Strain effects in phosphorus bound exciton transitions in silicon

Donor spin states in silicon are a promising candidate for quantum information processing. One possible donor spin readout mechanism is the bound exciton transition that can be excited optically and creates an electrical signal when it decays. This transition has been extensively studied in the bulk, but in order to scale towards localized spin readout, microfabricated structures are needed for detection. As these electrodes will inevitably cause strain in the silicon lattice, it will be crucial to understand how strain affects the exciton transitions. Here we study the phosphorus donor bound exciton transitions in silicon using hybrid electro-optical readout with microfabricated electrodes. We observe a significant zero-field splitting as well as mixing of the hole states due to strain. We can model these effects assuming the known asymmetry of the hole $g$ factors and the Pikus-Bir Hamiltonian describing the strain. In addition, we describe the temperature, laser power, and light polarization dependence of the transitions. Importantly, the hole mixing should not prevent donor electron spin readout, and using our measured parameters and numerical simulations, we anticipate that hybrid spin readout on a silicon-on-insulator platform should be possible, allowing integration into silicon photonics platforms.

Figure 1

(a) A schematic of our sample, the measurement geometry, and the optical setup. Light is guided to the sample in free space via a fixed lens. The magnetic field is in-plane and parallel to the direction of measured current and the [011] crystal direction. (b) Energy level diagram showing the Zeeman splitting of donor electron ($E_g$) and hole ($E_{D^{0}X}$) states. (c) Arrows showing the six allowed exciton transitions in a nonzero magnetic field. (d) Simulated current signal of the exciton transitions as a function of magnetic field in strain-free silicon. Transition brightness corresponds to the Clebsch-Gordan coefficient of the transition.

Figure 2

(a) Strain split hole eigenenergies and (b) exciton transition energies calculated using Eqs. (1, 2, 3). $B$ is the applied external magnetic field. The magnetic field lies in the [011] crystal direction. The labels show the dominating state at high magnetic fields. (c) Measured transition energies as a function of the magnetic field, in the [011] direction, overlaid with the calculated transition energies presented in (b). The center transition frequency is adjusted to match the data. Blue lines indicate transitions involving the donor electron spin-up state, and red lines indicate transitions coming from the donor electron spin-down state. Transitions that are forbidden without the hole mixing are shown with transparent lines.

Figure 3

(a) Half-wave-plate (HWP) angle dependence on the exciton signal at 427 mT. (b) Peak height profiles for each of the transitions. $\mathrm{\Delta }{R}_{\mathrm{tot}}$ is the change in total resistance. Color labels are given at the top of (a). Each line is normalized to start from zero in (b). The absence of proper polarization control is due to hole mixing. There is still a small trend visible where the middle transitions, transitions 3 and 4 ($\pi$ transitions in the absence of hole mixing), get weaker and the other transitions ($\sigma$ transitions in absence of hole mixing) get stronger from left to right as would be expected. This trend is partially obscured by some parasitic polarization-dependent power oscillations (the “fast” oscillations) in the data.

Figure 4

(a) Laser power dependence of the exciton signal at zero magnetic field. Current values are offset to show the difference in the peak heights. The base current difference between LED-on and LED-off measurements is $\approx 3.5$ nA. (b) Exciton transition at zero applied magnetic field for CZ and FZ silicon. Current values are normalized, and CZ data are offset to illustrate the difference in the linewidths. CZ data were captured without LED illumination, and FZ data were captured with LEDs. The laser power was the same in both measurements.

Figure 5

Measured magnetic field dependence of the exciton spectrum at millikelvin temperatures with (a) 30 $\mu \mathrm{W}$ (sample stage temperature 115 mK) and (b) 3 $\mu \mathrm{W}$ of laser power (sample stage temperature 45 mK). (c) and (d) show crosscuts at zero magnetic field. $R_n$ is the normalized resistance.

Figure 6

Simulated exciton transitions on a SOI platform in (a) natural silicon and (b) 99.991% enriched ${}^{28}\mathrm{Si}$. $B$ is the applied external magnetic field, and $\mathrm{\Delta }E$ is the transition energy offset from the value without strain or magnetic field. Simulation parameters are presented in Table 1, and strain is taken from simulations shown in Appendix pp2. The transition linewidth in (a) is the bulk value we have extracted from natural silicon, 4 $\mu \mathrm{eV}$, and the transition linewidth in (b) is 150 neV as reported in Ref. [18]. The white ring in (a) shows possible spin readout conditions in natural silicon at just below 200 mT, where the transition is isolated enough to be addressed optically, and similarly for the white line in (b) at 20 mT.

Figure 7

Simulated strain tensor elements on the sample surface (top) and at 1 $\mu \mathrm{m}$ depth (bottom) in the bulk sample. The black arrows show the coordinate directions, and the circle between the electrodes shows the location and size of the laser spot.

Figure 8

Simulated strain tensor elements at 100 nm depth in the suspended 220-nm silicon film. The black arrows show the coordinate directions. The stripes between the electrodes are narrow holes that allow the release of the Si film by hydrofluoric acid (HF) etching. Note how constant the strain components are within the released area.