Interplay of spin-orbit and hyperfine interactions in dynamical nuclear polarization in semiconductor quantum dots

We theoretically study the interplay of spin-orbit and hyperfine interactions in dynamical nuclear polarization in two-electron semiconductor double quantum dots near the singlet-triplet ($S-T_{+}$) anticrossing. The goal of the scheme under study is to extend the singlet-triplet ($S-T_0$) qubit decoherence time $T_2^{*}$ by dynamically transferring the polarization from the electron spins to the nuclear spins. This polarization transfer is achieved by cycling the electron spins over the $S-T_{+}$ anticrossing. Here, we investigate, both quantitatively and qualitatively, how this hyperfine-mediated dynamical polarization transfer is influenced by the Rashba and Dresselhaus spin-orbit interaction. In addition to $T_2^{*}$, we determine the singlet return probability $P_s$, a quantity that can be measured in experiments. Our results suggest that the spin-orbit interaction establishes a mechanism that can polarize the nuclear spins in the direction opposite to that of hyperfine-mediated nuclear spin polarization. In materials with relatively strong spin-orbit coupling, this interplay of spin-orbit and hyperfine-mediated nuclear spin polarizations prevents any notable increase in the $S-T_0$ qubit decoherence time $T_2^{*}$.

Figure 1

Geometry of the problem. The strength of spin-orbit interaction is tuned by varying the angle $\theta$ between the $\left[110\right]$ crystallographic axis and the interdot connection axis $p_{\xi }$. Spin-orbit interaction generates an effective magnetic field $\mathit{\Omega }$ along the $y$ axis. The external magnetic field is perpendicular to the $\left[110\right]-p_{\xi }$ plane.

Figure 2

Two-electron spectrum of a DQD in InAs as a function of the interdot bias $\varepsilon$, obtained by diagonalizing the Hamiltonian $H_0$ [Eq. (2)]. The energy $E$ and the detuning $\varepsilon$ are expressed in units of the Coulomb energy $U$. The parameters of the plot are the magnetic field $B=1\text{T}$, the Coulomb energy $U=4.86\text{meV}$, the extended tunneling hopping $t_H=0.11\text{meV}$, the triplet matrix element $V_{+}=2.16\mu \text{eV}$, the doubly occupied singlet matrix element $V_{-}=0.42$ $\mu \mathrm{eV}$, and half of the interdot separation $a=73.6\text{nm}$. Including hyperfine interaction and/or spin-orbit interaction opens up an avoided crossing $\Delta$ [34] (inset). The magnetic field is chosen to be large, as compared with the value in the remainder of the paper, for visualization purposes. $S(2,0)$ and $S(0,2)$ are singly occupied singlets, and $S(1,1)$ is the doubly occupied singlet. $T_{+}$, $T_0$, and $T_{-}$ are triplet states corresponding to $m_s=1$, $m_s=0$, and $m_s=-1$. $S_{-}$ and $S_{+}$ are the lower and the upper hybridized singlets [see Eqs. (4) and (5)].

Figure 3

Initial nuclear spin probability distribution with respect to the quantum number $j$ for $N=150$ nuclear spins $1/2$, where $j_{\mathrm{ml}}=\sqrt{N/2}$ and $j_{\max }=18$. Throughout our calculations we only consider the states $0\le j\le j_{\max }$ (blue diamonds) and do not consider the states $j>j_{\max }$ (black circles).

Figure 4

System initialization and measurement outcomes. (a) Initially, the quantum dots have an energy bias $\varepsilon$, and the two electrons rest in a singlet $(2,0)$ state on the left dot. (b) After slowly tuning $\varepsilon$ to zero and measuring a singlet outcome, due to the weak measurement, the spin of the nuclear bath decreases. (c) In the case of a spin-triplet outcome an electron spin flips, and the spin of the nuclear bath is changed accordingly. (d) The electronic spin can also be flipped due to spin-orbit coupling, and the spin of the nuclear bath is pumped in the opposing direction (up) due to the weak measurement. With $\varepsilon$ we denote the voltage bias, $\theta$ is the angle between the $\left[110\right]$ crystallographic axis and the interdot connection axis $p_{\xi }$, and $\mathit{\Omega }$ is the spin-orbit effective magnetic field.

Figure 5

(a) Probability distribution in the left quantum dot with respect to the nuclear spin projection quantum number $m$ for $j_L=14$ before and after 300 DNP cycles. (b) Probability distribution in the right quantum dot with respect to the nuclear spin projection quantum number $m$ for $j_R=7$ before and after 300 DNP cycles. The included spin-orbit interaction corresponds to the angle $\theta =\pi /2$. Here, $\theta$ is the angle between the $\left[110\right]$ crystallographic axis and the interdot connection axis $p_{\xi }$. The number of nuclear spins per quantum dot is $N=150$.

Figure 6

The singlet return probability $P_S$ as a function of the number of cycles across the $S-T_{+}$ anticrossing in ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$. Here, $\theta$ is the angle between the $\left[110\right]$ crystallographic axis and the interdot connection axis $p_{\xi }$.

Figure 7

Standard deviation of the nuclear difference field ${\sigma }^{\left(z\right)}$ with respect to the number of DNP cycles across the $S-T_{+}$ anticrossing and different values of angle $\theta$ in ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$. Here, $\theta$ is the angle between the $\left[110\right]$ crystallographic axis and the interdot connection axis $p_{\xi }$.

Figure 8

$S-T_0$ qubit decoherence time $T_2^{*}$ as a function of the number of DNP cycles across the $S-T_{+}$ anticrossing and strength of spin-orbit interaction in ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$. Here, $\theta$ is the angle between the $\left[110\right]$ crystallographic axis and the interdot connection axis $p_{\xi }$.

Figure 9

The ratio of the final $T_{2,f}^{*}$ and initial $T_{2,i}^{*}$ decoherence times in ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{As}$ for different values of the angle $\theta$ between the $\left[110\right]$ crystallographic axis and the interdot connection axis $p_{\xi }$.

Figure 10

$S-T_0$ electron spin coherence time $T_2^{*}$ as a function of the number of DNP cycles across the $S-T_{+}$ anticrossing for different abundances of indium $x$ in ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ and for $\theta =\pi /2$. Here, $\theta$ is the angle between the $\left[110\right]$ crystallographic axis and the interdot connection axis $p_{\xi }$.

Figure 11

$S-T_0$ electron spin coherence time $T_2^{*}$ for GaAs and InAs as a function of the number of DNP cycles across the $S-T_{+}$ anticrossing for $\theta =0$, i.e., the case where the $\left[110\right]$ crystallographic axis and the interdot connection axis $p_{\xi }$ are aligned.