Monday, February 18, 2008

FREE ELECTRON THEORY

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Classical free electron theory of metals
This theory was developed by Drude and Lorentz 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.
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.
Consider a conductor subjected to an electric field E in the x-direction. The force on the electron due to the electric field = -eE.
By Newton’s law, -eE = mdvd/dt
dvd = -eEdt/m
Integrating,
Vd = -eEt/m + Constant
When t = 0, vd = 0 Therefore Constant = 0
Vd = -eEt/m --------------- (1)
Electrical conductivity

Consider a wire of length ‘dl’ and area of cross section ‘A’ subjected to an electric field E. If ‘n’ is the concentration of the electrons, the number of electrons flowing through the wire in dt seconds = nAvddt.
The quantity of charge flowing in time dt = nAvddt.e
Therefore I = dq/dt = neAvd
Current density J = I/A = nevd
Subsittuting the value of vd from (1),
J = nee Et/m = ne2Et/m --------------- (2)
By Ohm’s law, J = s E
Therefore s = J/E = ne2t/m -------------- (3)
Mobility of a charge carrier is the ratio of the drift mobility to the electric field.
µ = vd/E m2/Volt-Sec
Substituting vd from (1),
µ = et/m -------------- (4)
Substituting this in equation (3),
s = neµ ------------- (5)
Relaxation time and mean free path
When the field E is switched off, due to the collision of the electrons with lattice ions and lattice defects, their velocity will start to decrease. This process is called relaxation. The relaxation time(t) is the time required for the drift velocity to reduce to 1/e of its initial value. The average distance traveled by an electron between two consecutive collisions is called mean free path (l) of the electron.
l = vdt -------------- (6)
Temperature dependence
The free electron theory is based on Maxwell-Boltzmann statistics.
Therefore Kinetic energy of electron = ½ mvd2 = 3/2 KBT
Vd = Ö 3KBT/m
Substituting this in equation (6),
t = lÖ m/3KBT -------------- (7)
Since s = ne2t/m, s is proportional to Ö1/T
Or r is proportional to ÖT.
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.
L = 3KB2/2e2
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 (Wiedmann-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 blackbody radiation.
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. In order to determine the actual number of electrons in a given energy range(dE), it is necessary to know the number of states(dNs) which have energy in that range. The number of states per unit energy range is called the density of states g(E).
Therefore, g(E) = dNs/dE
According to Fermi-Dirac statistics, the probability that a particular energy statewith energy E is occupied by an electron is given by,
f(E) = 1 / [1+e(E-EF/KT) ]
where EF is called Fermi level. Fermi level is the highest filled energy level at 0 K. Energy corresponding to Fermi level is known as Fermi energy. Now the actual number of electrons present in the energy range dE,
dN = f(E) g(E)dE
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.
Classification of semiconductors on the basis of fermi level and fermi energy
In intrinsic semiconductors, the fermilevel lie exactly at the centre of the forbidden energy gap. In n-type semiconductors fermilevel lie near the conduction band. In p-type semiconductors fermilevel lie near the valence band.

Impurity levels
In extrinsic semiconductors, addition of impurities introduces new allowed quantum energy states in the forbidden energy band. The quantum state which appears as a single energy level is known as impurity level.
Impurity level is called donor level (Ed) in n-type semiconductors and lie just below the conduction band. Impurity level is called acceptor level (Ea) in p- type semiconductor and lie just above the valence band.

Band theory of solids
The atoms in the solid are very closely packed. The nucleus of an atom is so heavy that it considered to be at rest and hence the characteristic of an atom are decided by the electrons. The electrons in an isolated atom have different and discrete amounts of energy according to their occupations in different shells and sub shells. These energy values are represented by sharp lines in an energy level diagram.
During the formation of a solid, energy levels of outer shell electrons got split up. As a result, closely packed energy levels are produced. The collection of such a large number of energy levels is called energy band. The electrons in the outermost shell are called valence electrons. The band formed by a series of energy levels containing the valence electrons is known as valence band. The next higher permitted band in a solid is the conduction band. The electrons occupying this band are known as conduction electrons.
Conduction band valence band are separated by a gap known as forbidden energy gap. No electrons can occupy energy levels in this band. When an electrons in the valence band absorbs enough energy, it jumps across the forbidden energy gap and enters the conduction band, creating a positively charged hole in the valence band.the hole is basically the deficiency of an electron.
Classification of solids on the basis of energy bands
Insulator
Insulators are very poor conductors of electricity with resitivity ranging 103 – 1017 Ωm. In this case Eg ≈ 6eV. For E.g. carbon.
Semiconductor
A semiconductor material is one whose electrical properties lie between that of insulators and good conductors. Their forbidden band is small and resistivity ranges between 10-4to 103Ωm. Ge and Si are examples with forbidden energy gap 0.7eV and 1.1eV respectively. An appreciable number of electrons can be excited across the gap at room temperature. By adding impurities or by thermal excitation, we can increase the electrical conductivity in semiconductors
Conductor
Here valence band and conduction band overlap and there is no forbidden energy gap. Resistivity ranges between 10=9 to 10-4 Ωm. Here plenty of electrons are available for electrical conduction. The electrons from valence band can freely enter the conduction band.
Intrinsic semiconductor
A pure semiconductor free from any impurity is called intrinsic semiconductor. Here charge carriers (electrons and holes) are created by thermal excitation. Si and Ge are examples. Both Si and Ge are tetravalent. I.e. each has four valence electrons in the outermost shell. Consider the case of Ge. It has a total of 32 electrons. Out of these 32 electrons, 28 are tightly bound to the nucleus, while the remaining 4 electrons (valence electrons) revolve in the outermost orbit. In a solid, each atom shares its 4 valence electrons with its nearest neighbors to form covalent bonds.The energy needed to liberate an electron from Ge atom is very small, of the order of 0.7 eV. Thus even at room temperature, a few electrons can detach from its bonds by thermal excitation. When the electron escapes from the covalent bond, an empty space or a hole is created. The number of free electrons is always equal to the number of holes.
Extrinsic semiconductor
Extrinsic semiconductors are formed by adding suitable impurities to the intrinsic semiconductor. This process of adding impurities is called doping. Doping increases the electrical conductivity in semiconductors. The added impurity is very small, of the order of one atom per million atoms of the pure semiconductor. The added impurity may be pentavalent or trivalent. Depending on the type of impurity added, the extrinsic semiconductors can be divided into two classes: n-type and p-type.
n-type semiconductor
When pentavalent impurity is added to pure semiconductor, it results in n-type semiconducutor. Consider the case when pentavalent Arsenic is added to pure Ge crystal. As shown in the figure, four electrons of Arsenic atom form covalent bonds with the four valence electrons of neighbouring Ge atoms. The fifth electron of Arsenic atom is not covalently bonded, but it is loosely bound to the Arsenic atom. Now by increasing the thermal energy or by applying electric field, this electron can be easily excited from the valence band to the conduction band. Thus every Arsenic atom contributes one conduction electron without creating a positive hole. Hence Arsenic is called donor element since it donates free electrons. Since current carriers are negatively charged particles, this type of semiconductor is called n-type semiconductor.
p-type semiconductor
When trivalent impurity is added to pure semiconductor, it results in p-type semiconducutor. Consider the case when trivalent Boron is added to pure Ge crystal. As shown in the figure, three valence electrons of Boron atom form covalent bonds with the three neighbouring Ge atoms. There is a deficiency of one electron (hole) in the bonding with the fourth Ge atom. The Ge atom will steal an electron from the neighbouring Ge atom to form a covalent bond. Due to this stealing action, a hole is created in the adjascent atom. This process continues. Impurity atoms that contribute hole in this manner are called acceptors. Since current carriers are positively charged particles, this type of semiconductor is called p-type semiconductor.

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