CLASSICAL FREE ELECTRON THEORY OF METALS
Free electron concept
All metal atoms consist of valence electrons which are responsible for electrical conduction. Valence electrons are loosely bound to the nucleus. When a large number of atoms join to form a metal, the boundaries of the neighbouring atoms overlap, and thus, valence electrons can move easily throughout the body of the metal. These electrons are called free electrons or conduction electrons which account for properties such as electrical conductivity, thermal conductivity, opacity, surface luster etc.
Classical free electron theory
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 body of the metal like molecules of a gas in a container.
Assumptions
1. A metal is imagined as a structure of 3-dimensional array of ions between which, there are freely moving valence electrons (conduction electrons) confined to the body of the material.
2. 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.
3. The free electrons are treated as equivalent to gas molecules and thus they are assumed to obey the laws of kinetic theory of gases. In the absence of field, the energy associated with each electron at a temperature T is given by 3/2 kT.
4. 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.
Mean collision time and mean free path
Mean collision time (t) is the average time that elapses between two consecutive collisions of an electron with lattice ions.
Average distance traveled by the conduction electrons between successive collisions is called mean free path (l). If ‘v’ is the total velocity of an electron, then mean free path,
l = v t
Relaxation time (tr)
When the field E is withdrawn, 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(tr) is the time required for the drift velocity to reduce to 1/e of its initial value.
If the probability of scattering of electrons by lattice ions is same in all directions, mean collision time can be treated as relaxation time.
Expression for drift velocity
Consider a conductor subjected to an electric field E in the x-direction. The force on the electron due to the electric field = eE.
In the steady state, let ‘vd’ be the drift velocity and ‘t’ the mean collision time.
By Newton’s law, eE = ma = m vd/t
vd = eEt/m --------------- (1)
Expression for electrical conductivity in metals
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 = ne eEt/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)
Effect of temperature and impurity on electrical resistivity of metals (Matthiessen’s rule)
The variation of electrical resistivity (r) with temperature T for a metal is shown below:
Resistivity arises due to scattering of conduction electrons. In metals, two types of scattering mechanisms exist..
1. Resistivity rph due to scattering of electrons by lattice vibrations (phonons) which is temperature dependent and is called ideal resistivity.
2. Resistivity ri due to the scattering of the electrons by the presence of impurities and imperfections. This resistivity is temperature independent and exists even at 0K. Hence it is called residual resistivity.
The total resistivity r of a material is given by,
r = rph + ri
This is called Matthiessen’s rule. Matthiessen’s rule states that the total resistivity of a metal is the sum of the resistivity due to phonon scattering (temperature dependent) and the resistivity due to scattering by impurities (temperature independent).
At low temperatures, lattice vibration is negligible and phonon scattering is very less.
\At low temperatures, r » ri
At high temperatures, lattice vibration becomes very significant and resistivity becomes linearly dependent on temperature.
\At high temperatures, r » rph
Failure of classical free electron theory
1. Temperature dependence of electrical resistivity
According to kinetic theory of gases,
Kinetic energy of electron, ½ mv2 = 3/2 kT
But electrical conductivity s = ne2t/m and mean free path, l = vt,
\s = ne2l/mv
But experimental observation is r µ T
Thus in this case, classical free electron theory is not agreeing with the experimental observations.
2. Specific heat
The theoretically predicted value of specific heat (Cv = 3/2 R) of a metal does not agree with the experimentally obtained value (10-4 RT). Experimentally observed value of specific heat is far lower than expected value.
3. Wiedmann-Franz law
According to Wiedmann-Franz law, the ratio of thermal conductivity to electrical conductivity of a metal is directly proportional to absolute temperature.
i.e, K/sT = constant for all temperatures.
But this is not true at low temperatures.
4. Dependence of electrical conductivity on electron concentration
As per the classical free electron theory, s = ne2t/m
Or, s µ n
But experimental observation disagrees with this.
5. Positive Hall co-efficient of Zinc
Metals are expected to exhibit negative Hall co-efficient since current carriers in them are electrons. The free electron theory cannot explain why Zinc and some other metals have positive Hall co-efficient.
6. The free electron theory cannot explain the classification of materials into conductors, semiconductors and insulators.
7. 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 classical free electron theory, we use Maxwell-Boltzman statistics which permits all the free electrons to gain energy. In 1928, Sommerfeld developed a new theory, in which he retained some of the features of classical free electron theory and included quantum mechanical concepts and Fermi-Dirac statistics to the free electrons in the metal. This theory is called quantum free electron theory. Quantum free electron theory permits only a fraction of electrons to gain energy. The main assumptions of this theory are:
1. The energy values of conduction electrons are quantized and are realized in terms of a set of energy levels.
2. The distribution of electrons in various allowed energy levels takes place according to Pauli’s exclusion principle.
3. The electrons move in a constant potential inside the metal and are confined within defined boundaries.
4. The attraction between the electrons and the lattice ions and the repulsion between the electrons themselves are ignored.
Fermi-Dirac statistics
According to Fermi-Dirac statistics, the probability that an electron occupies an energy level E at thermal equilibrium is given by,
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. Fermi energy is the maximum energy that a free electron can have in a conductor at 0K. The probability f(E) is known as Fermi factor.
Effect of temperature on Fermi factor
Fermi factor is given by,
At T=0K, for E <>F, f(E)=1
At T=0K, for E >EF, f(E)=0
At T=0K, for E = EF, f(E)= indeterminate
At T>0K, for E=EF, f(E)= ½. All these results are depicted in the figure.
Density of states
The density of states is defined as the number of available electron states per unit volume per unit energy range centered at a certain energy level E. It is given by,
Evaluation (Derivation) of density of states for electrons
We can consider the free electrons in a metal as particles in a 3-dimensional potential well. The equation for the energy of free electrons can be written as,
where nx, ny and nz are positive integers, ‘m’ the mass of the electron, ‘h’ the Planck’s constant and ‘L’ the length of the potential well.
---------- (1)
Since equation (1) represent the equation of a sphere, all points on the surface of a sphere of radius R have same energy. All points within the sphere represent quantum states with energies smaller than E. Since nx, ny and nz are positive integers, the number of quantum states with energy equal to or smaller than E is equal to the volume of the first octant (where nx, ny and nz are positive integers) of the sphere.
There can be two electrons per energy state. Therefore the number of allowed energy states is 2 ×pR3/6 = pR3/3 ---------- (3)
On differentiation, number of energy states in the energy range dE is obtained.
\ Density of states which is the number of available electron states per unit volume per unit energy range is given by,
Number of electrons present in the energy range dE
Number of electrons present in the energy range dE is given by,
dN = g(E) dE f(E) where f(E) is the probability of finding an electron in this energy range or Fermi factor.
Calculation of Fermi energy at 0K
Number of free electrons per unit volume or carrier concentration,
AtT = 0K, f(E) = 1
Expression for electrical resistivity/conductivity and its temperature dependence
According to Quantum free electron theory, electrical conductivity of a metal is given by,
s = ne2tF/m
Here tF = l/vF , where vF is the Fermi velocity which is the velocity of electrons occupying Fermi level and l the mean free path.
\EF = ½ mvF2
Or, vF = (2EF/m)½
Now electrical conductivity can be expressed as,
s = ne2l/mvF ---------(1)
Here vF is independent of temperature but l is temperature dependent. As the temperature increases, the lattice ions start vibrating with larger amplitudes and offer scattering of the electrons which results in the reduction in the value of mean free path of the electrons.
i.e., l µ 1/T
But from equation (1), s µ l
\s µ 1/T and r µ T
Classical free electron theory and quantum free electron theory-Comparison
Similarities
1. Valence electrons are treated as gas molecules are treated as gas molecules of an ideal gas which are free to move throughout the body of the solid.
2. The mutual repulsion between the electrons and the force of attraction between the electrons and ions are considered insignificant.
Differences
1. According to classical free electron theory, free electrons obey Maxwell-Boltzman statistics. According to quantum free electron theory, free electrons obey Fermi-Dirac statistics.
2. According to classical free electron theory, free electrons can possess any energy values and it is possible that many electrons possessing same energy. According to quantum free electron theory, free electrons can occupy certain energy levels with discrete energy values and they obey Pauli’s exclusion principle. Hence no two electrons possess same energy.
Merits of quantum free electron theory
1. Temperature dependence of electrical resistivity/conductivity
According to Quantum free electron theory, electrical conductivity of a metal is given by,
s = ne2tF/m
Here tF = l/vF , where vF is the Fermi velocity which is the velocity of electrons occupying Fermi level and l the mean free path.
\EF = ½ mvF2
Or, vF = (2EF/m) ½
Now electrical conductivity can be expressed as,
s = ne2l/mvF ---------(1)
Here vF is independent of temperature but l is temperature dependent. As the temperature increases, the lattice ions start vibrating with larger amplitudes and offer scattering of the electrons which results in the reduction in the value of mean free path of the electrons.
i.e., l µ 1/T
But from equation (1), s µ l
\s µ 1/T and r µ T
This result is in accordance with experimental observation.
2. Specific heat
According to classical free electron theory, specific heat Cv = 3/2 R. But experimentally observed value is around 10-4 RT which is very low. According to classical free electron theory, all electrons are capable of absorbing heat energy. Thus the theory predicts a large value of specific heat. But according to quantum free electron theory, only those electrons which occupy energy levels close to Fermi level EF absorb heat energy and hence specific heat value becomes very small.
3. Thermionic emission
When a metal surface is heated it emits electrons. This is called thermionic emission. Here heat provides enough energy for an electron to escape from the surface. The emitted electron current density is given by,
J = AT2 e –W/kT
where ‘T’ is the metal temperature in Kelvin, ‘W’ the work function of the metal, ‘k’ the Boltzman constant and ‘A = 1.20173 × 106 Am-2 k-2’ is the Richardson constant. This relation is called Richardson-Dushman equation which obeys well as per quantum free electron theory.
Demerits of quantum free electron theory
1. It fails to explain properties of metal alloys.
2. It fails to explain why only some solids are metals and others are either semiconductors or insulators.
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