Valley interference and spin exchange at the atomic scale in silicon

Abstract

Tunneling is a fundamental quantum process with no classical equivalent, which can compete with Coulomb interactions to give rise to complex phenomena. Phosphorus dopants in silicon can be placed with atomic precision to address the different regimes arising from this competition. However, they exploit wavefunctions relying on crystal band symmetries, which tunneling interactions are inherently sensitive to. Here we directly image lattice-aperiodic valley interference between coupled atoms in silicon using scanning tunneling microscopy. Our atomistic analysis unveils the role of envelope anisotropy, valley interference and dopant placement on the Heisenberg spin exchange interaction. We find that the exchange can become immune to valley interference by engineering in-plane dopant placement along specific crystallographic directions. A vacuum-like behaviour is recovered, where the exchange is maximised to the overlap between the donor orbitals, and pair-to-pair variations limited to a factor of less than 10 considering the accuracy in dopant positioning. This robustness remains over a large range of distances, from the strongly Coulomb interacting regime relevant for high-fidelity quantum computation to strongly coupled donor arrays of interest for quantum simulation in silicon.

Valley population of donor states in highly strained silicon

“…This onset can be measured away from a single donor as shown in Fig 3a-b. Here, it occurs at V b ∼0.2 V , which is offset compared to previous results 25,26,40 in unstrained silicon where it is found around V b ∼ − 0.8 V. This offset is attributed to an electrical short between the silicon epilayer and the handling wafer on the sample holder. Two extra resonances, at V b ∼0.6 V and V b ∼0.2 V respectively, can be observed when tunneling over a single dopant occurs (red line), which correspond to adding respectively the first (called D 0 state) and the second electron (called D − state) on the donor, the latter being close to the onset of direct transport from the valence band states 25,34 .…”

“…The objectives of this section are to spectroscopically evidence the presence of single donors addressable in the single electron tunneling regime, before spatially resolving their charge distribution and analysing these images in the context of the strain applied to the crystal. Donor states can be evidenced spectroscopically at 4.2 K as they yield resonances below the direct transport onset from silicon valence band states to the STM tip 25,26,40 . This onset can be measured away from a single donor as shown in Fig 3a-b.…”

“…We focus here on interactions based on the direct overlap between donor wave functions, such as tunnel and exchange couplings. These quantities can be very sensitive to the exact position of the dopants in the lattice because of the presence of interference at the valley frequency, an effect modulated by the anisotropy of the donor envelope that originates from the mass anisotropy in the conduction band of silicon 40,43 . Below, we investigate the effect of the valley population on these interactions in the context of atomically precise donor qubit devices, where donors are placed within the same crystallographic plane 13,27 .…”

“…Below, we investigate the effect of the valley population on these interactions in the context of atomically precise donor qubit devices, where donors are placed within the same crystallographic plane 13,27 . We start from the Heitler-London expression of the exchange coupling using an effective mass description of the single electron wave functions, and of the relationship between tunnel and exchange couplings 40,43 :…”

A silicon qubit platform for in situ single molecule structure determination

“…The results enable us to find the optimal qubit depth, which for phosphorus donors in silicon is around 2 − 2.5 nm. We note that recent studies based on STM fabrication and metrology of donor qubits have found phosphorus atoms at a similar depth range below the H-passivated surface [48,60].…”