Download unit 1 transport properties

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UNIT 1 TRANSPORT PROPERTIES
Structure
1.0
Introduction
1.1
Objectives
1.2
The Boltzman equation
1.3
Electrical conductivity
1.4
1.3.1
Factors affecting electrical conductivity
1.3.2
Calculation of relaxation time
1.3.3
Impurity scattering
1.3.4
Ideal resistance
1.3.5
Carrier mobility
General transport coefficients
1,4.1
Thermal conductivity
1.4.2
Thermoelectric effects
1.4.3
Lattice conduction & Phonon drag
1.5
The Hall effect
1.6
The two band model
1.7
Magneto resistance
1.8
Let Us Sum Up
1.9
Check Your Progress: The key
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1.0 INTRODUCTION
We have considered in the free electron theory of metals in which free electrons were
treated as an ideal gas of free particles which when in thermal equilibrium, obey
Maxwell-Boltzmann statistics. Now the problem is that what happens when a nonequilibrium state is established by allowing electric or thermal currents to flow, i.e.,
the problem is to investigate how the equilibrium distribution would be modified by
small electric or thermal current. It is also necessary to consider the kinetic behavior
of the electrons as being that of free particles subjected to instantaneous collisions
which serve to return the distribution to equilibrium condition, and to express the final
result for electrical and thermal conductivity in terms of mean free path between these
randomizing collisions. In a state of steady flow of heat or electricity, the distribution
function of velocity components and spatial coordinates of the electrons will be
Transport Properties, Semiconductor Crystals and Superconductivity
different from that in thermal equilibrium in the absence of flow. The theory of
transport phenomena is concerned with determining this distribution function for
given external fields.
In the calculation of this distribution function, two new, features appears which are of
no interest in thermal equilibrium. The first feature is that the external fields
accelerate the particles. This acceleration will be reflected in a change of the
distribution function away from its equilibrium value. The second feature is that the
effect of certain terms in the Hamiltonian of the crystal which have been neglected is
now more important. These terms, including the coupling between the electrons and
lattice, act to produce transitions, of scattering of electrons between one state and
another. The scattering terms have no net effect on the equilibrium state because in
equilibrium, the transition rates between any two states exactly balance. However in
the presence of external field, the equilibrium state is destroyed and the scattering
then does have some effect. In another words, we can say that scattering tends to
return the system to equilibrium while the fields to pull system away from
equilibrium.
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1.1 OBJECIVES
The main aim of this unit is to study the transport properties. After going through this
unit you should be able to:

Understand the Boltzman equation and Relaxation time.

Know the electrical conductivity, ideal resistance, carrier mobility, transport
coefficients and thermal conductivity.

Learn the various thermoelectric effects viz., Peltier effect, Tho mson effect
etc. and lattice conduction, Phonon drag.

Know the Hall effect and the two band model.

understand the magneto resistances.
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1.2 THE BOLTZMAN EQUATION
Transport phenomena such as the flow of electric current in solids, involve two
characteristic mechanisms with opposite effects: the driving force of the external
fields and the dissipative effect of the scattering of the carriers by phonons and
defects. The interplay between the two mechanisms is described by the Boltzman
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Transport Poperties
equation. With the help of this equation one may investigate how the distribution of
carriers in thermal equilibrium is altered in the presence of external forces and as a
result of electron scattering processes. In thermal equilibrium and with no external
fields, this distribution function is simply the Fermi distribution
f 0 E ( K ) 
1
(1)
e
1
Let us consider a system of particles that is in dynamical equilibrium under external
E ( K )  E F / kT
forces. For example, the system may consist of electrons in metal that is acted upon
by stationary external electric and magnetic fields. When the steady state current is
flowing, the system is in dynamical equilibrium of the type we wish to consider.
Suppose x, y, z, are the coordinates of an electron and vx, vy, uz are velocity
components, then the distribution function is given by
f (x, y, z, ux ,uy, uz,)
To derive the equation consider a region of space about the point (x, y, z, ux, uy, uz).
The number of particles having position co-ordinates in the range from x to x+dx, y to
y+dy, z to z+dz and velocity co-ordinates in the range u, to ux+duy uy+duy uz to uz +
duz can be represented by the function .
f (x, y, z, , ux ,uy, uz) dx dy dz dux duy duz.
(2)
There may be variation of the function with time due to the two independent ways:
(1) Drift variation: The function may vary because the particles are moving from
one region of space to another and are accelerated by external field during motion.
Consider the group of particles at an instant t + dt, that are drifted to a cell of phase
space corresponding to the co-ordinates (x, y, z, , ux ,uy, uz), The number of particles is
the
same
as
were
in
a
cell
located
at
x - u x dt , y  u y dt , z  u z dt , u x   x dt , u x   y dt , u z   z dt at a time t. Here  x ,  y and
 z are the components of acceleration. The relationship holds for a small time interval
dt for which the collisions have a negligible effect on the distribution. Thus the
change due to drift in number of particles having co-ordinates x, y, z and velocity ux,
uy and uz in time dt is .
(f )d  f ( x  u x dt , y  u y dt , z  u z dt , u x   x dt , u y dt , u z   z dt , t )  f ( x, y, z, u x , u y , u z , t )
(3)
Using Taylor,s expression and retaining only first order terms in the limit
dt - 0.the above eq. may be written as.
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Transport Properties, Semiconductor Crystals and Superconductivity

f
f
f
f
f
f 
   ux
 uy
 uz
x
 y
z
dt

dx
dy
z
u x
u y
u z 

Consequently the rate of change of f caused by drift is
f
f
f
f
f
f
 df 
 uy
 uz
x
 y
z
   u x
dx
dy
z
u x
u y
u z
 dt  d
(2) Collisions or scattering interactions: This is due to relatively discontinuous
changes in velocity that accompany collisions.
If
 u x ,u y , , u z ; u ' x , u ' y , u ' z du ' x du ' y du ' z
Represents the probability per unit time that a particle will change its velocity from ux,
uy, uz to a value having components in the range extending from u’x to u’x+du’z etc.
the total number the velocity of which alters from ux, uy, uz to some other value is
a = f(x, y, z, ux, uy, uz)
  u
x,
u y , , u z ; u ' x , u ' y , u ' z du ' x du ' y du ' z
similarly, the number the velocity of which changes to ux, uy, uz from another values is
b   f (u"x , u" y , u"z ) (u"x , u" y , u"z u x , u y , u z )du"x , du" y , du"z
Thus the rate of change of f caused by collisions is
 df 
  .  b  a.
 dt coll
The total rate of change is sum of drift variation and scattering interactions. Hence for
 df   df 
,     0
 dt  d  dt coll.
equilibrium the sum should vanish i.e.
Substituting the values from equations , we have
ux
Or
f
f
f
f
f
f
 uy
 uz
x
 y
z
 (b  a)  0
x
y
z
u x
u y
u z
 ux
f
f
f
f
f
f
 uy
 uz
x
 y
z
 (b  a)
dx
dy
dz
u x
u y
u z
Equation is Bolltzmann’s transport equation.
Now we shell consider two cases:
(A)
When the metal is homogeneous i.e., at the constant temperature in a field free
space, then
f f f
, ,
 0,
x y z
or
and
 x , y , z ,  0

 
df
dt coll.
0
a = b,
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Transport Poperties
Which shows that the number of particles that leave and enter a given volume of
momentum space as a result of collision are equal.
(B)
For a heterogeneous medium , i.e., if there is temperature gradient.
f f f
, ,
0
x y z
Hence
 
df
dt coll.
0
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1.3 ELECTRICAL CONDUCTIVITY
The electrical conductivity of solids was first demonstrated by Stephan Gray in 1729.
It is the ability of a material to conduct electricity. The resistance (R) offered by a
conductor to the flow of electric charge is found to be directly proportional to the
length (l) and inversely proportional to the area of cross section (a) of the conductor.
Therefore,
l
R  ( )
a
where  is the proportionality constant called the electrical resistivity. Then 1/  is
the electrical conductivity (  ). The magnitude of the electrical conductivity can be
determined by (1) the density of charge carriers, (2) the charge on the carrier, (3) the
average drift velocity of the carriers per unit electric field. The electrical conductivity
 may be defined as the quantity of electricity that flows in unit time per unit area of
cross section of the conductor per unit potential gradient.
If q is the quantity of electricity that flows through a conductor of cross sectional area
q  AEt
A in time t under a potential gradient E, then
Or when t=1,

q
i

AE E

ne 2
,
6 K BT
 
ne 2
6 K BT
This expression shows that different conductivities of different materials are due to
different number of free electrons.
1.3.1 Factors affecting electrical conductivity
The main factors affecting the electrical conductivity of solids are (a) temperature (b)
defects, e.g., impurities, and (c) electromagnetic radiation. In metals, the charge
carriers are electrons; the electronic concentration is large and constant, and is almost
unaffected by the presence of impurities. To understand the role of temperature, we
have to consider the effect of relaxation time  . At low temperature, the  value is
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Transport Properties, Semiconductor Crystals and Superconductivity
large since the metal ions vibrate simple harmonically. As the temperature is
increased, the metal ions vibrate more vigorously, and an harmonically and thus act as
better
scattering
canters
than
before;
as
a
result,
the
mean
free
path/mobility/relaxation time decreases and  decreases with increasing temperature.
On the other hand, semiconductors are insulators at temperatures <<Eg/kb. At high
temperatures, the thermal agitation promotes electrons into the upper, empty
conduction band and the conductivity rises exponentially with increasing temperature.
Introduction of some impurities (one part per billion) in an pure semiconductor gives
rise to a large increase in the conductivity. This type of conductivity is termed as
extrinsic semi conductivity.
The electromagnetic radiation can also affect the electrical resistivity of intrinsic
semiconductor. This phenomenon is called photoconductivity.
Check Your Progress 1
Note: a) Write your answers in the space given below.
b) Compare your answers with the ones given at the end of the unit.
(1) Derive the Boltzman equation.
(2) Explain the electrical conductivity. How does it vary with temperature ?
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1.3.2
Calculation of relaxation time
The relaxation time is closely related to the mean free time between two successive
collisions of the electron with lattice. We know from the Newton’s second law of
motion, the force on a particle of mass m is F  m  x , and the force on a charged
where m is the effective
particle of charge q will be qE x , then m x  qEx
mass.integration gives the velocity at time t,
x 
qE x t
 v x0
m
The second term is the initial velocity of the electron. We now assume that after a
certain time 2 the electrons suffer a collision and that the result of collision is to
decrease to zero the excess velocity of the electron acquired from the field.
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Transport Poperties
After the collision, the electron accelerates for another period of time 2 . The time 
is called relaxation time. The average velocity increment vx of the accelerating atom
during its time of flight is simply half its final velocity increment.
Jx = Nqvx ,
vx 
qE x
m
N is the no. of electrosns.
Jx 
The current can be written as
Then the conductivity is  
Nq 2 Ex
m
Nq 2
m
Fig. 1. Relaxation time
Here the quantity relaxation time is half the time between collisions. Remembering
that in the Fermi distribution only the electrons near the Fermi energy can participate
in collisions, the relaxation time is then the time between collisions for those
electrons.Then the shift of the Fermi sphere is given by
or the shift in the velocity of the electrons is given by
p0  qEx
v 
qE x
m
The relaxation time is also related to another parameter  , the mean free path of the
electrons capable of making collisions. If the velocity gained in the field is negligible
as compared with the Fermi velocity, then
  n f 2 
where nf is the speed of the electrons at the top of Fermi distribution.
1.3.3
Impurity scattering
By impurities we mean foreign atoms in the solid which are efficient scattering
centers, when they have a net charge. Ionized donors and acceptors in a
semiconductor are a common example of such impurities. The amount of scattering
due to electrostatic forces between the carrier and the ionized impurity depends on the
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Transport Properties, Semiconductor Crystals and Superconductivity
interaction time and the number of impurities. Larger impurity concentrations results
in a lower mobility. The dependence on the interaction time helps to explain the
temperature dependence. The interaction time is directly linked to the relative velocity
of the carrier and the impurity which is related to the thermal velocity of the carriers.
This thermal velocity increases with the ambient temperature so that the interaction
time increases, the amount of scattering decreases, which results in a mobility
increase with temperature. To first order the mobility due to impurity scattering is
proportional to, where N1 is the density of charged impurities.
Check Your Progress 2
Note: a) Write your answers in the space given below.
b) Compare your answers with the ones given at the end of the unit.
(2) What do you mean by relaxation time ?
(2) Describe the impurity scattering.
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1.3.4 Ideal resistance
If M is the mass of the atom, and (-fx) is the restoring force on the displaced atom, the
equation of the atomic oscillator is
M
d 2x  f 
  x  0
dt 2  M 
d 2x
 fx  0 ,
dt 2
1/ 2
where
 f 
.x is the amplitude of vibration.
 M 
  2  
The potential energy of the oscillator will be

1
1
( M 2 x 2 )  M 4 2 2 x 2
2
2

then the potential energy per degree of freedom will be
= 2 2 M 2 x 2 =
k BT
2
(1)
At high temperature, as the Einstein and Debye frequencies are essentially the same,
and most of the modes are the high frequency modes, then
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Transport Poperties
 2   E2   D2
(2)
with h D  kB D
Uing eq. (1) & (2) We have x 2 
 2T
Mk B D2
(3)
 1  cons tan t 
From the kinetic theory man free path is given by    
x2
n
We know that  

,

Obviously tT
At low temperature we use the Debye model. At low temperature since the lattice
vibrations begin to die out, the scattering cross section would fall and we would
therefore expect that their contribution to the resistivity, denote by  p will decrease at
low temperatures, eventually becoming zero at 0 K. The larger the amplitude of
vibration at any temperature, the greater will be  p . Since this amplitude depends on
the inverse of the Debye temperature  D  , it is to be expected that  p will be less for
metals with a high  D  , and vice-versa. The arrangement of point defects in a crystal
resistivity  0 , which they produce, would be expected to be constant. Their
contribution to the resistivity is temperature independent, but it does, of course,
increase with the impurity concentration.
The total resistivity  is therefore
   0   p T 
Fig. 2. Variation of the electrical resistivity with temperature.
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Transport Properties, Semiconductor Crystals and Superconductivity
This is shown in Fig. 2. it is very clear from here that  at first decreases linearly
with T, and at low temperatures , it flattens off to a constant value, equal to  0 , which
is called the residual resistivity. It is clear that, for a very pure sample,  0 will be
very small, whereas for an impure specimen it will have a high value.
The probabilities of electrons being scattered by photons and by impurities are
additive, therefore we may write
1


1
p

1
0
in this equation the first term on the right is due to phonons and the second is due to
impurities. The former is expected to depend on T and the latter on impurities, but not
on T. The    0   p T  simple addition of  0 and  p T  in this equation is often
referred to as Matthiessen’s rule. The thermally induced part of the resistivity,  p , is
sometimes known as the ideal resistivity and the resistance is called ideal resistance,
whereas the resistivity due to impurities and defects is summed up in the residual
resistivity (  0 ).
1.3.5 Carrier mobility
The conductivity of a solid has been expressed as   ne where the mobility  is
the average velocity by a carrier in a unit electric field. The electrons and holes in a
semiconductor are in a rapid random motion because of their thermal energies. It is
the additional velocity introduces by an external electric field which constitutes the
electric current observed. The net velocity due to the field is called drift velocity. Also
 eE 
an electron in a perfect crystal experiences an acceleration  
 in a field E which
 m
implies that its drift velocity continuously increases. This can not be the case because
according to Ohm’s law,
 nevd 
 = constant
 E 
  j/E 
This means that the average velocity vd must be a constant. Hence it is necessary to
assume that the electron loses energy in collisions with the crystal structure so that it
v 
has a constant average velocity or mobility,    d  Suppose that the average time
E
between such collisions is  and that at each collision the electron loses all the energy
it gained from the field subsequent to the previous collision. Assuming the 
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Transport Poperties
independent of electron velocity, then the number of collisions per second is 1/  and
v 
the rate of change of velocity is  d  . Under steady state conditions this rate of
 
change must be equal to the acceleration due to the field,
or
eE / m  vd /   E /  .
Therefore,   e / m . The parameter  is called relaxation time.
The imperfections disrupt the periodicity of atomic array and cause the electrons to be
scattered. The two important causes of scattering in semiconductors are (1) atomic
vibrations (phonons) (2) ionized impurity atoms. The effect that these imperfections
have on the nobilities of electrons or holes can be determined most easily by
calculating the corresponding relaxation time. An actual calculation of the mobility
 L due to scattering by phonons is the T 3 / 2 temperature dependence. Similarly the
mobility due to ionized impurities scattering  i is T 3 / 2 temperature dependence. The
actual mobility is given by
1


1
L

1
i
i.e.,
1


a
T
3/ 2
 bT 3 / 2
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1.4 GENERAL TRANSPORT COEFFICIENTS
1.4.1 Thermal conductivity
In order to discuss the thermal conductivity of metals we suppose that there exists a
temperature gradient across the specimen. The transport of energy in the metals is due
to conduction electrons and lattice waves. Here we shall consider the thermal
conductivity only due to conduction electrons although lattice conduction may
become important under certain circumstances such as low temperature, high
magnetic field, large impurity contents etc.
We will now consider the application of the Boltzmann equation to effects that
involve heat transport. Let there be a thermal gradient dT/dx in a metal and a current
density Q. In the measurement of thermal conductivity the specimen is electrically
insulated from its surroundings; thus the current vanishes but not the electric fields.
This is due to the fact that the temperature gradient produces a drift velocity of the
electrons, and a small electric fields. Is set up internally to counteract the drift
velocity. Thus the Boltzmann transport equation beside the thermal gradient dT/dx
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Transport Properties, Semiconductor Crystals and Superconductivity
includes a term containing an electric field  x . In this case the Boltzmann equation
can be written as
df
f e x f  f  f 0 
 0  u x


dT
x m u x
T
Or
 f  fo 
f
 f  T 
u x     e x

px
T
 T  x 
When the electric field and T / x are small, we can replace
(1)
f
f
f
and
by o
x
x
p x
f o
respectively. Substituting in above equation we have
p x
f  f o  e xT f o / px  u x f o x.
Now in view of fo being the Fermi-Dirac distribution function we write
and

 T
f o
f o T  
1



x
T x T 1  exp{( E  E F ) K BT }  X
 fo  
f o f o 
E  E F  T
 E F
.
e x  



p x E 
T
 T
 X

.

Now considering that EF is independent of T, we get
f 
 E E  T
f  f o   x o e x    F 
E 
 T T  X
(2)
We know that electric current density Ic and thermal current density Qx are defined as
 2e 
I x   3      x  f  f o dp x dp y dp z ,
h 
 2e 
Qx   3      x  f  f o Edpx dp y dp z
h 
Where E is the energy of an electron.
Substituting the value of (f-f0) from above equations, we get
and
 2e 
 E E  T
2 f 
I x   3      x o e x    F 
dp x dp y dp z
E 
h 
 T T  X
f 
 2e 
 E E  T 
2
Qx   3      x  E o e x    F 
 dpx dp y dpz
E 
h 
 T T  X 
Assuming that is only a function of the energy and not of the direction of motion, we
see that the integrals in equation are functions of energy alone. The triple integrals
2
may be transformed into single integrals by replacing  x by  2 / 3 and dpx dpy dpz
by 4p 2 dp . thus
12
Transport Poperties
Ix  
16e(2m)1/ 2
3h 3

16e(2m)1/ 2
Qx  
3h 3

E 3 / 2 ( E )
0


0
f o
E

 E E F  T 
e x   T  T  X  dE




f o
E

 E E F  T 
e x   T  T  X  dE




E 5 / 2 ( E )
Introducing a set of integral JN
16e(2m)1 / 2
3h 3
f o
dE , N  1,2,3....
0
E
 J E J  T
I x  e 2 F 1 
 e 2 J1 x
T
T

X


 J E J  T
Qx   2 F 2 
 e 2 J 2 x .
T
T

X


Now we calculate Qx under the condition Ix = o, because the thermal conductivity of
metals is defined as the rate of energy flow divided by thermal gradient when Ix=O.
 Qx 
K  
 I x O
 (T / x) 
JN 


From equation when Ix=O.,
 ( E ) E N 1
J
J 2  T
Qx   2  2 
 T J1T  x
Now
 J J  J 22 
Comparing the above equations K   1 3
.
 J1T 
The value of integral JN is given by
JN  

  2 E N 1/ 2
16 (2m)1/ 2  N 1  2
K BT 2 
EF 
6
E 2


Substituting equation in equation we get K 
 E  E



F 

 2 K B 2TN F
3
m
Where N is the density of electrons.
In most of the metals,  F varies about 1/T, and hence K is nearly temperature
independent. If the metal contains impurities, then electron-phonon scattering as well
as electron impurity scattering takes place. If the two scatterings are considered to be
independent to each other, then the total conductivity K can be represented by
1
1
1


K K1 K '
Where Kl is the contribution arising from electron-lattice scattering and Kl is the
contribution from electron impurity scattering. This expression shows that the
impurities decreases the thermal conductivity.
13
Transport Properties, Semiconductor Crystals and Superconductivity
1.4.2 Thermoelectric effects
The thermoelectric effect is the direct conversion of temperature differences to
electric voltage and vice versa. Generally, the term thermoelectric effect encompasses
three separately identified effects, the Seebeck effect, the Peltier effect, and the
Thomson effect.
Let us consider the two metals A and B having different electron density let the
electron density in A is greater than electron density in metal B. Now the electronic
pressure in A will be greater than in B. Due to the difference in electronic pressure, the
electrons diffuse from A to B. This makes A positive and B negative. Thus a potential
difference is created at the junction of two metals. When this potential difference
reaches a certain value, it prevents the migration of electrons from A to B and a state
of equilibrium is set up. This explains that how a potential difference is created at the
junction of two metals . We shall now apply this general conclusion to the three
effects of thermo electric phenomena.
Seebeck effect
The seebeck effect is the conversion of temperature differences directly into
electricity. In this effect the thermoelectric EMF is created in the presence of a
temperature difference between two different metals or semiconductors. The voltage
created is of order of several microvolts per kelvin difference.
T2
B
A

vV
_
B
T1
Fig. 3. Seebeck effect
In the circuit, the voltage developed can be derived from:
T2
V   S B T   S A T dT
T1
SA and SB are the Seebeck coefficients of the metals A and B as a function of
temperature, and T1 and T2
are the temperatures of the two junctions. If the
coefficients are constant for the measured temperature range, then the above formula
can be written as:
V  (SB  S A ).(T2  T1 )
14
Transport Poperties
Peltier effect
In case of Peltier effect an external potential difference is applied to the Junction i.e.,
current is allowed to flow from A to B. Due to this current flow, there will be a
transfer of electrons from B to A. As the electron density in A is greater than in B,
hence certain amount of work is done against the electronic pressure difference. This
involves the absorption of some energy at the junction which in consequence gets
cooled. When the direction of the current is reversed, the electrons flow from A to B
which make the energy available at junction in the form of heat i.e., the junction gets
heated. The Peltier coefficient  is defined as the amount of energy liberated or
absorbed when unit charge passes through the junction
The expression for  can be derived as follows:
1
According to the kinetic theory, the electronic pressure is given by p  mn 2 ,
3
Where m is the mass of electron, n is the number of electrons per unit volume of the
conductor and  2 is the mean square velocity of the electron .
We know that p 
1
1
mn 2Tor mn 2  T
2
2
Then the above equation Equation can be written as p 
2 1
 2
n m 2   nT
3 2
 3
(1)
Now charge per unit volume is ne where e is the charge on the electron. The volume V
corresponding to a unit charge is given by v 
1
.
ne
Now the work done when unit charge passes through the junction will be the amount
of work done by the electrons in moving from higher to lower electronic pressure.
V2
This is equal to Peltier coefficient  , hence    pdv
V1
Where V1 and V2 represent volumes corresponding to unit charge. These are equal to
1/n1e and 1/n2e by relation where n1 and n2 be the electron densities in the two metals
respectively. Substituting the values of p from equation (1) in equation we get .
V2
 
V1
2 1
Tdv
3 eV
 
Again
V2 1 / n2 e n1


V1 1 / n1e n2
2 T n1
.
3 e n2
15
Transport Properties, Semiconductor Crystals and Superconductivity
The relation shows that Peltier coefficient is directly proportional to the absolute
temperature. This agrees with the result obtain by thermo dynamical consideration.
Thomson effect
According to Thomson effect, there is absorption or evolution of heat due to the
passage of a current in a single unequally heated conductor. Copper, Silver, zinc,
antimony and cadmium have positive Thomson effect because heat is evolved when
current flows from hot to cold side while heat is absorbed when current flows from
cold to hot side. On the other hand, cobalt, bismuth and platinum have a negative
Thomson effect .
The electron theory only provides a partial explanation of this effect. Let us consider
the case of a conductor whose one end is at higher temperature than the other and a
current is passed from hot end to the cold end. Now there will be transfer of electrons
from the cold part to the hotter part. We know that the energy of the electron is
1

proportional to its absolute temperature  mv2T  , hence the energy of the electron
2

moving towards hotter part will be increased. This produces a cooling effect.
1.4.3 Lattice conduction & Phonon drag
At any finite temperature, the atoms vibrate about their equilibrium positions. These
lattice vibrations may be represented by waves. A lattice vibrational wave in a solid is
a repetitive and systematic sequence of atomic displacements viz., longitudinal,
transverse, or a combination of the two, which may be characterized by a propagation

velocity (v), a wavelength (  ), a wave-vector k , a linear frequency ( ).The energy
of lattice vibrational or elastic waves is quantized. This quantum is called a phonon.
Transmission of a displacement wave in a solid may be regarded as the movement of

one or more phonons, each carrying energy  and momentum k . Thermal
conduction in non-metallic solids is due to creation or annihilation of a phonon. The
of phonons is of the order of 0.1 eV. Phonons are not always in local thermal
equilibrium, they move along the thermal gradient. They lose momentum by
interacting with electrons and imperfections in the crystal. If the phonon-electron
interaction is predominant, the phonons will tend to push the electrons to one end of
the material, losing momentum in the process. Phonon drag is an increase in the
effective mass of conduction electrons or valence holes due to interactions with the
crystal lattice in which the electron moves. As an electron moves past atoms in the
16
Transport Poperties
lattice its charge distorts or polarizes the nearby lattice. This effect leads to a decrease
in the electronic/ hole mobility, which results in a decreased conductivity. However,
as the magnitude of the thermoelectric power increase with phonon drag, it may be
beneficial in a thermoelectric material for direct energy conversion applications. The
magnitude of this effect is typically appreciable only at low temperatures notable
below 200 K.
Check Your Progress 3
Note: a) Write your answers in the space given below.
b) Compare your answers with the ones given at the end of the unit.
(1) Write the various thermoelectric effects and obtain their coefficients ?
(2) What are phonons ? Write a short note on phonon drag.
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
..........................................................................................................................
____________________________________________________________________
1.5
THE HALL EFFECT
Definition : When a current carrying metallic wire is placed in an external magnetic
field in direction perpendicular to the direction of flow of current, an electric field is
produced in the wire in direction perpendicular to both the direction of current and
direction of magnetic field. This effect is called the Hall effect. The electric field
produced in the wire is called the Hall field and the potential difference developed at
the ends of wire is called the Hall voltage.
Calculation of Hall field and Hall voltage
Consider a metallic rod of rectangular cross-section through which current flows by
applying an external electric field Ex along its axis (i.e., X-direction). Obviously
electrons will drift with velocity Vx in direction opposite to the electric field . Now if
a magnetic field Hz is applied in direction perpendicular to the axis of rod. (k.e., in Zdirection), the electrons will experience Lorentz force and will get deflected towards
one end of the rod (i.e., in negative Y-direction) . These electrons will not drift in
17
Transport Properties, Semiconductor Crystals and Superconductivity
space outside the surface of rod, but they will accumulate on the surface of rod and
form the surface charge. Obviously the positive ions get accumulated on the opposite
face. Due to charges on the surface, a transverse electric field EH is produced in the
rod which is called the Hall field. Due to this electric field, compensating drift of
electrons begins in positive Y-direction and an equilibrium of electrons is maintained
within the rod (i.e., the force acting on electrons due to the Hall field EH balances the
Lorentz force acting on electrons due to the magneticfield).
Fig. 4. Creation of Hall field
Remember that if in a conductor the charge carriers are holes (or positive charges),
then the direction of Hall – electric field EH is reversed. Thus by measuring the Hallvoltage we can determine whether the charge carriers in conductor are electrons or
holes.Let the charge carriers in the rod be electrons (charge = -e). Hall electric field
EH (or Ey) will be in negative Y-direction.
Force on electron due to Hall field EH = eEH ( in positive Y-direction )
And Lorentz force on electron due to magnetic field HZ
^
^
^
 (e)(  iv z )  (k H Z )   J ev x H Z
(1)
i.e.,
Lorentz force =evxHz (in-negative Y-direction)

EH v x H Z
In steady state,
(2)
e EH  ev x H Z ,
The expression gives the intensity of Hall electric field produced in the rod. If d is the
thickness of the rod, Hall potential difference
VH = Ed = vxHZd
Hall coefficient
Let number of electrons (or charge carriers) per unit volume in
J
the rod be n. Then current density Jx = n(-e)Vx or drift velocity vx   x But from
ne
eqn.(2) The Hall electric field produced in the rod EH = vxHZ Substituting the value of
vx from eqn. (1) We get
j
1
EH   x H Z orE H   ( J x H Z )
ne
ne
1
RHall  
EH J x H Z orEH  RHall ( J x H Z ) , Where
Or
ne
18
Transport Poperties
The quantity R Hall is called the hall coefficient. Obviously, its unit is volt x metre 3/
amper × weber.
From the above expression, it is clear that the sign of Hall coefficient is same as that
of the charge carrier in the conductor i.e., tf the charge carrier is electron (negative
charge ) , the sign of RHall is negative and if the charge carrier is hole (positive
charge), the sign of RHall is positive.
Hall Voltage
If b is the width of the metallic rod and d is its thickness and a current I flows in the
rod, then current density in the rod along X-axis
Jx = I / bd
Then Hall electric field produced in the rod
EH  
1
1 I
JxHZ  
Hz
ne
ne bd
Hall voltage

VH = EH × d
1 I
IH Z
H z d  RH
ne bd
b
Importance of Hall effect:
(i)
Measuring the hall voltage in a metal or semi-conductor, we can find the
sign of charge carrier in it also we can find the number of charge carriers
per unit volume in it.
(ii)
Measuring the Hall coefficient of a metal, we can find the number of
charge carriers per unit volume in it.
(iii)
The mobility of charge carriers in a metal can be determined.
(iv)
It can be found that the given substance is a metal or semi-conductor or
insulator. It should be remembered that the Hall coefficient is not negative
for all the metals, but some metals have the positive Hall coefficient also
(i.e., in such metals the charge carriers are holes.) if there are both the
holes and electrons as charge carriers, their Hall coefficient can be either
positive or negative depending on their relative concentration and
mobility.
19
Transport Properties, Semiconductor Crystals and Superconductivity
Experimental Determination of Hall Coefficient
Fig. 5. Measurement of Hall coefficient
Fig.5. Shows the experimental arrangement of the apparatus used for the
measurement of Hall coefficient. It consists of a thin metallic rod of width nearly 2-3
cm and length nearly 5-10 cm placed horizontally and with its length along the Xdirection in between the pole pieces of a strong electromagnet. The two ends of the
rod C and D are connected with a battery S, a milliammeter (mA), a rheostat Rh and a
key K so that current flows in the rod along its length (i.e., in X-directin). A sensitive
calibrated potentiometer is connected between the points A and B along the width of
the rod (i.e., in Y-direction ) to measure the Hall voltage..
Let breadth of metallic rod be b and thickness be d. Current I flows in the rod due to
which current density in the rod along X-axis is Jx =
I
bd
But Hall electric field produced in the rod
1
1 I
EH   J x H z ,
EH  
Hz
Hence
ne
ne bd
1 I
IH z

H zd ,
 RHall
Hall voltage VHall = EH × d,
ne bd
b
V b
Hence Hall coefficient  RHall Hall
IH z
Thus knowing the Hall voltage VHall, width of the rod b, current flowing in the rod I,
and the magnetic field Hz, we can calculate the Hall coefficient RHall from the above
expression.
20
Transport Poperties
_____________________________________________________________________
1.6 TWO BAND MODEL
Band model
According to band model, the Sommerfeld’s assumption that the free electrons inside
the metal move in constant potential well, was considered to be wrong. Actually a free
electron inside the metal moves in the electric field of positive ions and of other free
electrons. Since the crystal structure itself is periodic, therefore the potential energy of
electron also changes periodically with the position coordinate i.e., the motion of
electron inside the metal is in a periodic potential well.
We know that inside a metal (atomic number Z), the potential energy of free electron
at a distance x in the potential field of an atom I is given as U ( x) 
 Ze 2
. Hence a
4ox
graph plotted for U(x) versus x is a rectangular hyperbola as shown in Fig. 6.
Fig.6. U(x)-x curve in the region of a single atom
From the graph, it is clear that for different values of x, the value of U(x) is negative
and at x=  , the value of U(x) is zero.But inside the metal, atoms are arranged in a
definite order, therefore the potential energy of electron in the combined field of
atoms I and j can be represented by the complete curve as shown in Fig. 7.
In Fig.7. The dotted curves represent the potential energy of electron in the field of
atom I and atom j separately, while the complete curve represents the resultant
potential energy.
But in a metal, there are number of atoms arranged in a definite order in any
direction, hence in the length L of the metal, the resultant potential energy of electron
in the fields of atoms i. J. k. l. etc. can be represented as shown in Fig. 8.
21
Transport Properties, Semiconductor Crystals and Superconductivity
Fig. 7. U(x)-x curve in the region of two atoms
From Fig.7. It is clear that at the boundaries of metal (i.e., at its free surfaces), the
potential energy suddenly rises and becomes zero, while inside the metal, the potential
well is not of uniform depth everywhere, but it is periodic.
Fig. 8. U(x)-x curve in the region of many atoms
Fig.8. Also shows the different energy levels inside the metal . All the electrons from
the lowest energy to Eb are bound with their atoms and they can vibrate only with a
very small amplitude, they cannot leave their atoms. The electrons of energy higher
than this, with energy in between Eb and EF can move anywhere within the metal, but
on reaching at the boundary (or metal surface), they have to face a surface barrier and
hence they cannot emerge out of the metal surface. Here EF is the Fermi energy level
(i.e., the level of maximum kinetic energy of electrons). Thus, according to band
model
22
Transport Poperties
(i)
Each electron in metal is in the electric field produced due to charge
distribution of positive ions and remaining electrons.
(ii)
Electron is associated with the entire crystal, and not only with an
atom.
(iii)
Electron moves in a periodic potential produced by the ion cores and
other electrons inside the crystal. This periodicity vanishes at the free
surface of the crystal. The motion of electron inside the crystals like
the elastic waves in a continues medium.
Kronig – Penny model
To explain the behavior of electrons in a solid under a periodic potential, kronig and
Penney assumed that the potential energy of an electron can be represented by a
periodic array of rectangular potential well as shown in Fig.9. Here the potential
peaks obtained from the hyperbolic curves have been assumed to be in form of
rectangular peaks.
Fig. 9.
Periodic array of rectangular potential well
Each potential well represents the potential near an atom. If the time period of
potential is (a+b), the potential energy is zero in 0<x<a and potential energy is
constant (=V0) in – b <x < 0. i.e.,
In region 0 < x < a,
V(x) =0
And in region – b < x < 0,
V(x) = V0 (constant )
(1)
In both these regions, the Schrodinger wave equation for the wave function
 n associated with n th energy state En of electron are
d 2 n 2m
 2 En n  0 (since V= 0 )
(2)
dx 2
h
d 2 n 2m
 2 En  V0  n  0 (since V = V0)
and in region – b < x < 0,
(3)
dx 2
h
Here, the energy of electron En is very small in comparison to the potential V0.
Assuming that as b tends to zero, V0 becomes infinite, Kronig and Penney obtained
the following condition for the allowed wave function on solving the above equation :
In region
0<x<a
23
Transport Properties, Semiconductor Crystals and Superconductivity
(mV0b / h 2 ) sin  a + cos  a = cos ka
(4)
Where  = 2mEn / h and k is the wave vector .
Substituting mV0ba/h2 =P in eqn. The condition for the allowed wave function is
P sin a
 cos a  cos a
a
Fig. 10. Energy bands and forbidden energy gaps
Fig.10. Shows a graph between the quantity on the left side of above equation and
a for P = 3  /2. Since the maximum and minimum values of the term cos a are
respectively +1 and –1, own by thick lines from p to  , from q to 2  from r to 3  ,
from s to 4  …….. in the positive directions and from p’ to -  , from q’ to –2  , from
r’ to – 3  , from s’ to –4  …… in the negative direction. It is clear that only in these
specific ranges of a , the allowed energy levels are continuously obtained i.e.,
energy bands are obtained in these specific ranges and not the discrete energy states.
Obviously there are no energy states possible for electrons in the ranges  to q, 2  to
r, 3  to s, ……….., i.e., these ranges represent the forbidden energy gap.
Fig. 11. Energy spectrum in a crystal
Thus from the Kronig–Penney model, we get the following conclusions :
24
Transport Poperties
(i)
In the energy spectrum of all the electrons present in the metal, there
are several energy bands separated by the forbidden energy region. The
energy band completely filled with electrons is called the
valance band and the energy band which is either completely empty or is partially
filled, is called the conduction band .
Fig. 12. Energy spectrum with P
(ii)
From Fig. It is clear that as the value of a increases (or as energy
increases because   2mEn / h , the width of the allowed energy
band increases (since the width of energy band from p to  is less, it is
more from q to2  and still more from r to 3  ,…….).
allowed energy band decreases and ultimately when the binding energy becomes
infinite, the allowed region becomes very narrow At the wave vector k=±n  /a
(where
n=
1,2,3,…)the
energy
is
discontinuous.
Fig. 13. E- k curve
These values to k give the boundaries of Brillouin zones. Fig shows the E-k
curve. For n=1, we get the first Brillouin zone from k=- 
/a to k=+  /a.
The energy in an energy band is a periodic function of k.
(iii) The number of total possible wave functions in an energy band is equal to the
number of unit cells.
(iv)
The velocity of free electron is zero at the top and bottom of an energy band
and the velocity of free electron is maximum at the point of inflexion of
25
Transport Properties, Semiconductor Crystals and Superconductivity
energy band After this point, the velocity of electron decreases with the increase in
energy.
(v)
At T=0 K, the effective number of electron in a completely filled band is zero,
_____________________________________________________________________
1.7 MAGNETO RESISTENCE
The magneto resistance is defined as the ratio of change in resistance of a substance
due to the application of magnetic field to the resistance in zero field. This effect is
due to the fact that, when the magnetic field is applied, the paths of the electrons
become curved and the electrons now do not follow the exact direction of the
superimposed electric field. When the magnetic field is applied normal to the current
flow, the effect is termed as transverse magneto resistance and when the field is
applied parallel to current flow, the effect is termed as longitudinal magneto
resistance.In case of a magnetic metal the external magnetic field increases the
alignment of magnetic moments opposite to the thermal vibrations which decreases
the alignment. If the temperature is below Curie point then magneto resistance is
decreased and if the temperature is above Curie point, there will be no change in it. At
low magnetic field there are changes in resistivity which are apparently associated
with magnetostriction ie., elongation or contraction of the metal depending on the
direction of magnetic moments. These changes are assumed due to the changes in
Fermi surface. Fig. shows the longitudinal and transverse magneto-resistances of
nickel at from temperature. The resistivity in the low fields region governed by
magnetostrictive effects while at higher field the decrease due to a figment of the
magnetic moments is predominant.
Fig. 14. Longitudinal and transverse magneto-resistance
26
Transport Poperties
The transverse magneto resistance measurements made on a single crystal predict
pronounced dependences on magnetic field direction relative to the crystal axes,
especially in pure metals. It is observed that the variation of magneto resistance with
magnetic field strength may be different for magnetic fields in different
crystallographic direction. In some direction it may increase with field, then fall off
and saturate at some constant value; on the other hand, in some other directions it may
continue to rise even at the highest fields.
Mathematical Analysis
Let us consider the case of a wire subjected to an electric field  x along the x-direction
and a magnetic field H normal to the axis of wire i.e., along the z-axis. Now the
Lorentz force is given by
1
d 2x
1




 e  x  v y H z  vH y 
 e x  v  H 
2
c
dt
c




(as the field is only in z- direction, Hy=0)
d 2x
1


 m 2  e x  .v y H z .
dt
c


2
d x
1 dy 

Or
(1)
m 2  e x  . H z .
dt
c dt


d2y
1


Similarly
(2)
m 2  e x  vz H x  vx H z .
dt
c


Similarly for z-direction
d 2z
(3)
m 2  0.
dt
Integrating equation we have
dx
eH z
m e z t 
y  C1
dt
c
dx
e
m
  H z x  C2
and
dt
c
Where C1 and C2 are constants of integration . The values of these constants can be
obtained from the following conditions:
dx
 ux
C1  mux
At
t=0, x=0,
dt
F m
dx
 uy
dt
dx
H
m
 e x t  e z y  mu
Thus
dt
c
dy
e

H z x  uy
Or
dt
mc
Integrating equation we have
t=0,
y=0,
 C2  mu y
(4)
27
Transport Properties, Semiconductor Crystals and Superconductivity
x
As
ex t 2 e H z

yt  u xt  C3 .
m 2 m c
x=0,
when t=0,
 C3=0.
ex t 2 e H z

yt  ut.
m 2 m c
Substituting the value of x from equation in equation we have
 x

dy
eH  e t 2 eH z
 y   z  x  
 yt  u xt   u y
dt
m  m 2 mc

dy
From equation , substituting value of
in equation (1)
dt


d 2x e 
H z  eH Z  e x t 2 eH z







yt

u
t

u


x
x 
y 
dt 2 m 
c  mc  m 2 mc



dx e 
H z  eH Z  e x t 3 eH z yt 2 ut 2 


Integrating it


t





u
t



y
dt 2 m 
c  mc  m 2 mc 2
2 

If  be the time between two successive collisions, we have
(5)
(6)
dx
1  dx
 
dt
dt
 0 dt
e   2 H z  eH z  e x 4 eH z y 3 u x 2
 2 

the

 
   u y  Here
 x 

m  2
c  mc  24m
6mc
6
2 
term Hzy/6 is very small as compared to  3 / 6 hence it is neglected. Average values
of ux and uy are also zero as the carriers have same probability of moving in positive
and negative directions.
e   2 e3 H 2 z x 4 

 x 
.
m  2
24m2c 2 
dx
Ne2 x  e 2 H 2 z 3 
,

 
.
dt
2m 
12m 2c 2 
The electrical conductivity is given by
I
Ne2  e 2 H 2 z 3 
 x 

.
 x 2m  12m2c 2 
Now current density I x  Ne
When magnetic field is zero, Hz=0, then electrical conductivity  0 is given by
Ne3
0 
 0.
2m
The change of resistively is given by
2
2
p
  1 H z  0



p0
 0  0 3 c 2 N 2e 2
28
Transport Poperties


1  
2
 
 Hz
 0 3  cNe 
2
  
2

 Hz
cNe


2
 AH z ,
2
p
1  
Where A   0  and
is very small.
3  cNe 
0
2
Check Your Progress 4
Note: a) Write your answers in the space given below.
b) Compare your answers with the ones given at the end of the unit.
(1) What is Hall effect. What important results are obtained with the help of Hall
coefficient ?
(2) Write a note on magnetoresistence.
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____________________________________________________________________
1.8 Let Us Sum Up
After going through this unit, you have achieved the objectives stated earlier in the
unit. Let us recall what we have discussed so far.

Transport phenomena such as the flow of electric current in solids, involve
two characteristic mechanisms with opposite effects: the driving force of the
external fields and the dissipative effect of the scattering of the carriers by
phonons and defects. The interplay between the two mechanisms is described
by the Boltzman equation. With the help of this equation one may investigate
how the distribution of carriers in thermal equilibrium is altered in the
presence of external forces and as a result of electron scattering processes.

In about 1900, long before an exact theory of the solid state was available,
Drude describe de metallic conductivity using the assumption of an ideal
29
Transport Properties, Semiconductor Crystals and Superconductivity
electron gas in the solid. For an ideal electron gas in an external field E, the dynamics
of the electrons is described by the classical equation of motion
mv 
m

vD  eE
The dissipative effect of scattering is accounted for by the friction term
mvD /  where vD=v-vthermal is the so called drift velocity.

The relaxation time  is the time constant with which the nonequilibrium
distribution relaxes via scattering to the equilibrium state when the external
perturbation is switched off.

If
one end of a long metallic bar or wire is heated, then heat flows
spontaneously from one end to the other because of the ensuing temperature
gradient. If the gradient is uniform, then the amount of thermal energy
crossing a unit area per second (Q) is directly proportional to the temperature
gradient (dT/dx).,
Q
dT
dx
and then
 dT 
Q  K

 dx 
where the proportionality constant (K) is called the thermal conductivity.

knowing the Hall Coefficient it can be found that the given substance is a
metal or semi-conductor or insulator. The magneto resistance is defined as the
ratio of change in resistance of a substance due to the application of magnetic
field to the resistance in zero field.
When the magnetic field is applied normal to the current flow, the effect is
termed as transverse magneto resistance and when the field is applied parallel
to current flow, the effect is termed as longitudinal magneto resistance. It is
observed that the variation of magneto resistance with magnetic field strength
may be different for magnetic fields in different crystallographic direction. In
some direction it may increase with field, then fall off and saturate at some
constant value; on the other hand, in some other directions it may continue to
rise even at the highest fields.
_____________________________________________________________________
1.9 Check Your Progress: The key
1. (1)
See the section 1.2.
30
Transport Poperties
(2) See the section 1.3.1.
2. (1) See the section 1.3.2.
(2) See the section 1.3.3.
3. (1) See the section 1.4.2.
(2) See the section 1.4.4.
4. (1) See the section 1.5
(2) See the section 1.7.
31
Transport Properties, Semiconductor Crystals and Superconductivity
REFERENCES AND SUGGESTED READINGS
1.Material Science by Narula, TMH.
2.Solid State Physics by C. Kittel, TMH.
3. Solid State Physics by Pillai.
4. Solid State Physics by Gupta.
*********
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