In the name of God
Energy bands and charge carriers
in semiconductors
Chapter 3
Mr. Harriry (Elec. Eng.)
By: Amir Safaei
2006
Outlines

3-1. Bonding Forces and Energy Bands
in Solids





3-1-1. Bonding Forces in Solids
3-1-2. Energy Bands
3-1-3. Metals, Semiconductors & Insulators
3-1-4. Direct & Indirect Semiconductors
3-1-5. Variation of Energy Bands with Alloy
Composition
2
Outlines

3-2. Carriers in Semiconductors





3-2-1. Electrons and Holes
3-2-2. Effective Mass
3-2-3. Intrinsic Material
3-2-4. Extrinsic Material
3-2-5. Electrons and Holes in Quantum
Wells
3
Outlines

3-3. Carriers Concentrations


3-3-1. The Fermi Level
3-3-2. Electron and Hole Concentrations at
Equilibrium
4
3-1. Bonding Forces & Energy
Bands in Solids


In Isolated Atoms
In Solid Materials
Core
1st Band
2nd Band
3rd Band
5
3-1-1. Bonding Forces in
Solids


Na (Z=11) [Ne]3s1
Cl (Z=17) [Ne]3s1 3p5
Na+ Cl
_
6
3-1-1. Bonding Forces in
Solids
Na+
e
_
7
3-1-1. Bonding Forces in
Solids
8
3-1-1. Bonding Forces in
Solids
<100>
Si
9
3-1-2. Energy Bands

Pauli Exclusion Principle
C (Z=6) 1s2 2s2 2p2
2 states for 1s level
2 states for 2s level
6 states for 2p level
For N atoms, there will be 2N, 2N, and 6N states
of type 1s, 2s, and 2p, respectively.
10
3-1-2. Energy Bands
Energy
4N States
Conduction
band
2p
2p
Eg
2s-2p
2s
4N States
Diamond
lattice
spacing
2s
Valence
band
1s
Atomic separation
1s
11
3-1-3. Metals, Semiconductors
& Insulators


For electrons to experience acceleration in an
applied electric field, they must be able to
move into new energy states. This implies
there must be empty states (allowed energy
states which are not already occupied by
electrons) available to the electrons.
The diamond structure is such that the
valence band is completely filled with
electrons at 0ºK and the conduction band is
empty. There can be no charge transport
within the valence band, since no empty
states are available into which electrons can
move.
12
3-1-3. Metals, Semiconductors
& Insulators

The difference between insulators and
semiconductor materials lies in the size
of the band gap Eg,
which is much smaller in semiconductors
than in insulators.
Empty
Empty
Eg
Eg
Filled
Filled
Insulator
Semiconductor
13
3-1-3. Metals, Semiconductors
& Insulators

In metals the bands
either overlap or are
only partially filled.
Thus electrons and
empty energy states
Partially
Filled
Filled
Metal
Overlap
Metal
are intermixed within the bands so that
electrons can move
freely under the influence of an electric
field.
14
3-1-4. Direct & Indirect
Semiconductors


A single electron is assumed to travel
through a perfectly periodic lattice.
The wave function of the electron is assumed
to be in the form of a plane wave moving.
 k ( x )  U ( k x , x )e
jk x x
x : Direction of propagation
 k : Propagation constant / Wave vector
  : The space-dependent wave function for
the electron

15
3-1-4. Direct & Indirect
Semiconductors

U(kx,x): The function that modulates the
wave function according to the periodically of
the lattice.

Since the periodicity of most lattice is
different in various directions, the (E,k)
diagram must be plotted for the various
crystal directions, and the full relationship
between E and k is a complex surface which
should be visualized in there dimensions.
16
3-1-4. Direct & Indirect
Semiconductors
E
E
Eg=hν
Eg
Et
k
Direct
k
Indirect
Example 3-1
17
3-1-4. Direct & Indirect
Semiconductors
Example 3-1:
Assuming that U is constant in  k ( x)  U (k x , x)
jk x x
e for an essentially free electron, show
that the x-component of the electron
momentum in the crystal is given by
 Px  h k x
Example 3-2
18
3-1-4. Direct & Indirect
Semiconductors
Answer:
 Px 



h  jkx x
(e )dx
j x
2  jk x x
U e


U 2 dx



h k x  U 2 dx



2
 h kx
U dx

The result implies that (E,k) diagrams such as shown
in previous figure can be considered plots of electron
energy vs. momentum, with a scaling factor h .
19
3-1-4. Direct & Indirect
Semiconductors
Properties of semiconductor materials
Eg(eV) n





Si
Ge
GaAs
AlAs
Gap
p

1.11 1350
480 2.5E5
0.67 3900 1900
43
1.43 8500 400 4E8
2.16
180
0.1
2.26
300
150
1
Lattice Å
D 5.43
D 5.66
Z 5.65
Z 5.66
Z 5.45
20
3-1-5. Variation of Energy
Bands with Alloy Composition
E
3.0
E

2.8
2.6
2.4
L

0.3eV
X
L
2.2
X
2.0
X
1.43eV
2.16eV
1.8
k 1.6
AlxGaAs
AlAs
Ga1-
L

1.4
0
0.2
0.4
X
0.6
0.8
1 21
3-2. Carriers in Semiconductors
Ec
300
18
14
15
20
19
11
12
13
17
16
1487652309ºKK
10
Eg
Ev
Electron H
Hole P
Pair
E
22
3-2-1. Electrons and Holes
E
-kj
kj
j`
k
j
0
NN
(qq))
q )V j
J J(
(VViiq)V(0j q)
ii
23
3-2-2. Effective Mass


The electrons in a crystal are not free,
but instead interact with the periodic
potential of the lattice.
In applying the usual equations of
electrodynamics to charge carriers in
a solid, we must use altered values of
particle mass. We named it Effective
Mass.
24
3-2-2. Effective Mass
Example 3-2:
Find the (E,k) relationship for a free
electron and relate it to the electron
mass.
E
k
25
3-2-2. Effective Mass
Answer:
From Example 3-1, the electron momentum
is:
p  mv  h k
1 2 1 p2 h 2 2
E  mv 

k
2
2 m 2m
d 2E h 2

2
dk
m
26
3-2-2. Effective Mass
Answer (Continue):
Most energy bands are close to parabolic at
their minima (for conduction bands) or
maxima (for valence bands).
EC
EV
27
3-2-2. Effective Mass

The effective mass of an electron in a band
with a given (E,k) relationship is given by
m* 

h2
d 2E
dk 2
Remember that in GaAs:
E
L

m ( )  m ( X
*
X
*
or
L)
1.43eV
k
28
3-2-2. Effective Mass


At k=0, the (E,k) relationship near the
minimum is usually parabolic:
h2 2
E
k  Eg
*
2m
d 2E
In a parabolic band, dk 2 is constant. So,
effective mass is constant.
m* 

h2
d 2E
dk 2
Effective mass is a tensor quantity.
29
3-2-2. Effective Mass
2
EV
d E
0
2
dk
m 0
*
m* 
2
EC
d E
0
2
dk
h2
d 2E
dk 2
m 0
*
Table 3-1. Effective mass values for Ge, Si and GaAs.
Ge
*
m
m
n
*
p
Si
GaAs
0.55m0
1.1m0
0.067m0
0.37m0
0.56m0
0.48m0
† m0 is the free electron rest mass.
30
3-2-3. Intrinsic Material

A perfect semiconductor crystal with no
impurities or lattice defects is called an
Intrinsic semiconductor.

In such material there are no charge
carriers at 0ºK, since the valence band is
filled with electrons and the conduction
band is empty.
31
3-2-3. Intrinsic Material
eEg
Si
h+
n=p=ni
32
3-2-3. Intrinsic Material

If we denote the generation rate of EHPs
as g i ( EHP
3
) and the recombination rate
cm s
as ri ( EHP 3 ) equilibrium requires that:
cm s
ri  g i

Each of these rates is temperature dependent. For example, g (T ) increases when
i
the temperature is raised.
ri   r n0 p0   n  g i
2
r i
33
3-2-4. Extrinsic Material


In addition to the intrinsic carriers generated
thermally, it is possible to create carriers in
semiconductors
by
purposely
introducing
impurities into the crystal. This process, called
doping, is the most common technique for
varying the conductivity of semiconductors.
When a crystal is doped such that the equilibrium
carrier concentrations n0 and p0 are different from
the intrinsic carrier concentration ni , the material
is said to be extrinsic.
34
3-2-4. Extrinsic Material
Ec
V
P
Ed
50
18
14
15
20
19
11
12
13
17
16
1487652309ººKKK
10
As
Ev
Sb
Donor
35
3-2-4. Extrinsic Material
ш
Ec
B
Al
50
18
14
15
20
19
11
12
13
17
16
1487652309ººKKK
10
Ga
In
Ea
Ev
Acceptor
36
3-2-4. Extrinsic Material
e- Sb
h+
Al
Si
37
3-2-4. Extrinsic Material

We can calculate the binding energy
by using the Bohr model results,
consider-ing the loosely bound
electron as ranging about the tightly
bound “core” electrons in a
hydrogen-like orbit.
4
mq
E
; n  1 , K  4  0 r
2 2
2K h
38
3-2-4. Extrinsic Material
Example 3-3:
Calculate the approximate donor binding
energy for Ge(εr=16, mn*=0.12m0).
39
3-2-4. Extrinsic Material
Answer:
*
n
4
mq
E
2 2
8( 0 r ) h
31
19 4
0.12(9.1110 )(1.6 10 )

12
2
34 2
8(8.85 10 16) (6.63 10 )
 1.02 10
 21
J  0.0064eV
Thus the energy to excite the donor electron from
n=1 state to the free state (n=∞) is ≈6meV.
40
3-2-4. Extrinsic Material


When a ш-V material is doped with Si
or Ge, from column IV, these impurities
are called amphoteric.
In Si, the intrinsic carrier concentration
ni is about 1010cm-3 at room temperature. If we dope Si with 1015 Sb Atoms/cm3,
the conduction electron concentration
changes by five order of magnitude.
41
3-2-5. Electrons and Holes in
Quantum Wells


One of most useful applications of MBE
or OMVPE growth of multilayer compound semiconductors is the fact that a
continuous single crystal can be grown
in which adjacent layer have different
band gaps.
A consequence of confining electrons
and holes in a very thin layer is that
42
3-2-5. Electrons and Holes in
Quantum Wells
these particles behave according to the
particle in a potential well problem.
Al0.3Ga0.7As
GaAs
Al0.3Ga0.7As
50Å
E1
1.85eV
0.28eV
1.43eV
Eh
0.14eV
43
3-2-5. Electrons and Holes in
Quantum Wells

Instead of having the continuum of states
2 2 2
n

h
as described by En 
, modified for
2
2mL
effective mass and finite barrier height.

Similarly, the states in the valence band
available for holes are restricted to discrete
levels in the quantum well.
44
3-2-5. Electrons and Holes in
Quantum Wells

An electron on one of the discrete conduction band states (E1) can make a transition
to an empty discrete valance band state in
the GaAs quantum well (such as Eh), giving
off a photon of energy Eg+E1+Eh, greater
than the GaAs band gap.
45
3-3. Carriers Concentrations

In calculating semiconductor electrical properties and analyzing device behavior, it is
often necessary to know the number of
charge carriers per cm3 in the material. The
majority carrier concentration is usually
obvious in heavily doped material, since
one majority carrier is obtained for each
impurity atom (for the standard doping
impurities).
The concentration of minority carriers is not
obvious, however, nor is the temperature
dependence of the carrier concentration.
46
3-3-1. The Fermi Level


Electrons in solids obey Fermi-Dirac statistics.
In the development of this type of statistics:




Indistinguishability of the electrons
Their wave nature
Pauli exclusion principle
must be considered.
The distribution of electrons over
these statistical arguments is that
ution of electrons over a range
energy levels at thermal equilibrium
a range of
the distribof allowed
is
47
3-3-1. The Fermi Level
1
f (E) 
( E E f )
kT
1 e



k : Boltzmann’s constant
f(E) : Fermi-Dirac distribution function
Ef : Fermi level
48
3-3-1. The Fermi Level
f (E f ) 
1
1 e
( E f E f )
kT
1
1


11 2
f(E)
1
T
>T1
T=0ºK
2
1>0ºK
1/2
Ef
E
49
3-3-1. The Fermi Level
E
f(Ec)
f(Ec)
Ec
Ef
[1-f(Ec)]
≈
f(E)
1
≈
1/2
0
Ev
Intrinsic
p-type
n-type
50
3-3-2. Electron and Hole
Concentrations at Equilibrium

The concentration of electrons
in the

conduction band is n0 
f ( E ) N ( E )dE

EC


N(E)dE : is the density of states (cm-3)
in the energy range dE.
The result of the integration is the
same as that obtained if we represent all of the distributed electron
states in the conduction band edge
EC.
n0  NC f ( EC )
51
3-3-2. Electron and Hole
Concentrations at Equilibrium
E
Electrons
N(E)f(E)
EC
Ef
EV
N(E)[1-f(E)]
Holes
Intrinsic
p-type
n-type
52
3-3-2. Electron and Hole
Concentrations at Equilibrium
f ( EC ) 
1
1 e
( EC  EF )
n0  NC e
e
 ( EC  EF )
kT
kT
 ( EC  EF )
kT
2 mn* kT 3 2
N C  2(
)
2
h
53
3-3-2. Electron and Hole
Concentrations at Equilibrium
p0  NV [1  f ( EV )]
1  f ( EV )  1 
1
1 e
p0  NV e
N V  2(
( EV  EF )
 ( EF  EV )
2 m*p kT
h2
)
e
 ( EF  EV )
kT
kT
kT
3
2
54
3-3-2. Electron and Hole
Concentrations at Equilibrium
ni  N C e
 ( Ec  Ei )
n0 p0  N c N v e
ni pi  N c N v e
pi  NV e
kT
 ( Ec  Ev )
kT
 Eg
 Nc Nve
 ( Ei  Ev )
 Eg
kT
ni  Nc N v e
kT
kT
 Eg
2 kT
n0 p0  n
2
i
n0  ni e
( EF  Ei )
kT
p0  ni e
( Ei  EF )
kT
55
3-3-2. Electron and Hole
Concentrations at Equilibrium
Example 3-4:
A Si sample is doped with 1017 As Atom/cm3.
What is the equilibrium hole concentration p0 at 300°K? Where is EF relative to
Ei?
56
3-3-2. Electron and Hole
Concentrations at Equilibrium
Answer:
Since Nd»ni, we can approximate n0=Nd
and
ni2 2.25 10 20
3
3
p0 


2
.
25

10
cm
17
n0
10
n0  ni e
( EF  Ei )
kT
n0
1017
EF  Ei  kT ln
 0.0259 ln
 0.407eV
10
ni
1.5 10
57
3-3-2. Electron and Hole
Concentrations at Equilibrium
Answer (Continue) :
1.1eV
0.407eV
Ec
EF
Ei
Ev
58
References:
Solid State Electronic Devices
Ben G. Streetman, third edition
Modular Series on Solid State Devices,
Volume I: Semiconductor Fundamentals
Robert F. Pierret
59