Hyperfine Stark effect of shallow donors in silicon

Abstract

We present a complete theoretical treatment of Stark effects in bulk doped silicon, whose predictions are supported by experimental measurements. A multi-valley effective mass theory, dealing non-perturbatively with valley-orbit interactions induced by a donor-dependent central cell potential, allows us to obtain a very reliable picture of the donor wave function within a relatively simple framework. Variational optimization of the 1s donor binding energies calculated with a new trial wave function, in a pseudopotential with two fitting parameters, allows an accurate match of the experimentally determined donor energy levels, while the correct limiting behavior for the electronic density, both close to and far from each impurity nucleus, is captured by fitting the measured contact hyperfine coupling between the donor nuclear and electron spin.We go on to include an external uniform electric field in order to model Stark physics: With no extra ad hoc parameters, variational minimization of the complete donor ground energy allows a quantitative description of the field-induced reduction of electronic density at each impurity nucleus. Detailed comparisons with experimental values for the shifts of the contact hyperfine coupling reveal very close agreement for all the donors measured (P, As, Sb and Bi). Finally, we estimate field ionization thresholds for the donor ground states, thus setting upper limits to the gate manipulation times for single qubit operations in Kane-like architectures: the Si:Bi system is shown to allow for A gates as fast as ≈10 MHz.

Quantum gates with donors in germanium

“…Effective mass theory (EMT) maintains that the donor quantum states can still be expanded in terms of packets of Bloch functions whose k-vectors concentrate around each minimum, since the wave function is known to spread across tens of lattice constants in space. While this approximation fails to accurately describe the behavior of the donor electron wave function within the crystal cell occupied by the binding substitutional impurity (the central cell), it does not impede a reliable picture outside of it, if one averages the short-range Hamiltonian with a pseudopotential 5,18 . Importantly, both single and two-qubit manipulations of donor spins rely on the long-range component of the donor wave function, since it is this part that is affected by external control fields and/or overlap with neighboring donor states.…”

“…Over the years, EMT has been improved from its original single-valley formulation 26,27 to a multi-valley framework 5,18,24,[28][29][30][31] , which aims at describing how all valleys are admixed in the electron eigenstates as a result of the donor-dependent impurity potential U (r). The most important consequences for the electron ground state are that it is non-degenerate, and its orbital energy and hyperfine coupling to the ion differ significantly from a single-valley state 14,32 .…”

“…We combine this multi-valley EMT with a pseudopotential method to describe the impurity potential U (r), an approach that has been applied successfully to understand some properties of donor electron states in Si 5 . The use of a smooth pseudopotential in place of the true impurity potential implies that the eigenstates of the corresponding Hamiltonian are not accurate on the scale of the central cell 18 , where strong oscillations impede the semi-analytic description we are aiming at.…”

“…In spite of this, the intrinsically weak spin-orbit interaction of the conduction electrons with the host Si nuclei 4 and the relatively tight wave functions of the donor electrons 5 can seriously restrict the extent to which the qubit states can be selectively controlled. Moreover, coupling two donor spins in Si requires the ions to be implanted very closely 6 , and with high precision 7 .…”

“…Nonetheless, a detailed picture is crucial for deciding whether Ge-donor spins could make promising qubits, and for anticipating the most desirable regimes for spin manipulation in such devices. In this paper, we provide for the first time a thorough description of shallow Ge-donors from group V, based on an improved multi-valley effective mass theory (MV-EMT) that includes the strong anisotropy of the conduction band and approximates phenomenologically the interaction between the donor electron and the nucleus 5,18 . Unlike in most of the traditional EMT papers, here the full plane-wave expansion of the Bloch states of the pure Ge crystal is included: this provides a crucial step for developing a unified picture across different donor chemical species.…”