Orbital and spin relaxation in single and coupled quantum dots

Phonon-induced orbital and spin relaxation rates of single electron states in lateral single and double quantum dots are obtained numerically for realistic materials parameters. The rates are calculated as a function of magnetic field and interdot coupling, at various field and quantum dot orientations. It is found that orbital relaxation is due to deformation potential phonons at low magnetic fields, while piezoelectric phonons dominate the relaxation at high fields. Spin relaxation, which is dominated by piezoelectric phonons, in single quantum dots is highly anisotropic due to the interplay of the Bychkov-Rashba and Dresselhaus spin-orbit couplings. Orbital relaxation in double dots varies strongly with the interdot coupling due to the cyclotron effects on the tunneling energy. Spin relaxation in double dots has an additional anisotropy due to anisotropic spin hot spots which otherwise cause giant enhancement of the rate at useful magnetic fields and interdot couplings. Conditions for the absence of the spin hot spots in in-plane magnetic fields (easy passages) and perpendicular magnetic fields (weak passages) are formulated analytically for different growth directions of the underlying heterostructure. It is shown that easy passages disappear (spin hot spots reappear) if the double dot system loses symmetry by an $xy$-like perturbation.

Figure 1

(Color online) Orbital and spin (labels $D$, $BR$, and $D3$ denote which spin-orbit interaction is present) relaxation rates in a single quantum dot, for the piezoelectric (solid) and deformation potential (dashed) phonons. The confining length is $32\mathrm{nm}$. Anticrossing of the unperturbed spin and orbital state occurs at $B_{\perp }=5.2\mathrm{T}$.

Figure 2

(Color online) Orbital relaxation rate (the sum of the deformation and piezoelectric contribution) in a single quantum dot as a function of magnetic field and the confinement length $l_0$/the confinement (excitation) energy $E_0$. The rate is given on the logarithmic scale in the units of ${\mathrm{s}}^{-1}$. The solid lines represent equal relaxation rates (equirelaxation lines) with values shown by the labels. The granular structure in the figure is an artifact of the limited data resolution.

Figure 3

(Color online) Spin relaxation rate in a single quantum dot as a function of magnetic field and the confinement length $l_0$/the confinement energy $E_0$. The rate is given on the logarithmic scale in the units of ${\mathrm{s}}^{-1}$. The solid lines represent equirelaxation lines.

Figure 4

(Color online) Spin relaxation rate in a double quantum dot as a function of in-plane magnetic field for $\gamma =0°$ and the interdot distance $d$/tunneling energy $\delta E_t$, for a confinement length $32\mathrm{nm}$. The relaxation rate is given on the logarithmic scale in the units of ${\mathrm{s}}^{-1}$. The double dot is oriented along [100] $(\delta =0°)$.

Figure 5

(Color online) Orbital relaxation rate in a double quantum dot as a function of in-plane magnetic field for $\gamma =0°$ and the interdot distance $d$/tunneling energy $\delta E_t$, for a confinement length $32\mathrm{nm}$. The relaxation rate is given on the logarithmic scale in the units of ${\mathrm{s}}^{-1}$. The double dot is oriented along [100] $(\delta =0°)$.

Figure 6

(Color online) Spin relaxation rate as a function of perpendicular magnetic field for $\gamma =0°$ and the interdot distance $d$/tunneling energy $\delta E_t$ (at zero magnetic field only), for a confinement length $32\mathrm{nm}$. The relaxation rate is given in logarithmic scale in the units of ${\mathrm{s}}^{-1}$. The double dot is oriented along [100] $(\delta =0°)$. The upper figure shows results when only the Bychkov-Rashba term is present in the Hamiltonian. In the lower figure, only the Dresselhaus terms are present.

Figure 7

(Color online) Spin relaxation rate as a function of $\gamma$ and the tunneling energy for $B_{\Vert }=1\mathrm{T}$, for [110] growing direction. The dot orientation is given by $\delta =\pi ∕2$. The relaxation rate is given in logarithmic scale in the units of ${\mathrm{s}}^{-1}$.

Figure 8

(Color online) Spin relaxation rate as a function of $\xi$ and tunneling energy for $B=1\mathrm{T}$, for [110] growing direction. The dot orientation is given by $\delta =\pi ∕2$. The relaxation rate is given in logarithmic scale in the units of ${\mathrm{s}}^{-1}$.

Figure 9

(Color online) Spin relaxation rate in a double dot as a function of the orientation of the in-plane magnetic field and tunneling energy for $B=5\mathrm{T}$, for [001] growing direction. The dot orientation is given by $\delta =\pi ∕4$. A small asymmetric term is added into the confinement potential (an electric field of ${10}^3\mathrm{V}∕\mathrm{m}$ in $y$ direction is applied in one of the dots). By this, the easy passage is turned into a weak passage—compare with Fig. 4 in Ref. 37.