Hyperfine Stark effect of shallow donors in silicon

We present a complete theoretical treatment of Stark effects in bulk doped silicon, whose predictions are supported by experimental measurements. A multivalley effective mass theory, dealing nonperturbatively 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 $\approx 10$ MHz.

Figure 1

Spatial electronic density of a Si:P bulk donor electron around the implanted nucleus it is bound to, in the plane (010), up to 5 nm away from donor nucleus along the vertical and the horizontal axes. The three panels show how the density changes under different electrostatic environments. The left panel shows the symmetric situation for the isotropic $\mathbf{E}=0$ case, the center and the right panel display how the density is driven off the central nucleus in the direction opposite to the vector $\mathbf{E}$, under an intermediate and a strong electric field, respectively. Red dots represent the positions of the silicon nuclei of the underlying lattice: their positions do not coincide with the local maxima and minima of the density because of the interference of different valleys contributing to the ground-state wave function.

Figure 2

Absolute hyperfine frequency shifts $\Delta A$ as calculated from Eq. (21), as a function of the applied uniform field $E$, for all donors considered here. The field range shown is typical of those required for executing quantum gates and corresponds to the range investigated experimentally in Sec. 4, but is well below the ionization thresholds discussed in Sec. 5.

Figure 3

Donor ${\eta }_a$ coefficients as a function of the zero-field ground binding energy ${\epsilon }_g^0$: both theoretical points (red) and experimental values with absolute errors (blue) are reported. The monotonic dependence displayed here is qualitatively explained in the text.

Figure 4

Measurement of the Stark shift in ${}^{28}\mathrm{Si}$:${}^{121}\mathrm{Sb}$ using dynamical decoupling. (a) Uhrig dynamical decoupling (UDD) sequence with four refocusing pulses. As each $\pi$ pulse reverses the phase acquisition (see signs in sequence), the dc electric field is applied in between alternating pairs of $\pi$ pulses. Using positive and negative voltage pulses, the linear Stark shift contribution, arising from local defects, is eliminated and only the quadratic part remains. (b) Electron spin phase evolution measured in the $m_{\mathrm{I}}=+5/2$ ESR transition of Sb in material 1 for different electric fields. (c) Fast Fourier transform (F.F.T.) showing the frequency shift distribution in the sample, fitted here with a Lorentzian.

Figure 5

Stark shift for P, As, Sb, and Bi measured at X band, for each possible ESR transitions. The quadratic ${\eta }_a$ and ${\eta }_g$ shift coefficients are the only parameters in the fits (lines). (a) ${}^{31}\mathrm{P}$, (b)${}^{75}\mathrm{As}$, (c) ${}^{121}\mathrm{Sb}$, and (d) ${}^{209}\mathrm{Bi}$.

Figure 6

Donor binding energies of the ground states of Si:P, Si:Sb, Si:As, and Si:Bi decrease very weakly as a function of the field, as detailed in the text. For each species, values are reported only up to the point $E_c$ where they become degenerate with the “$2p$” energy level, which is common to all donors, as it is not influenced by central cell corrections. Each crossing point corresponds to the ionizing field for each donor.