Energy bands and charge carriers
in semiconductors
UNIT-1
ELECTRONIC DEVICES (EC-301)
Outlines
1. Bonding Forces and Energy Bands in Solids
1-1. Bonding Forces in Solids
1-2. Energy Bands
1-3. Metals, Semiconductors & Insulators
1-4. Direct & Indirect Semiconductors
1-5. Variation of Energy Bands with Alloy Composition
Outlines
2. Carriers in Semiconductors
2-1. Electrons and Holes
2-2. Effective Mass
2-3. Intrinsic Material
2-4. Extrinsic Material
2-5. Electrons and Holes in Quantum Wells
Outlines
3. Carriers Concentrations
3-1. The Fermi Level
3-2. Electron and Hole Concentrations at Equilibrium
1. Bonding Forces & Energy Bands
in Solids
In Isolated Atoms
In Solid Materials Core
1st Band
2nd Band
3rd Band
1-1. Bonding Forces in Solids
Na (Z=11) [Ne]3s1
Cl (Z=17) [Ne]3s1 3p5
_
Na+ Cl
1-1. Bonding Forces in Solids
Na+
_
e
1-1. Bonding Forces in Solids
1-1. Bonding Forces in Solids
<100> Si
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.
1-2. Energy Bands
Conduction
band
4N States
2p
Energy
2p
Eg 2s-2p
2s
2s
4N States
Valence
Diamond band
lattice 1s
1s
spacing
Atomic separation
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 0K 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.
1-3. Metals, Semiconductors &
Insulators
Empty
The difference bet-ween
insulators and Empty
semiconductor mat-erials
Eg
lies in the size of the band Eg
gap Eg, which is much Filled
small-er in
semiconductors than in Filled
insulators.
Insulator Semiconductor
1-3. Metals, Semiconductors &
Insulators
In metals the bands
either overlap or are Overlap
only partially filled.
Thus electrons and
empty energy states Metal
Partially are intermixed with-
Filled in the bands so that
electrons can move
Filled freely under the infl-
uence of an electric
Metal field.
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.
jk x x
k ( x ) U ( k x , x )e
x : Direction of propagation
k : Propagation constant / Wave vector
: The space-dependent wave function for
the electron
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.
1-4. Direct & Indirect Semiconductors
E E
Eg=h
Eg Et
k k
Direct Indirect
Example 3-1
1-4. Direct & Indirect
Semiconductors
Example 3-1:
Assuming that U is constant in k ( x) U (k x , x)
for
jk x x an essentially free electron, show that the x-component of
e electron momentum in the crystal is given by
the
Px h kx
Example 3-2
1-4. Direct & Indirect Semiconductors
Answer: U e 2 jk x x h jk x x
(e )dx
j x
Px
U 2 dx
h
2
h kx U dx
h kx
2
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 .
1-4. Direct & Indirect
Semiconductors
Properties of semiconductor materials
Eg(eV) n p Lattice
Si 1.11 1350 480 2.5E5 D 5.43
Ge
0.67 3900 1900 43 D 5.66
GaAs
AlAs 1.43 8500 400 4E8 Z 5.65
Gap 2.16 180 0.1 Z 5.66
2.26 300 150 1 Z 5.45
1-5. Variation of Energy Bands with
Alloy Composition
E E
3.0
2.8
2.6
2.4
L
X 2.2
L
X
0.3eV 2.0
X
1.43eV
2.16eV 1.8
L
k 1.6
AlxGaAs
AlAs
Ga1- 1.4
0 0.2 0.4 0.6 0.8
1
X
2. Carriers in Semiconductors
Ec
1487652309KK
300
18
14
15
20
19
11
12
13
17
16
10 Eg
Ev
Electron H
E Hole P
Pair
2-1. Electrons and Holes
E
-kj kj
k
j` j
N
J ( q) Vi 0
i
0
N
J ( q) (Viq)V( j qq))V j
i
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.
2-2. Effective Mass
Example 3-2:
Find the (E,k) relationship for a free electron and relate it to
the electron mass.
k
2-2. Effective Mass
Answer:
From Example 3-1, the electron momentum
is:
p mv hk
1 2 1 p2 h2 2
E mv k
2 2 m 2m
d 2E h2
dk 2 m
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
2-2. Effective Mass
The effective mass of an electron in a band with a
given (E,k) relationship is given by
h2
m*
d 2E
dk 2
Remember that in GaAs:
E
X * *
L m( ) m (X or L)
1.43eV
k
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.
h2
m*
d 2E
dk 2
Effective mass is a tensor quantity.
2-2. Effective Mass
EV d 2E *
2
0 m 0
dk h2
m*
2 d 2E
d E dk 2
*
EC 0 m 0
dk 2
Table 3-1. Effective mass values for Ge, Si and GaAs.
Ge Si GaAs
*
m 0.55 m0 1.1m0 0.067 m0
n
*
m p
0.37 m0 0.56 m0 0.48 m0
m0 is the free electron rest mass.
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 0K,
since the valence band is filled with
electrons and the conduction band is
empty.
2-3. Intrinsic Material
e-
Eg
Si
h+
n=p=ni
2-3. Intrinsic Material
If we denote the generation rate of EHPs as
and gi ( EHP
the 3 ) recombination rate as
cm s
equilibrium
ri ( EHP requires
3 ) that:
cm s
ri gi
Each of these rates is temperature depe-
ndent. For example, g i (T ) increases when
the temperature is raised.
2
ri r n0 p0 n
r i gi
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.
2-4. Extrinsic Material
V Ec
Ed
P
50
14
15
20
191487652309KKK
18
11
12
13
17
16
10
As
Ev
Sb Donor
2-4. Extrinsic Material
Ec
B
50
14
15
20
191487652309KKK
18
11
12
13
17
16
10
Al
Ea
Ga Ev
In Acceptor
2-4. Extrinsic Material
e- Sb
h+
Al Si
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 2 2
; n 1, K 4 0 r
2K h
2-4. Extrinsic Material
Example 3-3:
Calculate the approximate donor binding energy for
Ge(r=16, mn*=0.12m0).
2-4. Extrinsic Material
Answer:
mn* q 4
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 )
21
1.02 10 J 0.0064 eV
Thus the energy to excite the donor electron from
n=1 state to the free state (n=) is 6meV.
2-4. Extrinsic Material
When a -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 tempera-ture. If we dope Si with 1015 Sb Atoms/cm3,
the conduction electron concentration changes by five order
of magnitude.
2-5. Electrons and Holes in Quantum
Wells
One of most useful applications of MBE or OMVPE growth
of multilayer compou-nd 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
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
50
E1
0.28eV
1.85eV 1.43eV
0.14eV
Eh
2-5. Electrons and Holes in
Quantum Wells
Instead of having the continuum of states as described
2 2 2
by , modified
n h
En for effective mass and finite
2mL2
barrier height.
Similarly, the states in the valence band available for
holes are restricted to discrete levels in the quantum
well.
2-5. Electrons and Holes in
Quantum Wells
An electron on one of the discrete condu-ction 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.
3. Carriers Concentrations
In calculating semiconductor electrical pro-perties 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.
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 a range of these
statistical arguments is that the distrib-ution of electrons
over a range of allowed energy levels at thermal
equilibrium is
3-1. The Fermi Level
1
f (E) (E Ef )
1 e kT
k : Boltzmanns constant
f(E) : Fermi-Dirac distribution function
Ef : Fermi level
3-1. The Fermi Level
1 1 1
f (E f ) (Ef Ef )
1 e kT 1 1 2
f(E)
1
T >T1
T=0K
1>0K
2
1/2
E
Ef
3-1. The Fermi Level
E
f(Ec) f(Ec)
Ec
Ef
[1-f(Ec)]
Ev
f(E) Intrinsic
p-type
n-type
1 1/2 0
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 repres-
ent all of the distributed electron
states in the conduction band edge
EC.
n0 N C f ( EC )
3-2. Electron and Hole Concentrations
at Equilibrium
E
Electrons
N(E)f(E)
EC
Ef
N(E)[1-f(E)] EV
Holes
Intrinsic
p-type
n-type
3-2. Electron and Hole Concentrations
at Equilibrium
1 ( EC EF )
f ( EC ) ( EC EF ) e kT
1 e kT
( EC EF )
n0 NC e kT
2 mn* kT 3 2
NC 2( 2
)
h
3-2. Electron and Hole Concentrations
at Equilibrium
p0 NV [1 f ( EV )]
1 ( EF EV )
1 f ( EV ) 1 ( EV EF ) e kT
1 e kT
( EF EV )
p0 NV e kT
2 m*p kT 3
NV 2( ) 2
h2
3-2. Electron and Hole Concentrations
at Equilibrium
( Ec Ei ) ( Ei Ev )
ni NC e kT
pi NV e kT
( Ec Ev ) Eg
n0 p0 Nc Nv e kT
Nc Nv e kT
Eg Eg
ni pi Nc Nve kT
ni Nc Nv e 2 kT
2
n0 p0 n i
( EF Ei ) ( Ei EF )
n0 ni e kT
p0 ni e kT
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 concentra-
tion p0 at 300 K? Where is EF relative to
Ei?
3-2. Electron and Hole Concentrations
at Equilibrium
Answer:
Since Ndni, we can approximate n0=Nd
and
ni2 2.25 1020
p0 17
2.25 103 cm 3
n0 10
( EF Ei )
n0 ni e kT
n0 1017
EF Ei kT ln 0.0259 ln 0.407eV
ni 1.5 1010
3-2. Electron and Hole Concentrations
at Equilibrium
Answer (Continue) :
Ec
EF
0.407eV
1.1eV Ei
Ev
References:
Solid State Electronic Devices
By Streetman and Banarjee, Sixth edition
