Transcript Slide 1

LECTURE 2
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CONTENTS
MAXWELL BOLTZMANN STATISTICS
FERMI- DIRAC STATISTICS & ITS
DISTRIBUTION
SEMICONDUCTORS AND ITS
CLASSIFICATION and
FERMI ENERGY LEVEL DISTRIBUTION
IN INTRINSIC SEMICONDUCTORS.
Maxwell Boltzmann Statistics (Classical law)
This law states that, the total fixed amount of energy
is distributed among the various members of an assembly
of identical particles in the most proable distribution.
The Maxwell Boltzmann law is
Where
ni 
gi
e (   E i
)
ni ─ number of particles having energy Ei.
gi ─ number of energy states.

 EF
kT
and
 
1
kT
(Here k ─ Boltzmann constant; T ─ Absolute temperature of the
gas, EF ─ Fermi energy)
Therefore, n 
i
e
gi
( Ei  E F ) / kT

 Particles are distinguishable.
 Classical particles can have any spin.
 Particles do not obey Pauli’s exclusion principle.
 Any number of particles may have identical energies.
Fermi-Dirac Statistics (Quantum law)
This statistics applicable to the identical, indistinguishable
particles of half spin.
These particles obey Pauli’s exclusion principle and are
called fermions (e.g.) Electrons, protons, neutrons …,
In such system of particles, not more than one particle can
be in one quantum state.
Fermi Dirac Distribution Law is
ni 
gi
(e 
 E i
) 1
or ni =
e
gi
( E i  E F ) / kT
1
Example
Let us consider two particles a and a. Let if, these two
particles occupy the three energy levels (1,2,3). The number of
ways of arranging the particles 31=3 (not more than one
particle can be in any one state)
Energy
level
Possible distribution in various
energy level
1
a
A
-
2
a
-
a
3
-
A
a
Fermi Energy (EF) and Fermi-Dirac Distribution
Function f(E)
Fermi Energy (EF)
Fermi Energy is the energy of the state at which the
probability of electron occupation is ½ at any temperature
above 0 K.
It is also the maximum kinetic energy that a free
electron can have at 0 K.
The energy of the highest occupied level at absolute
zero temperature is called the Fermi Energy or Fermi Level.
The Fermi energy at 0 K for metals is given by
 3N 
EF   
 
2/3
 h2 
 
 8m 
When temperature increases, the Fermi level or Fermi
energy also slightly decreases.
The Fermi energy at non–zero temperatures,
E F  E F0

2
1  
 12

 kT

 EF
 0




2




Here the subscript ‘0’ refers to the quantities at zero kelvin.
Fermi-Dirac Distribution Function f(E)
The free electron gas in a solid obeys Fermi-Dirac
statistics.
Suppose in an assemblage of fermions, there are M(E)
allowed quantum states in an energy range between E and
E+dE and N(E) is the number of particles in the same range.
Then,
The Fermi-Dirac distribution function is defined as,
N (E)
1

M ( E ) 1  exp(E  EF )/kT
N(E) / M(E) is the fraction of the possible quantum
which are occupied.
states
The distribution of electrons among the levels is
described by function f (E), probability of an electron occupying
an energy level ‘E’.
If the level is certainly empty, then f(E) = 0.
Generally the f(E) has a value in between zero and unity.

When E< EF (i.e.,) for energy levels lying below EF,
(E –EF) is a negative quantity and hence,
1
1
f(E)

1

1 0
1 e
That means all the levels below EF are occupied by
the electrons.
Fermi Dirac distribution function at different temperatures

When E > EF (i.e.) for energy levels lying above EF,
(E – EF) is a positive quantity
1
1
f(E)

0

1 
1 e
This equation indicates all the levels above EF are
vacant.
At absolute zero, all levels below EF are completely filled
and all levels above EF are completely empty.This level,
which divides the filled and vacant states, is known as the
Fermi energy level.
 When E = EF ,
f (E) 
1
1
1


1 e0 11 2
, at all temperatures
The probability of finding an electron with energy equal
to the Fermi energy in a metal is ½ at any temperature.
At T = 0 K all the energy level upto EF are occupied
and all the energy levels above EF are empty .
When T > 0 K, some levels above EF are partially filled
while some levels below EF are partially empty.
Semiconductors
Introduction
 The materials are classified on the basis of conductivity
and resistivity.Semiconductors are the materials which has
conductivity, resistivity value inbetween conductor and
insulator . The resistivity of semiconductor is in the order of
10−4 to 0.5 Ohm-metre.
 It is not that, the resistivity alone decides whether a
substance is a semiconductor (or) not , because some
alloys have resistivity which are in the
range of
semiconductor’s resistivity. Hence there are some properties
like band gap which distinguishes the materials as
conductors, semiconductors and insulators.

semi-conductor is a solid which has the energy band
similar to that of an insulator. It acts as an insulator at
absolute zero and as a conductor at high temperatures and
in the presence of impurities.
Semiconductors are materials whose electronic properties
are intermediate between those of metals and insulators.
These intermediate properties are determined by the
crystal structure, bonding characteristics and electronic energy
bands.
They are a group of materials having conductivities between
those of metals and insulators.
Classification of Semiconductors According
to their Structure
Amorphous semiconductors-have poor electrical
characteristics.
Polycrystalline semiconductors – have better electrical
characteristics and lower conductivity.
Single crystal semiconductor – have superior electrical
characteristics and higher conductivity. The majority of the
semiconductor devices, single-crystal materials are used.
Classification of Semiconductor According
to the nature of the current carriers
Ionic semi conductor, in which conduction takes
place through the movement of ions and.
Electronic semiconductor, in which conduction
takes place through the movement of electrons and
no mass transport, is involved
Classification of semiconductors According to
the constituent atoms
Elemental semiconductor: All the constituent atoms
are of the same kind (i.e) composed of single
species of atoms. (eg) germanium and silicon.
Compound semiconductor: They are composed of
two or more different elements (eg) GaAS, AlAs
etc.,
Crystal structure of silicon and germanium
The structure of Si and Ge, which are having
covalent bonding. Covalent bondings are stereo
specific; i.e. each bond is between a specific pair of
atoms.
The pair of atoms share a pair of electrons (of
opposite magnetic spins).
Three dimensional representation of the structures Si,
and Ge, with the bonds shown in below figure, the region of
high electron probability (shaded).
(a)
(b)
Structure of (a) silicon and (b) germanium crystals
All atoms have coordination number 4; each material has
an average of 4 valence electrons per atom, and two electrons
per bond.
Each atom of a material is coordinated with its
neighbours.
(a)
(b)
Structure of (a) silicon and (b) germanium crystals
The thermal vibrations on one atom influence the
adjacent atoms; the displacement of one atom by
mechanical forces, or by an electric field, leads to
adjustments of the neighbouring atoms.
The number of coordinating neighbours that each atom
has is important. Covalent bonds are very strong.
(a)
(b)
Structure of (a) silicon and (b) germanium crystals
Some important properties of elemental semiconductor
Property
Silicon (Si)
Germanium (Ge)
Atomic number
14
32
Atom/m3
5.02  1028
4.42  1028
Electronic shell configuration
1s22s22p63s23p2
1s22s22p63s23p63d104s24p2
Atomic weight
28.09
72.6
Crystal structure
Diamond
Diamond
Breakdown field (V/m)
~ 3.0107
~ 107
Density (gm/m3)
2.329106
at 298K
5.3234106 at 298 K
Energy gap (eV)
1.12 at 300 K
1.17 at 77K
0.664 at 291 K
0.741 at 4.2K
Dielectric constant
11.7 at 300K
16.2 at 300K
Intrinsic carrier concentration (m3) at 300
K
1.02  1016
2.33  1019
Lattice constant (Å)
5.43107 at 298.3K
5.65791 at 298.15K
Melting point (C)
1412
937.4
Thermal conductivity [Wm1(C1)]
131 at 300K
60 at 300K
Mobility of electrons (m2V1s1)
0.135 at 300K
0.39 at 300K
Mobility of holes (m2V1s1)
0.048 at 300K
0.19 at 300K
Intrinsic Semiconductors
In semiconductors and insulators, when an external
electric field is applied the conduction is not possible as there
is a forbidden gap, which is absent in metals.
In order to conduct, the electrons from the top of the
full valence band have to move into the conduction band, by
crossing the forbidden gap.
The field that needs to be applied to do this work will
be extremely large.
Eg: Silicon where the forbidden gap is about 1 eV.
The distance between these two locations is about 1 Å (1010
m).
A field gradient of approximately 1V/ (1010 m) = 1010Vm1
is necessary to move an electron from the top of the valence
band to the bottom of the conduction band.
 The other possibility by which this transition can be
brought about is by thermal excitation.
 At room temperature, the thermal energy that is available
can excite a limited number of electrons across the energy
gap. This limited number accounts for semi-conduction.
 When the energy gap is large as in diamond, the number
of electrons that can be excited across the gap is extremely
small.

In intrinsic semiconductors, the conduction is due to
the intrinsic processes (without the influence of impurities).

A pure crystal of silicon or germanium is an intrinsic
semiconductor. The electrons that are excited from the top of
the valence band to the bottom of the conduction band by
thermal energy are responsible for conduction.

The number of electrons excited across the gap can
be calculated from the Fermi-Dirac probability distribution.
1
f(E) =
1  {exp[ E  EF ) / k BT ]}
#
The Fermi level EF for an intrinsic semiconductor lies
midway in the forbidden gap.
#
The probability of finding an electron here is 50%, even
though energy levels at this point are forbidden.
#
Then (EEF) is equal to Eg /2,
where Eg is the magnitude of the energy gap.
# For a typical semiconductor like silicon, Eg = 1.1 eV, so
that (EEF) is 0.55 eV, which is more than twenty times
larger than the thermal energy kBT at room temperature
(=0.026 eV).
Conduction band
Eg
EF
E
Valence band
O
O.5
F(E)
1.0
The Fermi level in an intrinsic semiconductor lies
in the middle of the energy gap.
#
The probability f(E)of an electron occupying energy level
E becomes f(E) = exp(Eg / 2kBT ).
#
The fraction of electrons at energy E is equal to the
probability f(E). The number n of electrons promoted across the
gap,
n = N exp(Eg / 2kBT)
where N is the number of electrons available for excitation from
the top of the valence band.
The promotion of some of the electrons across the gap
leaves some vacant electron sites in the valence band. These
are called holes.
An intrinsic semiconductor contains an equal number of
holes in the valence band and electrons in the conduction
band, that is ne = nh.
Under an externally applied field, the electrons, which
are excited into the conduction band by thermal means, can
accelerate using the vacant states available in the conduction
band.
At the same time, the holes in the valence band also
move, but in a direction opposite to that of electrons.
The conductivity of the intrinsic semiconductor depends
on the concentration of these charge carriers, ne and nh.
In the case of metals, the drift velocity acquired by the
free electrons in an applied field.
The mobility of conduction electrons and holes, e and
h, as the drift velocity acquired by them under unit field
gradient.
The conductivity  of an intrinsic semiconductor as
i = ne e e + nh e h
where e is the electronic charge, ne and nh are
concentrations of electrons and holes per unit volume.
Fermi level
The number of free electrons per unit volume in an intrinsic
semiconductor is
3/ 2
*
me kT 
 EF

2
 exp
 2
n  2 
 h

 Ec 

 kT 
The number of holes per unit volume in an intrinsic
semiconductor is
p=
 2mh
2

3
 k T 2
 EV  E F 
.
exp



KT
h 2 


Since n = p in intrinsic semiconductors.
3
3
 2 me* k T  2
 2 mhk T 2
E

E


 Ev  EF 
F
c
 exp


2

2
exp



2
2




kT
kT
h
h








m 

e
3
2
E
exp F
 EC    3 2 
 Ev  E F 
  mh exp


kT
 KT 
3
or
e
2 EF
kT
 mh  2
 Ev  Ec 

 
exp


* 
kT
m


 e 
Taking log on both sides,
 mh
2 EF
3
 loge  *
kT
2
 me



  loge  exp  Ev  Ec 

 kT 


 mh   Ev  Ec 
2EF 3
 loge  *   
 m   kT 
kT
2
 e 
or Ef =
 m h
3kT
log e  *
m
4
 e
  Ev  Ec 


 
2


If we assume that,
 E  Ec 
EF   v

2


*
m*

m
e
h
[ since loge1 = 0]
Thus, the Fermi level is located half way between the
valence and conduction band and its position is independent
of temperature. Since mh* is greater than me*, EF is just
above the middle, and rises slightly with increase in
temperature
Conduction band
Ec
(Eg /2)
(b)
(a)
EF
Eg
Ev
Valence band
Position of Fermi level in an intrinsic semiconductor at various
temperatures
(a) at T = 0 K, the Fermi level in the middle of the forbidden gap
(b) as temperature increases, EF shifts upwards