A general treatment of deformation effects in Hamiltonians for inhomogeneous crystalline materials

In this paper, a general method of treating Hamiltonians of deformed nanoscale systems is proposed. This method is used to derive a second-order approximation both for the strong and weak formulations of the eigenvalue problem. The weak formulation is needed in order to allow deformations that have discontinuous first derivatives at interfaces between different materials. It is shown that, as long as the deformation is twice differentiable away from interfaces, the weak formulation is equivalent to the strong formulation with appropriate interface boundary conditions. It is also shown that, because the Jacobian of the deformation appears in the weak formulation, the approximations of the weak formulation is not equivalent to the approximations of the strong formulation with interface boundary conditions. The method is applied to two one-dimensional examples (a sinusoidal and a quantum-well potential) and one two-dimensional example (a freestanding quantum wire), where it is shown that the energy eigenvalu...

[1]  Tight-binding study of the influence of the strain on the electronic properties of InAs 'GaAs quantum dots , 2003, cond-mat/0306129.

[2]  E. O’Reilly,et al.  A simple method for calculating strain distributions in quantum dot structures , 1996 .

[3]  Eoin P. O'Reilly,et al.  Theory of the electronic structure of GaN/AlN hexagonal quantum dots , 2000 .

[4]  Zhang.  Motion of electrons in semiconductors under inhomogeneous strain with application to laterally confined quantum wells. , 1994, Physical review. B, Condensed matter.

[5]  Magnus Willander,et al.  Critical thickness and strain relaxation in lattice mismatched II–VI semiconductor layers , 1998 .

[6]  W. Ziemer.  Weakly differentiable functions , 1989 .

[7]  Forchel,et al.  Linear polarization of photoluminescence emission and absorption in quantum-well wire structures: Experiment and theory. , 1995, Physical review. B, Condensed matter.

[8]  V. Aldaya,et al.  Higher-order Hamiltonian formalism in field theory , 1980 .

[9]  E. O’Reilly.  Valence band engineering in strained-layer structures , 1989 .

[10]  J. Schwartz.  Nonlinear Functional Analysis , 1969 .

[11]  Israel Michael Sigal,et al.  Introduction to Spectral Theory , 1996 .

[12]  S. Lang.  Real and Functional Analysis , 1983 .

[13]  G. E. Pikus,et al.  Symmetry and strain-induced effects in semiconductors , 1974 .

[14]  Israel Michael Sigal,et al.  Introduction to Spectral Theory: With Applications to Schrödinger Operators , 1995 .

[15]  R. Bose,et al.  Nanoindentation effect on the optical properties of self-assembled quantum dots , 2003 .

[16]  Excitonic properties of strained wurtzite and zinc-blende GaN/AlxGa1−xN quantum dots , 2003, cond-mat/0310363.

[17]  Nikolai N. Ledentsov,et al.  Quantum dot heterostructures , 1999 .