Magnetic-field-induced binding of few-electron systems in shallow quantum dots

Binding of few-electron systems in two-dimensional potential cavities in the presence of an external magnetic field is studied with the exact diagonalization approach. We demonstrate that the magnetic field leads to the formation of bound few-electron states which are unbound in the absence of the field. The critical value of the depth of the cavity allowing the formation of a bound state decreases with the magnetic field in a nonsmooth fashion due to the increasing angular momentum of the first bound state. In the high-magnetic-field limit the binding energies and the critical values for the depth of the potential cavity, allowing the formation of a bound system, tend to the classical values.

Figure 1

(Color online) Binding energy of a two-electron system versus potential well depth for $B=0$ and $5\mathrm{T}$. Solid lines were calculated using the multicenter basis (the inset shows the choice of centers, $d$ is a variational parameter) and the dashed lines with the finite-difference scheme. The arrow indicates where the ground-state angular-momentum $L$ changes.

Figure 2

(Color online) Binding energy of the two-electron system versus the $B$ for depth of the potential from $V_0=7\mathrm{meV}$ (lowest curve) to $4.18\mathrm{meV}$ (highest curve). Horizontal dashed lines show the binding energies of the classical system. The energy cusps due to the angular-momentum transition are linked by the thin solid lines. The dotted curve shows the binding energy of the state with $L=3$ for $V_0=5.4\mathrm{meV}$.

Figure 3

(Color online) Binding energy of the two-electron system minus the classical binding energy versus the inverse of the magnetic field for depth of the potential from $V_0=7\mathrm{meV}$ (highest curve) to $4.18\mathrm{meV}$ (lowest curve). The thin dashed straight lines are given by $0.39∕B$, $0.78∕B$, and $1.28∕B$.

Figure 4

(Color online) Binding energy of the three-electron system versus the magnetic field for three values of the potential depth. Horizontal dashed lines show the binding energies of the classical system. The energy cusps corresponding to the angular-momentum transitions in the three- two-electron ground state are linked by thin solid (dotted) lines. The numbers below the lowest curve list $L$ for $N=3$.

Figure 5

(Color online) Same as Fig. 3, but for $N=3$. The thin dashed straight lines are given by $6.6∕B$, $5.4∕B$, and $4.1∕B$.

Figure 6

(Color online) The maximum number of bound electrons on the $V_0∕B$ plane. Below the lowest curve only one electron can be bound, and above the highest the third electron becomes bound; between the curves the maximum number of bound electrons is two. The horizontal dashed lines show the classical depths of the well, allowing binding of the second and third electron.

Figure 7

(Color online) Same as Fig. 6, but in $1∕B$ scale. The thin dashed straight lines are given by $V_N^{\mathit{class}}+A_N∕B$, with $V_2^{\mathit{class}}=4.073\mathrm{meV}$, $V_3^{\mathit{class}}=8.245\mathrm{meV}$, $A_2=0.7$, $A_3=5$.