Phase diagram of a three-electron quantum dot with an impurity

The ground-state property of a two-dimensional three-electron quantum dot subjected to an external magnetic field $B$ and affected by an impurity is studied. The impurity is located at the $z$ axis by a distance $d$ from the plane of the dot. The evolution of the angular momentum $L_0$ and the total spin $S_0$ of the ground states in accord with $B$ and $d$ has been calculated, and is visualized in a $d\text{−}B$ phase diagram. The physics underlying this diagram has been analyzed.

Figure 1

The $d\text{−}B$ phase diagram of the ground states. (a) The impurity has $q=1$; (b) $q=-1$. In each domain of parameters the associated ground state has a specific pair of $(L_0,S_0)$ marked inside the domain.

Figure 2

Evolution of $r_{rms}$ of the ground state as a function of $B$ in the vicinity of $P_1$ to $P_4$ when $q$ and $d$ are given at the values of the critical points [(a)–(d)], as a function of $d$ when $q$ and $B$ are given at the values of the critical points [(e)–(f)].

Figure 3

Contour plots of ${\rho }_2$ of the ground states in the vicinity of $P_2$ [(a)–(d)] and $P_3$ [(e)–(h)]. ${\mathbf{r}}_1$ is given at the $X$ axis marked by a black spot, and ${\rho }_2$ is plotted as a function of ${\mathbf{r}}_2.$ A darker area is associated with a larger ${\rho }_2$.

Figure 4

Evolution of $E_{L_0,S_0}$ and $⟨V_2⟩$ of the $(L_0,S_0)=(2,1∕2)$ and $(3,3∕2)$ ground states in accord with $d$. Solid lines are for the $(2,1∕2)$ state, dashed lines for $(3,3∕2)$. $B$ is given at the critical point $P_2$ with $q=1$ (a) and at $P_3$ with $q=-1$ (b). The black spot denotes the critical value of $d$. The dotted line is the same as the dashed line but shifted by a constant value to guide the eyes.