Opto electronics

Solid State Electronics

Text Book
Ben. G. Streetman and Sanjay Banerjee: Solid State
Electronic Devices, Prentice-Hall of India Private
Limited.
Chapter 3 and 4

Metals, Semiconductors, and Insulators
Insulator: A very poor conductor of electricity is
called an insulator.
In an insulator material the valance band
is filled while the conduction band is empty.
The conduction band and valance band in
the insulator are separated by a large forbidden
band or energy gap (almost 10 eV).
In an insulator material, the energy which
can be supplied to an electron from a applied field
is too small to carry the particle from the field
valance band into the empty conduction band.
Since the electron cannot acquire
sufficient applied energy, conduction is
impossible in an insulator.

Semiconductor: A substance whose conductivity lies between
insulator and conductor is a semiconductor.
A substance for which the width of the forbidden energy
region is relatively small (almost 1 eV) is called semiconductor.
In a semiconductor material, the energy which can be supplied to
an electron from a applied field is too small to carry the particle
from the field valance band into the empty conduction band at 0
K.
As the temperature is increased, some of the valance band
electrons acquire thermal energy. Thus, the semiconductors allow
for excitation of electrons from the valance band to conduction
band.
These are now free electrons in the sense that they can move
about under the influence of even a small-applied field.

Metal: A metal is an excellent conductor.
In metals the band either overlap or are only partially
filled.
Thus electrons and empty energy states are intermixed within the
bands so that electrons can move freely under the influence of an
electric field.

Direct and Indirect Semiconductors
Direct Material: The material (such as
GaAs) in which a transition of an electron
from the minimum point of conduction band
to the maximum point of valence band takes
place with the same value of K (propagation
constant or wave vector) is called direct
semiconductor material.
According to Eq. (3-1) the energy (E) vs
propagation constant (k) curve is shown in
the figure.
A direct semiconductor such as GaAs, an
electron in the conduction band can fall to an
empty state in the valence band, giving off
the energy difference Eg as a photon of light.

Indirect Material: The material (such as Si) in
which a transition of an electron from the
minimum point of conduction band to the
maximum point of valence band takes place
with the different values of K (propagation
constant or wave vector) is called indirect
material.
According to Eq. (3-1) the energy (E) vs
propagation constant (k) curve is shown in the
figure.

An electron in the conduction band minimum of an indirect
semiconductor cannot fall directly to the valence band maximum but
must undergo a momentum change as well as changing its energy.
It may go through some defect state (Et) within the band gap.
In an indirect transition which involves a change in k, the energy is
generally given up as heat to the lattice rather than as emitted photon.

Intrinsic Material
A perfect semiconductor with no impurities or lattice defect is
called an intrinsic material.
In intrinsic material, there are no charge carrier at 0K, since the
valence band is filled with electrons and the conduction band is
empty.
At high temperature electron-hole pairs are generated as valence
band electrons are excited thermally across the band gap to the
conduction band.
These EHPs are the only charge carriers in intrinsic material.
Since the electrons and holes are crated in pairs, the conduction
band electron concentration n (electron/cm3) is equal to the
concentration of holes in the valence band p (holes/cm3).
Each of these intrinsic carrier concentrations is commonly
referred to as ni. Thus for intrinsic material: n=p=ni (3-6)
At a temperature there is a carrier concentration of EHPs ni.

Recombination is occurs when an electron in the conduction
band makes transition to an empty state (hole) in the valence
band, thus annihilating the pair.
If we denote the generation rate of EHPs as gi (EHP/cm3-s)
and the recombination rate ri, equilibrium requires that
ri=gi (3-7a)
Each of these rates is temperature dependent.
gi(T) increases when the temperature is raised, and a new
carrier concentration ni is established such that the higher
recombination rate ri(T) just balances generation.
At any temperature, the rate of recombination of electrons
and holes ri is proportional to the equilibrium concentration
of electrons n0 and the concentration of holes p0:
ri=arn0p0= arni2=gi (3-7b)
The factor ar is a constant of proportionality which depends
on the particular mechanism takes place.

Extrinsic Material
When a crystal is doped such that the equilibrium carrier
concentrations n0 and p0 are different from carrier concentration
ni, the material is said to be extrinsic material.
In addition to the intrinsic carriers generated, 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 conductivity of semiconductor.
There are two types of doped semiconductors, n-type (mostly
electrons) and p-type (mostly holes).
An impurity from column V of the periodic table (P, As and Sb)
introduces an energy level very near the conduction band in Ge or
Si.

The energy level very near the conduction band is filled with electrons
at 0K, and very little thermal energy is required to excite these
electrons to the conduction band (Fig. 3-12a).
Thus at 50-100K virtually all of the electrons in the impurity level are,
“donated” to the conduction band.
Such an impurity level is called a donor level and the column V
impurities in Ge or Si are called donor impurities.
Semiconductors doped with a significant number of donor atoms will
have n0>>(ni,p0) at room temperature.
This is n-type material.

Fig. 3-12 (a) Donation of
electrons from donor level to
conduction band.

Similarly, an impurity from column III of the periodic table (B, Al, Ga
and In) introduces an energy level very near the valence band in Ge or
Si.
These levels are empty of electrons at 0K (Fig. 3-12b).
At low temperatures, enough thermal energy is available to excite
electrons from the valence into the impurity level, leaving behind holes
in the valence band.
Since this type of impurity level “accepts” electrons from the valence
band, it is called an acceptor level, and the column III impurities are
acceptor impurities in the Ge and Si.
Doping with acceptor
impurities can create a
semiconductor with a hole
concentration p0 much greater
that the conduction band
electron concentration n0.
This type is p-type material. Fig. 3.12b

Carrier concentration
The calculating semiconductor properties and analyzing device
behavior, it is often necessary to know the number of charge carriers
per cm3 in the material.
To obtain equation for the carrier concentration, Fermi-Dirac
distribution function can be used.
The distribution of electrons over a range of allowed energy levels at
thermal equilibrium is 1
f (E) 
1  e( E  EF ) / kT
where, k is Boltzmann’s constant (k=8.2610-5 eV/K=1.3810-23
J/K).
The function f(E), the Fermi-Dirac distribution function, gives the
probability that an available energy state at E will be occupied by an
electron at absolute temperature T.
The quantity EF is called the Fermi level, and it represents an
important quantity in the analysis of semiconductor behavior.

For an energy E equal to the Fermi level energy EF, the occupation
probability is 1 1
f ( EF )  ( E F  E F ) / kT

1 e 2
The significant of Fermi Level is that the probability of electron and
hole is 50 percent at the Fermi energy level. And, the Fermi function
is symmetrical about EF for all temperature; that is, the probability
f(EF +E) of electron that a state E above EF is filled is the same as
probability [1-f(EF-E)] of hole that a state E below EF is empty.

At 0K the distribution takes the
simple rectangular form shown in
Fig. 3-14.
With T=0K in the denominator of
the exponent, f(E) is 1/(1+0)=1
when the exponent is negative
(E<EF), and is 1/(1+)=0 when
the exponent is positive (E>EF).

This rectangular distribution implies that at 0K every available energy
state up to EF is filled with electrons, and all states above EF are empty.
At temperature higher than 0K, some probability exists for states above
the Fermi level to be filled.
At T=T1 in Fig. 3-14 there is
some probability f(E) that states
above EF are filled, and there is
a corresponding probability [1-
f(E)] that states below EF are
empty.
The symmetry of the
distribution of empty and filled
states about EF makes the Fermi
level a natural reference point
in calculations of electron and
hole concentration in
semiconductors.

For intrinsic material, the concentration of holes in
the valence band is equal to the concentration of
electrons in the conduction band.
Therefore, the Fermi level EF must lies at the middle
of the band gap.
Since f(E) is symmetrical
about EF, the electron
probability „tail‟ if f(E)
extending into the conduction
band of Fig. 3-15a is
symmetrical with the hole
probability tail [1-f(E)] in the
valence band.
Fig. 3-15(a) Intrinsic Material

In n-type material the Fermi level lies near Fig. 3.15(b) n-
the conduction band (Fig. 3-15b) such that type material
the value of f(E) for each energy level in the
conduction band increases as EF moves
closer to Ec.
Thus the energy difference (Ec- EF) gives
measure of n.
Fig. 3.15(c) p-
type material

In p-type material the Fermi level lies
near the valence band (Fig. 3-15c) such
that the [1- f(E)] tail value Ev is larger
than the f(E) tail above Ec.
The value of (EF-Ev) indicates how
strongly p-type the material is.

Example: The Fermi level in a Si sample at equilibrium is located at
0.2 eV below the conduction band. At T=320K, determine the
probability of occupancy of the acceptor states if the acceptor states
relocated at 0.03 eV above the valence band.

Solution:
From above figure, Ea-EF={0.03-(1.1-0.2)} eV= -0.87 eV
kT= 8.6210-5 eV/K320=2758.4 eV
we know that,
1 1
f ( Ea )  ( Ea  E F ) / kT
  1.0
1 e 0.87 /( 2758.4105 )
1 e

Electron and Hole Concentrations at Equilibrium
The concentration of electron and hole in the conduction band and
valance are

n0  E f ( E ) N ( E )dE (3.12a)
c

p0   [1  f ( E )] N ( E )dE
Ev
(3.12b)
where N(E)dE is the density of states (cm-3) in the energy range dE.
The subscript 0 used with the electron and hole concentration symbols
(n0, p0) indicates equilibrium conditions.
The number of electrons (holes) per unit volume in the energy range
dE is the product of the density of states and the probability of
occupancy f(E) [1-f(E)].
Thus the total electron (hole) concentration is the integral over the
entire conduction (valance) band as in Eq. (3.12).
The function N(E) is proportional to E(1/2), so the density of states in
the conduction (valance) band increases (decreases) with electron
(hole) energy.

Similarly, the probability of finding an empty state (hole) in the
valence band [1-f(E)] decreases rapidly below Ev, and most hole
occupy states near the top of the valence band.
This effect is demonstrated for intrinsic, n-type and p-type materials
in Fig. 3-16.

Fig. 3.16 (a) Concentration of electrons and holes in intrinsic material.

Fig. 3.16 (b) Concentration of electrons and holes in n-type material.

Fig. 3.16 (a) Concentration of electrons and holes in p-type material.

The electron and hole concentrations predicted by Eqs. (3-15) and (3-
18) are valid whether the material is intrinsic or doped, provided
thermal equilibrium is maintained.
Thus for intrinsic material, EF lies at some intrinsic level Ei near the
middle of the band gap, and the intrinsic electron and hole
concentrations are
ni  Nce( Ec  Ei ) / kT , pi  Nve( Ei  Ev ) / kT (3.21)

From Eqs. (3.15) and (3.18), we obtain

( Ec  EF ) / kT ( EF  Ev ) / kT
n0 p0  Nce Nve

( Ec  Ev ) / kT  E g / kT
n0 p0  Nc Nve  Nc Nve (3.22)

From Eq. (21), we obtain ni pi  Nc e( Ec  Ei ) / kT Nv e( Ei  Ev ) / kT
( Ec  Ev ) / kT  E g / kT
ni pi  Nc Nve  Nc Nve (3.23)
From Eqs. (3.22) and (3.23), the product of n0 and p0 at equilibrium is
a constant for a particular material and temperature, even if the
doping is varied.
The intrinsic electron and hole concentrations are equal, ni=pi; thus
from Eq. (3.23) the intrinsic concentrations is
 E g / 2 kT
ni  Nc Nv e (3.24)
The constant product of electron and hole concentrations in Eq. (3.24)
can be written conveniently from (3.22) and (3.23) as
n0 p0  ni2 (3.25)

At room temperature (300K) is: For Si approximately ni=1.51010
cm-3; For Ge approximately ni=2.51013 cm-3;

From Eq. (3.21), we can write as N c  ni e( Ec  Ei ) / kT
( Ei  Ev ) / kT
N v  pi e (3.26)
Substitute the value of Nc from (3.26) into (3.15), we obtain
n0  ni e( Ec  Ei ) / kT e( Ec  EF ) / kT  ni e( Ec  Ei  Ec  EF ) / kT
( Ei  EF ) / kT ( EF  Ei ) / kT
n0  ni e  ni e (3.27)
Substitute the value of Nv from (3.26) into (3.18), we obtain
p0  pi e( Ei  Ev ) / kT e( EF  Ev ) / kT  ni e( Ei  Ev  EF  Ev ) / kT
( EF  Ei ) / kT ( Ei  EF ) / kT
p0  ni e  ni e (3.28)
It seen from the equation (3.27) that the electron concentrations n0
increases exponentially as the Fermi level moves away from Ei
toward the conduction band.
Similarly, the hole concentrations p0 varies from ni to larger values as
EF moves from Ei toward the valence band.

Temperature Dependence of Carrier
Concentrations
The variation of carrier concentration with temperature is indicated
by Eq. (3.21)
( Ec  Ei ) / kT ( Ei  Ev ) / kT
ni  Nce , pi  Nve (3.21)
The intrinsic carrier ni has a strong temperature dependence (Eq.
3.24) and that EF can vary with temperature.

 E g / 2 kT
ni  Nc Nv e (3.24)

The temperature dependence of electron concentration in a doped
semiconductor can be visualized as shown in Fig. 3-18.

In this example, Si is doped
n-type with donor
concentration Nd of 1015 cm-3.
At very low temperature
(large 1/T) negligible intrinsic
EHPs exist, and the donor
electrons are bound to the
donor atoms.
As the temperature is raised,
these electrons are donated to
the conduction band, and at
about 100K (1000/T=10) all
the donor atoms are ionized. Figure 3-18 Carrier concentration vs.
This temperature range is inverse temperature for Si doped with
called ionization region. 1015 donors/cm3.
Once the donor atoms are ionized, the conduction band electron
concentration is n0Nd=1015 cm-3, since one electron is obtained for each
donor atom.

When every available extrinsic electron has been transferred to the
conduction band, no is virtually constant with temperature until the
concentration of intrinsic carriers ni becomes comparable to the
extrinsic concentration Nd.
Finally, at higher temperature ni is much greater than Nd, and the
intrinsic carriers dominate.

In most devices it is desirable to
control the carrier concentration by
doping rather than by thermal EHP
generation.
Thus one usually dopes the material
such that the extrinsic range extends
beyond the highest temperature at
which the device to be used.

Excess Carrier in Semiconductors
The carriers, which are excess of the thermal equilibrium
carries values, are created by external excitation is called excess
carriers.
The excess carriers can be created by optical excitation or
electron bombardment.
Optical Absorption
Measurement of band gap energy: The band gap energy of a
semiconductor can be measured by the absorption of incident photons
by the material.
In order to measure the band gap energy, the photons of selected
wavelengths are directed at the sample, and relative transmission of the
various photons is observed.
This type of band gap measurement gives an accurate value of
band gap energy because photons with energies greater than the band
gap energy are absorbed while photons with energies less than band gap
are transmitted.

Excess carriers by optical excitation: It
is apparent from Fig. 4-1 that a photon
with energy hv>Eg can be absorbed in a
semiconductor.
Since the valence band contains
many electrons and conduction band has
many empty states into which the
electron may be excited, the probability
of photon absorption is high. Figure 4-1 Optical absorption of a photon with
hv>Eg: (a) an EHP is created during photon
Fig. 4-1 indicates, an electron
absorption (b) the excited electron gives up
excited to the conduction band by optical energy to the lattice by scattering events; (c)
absorption may initially have more the electron recombines with a hole in the
energy than is common for conduction valence band.
band electrons.
Thus the excited electron losses energy to the lattice in scattering events until
its velocity reaches the thermal equilibrium velocity of other conduction band
electrons.
The electron and hole created by this absorption process are excess carriers:
since they are out of balance with their environment, they must even eventually
recombine.
While the excess carriers exit in their respective bands, however, they are free
to contribute to the conduction of material.

I0 It

If a beam of photons with hv>Eg falls on a
semiconductor, there will be some predictable amount of
absorption, determined by the properties of the material.
The ratio of transmitted to incident light
intensity depends on the photon wavelength and the
thickness of the sample.
let us assume that a photon beam of intensity I0 (photons/cm-2-s) is directed
at a sample of thickness l as shown in Fig. 4-2.
The beam contains only photons of wavelength  selected by
monochromator.
As the beam passes through the sample, its intensity at a distance x from
the surface can be calculated by considering the probability of absorption with in
any increment dx.
The degradation of the intensity –dI(x)/dx is proportional to the intensity remaining
at x:  dI( x)  aI( x) (4.1)
dx

The solution to this equation is
I( x) I eax (4.2)
0
and the intensity of light transmitted
through the sample thickness l is
It I eal (4.3)
0
The coefficient a is called the
absorption coefficient and has units of
cm-1. Figure 4-3 Dependence of optical
absorption coefficient a for a
This coefficient varies with the photon
semiconductor on the wavelength
wavelength and with the material. of incident light.

Fig. 4-3 shows the plot of a vs. wavelength.
There is negligible absorption at long wavelength (hv small) and
considerable absorptions with energies larger than Eg.
The relation between photon energy and wavelength is E=hc/. If E is
given in electron volt and  is micrometers, this becomes E=1.24/.

Steady State Carrier Generation
The thermal generation of EHPs is balanced by the recombination rate that means
[Eq. 3.7] g (T ) ar n2 ar n p (4.10)
i 0 0
If a steady state light is shone on the sample, an optical generation rate gop will be
added to the thermal generation, and the carrier concentration n and p will increase to
new steady sate values.
If n and p are the carrier concentrations which are departed from equilibrium:
g (T )  gop  a r np  a r (n0  n)( p0  p) (4.11)
For steady state recombination and no traping, n=p; thus Eq. (4.11) becomes
g (T )  gop  a r n0 p0  a r [(n0  p0 )n  n 2 ] (4.12)
Since g(T)==arn0p0 and neglecting the n2, we can rewrite Eq. (4.12) as
gop  a r [(n0  p0 )n]  (n /  n ) (4.13)

1
where,  n  is the carrier life time.
a r (n0  p0 )

The excess carrier can be written as n  p  gop n (4.14)

Quasi-Fermi Level

The Fermi level EF used in previous equations is meaningful only when no excess
carriers are present.
The steady state concentrations in the same form as the equilibrium expressions by
defining separate quasi-Fermi levels Fn and Fp for electrons and holes.
The resulting carrier concentration equations
( Ei  F p ) / KT
n  ni e( Fn  Ei ) / KT ; p  ni e (4.15)

can be considered as defining relation for the quasi-
Fermi levels.

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