Chapter 4: Semiconductor Physics
Crystal structures of solids
Energy band structures of solids
Charge carriers in semiconductors
Carrier transport
PHYS5320 Chapter Four (II)
1
Equilibrium Distribution of Electrons and Holes
The densities of electrons and holes are related to the density
density-of-states
of states
function and the Fermi distribution function.
nE   gc E  fFE 
pE   g v E 1 f FE 
The total electron/hole concentration per unit volume is found by integrating
the corresponding function over the entire conduction/valence band energy.
We need to determine the Fermi energy in order to find the thermalequilibrium electron and hole concentrations.
We first
W
fi t consider
id an intrinsic
i t i i semiconductor.
i
d t An
A id
ideall intrinsic
i ti i
semiconductor is pure semiconductor without impurities or lattice defects. At
T = 0 K,, all the energy
gy states in the valence band are filled with electrons and
all the energy states in the conduction band are empty. The Fermi energy
must be somewhere between Ec and Ev. At T > 0 K, the number of electrons
in the conduction band is equal to that of holes in the valence band.
band
PHYS5320 Chapter Four (II)
2
Equilibrium Distribution of Electrons and Holes
gc E  
g v E  
4
4
 
* 3/ 2
2mn
h3
 
* 3/ 2
2mp
h3
E E c
Ev  E
If mn* = mp*, the Fermi energy is at
th midgap
the
id
energy. If mn*  mp*, the
th
Fermi level for an intrinsic
semiconductor will slightly shift
away from the midgap energy.
PHYS5320 Chapter Four (II)
3
The n0 and p0 Equations
n0   gc E  fFE dE
d
Because the Fermi distribution rapidly approaches zero with increasing
energy, therefore

n0   gc E  fFE dE
Ec
If (EcEF) >> kT, then (EEF) >> kT, so that
 E  EF 
1
fFE  
E  EF   expp kT
1  exp
kT

n0  
Ec
If we let
4
 
* 3/ 2
2mn
h3

  E  EF 
E  Ec exp
dE

 kT
E  Ec
kT
PHYS5320 Chapter Four (II)
4


  Ec  EF   1 / 2
n0 
expp
 expp  d



0
h3
kT
The integral is a gamma function.
Then
4
3/ 2
*
2mn kT


0
So

1/ 2
1
exp  d 

2
3/ 2
*
 2mn kT 


n0  2


h2
We define a parameter Nc as


3/ 2
*
 2mn kT 


Nc  2


Then
  Ec  EF 
exp



kT
h
2


Nc is called the effective
density-of-states function
in the conduction band.
  Ec  EF 
n0  Nc exp



kT
PHYS5320 Chapter Four (II)
5
Nc is called the effective density-of-states function in the conduction
band. If we were to assume that mn* = m0, then at T = 300 K,


 2 9.109 10 1.38110
Nc  2
34 2

6.626 10
F hholes:
For
l
31


 23
300

3/ 2
 2.5 1019 cm3
p0   g v E 1  fFE dE
 E  EF 

exp
1
1


kT
1  f FE   1 


 E  EF 
 E  EF 
 EF  E 
 1  exp
 1  exp

1  exp
 kT 
 kT 
 kT 
If (EFEv) >> kT,
kT then (EFE) >> kT.
kT
  EF  E 
1  f FE   exp

 kT

PHYS5320 Chapter Four (II)
6
p0  
Ev
4
 

h3
' 
If we let
Then
p0 
  EF  E 
Ev  E exp
dE
 kT

* 3/ 2
2mp
 4


3/ 2
*
2mpkT
h3
Ev  E
kT
  EF  Ev  0
1/ 2
exp


'

exp ' d'



kT
  EF  Ev 
exp




kT

then
3/ 2
*
 2mpkT 


p0  2


If we let
h2
3/ 2
*
 2mpkT 


  EF  Ev 
Nv  2
p0  N v exp



2


kT
 h

Nv is
i called
ll d the
h effective
ff i density-of-states
d i
f
f
function
i in
i the
h valence
l
band.
b d
PHYS5320 Chapter Four (II)
7
The Intrinsic Carrier Concentration
We use ni and pi to denote electron and hole concentrations in an intrinsic
semiconductor. Since ni = pi, we usually use ni to denote either the intrinsic
electron or hole concentration. The Fermi level for an intrinsic semiconductor
i called
is
ll d the
th intrinsic
i t i i Fermi
F
i energy, EFi.
  Ec  EFi 
n0  ni  Nc expp



kT
  EFi  Ev 
p0  pi  ni  N vexp



kT
  Ec  EFi 
  EFi  Ev 
2
ni  Nc N v exp
 exp





kT
kT
ni2
  Ec  Ev 
  Eg 
 Nc N v exp
  Nc N v exp



 kT 
kT
PHYS5320 Chapter Four (II)
8
At T = 300 K
Nc, Nv, and ni are constant for a given semiconductor material at a
given temperature.
PHYS5320 Chapter Four (II)
9
3/ 2
*
 2mn kT 


Nc  2


h
2


3/ 2
*
 2mpkT 


Nv  2


ni2
h2


  Eg 
 Nc N v exp

 kT 
PHYS5320 Chapter Four (II)
10
The Intrinsic Fermi Level Position
We can calculate the Fermi level pposition since the electron and hole
concentrations are equal for an intrinsic semiconductor:
  Ec  EFi 
  EFi  Ev 
n0  Nc exp
  p0  N v exp





kT
kT
 Nv 
 Ec  E Fi EFi  Ev  kT ln 
N 
 c
 Nv 
EFi  Ec  Ev   kT ln 
N 
2
2
 c
1
1
 mp* 
EFi  Emidgap  kT ln 
 m* 
4
 n
3
 mp* 
EFi  Emidgap  kT ln 
 m* 
4
 n
3
If mp* = mn*, the intrinsic Fermi
level will be in the center of the
bandgap. If mp* > mn*, the
intrinsic Fermi level will be
slightly above the center. If mp*
< mn*, the
th intrinsic
i t i i Fermi
F
i level
l l
will be slightly below the center
of the bandgap.
bandgap
PHYS5320 Chapter Four (II)
11
Donor Atoms and Energy Levels
Real ppower of semiconductors is realized byy addingg controlled amounts of
specific dopant, or impurity atoms. The doped semiconductor is called an
extrinsic material. Doping is the primary reason that we can fabricate various
semiconductor
i d t devices.
d i
Add a ggroup
p V element,, such as phosphorus,
p p
, to silicon as a substitutional
impurity. The group V element has five valence electrons. Four of these will
contribute to the covalent bonding with the silicon atoms, leaving the fifth
more loosely bound to the phosphorus atom
atom.
PHYS5320 Chapter Four (II)
12
Donor Atoms and Energy Levels
At very low temperatures
temperatures, the extra electron
is bound to the phosphorus atom. However, it
should be clear that the energy required to
elevate the extra electron into the conduction
band is considerably smaller than that for the
electrons involved in the covalent bonding.
bonding
The electron elevated into the conduction
band can move through the crystal to
generate a current, while the positively
charged phosphorous atoms are fixed in the
crystal.
l This
hi type off atom is
i called
ll d a donor
impurity atom. The donor atoms add
electrons to the conduction band without
creating holes in the valence band. The
resulting material is referred to as an n-type
semiconductor.
i
d t
PHYS5320 Chapter Four (II)
13
Acceptor Atoms and Energy Levels
Consider adding a group III element, such as boron, which has three valence
electrons. One covalent bonding position is empty. If an electron were to
occupy this “empty” position, its energy would have to be greater than that of
th valence
the
l
electrons,
l t
since
i
the
th nett charge
h
off the
th B atom
t
would
ld become
b
negative. However, the electron occupying this “empty” position does not
have sufficient energy
gy to be in the conduction band,, so its energy
gy is far
smaller than the conduction band energy. The “empty” position associated
with the B atom can be occupied and other valence electron positions become
vacated These other vacated electron positions can be thought of as holes.
vacated.
holes
PHYS5320 Chapter Four (II)
14
Acceptor Atoms and Energy Levels
The hole can move through the crystal to
generate a current, while the negatively
charged
h
d boron
b
atoms are fixed
fi d in
i the
h crystal.
l
The group III atom accepts an electron from
the valence band and so is referred to as an
acceptor impurity atom. The acceptor atom
can generate holes in the valence band
without
ith t generating
ti electrons
l t
in
i the
th
conduction band. This type of semiconductor
material is referred to as a p-type
p yp
semiconductor.
An extrinsic semiconductor will have either
a preponderance of electrons (n-type) or a
preponderance of holes (p-type).
PHYS5320 Chapter Four (II)
15
Ionization Energy
We can calculate the approximate distance of the donor electron from the
donor impurity ion, and also the approximate energy required to elevate the
donor electron into the conduction band. This energy is referred to as the
ionization energy. We will use the Bohr model of the atom for these
calculations.
The permittivity
Th
i i i off the
h semiconductor
i d
material
i l instead
i
d off the
h permittivity
i i i off
free space will be used, and the effective mass of the electron will be used.
From the
F
h Coulomb
C l b force
f
off attraction
i being
b i equall to the
h centripetal
i
l
force of the orbiting electron:
e
2
* 2
mv

2
4rn
rn
Assume that the angular momentum is quantized,
PHYS5320 Chapter Four (II)
m vrn  n
*
16
n
v *
m rn
The Bohr radius is
n = 1,2,3,
n 2  2 4
rn 
m *e 2
4 0  2
 0.053nm
a0 
2
m0 e
 m0 
 n  r  
 m* 
a0
rn
Then
2
If we consider n = 1 state, and if we consider silicon, for which r =
11.7, m*/m
/ 0 = 0.26, then
h we hhave
r1
 45
a0
r1  2.39nm
For silicon,, a = 0.543 nm.
PHYS5320 Chapter Four (II)
17
e2
v
4n
* 4
1
m
e
* 2
The kinetic energy is
T mv 
2
2
2
2n  4 
2
* 4

e

m
e
The p
potential energy
gy is

V
4rn n 2 4 2
 m *e 4
The total energy is
E  T V 
2
2
2n  4 
The lowest energy state of the
hydrogen atom is E = 13.6
eV. For silicon,, the lowest
energy is E = 25.8 meV (Eg =
1.12 eV at T = 300 K).
PHYS5320 Chapter Four (II)
18
Quantum control of single
atoms:
t
the
th groundd state
t t off
the donor is within 0.1 nm
and non-interacting,
g, but the
excited states expand to
several nanometers and
strongly interacting
interacting.
Solid-state implementation
of coherent control of
impurity wavefunctions
Nature 2010, 465, 1057.
PHYS5320 Chapter Four (II)
19
Group III
III--V Semiconductors
The donor and acceptor impurities in III
III-V
V compound semiconductors is
more complicated than those in Si. When we talk about donors or acceptors in
III-V semiconductors, we need to know for which atoms (III or V) impurity
atoms are substituted.
b i d For example,
l for
f Si atoms in
i GaAs
G A semiconductor,
i d
if Si
atoms replace Ga atoms, Si impurities will act as donors. But if Si atoms
replace As atoms, they will act as acceptors.
For GaAs
PHYS5320 Chapter Four (II)
20
Equilibrium Distribution of Electrons and Holes in
Extrinsic Semiconductors
Adding donor or acceptor impurity atoms
to a semiconductor will change the
distribution of electrons and holes in the
material. Since the Fermi energy is
related
l t d to
t the
th distribution
di t ib ti function,
f ti the
th
Fermi energy will change as dopant
atoms are added.
In general,
general when EF > Emidgap, the density
of electrons is larger than that of holes,
and the semiconductor is n-type.
PHYS5320 Chapter Four (II)
21
Equilibrium Distribution of Electrons and Holes in
the Extrinsic Semiconductor
In general, when EF < Emidgap, the density
of electrons is smaller than that of holes,
holes
and the semiconductor is p-type.
  Ec  EF 
n0  Nc exp



kT
  EF  Ev 
p0  N v exp



kT
The above are ggeneral equations
q
for n0
and p0 in terms of the Fermi energy.
The values of n0 and p0 will change with
th Fermi
the
F
i energy, EF.
PHYS5320 Chapter Four (II)
22
Example: consider silicon at T = 300 K so that Nc = 2.8  1019 cm3 and Nv =
1.04  1019 cm3. If we assume the Fermi energy is 0.25 eV below the
conduction band, calculate the thermal equilibrium concentrations of
electrons and holes. The bandgap energy of silicon is 1.12 eV.
Solution: Ec  EF  0.25eV
EF  Ev  0.87eV


19 


0
.
25

1
.
602
10


n0  2.8 1019 exp
 1.8 1015cm3

 1.38110 23 300 
19 


0
.
87

1
.
602
10


p0  1.04 1019 exp
 2.6 104 cm3

 1.38110 23 300 










Comment: electron and hole concentrations change
g by
y orders of magnitude
g
from the intrinsic carrier concentrations (at 300 K, ni = 1.5  1010 cm3) as
the Fermi energy changes by a few tenths of an eV.
In an n-type semiconductor,
i d
n0 > p0, electrons
l
are referred
f
d to as majority
carriers and holes as minority carriers. In a p-type semiconductor, p0 > n0,
holes are referred to as majority carriers and electrons as minority
carriers.
PHYS5320 Chapter Four (II)
23
We can derive another form of the equations for the thermalequilibrium concentrations of electrons and holes:
  Ec  EF 
  Ec  EFi   EF  EFi 
n0  Nc exp
  Nc exp





k
kT
k
kT
 EF  EFi 

n0  ni exp
 kT 
  EF  Ev 
  EFi  Ev   EFi  EF 
p0  N v exp
  N v exp





kT
kT
 EFi  EF 
  EF  EFi 
p0  ni exp
  ni exp

 kT 


kT
PHYS5320 Chapter Four (II)
24
The n0p0 Product
  Ec  EF 
  EF  Ev 
n0 p0  Nc N v exp
 exp





kT
kT
  Eg 
2
n0 p0  Nc N v exp
  ni
 kT 
The product of n0 and p0 is always a constant for a given semiconductor
material at a given temperature. It is one of the fundamental principles of
semiconductors in thermal equilibrium
equilibrium.
It is important to keep in mind that the above equation is derived using the
Boltzmann approximation.
approximation
We may think of the intrinsic concentration ni simply as a parameter of the
semiconductor material.
material
PHYS5320 Chapter Four (II)
25
The Fermi
Fermi--Dirac Integral

n0   gc E  fFE dE
Ec
n0 
4
h

If we define
  
* 3/ 2 
2mn
Ec
3
E  Ec

F1 / 2  F   
0
3/ 2
*
 2mn kT 





h
2
F 
and
kT
n0  4
E  Ec dE
 E  EF 

1  exp
 kT 


EF E c
kT
1 / 2d
0 1  exp  
F
1 / 2d
1  exp  F 
PHYS5320 Chapter Four (II)
Fermi-Dirac integral
26
For holes:
p0   g v E 1  fFE dE
Ev

p0 
4
  
* 3 / 2 Ev
2mp

3
h
If we define
Ev  E
' 
kT
We need
W
d tto use th
the F
Fermii
Dirac integral when EF is
above Ec or below Ev.
p0  4
Ev  E dE
 EF  E 

1  exp
 kT 
and 'F 
3/ 2
*
 2mpkT 





h
PHYS5320 Chapter Four (II)
2


0
Ev E F
kT
' 1 / 2 d'
1  exp
p'  'F 
27
Degenerate and Nondegenerate Semiconductors
When discussing donors and acceptors, the concentration of dopant atoms is
assumed to be small compared to the density of host atoms. Impurities
introduce discrete, non-interacting donor and acceptor energy states in an ntype andd p-type semiconductor,
i d
respectively.
i l These
Th
types off semiconductors
i d
are referred to as nondegenerate semiconductors.
As the
A
h iimpurity
i concentration
i is
i increased,
i
d the
h distance
di
between
b
i
impurity
i
atoms decreases and the electrons from impurity atoms will begin to interact.
When this occurs,, the discrete donor or acceptor
p energy
gy level will combine
into a band of energies. As the impurity concentration is further increased, the
band of donor or acceptor states widens and overlaps with the conduction
b d bottom
band
b tt
or the
th valence
l
band
b d top.
t When
Wh the
th concentration
t ti off electrons
l t
in
i
the conduction band exceeds the density of states Nc, the Fermi energy lies
within the conduction band. This type
yp of semiconductors is called degenerate
g
n-type semiconductors. When the concentration of holes exceeds the density
of states Nv, the Fermi energy lies in the valence band. This type of
semiconductors is called degenerate p
p-type
type semiconductors.
semiconductors
PHYS5320 Chapter Four (II)
28
Degenerate and Nondegenerate Semiconductors
Nondegenerate
Degenerate
PHYS5320 Chapter Four (II)
29
Statistics of Donors and Acceptors
The probability of the donor energy level being occupied is
1
1
 Ed  EF 

1  exp
 kT 
2
Each donor level can be empty, contain one electron of
either spin, or two electrons of opposite spins. However, the
C l b repulsion
Coulomb
l i off two
t localized
l li d electrons
l t
raises
i the
th
energy of the doubly occupied level so high that double
occupation is essentially prohibited. This is the reason for
the factor of ½ appearing in the probability function.
Nd
nd 
1
 Ed  EF 

1  exp
 kT 
2
nd  Nd  Nd
Na
pa 
1
 EF  Ea 

1  exp
 kT 
g
pa  Na  Na
nd is the density of electrons occupying the
donor level.
Nd is the concentration of donor atoms.
Nd+ is the concentration of ionized donors.
donors
g is a degeneracy factor. The ground state g is
normally taken as 4 for the acceptor level in Si and
GaAs because of the detailed band structure.
PHYS5320 Chapter Four (II)
30
Complete Ionization and Freeze
Freeze--Out
If ((Ed  EF) >> kT,, we then have
Nd
  Ed  EF 
 2Nd exp
nd 

1
 Ed  EF 


kT

1  exp
 kT 
2
Because
  Ec  EF 
n0  Nc exp



kT
We can determine the
percentage off electrons
l
in the donor state
compared with the
total number of
electrons:
  Ed  EF 
2Nd exp

nd


kT

nd  n0 2N exp  Ed  EF   N exp  Ec  EF 
d
c








kT
kT
nd
1

nd  n0 1  Nc exp  Ec  Ed 




2Nd
kT
PHYS5320 Chapter Four (II)
31
For phosphorus-doped silicon at T = 300 K, Nc = 2.8  1019 cm3, Nd = 1016
cm3, and the ionization energy is 0.045 eV:
1
nd

 0.004  0.4%
19
nd  n0 1  2.8 10 exp  0.045 


16
 0.0259 
2 10
 
At room temperature, the donor states are almost completely ionized,
which is also true for the acceptor states at room temperature.
C
Complete
l t ionization
i i ti
PHYS5320 Chapter Four (II)
32
At T = 0 K, all electrons are in their lowest energy state. For an n-type
semiconductor, each donor state must contain an electron, therefore, nd = Nd.
expEd  EF  / kT   0
 EF  Ed

T  0
The Fermi level is above the donor level for an n-type semiconductor at T
=0K
K. Similarly,
Similarly the Fermi level will be below the acceptor level for a ptype semiconductor at T = 0 K.
Freeze-out (T = 0 K)
PHYS5320 Chapter Four (II)
33