Voltage-tunable ferromagnetism in semimagnetic quantum dots with few particles: Magnetic polarons and electrical capacitance

Magnetic semiconductor quantum dots with a few carriers represent an interesting model system where ferromagnetic interactions can be tuned by voltage. By designing the geometry of a doped quantum dot, one can tailor the anisotropic quantum states of magnetic polarons. The strong anisotropy of magnetic polaron states in disklike quantum dots with holes comes from the spin splitting in the valence band. The binding energy and spontaneous magnetization of quantum dots oscillate with the number of particles and reflect the shell structure. Due to the Coulomb interaction, the maximum binding energy and spin polarization of magnetic polarons occur in the regime of Hund’s rule when the total spin of holes in a quantum dot is maximum. With increasing number of particles in a quantum dot and for certain orbital configurations, the ferromagnetic state becomes especially stable or may have broken symmetry. In quantum dots with a strong ferromagnetic interaction, the ground state can undergo a transition from a magnetic to a nonmagnetic state with increasing temperature or decreasing exchange interaction. The characteristic temperature and fluctuations of magnetic polarons depend on the binding energy and degeneracy of the shell. The capacitance spectra of magnetic quantum dots with few particles reveal the formation of polaron states.

Figure 1

(Color online) (a) Model of a Mn-doped self-assembled QD. (b) Transistor structure with magnetic QDs; the gate voltage controls the number of holes in the QD layer and therefore the magnetic state of the QDs. The capacitance of this structure can reveal the magnetic state of the QDs.

Figure 2

(Color online) Calculated energy of a MP with one hole for different Mn distributions; $x_{\mathrm{Mn}}=0.04$ and $R_{\mathrm{Mn}}=2$ and 3 nm and $\infty$.

Figure 3

(Color online) Calculated magnetization of Mn ions as a function of temperature for the QD with one hole and different Mn distributions; $x_{\mathrm{Mn}}=0.04$ and $R_{\mathrm{Mn}}=2$ and 3 nm and $\infty$.

Figure 4

Spin anisotropy of the MP energy as a function of the probability to find the hole in the state $j_z=+3∕2$; $x_{\mathrm{Mn}}=0.04$ and $R_{\mathrm{Mn}}=\infty$.

Figure 5

(Color online) (a) Ground-state configuration of a cylindrically symmetric QD with few holes and without Mn ions; the state with $n_h=4$ is constructed according to Hund’s rule. Their configurations remain ground states if the Mn-hole interaction is weak or the QD confinement is strong enough. (b) Ground-state configuration for $n_h=9$ in the regime of Hund’s rule; again this configuration represents a ground state if the Mn-hole interaction is not very strong.

Figure 6

(Color online) Ground-state configurations of a QD with cylindrical symmetry and two holes. The configuration (a) is realized in the case of a relatively weak Mn-hole interaction (high temperature or small Mn density) and the state (b) corresponds to the system with a strong Mn-hole interaction (low temperature or high Mn density).

Figure 7

(Color online) (a) Orbital anisotropy of the MP energy with $n_h=2$ and $j_{z,\mathit{tot}}=3$. The minimum energy corresponds to the state with ${\mid \alpha \mid }^2=1∕2$ in which the angular part of the spatial probability distribution is most inhomogeneous, i.e. ${\psi }_p^2\left(\varphi \right)\propto \mathit{cos}{(\varphi +{\varphi }_0)}^2$. (b) The energy of the states $j_{z,\mathit{tot}}=3$ and $j_{z,\mathit{tot}}=0$ (figs. 6a, 6b) as a function of temperature; the ground state of the system changes with temperature. $R_{\mathit{Mn}}=\infty$ and $x_{\mathit{eff}}=0.04$

Figure 8

Calculated binding energy of a MP as a function of temperature for various charged states of QD: $n_h=1$, 3, 4, and 5. The binding for the states with completely filled shells ($n_h=2$, 6) is zero in our approach; $R_{\mathrm{Mn}}=\infty$.

Figure 9

(Color online) Calculated magnetization of Mn ions as a function of temperature for various charged states of the QD: $n_h=1$, 3, 4, and 5. The magnetization for the states with completely filled shells ($n_h=2$, 6) is zero in our approach; $x_{\mathit{eff}}=0.01$ and $R_{\mathrm{Mn}}=\infty$.

Figure 10

(Color online) (a) Calculated average spin of the hole subsystem (solid curves) and its relative fluctuations (dashed lines) for the states $n_h=1$,4, and 9. One can see that the ferromagnetic state becomes more stable with increasing number of holes. (b) The solid curves show the calculated binding energy of a MP as a function of temperature for three charged states of a QD ($n_h=1$, 3, and 9) with maximum binding energy; $x_{\mathrm{Mn}}=0.01$ and $R_{\mathrm{Mn}}=\infty$. The dashed curves show the results of self-consistent mean-field theory which demonstrates critical temperature. The dashed straight line is the thermal energy.

Figure 11

(a) Calculated energies of different charged states in a layer of semimagnetic quantum dots with $x_{\mathrm{Mn}}=0.01$; $R_{\mathrm{Mn}}=\infty$. The labels show the charged states of the system.

Figure 12

(Color online) (a) Calculated capacitance spectra for nonmagnetic (a) and magnetic (b) QDs; $x_{\mathit{eff}}=0.01$. The broadening of the peaks was taken as 16 meV.