To explain the observed spin-relaxation anisotropy, we turn to the full effective mass Hamiltonian for donor electron spins in an electric field, expanded up to fourth order in perturbation theory29

$$H = H_{\mathrm{Z}} + H_{\mathrm{R}} + H_{EB}$$ (2)

Here, \(H_{\mathrm{Z}} = {\textstyle{{\mu _{\mathrm{B}}} \over 2}}\sigma \cdot {\bf{g}} \cdot B\) is the Zeeman Hamiltonian (second order perturbation) with the g-tensor according to Roth,17,18 H R = σ · R · (E × k) is the bulk Rashba Hamiltonian (third order), and H EB the Hamiltonian introduced in Eq. (1) (fourth order). This terms arises in an inversion asymmetric system in the presence of both electric and magnetic fields, and in our specific case, is influenced by the anisotropic multivalley nature of the conduction band. Different from bulk donors, both H R and H EB can be shown to enter valley-repopulation theory for donors in nanoelectronic devices (see Supplementary Information), dominating spin-relaxation rates and anisotropy in the presence of external electric fields.

Donor qubits in electric fields are thus different from other, inversion asymmetric, electronic systems—such as semiconductor quantum dots11,31—in which electronic wave functions are confined by interfaces, and in which the effect of any higher-order magnetoelectric coupling is dwarfed by lower order Rashba and Dresselhaus SOC terms. Atomically confined spins in low-SOC materials—such as P donors in silicon—typically have spherically symmetric ground state wave functions, and hence negligible Rashba and Dresselhaus SOC due to inversion symmetry. Although electric fields from nearby gate electrodes can introduce a non-negligible bulk Rashba contribution in donors, its effects are much weaker compared to quantum dots as the donor wave function remains strongly bound to the Coulomb potential over a wide range of realistic electric field strengths. Indeed, our calculation shows that the ratio of the coupling constants is given by (see Supplementary Information).

$$\frac{C}{R} = \frac{e}{{m_0}}\frac{{p_{\mu

u }}}{{E_{\alpha \mu }}} \simeq 6 \times 10^4 \frac{{\mathrm{m}}}{{{\mathrm{Vs}}}}{.}$$ (3)

Here, (ħ/m 0 )p μν = 1.5 × 10−10 eV is the momentum matrix element between the silicon valence bands, evaluated at the conduction band minimum, and E αμ ~ 4 eV is the energy difference between the conduction band minimum and the valence band, both at k 0 . μ and ν are the indices of the two highest valence bands, α is the index of the lowest conduction band, and k 0 is the momentum vector where the conduction band minimum occurs.

Single-shot spin readout of donor-bound electrons requires the application of considerable gate electric fields20 to bring the Zeeman-split donor ground state into resonance with the electrochemical potential of a single-electron transistor charge sensor. Such electric fields, together with the external magnetic field thus induce an effective magnetic field, |B eff |, that is oriented normal to the external magnetic field orientation when aligned along one of the main symmetry axes (compare Fig.1b). In this case, the magnitude of this field maximises whenever B⊥E and is reduced to zero when they are aligned. From an electrostatic model of our device (Fig. 3a, b, see Methods) we find that E y ≃ 5 MV/m along the G T − G SET axis (green arrow in Fig. 1a) at the readout point (Supplementary Information), confirming that we expect rapid spin relaxation for B||[001] and B||[100] where B⊥E (Fig. 2a–c).

Fig. 3 Evolution of spin-relaxation rates with electric field. a, b Modelled electrostatic potential and electric field in the nanoelectronic device, showing a dominant field (E y ≃ 5 MV/m) along G SET − G T . c Electric field dependence of spin-relaxation rates, consistently describing donor qubits in the bulk (E y = 0) and in nanoelectronic devices (E y > 0) Full size image

Following the same argument, we show that the lifetime-limiting effects of H EB can be eliminated by aligning E||B, such that B eff vanishes. Indeed, at both B||[010] and \(B{\mathrm{||}}[0\bar 10]\) (Fig. 2a–c), we measure T 1 = 1.25 s (1/T 1 ≃ 0.8 1/s) at B = 3.5 T. This value is an enhancement by a factor ~6–7 compared to previous measurements in nanoelectronic devices5 and identical to spin lifetimes measured in bulk donors along the equivalent B||[001] axis.28 Such consistency in spin-relaxation times for equivalent crystal directions thus give a strong indication that we have reached the bulk limit of donor spin lifetimes within a nanoelectronic device. Given that spin lifetimes as long as 1111 s (18 min) at 1.25 K and B = 0.8 T have been measured in bulk,28 such timescales should be attainable in nanoelectronic devices by careful design of the electromagnetic environment and at low enough magnetic fields.12 A further enhancement of spin-relaxation times beyond the bulk limit may be achieved using single spins confined to donor clusters instead of single donors.19

For an estimate of the respective bulk Rashba SOC and H EB coupling strengths, we can fit the spin-relaxation anisotropy (solid red and blue lines in Figs. 1e and 2a, b, see Supplementary Information for detail), from which we extract R = 9.05 × 10−19e m2 and C = 5.86 × 10−14e m/T (out-of-plane), as well as R = 9.53 × 10−19e m2 and C = 6.01 × 10−14e m/T (in-plane). Not only are the extracted coupling coefficients in remarkable agreement across both measurement cooldowns, we also confirm their theoretically predicted ratio as expressed in Eq. (3). The extracted H EB coupling strength now allows us to to calculate a maximum effective magnetic field strength |B eff | = CE y B z /μ B ~ 18.9 mT for E y = 5 MV/m and B = 3.5 T—more than two orders of magnitude larger than the vanishing bulk Rashba SOC field B R = Rk 0m E y /μ B ~ 0.1 mT.

The evolution of the spin-relaxation rates with increasing electric field strength E y from bulk donors (circles) to donors in electric fields (squares) is shown in Fig. 3c. For simplicity, here we have assumed E x = E z = 0, consistent with our COMSOL calculations (<1 MV/m), accounting for a small phase shift in θ. As evident from Fig. 3c, our theory describes both donors in zero and finite electric fields, providing a consistent theoretical framework for spin relaxation of donor-bound electrons.

To conclude, we have demonstrated a previously unreported spin–orbit induced magnetoelectric coupling for donor-bound electron spins in silicon. In contrast to bulk donors, we find that this higher-order coupling provides a spin-relaxation pathway, that can dominate spin relaxation for certain orientations of external electric and magnetic fields. By careful alignment of an external vector-magnetic field, we can eliminate the lifetime-limiting effect, thereby recovering the bulk-regime of donor spin relaxation. Thus, we resolve the long-standing puzzle as to the significant differences of spin lifetimes in bulk donors to those in nanoelectronic devices. Given the strength of the observed magnetoelectric coupling—two orders of magnitude larger than bulk Rashba SOC—it should allow for electrically driven spin resonance,32,33,34 not previously thought possible in donors.

Finally, we note that this higher-order coupling could be observable for atomically confined spins in inversion asymmetric semiconductors, particularly when multiple valleys are present. A similar prediction of such magnetoelectric coupling has recently been made for bilayer graphene.35 We thus expect that our discovery will stimulate future theoretical and experimental investigations of spin control via higher-order electromagnetic coupling.