Previous work employing an effective mass theory of Si:P [ABG+81, KHDS02b, KHDS02a]
predicts that the oscillations in the dependence of the coupling on distance, that have just been shown to derive from valley-mixing in the multi-valley silicon conduction band, are so strong as to limit the hope of implementing two-qubit gates that depend precisely on the magnitude of J. The first attempt at an estimate of inter-donor J was devised al- most 25 years ago [ABG+81], within the apparently different context of the analysis of the magnetic susceptibility of the Si:P system. This seminal approach relied on Kohn- Luttinger-type donor wavefunctions, based on the EMT discussed in section 3.3, which does not include any account of the central cell effects or the valley-orbit coupling; further simplification came from assuming isotropic electronic clouds. Finally, in their analysis the exchange,J, is approximated by the two-particle integralCI in Eq. 4.6. The resulting
predictions of very large J(d) oscillations (where d is the inter-donor separation) were averaged statistically over the different phase mismatches due to even very small differ- ences in the nominal donor implantation positions, as evident for example from Eq. 4.19. Andreset al.conclude that the overall strength of theJcoupling is reduced by a factor of
1
6 as compared to the case of lack of valley degeneracy.
This theory is refined in Ref. [KHDS02a], where a full Heitler-London calculation and anisotropic envelopes are employed to estimate the inter-donor exchange couplings as a function of their separations along different high symmetry crystallographic directions. While the first improvement does not lead to qualitative discrepancies as compared to the simpler J → C1 calculation, as the valley structure of the two terms is essentially
the same (as discussed in the previous section), the inclusion of anisotropy leads to the important distinction that oscillations in J(d) are significantly reduced if d k [100]. It
CHAPTER4: EXCHANGE COUPLING BETWEEN DONORS IN SILICON
is also shown how donors within a semiconductor with lower valley degeneracy, namely germanium, would experience interactions that oscillate less violently, with even mono- tonicJ(d)trends in the case ofdk[100].
The same framework is then extended to include the effects of external strain on the silicon layer: applying large tensions or compressions on the bulk structure, or implanting donors in heterostructures like epitaxial Si/Si1−xGex, will shift some of the valleys with respect
to their energy in relaxed silicon, more or less uniformly across the device. This is due to the lower symmetry that the strain induces on the bulk system discussed so far, whose ef- fects can be treated with the theory of deformation potentials introduced in Ref. [HV56]. The main consequence is that the ground donor state is not any longer an equal superpo- sition of the Bloch functions of all the six valleys, but the valleys shifted to lower energies contribute more to its wavefunction. As a result, the oscillations inJ(d)can be strongly suppressed, if the lower valleys are transverse to the direction defined by a line connecting the donor pair, since their contribution to the exchange will not vary strongly with small displacements along theddirection (cf. for example Eq. 4.19). Such suppression is more efficient if the strain is stronger. Nonetheless, scattered J values will still be present if the donors are displaced along the plane perpendicular tod, in other words if they are not exactly aligned along one crystallographic direction. A richer electrostatic environment is investigated in Ref. [WHP+03], where externally applied biases on the donor pair al- low, in principle, the tuning of the exchange interaction. It is shown, moreover, that a full evaluation of the lattice-periodicucomponent of the Bloch function is not necessary for calculations ofJ, presenting the argument that takes Eq. 4.16 to Eq. 4.18.
In all the works cited so far multi-valley effects are only taken into account in somead hocmanner: the donor wavefunction does include a multi-valley structure, but one which is dictated only by the symmetry arguments based on the tetrahedral point group (see Ap- pendix B). No treatment is given of the physical source of the removal of the valley de- generacy, which indeed will also affect the weight of each valley within the ground state, represented by the envelopesFµ(r). A more recent numerical calculation [WH05] going beyond effective mass theory, in fact, found that such oscillations were overestimated. Wellard and Hollenberg propose a numerical solution of the full Hamiltonian describing the donor electrons, performed though exact diagonalization in the basis of the undoped crystal Bloch functions (their technique is called BMB: Band Minima Basis). Thus, they are not limited by any of the EMT approximations, but only by the convergence and ac- curacy properties of their calculations and by the validity of the pseudopotential they use, taken from Ref. [PS74a]. As compared to previous benchmark J predictions, such de- tailed and numerically intensive microscopic calculations predict that the strength of the coupling is reduced, and its oscillations have their amplitude decreased. Going beyond EMT is traded, however, with exponentially increased computing times: in fact only donor