References

1

H. Kroemer, Proc. IEEE 63, 988 (1975).

2

W. A. Harrison, Phys. Rev. 123, 85 (1961).

3

D. J. BenDaniel and C. B. Duke, Phys. Rev. 152, 683 (1966).

4

T. Ando and S. Mori, Surf. Sci. 113, 124 (1982).

5

Q.-G. Zhu and H. Kroemer, Phys. Rev. B 27, 3519 (1983).

6

The form of the effective Hamiltonian cannot be completely determined at the mesoscopic level. One requires a realistic microscopic model from which the mesoscopic approximation may be derived. Nevertheless, the commonly used forms, obtained by placing material-dependent quantities in the center of the derivatives or taking average values for inter-atomic-plane matrix elements, appear to be adequate for many systems.

7

G. H. Wannier, Phys. Rev. 52, 191 (1937).

8

J. C. Slater, Phys. Rev. 76, 1592 (1949).

9

By ``apparently hermitian,'' I mean an operator in which the z-dependent parameters appear symmetrically with respect to the derivatives. The rigorous demonstration of hermiticity requires a specification of the boundary conditions applied to the operator. For an extensive discussion of the physical consequences of boundary conditions, see [20].

10

J. M. Luttinger and W. Kohn, Phys. Rev. 97, 869 (1955).

11

G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (Halsted Press, New York, 1989), ch. 3.

12

M. G. Burt, Semicond. Sci. Technol. 2, 460 (1987).

13

Bastard's Hamiltonian [equation (20) of chapter 3] contains a term of the form , which is hermitian only if this matrix element is independent of z. Burt [12] has pointed out that a rigorous derivation of the envelope-function approach requires that the same Bloch functions be used throughout the heterostructure, which would assure the hermiticity. In practice this condition is often violated as the Bloch functions are presumed to vary with the local material composition.

14

M. Altarelli, Phys. Rev. B 28, 842 (1983).

15

J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954).

16

J. N. Schulman and T. C. McGill, Phys. Rev. B 19, 6341 (1979).

17

D. Z.-Y. Ting, E. T. Yu, and T. C. McGill, Phys. Rev. B 45, 3583 (1992).

18

A similar concept and notation for the current density appears in finite-difference numerical computations. See, for example, S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer, Vienna, 1984), ch. 6.

19

D. Z.-Y. Ting and Y.-C. Chang, Phys. Rev. B 36, 4359 (1987).

20

W. R. Frensley, Rev. Mod. Phys. 62, 745 (1990).