Anisotropy of spin splitting and spin relaxation in lateral quantum dots

Inelastic spin relaxation and spin splitting $\epsilon_{\mathrm{s}}$ in lateral quantum dots are studied in the regime of strong in-plane magnetic field. Due to both g-factor energy dependence and spin-orbit coupling $\epsilon_{\mathrm{s}}$ demonstrates a substantial non-linear magnetic field dependence similar to that observed by R.Hanson \textit{et al} [Phys. Rev. Lett. \textbf{91}, 196802 (2003)]. It also varies with the in-plane orientation of magnetic field due to crystalline anisotropy of spin-orbit coupling. Spin relaxation rate is also anisotropic, the anisotropy increasing with the field. When the magnetic length is less than the 'thickness' of GaAs dot, the relaxation can be order of magnitude faster for $\mathbf{B}$ than for $\mathbf{B\Vert}[110]$.