unit i conducting materials -simple form

Unit I - Conducting Materials

UNIT ICONDUCTING MATERIALS

1.1 Introduction :

Metals: good conductors ,absence of energy ,presence of free electrons , low resistivity.

Resistance factors : Impurities ,Temperature , Number of free electrons.

The free electrons are flowing by the application of electrical field ( E ) or

potential difference ( V ) across the length of the wire ( l ) and it produces a current

density ( J ).

1.2 Mobility and Conductivity :

According to classical free electron theory the electrons in a metal make

random elastic collisions in all directions and the net current is zero. When a constant

electric field is applied to the metal, the electrons are accelerated towards the positive

of the field .

During their movement , the random collisions are produced and it has attained

a drift velocity Vd in the opposite direction of the applied electric field . The

electrons move with an average velocity is called drift velocity .The drift velocity

Vd is proportional to the applied electric field E. ie., Vd ∞ E , Vd = μ E

Where μ is called mobility of the electron . Mobility is defined as the drift

velocity gained by the electron per unit electric field. μ = Vd / E , m2 V-1 s-1

The steady state drift velocity of the electrons produces a current ( I ) .

{ Vd =

If ‘ n ’ is the concentration of free electrons then the current density ‘J ’

can be written as


Where σ is electrical conductivity . It can be written as ,

Thus the electrical conductivity is directly proportional to the mobility of

electrons. The Mobility of the electron depends on temperature μ ∞ 1 / T 3/2 . When

temperature of the metal increases, the mobility of the electron decreases and hence the

electrical conductivity decreases .The addition of impurities in the metal decreases the

electrical conductivity.

1.3 Classical free electron theory ( Drude - Lorentz Theory ):

It is a Macroscopic theory developed by Drude and Lorentz in 1900.

A solid metal has nucleus with revolving electrons.

The electrons move freely like molecules in a gas.

The free electrons in a metal are moving in a uniform potential field due to fixed

ions.

In the absence of electric field ( E = 0), the electrons have random collisions

called elastic collisions (no loss of energy ).

In the presence of electric field (E ≠ 0 ), the free electrons are accelerated

opposite to the applied electric field.

The electron obeys Max well – Boltzmann distribution of velocities .

The electron obeys the laws of kinetic theory of gases.


Mean collision time (τc ) is the time taken by the electron between two

successive collisions.

Relaxation time (τ) is the time taken by the electron to reach equilibrium

position from its disturbed position in the presence of electric field. It is

approximately is equal to 10 -14 sec.

Drift velocity ( Vd ) is the velocity acquired by an electron in a perpendicular

direction due to the application of electric field.

Uses of Classical free electron theory :

It is used to verify the Ohm’s law.

It is used to explain electrical conductivity (σ ) and thermal conductivity (K) of

metals.

It is used to derive Widemann- Franz law.

It is used to explain the optical properties of metal.

Drawbacks of Classical free electron theory

1. According to classical free electron theory, all free electrons will absorb energy but

quantum free electron theory states that only few electrons will absorb energy.

2. This theory cannot explain the Compton effect , photo-electric effect , para-

magnetism and ferromagnetism, etc.,

3. The theoretical and experimental values of specific heat and electronic specific

heat are not matched.

4.By classical theory K / σT is constant for all temperatures, but by quantum

theory K / σT is not a constant for all temperatures.

5.The Lorentz number obtained by classical theory does not have good agreement

with experimental value and it is rectified by quantum theory.

1.4 Expression for electrical conductivity and thermal conductivity of metals

on the basis of free electron theory :

Electrical Conductivity :

The electrical conductivity is defined as the quantity of electricity flowing per unit area per unit time at a constant potential gradient.


Let ‘n’ be the no. of electrons per unit volume , ‘ e ‘ be the charge of the electrons , Vd be the drift velocity and τ be the relaxation time. Then, We can write the Current density J = n Vd (- e ) ---------(1)

Due to electric field E, electron gains acceleration ‘a’ =

drift velocity Vd = a τ -----------------(2)

The force experienced by the free electron , F = - eE --------------(3)

From Newton’s second law, we can write F = m a ---------------(4)

On comparing (3) and (4) , we have, - e E = m a

a = ------------ (5)

equation (2) becomes Vd = -----------(6)

substitute equation (6) in (1) J = n Vd (-e )

J =

J =

-------- ( 7 )

The electrical conductivity, σ =

σ =


Correct expression for electrical conductivity of conductors :

By using the classical free electron theory , quantum free electron theory and band theory of solids we can get ,



where m* is called as effective mass of free electron

Thermal conductivity of metals :

The Thermal conductivity is defined as the amount of heat flowing per

unit area per unit temperature gradient. K =

Expression for thermal conductivity (K) of an electron:

Consider a metal bar with two planes A and B separated by a distance ‘λ’ from C

Here T1 is hot end and T2 is cold end. ie., T1 > T2 .

A C B

Direction of flow of heat

T1 2 T2

Let ‘ n’ be the number of conduction electrons and ‘v’ be the velocity of the electrons. KB is the Boltzmann constant ( 1.38 x 10 -23 )

From kinetic theory of gases,

The kinetic energy of an electron at A = ½ m v2

= --------- [1]

The kinetic energy of an electron at B = ½ m v2

= ----------[2]

The net energy transferred from A to B = --------[3]


No. of electrons crossing per unit area = ---------------[4]

in unit time from A to B

The energy carried by the electrons from A to B, Q =

Q = ------- [5]

we know that the thermal conductivity, K =

The heat energy transferred /unit sec / unit area Q = [ A =1 unit area

------------ [6]

comparing equations (5) and (6) , =

Thermal conductivity

WIEDEMANN – FRANZ LAW :

Statement : It states that the ratio of thermal conductivity K to the electrical conductivity

σ is constant at all temperature .

Derivation:

We have

Thermal conductivity



but ,

L is called Lorentz number

2.44 x 10-8 W Ω K - 2

Lorentz number L =

L =

L = 1.12 x10 -8 w Ω K - 2

It is found that the classical value of Lorentz number is only one half of the experimental value ( 2.44 x10 -8 ) .

The discrepancy of L value is the failure of the classical theory. This can be rectified by quantum theory.


By Quantum theory :Here, mass ‘m’ is replaced by effective mass m*

The electrical conductivity σ =

By substituting the electronic specific heat , the thermal conductivity K can be written as

K =

=

Where

L = 2.44 x10 -8 w Ω K - 2

Thus the quantum theory verifies the Wiedmann – Franz law . i.e., it has good agreement with the experimental and theoretical values.

According to quantum free electron theory , the electrons in a metal were assumed to be moving in a region of constant potential but it fails to explain why some solids behave as conductors , some as insulators and some as semiconductors.

Therefore instead of considering an electron to move in a constant potential,the Zone theory of solids tells that the electrons are assumed to move in a field of periodic potential.

1.5 Fermi – Dirac distribution function.

It is an expression for the distribution of electrons among the energy levels as

a function of temperature.


The probability of finding an electron in a particular energy state of energy E is given by,

Where , E F - Fermi energy ( highest energy level of an electron )

K - Boltzmann’s constant

T - Absolute temperature.

Effect of Temperature on Fermi Function :e0

= 1Case 1: At T = 0 K and E < EF e -∞ = 0

e ∞ = ∞

F ( E ) = 1 = 100 % It means that 100 % chance to find the particle [ electron ] At zero Kelvin all energy states below E F are occupied by electrons .

Case 2: At T = 0 K and E > EF

F ( E ) = 0 = 0 %

It means that 0 % chance to find the particle [electron ] At zero Kelvin all energy states above E F are empty.

Case 3 : At T > 0 K and E = EF

= = 0.5

F ( E ) = 0.5 = 50 %

It means that 50 % chance to find the particle [electron ] At zero Kelvin energy states above E F are empty and below EF are filled.

E EF


Where EF is Fermi energy or Fermi Level .

It is defined as the highest energy level filled by the electrons

in that energy level with higher energy values.

Fermi level :The Fermi level is the highest reference energy level of a particle at

absolute zero.

Importance : It is the reference energy level which separates the filled energy

levels and vacant energy levels.

Fermi energy [ EF] : It is the maximum energy of the quantum state corresponding

to Fermi energy level at absolute zero.

Importance :Fermi energy determines the energy of the particle at any temperature.

Variation of Fermi function with temperature :

EF at O K 1

F(E) 0.5 EF at T K

0 Energy

The Fermi function varies with respect to the temperature . At 0K all the

energy states below EF are filled and all those above EF are empty .When the temperature is increased, the electron takes an energy KT and

hence the Fermi function falls to zero.

1.6 Density of states for a metal :

Definition : It is defined as the number of energy states per unit volume in an energy interval of a metal

nz

dn

E+dE

ny

nx


Number of energy states between E and E + dE D(E) dE = Volume of the metal

The Number of electrons in the energy interval E and E + dE ,

N (E) dE = D(E) dE . F ( E )

Derivation for Density of States :

Consider a metal of cubical shape with sides ‘L’.

Consider a sphere with nx , ny ,nz as three coordinate axes.

Let n2 = nx2 + ny

2 + nz

2 be the radius of the sphere.

The sphere contains a series of shells with radius ‘n’ and energy ‘E’.

No. of states in shell of thickness ‘dn’ in the energy interval E and E + dE , is

D(E)dE = ----------------------(1)

In equation (1) the Value represents the positive integers in the octant of the

sphere.

We know that the allowed energy values is ----------(2)

Differentiating equation (2) with respect to ‘ n ‘

--------(3)

E n o


from equation (2)

------- (4)

On substituting eqs (4) and (3) in eq. (1) we get ,

If volume of the metal , V = L3

For unit volume of a metal , V = 1 m3 :

Density of States , --------------(5)

1.7 Carrier Concentration in Metals :

Carrier concentration in metals is the number of electrons N(E) per unit

volume in the energy interval E and E + dE .

Each electron energy level can accommodate two electrons as per Pauli’s

exclusion principle.

We have, N (E) dE = D(E) dE . F ( E )

F(E)

The actual number of electrons in dE ,


Calculation of density of electrons and Fermi energy at 0K :

At T = 0K, F(E) = 1

=

Normally,

N = number of free electrons per atom x Density x Avogadro

Number

Atomic Weight

Hence the Fermi energy of a metal depends only on the density of electrons of

that metal.

Energy Distribution of electrons in metals:

The energies possessed by electrons in a metal is given by the energy

distribution function.

Density of energy states

Energy Distribution graph:

0o K

N( E ) 2500 o K

E


EF

The energy distribution for the tungsten for T = 0oK and T = 2500 o K.

The area of the curve gives the total number of particles per unit volume .

Form the graph , it is clear that even for larger variation of temperature

(2500oK) the distribution function changes very slightly.

When temperature increases , only the electrons with near EF are moved to

the higher energy level and lower energy electrons are not disturbed.

Appendix:

Work Function :

It is defined as the minimum energy required to remove an electron from the

metal surface at 0 K.

Explanation :

Let EF be the maximum energy of an electron called Fermi energy and EB be

the energy of the metal barrier surface .

If we supply energy EB greater than EF , then no electron escapes from

the metal .

In order to make it to escape, an additional amount of energy equal to

(EB - EF ) is required . i.e., EB + ( EB - EF )

This difference in energy EW = EB - EF is called Work function.

This concept is shown in energy diagram.

Different metals have different work function.

Metal surface

EB - EF

Energy EB

eV EF


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SOLVED PROBLEMS

1 .The following datas are given for copper (i) Density =8.92x103 kg/m3 (ii) Resistivity =1.73 x108 Ωm (iii) Atomic weight =63.5. Calculate the mobility and the average time collision of electrons in copper obeying classical laws. Solution:

n = 6.023 x 1023 x

n = 8.46 x 1025/m3

σ = =

We know


ASSIGNMENT PROBLEMS

1. A Copper wire whose is 0.16 cm carries a steady current of 10A. What is the current density of wire? Also calculate the drift velocity of the electrons in copper (Ans: J =497.6 A/m2, and vd=3.6x10-4m/s)

2 .The thermal and electrical conductivities of Cu at 200 C are 390 Wm-1 K-1 and 5.87x10-7Ω-1 m-1 respectively. Calculate the Lorentz number. (Ans: 2.267x10-8WΩ -2)

3. Calculate the electrical and thermal conductivities of a metal rod with relaxation time 10-14 second at 300K.Also calculate the Lorentz number. Given :density of electrons=6x1028/m3

4. Calculate the drift velocity and mean free path of copper when it carries a steady current of 10 amperes and whose radius is 0.08 cm. Assume that the mean thermal velocity = 1.6x106 m/s and the resistivity of copper = 2x10-8 ohm m. {Ans. (i) 36.6x10-5 m/s (ii) 3.94x10-8 m}

5. The resistivity of aluminium at room temperature is 2x10-8 ohm m. Calculate i) The drift velocity ii) their mobility and iii) their relaxation time and iv) mean free path on the basis of classical free electron theory. { Ans. (i) 0.396 ms-1; ii) 3.96x10-3 m 2 V -1 s-1 iii) 2.3x10-14 s: iv) 2.65 nm}

6. Using the Fermi function, evaluate the temperature at which there is 1% probability in a solid will have an energy 0.5 e V above EF of 5 e V. {Ans; (1260 K)

7. Use the Fermi distribution function to obtain the value of F(E) for E - EF = 0.01eV at 200 K. (Ans:F(E) = 0.36)

8. Assuming the electrons to be free, calculate the total number of states below E = 5eV


in a volume 10-5 m3. (Ans: 5.1 * 1023)

Part AQuestions & Answers

1.What is a conducting material ?Metals are good conductors due to the absence of energy gap and the

presence of free electrons with low resistivity.

2.Define drift velocity of electrons.

Drift velocity ( Vd ) is the velocity acquired by an electron in a perpendicular

direction due to the application of electric field.

3.Define mobility of electrons.

It is the drift velocity per unit electric field , μ = and also we can written

the mobility μ = , μ =

4..Define mean free path. It is the average distance travelled by an electron between two successive

collisions in the presence of electric field.

5. Define Mean Collision time (τc ) ?

It is the time taken by the electron between two successive collisions.

6.Define relaxation time.

Relaxation time (τ) is the time taken by the electron to reach equilibrium position


from its disturbed position in the presence of electric field. It is approximately is

equal to 10 -14 sec.

7.Define electrical conductivity.

The electrical conductivity is defined as the quantity of electricity flowing per

unit area per unit time at a constant potential gradient.

8. Define Thermal Conductivity .

Thermal conductivity of material is defined as the amount of heat flowing

through an unit area of a material per unit temperature gradient.

9. State Widemann- Franz law .

It states that the ratio of thermal conductivity to electrical conductivity is

constant at all temperatures . a constant

10.Mention any four postulates of classical free electron theory.

A solid metal has nucleus with revolving electrons.

The electrons move freely like molecules in a gas.

The free electrons in a metal are moving in a uniform potential field due to fixed

ions.

In the absence of electric field ( E = 0), the electrons have random collisions

called elastic collisions (no loss of energy ) and in the presence of electric field

(E ≠ 0 ), the free electrons are accelerated opposite to the applied electric field.

11. What are the Sources of resistance in metals ?

The resistance in metals is due to i). Presence of impurities in the metals.

ii). Temperature of the metal. iii). Number of free electrons.

12.What is the effect of temperature on metals ?

When temperature of the metal increases, the mobility of the electron decreases

and hence the electrical conductivity decreases . The addition of impurities in the metal

decreases the electrical conductivity.

13. What are the uses of classical free electron theory ?

It is used to verify the Ohm’s law.


It is used to explain electrical conductivity (σ ) and thermal conductivity (K) of metals.

It is used to derive Widemann- Franz law. It is used to explain the optical properties of metal.

14. What are the drawbacks of classical free electron theory ?

There is a difference in theoretical and experimental values of specific heat of metals and electronic specific heat of metals

It cannot explain the electrical conductivity (σ ) of semiconductors and insulators.

As per classical theory is constant at all temperatures. But at low

temperature is not a constant.

The theoretical paramagnetic susceptibility ( χ ) is greater than the experimental value.

Photoelectric effect , Compton effect and Black body radiation phenomenons cannot be explained by this theory.

15. What is Lorentz Number ?

Lorentz number L =

L =

L = 1.12 x10 -8 W Ω K - 2

It is found that the classical value of Lorentz number is only one half of the experimental value ( 2.44 x10 -8 ) .The discrepancy of L value is the failure of the classical theory. This can be rectified by quantum theory.

16 . What is the basic assumption of Zone theory or Band theory of solids ?According to quantum free electron theory , the electrons in a metal were

assumed to be moving in a region of constant potential but it fails to explain why some solids behave as conductors , some as insulators and some as semiconductors.

Therefore instead of considering an electron to move in a constant potential,the Zone theory of solids tells that the electrons are assumed to move in a field of periodic potential.

17. Define Fermi level and Fermi energy with its importance .


Fermi level :The Fermi level is the highest reference energy level of a particle at

absolute zero.

Importance : It is the reference energy level which separates the filled energy

levels and vacant energy levels.

Fermi energy [ EF] : It is the maximum energy of the quantum state corresponding

to Fermi energy level at absolute zero.

Importance :Fermi energy determines the energy of the particle at any temperature.

18.Define Fermi Distribution function .

It is an expression for the distribution of electrons among the energy levels as

a function of temperature and it is the probability of finding an electron in a

particular energy state of energy E is given by,

Where E F - Fermi energy ( highest energy level of an electron )

K - Boltzmann’s constant and T - Absolute temperature

19. Draw the Fermi distribution curve at 0 K and at any temperature TK [or] How does the Fermi function varies with temperature.

EF at O K 1

F(E) 0.5 EF at T K

0 Energy

The Fermi function varies with respect to the temperature . At 0K all the

energy states below EF are filled and all those above it are empty .


When the temperature is increased, the electron takes an energy KT and hence the Fermi function falls to zero.

42. Define density of states . Density of states is defined the as the number of energy states per unit volume in

an energy interval of a metal. It is use to calculate the number of charge carriers per unit volume of any solid.

Number of energy states between E and E + dE D(E) dE = Volume of the metal

The Number of electrons in the energy interval E and E + dE N (E) dE = D(E) dE . F ( E )

43.Define work function .

It is defined as the minimum energy required to remove an electron from the

metal surface at 0 K. In order to make it to escape, an additional amount of energy

equal to (EB - EF ) is required . i.e., EB + ( EB - EF ) . This difference in energy EW

= EB - EF is called Work function.

Part - B questions

1. Explain the mobility and conductivity in metals

2. Write down the postulates of classical free electron theory with its uses and

drawbacks .

3. Derive an expression for Thermal conductivity of metals and hence derive

Widemann- France law .

4. Write Fermi – Dirac distribution function. Explain how Fermi function varies with

temperature

5. Obtain an expression for the density of states for a metal and hence obtain the Fermi

energy in terms of free electrons.

6. Explain the energy distribution of electrons in terms of number of electrons . Also

explain the work function in a metal.

unit i conducting materials -simple form

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