Oscillatory spin relaxation rates in quantum dots

Phonon-induced spin relaxation rates in quantum dots are studied as function of in-plane and perpendicular magnetic fields, temperature, and electric field, for different dot sizes. We consider Rashba and Dresselhaus spin-orbit mixing in different materials and show how Zeeman sublevels can relax via piezoelectric and deformation potential coupling to acoustic phonons. We find that strong lateral and vertical confinements may induce minima in the rates at particular values of the magnetic field, where spin relaxation times can reach even seconds. We obtain good agreement with experimental findings in GaAs quantum dots.

Figure 1

(Color online) (a) Zeeman sublevel spin-flip rates for GaAs QDs (Ref. 16) under in-plane magnetic field for different QD confinements (solid lines with symbols). Dashed-dotted, dotted, and dashed lines respectively show influence of higher temperature, well width, and electric field on the $3.5\mathrm{meV}$ QD. Curves with open symbols show isolated contributions from the distinct phonon mechanisms for the $5.0\mathrm{meV}$ QD. (b) Spin average values of the Zeeman sublevels after diagonalization of $H$. Curves with filled (open) symbols refer to QD ground (first excited) state. (c) Energy splitting of Zeeman sublevels after diagonalization of $H$. Dashed-dotted lines show results without SO coupling. Color, symbol, and line form schemes follow from (a).

Figure 2

(Color online) Same as Fig. 1, but for GaAs QDs under perpendicular magnetic field. Vertical lines in (b) show the field where a level crossing occurs and the normal spin character of states is restored. Arrows in (c) indicate where suppression of SO coupling is induced by the field, which is at the origin of the respective minima in (a). Color, symbol, and line form schemes, as well as QD parameters of each plot, are exactly the same used in Fig. 1.

Figure 3

(Color online) InSb QDs (Ref. 19) under in-plane magnetic field. The confinements are such that the same QD size-range used for GaAs is covered. In (a), the 15.0 $\left(20.0\right)\mathrm{meV}$ QD is chosen for the study of different QD parameters (distinct phonon mechanisms); dashed arrows show that both LA modes have the same number of minima and happen at the same fields. Color, symbol, and line form schemes in each panel follow from the respective panel in Fig. 1.

Figure 4

(Color online) Same as Fig. 3, but for InSb QDs under perpendicular magnetic field. The only minimum of the $3.0\mathrm{meV}$ QD around $1\mathrm{T}$ in (a) indicates a sublevel crossing, as shown by the vertical line in (b) and the vanishing of $\Delta E$ in (c). A larger $z$ well introduces an oscillatory rate. Color, symbol, and line form schemes, as well as QD parameters of each plot, are exactly the same used in Fig. 3.