Anisotropic Stark Effect and Electric-Field Noise Suppression for Phosphorus Donor Qubits in Silicon

We report the use of novel, capacitively terminated coplanar waveguide resonators to measure the quadratic Stark shift of phosphorus donor qubits in Si. We confirm that valley repopulation leads to an anisotropic spin-orbit Stark shift depending on electric and magnetic field orientations relative to the Si crystal. By measuring the linear Stark effect, we estimate the effective electric field due to strain in our samples. We show that in the presence of this strain, electric-field sources of decoherence can be non-negligible. Using our measured values for the Stark shift, we predict magnetic fields for which the spin-orbit Stark effect cancels the hyperfine Stark effect, suppressing decoherence from electric-field noise. We discuss the limitations of these noise-suppression points due to random distributions of strain and propose a method for overcoming them.

Figure 1

(a) Optical micrograph of the CPW resonator. Microwaves excite the resonator from the transmission line on the left and a dc bias is applied through the wire on the right. Electric potential lines are shown at an antinode in ${\overrightarrow{B}}_1$ for $\overrightarrow{E}\perp {\overrightarrow{B}}_0$ (b) and $\overrightarrow{E}\parallel {\overrightarrow{B}}_0$ (c). The CPW cross section is depicted by a cartoon at the top of the plot. The 0.9 V line is labeled (given a 1 V bias) and each subsequent line represents a 0.1 V decrease in potential. Red regions denote where 50% of the ESR signal originates. Microwave power was optimized to enhance sensitivity to spins at the center of the CPW gap where $\overrightarrow{E}$ is most uniform. (d) $\overrightarrow{E}$ distribution in the CPW (for a 4 V bias) weighted by the signal contribution.

Figure 2

Measured ESR frequency shift as a function of CPW center pin voltage for ${\overrightarrow{B}}_0$ oriented along the [100] (a) and [110] (b) crystal axes. Data are plotted for $\overrightarrow{E}$ either parallel (red diamond) or perpendicular (black square) to ${\overrightarrow{B}}_0$. Smooth curves are fits to the data using Eq. (2). Note that the hyperfine Stark shift is symmetric whereas the spin-orbit Stark shift is asymmetric with respect to zero as a function of $M_I$. The error bars for the data are smaller than the points used in the plot. Data were taken at 1.7 K with a $B_0$ of $0.26\mathrm{T}$.

Figure 3

(a) Plot of the electric-field sensitivity of P donors in Si for various orientations of $\overrightarrow{E}$ and ${\overrightarrow{B}}_0$ as a function of $B_0$. This plot assumes an internal strain of $1\mathrm{V}/\mu \mathrm{m}$ directed along $\overrightarrow{E}$. Dashed lines indicate $M_I=-1/2$ whereas solid lines indicate $M_I=+1/2$. $\overrightarrow{E}$ noise is suppressed at 19.12, 27.4, and 45.32 GHz. Another minima for the [110] oriented $\overrightarrow{E}\perp {\overrightarrow{B}}_0$ occurs at 56.65 GHz but is not shown. (b) Plot of the maximum (worst case) $\overrightarrow{E}$ sensitivity of spins subject to random strain (with magnitude $1\mathrm{V}/\mu \mathrm{m}$) as a function of externally applied strain. This plot assumes the external strain is oriented perpendicular to the $\overrightarrow{E}$ noise.