Interaction of Strain and Nuclear Spins in Silicon: Quadrupolar Effects on Ionized Donors

The nuclear spins of ionized donors in silicon have become an interesting quantum resource due to their very long coherence times. Their perfect isolation, however, comes at a price, since the absence of the donor electron makes the nuclear spin difficult to control. We demonstrate that the quadrupolar interaction allows us to effectively tune the nuclear magnetic resonance of ionized arsenic donors in silicon via strain and determine the two nonzero elements of the $S$ tensor linking strain and electric field gradients in this material to $S_{11}=1.5\times {10}^{22}\mathrm{V}/{\mathrm{m}}^2$ and $S_{44}=6\times {10}^{22}\mathrm{V}/{\mathrm{m}}^2$. We find a stronger benefit of dynamical decoupling on the coherence properties of transitions subject to first-order quadrupole shifts than on those subject to only second-order shifts and discuss applications of quadrupole physics including mechanical driving of magnetic resonance, cooling of mechanical resonators, and strain-mediated spin coupling.

Figure 1

Manipulation and electrical detection of ${}^{75}\mathrm{As}$ nuclear spins based on spin-dependent recombination. (a) Charge states ${\mathrm{As}}^0$ and ${\mathrm{As}}^{+}$ of an arsenic donor with nuclear and electron spin states denoted by green and blue arrows, respectively. The ionization is governed by a Pauli blockade of the donor electron spin with a paramagnetic defect (SL1). (b) Schematic spectrum showing the enhanced conductivity caused by electrons ejected to the conduction band during the ionization process [14]. (c) Evolution of the charge and spin states of the ${}^{75}\mathrm{As}$ donors during the electrically detected ENDOR sequence. See text for details.

Figure 2

Demonstration of the strain-induced tuning of ionized donors ${\mathrm{As}}^{+}$. (a) Effects of the first- and second-order quadrupole interaction on the magnetic sublevels of ${\mathrm{As}}^{+}$. (b) Generation of strain via the differences in the thermal expansion coefficients of Si and a sapphire substrate. As donors are generated by shallow implantation. (c) EDMR spectrum of ${\mathrm{As}}^0$ in Si. (d) NMR transitions of ${\mathrm{As}}^{+}$ in unstrained Si after ionization on each of the four ${\mathrm{As}}^0$ EDMR peaks. (e) Corresponding NMR transitions for ${\mathrm{As}}^{+}$ in strained Si showing first-order quadrupole shifts. The inset shows a high-resolution spectrum revealing the second-order quadrupole shift of the $m_I=1/2\leftrightarrow -1/2$ transition.

Figure 3

Angular dependences of the (a) first-order and (b) second-order quadrupole shifts on the relative orientation of the magnetic field $B_0$ and the electric field gradient $V_{\perp }$ generated in [111] (black) and [100] (blue) direction.

Figure 4

(a) Pulse sequence used for coherence time measurements. Nuclear spin echo decays of the (b) $3/2\leftrightarrow 1/2$ and (c) $1/2\leftrightarrow -1/2$ subsystems of the ${\mathrm{As}}^{+}$ nuclear spins. Colors denote different numbers of refocusing pulses $n$ in the dynamical decoupling sequence as indicated in the insets, dashed and solid lines correspond to fits with superexponential decays $\exp [-(t/T_2{)}^{\alpha }]$ with different exponents $\alpha$. The insets show the extracted decay constants $T_2$ as a function of $n$.