  Strain is extensively used to controllably tailor the electronic properties
of materials. In the context of indirect band-gap semiconductors such as
silicon, strain lifts the valley degeneracy of the six conduction band minima,
and by extension the valley states of electrons bound to phosphorus donors.
Here, single phosphorus atoms are embedded in an engineered thin layer of
silicon strained to 0.8% and their wave function imaged using spatially
resolved spectroscopy. A prevalence of the out-of-plane valleys is confirmed
from the real-space images, and a combination of theoretical modelling tools is
used to assess how this valley repopulation effect can yield isotropic exchange
and tunnel interactions in the $xy$-plane relevant for atomically precise donor
qubit devices. Finally, the residual presence of in-plane valleys is evidenced
by a Fourier analysis of both experimental and theoretical images, and
atomistic calculations highlight the importance of higher orbital excited
states to obtain a precise relationship between valley population and strain.
Controlling the valley degree of freedom in engineered strained epilayers
provides a new competitive asset for the development of donor-based quantum
technologies in silicon.
