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 eigenvalues of the second-order approximations lie within 1% of the exact energy eigenvalues for a linear strain of up to 9.8%, whereas the first-order approximation has an error of less than 1% for a linear strain of up to 5.5%.
Lassen, Benny; Willatzen, Morten; Melnik, Roderick V.N.; and Yan Voon, L.C. Lew, "A General Treatment of Deformation Effects in Hamiltonians for Inhomogeneous Crystalline Materials" (2005). Mathematics Faculty Publications. 41.