Spin polaron and bistability in ferromagnetic semiconductor quantum structures

This work shows that the in-plane localization of a hole confined in a ferromagnetic semiconductor quantum well (QW) can lead to significant energy gain if spontaneous easy-plane magnetization is mediated by the mechanisms other than itinerant carriers. The hole spin normal to the QW plane reorients the in-plane magnetization of the ferromagnetic layer at the location of polaron formation, resulting in an exchange potential with a discrete level of localization. A flexible model that incorporates the magnetization gradient term, as well as magnetic anisotropy, is proposed. In contrast to the calculations of magnetic polaron in the paramagnetic semiconductors, the energy of spin polaron in a ferromagnetic semiconductor is almost independent of the temperature in a wide range below the critical temperature of phase transition. Our calculation also demonstrates the existence of bistability in the hole state when the structure consists of appropriate ferromagnetic and nonmagnetic QWs separated by a finite barrier. Hence, a memory element that can be scaled down to a single hole may be achieved through polaron formation.

Figure 1

Schematic diagram of polaron formation in a FM QW; $\widehat{x}$ and $\widehat{z}$ are the directions of easy magnetization and normal to the QW plane. Small arrows show the local magnetization direction of the FM layer and ${\theta }_0$ the maximal angle of its deviation. Curves 1 and 2 depict the in-plane effective magnetic potential well and the hole wave function after polaron formation. The hole spin ${\vec{S}}_h$ can be either parallel or antiparallel to the $z$ axis. Only one case is considered as they are equivalent.

Figure 2

Polaron energy $E_{\mathit{pol}}$ as a function of QW width $L_w$ for lateral potential parameters $\{u_{\mathit{loc}},r_0\}=\{0.01,2\}$, {0.1, 10}, {1, 20} in units of $U_s$ and $d_s$. The other parameters of the FM QW are as discussed in the text.

Figure 3

Bistability with respect to hole localization (a) in the NM QW (N) or (b) in the FM QW (M) with polaron formation. A nonmagnetic barrier (B) separates the NM and FM QWs; $E_N$ and $E_M\left(E_P\right)$ are the eigenenergies of hole localization in the NM QW and the FM QW without (with) polaron formation, respectively; and $\Delta E_P$ is the energy of polaron formation. Curves depict the hole $\psi$ function in the NM and FM QWs. Small arrows indicate the local magnetization directions; the large arrow is the hole spin directed normal to the QW plane. It is important to note that the device does not require a polarized hole gas in the NM QW. Once a hole is injected into the FM QW with the spin directed either parallel or antiparallel to the growth direction (a 50-50 split in the unpolarized gas), it provides an effective field normal to the QW plane. Only one case is shown schematically for simplicity as in Fig. 1.

Figure 4

Dependence of hole energy $\Delta F_{2W}$ as a function of $\phi$ and $E_N$. ${\cos }^2\phi \left({\sin }^2\phi \right)$ is a probability of hole localization in the FM QW (NM QW). The energy scale is in units of $U_s$ assuming $E_M=0$. $\Delta U$ is as defined in Fig. 3. The calculation is conducted for $L_w=50\mathrm{\AA }$ and $\{u_{\mathrm{loc}},r_0\}=\{0.1,10\}$; the other parameters are the same as in Fig. 2. The bistability is observed for $\Delta U$ in the range 0.2–0.8.