10. Appendix

Solution to Problem 2.16
Solution to Problem 2.16 595
(a) The matrix elements of the tight-binding Hamiltonian are given by (2.78).
Examples of the calculation of these matrix elements, such as Hs2,s2, are given
in the book for the diamond structure. The results for the zincblende structure
differ only slightly from that of the diamond structure in that the two atoms
in the unit cell are different. Since the nearest neighbor tetrahedral arrangement
is the same in the two structures, one would expect that the definition
of the vectors d1, ..., d4 would be the same. The main difference would be the
matrix elements 〈S1|int|S1〉 and 〈S2|int|S2〉 which are no longer equal. Similarly
〈X1|int|X1〉 would not be equal to 〈X2|int|X2〉 etc. This means that
in Table 2.25 it would be necessary to introduce an energy ES1 for the matrix
element 〈S1|int|S1〉 and a different energy ES2 for the matrix element
〈S2|int|S2〉. Similarly, one can introduce energies Ep1 and Ep2.
(b) At k 0, g0(k) 1 and g1(k) g2(k) g3(k) 0. Hence, the s-band
determinant derived from the modified Table 2.25 for the zincblende crystal is
given by:
Es1 Es(0) Vss
V∗ ss
Es2 Es(0) 0
Evaluating this determinant one obtains a quadratic equation in Es(0):
[Es(0)] 2 Es(0)(Es1 Es2) (Es1Es2) |Vss| 2 0
The solutions for the s-band energies at k 0 are therefore:
Es(0) [(Es1 Es2)]/2] ± (1/2)[(Es1 Es2) 2 4|Vss| 2 ] 1/2
The solutions for the p-band energies at k 0 can be obtained similarly.
Solution to Problem 2.18
In this case the overlap parameter Vxx can be evaluated in the same way as
〈px|H|px〉 in Fig. 2.23 of the book and the result is the same, except for replacing
the angle £ by £x.
The overlap parameter Vxy can be evaluated, in principle, by drawing the
following diagram and decomposing the figure into a sum of two figures as
shown in Prob. 2.18-Fig. 1.
The first figure on the right is rather simple and easy to calculate since
both orbitals are projected along the vector d. The projections of px and py
along d are simply px cos £x and py cos £y, respectively. The resulting overlap
integral is VppÛ cos £x cos £y.
The second overlap integral is more difficult to evaluate from the second
figure on the right, since the components of px and py perpendicular to d are
not parallel to each other. Instead, we will decompose the unit vector along