Spin relaxation at the singlet-triplet crossing in a quantum dot

We study spin relaxation in a two-electron quantum dot in the vicinity of the singlet-triplet crossing. The spin relaxation occurs due to a combined effect of the spin-orbit, Zeeman, and electron-phonon interactions. The singlet-triplet relaxation rates exhibit strong variations as a function of the singlet-triplet splitting. We show that the Coulomb interaction between the electrons has two competing effects on the singlet-triplet spin relaxation. One effect is to enhance the relative strength of spin-orbit coupling in the quantum dot, resulting in larger spin-orbit splittings and thus in a stronger coupling of spin to charge. The other effect is to make the charge density profiles of the singlet and triplet look similar to each other, thus diminishing the ability of charge environments to discriminate between singlet and triplet states. We thus find essentially different channels of singlet-triplet relaxation for the case of strong and weak Coulomb interactions. Finally, for the linear in momentum Dresselhaus and Rashba spin-orbit interactions, we calculate the singlet-triplet relaxation rates to leading order in the spin-orbit interaction and find that they are proportional to the second power of the Zeeman energy, in agreement with recent experiments on triplet-to-singlet relaxation in quantum dots.

Figure 1

Sketch of the singlet-triplet crossing in a quantum dot with spin-orbit interaction. The triplet level splits in three due to the Zeeman interaction. The singlet $\mid {\Psi }_S⟩$ undergoes avoided crossing with the triplets $\mid {\Psi }_{T_{\pm }}⟩$, with the splittings ${\Delta }_{\pm }$ given in Eq. (65). At the same time, the degeneracy of $\mid {\Psi }_S⟩$ and $\mid {\Psi }_{T_0}⟩$ at the crossing point is not lifted.

Figure 2

(Color online) Angular dependence of the splitting ${\Delta }_{+}$ as given by Eq. (65), evaluated for ${\lambda }_{+}∕{\lambda }_{-}=2$. The direction in space corresponds to the spin-quantization axis, given by $\mathit{l}=\mathit{B}∕B$. At the magic angles, the value of ${\Delta }_{+}$ equals zero. The angular dependence of the splitting ${\Delta }_{-}$ in Eq. (65) is obtained from this figure by reflecting it in the $(x^{\prime },y^{\prime })$ plane. Note that the circular symmetry of the quantum dot was essential for obtaining this angular dependence.

Figure 3

(Color online) Opposite to Fig. 2, we plot here the angular dependence of the splittings ${\Delta }_{\pm }$ (with ${\Delta }_{+}={\Delta }_{-}$) as given by Eq. (72), which describes a quantum dot without circular symmetry. The degree of asymmetry needs to be large enough, such that in a magnetic field causing the singlet-triplet crossing the imaginary part of $\overline{\mathbf{r}}=⟨{\psi }_T\mid \mathbf{r}\mid {\psi }_S⟩$ to be negligible. For the plot, we assumed ${\overline{r}}_{y^{\prime }}{\lambda }_{+}∕{\overline{r}}_{x^{\prime }}{\lambda }_{-}=2$.

Figure 4

(Color online) Spin relaxation processes (a) accounted for and (b) neglected in our treatment. The arrows show transitions between states of the lowest in energy singlet-triplet subspace. The excited states (upper row) are involved only in virtual transitions as intermediate states. (a) We take into account the strong matrix elements, which mix the singlet-triplet subspace inside itself, allowing phonon-induced transitions to occur between the singlet and two triplet states. We denote these transitions by the lowermost arrows, where the product $H_Z^{\mathit{SO}}{U}_{\mathrm{ph}}$ next to the arrows denotes that the transition occurs under the simultaneous action of $H_Z^{\mathit{SO}}$ and $U_{\mathrm{ph}}$. Additionally, we take into account the weak matrix elements for the triplet-triplet transition. In contrast to the singlet-triplet transition, a triplet-triplet transition requires the participation of an excited state. During the virtual process, $H_Z^{\mathit{SO}}$ and $U_{\mathrm{ph}}$ act separately on the two halves of the process (e.g., going up with $H_Z^{\mathit{SO}}$ and coming down with $U_{\mathrm{ph}}$). We denote this difference by placing $H_Z^{\mathit{SO}}$ and $U_{\mathrm{ph}}$ on opposite sides of the arrow. Note that $H_Z^{\mathit{SO}}$ connects states with different spins, whereas $U_{\mathrm{ph}}$ with the same. (b) Transitions between the singlet and triplet states due to the weak matrix elements are neglected. These processes are relevant only in special cases mentioned in the text (see Sec. 5C).

Figure 5

(Color online) The form factor ${\mathcal{F}}_{ST}\left(q_{\parallel }\right)$ for $\lambda ∕a_B^{*}=0,1,3$, as well as ${10}^3{\mathcal{F}}_{ST}\left(q_{\parallel }\right)$ for $\lambda ∕a_B^{*}={10}^3$.

Figure 6

(Color online) (a) The factors $a_0^2{a}_{+}^2$ (solid line) and $a_0^2{a}_{-}^2$ (dashed line) in Eq. (105) as functions of the magnetic field $B_z=B$. (b) Similarly, the factor $a_{+}^2{a}_{-}^2$ in Eq. (106) as a function of $B$. For the plot, we used the following model (arbitrary units): ${\Delta }_{+}=0.15\mid E_Z\mid$, ${\Delta }_{-}=0.1\mid E_Z\mid$, $E_S=0$, $E_{T_{\pm }}=10-2B\pm E_Z$, and $E_Z=-0.2B$. The exact solutions for $a_{\alpha }^2{a}_{\beta }^2$ (dotted line) coincide with the approximated ones [see Eqs. (105, 106)] on the scale of the plot. Note the different scales on the left and right panels.

Figure 7

(Color online) The factors ${\mathcal{A}}_0(\lambda ∕a_B^{*})$ (solid curve), ${\mathcal{A}}_2(\lambda ∕a_B^{*})$ (dashed curve), and ${\mathcal{A}}_5(\lambda ∕a_B^{*})$ (dot-dashed curve) as functions of the Coulomb interaction strength $\lambda ∕a_B^{*}$. Inset: the factor ${\mathcal{A}}_5(\lambda ∕a_B^{*})$ on a larger scale.

Figure 8

(Color online) Schematic of ${\tilde{\Gamma }}_{\mathit{LA}}^{\mathit{DP}}\left(w\right)$, showing the relevant scales of $w$: $s∕\lambda$, $s∕d$, and $s∕⟨r⟩$. The curve was obtained using the form factor in Eq. (100) for a fairly large value of the Coulomb interaction strength, $\lambda ∕a_B^{*}={10}^3$.

Figure 9

(Color online) The factors ${\mathcal{B}}_1$, ${\mathcal{B}}_2$, and ${\mathcal{B}}_3$ as functions of the Coulomb interaction strength $\lambda ∕a_B^{*}$. Inset: the factors in the main figure multiplied by ${({\lambda }_{1,0}∕\lambda )}^2$.

Figure 10

(Color online) Relaxation rates ${\Gamma }_{+,0}$, ${\Gamma }_{+,-}$, and ${\Gamma }_{0,-}$ as functions of an in-plane magnetic field $\mathit{B}=(0,B_{\parallel },0)$. An avoided crossing of the singlet $\mid S⟩$ and triplet $\mid T_{+}⟩$ occurs at $E_{TS}=-E_Z\approx 0.75\mathrm{meV}$ $(B_{\parallel }\approx 8.2\mathrm{T})$. We used parameters typical for a GaAs quantum dot: $\hslash {\omega }_0=2\mathrm{meV}$, ${\lambda }_{\pm }=8\mu \mathrm{m}$, $d=5\mathrm{nm}$, $g=-0.44$, and $\kappa =13$. Inset: Schematic of level crossing for applying an in-plane magnetic field.

Figure 11

(Color online) Relaxation rates ${\Gamma }_{+,0}$, ${\Gamma }_{+,-}$, and ${\Gamma }_{0,-}$ as functions of the transverse magnetic field $\mathit{B}=(0,0,B_{\perp })$. Two avoided crossings occur at $E_{TS}=\pm E_Z$, as shown in Fig. 1. We use the same parameters as in Fig. 10, except for ${\lambda }_{+}={\lambda }_{-}∕2=8\mu \mathrm{m}$. For these parameters, we obtain $E_{TS}\approx 0.75\mathrm{meV}$ at $B_{\perp }=0$ and $E_{TS}=0$ at $B_{\perp }\approx 0.95\mathrm{T}$.

Figure 12

(Color online) (a) The variational parameters $\gamma$ as a functions of the Coulomb interaction strength $\lambda ∕a_B^{*}$ for the singlet $m=0$ and the triplet $\mid m\mid =1$ states. (b) The ratio $\tilde{\omega }∕\omega$ as a function of the decimal logarithm of $\lambda ∕a_B^{*}$ for $m=0$ and $\mid m\mid =1$. To find $\gamma$ and $\tilde{\omega }∕\omega$, we minimize the ground state energy in Eq. (13); see Appendix for details.

Figure 13

(Color online) (a) The parameter $\delta$ as a function of the Coulomb interaction strength $\lambda ∕a_B^{*}$. The dotted curve shows $\delta$ calculated according to Eq. (B2). (b) The singlet-triplet degeneracy occurs at a magnetic field $B_z^{*}$ $(E_{TS}=0)$. This figure shows the corresponding ${\omega }_c^{*}∕{\omega }_0$ as a function of ${\lambda }_0∕a_B^{*}$, where ${\omega }_c^{*}=e{B}_z^{*}∕m^{*}c$ and ${\lambda }_0=\sqrt{2\hslash ∕m^{*}{\omega }_0}$. The dotted curve corresponds to the fitting function in Eq. (B3).

Figure 14

(Color online) Infidelity $1-F$ of the ground state of ${\mathcal{H}}_m$ for $\mid m\mid =0,1$. We used Eq. (C5) with the matrix elements of $V$ evaluated in appendix . Note that both axes are logarithmic (with decimal base).