1. Bloch theorem: - A crystalline solid consists of a lattice, which is composed of a large number of ion cores at regular intervals, and the conduction electrons that can move freely through out the lattice. The conduction electrons move inside periodic positive ion cores. Hence instead of considering uniform constant potential as we have done in the electron theory, we have to consider the variation of potential inside the metallic crystal with the periodicity of the lattice as shown fig. The potential is minimum at the positive ion sites and maximum between the two ions. The one-dimensional Schrödinger equation corresponding to this can be written as d 8 m E V x dx ( ) 0 ---------------------------- (1) h The periodic potential V(x) may be defined by means of the lattice constant as Bloch has shown that the one-dimensional solution of the Schrödinger equation is of the form. ( x) exp ikx u ( x) ------------------------------- (3) k 1
In the above equation u k (x) is called modulating function. Because free electron wave is modulated by u k (x) is periodic with the periodicity at the crystal lattice. Let us now consider a linear chain of atoms of length L in one-dimensional case with N number of atoms in the chain. Then Where a is a lattice distance. u ( x) u x Na ----------------------------- (4) k k From equation (3) and (4), we have x Na uk x Na.expik x Na written as ikna. exp ikna. u ( x).exp( ikx) ( x). e k x Na ( x). e ikna ------------------------------- (5) This is referred to as Bloch condition. Similarly, the complex conjugate of (5) can be * From equation (5) and (6) ikna x Na *( x). e ------------------------ (6) x Na. x Na ( x). x x Na x This means that the electron is not localized around ones particular atom and the probability of finding the electron is same throughout the crystal.. Kronig Penny Model: - According to this theory the electrons moves in a periodic potential produced by the positive ion cores. The nature of the energies of the electron is determined by solving Schrödinger wave equation. For simplicity, the periodic potential is taken in the form of a regular one-dimensional array of square well potentials. This is called Kronig Penny model. It consists of two series of regions labeled I and II.
Kronig Penny Model. The corresponding Schrödinger equations for the two regions I and II are the form Regions of type I: - V = 0 for 0 < x < a d dx 1 me 1 0 d 1 1 0 ------------------ (1) Where dx me Regions of type II: - V = 0 for -b < x < 0 d me E Vo dx 0 d 0 ------------------- () Where dx Assuming E < V o. Let us suppose that the general solution of equation (1) and () are x A i x B i x 1.exp.exp ------------------- (3) x C x D x m Vo.exp.exp ------------------- (4) For E < V 0 Where A, B and C, D are constants in the regions I and II, their values can be obtained by applying the boundary conditions. E 3
x x 1 x0 x0 d1 d dx dx x0 x0 ----------------- (5) And x x 1 xa xb d 1 d d dx xa xb ------------------ (6) Since, for a periodic lattice with V(x) = V(x + a), it is expected that the wave function will also exhibit the same periodicity. Therefore the expected solutions of the above Schrödinger equation must have the same form as that of the Bloch function. x a b x.expik a b Now at x = -b the equation (6) becomes exp x x a b ik a b b 1 a.exp ik a b ---------------------- (7) d d1 exp -----------------(8) dx dx and ik a b xb xa Now applying boundary conditions (5), (7) and (8) to equation (3) and (4), we obtain the following modified equations. A B C D ---------------- (9) i A B C D -------------------- (10) b b ikab i a i a Ce De e Ae Be ----------------- (11) b b ika b i a i a Ce De ie Ae Be ----------------- (1) Equations (9), (10), (11) & (1) will have non-zero solutions if and only if the determinant at the coefficients A, B, C and D zero i.e., 4
1 1-1 -1 i -i - = 0 e ik ab ia e ik ab ia e b e b ik a b ia i e ie ikabi a e b e b On simplifying this determinant, we obtain Equation (13) is complicated Kronig and Penny simplify this without any loss of its physical significance. Let V 0 is tending to infinite and b is approaching to zero. Such that V 0 b remains finite. Therefore, Sin hb b and Cos hb 1 Where b 0. Now m m V0 E E m V 0 E Since V 0 >> E, so that m V 0 Substituting all these values in equation (13) it modifies as mv0. b.sina cosa cos ka mv0ab sina cosa cos ka a 5
Where mv ab p 0 p sina cosa cos ka ---------------------- (14) a 4 ma V0b h and is a measure of potential barrier strength. The left hand side of the equation (14) is plotted as a function of a. This is shown in fig. The cosine form on the Right hand side of the equation can only have values between 1 and +1 as indicated by horizontal lines in the fig. Therefore, the equation (14) is satisfied only for those values a for which left hand side lies between +1. This means some ranges of energies are allowed and some other ranges are not allowed. From the above figure the conclusions are 1) The energy spectrum of the electron consists of a number of allowed and forbidden energy bands. ) The width of the allowed energy band increases with increase of energy values i.e., increasing the values of a. This is because the first term of equation (14) decreases with increase of a. 3) With increasing P i.e., with increasing, potential barrier, the width of an allowed band decreases. As P, the allowed energy region becomes infinitely narrow and the energy spectrum is a line spectrum as shown in fig. 6
If P, then equation (14) has the only solution. i.e., Sin a 0 (Or) nh E ------------------------ (15) 8ma h This expression shows that the energy spectrum of the electron contains discrete energy levels separated by forbidden regions. 4) When P 0, then 7
Therefore me K K E me E h 1 m h and, k h. p p m h m 1 E mv ------------------------- (16) The equation (16) shows all the electrons are completely free to move in the crystal without any constraints. Hence, no energy level exists. This case supports the classical free electron theory. The electrons first occupy the lower energy bands and are of no importance in determining many of electrical properties of solids. Instead the electrons in the higher energy bands of solids are important i n determining many of the physical properties of solids. Hence, we are interested in those two allowed energy bands called Valence and Conduction Bands. The band corresponding to the outer most orbits is called Conduction Band and the next inner band is called Valence Band. The gap between these two allowed bands is called forbidden energy gap (or) Band gap. Normally we are interested in the valence band occupied by valence electrons. Since they are responsible for the origin of energy Band Formation. 3. Origin of energy band formation:- In case of a single isolated atom, there are single energy levels. But when two identical atoms are brought closer the outer most orbits of these atoms overlap and interact. When the wave function of the electrons of the two different atoms begin to overlap considerably. The energy levels corresponding to those wave functions split into two. 8
If more atoms are brought together, more levels are formed and for a solid of N atoms. Each of the energy levels of an atom splits into N levels of energy. The levels are so close together that they form an almost continuous band. The width of this band depends on the degree of overlap of electrons of adjacent atoms and is largest for the outermost atomic electrons. Hence, in case of a solid instead of single energy levels associated with the single atom, there will be bands of energy levels for the entire solid. 4. classification of Solids into metals, semiconductors and Insulators: - By using Kronig-Penny model, we discussed the origin of allowed and forbidden bands in solids, as a result of electrons in periodic potential. Each allowed energy band was found to contain a limited number of energy levels. Then by applying Pauli s exclusion principle each energy level must be occupied by no more than two electrons. However, with a limited number of electrons in the atoms of a solid it is expected that only the lower energy bands will be filled. The outermost energy band that is completely (or) partially filled is called the valence band in solids. The band that is above the valence band and that is empty at 0k is called the conduction band. According to the nature of band occupation by electrons all solids can be classified broadly into two groups. I) The first group solids are called metals in which there is a partially filled band immediately above the uppermost filled band. This is possible in two ways, in the first case; the valence band is only partially filled. In the second case, a completely filled valence band overlaps the partially filled conduction band as shown in fig. 9
II) The second group includes solids with empty bands lying above completely filled bands. The solids of this group are conveniently sub divided into insulators and semi-conductors depending on the width of the forbidden band. Insulators include solids with relatively wide forbidden bands. For a typical insulators the band gap E g > 3 ev. Ex: - For diamond Eg = 5.4 ev On the other hand, semi-conductors include solids with relatively narrow forbidden bands. For typical semi-conductor Eg < 1eV Ex: - For Germanium with Eg = 0.7 ev Silicon Eg = 1 ev. 10
Brillouin zones:- The discontinuities in the energy versus K curve resulting from electron diffraction mark the boundaries of Brillouin zones. 5. Concept of effective mass of an electron: - When an electron in a periodic potential of lattice is accelerated by an electric field (or) magnetic field, then mass of the electron is called Effective Mass (m*). If the mass of electron is effective mass (m*) then force acting on it is F * m a ---------------------- (1) Considering the free electron as wave packet the group velocity Vg. Corresponding to the particles velocity can be written as: V V g g dw dv W v = de h E hv 1 de. --------------------- () Where h Now, acceleration dv a g dt 1 de. dt. 11
Now according to Newton s second Law 1 d E.. dt --------------------- (3) dp F dt And relation between momentum (P) and wave vector (k) is p k d F ( k) dt From equation (4) and (3) we have Comparing (1) and (5) we get = dt F ----------------------- (4) dt a F d E. F. a -------------------------------- (5) de m* ------------------------------- (6) d E d E This equation indicates that effective mass is determined by. Now we study the variation of the effective mass when the electron is in an energy band of a crystal. This is shown in fig. Which shows the plots of de d E E,,, m* Vs K. 1
. Questions:- 1. (a) Explain the origin of energy bands in solids. (b) Assuming the electron - lattice interaction to be responsible for scattering of conduction electrons in a metal, obtain an expression for conductivity in terms of relaxation time and explain any three draw backs of classical theory of free electrons. 13
. Explain the Bloch s theorem 3. Explain the concept of effective mass of an electron moving in a one-dimensional periodic potential. 4. a) Explain the concept of Effective Mass. b) Discuss the motion of an electron in a periodic potential field and explain the formation of energy bands. c) An electron is confined in one-dimensional potential wall of width 310-10 m. Find the kinetic energy of electron when it is in the ground state. 5. Distinguish between metals, semiconductors and insulators 6. Discuss the motion of an electron in a periodic lattice. 7. Discuss the kronig-penny model for the motion of an electron in a periodic potential. 14