SEMICONDUCTORS
Semiconductors
Semiconductor devices
Electronic Properties
Robert M Rose, Lawrence A Shepart, John Wulff
Wiley Eastern Limited, New Delhi (1987)
Energy gap in solids
In the free electron theory a constant potential was assumed inside the solid
In reality the presence of the positive ion cores gives rise to a varying
potential field
The travelling electron wave interacts with this periodic potential
(for a crystalline solid)
The electron wave can be Bragg diffracted
Bragg diffraction from a 1D solid
n
= 2d
n
= 2d Sin
1D
=90
o
The Velocity of electrons for the above values of
k
are zero
These values of
k
and the corresponding E are forbidden in the solid
The waveform of the electron wave is two standing waves
The standing waves have a periodic variation in amplitude and hence the
electron probability density in the crystal
The potential energy of the electron becomes a function of its position
(cannot be assumed to be constant (and zero) as was done in the
free electron model)
k
→
E
→
Band gap
The magnitude of the Energy gap between two bands is the difference
in the potential energy of two electron locations
k
→
E
→
K.E of the electron increasing
Decreasing velocity of the electron
ve effective mass (m
*
) of the electron
Within a band
Effective energy gap
→ Forbidden gap →
Band gap
k
→
E
→
E
→
[100]
[110]
k
→
Effective gap
The effective gap for all directions of motion is called the forbidden gap
There is no forbidden gap if the maximum of a band for one direction of
motion is higher than the minimum for the higher band for another
direction of motion
this happens if the potential energy of the electron
is not a strong function of the position in the crystal
Energy band diagram: METALS
Monovalent metals
Divalent metals
Monovalent metals: Ag, Cu, Au → 1 e
in the outermost orbital
outermost energy band is only half filled
Divalent metals: Mg, Be → overlapping conduction and valence bands
they conduct even if the valence band is full
Trivalent metals: Al → similar to monovalent metals
!!!
outermost energy band is only half filled
!!!
Energy band diagram: SEMICONDUCTORS
2

3 eV
Elements of the 4
th
column (C, Si, Ge, Sn, Pb) → valence band full but no
overlap of valence and conduction bands
Diamond → PE as strong function of the position in the crystal
Band gap is 5.4 eV
Down the 4
th
column the outermost orbital is farther away from the nucleus
and less bound
the electron is less strong a function of the position
in the crystal
reducing band gap down the column
Energy band diagram: INSULATORS
> 3 eV
P(E)
→
E
→
E
g
E
g
/2
Intrinsic semiconductors
At zero K very high field strengths (~ 1010 V/m) are required to move an
electron from the top of the valence band to the bottom of the
conduction band
Thermal excitation is an easier route
T > 0 K
n
e
→ Number of electrons promoted
across the gap
(= no. of holes in the valence band)
N → Number of electrons available
at the top of the valance band
for excitation
Unity in denominator can be ignored
Under applied field the electrons
(thermally excited into the conduction
band)
can move using the vacant sites in the conduction band
Holes move in the opposite direction in the valence band
The conductivity of a semiconductor depends on the concentration of
these charge carriers (
n
e
&
n
h
)
Similar to drift velocity of electrons under an applied field in metals in
semiconductors the concept of mobility is used to calculate conductivity
Conduction in an intrinsic semiconductor
Mobility of electrons and holes in Si & Ge (at room temperature)
Species
Mobility (m
2
/ V / s)
Si
Ge
Electrons
0.14
0.39
Holes
0.05
0.19
Conductivity as a function of temperature
Ln(
)
→
1/T (/K)
→
Extrinsic semiconductors
The addition of doping elements significantly increases the conductivity
of a semiconductor
Doping of Si
V column element
(P, As, Sb)
→ the extra unbonded electron
is practically free
(with a radius of motion of ~ 80 Å)
Energy level near the conduction band
n

type semiconductor
III column element
(Al, Ga, In)
→ the extra electron for bonding
supplied by a neighbouring Si atom → leaves a hole in Si.
Energy level near the valence band
p

type semiconductor
E
g
Donor level
n

type
Ionization Energy
→
Energy required to promote an
electron from the Donor level to
conduction band
E
Ionization
< E
g
even at RT large fraction of
the donor electrons are exited
into the conduction band
Electrons in the conduction band are the majority charge carriers
The fraction of the donor level electrons excited into the conduction band
is much larger than the number of electrons excited from the valence band
Law of mass action:
(n
e
)
conduction band
x (n
h
)
valence band
= Constant
The number of holes is very small in an n

type semiconductor
Number of electrons ≠ Number of holes
E
Ionization
Acceptor level
E
g
p

type
At zero K the holes are bound to the dopant atom
As T↑ the holes gain thermal energy and break away from the dopant atom
available for conduction
The level of the bound holes are called the acceptor level
(which can accept
and electron)
and acceptor level is close to the valance band
Holes are the majority charge carriers
Intrinsically excited electrons are small in number
Number of electrons ≠ Number of holes
E
Ionization
Ionization energies for dopants in Si & Ge (eV)
Type
Element
In
Si
In
Ge
n

type
P
0.044
0.012
As
0.049
0.013
Sb
0.039
0.010
p

type
B
0.045
0.010
Al
0.057
0.010
Ga
0.065
0.011
In
0.16
0.011
(/ Ohm / K
)
→
1/T (/K)
→
0.02
0.04
0.06
0.08
0.1
Intrinsic
Exhaustion
Exponential
function
Slope can be used
for the calculation
of E
Ionization
10 K
50 K
+ve slope due to
Temperature dependent
mobility term
All dopant atoms have been excited
slope
Semiconductor device
chose the flat region where the conductivity does
not change much with temperature
Thermistor
(for measuring temperature)
maximum sensitivity is
required
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