Magnetic ordering in quantum dots: Open versus closed shells

In magnetically doped quantum dots, changing the carrier occupancy from open to closed shells leads to qualitatively different forms of carrier-mediated magnetic ordering. While it is common to study such nanoscale magnets within a mean-field approximation, excluding the spin fluctuations can mask important phenomena and lead to spurious thermodynamic phase transitions in small magnetic systems. By employing coarse-grained, variational, and Monte Carlo methods on singly and doubly occupied quantum dots to include spin fluctuations, we evaluate the relevance of the mean-field description and distinguish different finite-size scaling in nanoscale magnets.

Figure 1

Qualitative behavior of magnetic ordering in nanoscale systems. An example of a system that (a) does not and (b) does display a phase transition. The free energy, Eq. (1), is shown as a function of the order parameter at various temperatures applicable to both (a) and (b). (c) The temperature evolution of the order parameter for the free energy described in (a) and (b). (d) For high temperature, there are two qualitatively different finite-size scalings for the normalized order parameter, which also extrapolate to distinct thermodynamic limits (circles).

Figure 2

(a) A scheme of QDs grown on a two-dimensional wetting layer (WL) in which electron-hole pairs are created by interband absorption of light. Holes are subsequently captured in the QD. Type-II conduction/valence band (CB/VB) profile [61] of a II-VI QD doped with Mn spins. [34] $\varepsilon$ and $E_{CV}$ are the confinement and band-gap energies.

Figure 3

(a) The free energy as a function of magnetization, $\xi$, reveals a minimum at low $T$ and vanishes at high $T$. MFA will give a second-order phase transition. (b) Corrected free energy of a nanoscale magnet as a function of observable exchange energy, $X$, giving a finite value to the order for any finite $T$.

Figure 4

(a) A magnetic QD doped with random paramagnetic Mn ions (green arrow) with one carrier spin density (red arrow). (b) The system lowers its energy through the exchange interaction and results in an antiferromagnetic alignment between the hole spin and the Mn spins producing the MP. (c) The doubly degenerate QD energy level (1) splits with formation of the MP (2). The difference between the nonmagnetic QD energy and the ground-state energy is the average exchange energy, $E_{\text{MP}}$.

Figure 5

A mean-field solution for MP blue/dashed curve and the result obtained through the fluctuation approach red/solid. For simplicity, we approximate the hole wave function as uniform throughout the QD volume $V,\varphi \left(r\right)=1/\sqrt{V}$ [83]. The parameters are $N_0\beta =-1.05\mathrm{eV}$, the cation density, $N_0=4/a_0^3$ with $a_0\simeq 6.1$ Å, $V=\pi r^2{h}_z,r=5\mathrm{nm}, h_z=2.5\mathrm{nm}, N_{\text{Mn}}=90$.

Figure 6

Illustration of the finite-size effect for the MP. (a) Spin density (blue/dashed) for fully aligned Mn spins (green). (b) A flipped Mn spin does not effect the carrier spin density (red/solid).

Figure 7

(a) The MP average exchange energy normalized by its $T=0$ K value for 10 Mn (red/solid) and 20 Mn (black/dashed) as function of $T$. (b) Normalized MP average exchange energy as a function of the total number of Mn at fixed $T$.

Figure 8

(a) A magnetic QD doped with random paramagnetic Mn ions spins (green arrows). (b) The system lowers its energy through the exchange interaction leading to a nonvanishing spin density and formation of the MBP. Red arrows and lines in (a) and (b) show how the two-carrier spin density changes due to the presence Mn. (c) The doubly occupied QD energy level (1) lowers its energy through the formation of the MBP (2). The difference between the nonmagnetic QD energy and the MBP ground-state energy is the average exchange energy $E_{\text{ex}}$.

Figure 9

Illustration of the finite-size effect for the MBP. (a) Spin density (blue/dashed) for fully aligned Mn spins (green). (b) A flipped Mn spin changes the carriers spin density (red/solid).

Figure 10

(a) $T$ dependence of the average exchange energy and the removal of the phase transition for the two-site model of the MBP, [Eq. (40)] with a nonfluctuating carrier spin density (green/dashed) and a fluctuating spin density (red/solid) [Eq. (41)]. (b) $T$ dependence of the product of magnetization of each site, $\langle m_2{m}_1\rangle$, for a fluctuating spin density (red/solid) and nonfluctuating spin density (green/dashed). The parameters are $h_z=2.5\text{nm}, \hslash \omega =30\text{meV}, m_h^{*}=0.21, N_0\beta =-1.05\mathrm{eV}, U=30\mathrm{meV}, R_{1,2}=\pm 2\mathrm{nm}$, and $N_{\mathrm{Mn}}=10$.

Figure 11

The finite-size effect in a doubly occupied QD. (a) $T$ dependence of the normalized MBP average exchange energy for 10 Mn (red/solid), 20 Mn (black/dashed), and 40 Mn (blue/dash-dotted), thus 5, 10, and 20 Mn per site, respectively. Mean-field solution with 10 Mn (black/dotted) and 40 Mn (green/dashed). (b) The normalized mean-field solution with varying number of Mn for a fixed $T/N_{\mathrm{Mn}}$.

Figure 12

(a) $T$ dependence of the MP energy [Eq. (B3)] with the variationally-obtained (renormalized) (red/solid) and the fixed wave function for a harmonic QD confinement without Mn spins. (b) The corresponding $T$ dependence of the most probable width for a renormalized wave function, $L_{\text{MP}}$ (red/solid) and fixed wave function (blue/dashed), $L_{\text{MP}}\equiv L_0$, from Eq. (B1). (c) $T$ dependence of the $L_{\text{MP}}$ from a variational method (red/solid) and Monte Carlo simulations (blue/dotted).