Anisotropic electron $g$ factor in quantum dots with spin-orbit interaction

$g$-factor tuning of electrons in quantum dots is studied as a function of in-plane and perpendicular magnetic fields for different confinements. Rashba and Dresselhaus effects are considered, and comparison is made between wide and narrow-gap materials. The interplay between external magnetic fields and intrinsic spin-orbit couplings is analyzed, with two distinct phases found in the spectrum for GaAs in perpendicular field. The anisotropy of the $g$ factor is reported, and good agreement with available experimental findings is obtained.

Figure 1

(Color online) InSb QD spectrum (Ref. 19). Left (right) panels refer to in-plane (perpendicular) magnetic field. $A$ and $B$ show spectra (dotted lines refer to QD without SO interaction). $C$ and $D$ ($E$ and $F$) show field evolution of spin (orbital) angular momentum expectation values for the seven lowest levels; ground state is spin-up. Labels in $C$ and $D$ show which SO term is responsible for the lowest ACs. Colors (symbols) indicate different levels: black, red, green, … (square, circle, triangle, …) stand for ground, first excited, second excited, … states, in all panels. A sudden change in line color (symbol) indicates a level crossing.

Figure 2

(Color online) Level spectrum for GaAs QD (Ref. 20). $B_{\perp }$ scale of $B$ is different from $D$ and $F$. No AC occurs under $B_{\Vert }$. Ground state at low $B_{\perp }$ fields is spin down, and it is flipped back to spin up around $13.7\mathrm{T}$, as shown in $D$.

Figure 3

(Color online) Electron $g$ factor of InSb QDs with different confinement energies $E_0$ (or lateral sizes $l_0$). Left (right) panels refer to in-plane (perpendicular) field. $A$ and $C$ ($B$ and $D$) show $g_{\Vert }$ $\left(g_{\perp }\right)$, while $E$ and $F$ show sublevel splitting $\Delta E$ for the two lowest QD states. $A$ and $B$ ($C$ and $D$) show $g$ as obtained from the $⟨\Delta {\sigma }_z⟩$ $\left(\Delta E\right)$ calculation; values of $⟨\Delta {\sigma }_z⟩$ for the $15\mathrm{meV}$ QD are obtained from Fig. 1. Dotted lines in all panels refer to corresponding QDs without SO coupling. $\Delta E$ in $F$ is multiplied by 2. Color (symbol) legend is shown in $B$; dotted lines follow same scheme.

Figure 4

(Color online) Same as Fig. 3 but for GaAs QDs. $g_{\perp }^{{\sigma }_z}$ has inverted sign at low fields indicating that SO coupling is stronger than the Zeeman splitting ($⟨\Delta {\sigma }_z⟩$ of the $1.1\mathrm{meV}$ QD from Fig. 2). $g_{\perp }^E$ has large values at low fields. Dashed lines connecting $D$ and $F$ show fields $B_g$ for which $g_{\perp }^E∕g_0=1$, and the SO effect is effectively cancelled. Color (symbol) legend is in $A$. $g$ in $D$ is divided by 10.