Download electron theory of metals

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2
ELECTRON THEORY OF METALS
2.1 CONDUCTIVITY
2.1.1 Introduction
In an ionic solid, the atom forming a cation gives up its valence electron that is captured by the
atom forming an anion. Here the electrons are strictly localized. However in metals, the core
electrons are localized at the atoms but the valence electrons are delocalized and belong to the
entire solid. The behaviour and energy states of these delocalized electrons determine many
properties of these solids. We shall try to understand the behavious of these electrons and see how
they influence the property of the solid. The simplest approach is the classical free electron model
that was successful in explaining some of the properties of metals. This was succeded by the more
sophisticated band theory of solids.
The general characteristics of electrical conductors are:
(i) The current density in the steady state is proportional to the electric field strength
(Ohm’s law).
(ii) For pure specimens, the electrical conductivity (σ) and the thermal conductivity (k)
vary with temperature as follows:
σ ∝ T−1 and k = constant (for T > θD)
so that
σT
is a constant independent of temperature (Wiedemann−Franz law);
KT
σ ∝ T−5 and K ∝ T−2 (for T < θD)
where θD is the characteristic Debye temperature. The relation ρ α T5 is known as Bloch
Gruneisen T5 law.
(iii) For metals that exhibit the phenomenon of superconductivity, their resistivity disappears
at temperature above 0 K and below the critical temperature (Tc) for the superconducting
phase transition (Tc = 4.15 k) for mercury.
31
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ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
(iv) For metals containing small amounts of impurities, the electrical resistivity (ρ) may be
written as
ρ = ρ0 = ρp (T)
where ρ0 is a constant that increases with increasing unpurity content and ρ(T) is the
temperature dependent part of the resistivity. This is known as Matthiessen’s rule.
(v) For most metals the electrical resistivity decreases with increase of pressure.
(vi) The resistivity of alloys that exhibit order-disorder transitions shows pronounced minimum
corresponding to the ordered phase.
Table 2.1: Electrical conductivity and concentration of electrons in some selected
metals
Metal
Copper
Lead
Zinc
Gold
Aluminium
Cadmium
Iron
Number of
valence
electrons (Z)
Concentration of
electrons
in 1028/m3
1
4
2
1
3
2
2
8.47
13.2
13.2
5.90
18.1
9.27
17.0
Electrical
conductivity
at 100 K
in W−1 m−1
2.9
1.5
6.2
1.6
2.1
4.3
8.0
×
×
×
×
×
×
×
108
108
108
108
108
108
108
Electrical
conductivity
at T = 300K
6.5
5.2
1.8
5.0
4.0
1.5
1.1
×
×
×
×
×
×
×
107
106
107
107
107
107
107
2.1.2 Electron Drift in an Electric Field
When an electric field E is applied to a conductor an electric current is established in the conductor.
The density j of this current is given by
j = σE
The constant of proportionality σ is called the specific conductance or electrical conductivity of
1
. The flow of current in a conductor
σ
is an indication that the electrons in it, under the action of the applied field, move in a specific
direction. As a result the distribution function of the electrons in the conductor undergoes a
change. This directional motion of the electrons is called a drift. The average velocity of this drift
motion is called drift velocity.
the conductor. Its unit is ohm−1 m−1. The resistivity, ρ =
The force F acting on an electron in an electric field E is F = − eE where e is the electronic
charge. The minus sign introduced because the charge on the electron is negative. The electron
drift is in a direction opposite to that of the applied field. Acted on by the applied field the electron
will have to accelerate continuously. But in its accelerated motion, the electron collides with the
defects in the lattice. As a result of the consequent scattering the electron loses the velocity it gains
from the field. The effect of the lattice may be reduced formally to the action of a retarding force.
This force is proportional to the velocity v and mass m of the electron. Thus the retarding force may
be taken to be − Kmv where K is a constant. We can then write the equation of motion of the electron
as
ELECTRON THEORY OF METALS
33
dv
= − eE − Kmv
...(2.1.1)
dt
After the field is applied the velocity of the electron rises till the retarding force which is
proportional to the velocity equals the force due to applied field. When these forces become equal
the acceleration ceases. Thereafter the electron moves with the drift velocity. If this drift velocity is
vd, then
m
0 = − eE − K mvd
vd = −
i.e.,
eE
Km
...(2.1.2)
The ratio of the drift velocity to the applied field intensity is called carrier mobility.
Thus
Carrier mobility, m =
e
Km
...(2.1.3)
Suppose that as soon as the velocity of the directional motion of the electrons attains its steady
value, the field is cut off. Because of the collisions of the electrons with the lattice defects the velocity
starts decreasing. After some time the electron gas resumes its equilibrium condition. Such a process
which leads to the establishment of equilibrium in a system from which it was previously disturbed
is called a relaxation process. When the applied filed is cut off the equation of motion of the electron
becomes:
m
dv
= − Kmv
dt
...(2.1.4)
dv
= − Kv
dt
This can be written as
dv
= − Kdt
v
Integrating
log v = − Kt + C
...(2.1.4a)
Let us now find the value of integration constant C using the known conditions.
When t = 0, v = vd and hence C = |log vd |C = log vd. Now Eqn. 2.1.4a becomes
log v = − Kt + log vd
or
v = vd exp (− Kt)
The time taken by an electron to reach a directional velocity which is
is called the relaxation time denoted by τ.
vd
= vd exp (− Kτ)
e
1
of the drift velocity
e
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ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
This is possible or this leads to the result K =
vd = −
or
1
eE
. Thus Eqn. (2.1.2) becomes −
τ
τ
m
eE
τ
m
...(2.1.5)
V
A
Fig. 2.1.1 Drifting electrons
e
–1
Fig. 2.1.2 Scattering electron at ion cores
Let n be the density of electrons. The charge flowing per unit area per unit time is the current
density.
i.e., j = nevd [Referring Eqn. 2.1.2]
or
j =
ne 2 τ
E
m
...(2.1.6)
Defining the current density for unit electrial field as electrical conductivity, we get
j
= σ or
E
σ =
ne2 τ
m
...(2.1.7)
ELECTRON THEORY OF METALS
35
If we take the experimental value of s for copper as 6 × 107 (Ω-m)–1 and n = 8.5 ×
10 /m3, we find t = 2 × 10-14 see. In discussing electrical conductivity it is convenient to introduce
the mobility of charge carriers. The mobility m is the magnitude of the ratio of the drift velocity
to the applied electric field. Thus (Refer Eqn. 2.1.5).
28
µ =
The unit of µ is
vd
σ
eτ
=
=
m
ne
E
...(2.1.7a)
m×m
= m2 V−1 s−1
sec × volt
The electrical resistivity, ρ =
m
1
=
σ
ne 2 τ
2.1.3 Temperature Dependence of Conductivity
In most solids the two main types of internal energy are (i) the vibrational energy of the atoms about
their mean lattice positions and (ii) the kinetic energy of the free electrons. If heat is supplied to the
body its temperature rises and the internal energy increases. The important thermal properties of
matter such as heat capacity, thermal expansion and thermal conductivity depend upon the changes
in the energy of the atoms and free electrons. If an electrical field is applied to a solid the free
electrons are accelerated. Their kinetic energy increases. Some of this kinetic energy is of course lost
by collisions with the atoms in the lattice. The resulting current is proportional to the average
velocity of the free electrons. This velocity is determined by the applied electrical field and also the
collision frequency.
We should make a distinction here between the two different velocities associated with the
electrons. The velocity appearing in the Eqn. 2.1.5 is called the drift velocity (vd) which is superimposed
—
on a much higher velocity called root mean square velocity (c ) due to the random motion of the
electron which is possible even in the absence of the electric field. The random motion, which
contributes zero current, exists also in the presence of the field.
It is evident now that in the absence of an electric field the free electrons in a metal will be
moving about at random in all directions and will be in temperature equilibrium with it. If the
mass of an electron is m then at absolute temperature T it will possess an average random velocity
given by the kinetic theory of gases. Thus
1
3
mc 2 = kT
...(2.1.8)
2
2
where k is the Boltzmann constant. Suppose an electric field E is applied. Under the influence of this
eE
where eE
m
is the force acting on the electron and m is the mass of the electron. The drift velocity is much
smaller compared to random velocity —
c . Further the drift velocity is not retained after a collision
with an atom because of the relatively large mass. Hence just after a collision the drift velocity is
zero. If the mean free path is λ then the time that elapses before the next collision takes place is
field the electron acquires a drift velocity. The resulting acceleration of the electron is
λ
. Hence the drift velocity acquired just before the next collision takes place is
c
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ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
u = acceleration × time
=
LM eE OP × λ
NmQ c
Thus the average drift velocity of the electron is
u
eE λ
=
×
2
2m c
If the number of electrons per unit volume is n, then the number of electrons crossing unit area
for unit time is nothing but current density, j.
vd =
Thus
j = nevd
j = nevd = ne
i.e.,
c =
3kT
m
j =
ne 2Eλ
2m
—
with
Thus
RS eEλ UV
T 2mc W
m
3kT
Thus electrical conductivity,
σ =
ρ =
j
=
E
ne 2 λ
12mkT
12mkT
ne 2λ
...(2.1.9)
These two expressions are independent of current. Hence at constant temperature the electrical
conductivity (i.e., the ratio of the current density to the applied field) is a constant. In other words
at constant temperature the applied p.d is directly proportional to the current. This is Ohm’s law.
2.1.4 Impotentialness of Classical Free Electron Theory
Some of the drawbacks of classical free electron theory are listed below:
1. Classical theory failed to explain the variation of electronic specific heat at low temperatures,
on the basis of Drude’s model, it was concluded that the molar specific heat of electron
gas for any metal was 3R (12.5 kJ/kmol). This value is many times the magnitude of
specific heats found experimentally.
2. The mean free path of electrons in any metal, calculated on the basis of Drude’s model
was ten times less than the experimentally observed value.
3. The Lorenz number, L calculated for many metals using the standard relation
FG IJ
H K
π2 k
3 e
W
=L=
σT
2
is equal to 2.45 × 10−8 watt −m/K2 comes to be a constant. However, for many
ELECTRON THEORY OF METALS
37
metals, the Lorentz number varies with temperature at low temperatures. This is due to the
fact all the electrons may not be participating in conduction process. W is the thermal
conductivity of the metal.
4. Drude’s classical free electron theory totally failed to explain the conduction mechanism in
semiconductors and insulators.
5. The classical model could not explain the origin of Pauli’s paramagnetism.
We shall now discuss one or two failures of classical free electron theory of metals with
necessary theory.
(i) Heat capacity of the electron gas: Classical free electron theory assumes that all the
valence electrons can absorb thermal energy, move to higher energy states and contribute to heat
capacity. According to the law of equipartition of energy, every free electron in a metal has an
3
KT. For a manovalent metal (copper) these will be NA (Avagadro’s number
2
of electrons = 6.02 × 1026) in one kmol. Hence the molar electronic specific heat is
average kinetic energy
[Cv]el =
with
U =
Thus
[Cv]el =
i.e.,
dU
dT
3
N kT
2 A
dU
= 1.5 NA k = 1.5 Ru
dT
[Cv]el = 1.5 R = 1.5 × 1.38 × 10−23 × 6.02 × 1026
...(2.1.10)
= 1.5 R or 12.5 × 103 J/kmol/k
This classical value of 1.5 R is about hundred times greater than the experimentally
predicted value.
(ii) Computation of mean free path: The microscopic expression for the resistivity
(Eqn. 2.1.7) is
ρ=
m
τ =
m
...(2.1.11)
ne 2 τ
The resistivity of the most useful metal copper at 20°C is 1.69 × 10−8 ohm-m and the
concentration of free electrons in copper, n = 8.5 × 1028/m3. Thus
ρne 2
=
9.1 × 10 −31
e
1.69 × 10 −8 × 8.5 × 10 28 × 1.6 × 10 −19
τ = 2.47 × 10−14 sec
But
τ =
λ
c
with τ =
3kT
m
λ = τ c = 2.47 × 10−14 × 1.154 × 105
λ = 2.85 nm
j
2
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ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
The experimentally found value for λ is about ten times above this value. Classical theory
could not explain this value.
2.1.5 Elementary Treatment of Quantum Free Electron Theory of Metals
The general expression for the drift velocity is
eE
τ
...(2.1.12)
m
where τ is the average time elapsed after collision. Thus it is obvious that the average deift velocity
of free electrons is proportional to force eE.
vd = −
Therefore current density,
J = ne v b = −
But
ne 2 τ E
m
J = −σE
...(2.1.13)
...(2.1.14)
Comparing these two equations, one gets
ne2 τ
m
If λ is the mean free paths and vF the speed of free electrons whose kinetic energy is equal to
Fermi energy since only electrons near Fermi level contribute to the conductivity. The average time
τ between collisions is given by
λ
τ =
vF
σ =
i.e.,
σ =
ne 2λ
m × vF
...(2.1.15)
The drift velocity for unit electric field is called mobility.
i.e.,
vd
eEτ
eτ
=
=
mE
m
E
Thus the electric conductivity,
µ =
ne 2 τ
ne 2λ
= ne µ =
...(2.1.16)
m
mv F
From expression (2.1.16) it is obvious at a given temperature the only factor which varies from
one metal to the other is densities of free electron.
σ =
One must also note that the energy kT (where T is of the order of 300 k) can activate only
the free electrons near the Fermi level to move to unoccupied states and contribute to specific heat.
We may therefore require an energy EF called (very high compared with kT = 0.025 eV at 300 k)
Fermi energy to make all the electrons to more to the unoccupied states corresponding to a
temperature TF called Fermi temperture. The unique relation connecting the various parameters
in quantum theory of free electron is
1
mv F2 = kTF
EF =
...(2.1.17)
2
ELECTRON THEORY OF METALS
39
2.1.6 Relation Between Thermal Conductivity and Electrical Conductivity
(Wiedemann-Franz Law)
Figure 2.1.3 shows the view of a copper rod of appreciable length with unit area of cross-section
in the steady state.
T+
dT
dx
λ
T–
T
dT
dx
λ
Ste am
A
B
C
H e a t flo w
E
E +
dE
dx
λ
E –
dE
dx
λ
Fig. 2.1.3 Flow of heat through a copper rod at the steady state
Let λ = AB = BC be the mean free path of the electron.
The excess of energy carried by an electron from A to B is
LM dE OP λ. Hence the excess of energy
N dx Q
transported by the process of conduction through unit area in unit time at the middle layer B is
LM OP
N Q
LM OP
N Q
assuming
1
nc is the number of free electrons flowing in a given direction through unit area in unit
6
nc λ dE
1
dE
. Similarly the deficit of energy transported through B in the opposite direction is nc λ
6 dx
dx
6
time.
Thus the net energy transported through unit area in unit time from A to B is:
LM OP LM
N Q N
LM OP =
N Q
1
dE
1
dE
− − nc λ
nc λ
6
dx
6
dx
FG IJ FG dT IJ OP
H K H dx K Q
nc λ dE
3 dT
The general expression for the quantity of heat energy transported through unit area for unit
dT
. Equating the two equations, one gets
time is σ T
dx
FG IJ
H K
σT =
But
Now
FG IJ
H K
nc λ dE
3 dT
dE
is the energy required to raise the temperature by one degree and hence it is [cv]el.
dT
σT =
nc λ
cv
3
el
=
nλ cv
3
el
3k T
m
...(2.1.18)
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ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
But
Thus
[cv]el =
σT =
3
k
2
with n = 1 electron
FG IJ
H K
nλ 3
k
3 2
3k T
nλk
=
m
2
3k T
m
The general expression for electrical conductivity by referring Eqn. 2.1.9.
ne 2λ
σ =
Thus
12mk T
nλk
σT
=
2
σ
3k T
m
LM OP T
NQ
k
σT
= 3
e
σ
|RS
|T
12mk T
2
ne λ
|UV
|W
2
...(2.1.19)
This is called Wiedemann Franz law and the multiplying factor 3 (k/e)2 is called Lorentz number.
2.1.7 Heat Developed in Current Carrying Conductor (Joule’s Law)
The heat developed in a current carrying conductor is given by
V2
R
H = I2 R =
H =
bElg
ρ
i.e.,
2
l
A
=
E 2 Al
ρ
H = σ E2 Al
Thus heat developed per unit volume per second is
W = σ E2
But current density, j = σ E
jE 2
= jE = σ E2
E
W will be found to be in watt/m3 (2.1.17)
W =
...(2.1.20)
In an isotropic medium, consider a particular electron which at the instant t = 0, has carried
out a collision with the lattice, and let the velocity components of the electron be vx, vy and vz.
Now at the instant (t > 0), the electron has yet to collide with the lattice again, and assuming
an electric field of intensity E is applied along the negative x-direction, the velocity components
of the electron are
vx +
FG e IJ Et
H mK
ELECTRON THEORY OF METALS
41
LM e EtOP in velocity is due to the acceleration due to the field on the electron.
Nm Q
Thus the increase in energy of the electron over the field is
O
e U
1 LR
m M Sv +
Et V − v P
(∆W) =
m W
2 MN T
PQ
1 L L 2e O
LeO O
m MM P E v t + M P E t P
(∆W) =
2 NM N m Q
N m Q QP
The increment
2
2
x
x
t
2
2 2
x
t
The above expression may be averaged over a large number of electrons which have all lived
through the period without having suffered a collision, but which presumbly have a random distribution
of their velocities. Thus are finds
(∆W)t =
FG IJ
H K
1
e
m
2
m
2
E 2t 2 =
e 2Et 2
2m
...(2.1.21)
If P (t) represents the probability that an electron moves for a ture t without suffering a
collision, then P (t) = exp (−t/τc). For isotropic scattering, the average time between collisions, is
equal to the relaxation time τ. Also, the probability that an electron will suffer a collision during
a time dt is given by
dt
dt
=
τ
τc
So, the probability that the electron makes a collisions between t and t + dt is given by
LMexp FG −t IJ OP FG dt IJ
N H τ KQ H τ K
Thus, the average energy increase of the electrons during the period between two collisions
is equal to
zb
∞
(∆W) =
t=0
LM e E OP
MN 2mτ PQ
2 2
=
g
∆W t exp
z
∞
t
2
−t
eτ
FG −t IJ dt
H τK τ
dt
0
We known that
z
∞
x n e − ax dx =
0
∠n
an + 1
e2 E 2τ 2
m
If n is the number of electrons present in unit volume, the total energy dissipated per unit
volume per second
or
∆W =
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ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
W =
n
∆W
τ
n e 2E 2 τ 2
= σ E2
m
τ
This agrees with the experimental work.
W =
...(2.1.22)
2.1.8 Thermoelectric Effect
It is a phenomenon related to the thermal conductivity of electrons in that it too is a consequence
of the drift of electrons under a thermal gradient. These are two aspects to thermoelectric
phenomena. The first is the Seeback effect in which a temperature between the two junctions of two
dissimilar materials gives rise to an emf in the circuit. The second is the peltier effect in which, if
a current is circulated in a circuit consisting of two dissimilar materials, is liberated at one junction
and absorbed at the other. Both effects are the subject of considerable study because of their practical
applications. The generation of electrical power using the Seeback effect offers the desirable freedom
from removing parts, and similar advantages are available in the use of the Peltier effect in
refrigeration. In either case one of the chief considerations is the size of the effect, which is a
consequence of the detailed electronic structure of the two materials, and a simple theoretical
treatment is not available. If emerges, however, that the parameters of the material which are
significant in this respect are the same as for the conductivity, namely the effective mass of the
electrons and the relaxation time. Thermoelectric effects of either sign can be observed, depending
mainly on whether electron effects or hole effects are predominant. By careful selection of the
parameters using doped semiconductors, thermoelectric power supplies now have efficiencies
approacting 20% and thermoelectric refrigerators have been built which can maintains a temperature
50°F below room temperature.
The most familiar example of a metal-metal junction is the thermocouple, which is made of
a configuration of the type A-B-A. If a current is passed through such a combination, the temperature
of one of the functions is found to rise and that of the other to fall. This is called the Peltier effect.
Conversely, if one junction is heated and the other cooled, a p.d develops across the combination.
This is called Seeback effect.
Apart from their normal use as temperature sensitive elements, in recent years attention has
been paid to the possibility of using such devices as refrigerators and heat pumps (using the Peltier
effect) or as electrical generators (using the Seeback effect). With metal-metal combinations, both
effects are too small to be of any practical use (for example, the thermo-electric e.m.f developed
for a copper-constantan thermocouple is 40 µV per Kelvin temperature difference), but since the
advant of semiconductors and the possibility of manufacturing metal-semiconductor junctions, the
constriction of practical devices based on these effects has become a reality.
2.1.9 Thermionic Emission
When metals are heated to sufficiently high temperatures, they emit electrons. This is known as
thermionic emission. The free electrons in the metal should be supplied with a minimum amount
energy before they can escape from the metal by thermionic emission. This miniumum energy is
called thermionic work function. The work function is different for different metals.
Richardson and Dushmans equation. Consider a pure metal situated in an evacuated
endosure. Let the metal be heated. It then emits electrons just as a heated liquid emits vapours. These
ELECTRON THEORY OF METALS
43
electrons form a cloud over the metal just as a cloud of vapour is formed over an evaporating liquid.
Just as in the case of vapour, the electrons in the electron clould will be returning into the metal.
When equilibrium sets in, the number of electrons emitted from the metal per second will be equal
to the number of electrons returning to the metal per second.
Consider one mole of electrons remembering that one mole of electrons will contain Avagadro’s
number of electrons (NA). Let the pressure exerted by these electrons on the metal be P. Let the
specific volumes of electrons in the metal be u and specific volume in the cloud by v. Let the heat
energy absorbed by the NA electrons when they escape from the metal be W. Let us assume that
Clausius-Clapeyron’s equation in thermodynamics can be applied to the electrons.
We then have
b
g
dP
v−u
...(2.1.23)
dT
where T is the temperature of the metal and the enclosure. In reality, u << v. We can, therefore,
write the above quation as
W = T
dP
...(2.1.24)
dT
The energy W consists of two parts: (i) energy required to provide the work function, which
may be written as NA φ, where φ is the work function per electron, and (ii) the work done by the
ejected electrons is moving against the pressure Pexerted by the electron cloud over metal, which
W = Tv
z
V
can be written as Pv [just as W =
P dV = PV for a gas expanding at constant pressure from
0
zero volume to a volume V].
Therefore
W = NA φ + Pv
...(2.1.25)
Let us ignose the electric repulsion between electrons and assume that the cloud behaves as
a perfect gas. We have
Pv = RT = NA kT
where k is Boltzmann’s constant.
Therefore
W = NA φ + NA KT = NA (φ + kT)
Therefore (Referring Eqn. 2.1.23)
NA (φ + kT) = Tv
But
FG dP IJ
H dT K
Pv = RT = NA kT
N A kT
P
Substituting this in Eqn (2.1.26)
v =
NA (φ + kT) = T
N A kT dP
dT
P
...(2.1.26)
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ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
kT 2 dP
P dT
φ + kT = T
i.e.,
dP
=
dT
or
FG φ + kT IJ dT
H KT K
2
Integrating both sides, we get
loge P =
loge P =
i.e.,
z
z
z
φ + kT
T2
φ dT
kT 2
dT + log e A
...(2.1.27)
+ log e T + log e A
φ dT
+ log e T + log e A
kT 2
where loge A is a constant of integration
Let
Then
z
φ dT
= I
kT 2
loge P = I + loge T + loge A
loge P − loge T − loge A = 1
log e
FG P IJ
H AT K
= I
P
= eI, or P = AT eI
...(2.1.28)
AT
If these are n electrons per unit volume of the gas, we have from the following general
equation.
PV = RT = kA NA T
if
V = 1, NA = n and hence
P = knT
n =
i.e.,
AT e I
P
A I
e
=
=
kT
kT
k
P
A I
A
e = BeI with B =
=
kT
k
k
According to kinetic theory of gases, we have
n =
1
3
mc 2 =
kT
2
2
According to Knudsen’s cosine law, the number of electrons n0 crossing unit area per second
is given by
n0 =
nc
6π
ELECTRON THEORY OF METALS
45
1
=
6π
BeI
3kT
m
where m is the mass of the electron
1
n0 =
3K
Be I T
m
6π
n0 = c e I T
i.e.,
where
1
c =
3k
B = constant
m
6π
Richardson first assumed that the work function φ is independent of temperature, then
I =
z
ψ dT
2
kT
n0 = c e−(φ/KT)
= −
φ
kT
Thus current density,
j = n0e = c e−(φ/KT)
T
j = D e−φ/kT
where
T
D = c e = constant
...(2.1.29)
Equation (2.1.29) is the simplest form of Richardson and Dushman of themionic emission.
2.2 PROPERTIES AND APPLICATIONS OF METALS
As was pointed out in the discussion of the free electron theory of metals, the valence electrons is
metallic atoms, being loosdy bound, escape from the atom. These essentially free electrons provide
a medium of negative charge which helps to bind the positive ions. That this leads to a lower energy
state which can be seen from the following arguments.
An electron in isolated atom is confined to a small volume around the nucleus. This confinement
D
, where r is the radius of the atoms.
r
Consequently, the electron has a fairly substantial amount of kinetic energy, of the order of several
eV. However, in the crystalline, metallic state, the electrons are essentially free to be anywhere in
the entire crystal. As a result, these is considerable reduction in kinetic energy. This is the source
of metallic bonding. The bond between two metallic atoms is somewhat weaker than ionic or
covalent bonds. This leads to relatively low melting points, for example, 63°C for K. However, the
cohesive energy of the metals is fairly large since each valence electron interacts with several ions.
The metallic bonds are not directional where allows the planes of atoms to slide over each other
quite easily. Hence metals are found to be ductile and malleable rather than brittle. The existence
of essentially free electrons gives rise to high electrical and thermal conductivity for metals.
gives rise to an uncertainty in the momentum ∆p ≈
Because of these favourable properties metals are widely used in many areas mechanical
engineering and electrical engineering.
46
ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
2.2.1 Properties of Insulating Materials
Insulating materials are those which offer very high resistance to the flow of electric current. They
require extremely high voltage of the order of kilovolt or megavolt to send a few milli-ampere of
current through them. They are non-metallic materials and are used to prevent leakage of current
from are conductor to another or to earth. Almost all insulating materials have negative temperature.
Co-efficient of resistance and as such their resistivity is reduced with increase in temperature. The
function of any insulator is very important, for without which no electrical machines can work.
Ideal insulators are those substances into which static electricfield penetrates uniformly. Primarily
they are used for storage of electrostatic energy as in capacitor which are known as dielectrics. But
here we prefer to deal with that part of its function which prevents flow of electric current to any
desired conducting medium. The insulators provide mechanical strengths as well as insulation to
the conductors. Wood, paper, mica, oil, asbestos, porcelain are some examples of insulating materials.
Some important applications of insulators in the field of electrical engineering are:
(a) To isolate the overhead conductors from earth in the transmission and distribution of
electrical energy
(b) In all rotary machines to isolate the winding from the body and other unwanted currenty
path.
(c) In transformers to isolate the winding from the core or magnetic path.
(d) In industrial and house wiring to isolate the conductor from earthing.
(e) In all domestic, commercial electrical equipments to isolate the conductor from earth or
body.
Majority of the breakdown in the field of electrical engineering is due to the falure of
insulation. The importance of insulating material is ever increasing day by day and there are
innumerable number of types of insulations are available in the market. The selection of the right
type of insulating materials for a particular job is very important because the life of equipment will
depend on the type of insulating materical used. All insulators when used should not only behave
as insulators over range of electrical voltage but also should be strong enough mechanically, should
not affected by heat, atmosphere, chemical and should be free from deformation due to aging.
Therefore, the properties to be studied while selecting an insulator for a particular job are:
(a) Electrical properties
(b) Mechanical properties
(c) Visual properties
(d) Chemical properties
Some of the important electrical properties of insulators are
(a) Insulation resistance
(b) Volume and surface resistance
(c) Dielectric strength and dielectric loss
(d) Breakdown voltage i.e., the voltage above which the material ceases to act as insulator.
ELECTRON THEORY OF METALS
47
2.3 SUPERCONDUCTIVITY
2.3.1 Introduction
Superconductivity is a phenomenon observed in some materials: there is disappearance of the electrical
resistivity at temperatures approaching 0K. The discovery of this phenomenon was not an attempt
of any body; but by charm of face.
The phenomenon of superconductivity was discovered by Kamerlingh Onnes (1911), when he
was measuring the resistivity of mercury at low temperature. He observed that the electrical resistivity
of pure mercury drops abruptly to zero at about 4.2 K. He concluded that the mercury has passed
into a new state which is called the superconductivity state.
The materials that display this behaviour are called superconductors and the temperature at
which they attain superconductivity is called critical temperature Tc.
The resistivity-temperature behaviours are contrasted in Fig. 2.3.1. The critical temperature
varies from superconductor to superconductor but lies between less than 1 K and approximately
20 K for metals and alloys. Recently it has been demonstrated that some complex ceramics have
critical temperature approaching 100 K and above. Superconductivity is Adonis and the adolescence
is retained even after one hundred years. Four times Nobel Prize was awarded for discoveries and
inventions in this area, and many more are aiming.
S uperconductor
Im pure m etal
P erfectly
pure m etal
ρ
ρo
0
Tc
T(in K )
Fig. 2.3.1 Variation of resistivity with temperature
The resistivity ρ of metal falls with temperature T as per the law ρ α T when T > θD ρ α
T5 when T < θD, θD being the Debye temperature, but the experiments do not lead to the conclusion
ρ → 0 as T → 0. It is because any trace of impurity or crystal defects prevent the metals becoming
perfect conductors and gives rise to a residual resistivity, ρ0. Then the question is whether
superconductivity is related to purity of the metal. An emphatic ‘no’ is the answer. Superconductivity
is a different phenomenon. If impurities are present in a superconductor, then fall of ρ takes place
in a very small region of temperature instead of sharply at Tc.
2.3.2 Some More Aspects of Superconductivity
The phenomenon superconductivity can be explained satisfactorily on the basis of wave mechanics.
In an ordinary metal, the electrical resistance is the result of collisions of the conduction electrons
48
ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
with the vibrating ions in the crystal lattice. In the superconductivity state, the electrons tend to be
scattered in pairs rather than individually. This gives rise to an ‘exchange force’ (similar to the force
between the atoms in a hydrogen molecule and the force between nucleans (in a nucleus), between
the electrons. The force is attractive; and is very strong if the electrons are of opposite spins and
momenta. In the superconducting state, the forces of attraction between the conduction electrons
exceed the forces of electrostatic repulsion. The entire system of conduction electrons then becomes
a bound system. No transfer of energy takes place from the system to the lattice ions. If an electric
field is established the bound system of electrons gain additional kinetic energy and give rise to a
current. But they do not transfer this energy to the lattice, so that they do not get slowed down. As
a consequence of this, the substance does not possess any electrical resistivity. This theory was put
forward, by John Bardeen, L. N. Cooper and J. R. Schriffer in 1957. The bound pairs of electrons
is called Cooper pairs.
Normally, the resistivity of a superconductor is measured by causing a current to flow in a ring
shaped sample (one can start the current by induction after removing a magnetic flux linking the
ring), and observing the current as a function of time. If the sample is in a normal state, the current
damps out quickly because of the resistance of the ring. But if the ring has zero resistance, the
current, once set up, flows indefinitely without any decrease in value. According to experiments
conducted, the current remained constant in the ring even for several years.
2.3.3 Magnetic Properties of Superconductors
Effect of magnetic field: One should also know that the superconducting state can be brought
to the normal state by applying a magnetic field at which superconductivity vanishes is called critical
magnetic field. Its value depends upon the nature of the material and its temperature.
H c (0)
H c (0)
N ot
superconductor
H c1
N orm al
state
Hc
H c2
S uperconductor
Hc
0
S uperconductor
Tc
T
0
T c1
Tem perature
T c2 T c
T
Fig. 2.3.2 Variation of Hc with temperature
The equation used in this connection is
b g LMM
N
Hc(T) = H c 0 1 −
T2
Tc2
OP
PQ
...(2.3.1)
2.3.4 Critical Currents
The readers should recognize that the magnetic field which destroy the superconducting property
need not be the external electrical field, it can be due to the current flowing through your
ELECTRON THEORY OF METALS
49
superconductor. Hence the maximum current flowing through the specimen at which this property
is destroyed is called critical current.
If a superconducting wire of radius r carries a current I, then as per
M a g n e tic
lin es of fo rc e
I
r
Fig. 2.3.3 Superconducting wire carrying current
Ampere’s law,
i.e.,
z
H ⋅ dl = I
H2πr = I
at H = Hc, I = Ic
Hence
Ic = 2π rHc
...(2.3.2)
If I becomes Ic superconductivity will be destroyed. If in addition to current, transverse
magnetic field H is applied, the value of critical current decreases.
I
H
Fig. 2.3.4 Current carrying superconductor in transverse magnetic field
Now
Hc = HI + 2H where HI is the field due to current
HI = Hc − 2H
HI =
Ic
= Hc − 2H
2πr
Ic = 2πr (Hc − 2H)
...(2.3.3)
This is called Silsbee’s rule.
2.3.5 The Meissner Effect
When superconducting material is cooled below its critical temperature, it not only becomes
resistanceless but perfectly diamagnetic also. That is to say that there is no magnetic field inside
superconductor, whereas inside a merely resistanceless metal there may or may not be a magnetic
field, depending on the circumstances.
This interesting observation, when superconductor placed inside a magnetic field cooled
below its critical temperature, all the magnetic flux is expelled out of it, called Meisnner effect.
50
ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
T < Tc
T > Tc
C ooling
B=0
B ≠0
B=0
Fig. 2.3.5 Meisnner Effect
The perfect diamagnetism is an account of some special bulk magnetic property of the
superconductor. If there is no magnetic field inside the superconductor, it can be said that its relative
permeability µr is zero. Here the mechanism of diamagnetism is not considered.
0
H
M =–H
M
Fig. 2.3.6 Versus H in a superconductor
The general equation connecting magnetic induction and magnetic field is
B = µo (H + M)
or
H + M =
and also
i.e.,
B
0
=
= 0 (under superconducting state B = 0)
µo
µo
H + M = 0
M = − H
shows the magnetization curve for a superconductor.
The magnetic susceptibility is
χM =
M
= −1
H
...(2.3.4)
It must be noted that superconductivity is not only a strong diamagnetism but also a new type
of diamagnetism.
Critical field: Superconductivity can be destroyed by the application of a strong magnetic
field called critical field, Hc even at T < Tc. The critical field depends on temperature and decreases
as the temperature rises from T = 0 K to T = Tc.
ELECTRON THEORY OF METALS
51
O
L
b g MM FGH TT IJK PP
Q
N
2
Hc (T) = H c 0 1 −
i.e.,
...(2.3.5)
c
C ritical m agn etic field
A p plied m agnetic field strength
Figure 2.3.7 shows Hc vs Tc for some selected superconductors and the phase diagram.
Hc
N orm al
P
S uperconductivity
6 × 10
Pb
3 × 10
2 × 10
Sn
1 × 10
Al
0
0
Tc
T(K )
4
1
3
4
7
4
4
4
8
T(K )
(a)
(b)
Fig. 2.3.7 (a) Phase diagram of a superconductor (b) Critical fields of superconductors
2.3.6 Superelectrons
According to Two-fluid Model (a model introduced for explaining the electrodynamic property of
superconductors), the conduction electrons in a superconducting substance fall into two classes:
Superelectrons and normal electrons: The superelectrons experience no scattering, have zero
entropy (perfect order), etc. The normal electrons behave in the usual fashion discussed in the free
electron study chapter.
The number of superelectrons depends on the temperature as stated below:
LM F T I OP
MN GH T JK PQ
4
ns = n 1 −
c
1
ns
n
0
Te
T
Fig. 2.3.8 Dependence of superelectrons on temperature
...(2.3.6)
52
ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
This is plotted in Fig. 2.3.7. Thus at 0 K, all the electrons are superelectrons, but as T increases,
the superelectrons decrease in number, and they become normal electrons at T = Tc.
2.3.7 Current Trends in Superconductivity
The failure to raise the transition temperature (Tc) of a metallic system beyond 23.2 K forced
research to look into other kinds of systems for possible high temperature superconductivity. This
effort seems to be paying off with the recent discovery of ceramic oxides found to be superconducting
well beyond 100 K.
Superconductivity around 90 K (High temperature superconductors): In March
1987, the Texas-Alabama group reported on the observation of stable and reproducable
superconductivity between 93 K and 80 K in Y-Ba-Cu-O system. For the first time in history,
superconductivity above the liquid nitrogen boiling point (77 K) was discovered. Research groups
in India soon confirmed the existence of superconductivity in a system of normal composition Y1.2
Ba0.8 Cu2 O4, in the 90 K range.
Superconductivity at 155 K: In Michigan, U.S., a research group synthesized fluorinated
Y-Ba-Cu-O compounds of nominal composition Y1 Ba2 Cu3 F2 Oy (bulk material of multiphase) to
achieve zero resistance at −155°C.
Superconductivity above 200 K: Recently some researches have observed a drop in the
electrical dc resistance of their samples in the 200 K range. Unfortunately, in these cases repeated
cooling and warming (thermal cycling) appears to disrupt superconductivity which is perhaps
present only in tiny filaments embedded in a mostly non-superconducting sample.
Researches at the national physical laboratory, New Delhi and at Wayne State University,
Detroit, U.S. have observed in multiphase samples of Y-Ba-Cu-O a drop in electrical resistance in
200 K range. It has become possible to fabricate superconducting wires from ceramic materials,
but the mechanical properties such as ductility, flexibility may not be so good as aluminium.
A nightmare: Expressing concern over mounting transmission and distribution (T and D)
losses in the power sector, the former president of India Prof. A.P.J. Abdul Kalam asked the Union
Power Ministery to bring down the power loss from 31000 MW to 12000 MW in a couple of years
comparing with that in developed countries. Mr. Kalam said if the country were to set up fresh
capacity to cover of this loss, it would cost over Rs. 76000 crores; this is the magnitude of the
problem we are faced with; he added while addressing a group of Engineers attached to Central
Electricity Authority.
Another method of solving this problem is summarized now:
The phenomenon of superconductivity was observed in mercury about 100 years ago (1911).
The transition temperature of the metal aluminium used liberally in transmission lines on account
of their mechanical and electrical properties is 1.2 K. i.e., below this temperature electron-phononelectron interaction becomes stronger and cooper pairs are formed. All of them have the same de
Broglie wavelength with very small momentum and hence scattering of cooper pairs becomes
absent, mobility becomes infinity. This may brought down power loss almost nil. However this is
possible only at 1.2 K.
Huge amount of fund was made available in U.S. and in India for research in high temperature
superconductors hoping aluminium will be made superconductor in the wide range say 0°C to 45°C
by discovering new process, mechanism and techniques though the phenomenon was discovered
about 100 years ago, but in vain. I hope we will succeed in this attempt in the years to come.
ELECTRON THEORY OF METALS
53
2.3.8 Field Applications of Superconductivity
Superconductivity as it is, does not find much applications. This is on account of the difficulty
involved in reaching a low temperature of − 273°C and maintaining the equipment in that state but
it is likely to find applications in the following main areas:
(a) Electrical machines: By using superconductivity it is possible to manufacture electrical
generators and transformers in exceptionally small sizes, having an efficiency nearly
equal to 100%.
(b) Power cables: A 200 kV cable with superconducting material will enable transmission
of power over long distances using thin conductors without any significant power loss
or voltage drop.
(c) Electromagnets: Superconducting solenoids have been made which do not produce
any heat during operation. Superconductivity gets destroyed if the magnetic field exceeds
a critical value. In superconducting state, the magnetic field does not penetrate into the
superconductor. It behaves as a substance with zero magnetic permeability. Therefore
the external magnetic field in the superconductor repel each other. This property can be
utilized in developing frictionless bearings with magnetic lubrication for use in gyroscopes
and electric machines.
Helium, an expensive gas is now used to attain a low temperature, required for
superconductivity. Efforts have already being made to develop compounds which are superconductors
at temperatures possible to obtain by easily available hydrogen gas.
2.3.9 Squid
Squid is an acronym for superconducting quantum interference device and the arrangement is as
shown in Fig. 2.3.9. All squids make use of the fact that the maximum current in superconducting
ring that contains a Josephson junction varies periodically as the magnetic flux through the ring
changes. This periodicity is interpreted as an interference effect involving the wave functions of the
cooper pairs. It consists of a ring superconducting material having two side arms A and B which
act as entrance and exist for the supercurrent respectively.
P
l1
B
A
B
l2
Q
Josephson
junction
Fig. 2.3.9 A squid
54
ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
The insulating layers P and Q may have different thicknesses and let the currents through
these layers be I1 and I2 respectively. The variations I1 and I2 versus the magnetic field is shown
in Fig. 2.4.1. Both I1 and I2 vary periodically with the magnetic field, the periodicity of I1 being
greater than that of I2. The variation of I2 is an interference effect of the two junctions while I1
is the diffraction effect that arises from the finite dimension of each junction. Since the current is
sensitive to very small changes in the magnetic field, the SQUID can be used as a very sensitive
galvanometer.
OBJECTIVE TYPE QUESTIONS
2.1 Which one of the following is a trivalent metal?
(a) Na
(b) Cu
(c) Hg
(d) Al
2.2 The temperature coefficient of resistivity is
(a) σ
dσ
dT
(c) − σ
dσ
dT
(b)
1 dσ
σ dT
(d) −
1 dσ
σ dT
2.3 If j is the current density through a conductor in a field E, then
(a) j ∝
1
E
(c) j ∝ E
(b) j ∝ E
(d) j ∝
1
E2
2.4 R1 is the resistance of a copper wire of length l cm and radius r cm. If R2 is the resistance of another
R
copper wire of length l2 cm and radius 2r cm. Now 2 is
R1
(a)
l
2
(b)
l2
4
(c)
l
4
(d)
l
r
2.5 The length of copper wire is halved and diameter is doubled, the resistivity of copper
(a) reduces by 50%
(b) increases by 25%
(c) decreases by 60%
(d) remains the same
–
2.6 In the absence of an external field, the root mean square velocity c follows:
(a) c ∝ T
(b) c ∝
(c) c ∝ T 2
(d) c ∝
1
T
1
T4
2.7 The resistance of an aluminium wire of length l1 and radius r1 is R1 ohm. If the length is halved and
the radius doubled, the corresponding resistance R2 is
ELECTRON THEORY OF METALS
(a) 0.5R1
(c)
55
(b) 4R1
R1
8
(d) 16R1
2.8 The mobility of electron in a metal is 0.00132 m2/V-sec subjected to a field 50 V/m. The drift velocity
of the electron is
(a) 0.066 m/sec
(b) 6.66 m/s
(c) 66 m/sec
(d) 6.6 × 107 m/sec
2.9 At Fermi temperature the number of electrons excited from the occupied states to the unoccupied states
is
(a) 10%
(b) 20%
(c) 50%
(d) 100%
2.10 The formula for Fermi velocity is
(a) vF =
kTF
2m
(c) vF =
LM kT OP
N 2m Q
2kTF
m
(b) vF =
12
F
(d) vF =
m
2kTF
2.11 If the mobility of the electrons in a metal increases, the resistivity
(a) decreases
(b) increases
(c) first increases and then decreases
(d) first decreases and then increases
2.12 In thermionic emission the work function for a given current
(a) is the same for all metals
(b) it varies exponentially
(c) it is different for different metals
(d) depending on the electronic structure of the metal
2.13 Depositing a layer of thorium on tungsten will
(a) lower the work function
(b) raise the work function
(c) no effect on the work function
2.14 The Fermi energy of a metal is 1.4 eV, the Fermi temperature of the metal is approximately
(a) 1.6 × 103 K
(c) 1.6 ×
105
K
(b) 1.6 × 104 K
(d) 1.6 × 106 K
2.15 Superelectrons become normal electrons at
(a) zero degree kelvin
(b) 0°C
(c) the critical temperature
(d) Debye temperature
2.16 If the temperature of a metal is reduced to a value below the critical temperature, the value of the
critical magnetic field will
(a) increase
(b) decrease
(c) may increase or decrease
(d) keeps a constant value
56
ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
HINTS FOR SELECTED OBJECTIVE TYPE QUESTIONS
2.4
R1 =
ρl
and
πr 2
R2
ρl 2
=
R1
4 πr 2
F πr I
GH ρl JK
2
R2 =
=
ρl 2
b g
π 2r
2
=
ρl 2
4 πr 2
l
4
R2
l
=
Ans.
R1
4
2.6
P =
1 2
ρc
3
P =
1 mN A 2
c
3 Vm
if the volume is 1 m3
For molar volume
PVm =
c2 =
c– =
Hence
2.8
–
1
mN A c 2 = RT
3
3RT
3k T
=
mN A
m
3kT
m
c ∝
T
µ =
vd
E
vd = µE = 0.00132 × 50
vd = 0.066 m/s
2.14
EF = kTF
1.4 × 1.6 × 10 −19
EF
= TF =
1.38 × 10−23
k
TF = 1.62 × 104 K.
SHORT QUESTIONS WITH ANSWERS
2.1 Explain free electron gas model for metals.
Ans. In most solids the two main types of internal energy are (i) the vibrational energy of the atoms
about their mean lattice positions and (ii) the kinetic energy of the free electron. If heat is supplied
to a body its temperature rises and the internal energy increases. The important thermal properties
of matter such as heat capacity, thermal expansion and thermal conductivity depend upon the
changes in the energy of the atom and free electrons. If an electric field is applied to a metal the
free electrons are accelerated. Their kinetic energy increases. Some of the K.E is of course lost by
ELECTRON THEORY OF METALS
57
collisions with the atoms in the lattice. The resulting current is proportional to the average velocity
of the free electrons. The velocity is determined by the applied electric field and also the collision
frequency. Thus both thermal and electrical behaviour of matter depend on the free electrons in it.
2.2 Discuss some of the important factors which reduce electron mobility or scattering which in turn reduce
the mobility.
Ans.
(i) Thermal agitation: Increase in temperature increases the resistance or scattering which in
turns reduce the mobility.
(ii) Plastic deformation: Internal stresses are caused by cold working which distort the energy
levels at which electron transfer takes place. This is the other reason.
(iii) Impurities: Lattice distortion and impurities cause irregularities and hence reduce the mean
free path of the electrons and thus reduce the mobility of the electron.
2.3 Distinguish between the two velocities associated with the electron in metals.
Ans. The velocity, vd =
eEτ
is called drift velocity. This is super imposed on a much higher velocity
m
known as random velocity ( –c ) due to the random motion of the electron (thermal motion). Just as
in an ordimary gas, the electrons have random motion even in the absence of the field. This is due
to the fact that the electrons move about and occasionally scatter and change direction. The random
motion, which contributes zero current, exists also in the presence of the field, but in this case there
is an additional velocity opposite to the field.
2.4 Where metals and alloys are preferred?
Ans. Metals are chosen in places where high conductivity and large temperature coefficient of resistance
is desired. Alloys are chosen for heating purposes by I2R dissipation and for low temperature
coefficient of resistance.
2.5 Why electrons and neutrons are diffracted easily by metallic crystals?
Ans. Both neutrons and electrons have wave characteristics and the wavelength is of the order of grating
elements in crystals.
2.6 Discuss briefly the important properties of superconducting materials.
Ans.
(i) At room temperature, superconducting materials have greater resistivity than other elements.
(ii) The transition temperature Tc is different for different isotopes of an element. If decreases
with increasing atomic weight of the isotopes.
(iii) The superconducting property of a superconducting element is not lost by impurities to it but
the critical temperature is lowered.
(iv) There is no change in the crystal structure as revealed by x-ray diffraction studies. This means
that superconductivity may be more concerned with the conduction electrons than with the
atoms themselves.
(v) The thermal expansion and elastic properties do not change in the transition.
(vi) All thermoelectric effects disappear in superconducting state.
(vii) When a sufficiently strong magnetic field is applied to a superconductor below the critical
temperature, its superconducting property is destroyed. At any given temperature below Tc,
there is a critical magnetic field Hc such that the superconducting property is destroyed by the
application of a magnetic field. The value of Hc decreases as the temperature increases.
2.7 How will you know that a metal has become a superconductor?
Ans. The simple method to study this problem is discussed below:
A superconducting metal is connected with a source of electricity. A voltmeter is connected across
as shown in figure. When the material is in normal conducting state a p.d is maintained at the ends
58
ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
of the wire or conductor. When the material is cooled below its critical temperature Tc, the potential
disappears as shown in the figure below:
V
V=0
V
V
(b)
(a)
2.8 Show that the following two definitions of the resistivity ρ are equivalent.
(a) resistance of a bar =
(b) ρ =
ρ × length of bar
area of cross-section
electric field
current density
Ans. (a)
ρ =
R×a
l
...(1)
(b) Ohm’s law:
V = IR
R =
V
ρl
=
I
a
V =
Iρl
a
E =
V
Iρ
=
= ρj
l
a
ρ =
E
j
or
σ =
j
E
...(2)
Both equations are the same.
REVIEW QUESTIONS
2.1 Outline free electron model of metals. Derive an expression for the electrical conductivity of a metal
on the basis of classical free electron theory of metals.
2.2 What are the assumptions introduced by Drude-Lorentz to explain classical free electron theory of
metals? Discuss the achievements and failures of this model.
2.3 Discuss the failures of classical theory of free electrons with special reference to specific heat and
mean free path.
2.4 Obtain Wiedemann-Franz law. Give the significance of Lorentz number.
2.5 Enumerate the heat developed in a current carrying conductor and get the expression for the total
energy dissipated per unit volume per second.
ELECTRON THEORY OF METALS
59
2.6 Discuss the properties of metals and insulators bringing in the important applications of them.
2.7 Give an account of the phenomenon of superconductivity.
2.8 Discuss the significance of critical temperature, critical magnetic field and critical current density in
superconductors.
2.9 Distinguish between type I and type II superconductors with suitable diagrams.
2.10 List out the various applications of superconductors.
PROBLEMS AND SOLUTIONS
2.1 A copper wire of radius 1mm and length 10 metre carries a direct current of 5 ampere. Calculate the
drift velocity of electrons in copper if n = 5 × 1028/m3.
Sol. The expression for drift velocity is
vd =
eEτ
m
and
σ =
ne2 τ
m
i.e.,
τ =
or
...(1)
σm
ne2
eτ
σ
=
m
ne
Substituting this in Eqn. (1),
vd =
σE
ne
E =
V
IR
=
=
l
l
E =
I
σa
vd =
=
FG σ IJ FG I IJ
H ne K H σa K
FG I IJ ρl
H lK a
=
I
nea
5
e
5 × 1028 × 1.6 × 10−19 × π × 10−3
j
2
vd = 1.99 × 10−4 m/s.
2.2 In the first case a copper wire of cross-sectional area 10−4 m2 and in which there is a current of 200
ampere is considered. If the free electron density of copper is 8.5 × 1025/m3, compute the drift velocity.
If a p.d. of 1 volt is applied across 20 metre of this wire, express the new drift velocity in terms of the
one calculated in the previous case. Relaxation time is 10−14 sec.
Sol. Case 1: If I is the current flowing, then
I = n A vd e
vd =
I
200
=
nAe
8.5 × 1028 × 1.6 × 10 −19 × 10 −4
60
ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
vd = 1.47 × 10−4 m/s
Case 2:
E =
1
= 0.05 V/m
20
eEτ
1.6 × 10 −19 × 0.05 × 10 −14
1
= vd =
= 8.8 × 10−5
m
9.1 × 10 −31
vd
v1d
= 1.67 Ans.
2.3 Calculation of Fermi energy for some monovalent element yield the following results:
Metal
EF (eV)
Cu
Li
Rb
Cs
Ag
K
7.04
4.72
1.82
1.53
5.51
2.12
If the Fermi velocity of the electron in one of the metals of the above series is 0.73 × 106 m/s, identify
the metal. Also compute its Fermi temperature.
1
m vF2 = E
F
2
Sol.
i.e.,
EF =
FG 1 IJ
H 2K
9.1 × 10−31 × (0.73 × 106)2 joule
= 4.55 × 10−31 × 1012 × 0.732
=
4.55 × 0.732 × 10−19
1.6 × 10−19
EF = 1.5 eV Ans.
and
EF = k TF
or
TF =
1.5 × 1.6 × 10−19
EF
=
k
1.38 × 10 −24
TF = 1.75 × 104 K
Ans.
2.4 For a specimen of V3 Ga, the critical fields are respectively 1.4 × 105 and 4.2 × 105A/m for 14 K and
13 K. Calculate the transition temperature and critical fields at 0 K and 4.2 K.
Sol. The general formula is
L
b g MM FGH TT IJK
N
L
b0g MM1 − FGH 14T IJK
N
L
I
b0g MM1 − FGH 13
T JK
N
Hc (T) = H c 0 1 −
1.4 × 105 = H c
2
c
2
c
4.2 ×
105
= Hc
c
2
OP
PQ
OP
PQ
OP
PQ
ELECTRON THEORY OF METALS
61
1
=
3
eT
eT
j
− 13 j
2
c
− 14 2
Tc2
2
c
2
Tc2
3 Tc2 − 3 × 142 = Tc2 − 132
2 Tc2 = 3 × 142 − 132
Tc = 14.47 K
b g LMM
N
1.4 × 105 = H c 0 1 −
Hc =
14 2
14.47
2
OP
PQ
= Hc (0) × 0.16
1.4 × 105
= 2.3 × 106.
0.06
2.5 The resistivity of a superconductor becomes zero. Consequently the flux density is zero due to this abrupt
change. Prove that the superconductor behaves as perfect diamagnetic.
Sol.
The general equation is
B = µ0 (H + M)
Since
B = 0
H = − M or
χ =
M
= (µr
H
Thus
−1 = µr
or
µr = 0
−M
= 1
H
or
Μ
= −1
H
...(1)
− 1)
− 1
...(2)
Equations 1 and 2 show the possibility of perfect diamagnetism.
Some Important Tables
Table I: Critical field of some elements
Element
Al
BC (0) in 10−2 Wb/m2
0.99
Cd
0.30
Ga
0.51
Pb
0.51
Hg (α)
4.13
Hg (β)
3.40
Ta
8.30
Sn
3.06
62
ELECTRICAL AND ELECTRONICS ENGINEERING MATERIALS
Table II. Properties of some selected superconducting elements
Element
Tc (K)
Ho (A/m)
Type
Aluminium
1.196
0.79 × 104
1
Cadmium
0.52
0.22 × 104
Gallium
1.09
0.4 × 104
3.4
104
1
3.72
2.4 × 104
1
Mercury
4.12
104
1
Iridium
0.11
0.13 × 104
1
7.175
104
Indium
Tin
Lead
2.2 ×
3.3 ×
6.5 ×
Tc K
Ho A/m
Zinc
0.9
0.42 × 104
1
1
Tantalum
4.5
6.6 × 104
11
1
Thlium
2.4
1.4 × 104
11
Niobium
9.3
104
11
Rhenium
1.7
1.6 × 104
11
Thorium
1.4
104
11
Zirconium
0.8
0.37 × 104
11
0.7
104
11
1
Element
Osmium
15.6 ×
1.3 ×
0.5 ×
Type
ANSWERS TO OBJECTIVE TYPE QUESTIONS
2.1
2.6
2.11
2.16
(d)
(d)
(a)
(a)
2.2 (d)
2.7 (b)
2.12 (c)
2.3 (b)
2.8 (a)
2.13 (a)
2.4 (c)
2.9 (d)
2.14 (b)
2.5 (d)
2.10 (b)
2.15 (c)
GGG