Spin Echoes in the Charge Transport through Phosphorus Donors in Silicon

The electrical detection of spin echoes via echo tomography is used to observe coherent processes associated with the electrical readout of the spin state of phosphorus donor electrons in silicon near a ${\mathrm{SiO}}_2$ interface. Using the Carr-Purcell pulse sequence, an echo decay with a time constant of $1.7\pm 0.2\mu \mathrm{s}$ is observed and discussed in terms of decoherence and recombination times. Electrical spin echo tomography thus can be used to study the dynamics of the spin-dependent transport processes, e.g., in realistic spin qubit devices for quantum information processing.

Figure 1

(a) Spin-dependent transition involving the electron spin of a ${}^{31}\mathrm{P}$ donor and a paramagnetic state ${\mathrm{P}}_{b0}$ at the interface between crystalline Si ($c$-Si) and ${\mathrm{SiO}}_2$. (b) Carr-Purcell echo for an ensemble of identical spins plotted on a Bloch sphere. The echo in the $x\mathrm{\text{−}}y$ plane is rotated to an eigenstate by the final ”echo tomography” $\pi /2$ pulse, necessary for the electrical detection of the echo. (c) The effect of a change in the second evolution time ${\tau }_2$ with respect to ${\tau }_1$ on the same spin ensemble before and after the final $\pi /2$ pulse. (d) Charge $Q$ obtained from the integration of the current transients after microwave pulses of length ${\tau }_{\mathrm{Rabi}}$ resonant with the high-field resonance of the ${}^{31}\mathrm{P}$ donor spin ensemble (Rabi oscillations). The color code for the $\mathrm{P}\mathrm{\text{−}}{\mathrm{P}}_{b0}$ pair is identical to (a). (e) Charge $Q$ after application of a Carr-Purcell pulse sequence including the final tomography pulse, again detected on the high-field ${}^{31}\mathrm{P}$ resonance. The sign and the amplitude of the echo signature agree well with the corresponding Rabi oscillations as discussed in the text.

Figure 2

Electrically detected Carr-Purcell echoes at 6.5 K for a first free evolution period of ${\tau }_1=200\mathrm{ns}$. A clear signature in the integrated charge $\Delta Q$ appears at ${\tau }_1={\tau }_2$ for all spin resonance lines, while no further features are resolved up to 900 ns. The features at ${\tau }_2\lesssim 60\mathrm{ns}$ are so-called Ramsey fringes, which are observable for evolution periods ${\tau }_2$ below the dephasing time of the system.

Figure 3

Electrically detected Carr-Purcell echoes observed on the isolated high-field P resonance at $B_0=351.2\mathrm{mT}$ for different evolution times ${\tau }_1$. The solid line represents a monoexponential decay with a time constant of $1.7\pm 0.2\mu \mathrm{s}$.