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

Abstract

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 potential͒ and one two-dimensional example ͑a freestanding quantum wire͒, 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%.

Geometric treatment of conduction electron scattering by crystal lattice strains and dislocations

“…For perfect crystals, the use of the idea of an electron gas is justified within such a continuum model (referred to as the 'jellium model' [35]). Hence, in the presence of inhomogeneous strains, generalizing this method by use of the covariant derivative is not a priori justified [18,7,19]. Indeed, as our microscopic derivation shows, the assumptions involved are obscured by such a method.…”

“…Lassen et al [7]. When the strains are constant and purely elastic, it reduces to the Hamiltonian used for homogeneous strains [2], upto a change inm e (see earlier footnote).…”

“…To treat purely elastic inhomogeneous strains, several models have been proposed for both metals and semiconductors [5,6,7]. The case of strains due to dislocations in metals was treated in detail by Hunter and Nabarro [8], who calculated the electrical resistivity due to edge and screw dislocations.…”

“…The effect of inhomogeneous elastic strains on the electronic band structure in semiconductors can be modeled [6,7] as an extension of the coordinate transformation method [2] for homogeneous strains. This is the most commonly used technique to estimate the shift in band degeneracies [17].…”

Coupled Magneto-Thermo-Electromechanical Effects and Electronic Properties of Quantum Dots

“…[28,[51][52][53][54]) and mechanical displacement, electric field and electric potential, the thermal field and temperature change, are referred to as gradient equations which can be written as,…”