Unique Spin Vortices and Topological Charges in Quantum Dots with Spin-orbit Couplings

Spin textures of one or two electrons in a quantum dot with Rashba or Dresselhaus spin-orbit couplings reveal several intriguing properties. We show here that even at the single-electron level stable spin vortices with tunable topological charges exist. These topological textures appear in the ground state of the dots. The textures are stabilized by time-reversal symmetry breaking and are robust against the eccentricity of the dot. The topological charge is directly related to the sign of the z component of the spin in a large dot, allowing a direct probe of its topological properties. This would clearly pave the way to possible future topological spintronics. The phenomenon of spin vortices persists for the interacting two-electron dot in the presence of a magnetic field.


Single-electron dot: Equal Rashba and Dresselhaus couplings
The Hamiltonian of an electron in an isotropic dot Ω x,y = Ω, ℓ x,y = ℓ and g 1 = g 2 = g without magnetic field is equivalent to a two-component quantum Rabi model. In this case the Hamiltonian (2) in the main text reads, We now define the ladder operators of the quantum harmonic oscillator and use a x = −ib x , a y = ib y , to transform the Hamiltonian to which is a two-component quantum Rabi model with zero splitting [4]. The case of g 1 = −g 2 can be solved in a similar manner.
In the main text, we show in Eq. (5) that the ground states are a degenerate Kramers pair. If the magnetic field is infinitesimal then the degeneracy is lifted. Since ± GS| L z |GS ± = 0, the unique ground state can be found to lowest order by minimizing the Zeeman energy only. We use the ansatz with coefficients A, B. The Zeeman energy is then proportional to For ∆ > 0 the minimization of the Zeeman energy requires The other calculations presented in the main text are the standard first-order perturbative calculations.
2 Single-electron dot: size effects For a small dot (e.g. the case R x = R y = 15nm for an InAs dot), the sign of σ z in the ground state cannot be changed by the SOCs. However, if the size is enlarged, then the sign of σ z may be changed due to SOCs. We compute the perturbative energies for different spin projections. The first-order correction of the energy is zero, so we consider the second-order corrections, If there is no magnetic field, both states are still degenerate due to time reversal symmetry, E + = E − . In general, where E + is for the state with dominant spin |+ while E − is for dominant spin |− . We have to compare these two energies to figure out the true ground state. If there is only Rashba coupling present, then If, on the other hand, there is only Dresselhaus present, then (S.14) We note that the perturbation theory becomes invalid if the denominators in the perturbative corrections are close to zero. In most cases, however, the perturbation theory works fine, especially if the magnetic field is not too large.
If the dot is very large (approaching infinity) then Ω → ω c /2 and Let us consider, in particular, an InAs dot. If only Rashba SOC is present, then the ground state is always the state with energy E R+ (E R+ − E R− < 0 because g L < 0). Therefore σ z > 0 and σ z → 1 when the magnetic field is strong. However, when only the Dresselhaus SOC is present, the sign of σ z can be reversed. This happens because for weak magnetic field the ground state can change to the state with energy E D− (E D+ − E D− > 0 leads to Eq. (10) in the manuscript) with σ z < 0. Consider, on the other hand, a material with positive Landé factor, g L > 0, for example ZnO. If only Dresselhaus SOC is present, then the ground state is the state with energy E D− , since E D+ −E D− > 0. Therefore we always have σ z < 0. When only the Rashba SOC exists, then the sign of σ z can be reversed. With regard to a possible sign change of σ z in the ground state of the dot the roles of Dresselhaus and Rashba SOC are thus reversed if the sign of the Landé factor g L is changed. Note that in both cases (for the g L < 0 Dresselhaus dot and for the g L > 0 Rashba dot), the effective mass m * should not be too large, m * < 2m e /|g L |, otherwise a sign reversal is not possible.
In Fig. S1, we compare the spin textures in InAs dots with different sizes (a) R = 15nm and (b) R = 50nm. The Dresselhaus SOC is the same in both cases, g 2 = 20 nm· meV, and the Rashba SOC is zero. We find that the topological charge is the same while σ z is reversed by the size effect.
In other materials we can do the same analysis, i.e. compare the energies between E R+ and E R− or between E D+ and E D− to find if there is a sign change of σ z . The perturbative energies may not be very accurate for a large dot in strong magnetic fields, but the numerical calculations directly yield reliable results for σ z .

Single-electron dot: Numerical results
In Fig. 1(a) of the main text we have shown that the in-plane spin texture in a weak magnetic field for the case g 1 = g 2 is mirror symmetric around the line x = y, consistent with the perturbative calculation. For larger fields the perturbative ground state given above is, however, no longer a good starting point. In Fig. S2, we show how the in-plane spin textures evolve with increasing magnetic field. All these results show a mirror symmetry about the line x = ±y. However, when the magnetic field becomes stronger the spins start to rotate leading to a spin texture similar to the case of Rashba SOC only. Note that the in-plane spin components are weaker than in the Rashba case though because the spin becomes more and more polarized along the z-direction.
In Fig. 2 of the main text we have shown the spin vortices in a single-electron dot if only the Rashba or the Dresselhaus SOC is present. In Fig. S3 we show that these results are indeed representative for the regimes g 1 ≫ g 2 and g 2 ≫ g 1 . We also note that even for larger magnetic fields the topological properties are not changed, although the spin textures are weakened. States with higher topological charge |q| > 1 may exist in the excited states. Contrary to the spin textures in the ground state they are, however, fragile due to their Kramers partner.
Finally, we also show that the spin texture is robust against the eccentricity of the dot. We consider an elliptical InAs dot with R x = 15nm and R y = 10nm. From Fig. S4 it is obvious that the distortion of the dot does not qualitatively change the structure of the spin vortex.