Shallow-donor wave functions and donor-pair exchange in silicon: Ab initio theory and floating-phase Heitler-London approach

Electronic and nuclear spins of shallow donors in silicon are attractive candidates for qubits in quantum computer proposals. Shallow donor exchange gates are frequently invoked to perform two-qubit operations in such proposals. We study shallow donor electron properties in $\mathrm{Si}$ within the Kohn-Luttinger envelope function approach, incorporating the full Bloch states of the six band edges of the $\mathrm{Si}$ conduction band, obtained from ab initio calculations within the density-functional and pseudopotential frameworks. Intervalley interference between the conduction-band-edge states of $\mathrm{Si}$ leads to oscillatory behavior in the charge distribution of one-electron bound states and in the exchange coupling in two-electron states. The behavior in the donor electron charge distribution is strongly influenced by interference from the plane wave and periodic parts of the Bloch functions. For two donors, oscillations in the exchange coupling calculated within the Heitler-London (HL) approach are due to the plane-wave parts of the Bloch functions alone, which are pinned to the impurity sites. The robustness of this result is assessed by relaxing the phase pinning to the donor sites. We introduce a more general theoretical scheme, the floating-phase HL, from which the previously reported donor exchange oscillatory behavior is qualitatively and quantitatively confirmed. The floating-phase formalism provides a “handle” on how to theoretically anticipate the occurrence of oscillatory behavior in electronic properties associated with electron bound states in more general confining potentials, such as in quantum dots.

Figure 1

(Color) (a) and (b) Probability density for the single Bloch state ${\rho }_x={\mid {\varphi }_x\mid }^2$ in two different crystal planes. Notice the $\mid p_x⟩$ atomiclike signature. (c) and (d) Total probability density for the six conduction-band minima, showing a more symmetric structure. White dots represent $\mathrm{Si}$ sites in the diamond structure and the color scheme runs from purple (low density) to red (high density).

Figure 2

(Color) Frames (a) and (b) give the electron probability densities on the (001) plane of bulk $\mathrm{Si}$ for the bottom of the conduction band eigenstate corresponding to a symmetric combination of the six degenerate Bloch states at the conduction band edge, calculated within models (i) and (ii), respectively. Frames (c) and (d) give the corresponding probabilities for the ground state of a donor in $\mathrm{Si}$ within the envelope function approximation. The white dots give the in-plane atomic sites and the color scheme runs from purple (low density) to red (high density).

Figure 3

(Color online) Calculated exchange coupling for a donor pair versus interdonor distance in (a) unstrained and (b) uniaxially strained (along $z$) $\mathrm{Si}$. The open squares correspond to substitutional donors placed exactly along the [100] axis, the lines give the calculated values for continuously varied interdonor distance along this axis, assuming the envelopes do not change. The open triangles give the exchange for a substitutional pair almost along [100], but with one of the donors displaced by $d_{NN}\sim 2.3\mathrm{\AA }$ into a nearest-neighbor site. The lower frames give the same data in a logarithmic scale. When the floating-phase HL approach is adopted, the results change negligibly; the filled symbols on the lower left frame give examples of calculated corrections (see text).

Figure 4

(Color online) Calculated corrections to the total energy for a P donor pair in $\mathrm{Si}$. The donors are $87\mathrm{\AA }$ apart, along the [100] direction. The parameter $\delta R$ is the amplitude of the individual phase shifts from the donor sites, $\pm \delta \mathbf{R}$, along the interdonor line. The solid line gives ${\Delta }_0$, the single-center contribution, while the dotted (dashed) line gives ${\Delta }_s\left({\Delta }_t\right)$, the two-center singlet and triplet contributions, respectively.