states above the Fermi level are empty. Since the donor states is fully
occupied by donor electrons, the Fermi level must be well above the donor
energy state.
At T= 0K, no electrons from the donor state are thermally elevated into
conduction band ; this effect is called freeout.
W.K. Chen
Electrophysics, NCTU
44
Example 4.8 90% ionization temperature
kT
EE
N
N
pp
p
a
a
oa
a
)(
exp
4
1
1
υυ
−−
+
=
+
ionized are acceptors of 90%at which re temperatu theFind
cm10
(B)
b
oron with dpoedtor semiconduc Si t
y
pe
p
316
=
a
N
Solution:
K 193
)
300
(0259.0
045.0
exp
)10(4
)
300
)(1004.1(
1
1
1.0
16
2/319
=⇒
−
×
+
==
+
T
T
T
pp
p
a
oa
a
W.K. Chen
Electrophysics, NCTU
45
4.5 Charge neutrality
Charge neutrality
In thermal equilibrium, the semiconductor crystal is electrically neutral
The chargeneutrality condition is used to determine the thermal equilibrium
electron and hole concentration as a function of impurity doping
concentration
Compensated semiconductor
A semiconductor contains both donor and acceptor impurity atoms in the
same region
Ntype compensated semiconductor ( Nd>Na)
Ptype compensated semiconductor (Na>Nd)
Completely compensated semiconductor (Na=Nd)
W.K. Chen
Electrophysics, NCTU
46
4.5.2 Equilibrium electron and hole concentration
+−
+=+
doao
NpNn
)()(
ddoaao
nNppNn −+=−+
Charge neutrality:
Under complete ionization
0 ,0 ==
ad
pn
0)(
22
2
=−−−
+=+
+=+
ioado
d
o
i
ao
doao
nnNNn
N
n
n
Nn
NpNn
2
2
22
)(
typen
i
adad
o
n
NNNN
n +
⎟
⎠
⎞
⎜
⎝
⎛
−
+
−
=
W.K. Chen
Electrophysics, NCTU
47
o
i
o
i
dada
o
p
n
n
n
NNNN
p
2
2
2
22
)(
tor semiconduc typep
=
+
⎟
⎠
⎞
⎜
⎝
⎛
−
+
−
=
o
i
o
i
adad
o
n
n
p
n
NNNN
n
2
2
2
22
)(
tor semiconduc typen
=
+
⎟
⎠
⎞
⎜
⎝
⎛
−
+
−
=
W.K. Chen
Electrophysics, NCTU
48
Example 4.9 thermal equilibrium carrier conc.
34
16
210
2
316210
2
1616
cm1025.2
10
)105.1(
carrierminority
cm10)105.1(
2
10
2
10
carrie
r
ma
j
orit
y
−
−
×=
×
==
≈×+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+=
o
i
o
o
n
n
p
n
Solution:
The thermal equilibrium majority carrier concentration is essentially equal to
impurity concentration
310316
cm105.1 and 0 ,cm10
doping sphosphorou 300K,T tor,semiconduc Si typen
×===
=
iad
nNN
W.K. Chen
Electrophysics, NCTU
49
Redistribution of electrons when donors are
added
When add shallow donors
Most of donor electrons will gain
thermal energy and jump into the CB
A few of the donor electrons will fall
into the empty states in the VB and will
annihilate some of the intrinsic holes
W.K. Chen
Electrophysics, NCTU
50
Electron concentration vs. temperature
typenfor )(
ddio
nNnn −+≈
Freezeout (0 K)
Partial ionization (0 ∼100K)
Extrinsic region (∼100K400K)
Intrinsic region (>∼400 K)
kT
EE
N
N
nn
n
dC
d
C
od
d
)(
exp
2
1
1
−−
+
=
+
kT
E
NNn
g
Ci
−
⋅= exp
2
υ
0 ,
=
=
odd
nNn
The above regions are determined by
type of semiconductor and doping
concentration
W.K. Chen
Electrophysics, NCTU
51
Example 4.11 Compensated psemiconductor.
34
15
210
2
315210
2
15161516
cm1021.3
107
)105.1(
carrierminority
)(
cm107)105.1(
2
10310
2
)10310(
carrie
r
ma
j
orit
y
−
−
×=
×
×
==
−≈
×≈×+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
×−
+
×−
=
o
i
o
da
o
p
n
n
NN
p
Solution:
Because of complete ionization at 300K, the majority carrier hole concentration
is just the difference between the acceptor and donor concentrations
310315316
cm105.1 and cm105.1 ,cm10
300K,T tor,semiconduc Si typep
×=×==
=
ida
nNN
2
2
22
)(
i
dada
o
n
NNNN
p +
⎟
⎠
⎞
⎜
⎝
⎛
−
+
−
=
W.K. Chen
Electrophysics, NCTU
52
4.6 Position of Fermi energy level
kT
EE
Nn
fC
Co
)(
exp
−−
=
kT
EE
Np
f
o
)(
exp
υ
υ
−−
=
When Boltzmann approximation holds,
The position of Fermi energy level moves through the bandgap energy,
depending on the electron and hole concentrations and temperature
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=−
o
C
fC
n
N
kTEE ln typen
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=−
o
F
p
N
kTEE
υ
υ
ln typep
W.K. Chen
Electrophysics, NCTU
53
iaoda
da
iada
a
f
nNpNN
NN
N
kT
nNNN
N
N
kTEE
>>=−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
>>>>
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=−
, when ln
, when ln typep
υ
υ
υ
idoad
ad
C
idad
d
C
fC
nNnNN
NN
N
kT
nNNN
N
N
kTEE
>>=−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
>>>>
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=−
, when ln
, when ln typen
W.K. Chen
Electrophysics, NCTU
54
Example 4.13 Fermi energy
3161616
31619
cm1024.2101024.1
cm1024.1)
0259.0
20.0
exp(108.2
carrierma
j
orit
y
−
−
×=+×=
−=
×=
−
×=
d
ado
o
N
NNn
n
Q
Solution:
edge band conduction thebelow eV 0.20 isenergy Fermi
cm105.1 and cm10
300KTtor,semiconduc Si t
y
pen
310316
×==
=
ia
nN
kT
EE
Nn
fC
Co
)(
exp
−−
=
W.K. Chen
Electrophysics, NCTU
55
Variation of Fermi level with doping conc at 300K
For silicon, at 300K, N
d
, N
a
>>n
i
= (1.5x10
10
cm
3
)
The carrier concentrations are mainly determined by the impurity
concentration.
As the doping levels increase, the Fermi level moves closer to the
conduction band for ntype material and closer to valence band for the
ptype material
kT
EE
Nn
fC
Co
)(
exp
−−
=
kT
EE
Np
f
o
)(
exp
υ
υ
−−
=
W.K. Chen
Electrophysics, NCTU
56
E
f
versus temperature
The intrinsic carrier concentration ni is a strong function of temperature,
so the Ef is a function of temperature also
As high temperature, the semiconductor material begins to lose its
extrinsic characteristics and begins to behave more like an intrinsic
semiconductor
At the very low temperature, freeout occurs; the Fermi level goes
above Ed for the ntype material and below Ea for the ptype material
kT
E
NNn
g
Ci
−
⋅= exp
2
υ
W.K. Chen
Electrophysics, NCTU
57
υ
E
f
E
i
f
E
c
E
d
E
Freeout
υ
E
f
E
i
f
E
c
E
d
E
Extrinsic
υ
E
f
E
i
f
E
c
E
d
E
Intrinsic
W.K. Chen
Electrophysics, NCTU
58
4.6.3 Relevance of the Fermi energy
Thermal equilibrium occurs when the distribution of electrons, as a
function of energy, is the same in the two materials.
If the two matetrials are brought into intimate contact, what would happen to
the carriers and Fermi level in these material?
The electrons will
tend to seek the
lowest possible
energy
In this example, the
electrons in
material A will flow
into the lower
energy states of
material B
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