In 1928, by applying quantum mechanical principles, Arnold Sommerfeld succeeded in overcoming many of drawbacks of the classical free electron theory, while retaining all the essential features of the classical free electron theory. His approach is based upon quantization of electrical energy levels. He realised the role played by Pauli exclusion principle in restricting the energy values of electron. The theory proposed known as Quantum Free Electron Theory
The main assumptions of quantum free electron theory are,
- The energy levels of the conduction electrons are quantised.
- The distribution of electrons in various allowed energy levels occur as per Pauli exclusion principle.
- However, the following assumptions of classical electron theory continue to be applicable in quantum free electron theory also.
- The electrons travel in a constant potential inside the metal but stay confined within its boundaries.
- Both the attraction between the electrons and the lattice ions, and the repulsion between the electrons themselves are ignored.
Due to the quantization rules, a material in solid state possess a set of allowed energy levels. For a metal containing N free electrons, there will be N such allowed energy levels, which are separated by energy differences that are characteristic of the material. The allowed energy values for the conduction electrons in the metal are also quantised, are related to the energy levels of the metal. As per the Pauli
exclusion principle, each energy level can accommodate a maximum of two electrons, one with spin up and the other one with spin down (Figure 11.1).
FIGURE 11.1 Fermi Energy
The allowed energy values of the electron are realised in terms of occupation of the allowed energy levels, under certain rules. When the filling up the energy levels is undertaken the universal rules that-any system which is free tends to go state of lowest energy-comes into picture.
Thus a pair of electrons, one with spin up, and other with spin down occupy the lowers level. The next pair of electrons occupy the next higher level, and so on, till all the electrons in metal are accommodated.
However, there will be many more allowed energy levels available for occupation by the electrons. The energy of the highest occupied level at zero degree absolute is called Fermi energy, and the energy level is referred to as the Fermi level. The Fermi energy is denoted as EF.
When there is no external energy supply for the electrons, such as thermal energy or electrical energy, the electrons are free, and thus settle in the lowest allowed energy state available. Thus at a temperature of absolute zero, and when the metal is not in the influence of any external field, all the energy levels lying above the Fermi level are empty, and those lying below are completely filled. Since there are two electrons in each energy level, out of the N allowed energy levels N/2 of them will be occupied.
The Fermi level is an energy level mid way between top surface of valence band and bottom surface of conduction-band as shown in Figure 11.2.
If EV is the energy corresponding to top surface valence band, EC is the energy corresponding to bottom surface of conduction band and EF is the Fermi energy corresponding to Fermi level, then
When the temperature is not at absolute zero, the material will be receiving thermal energy from surroundings. At ordinary temperature, the amount of energy will be quite small”, because of which the electrons occupying energy levels below the Fermi level cannot absorb the thermal energy. It is because, there is no availability of unoccupied higher energy levels into which the electron can come into, due to the increase in energy by the absorption of thermal energy. However, there are unoccupied higher energy levels which are above the occupied energy levels at small energy differences. They are located near Fermi level. Those are the energy levels into which, the electrons in the energy levels near Fermi level, are capable of being excited**.
During thermal excitation (i.e., T >0), the electrons which absorb the thermal energy move into higher energy levels which were unoccupied at zero degree absolute (i.e., T = 0). Though such excitations seem to be random, the occupation of various energy level obey a statistical distribution*** called Fermi-Dirac distribution, once the system is in thermal equilibrium (i.e., at a steady temperature state).
The probability f (E) that a given energy state with energy E is occupied at a steady temperature T, is given by
* The quantity of thermal energy is given by kbT, where kB is Boltzmann constant. A room temperature kBT=0.0025eV. Compare this with Fermi energy which is of the order of 5 eV.
** Since such electrons constitute only a small fraction of the total number of electrons in the metal as a whole, the specific heat of the material becomes low. Thus, reason for failure of classical free electron theory to account for low specific heat of metals is now understood.
*** The statistical distribution deals with the question as to what is the probability occupation of a particular energy level.
f (E) is called the Fermi factor and is defined as follows:
Fermi factor is the distribution function which gives the probability of occupation of a given energy state for material in thermal equilibrium in terms of the Fermi energy, Boltzmann constant and the temperature.
The dependence of Fermi factor on temperature, and the effect on occupancy of energy level is shown in Figure 11.3.
Let us consider the different cases of distribution
(i) Probability of occupation for E < EF at T = 0
When T = 0 and E < EF
At T = 0 K, all the energy levels below the Fermi level are occupied.
(ii) Probability of occupation for E > EF at T= 0 K
When T = 0, and E > EF
The Fermi energy is the most probable, or the average energy of the electron across which the energy transitions occurs at temperature above zero degree absolute, which result in the conductivity of the metal. This may be considered as the physical basis for the concept of Fermi energy.
Calculation of Fermi Energy at T = 0 K
Let the value of EF of 0 K be denoted as EF0 . Also, if we denote the number of electrons per unit volume, which possess energy only in the range E and (E + dE), by N(E) dE, then, N(E) dE is given by the product
density of available states in the energy range E and E + dE × probability of occupation of those energy levels by the electron.
But, probability of occupation of any given energy state by the electron is given by the Fermi factor f (E).
N(E) dE = g (E) dE × f (E)
The number of electrons per unit volume of the material n can be evaluated by integrating the above expression from E = 0 to E = Emax, where Emax is the maximum energy possessed by the electrons
Fermi energy at T >0K
The Fermi energy EF, at any temperature T in general can be expressed in terms of through the relation
(Fermi Temperature (TF)
Fermi temperature (TF) is the temperature at which the average thermal energy of the free electron in a solid becomes equal to the Fermi energy at 0K. But the thermal energy possessed by electrons is given by the product kBT.
which is quite an exaggerated temperature to be realized in practice.
Fermi Velocity (VF)
The energy of the electron, which are at the Fermi level is EF. The velocity of the electrons which occupy the Fermi level is called the Fermi Velocity vF.
Comparison Classical Free Electron Theory and Quantum Free Electron Theory
Similarities between the two theories
The following assumptions apply to both the theories:
- The valence electrons are treated as though they constitute an ideal gas.
- The valence electrons can move freely throughout the body of the solid.
- The mutual repulsion between the electrons, and the force of attraction between the electrons and ions are considered insignificant.
Difference between the two theories
Classical Free Electron Theory
Quantum Free Electron Theory