Donor electron wave functions for phosphorus in silicon: Beyond effective-mass theory

C. J. Wellard and L. C. L. Hollenberg

Phys. Rev. B 72, 085202 – Published 1 August 2005

Abstract

We calculate the electronic wave function for a phosphorus donor in silicon by numerical diagonalization of the donor Hamiltonian in the basis of the pure crystal Bloch functions. The Hamiltonian is calculated at discrete points localized around the conduction band minima in the reciprocal lattice space. Such a technique goes beyond the approximations inherent in the effective-mass theory, and can be modified to include the effects of altered donor impurity potentials and externally applied electrostatic potentials, as well as the effects of lattice strain. Modification of the donor impurity potential allows the experimentally known low-lying energy spectrum to be reproduced with good agreement, as well as the calculation of the donor wave function, which can then be used to calculate parameters important to quantum computing applications.

Received 22 December 2004

DOI:https://doi.org/10.1103/PhysRevB.72.085202

Authors & Affiliations

C. J. Wellard and L. C. L. Hollenberg

Centre for Quantum Computer Technology, School of Physics, University of Melbourne, Victoria 3010, Australia

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Issue

Vol. 72, Iss. 8 — 15 August 2005

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Images

Figure 1
(Color online) Probability density of the phosphorus donor electron in the [001] plane for the case $N = 11$ , with $η = 0$ and 5.8. The white dots mark atomic sites of the silicon lattice.Reuse & Permissions
Figure 2
(Color online) The strength of the exchange coupling between neighboring donors located in the same [001] plane as a function of their separation along the crystallographic [110] axis. The plot compares results calculated using the Kohn-Luttinger effective-mass wave function, as well as wave functions obtained using the numerical technique described in this paper, both with and without a corrected impurity potential. Note that the points refer to substitutional sites in the silicon matrix; the lines between these points are a guide to the eye only as the wave functions used are only valid for substitutional donors.Reuse & Permissions
Figure 3
(Color online) The strength of the exchange coupling between neighboring donors located in the same [001] plane as a function of their separation along the crystallographic [110] axis. The plot compares results for strained and unstrained silicon, calculated using both the effective-mass theory and the wave functions obtained as described in this paper. The tensile strain is obtained by overgrowth on a relaxed [001] ${Si}_{0.8} {Ge}_{0.2}$ surface. Note that the points refer to substitutional sites in the silicon matrix; the lines between these points are a guide to the eye only and do not have any physical meaning, as the wave-functions are only valid for substitutional donorsReuse & Permissions
Figure 4
(Color online) The strength of the exchange coupling between donors that are displaced by an amount $Δ R$ from a separation $R$ of 18 lattice spacings, $9.8 nm$ , along a [100] axis. Results are shown for the case of an unstrained silicon substrate, and the case in which the silicon substrate is grown on the [001] surface of a ${Si}_{0.8} {Ge}_{0.2}$ substrate, applying a tensile strain in this plane. Only displacements perpendicular to $R$ are considered; in the strained case these can be either within (‖) or perpendicular to (⊥) the plane of the tensile strain.Reuse & Permissions
Figure 5
(Color online) A plot of the field dependence of the electron-nuclear contact hyperfine coefficient $(A)$ , in units of the zero-field coupling $(A_{0})$ , for donors located at different distances from an oxide barrier $(Z_{0})$ ; the lines between points are a guide to the eye. Donors closer to the barrier require a stronger field to significantly deform the wave function.Reuse & Permissions

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