Question

We consider a two dimensional crystal organized as a square lattice of parameter a. We consider only the interaction with the first neighbors. We recall the main result of LCAO model 1212 2 = 80 - Nunc - Beki -1 Where nn represent the neighbor and the summation is done over all the position of the nearest neighbors 1. Recall the physical meaning of a and B. 2. Draw the lattice in the real space as well as in the reciprocal space (k-space). Evaluate the number and the position R, of the neighbors. 3. Give the expression of the energy in the following direction IX, XK and KT. Plot the dispersion relationship

Answer 1

DECK) - ES - B + y(R) eike Given E = &- Nanox - zik.R; j=1 It [X(R) eike Where Es is the energy of the atomic S-level. B= - S do OU (O)[0(0)|? K(R) - 5 do q*(R) +(0-R) and VCR) = - Sdo q *(8) AU (0) $(0-R) Since d is an S-level, PCO) is real and depends only on the magnitude of R. We have a (-R)= XCR). This and the inversion symmetry of the Bravais lattice which requires that Au(-0) - AU(0). =) YC-R)=Y(R). Now, E(K) = Es-B-EVCR) CosK.R - nn where the sum runs only over those R in Bravais lattice that connect the origin to its nearest neighbours. To be explicit let us apply ear 6 to a fcc crystal The 12 nearest neighbors of origin (see figure below) are at Rs a (+1, +1, 0) q (11,0, 11) a (0, 11, +1)

of K. R are K.R= If K = (Kx, Ky, Kz) then the corresponding 12 Values a (+ki, + ki) i, j = x,y; y, 2; 2x Now suco) = 0(x,y,z) So, ECK) = Es - B-48 (cosi ky a cos1 ky a t costky a costkza + costka cos | kxa) Where Y = -for q*(x,y,z) AU(4,9,2) P(x-4,4-2,2)

VO) Energy levels (Spacing) a = 2 Bands, 4 N' values (a) (b) Nfold degenerate