Nonvolatile bistability effect based on the electrically controlled phase transition in scaled magnetic semiconductor nanostructures

We explore the bistability effect in a dimensionally scaled semiconductor nanostructure consisting of a dilute magnetic semiconductor quantum dot (QD) and a reservoir of itinerant holes separated by a barrier. The bistability stems from the magnetic phase transition in the QD mediated by the changes in the hole population. Our calculation shows that when properly designed, thermodynamic equilibrium of the scaled structure can be achieved at two different configurations; i.e., the one with the QD in a ferromagnetic state with a sufficient number of holes and the other with the depopulated QD in a paramagnetic state. The parameter window suitable for this bistability formation is discussed along with the conditions for maximum robustness/nonvolatility. To examine the issue of scaling, an estimation of the bistability lifetime is made by considering the thermal fluctuation in the QD hole population via the spontaneous transitions. A numerical evaluation is carried out for a typical carrier-mediated magnetic semiconductor [e.g., (Ga,Mn)As] as well as for a hypothetical case of high Curie temperature for potential room-temperature operation.

Figure 1

Schematic diagram of the structure containing a DMS QD (in two different phases) and a nonmagnetic quantum well (NQW) reservoir separated by a barrier. The energy orientation is provided from the hole point of view in the valence band. Left: The PM phase corresponds to disordered magnetic ion spins (small arrows) and the lack of a magnetic contribution to the hole energy. This is achieved when the holes (small circles) only weakly populate the QD with a discrete energy spectrum and, thus, cannot change its magnetic state. The ground state $U$ of the QD is high relative to the chemical potential ${\mu }_0$ of the hole reservoir. Right: Another thermodynamically stable state (at the same external conditions) is possible when the magnetic ions are ferromagnetically ordered. Magnetic interactions can decrease the hole potential so that the ground state of the DMS QD is now substantially below ${\mu }_0$; i.e., the equilibrium hole population is high enough to stabilize the FM phase. Switching between the PM and FM states can be achieved by applying a gate bias $V_g$ that populates or depopulates the DMS QD.

Figure 2

Chemical potential of the DMS QD with a thickness of $5\mathrm{nm}$ and cross section of $25\times 25{\mathrm{nm}}^2$ as a function of the QD hole population [Eq. (10)]. The parameters of ${\mathrm{Ga}}_{0.95}{\mathrm{Mn}}_{0.05}\mathrm{As}$ are assumed with $T=77\mathrm{K}$ as discussed in the text. The solutions of Eq. (10) can be found as intersections of the solid curve with the horizontal line corresponding to a certain value of $\Delta \mu (={\mu }_0-U)$. Two cases $\Delta \mu ∕T_c^0=6$ and 0.5 (dashed lines 1 and 3) provide monostable FM and PM states, while dashed line 2 $(\Delta \mu ∕T_c^0=2.7)$ depicts the bistable state. Stable solutions are indicated by single-head arrows and the unstable one by the horizontal double-head arrow.

Figure 3

Phase diagram of the parameter space indicating the potential bistability region (the shaded area). The PM and FM labels denote the monostable areas corresponding to the PM and FM QD states, respectively. The same parameters as in Fig. 2 are assumed $(T_c^0=110\mathrm{K})$.

Figure 4

Free energy of the QD calculated as a function of hole population for three different values of $\Delta \mu (={\mu }_0-U)$: $\Delta \mu ∕T_c^0=6$ (curve 1, dotted line); 2.7 (curve 2, solid line); 0.5 (curve 3, dashed line). The single minima of curves 1 (FM phase) and 3 (PM phase) correspond to the vicinities of the right and left boundaries of the bistable area in Fig. 3; curve 2 represent the bistable case with the optimal free energy barrier height separating two local minima. The same parameters as in Fig. 2 are assumed ($T=77\mathrm{K}$, $T_c^0=110\mathrm{K}$).

Figure 5

Maximal free energy barrier height (in units of $k_{B}T$) between the local maximum and the closest local minimum in the bistable case (curve 2) and the corresponding optimal chemical potential (curve 1) as a function of temperature. Curves 3–5 in the bottom pane shows the hole population at the FM minimum (curve 3), the PM minimum (curve 5), and the local maximum (curve 4) of $F\left(j\right)$ for the bistable case shown in the top pane at the corresponding temperature. The same parameters as in Fig. 2 are assumed $(T_c^0=110\mathrm{K})$.

Figure 6

Bistability lifetime vs temperature at $T_c^0=110\mathrm{K}$. Two different QD dimensions are considered: (1) $25\times 25\times 5{\mathrm{nm}}^3$ and (2) $15\times 15\times 5{\mathrm{nm}}^3$. The mean time ${\tau }_0$ of particle exchange between the QD and reservoir via thermal processes is assumed to be $1\mathrm{ns}$. Other parameters are the same as in Fig. 2.

Figure 7

Bistability lifetime vs temperature for a hypothetical DMS material with characteristics similar to (Ga,Mn)As, in which $T_c^0$ in Eq. (9) is treated as a variable over a wide temperature range. $T$ is fixed at $300\mathrm{K}$ and three different QD sizes are considered: (1) $25\times 25\times 5{\mathrm{nm}}^3$, (2) $15\times 15\times 5{\mathrm{nm}}^3$, and (3) $25\times 25\times 3{\mathrm{nm}}^3$. The mean time ${\tau }_0$ of particle exchange between the QD and reservoir via thermal processes is assumed to be $1\mathrm{ns}$ as in Fig. 6.