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Chapter
2
SEMICONDUCTOR
BANDSTRUCTURE
HOLES IN SEMICONDUCTORS: HOW DO HOLES MOVE?
Holes behave as if they carry a positive charge.
t2 > t1>0
t=0
t = t1
Ε
Ε
Fx
F
E
D
G
C
B
A
t = t2
F
kx
D
H
I
J
K
Ε
Fx
E
G
C
B
A
kx
H
I
J
K
E
D
G
C
B
A
kx
H
I
J
K
(c)
(b)
(a)
Fx
F
F
ve
e
je
vh
h
jh
(d)
The movement of an empty electron state, i,e,. a hole under an electric field.
The electrons move in the direction opposite to the electric field so that the
hole moves in the direction of the electric field thus behaving as if it were
positively charged, as shown in (a), (b), and (c). (d) The velocities and
currents due to electrons and holes. The current flow is in the same direction,
even though the electron and holes have opposite velocities. The electron
effective mass in the valence band is negative, but the hole behaves as if it has
a positive mass.
© Prof. Jasprit Singh www.eecs.umich.edu/~singh
FREE CARRIERS IN SEMICONDUCTORS: INTRINSIC CARRIERS
In semiconductors, at finite temperatures, there are electrons in the conduction band and
holes in the valence band.
METALS
Evac
ENERGY
EF
Ec
Electrons in the
conduction band
can carry current
(a)
SEMICONDUCTORS
Evac
ENERGY
–
–
Electrons in the conduction band
(density, n) carry current
Ec
Mobile carrier density = n + p
+ + + EV
Holes in the valence band (density, p)
carry current
Valence band
(b)
(a) A schematic showing allowed energy bands in electrons in a metal. The electrons
occupying the highest partially occupied band are capable of carrying current. (b) A
schematic showing the valence band and conduction band in a typical semiconductor. In
semiconductors only electrons in the conduction band holes in the valence band can carry
current.
For small electron (n), hole (p) densities we can use Boltzmann approximation:
n = Nc exp [(EF– Ec)/kBT]
where
Nc = 2
p=2
(
m*ekBT 3/2
2πh2
)
m*hkBT 3/2
exp [(Ev– EF)/kBT]
2πh2
(
)
= Nv exp [(Ev– EF)/kBT]
Intrinsic case:
ni = pi = 2
kBT 3/2
(m*em*h)3/4 exp (–Eg/2kBT)
2πh2
( )
© Prof. Jasprit Singh www.eecs.umich.edu/~singh
INTRINSIC CARRIER DENSITIES FOR SOME SEMICONDUCTORS
1019
1000 500
T(°C)
200 100
27 0 –20
INTRINSIC CARRIER DENSITY ni (cm–3)
1018
1017
1016
Ge
1015
1014
Temperature dependence of
ni, pi in Si, Ge, GaAs
Si
1013
1012
11
10
10
10
109
GaAs
10 8
10 7
106
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1000/T (K–1)
MATERIAL
CONDUCTION BAND
EFFECTIVE DENSITY (NC )
VALENCE BAND
EFFECTIVE DENSITY (NV )
INTRINSIC CARRIER
CONCENTRATION
Si (300 K)
2.78 x 1019 cm–3
9.84 x 1018 cm–3
1.5 x 1010 cm–3
Ge (300 K)
1.04 x 1019 cm–3
6.0 x 1018 cm–3
2.33 x 1013 cm–3
GaAs (300 K)
4.45 x 1017 cm–3
7.72 x 1018 cm–3
1.84 x 106 cm–3
(ni = pi)
Effective densities and intrinsic carrier concentrations of Si, Ge and GaAs. The numbers for
intrinsic carrier densities are the accepted values even though they are smaller than the values
obtained by using the equations derived in the text.
© Prof. Jasprit Singh www.eecs.umich.edu/~singh
DOPING OF SEMICONDUCTORS: DONORS AND ACCEPTORS
If an impurity atom replaces a host semiconductor atom in a crystal it could donate (donor) an
extra electron to the conduction band or it could accept (acceptor) an electron from the valence
band producing a hole.
All 4 outer
electrons go into
the valence band
EC
EV
Silicon
host atom
=
+
+
Electron-ion
Coulombic
Pentavalent
donor impurity = Silicon-like + attraction
A schematic showing the approach one takes to understand donors in semiconductors. The
donor problem is treated as the host atom problem together with a Coulombic interaction term.
The silicon atom has four “free” electrons per atom. All four electrons are contributed to the
valence band at 0 K. The dopant has five electrons out of which four are contribted to the
valence band, while the fifth one can be used for increasing electrons in the conducton band.
E
Si
Si
Si
Ec
Excess
Donor
Si
positive
level
charge
–
Si
Si
Ev
Excess electron
+
As
Si
n-type
silicon
Arsenic (As) atom donates one
electron to the conduction band to
produce an n-type silicon
Conduction
band
Ed
AAAA
AAAA
AAAA
( )( εεso )2 eV
m*
Ed = Ec –13.6 m
o
Valence
band
Charges associated with an arsenic impurity atom in silicon. Arsenic has five valence electrons,
but silicon has only four valence electrons. Thus four electrons on arsenic form tetrahedral
covalent bonds similar to silicon, and the fifth electron is available for conduction. The arsenic
atom is called a donor because when ionized it donates an electron to the conduction band.
© Prof. Jasprit Singh www.eecs.umich.edu/~singh
FREE CARRIERS IN DOPED SEMICONDUCTORS
If electron (hole) density is measured as a function of temperature in a doped semiconductor, one
observes three regimes:
Freezeout: Temperature is too small to ionize the donors (acceptors), i.e.,
kBT < EC – ED (kBT< ED – EV).
Saturation: Most of the donors (acceptors) are ionzed.
Intrinsic: Temperature is so high that ni > doping density.
TEMPERATURE (K)
10 17
500
1000 300
200
100
75
50
Si
ELECTRON DENSITY n (cm–3)
Nd = 1015cm–3
Intrinsic range
10
16
Saturation range
10 15
10
Freeze-out
range
14
ni
10 13
0
4
8
12
16
20
1000/T (K–1)
Electron density as a function of temperature for a Si sample with donor impurity
concentration of 1015 cm–3. It is preferable to operate devices in the saturation region
where the free carrier density is approximately equal to the dopant density.
It is not possible to operate devices in the intrinsic regime, since the devices always have a high
carrier density that cannot be controlled by electric fields.
every semiconductor has an upper temperature beyond which it cannot be used in devices.
The larger the bangap, the higher the upper limit.
© Prof. Jasprit Singh www.eecs.umich.edu/~singh