T.Y. P 2 / 1 Quantum free electron theory N.Kapoor 1 Sommerfelds Free Electron Theory of Metals :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. Classical free electron theory permits all electrons to gain energy. But quantum free electron theory permits only a fraction of electrons to gain energy.The main assumptions of quantum free electron theory are,1. The energy levels of the conduction electrons are quantised.2. The distribution of electrons in various allowed energy levels occur as per Pauli exclusion principle.3. However, the following assumptions of classical electron theory continue to be applicable in quantum free electron theory also.4. The electrons travel in a constant potential inside the metal but stay confined within its boundaries.5. Both the attraction between the electrons and the lattice ions, and the repulsion between the electrons themselves are ignored.

Before we can calculate the density of carriers in metal, we have to find the number of available states at each energy. The number of electrons at each energy is then obtained by multiplying the number of states with the probability that a state is occupied by an electron. Since the number of energy levels is very large and dependent on the size of the metal, we will calculate the number of states per unit energy and per unit volume.

Calculation of the density of states

The density of states refers to the number of quantum states per unit energy. In other words, the density of states, denoted by, indicates how densely packed quantum states in a particular system. We will assume that the metal can be modeled as an infinite quantum well in which electrons with effective mass, m*, are free to move. The energy in the well is set to zero. The metal is assumed a cube with side L. This assumption does not affect the result since the density of states per unit volume should not depend on the actual size or shape of the metal. and E can be found by solving Schrdinger equation

The energy for an infinite walled box is

T.Y. P 2 / 1 Quantum free electron theory N.Kapoor 2 Treating the "quantum numbers" n as a space such that a given set of n values determines a point in that space, we can argue that the number of possible states is proportional to the "volume" in "n-space".Thus number of state between energy E & E+dE is the volume of the shell having radius n & thickness dnIt is convenient to define a radius n in n-space:

The Rayleigh scheme for counting modes.The n-space associated with the particle-in-a-box involves only positive values of n, so the volume must be divided by 8 (octant because ni 0). It then must be multiplied by 2 to account for the two possible spin values of the electron. The number of values is then Z(E) dE =But dE

Thus Z(E) dE =. dE = . dE

Thus number of states per unit volume ie density of state is g(E) dE =Z(E) dE/ volume = .dE= dEThis represents the number of electron states per unit volume per unit energy between energy E & E+dE. This energy density is a factor in many of the electical properties of solids. Note that the result is independent of the dimension L which was chosen above, showing that the expression can be applied to the bulk material. 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 f (E) is called the Fermi factor and is defined as follows:

T.Y. P 2 / 1 Quantum free electron theory N.Kapoor 3 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

T3 >T2 >T1=0Let us consider the different cases of distribution(i) Probability of occupation for E = EF at T =0 0 K At E =EF Thus, Fermi level represents the energy state with 50 probability of being occupied.(ii) Probability of occupation for E < EF at T = 00K 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 At T = 0 K, all the energy levels above the Fermi level are unoccupied. (iii)Probability of occupation at ordinary temperatures: At ordinary temperatures the value of f(E) starts reducing from 1 for values of E close to but less than EF. At E =EF at any temperature.Thus all states are filled up to an energy EF, known as Fermi energy,

T.Y. P 2 / 1 Quantum free electron theory N.Kapoor 4 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 productdensity of available states in the energy range E and E + dE probability of occupation of those energy levels by the electron.

g(E) 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

T.Y. P 2 / 1 Quantum free electron theory N.Kapoor 5 EF0= 3.65X10-19 n2/3 eVn density of e-s (varies from metal to metal). Hence EF also varies from metal to metal knowing n, we may calculate the Fermi level EF (EF numerically less than 10 eV).Fermi energy at T >0K: The Fermi energy EF, at any temperature T in general can be expressed in terms of T 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.

T.Y. P 2 / 1 Quantum free electron theory N.Kapoor 6which 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.

ElementNo. of ValenceelectronsFermi EnergyeVFermi Temperaturex 10^4 KFermi Velocityx 10^6 m/s

Li 14.745.511.29

Na 13.243.771.07

K12.122.460.86

Cu17.008.161.57

Ag15.496.381.39

Au15.536.421.40

Be214.316.62.25

Mg27.088.231.58

Ca24.695.441.28

Al311.713.62.03

Ga310.412.11.92

In38.6310.01.74

Pb49.4711.01.83

Average kinetic energy of electron at T=00K

Above EF0 f(E) =0 at 00K & below it is 1g(E) dE=

T.Y. P 2 / 1 Quantum free electron theory N.Kapoor 7But C= & Thus Thus K.E. at absolute zero is not zero.At any other temperature the average K.E. is given by Comparison Classical Free Electron Theory and Quantum Free Electron TheorySimilarities between the two theories The following assumptions apply to both the theories:1. The valence electrons are treated as though they constitute an ideal gas.2. The valence electrons can move freely throughout the body of the solid.3. The mutual repulsion between the electrons, and the force of attraction between the electrons and ions are considered insignificant.Difference between the two theoriesClassical Free Electron TheoryQuantum Free Electron Theory

1. The free electrons, which constitute the electron gas can have continuous energy values.1. The energy values of the free electrons are discontinuous because of which the energy levels are discrete.

1. It is possible that many electron may possess same energy.1. The free-electrons obey the Pauli exclusion principle. Hence no two electrons can possess same energy.

1. The pattern of distribution of energy among the free electron obey Maxwell-Boltzmann statistics.1. The distribution of energy among the free electrons is according to Fermi Dirac statistics which imposes a severe restriction on the possible way in which the electrons absorb energy from an external source.

T.Y. P 2 / 1 Quantum free electron theory N.Kapoor THERMIONIC EMISSION : RICHARDSON EQUATIONThe emission of the electrons from the metal surface under the effect of thermal energy is called thermoionic emission. The electrons so emitted are called the thermions.It is well known that the electrons in a metal are not completely free, but are bound to the metal surface by the attractive forces of atomic nuclei. Experimental observations show that the metals do not emit electrons till their temperature is sufficiently high. At ordinary temperatures, the electrons do not possess sufficient energy to overcome the attractive forces and to escape metallic surfaces.According to to Fermi-Dirac statistics, an electron at absolute zero possess a maximum energy equal to the Fermi energy EF which is not sufficient to escape out an electron from the metal. Further the free electron theory assumes that the potential within the metal is constant, being positive relative to the free space. If this constant potential is W, then W is the minimum energy of the electron for its emission from the metallic surface. Therefore the minimum energy to be supplied to the electron for its emission is (W-EF) and is defined as of the metal.When the metal is heated, the free electrons gain the thermal energy and when this thermal energy is more than , the electrons are emitted from the metallic surface. The thermal energy in excess of will be converted into the kinetic energy of the electrons. Richardson Equation:Let us now calculate the emission current density of the emitted electrons:In the free electron model of a metal the electrons are free to move throughout the volume V. According to Fermi-Dirac Statistics the number of electrons per unit volume having energy between E to E+dE is

The number of electrons per unit volume having the momentum range p to p+dp is

]

If px , py, pz be the components of momentum of the electron along X, Y and Z-axes such that p 2 = px2 + py2+ pz2 then the volume in the momentum space lying between px and px+dpx , py and py+dpy , pz and pz+dpz is dpx dpy dpz= 4p2 dpNumber of electrons per unit volume having momentum components lying between px and px+dpx, py and py+dpy, pz and pz+dpz is

[At 270C (300K), (E-EF) >> kT, hence 1 in denominator of Fermi Function is neglected]Number of electrons per unit volume having velocity components lying between vx and vx+dvx, vy and vy+dvy, vz and vz+dvz is

Now, suposing that the surface of the metal lies in Y-Z plane and X-axis is normal to this surface, the number of electrons per unit volume in velocity range vx and vx +dvx is

When the metal is heated only those electrons can escape from the surface along X axis which fulfil the condition Therefore, the number of electrons leaving unit area of the surface in unit time is

The emission current per unit area (i.e emission current density, J) is obtained by multiplying the number of electrons leaving unit area of the surface in unit time with the electric charge.

This equation is commonly known as Richardson-Dushman Equation and also known as Richardson Equation. This equation indicates that the emission current is exponentially dependent upon the work function (which is characteristics of the material because it is determined by the interatomic spacing) and inversely dependent upon the absolute temperature.

A plot of vs. is a straight line having a negative slope and an intercept equal to logA.