Engineering Physics - II - Unit 1 CONDUCTING MATERIALS Notes

Topic Covered :
            Conductors – classical free electron theory of metals – Electrical and thermal  conductivity – Wiedemann – Franz law – Lorentz number – Draw backs of  classical theory – Quantum theory – Fermi distribution function – Effect of  temperature on Fermi Function – Density of energy states – carrier concentration
in metals.


UNIT   I
CONDUCTING MATERIALS

INTRODUCTION
Materials can be broadly classified into three types based on conductivity. They are,         
1.   Conductors (Example: metals),
2.   Semi – conductors (Example: germanium, silicon) and
3.   Insulators (Example: wood, mica, glass).
CONDUCTORS:
  • Conductivity is the ability or power to conduct or transmit heat, electricity, or sound.
  • Conductors are materials that electricity easily passes through, that do not resist the flow of electricity.
  • Examples are copper, aluminum, steel, silver, gold, electrolytes. Not all materials conduct electricity equally well.
  
CLASSIFICATION OF CONDUCTORS:

                              Conducting materials are classified into three categories
           
             1.   Zero Resistivity materials:

                                 These materials conduct electricity with zero resistance below transition temperature.

 2.       Low Resistivity materials:

                                   These materials have very high electrical conductivity.

 3.       High Resistivity materials:
                            
                                    These materials have high resistivity and low temperature co-efficient of resistance.



Basic Terminologies

1.    Bound Electrons:
                     All the valence electrons in an isolated atom are bound to their parent nuclei are called as bound electrons.
2.     Free electrons:
                    Electrons which moves freely or randomly in all directions in the absence of external field.
3.     Drift Velocity
                    If no electric field is applied on a conductor, the free electrons move in random directions. They collide with each other and also with the positive ions. Since the motion is completely random, average velocity in any direction is zero. If a constant electric field is established inside a conductor, the electrons experience a force F = -eE due to which they move in the direction opposite to direction of the field. These electrons undergo frequent collisions with positive ions. In each such collision, direction of motion of electrons undergoes random changes. As a result, in addition to the random motion, the electrons are subjected to a very slow directional motion. This motion is called drift and the average velocity of this motion is called drift velocity (vd).
 
4.    Electric Field (E):
                       The electric field E of a conductor having uniform cross section is defined as the potential drop (V) per unit length (l).
                 i.e., E = V/ l    V/m
           
       5. Current density (J):
                         It is defined as the current per unit area of cross section of an imaginary plane holded normal to the direction of the flow of current in a current carrying conductor.
                               J = I/ A   Am-2
         
         6. Fermi level :
                           Fermi level is the highest filled energy level at 0 K.
             
       7. Fermi energy:
                           Energy corresponding to Fermi level is known as Fermi energy.

Electron Theory of metals:
  •        The electron theory of metals explain the following concepts. 
  •        Structural, electrical and thermal properties of materials.
  •        Elasticity, cohesive force and binding in solids.
  •        Behaviour of conductors, semi conductors, insulators etc.
So far three electron theories have been proposed.
  1. Classical Free electron theory:
        - It is a macroscopic theory.
        - Proposed by Drude and Loretz in 1900.
        - It explains the free electrons in lattice.

         - It obeys the laws of classical mechanics.
     
    2. Quantum Free electron theory:  
         
         -        It is a microscopic theory. 
      
        -        Proposed by Sommerfield in 1928. 

        -        It explains that the electrons moves in a constant potential. 

         -        It obeys the Quantum laws.
  
  3.     Brillouin Zone theory or Band theory: 

           -        Proposed by Bloch in 1928. 

           -        It explains that the electrons moves in a periodic potential. 
      
           -        It also explains the mechanism of semiconductivity , based on bands and hence called band theory.



CLASSICAL FREE ELECTRON THEORY OF METALS:
                                                                   This theory was developed by Drude and Lorentz  during 1900 All the atoms are composed of and hence is also known as Drude-Lorentz theory. According to this theory, a metal consists of electrons which are free to move about in the crystal like molecules of a gas in a container. Mutual repulsion between electrons is ignored and hence potential energy is taken as zero. Therefore the total energy of the electron is equal to its kinetic energy.
Postulates of Classical free electron theory:

 1.    All the atoms are composed of atoms. Each atom have central nucleus around which there are revolving electrons.
2.    The electrons are free to move in all possible directions about the whole volume of metals.
3.    In the absence of an electric field the electrons move in random directions making collisions from time to time with positive ions which are fixed in the lattice or other free electrons. All the collisions are elastic i.e.; no loss of energy.
4.    When an external field is applied the free electrons are slowly drifting towards the positive potential.
5.    Since the electrons are assumed to be a perfect gas they obey classical kinetic theory of gasses.
     6.    Classical free electrons in the metal obey  Maxwell-Boltzmann statistics. 

Drawbacks of Classical free electron theory:

      1) According to this theory, r is proportional to ÖT. But         experimentally it was found that r is proportional to T.
 2)   According to this theory, K/sT = L, a constant (Wiedemann-Franz law) for all temperatures. But this is not true at low temperatures.
 3)   The theoretically predicted value of specific heat of a metal does not agree with the experimentally obtained value.
 4)   This theory fails to explain ferromagnetism, superconductivity, photoelectric effect, Compton Effect and black body radiation.
5)   It is a macroscopic theory.
6)   Dual nature is not explained.
7)   Atomic fine spectra could not be accounted.
  
Merits of Classical Free Eletron Theory:
  1. It is used to verify Ohm's law.
  2. The electrical and thermal conductivities of metals can be explained.
  3. It is used to derive Wiedemann- Franz law
  4. It is used to explain the optical properties of metals.
Drawbacks of Classical Free Electron Theory:
  1.  It is a macroscopic theory.
  2. It cannot explain the electrical conductivity of semiconductors and insulators properly.
  3. Dual nature is not explained.
  4. It cannot explain the Compton effect,Photo-electric effect.
  5. The theoritical and experimental values of specific heat are not matched.
  6. Atomic fine spectra could not be accounted.
  7. Different types of magnetisms could not be explained satisfactorily by this theory.
Wiedmann-Franz law:

The ratio of thermal conductivity to electrical conductivity of a metal is directly proportional to absolute temperature.

                                          K/s is proportional to T

                                                   Or, 
                                                 K/sT = L, a constant called Lorentz number.
 

Quantum free electron theory:

  • Classical free electron theory could not explain many physical properties. 
  •  In 1928, Sommerfeld developed a new theory applying quantum mechanical concepts and Fermi-Dirac statistics to the free electrons in the metal. This theory is called quantum free electron theory. 
  • Classical free electron theory permits all electrons to gain energy. But quantum free electron theory permits only a fraction of electrons to gain energy. 
 
 
          Derivation of Electrical conductivity of a metal based on Drude & Lorentz theory:



Derivation of Thermal conductivity:







 
Fermi-Dirac Statistics:

           There are three statistics

1.     Maxwell- Boltzmann statistics
                               - Deals with particles which has no spin
                               - Eg: Gaseous particles

2.     Bose-Einstein statistics
                                - Deals with particles which has integral spin
                                - Eg: Photons
  
3.     Fermi-Dirac statistics
- Deals with particles which has half integral spin 
- Also known as Fermions 
-     Eg: Electrons


 Fermi-Dirac distribution function:




 
Effect of temperature on Fermi-Dirac distribution function:

             Fermi-Dirac distribution function is given by,
                                    
                                        f(E) = 1 / [1+e(E-EF/KT) ]

                        At T=0K, for EEF, f(E)=0
                        At T=0K, for E=EF, f(E)=indeterminate
                       At T>0K, for E=EF, f(E)=1/2

                All these results are depicted in the figure.



Importance of Fermi Energy: 
Fermi energy is used to seperate the vacant and filled states at
0 K.
  
It is used to know the status of the electrons. 

Electrons are completely filled below fermi energy level and completely empty above the fermi level at 0 K

Above 0 K some electrons absorb thermal energy and they jumps to the higher energy levels.

Density of States:
                      Density of states N(E)dE is defined as the number of energy states present per unit volume of a metal in an energy interval E and E+dE.

                                            
                                                         
 Density of states [N(E)dE] =
    No. of energy states available between E and E+dE in a metal piece
       ---------------------------------------------------------
                              Volume of that metal piece

 Derivation of Density of states:





 
Work Function:
           The minimum energy given to an electron in a metal to liberate it from the surface of that metal at absolute zero is called work function. It depends upon.
 1.The nature of the metal.
2.The surface conditions.
 There are four different ways of supplying energy to the electrons of a metal
* When the energy is supplied to the electrons thermally by heating the metal,then the work function is called thermionic work function.
* When the energy is supplied to the electrons optically by exposing it with the incident light , then the work function is called photoelectric work function.
* When the energy is received from electrons or ions that strike the metal surface from outside, then the work function is called secondary emission work function.
    * When the energy is received from the applied electric field, then the work function is called field emission work function.