Spin ordering in magnetic quantum dots: From core-halo to Wigner molecules

The interplay of confinement and Coulomb interactions in quantum dots can lead to strongly correlated phases differing qualitatively from the Fermi liquid behavior. We explore how the presence of magnetic impurities in quantum dots can provide additional opportunities to study correlation effects and the resulting ordering in carrier and impurity spins. By employing exact diagonalization we reveal that seemingly simple two-carrier quantum dots lead to a rich phase diagram. We propose experiments to verify our predictions; in particular, we discuss interband optical transitions as a function of temperature and magnetic field.

Figure 1

Ground-state phase diagram (PS vertical and diagonal, T horizontal hatching) as a function of Mn content $x_{\mathrm{Mn}}$ and Mn doping radius $R_{\mathrm{eff}}$, for a double-occupied circular dot, $\hslash {\omega }_{x,y}=25$ meV, at zero temperature. Insets (a)–(c): QD top view (in-plane coordinates in nanometers) of Mn spin patterns: spins up, light; down, dark. (a) Core-halo (CH), $R_{\mathrm{eff}}=2$ nm, $x_{\mathrm{Mn}}=2\%$; (b) triplet (T), $R_{\mathrm{eff}}=3.5$ nm, $x_{\mathrm{Mn}}=3\%$; (c) spin Wigner molecule (sWM), $R_{\mathrm{eff}}=3.5$ nm, $x_{\mathrm{Mn}}=1\%$; (d) hole spin density for the pattern in (a) (Ref. 21).

Figure 2

Temperature evolution of pseudosinglet $E_{\mathrm{PS}}$ (green) and triplet $E_{\mathrm{T}}$ (red) energies for a circular QD, $\hslash {\omega }_{x,y}=25$ meV, and $x_{\mathrm{Mn}}=1\%$ (homogeneous distribution). Bold ticks show the high-$T$ (zero $p-d$ exchange) limit. (a) Mn-induced energy gains at $T=0$ K. Dots (solid line), numerical (variational) results (Ref. 31) for the PS, $\Delta E_{\mathrm{PS}}=E_{\mathrm{PS}}(x_{\mathrm{Mn}}=0)-E_{\mathrm{PS}}$. Dash-dotted line, single magnetic polaron, $\Delta E_{\mathrm{MP}}=x_{\mathrm{Mn}}{N}_0\left|\beta \right|S/2$. $\Delta E_{\mathrm{MP}}<\Delta E_{\mathrm{PS}}$ for $x_{\mathrm{Mn}}\ge 0.7\%$. Insets (b)–(d) $m_k$ in units of 1/4 as a function of position (nm); (b) triplet at 1 K, (c), (d) PS at 1 K, 3 K.

Figure 3

Ground-state phase diagram (PS vertical, T horizontal hatching) as a function of confinement asymmetry $d$ and $x_{\mathrm{Mn}}$, at $T=0$. Insets: hole spin density of the PS (a) and T (b), $\hslash {\omega }_0=25$ meV, $d=1$, and $x_{\mathrm{Mn}}=1\%$.

Figure 4

Temperature dependence of the PS $\to 1$ hole (solid) and $1\to 0$ (dash-dotted) transition energies $E_{\mathrm{ph}}$, for $B=0$. To better compare the dependencies, each line is shifted: $\Delta E_{\mathrm{ph}}\equiv E_{\mathrm{ph}}\left(T\right)-E_{\mathrm{ph}}(T\to \infty )$. Standard QD parameters, homogeneous Mn doping.

Figure 5

$B\parallel z$ dependence of the PS $\to 1$ (solid lines) and T $\to 1$ (dashed) transitions, for $T=1$ K. To avoid PS-T crossing, we set $x_{\mathrm{Mn}}=0.5\%$. Each line is shifted: $\Delta E_{\mathrm{ph}}\equiv E_{\mathrm{ph}}\left(B\right)-E_{\mathrm{ph}}(B=0)$. Insets: PS Mn patterns for 0.4 (a), and 1.4 (b) tesla; light (dark) for $m_k>0$ ($m_k<0$).