Chapter 4 The semiconductor in equilibriumocw.nctu.edu.tw/upload/classbfs1209043550157212.pdf ·...
Transcript of Chapter 4 The semiconductor in equilibriumocw.nctu.edu.tw/upload/classbfs1209043550157212.pdf ·...
W.K. Chen Electrophysics, NCTU 1
Chapter 4The 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 flowElectrons in conduction band
Holes in valence band
The density of electrons and holes is related to the density of state function and Fermi-Dirac 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
no equation
∫∫ ==top
Ec Fc
top
Eco dEEfEgdEEnn )()()(
The thermal equilibrium concentration of electrons in CB
kT
EEEf
fF )(
exp1
1)(
−+
=Q
For electrons in CB,
⇒
>>−⇒> kTEEEE fC )(
on)aproximati (Boltzmann )(
exp)(
exp1
1)(
kT
EE
kT
EEEf f
fF
−−≈
−+
=Q
cn
c EEh
mEg −=
3
2/3* )2(4)(
π
W.K. Chen Electrophysics, NCTU 7
∫−−
⋅−=top
Ec
fc
no dE
kT
EEEE
h
mn
)(exp
)2(43
2/3*π
kT
EE
h
kTmn fCn
o
)(exp
22
2/3
2
* −−⎟⎟⎠
⎞⎜⎜⎝
⎛=
π
We define effective density of states in CB,
The thermal-equilibrium electron concentration in CB
2/3
2
*22 ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
h
kTmN n
C
π
kT
EENn fC
Co
)(exp
−−=
W.K. Chen Electrophysics, NCTU 8
Example 4.1 electron concentration
31519 cm108.1)0259.0
25.0exp()108.2(
)(exp
−×=−
×=
−−≈
kT
EENn fc
co
300Kat CBin ion concentratelectron theFind
below eV 0.25 is
cm108.2)300( 319
⇒
×= −
cf
c
EE
KN
Solution
W.K. Chen Electrophysics, NCTU 9
po equation
∫∫ −==top
Ec F
top
Eco dEEfEgdEEpp ))(1)(()( υ
The thermal equilibrium concentration of holes in VB
kT
EE
kT
EEEf
ffF )(
exp1
1)(
exp1
11)(1
−+
=−
+−=−Q
For holes in VB,
⇒
>>−⇒< kTEEEE f )( υ
on)aproximati (Boltzmann )(
exp)(
exp1
1)(1
kT
EE
kT
EEEf f
fF
−−≈
−+
=−Q
EEh
mEg p −= υυ
π3
2/3* )2(4)(
W.K. Chen Electrophysics, NCTU 10
∫−−
⋅−= υ
υ
πE fpo dE
kT
EEEE
h
mp
0 3
2/3* )(exp
)2(4
kT
EE
h
kTmp fp
o
)(exp
22
2/3
2
*υπ −−
⎟⎟⎠
⎞⎜⎜⎝
⎛=
We define effective density of states in CB,
The thermal-equilibrium electron concentration in CB
2/3
2
*22 ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
h
kTmN pπ
υ
kT
EENp f
o
)(exp υ
υ
−−=
W.K. Chen Electrophysics, NCTU 11
2/3
2
*22 ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
h
kTmN n
C
π
Nc & Nυ values
Nc & Nυ 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 Nc & Nυ will increase in values with power of 3/2 with increasing temperature
Assume mn*=mo, 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 *319onC mmN =×= −
2/3
2
*22 ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
h
kTmN pπ
υ
W.K. Chen Electrophysics, NCTU 12
Example 4.2 hole concentration
31519 cm1043.6)03453.0
27.0exp()1060.1(
)(exp
−×=−
×=
−−=
kT
EENp f
oυ
υ
3192/3
192/3
cm1060.1300
400)1004.1()400(
300
400
)300(
)400( −×=⎟⎠⎞
⎜⎝⎛×=⇒⎟
⎠⎞
⎜⎝⎛= KN
KN
KNυ
υ
υ
Solution:400Kat VBin ion concentrat hole theFind
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 concentrationFor intrinsic semiconductor
The concentration of electrons in CB is equal to the concentration of holes in VB
W.K. Chen Electrophysics, NCTU 14
kT
EENnn ifC
Cio
)(exp
−−==
kT
EENpp if
io
)(exp υ
υ
−−==
torsemiconduc intrinsicfor ii pn =
kT
EENN
kT
EEN
kT
EENn
CC
ffCCi
ii
)(exp
)(exp
)(exp2
υυ
υυ
−−=
−−⋅
−−=
kT
ENNn g
Ci
−= exp2
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
42.1exp()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
ENNn g
Ci
−= exp2
υ
310
212
cm1085.3)400(
1048.1)03885.0
42.1exp()400()400()400(
−×=
×=−
=
Kn
KNKNKn
i
ci υ
W.K. Chen Electrophysics, NCTU 16
kT
ENNn g
Ci
−= exp2
υ
The intrinsic carrier concentration is nearly increase exponentially with temperature
W.K. Chen Electrophysics, NCTU 17
4.1.4 The intrinsic Fermi-level position
torsemiconduc intrinsicfor ii pn =
2/3
*
*
ln2
1)(
2
1
ln2
1)(
2
1
)(exp
)(exp
⎟⎟⎠
⎞⎜⎜⎝
⎛++=
⎟⎟⎠
⎞⎜⎜⎝
⎛++=
−−=
−−
p
pcf
ccf
ffCC
m
mkTEEE
N
NkTEEE
kT
EEN
kT
EEN
i
i
ii
υ
υυ
υυ
2/3
*
*
ln4
3⎟⎟⎠
⎞⎜⎜⎝
⎛+=
p
pmidgapf m
mkTEE
i
midgapEcE
υE
fiE
)(2
1υEEE cmidgap +=
W.K. Chen Electrophysics, NCTU 18
Example 4.3 Intrinsic Fermi level
Solution:
level Fermi intrinsic theFind
56.0
08.1*
*
⇒
=
=
op
on
mm
mm
2/3
*
*
ln4
3⎟⎟⎠
⎞⎜⎜⎝
⎛+=
p
pmidgapf m
mkTEE
i
meV8.12eV0128.008.1
56.0ln)0259.0(
4
32/3
−=−=⎟⎠⎞
⎜⎝⎛+= midgapf EE
i
W.K. Chen Electrophysics, NCTU 19
4.2 Dopant atoms and energy levelsThe 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.
n-type 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
n-type 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
p-type 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 Ea is the energy state of group III element in Si..
VB
Acceptor energy level
W.K. Chen Electrophysics, NCTU 22
p-type 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 energyIonization energy
The energy required to elevate the donor electron into the conduction band
nnc r
m
r
eF
2*
2
2
4
υπε
==
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
)( *υmrn n ⋅== hl
oo
rn am
mn
em
nr ⎟
⎠⎞
⎜⎝⎛==
*2
2*
224 επε hao: Bohr radius
o
oo em
a A53.04
2
2
==hπε
2nrn ∝
2
2
*υ=⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅mr
n
n
h
*3
222
*
*
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
oo
rn am
mn
em
nr ⎟
⎠⎞
⎜⎝⎛==
*2
2*
224 επε h
o
o
o
or
aar
m
mn
>>==⇒
===
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
πευ
hn
emmT ==
22
4*2
)4()(4 πεπε hn
em
r
eV
n
−=
−=
22
4*
)4()(2Energy Ionization
πεhn
emVTE
−=+=
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 πεhn
emVTE
−=+=
Ordinary impurity
For ordinary impurities, such as P, As, and Sb in Si and Ge, the hydrogen model works quite well.
Amphoteric impurity
For III-V semiconductor, the group IV such as Si and Ge is amphotericimpurity. 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 semiconductorIntrinsic 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 thermal-equilibrium electron and hole concentrations are different the intrinsic carrier concentrations.
W.K. Chen Electrophysics, NCTU 28
4.3.1 Equilibrium distribution of electrons & holesAdding 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
EENn fC
Co
)(exp
−−=
kT
EENp f
o
)(exp υ
υ
−−=
W.K. Chen Electrophysics, NCTU 29
n-type semiconductor
When the density of electrons in CB is greater than the density of holes in VB
p-type semiconductor
When the density of holes in VB is greater than the density of electrons in CB
kT
EENn fC
Co
)(exp
−−=
kT
EENp f
o
)(exp υ
υ
−−=
fifoo
fifoo
EEnp
EEpn
<⇒>
>⇒>
type-p
type-n
W.K. Chen Electrophysics, NCTU 30
Example 4.5 thermal equilibrium concentrations
3-419
3-1519
cm107.20259.0
87.0exp)1004.1(
cm108.10259.0
25.0exp)108.2(
×=−
×=
×=−
×=
o
o
p
n
kT
EENp 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 −− ×=×= υNNC
Solution:
kT
EENn fC
Co
)(exp
−−=
The electron and hole concentrations change by order of magnitude as the Fermi energy changes by a few tenths of an electron-volt
W.K. Chen Electrophysics, NCTU 31
Another form of equations for no and po
⎥⎦
⎤⎢⎣
⎡ −⎥⎦
⎤⎢⎣
⎡ −−=
⎥⎦
⎤⎢⎣
⎡ −+−−=⇒
−−==
−−=
kT
EE
kT
EENn
kT
EEEENn
kT
EENnn
kT
EENn
fiFfiCCo
fiFfiCCo
fCCio
fCCo
i
)(exp
)(exp
)()(exp
)(exp intrinsic
)(exp type-n
)(
exp ⎥⎦
⎤⎢⎣
⎡ −=
kT
EEnn fiF
io ) , type-(n fifio EEnn >>
)(
exp ⎥⎦
⎤⎢⎣
⎡ −−=
kT
EEnp fiF
io ) , type-(p fifio EEnn >>
W.K. Chen Electrophysics, NCTU 32
4.3.2 The nopo product
kT
EENn fC
Co
)(exp
−−=
kT
EENp f
o
)(exp υ
υ
−−=
kT
EEN
kT
EENpn ffC
Coo
)(exp
)(exp υ
υ
−−⋅
−−=
2exp ig
Coo nkT
ENNpn =
−⋅= υ
The product of no and po 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 nopo product is one of the fundamental principles of the semiconductor in thermal equilibrium
W.K. Chen Electrophysics, NCTU 33
4.3.3 The Fermi-Dirac IntegralIf the Boltzmann approximation does not hold, the thermal equilibrium carrier concentrations is expressed as Fermi-Dirac distribution function
on distributi Dirac-Fermi elsewhere
holdsion approximatBoltzmann 3 typep
3 typen
⇒
⇒⎪⎭
⎪⎬⎫
>−−
>−−
kTEE
kTEE
f
fC
υ
cE
υE
fEkT3
kT3
Boltzmannapproximation
Fermi-Dirac
Fermi-Dirac
W.K. Chen Electrophysics, NCTU 34
∫ −+
−=
cEf
cno dE
kT
EEEE
mh
n)exp(1
)()2(
4 2/12/3*
3
π
2/3
2
*22 ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
h
kTmN n
C
π
kT
EE
kT
EE cFF
c −=
−= ηη and let
∫∞
−+=
0
2/12/3
2
*
)exp(1)
2(4 η
ηηηπ d
h
kTmn
F
no
∫∞
−+=
0
2/1
2/1 )exp(1)( η
ηηηη dF
FF
)(2
2/1 Fco FNn ηπ
=
)(2
2/1 Fo FNp ηπ υ=
2/3
2
*22 ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
h
kTmN pπ
υ
W.K. Chen Electrophysics, NCTU 35
4.3.4 Degenerate and nondegeneratesemiconductorsNondegenerate 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 (Nc or Nυ), the Fermi level lies with the band. This type of semiconductors are called degenerate semiconductors.
f
cfc
EENp
EENn
>>
>>
υυ ,tor semiconduc degenerate type-p
,tor semiconduc degenerate type-n
o
o
W.K. Chen Electrophysics, NCTU 37
4.4 Statistics of donors and acceptorsSuppose we have Ni electrons and gi quantum states at ith impurity energy levelEach donor level at least has two possible spin orientations for donor electron, thus each donor has at least two quantum statesThe distribution function of donor electrons in the donor energy states is slightly different than the Fermi-Dirac function
bands allowedfor )exp(1
1)(
kT
EEEf
fF −
+=
statesdonor for )exp(
21
1
1)(
kT
EEN
nEf
fd
dF −
+==
nd: the density of electrons occupying the donor levelNd: the concentration of donor Nd
+ : the concentration of ionized donorEd: the energy level of the donor
+−= ddd NNn
W.K. Chen Electrophysics, NCTU 38
statesacceptor for )exp(
11
1)(
kT
EE
gN
pEf
afa
aF −
+==
+−= 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 free-out
⎥⎦
⎤⎢⎣
⎡ −−=
−≈
−+
==
>>−
kT
EEN
kT
EEN
kT
EEN
EfNn
kTEE
fdd
fd
d
f
dFdd
fd
)(exp2
)exp(21
)exp(
21
1)(
If
For non-degenerate semiconductor,
kT
EENn 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
type-nfor )( 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
EEN
kT
EEN
kT
EEN
nn
n
fCC
fdd
fdd
od
d
)(exp
)(exp2
)(exp2
−−+⎥
⎦
⎤⎢⎣
⎡ −−
⎥⎦
⎤⎢⎣
⎡ −−
=+
kTEE
NNnn
n
dC
d
Cod
d
)(exp
21
1−−
+=
+
cE
υE
fEkT3
dE + + + + +
W.K. Chen Electrophysics, NCTU 41
Example 4.7
%41.00041.0
0259.0)045.0(
exp)10(2
108.21
1
16
19 ==−×
+=
+ od
d
nn
n
3-16 cm10
doping sphosphorou 300K,T tor,semiconduc Si
=
=
dN
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
Nnn
n
dC
d
Cod
d
)(exp
21
1−−
+=
+kT
EEN
Npp
p
a
a
oa
a
)(exp
41
1
υυ −−+
=+
At T=300K
W.K. Chen Electrophysics, NCTU 43
At T=0 K
All electrons are in their lowest possible energy states
dffd
fdd
dF EE
kT
EE
kT
EEN
nEf >⇒−∞==
−⇒=
−+
== )exp(0)exp( 1)exp(
21
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 free-out.
W.K. Chen Electrophysics, NCTU 44
Example 4.8 90% ionization temperature
kTEE
NNpp
p
a
a
oa
a
)(exp
41
1
υυ −−+
=+
ionized are acceptors of 90%at which re temperatu theFind
cm10
(B)boron with dpoedtor semiconduc Si type-p3-16=aN
Solution:
K 193
)300
(0259.0
045.0exp
)10(4
)300
)(1004.1(1
11.0
16
2/319
=⇒
−×+
==+
T
T
Tpp
p
a
oa
a
W.K. Chen Electrophysics, NCTU 45
4.5 Charge neutralityCharge neutrality
In thermal equilibrium, the semiconductor crystal is electrically neutral
The charge-neutrality condition is used to determine the thermal equilibrium electron and hole concentration as a function of impurity dopingconcentration
Compensated semiconductor
A semiconductor contains both donor and acceptor impurity atoms in the same region
N-type compensated semiconductor ( Nd>Na)
P-type 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
do
iao
doao
nnNNn
Nn
nNn
NpNn
22
22
)( type-n i
adado n
NNNNn +⎟
⎠⎞
⎜⎝⎛ −
+−
=
W.K. Chen Electrophysics, NCTU 47
o
io
idada
o
p
nn
nNNNN
p
2
22
22
)(tor semiconduc type-p
=
+⎟⎠⎞
⎜⎝⎛ −
+−
=
o
io
iadad
o
n
np
nNNNN
n
2
22
22
)(tor semiconduc type-n
=
+⎟⎠⎞
⎜⎝⎛ −
+−
=
W.K. Chen Electrophysics, NCTU 48
Example 4.9 thermal equilibrium carrier conc.
3416
2102
316210
21616
cm1025.210
)105.1(
carrierminority
cm10)105.1(2
10
2
10
carriermajority
−
−
×=×
==
≈×+⎟⎟⎠
⎞⎜⎜⎝
⎛+=
o
io
o
n
np
n
Solution:
The thermal equilibrium majority carrier concentration is essentially equal to impurity concentration
3-103-16 cm105.1 and 0 ,cm10
doping sphosphorou 300K,T tor,semiconduc Si type-n
×===
=
iad nNN
W.K. Chen Electrophysics, NCTU 49
Redistribution of electrons when donors are added
When add shallow donorsMost 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
type-nfor )( ddio nNnn −+≈
Freeze-out (0 K)
Partial ionization (0- ∼100K)
Extrinsic region (∼100K-400K)
Intrinsic region (>∼400 K)
kTEE
NNnn
n
dC
d
Cod
d
)(exp
21
1−−
+=
+
kT
ENNn g
Ci
−⋅= exp2
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 p-semiconductor.
3415
2102
315210
215161516
cm1021.3107
)105.1(
carrierminority
)(
cm107)105.1(2
10310
2
)10310(
carriermajority
−
−
×=×
×==
−≈
×≈×+⎟⎟⎠
⎞⎜⎜⎝
⎛ ×−+
×−=
o
io
da
o
p
nn
NN
p
Solution:
Because of complete ionization at 300K, the majority carrier hole concentration is just the difference between the acceptor and donor concentrations
3-103-153-16 cm105.1 and cm105.1 ,cm10
300K,T tor,semiconduc Si type-p
×=×==
=
ida nNN
22
22
)(i
dadao n
NNNNp +⎟
⎠⎞
⎜⎝⎛ −
+−
=
W.K. Chen Electrophysics, NCTU 52
4.6 Position of Fermi energy level
kT
EENn fC
Co
)(exp
−−= kT
EENp 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
CfC n
NkTEE ln type-n ⎟⎟
⎠
⎞⎜⎜⎝
⎛=−
oF p
NkTEE υ
υ ln type-p
W.K. Chen Electrophysics, NCTU 53
iaodada
iadaa
f
nNpNNNN
NkT
nNNNN
NkTEE
>>=−⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
>>>>⎟⎟⎠
⎞⎜⎜⎝
⎛=−
, when ln
, when ln type-p
υ
υυ
idoadad
C
idadd
CfC
nNnNNNN
NkT
nNNNN
NkTEE
>>=−⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
>>>>⎟⎟⎠
⎞⎜⎜⎝
⎛=−
, when ln
, when ln type-n
W.K. Chen Electrophysics, NCTU 54
Example 4.13 Fermi energy
3161616
31619
cm1024.2101024.1
cm1024.1)0259.0
20.0exp(108.2
carriermajority
−
−
×=+×=
−=
×=−
×=
d
ado
o
N
NNn
n
Q
Solution:
edge band conduction thebelow eV 0.20 isenergy Fermi
cm105.1 and cm10
300KTtor,semiconduc Si type-n3-103-16 ×==
=
ia nN
kT
EENn fC
Co
)(exp
−−=
W.K. Chen Electrophysics, NCTU 55
Variation of Fermi level with doping conc at 300K
For silicon, at 300K, Nd, Na>>ni = (1.5x1010cm-3 )
The carrier concentrations are mainly determined by the impurityconcentration.
As the doping levels increase, the Fermi level moves closer to the conduction band for n-type material and closer to valence band for the p-type material
kT
EENn fC
Co
)(exp
−−=
kT
EENp f
o
)(exp υ
υ
−−=
W.K. Chen Electrophysics, NCTU 56
Ef 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, free-out occurs; the Fermi level goes above Ed for the n-type material and below Ea for the p-type material
kT
ENNn g
Ci
−⋅= exp2
υ
W.K. Chen Electrophysics, NCTU 57
υE
fE
ifE
cEdE
Free-out
υE
fEif
E
cEdE
Extrinsic
υEfEif
E
cEdE
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