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Exchange in phosphorus donor spin based quantum computers

The long coherence times of both nuclear and electron phosphorus donor spins in silicon make them promising qubits for a scalable quantum computer. In such a system, the single qubit operations could me implemented via the application of resonant RF fields, while the inter-qubit interactions are mediated via the exchange interaction between neighboring donor electrons. This interaction could be controlled, to some extent, via the application of potential biases to control electrodes, which would perturb the electron wavefunctions. The degeneracy of the conduction band minima in silicon leads to interference effects between the donor wavefunctions resulting in a strong dependence of the strength of the exchange coupling on the relative separation of the phosphorus donors, this can have significant consquences to the fabrication of a donor spin based quantum computer. We present calculations of the donor electron exchange coupling as a function of the relative donor position, using donor wavefunctions that have been calculated by direct diagonalisation of the system Hamiltonian, in a basis of the silicon conduction band Bloch functions, beyond the standard effective mass approximation. We consider the effects of non-Coulombic donor potentials, as well as an applied lattice strain, which breaks the degeneracy of the conduction band minima and can be used to damp the oscillatory behavior of the exchange energy. We discuss these results and their implications with respect to fabrication of a scalable donor spin quantum computer.