Inter- and intra-band Coulomb interactions between holes in silicon nanostructures

Abstract

We present a full derivation of the interaction Hamiltonian for holes in Silicon within the sixband envelope-function scheme, which appropriately describes the valence band close to the Γ point. The full structure of the single-hole eigenstates is taken into account, including the Bloch part, and the scattering processes caused by the Coulomb interaction are shown to be both intraband and interband; the interband terms are mostly short-ranged. In the asymptotic long-range limit, the effective potential tends to the screened Coulomb potential, and becomes purely intraband, as assumed in previous models. Our findings can be directly used for realistic exact-diagonalization calculations related to systems of interacting holes in Silicon nanostructures, such as quantum dots.

Interacting holes in Si and Ge double quantum dots: from a multiband approach to an effective-spin picture

“…As we have shown in Ref. 38, the Coulomb interaction also includes short-ranged interband processes, which can play a role in strongly confined systems. However, Eq.…”

“…This Hamiltonian acts on vectors, each component of which corresponds to a Bloch state with crystal momentum Γ ≡ 0. In the cases of Si and Ge, the relevant Bloch states for the valence bands are built from p-type atomic orbitals 29,38 , carrying an angular momentum l = 1. Combining this with the electron s = 1/2 spin, one can write the Bloch basis set at Γ as a quartet of states with j = 3/2 and a doublet of states with j = 1/2.…”

“…is the Bloch state 29,38 for band b, that combines p-atomic orbitals (φ p ξ , with ξ ∈ {x, y, z}) centered at all the N a atomic positions R k of the crystal (where k ∈ {0, 1} labels the two atoms in each unit cell) with spin states (s z = ±1/2), through the Clebsch-Gordan coefficients S b,ξ,sz .…”

Two-body Wigner molecularization in asymmetric quantum dot spin qubits

“…Terms µ = ν in Eq. (D2) therefore integrate to zero on the scale of the unit cell and give rise to mostly short-range multipolar corrections to the Coulomb integrals that we neglect in the present work (an usual assumption in k • p-based CI [83,84], essentially valid when the physics is dominated by long-range interactions, as is the case here). In this approximation,…”