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Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

Copyright 2004 by Marcel Dekker, Inc. All Rights Reserved.

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a better description of the bulk bands must be incorporated into the theory.Although a variety of computational methods could be used, this route doesnot provide analytical expressions for the description of the bands. Thus, amore sophisticated effective mass approach, the k p method, is typically used[29]. In this case, bulk bands are expanded analytically around a particularpoint in k-space, typically k = 0. Around this point, the band energies andwave functions are then expressed in terms of the periodic functions u nk andtheir energies E nk .General expressions for u nk and E nk can be derived <strong>by</strong> considering theBloch functions in Eq. (6). These functions are solutions of the Schro¨dingerequation for the single-particle HamiltonianH 0 ¼ p22m 0þ VðxÞð17Þwhere V(x) is the periodic potential of the crystal lattice. Using Eqs. (6) and(17), it is simple to show that the periodic functions, u nk , satisfy the equationH 0 þ 1 ðk pÞ u nk ¼ E nk u nkð18Þm 0wherek 2E nk ¼ E nk2m 0ð19ÞBecause u n0 and E n0 are assumed known, Eq. (18) can be treated inperturbation theory around k = 0 withHV¼ k pð20Þm 0Then, using nondegenerate perturbation theory to second order, one obtainsthe energiesE nk ¼ E n0 þ k22m 0þ 1 m 2 0and functionswithu nk ¼ u n0 þ 1 m 0Xm p nu m0Xm p n ! k ! p nm 2E n0 E m0ð21Þ! k ! pnmE n0 E m0ð22Þ! pnm ¼ hu n0 ! pum0 i ð23Þ<strong>Copyright</strong> <strong>2004</strong> <strong>by</strong> <strong>Marcel</strong> <strong>Dekker</strong>, <strong>Inc</strong>. <strong>All</strong> <strong>Rights</strong> <strong>Reserved</strong>.

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