Donor Spin Qubits in Ge-based Phononic Crystals

We propose qubits based on shallow donor electron spins in germanium. Spin-orbit interaction for donor spins in germanium is in many orders of magnitude stronger than in silicon. In a uniform bulk material it leads to very short spin lifetimes. However the lifetime increases dramatically when the donor is placed into a quasi-2D phononic crystal and the energy of the Zeeman splitting is tuned to lie within a phonon bandgap. In this situation single phonon processes are suppressed by energy conservation. The remaining two-phonon decay channel is very slow. The Zeeman splitting within the gap can be fine tuned to induce a strong, long-range coupling between the spins of remote donors via exchange by virtual phonons. This, in turn, opens a very efficient way to manipulate the quits. We explore various geometries of phononic crystals in order to maximize the coherent qubit-qubit coupling while keeping the decay rate minimal. We find that phononic crystals with unit cell sizes of 100-150 nm are viable candidates for quantum computing applications and suggest several spin-resonance experiments to verify our theoretical predictions.