Magnetism in Closed-Shell Quantum Dots: Emergence of Magnetic Bipolarons

Similar to atoms and nuclei, semiconductor quantum dots exhibit the formation of shells. Predictions of magnetic behavior of the dots are often based on the shell occupancies. Thus, closed-shell quantum dots are assumed to be inherently nonmagnetic. Here, we propose a possibility of magnetism in such dots doped with magnetic impurities. On the example of the system of two interacting fermions, the simplest embodiment of the closed-shell structure, we demonstrate the emergence of a novel broken-symmetry ground state that is neither spin singlet nor spin triplet. We propose experimental tests of our predictions and the magnetic-dot structures to perform them.

Figure 1

Two-fermion states described by the symmetry of the orbital and spin part of their wave functions. Arrows show spin projections (up or down), while radii of the spheres indicate the extent of orbitals. For the $S$ state, the radii of the orbitals corresponding to spin-up or spin-down are the same. For PS, the spin-down orbital is larger [Eq. (3)], leading to finite spin density and a magnetically active state.

Figure 2

Emergence of magnetic bipolarons in QDs. (a) Without carriers, Mn spins (light arrows) are randomly oriented. Double arrows in (b),(c) show carrier’s spin projection associated with the orbitals (solid and dashed lines). (b) With one carrier, a magnetic-polaron forms, lowering the total energy of the system, due to the coupling of the carrier’s spin with the induced Mn magnetization. Mn spins align in one direction. (c) Two carriers assemble in a PS state, forming what we term a magnetic bipolaron. The sign of the Mn-spin projection depends on the sign of the carrier-spin density (difference between dashed and solid curves). The extent of the orbitals (length of dotted lines) is different.

Figure 3

Spin corral. Colored surface: The hole-spin density ${\rho }_{\mathrm{PS}}$ (arb. units) of PS. Black circle indicates ${\rho }_{\mathrm{PS}}(r=R_0)=0$. Green arrows: Mn spins, placed at a radius $R_C$, which maximizes the stability of the ferromagnetic alignment. Red and blue arrows: The more probable hole-spin projections at two positions. The parameters are $\hslash {\omega }_0={\Delta }_0=30\mathrm{meV}$, ${\mathrm{Ry}}^{*}=\hslash {\omega }_0/10$, $m^{*}=0.5m_0$, $w=1\mathrm{nm}$ [25].

Figure 4

(a) Phase diagram of two-hole ground states for homogeneous distribution of Mn. Horizontally (vertically) hatched area indicates the ranges of ${\mathrm{Ry}}^{*}$ and ${\Delta }_0$, for which the ground state is PS ($T$). PS reduces to singlet only for ${\Delta }_0=0$. Panels (b) for PS and (c) for $T$ show the nonmagnetic (dashed line) and total (solid line) energies as a function of $\delta$. Panels (d) and (e) show the hole-spin densities (at the corresponding variational minima), along any direction in the $x\mathrm{\text{−}}y$ plane, with arrows indicating the Mn-spin profiles. The parameters for (b)–(e) are $\hslash {\omega }_0=30\mathrm{meV}$, ${\Delta }_0=\frac{3}{4}\hslash {\omega }_0$, ${\mathrm{Ry}}^{*}=\hslash {\omega }_0/10$, $m^{*}=0.5m_0$. Vertical line in (d): $R_0=L_S^0/\sqrt{2}$.