44. INTRODUCTION inthis chapter the band theory ory of solids, the distinction further, based on FermiDirac of solids is brief among metals, Statistics the expressi A : prained. The effect of Tession for density of energy states in a metal isobtained. The ae temperature on the Fermi distribution function is also discussed. Finally, abrief account of work function is given, : ° y discussed qualitatively. Based on the band » Semiconductors and insulators is accounted for. 92 BAND THEORY OF SOLIDS (QUALITATIVE) Earlier in Section 8.5.9 we discussed the quantum free electron theory of metals, wherein it was assumed that free electrons in a metal experience a constant potential and for simplicity the same was considered zero. However, in solids the potential experienced by free electrons is no longer regarded constant since it varies rapidly, particularly near to the Positive ion cores in the lattice. Kronig and Penny later considered that the conduction electrons move in a perfectly periodic potential of a crystal. Such a distribution of the potential arises because of the regular spacing ofion cores in the lattice and its periodicity is equal to the lattice constant. The outcome of this model showed that the energy of the electrons can be situated only in allowed bands which are separated by forbidden energy regions. Within a particular allowed band, electrons are allowed to move freely; they can again interact with extemally applied electric fields to produce conduction effects but the interaction parameters change due to the presence of the lattice. The band theory of solids that has originated from the periodic potential model has been most suecessful in explaining why certain materials behave as good conductors while some are semiconductors and insulators. The factor that determines the conduction properties of a particular material is whether the electronic states within an allowed energy band are empty or full. §3 DISTINCTION OF MATERIALS BASED ON ENERGY BANDS Ithas tum theory that in crystalline materials there are bands of allowed nee » ons separated by bands of forbidden energy levels. If we consider a single 201202 Engineering Physics atom the energy levels are discrete but for a solid having several atoms it is foung proximity of other atoms modifies the discrete energy levels of the individual atoms to the where the energy distribution takes on the appearance of bands. The energy band descrip a bad conductor (insulator) is shown in Figure 9.1(a). on of In insulators, the conduction band is separated from valence band by more than 3 eV ap temperature; no covalent bonds are broken to provide free electrons in the conduction Applying a large electric field will not rupture the covalent bonds to liberate the free elegy” and thus the conductivity of such materials is practically zero. trons In a semiconductor material without impurities the energy band description is given ; Figure 9.1(b). Iti clear that the empty conduction band is separated from the valence band” a forbidden energy gap of 1 eV. As the temperature of the semiconductor is increased the valene, electrons gets liberated and move to the conduction band. At room temperature, approximately one in 10'6 atoms has electrons so liberated. ¥ In the case of good conductors like metals, as illustrated in Figure 9.1(c), the valence and conduction bands are superimposed so that free electrons are available in plenty. Thus, if, this material is subjected to an external field, conduction is found to occur with the electrons in the conduction band. Empty conduction band > ee oe E,>3eV E,rleV Electron energy (eV) @ (b) © Figure 9.1 (a) An insulator, (b) a semiconductor, and (c) good conductor. 9.4 FERMI-DIRAC DISTRIBUTION FUNCTION Particles which are indistinguishable and have a spin of half, obeying the Pauli exclusion principle are called fermions. Electrons are examples of fermions. The energy distribution of these particles at any temperature T may be expressed by the Fermi distribution function as 1 FE) = =— (9.1) oF 41 where E is the energy of an allowed state and E; is known as the Fermi energy. At T=0K, for E Ep 92)Ces eT nn i K, oe on, Ths, the oni os Below Ex ate completely flled-up while those above Ee etoile occupied at T= 0K, "ey is defined as the maximum energy level up to which oer ‘any other temperature T> Q K FE) = 3 when B= Ey 3) i level is also defi Oo - us, the Fermi 0 defined as the ability of : tion is half for T > 0 K and E = E,, energy level for which the pro! ity a very high temperature, i.e. at T > co when kT >> Ep, the electrons lose their quantum cal character and the Fermi distribution function reduces to the classical Maxwell— escan distribution function. The variation of f(E) with E is shown in Figure 9.2. A T=0K gs g T=0K = Ss { 7 T>0K t —_ Ep —+E Ep ——E Figure 9.2 The Fermi distribution function at various temperatures. 4XAMPLE 9.1 At what temperature can we expect a 10% probability that electrons in a metal | have an energy which is 1% above Ey. The Fermi energy of the metal is 5.5 eV. Given: F(E) = 10%, E = Ep + 1% of Er Ep =5.5eV Required: T=? will Forma: ~F(E)= — 2 5 (e *T +1) Solution: E = Ep + 1% of Ep= 5.5 + 5 = 5.5 + 0,555 E- Ep = 0.555 | 0.555 a 10-2 . 63 555 x 16 x exp( (+ 138x10>T Ja T=290K Result: The given temperature T = 290 K EXAMPLE 9.2 Find the value of F(E) for E - Ey = 0.01 eV at 200 K. Given: E— Ep = 0.01 eV = 0.01 1.6 x 107 J 1.6 x 1071 5SSeS SESE Ese sess se SESE seSsesesE SESE se sEsESESESES SESESESESESESESESESE SESE SE seSESSe Sess sss ROBE SESE SE SE SESE SESS SESE HEHE teeter ies ite eee ea 204 Engineering Physics 4 T = 200K Required: FE) Formula: [ FO = ear Solution: Result: The value of F(E) = 0.36 9.5 DENSITY OF ENERGY STATES To estimate the total number of electrons in a metallic conductor, it is essential to know the number of energy levels available and the probability of electrons available in each less energy level. Thus if s(E)d£, represents the number of quantum energy levels avail energy range of. E and E + dE and Fermi-Dirac function f(£) gives the probability of electrons occupying an energy level'E, then the total number of electrons in the system is given by N(E)dE = s(E)dE F(E) (9.4) To obtain an expression for the number of energy levels between E and E + dE, let us consider two spherical surfaces with radii m and n + dn in the n-space (see Figure 9.3). It is considered that any point with integer values of the coordinates represents an energy state. Thus all the points on the surface of the sphere 7 will have the same energy. It must be remembered that a+ mtb where n,, n, and n, can take positive integral non-zero values. lable in te Figure 9.3 Spheres showing density of states in n-space. =i (den!) 09 ighth of the octant will have'positive values of n, n, and f energy less than E + dE with a sphere of radius (n + dn) Cr) The number of energy states available within a sphere of radius. It should be noted that only o1 n,, Similarly, the number of statés 07 Energy Bands in Solids 205 ber of energy states in , Nom the oe imerval ge SEE = 55 mn + dnp i 4 ae Qu. x = © Gr?dn), consi a } considering dy Very small the higher powers of dn are neglected. = Fitdn = = fe ndn (9.7) we know from quantum free el lectron, theory of metals (Section 8.5.9) m= 8ma® wee (9.8) = (8ma? \” oS ( ma) ge 9) Differentiating Eq. (9.8) once, we get 8 ma? 2ndn = ( a Je Rw )2n : Using Eqs. (9.9) and (9.10) in Eq. (9.7), we have Bde = Sma?" sa 1 (8ma*) Tro 2\ = 2(8ma?\” pan ay ee 7 Since Pauli’s exclusion principle allows the existence of only two electrons of opposite spins inan electronic energy level a s(B)dE = 2.4( dae) E!? dE = + Vom)? E'dE (9.12) Where V is the volume enclosed in which He density of energy states are available = a°. The number of energy states per unit volume in energy interval dE i is s(EME = - (my? EYE (9.13) From Eq. (9.13), we get : s(E) = B'?206 Engineering Physics i.e. the density of states curve is a parabola as shown in Figure 9.4. —s(6 —F Figure 9.4 Variation of the density of states with electron energy, The total number of electrons available is obtained by multiplying the density of. energy interval dE given by Eq. (9.12) with the Fermi distribution function F(£), ab ie. N(E)E = s(E)4E F(E) -* 32 pla, ora (2.14 . (+n. This distribution is shown in Figure 9.5. Figure 9.5 The population density of free electron gas. At T= 0K, all the energy states up to the Fermi level are filled while the states above the Fermi level are empty. As the temperature increase, excitation of electron takes place and energy levels beyond the Fermi level are occupied depending on the operating temperature. At T= 0 K, F(E) = 1, when E < Ef and energy density is given by JN® ae = 4tY omy? fora @.15) ec. N= ome 2 SEP : 0.16 | Number of electrons per unit volume : : . n= # = 35mm 7 a on