Spin relaxation of a donor electron coupled to interface states

An electron spin qubit in a silicon donor atom is a promising candidate for quantum information processing because of its long coherence time. To be sensed with a single-electron transistor, the donor atom is usually located near an interface, where the donor states can be coupled with interface states. Here we study the phonon-assisted spin-relaxation mechanisms when a donor is coupled to confined (quantum-dot-like) interface states. We find that both Zeeman interaction and spin-orbit interaction can hybridize spin and orbital states, each contributing to phonon-assisted spin relaxation in addition to the spin relaxation for a bulk donor or a quantum dot. When the applied magnetic field $B$ is weak (compared to orbital spacing), the phonon assisted spin relaxation shows the $B^5$ dependence. We find that there are peaks (hot spots) in the $B$-dependent and detuning dependent spin relaxation due to strong hybridization of orbital states with opposite spin. We also find spin relaxation dips (cool spots) due to the interference of different relaxation channels. Qubit operations near spin relaxation hot spots can be useful for the fast spin initialization and near cool spots for the preservation of quantum information during the transfer of spin qubits.

Figure 1

Schematic diagram of system potential and energy levels of a donor coupled to a QD-like interface state under an electric field from metallic gates. The energy levels are shown for a donor and a QD confinement in the absence of tunnel coupling. $\epsilon$ is the detuning of QD and donor ground states, $E_{VS}$ is the valley splitting of the lowest two QD states, $t_{01}$ and $t_{02}$ are the tunnel coupling amplitude between the donor ground state and the two lowest QD states.

Figure 2

Spin relaxation as a function of magnetic field when detuning $\epsilon =-1$ meV. We show spin relaxation due to intra-QD mechanism (black solid line), intradonor mechanism (red dashed line), and inter-donor-QD mechanisms due to ZI (blue dotted line) and SOI (purple dot-dashed line).

Figure 3

Energy diagram as a function of magnetic field when detuning $\epsilon =-1$ meV. The ground orbital spin-up state crosses twice (marked as filled black dots) with two excited orbital spin-down states, which is responsible for the hot spots of the inter-donor-QD spin relaxation in Fig. 2. At the spin relaxation cool spot (empty circle), a schematic diagram shows the two possible spin relaxation channels.

Figure 4

A general schematic energy diagram at the spin relaxation cool spot. There are two possible spin relaxation channels (dashed and dotted lines) indicated by lines with arrows, whose interference leads to the suppression of spin relaxation.

Figure 5

Spin relaxation as a function of magnetic field when detuning $\epsilon =1$ meV.

Figure 6

Spin relaxation as a function of magnetic field when detuning $\epsilon =0$ meV.

Figure 7

Spin relaxation as a function of magnetic field when detuning $\epsilon =-0.1$ meV and $E_{VS}=0.1$ meV.

Figure 8

Spin relaxation as a function of detuning when $B=0.5$ T.

Figure 9

Spin relaxation as a function of detuning when $B=5$ T.

Figure 10

Energy diagram as a function of detuning when $B=5$ T. The ground orbital spin-up state crosses three times with excited orbital spin-down states, which is responsible for the spin relaxation hot spots in Fig. 9. A schematic diagram shows two possible spin relaxation channels (dashed and dotted lines) that can interfere destructively.

Figure 11

Spin relaxation as a function of detuning when $B=5$ T and $E_{VS}=0.1$ meV.

Figure 12

Spin relaxation as a function of the polar angle ${\theta }_B$ of the applied magnetic field when detuning $\epsilon =0$ meV and $B=0.5$ T. We show spin relaxation due to ZI (black solid line) with ${\theta }_B=\pi /4$ and ${\varphi }_B=0$, and spin relaxation due to SOI when $d_{\parallel }=2$ nm with ${\varphi }_B-{\varphi }_d=0$ (red dashed line), ${\varphi }_B-{\varphi }_d=\pi /4$ (blue dotted line), and ${\varphi }_B-{\varphi }_d=\pi /2$ (purple dot-dashed line).

Figure 13

Spin relaxation as a function of the azimuthal angle ${\varphi }_B$ of the applied magnetic field when detuning $\epsilon =0$ meV, $B=0.5$ T, and ${\theta }_B=\pi /4$. We show spin relaxation due to ZI (black solid line) and spin relaxation due to SOI when $d_{\parallel }=2$ nm with ${\varphi }_d=0$ (red dashed line), ${\varphi }_d=\pi /4$ (blue dotted line), and ${\varphi }_d=\pi /2$ (purple dot-dashed line).

Figure 14

Spin relaxation as a function of the azimuthal angle ${\varphi }_B$ of the applied magnetic field when detuining $\epsilon =0$ meV, $B=0.5$ T and ${\theta }_B=\pi /2$. We show spin relaxation due to ZI (black solid line), and spin relaxation due to SOI when $d_{\parallel }=2$ nm with ${\varphi }_d=0$ (red dashed line), ${\varphi }_d=\pi /4$ (blue dotted line), and ${\varphi }_d=\pi /2$ (purple dot-dashed line).