Combined asymmetry-induced and exchange-induced spin splitting in antimonide type-II quantum wells

We incorporate bulk and structure inversion asymmetries into an eight-band effective bond-orbital method to study the effects of an adjacent ferromagnetic GaMnSb layer on the transport and spin properties of high-mobility electrons in an asymmetric quantum well. An appreciable exchange interaction for electrons results, due to their hybridization with holes in the GaSb layer, which in turn overlap the GaMnSb. Interactions between the asymmetry and exchange effects cause these structures to have complicated spin splittings and spin orientations. When the GaMnSb magnetization points parallel to the growth axis, the spin polarization for electrons conducting in the plane can be as high as 71%. The conductivity decreases by two orders of magnitude when turning off the magnetization or rotating it toward the plane induces a transition from a gapless state to a more conventional type-II semimetal with a “hybridization gap.”

Figure 1

Band profiles for a $\mathrm{Ga}\mathrm{Mn}\mathrm{Sb}∕\mathrm{Al}\mathrm{Sb}∕\mathrm{Ga}\mathrm{Sb}∕\mathrm{In}\mathrm{As}∕\mathrm{Al}\mathrm{Sb}$ structure of the type studied in this paper. Wave functions for the lowest electron and hole subbands (excluding the GaMnSb valence subbands) are shown as the dash-dotted and dotted curves, respectively.

Figure 2

Polar plots of the spin splitting in the conduction band at $k=0.031{\mathrm{\AA }}^{-1}$, as calculated without magnetization $(T>T_c)$ under flat-band conditions for (a) an InAs QW of thickness $d=85\mathrm{\AA }$ and no BIA; (b) the same QW but including BIA, where the dotted curve is a fit to Eq. (2); and (c) a QW with $d=110\mathrm{\AA }$, for which the SIA and BIA spin splittings are nearly equal.

Figure 3

Conduction- and valence-band in-plane dispersion relations along [10] for a semiconducting structure $(d=49\mathrm{\AA })$ without magnetization $(T>T_c)$. The up and down arrows show the dominant spin directions perpendicular to $\mathit{k}$, while ↕ denotes mixed spin states.

Figure 4

Conduction-band (a) and valence-band (b) dispersions for directions parallel and perpendicular (conduction only) to the in-plane magnetization direction [10] at $T⪡T_c$, for the same structure as in Fig. 3 $(d=49\mathrm{\AA })$. Except at the $\Gamma$ point, all of the spins are oriented in the plane. The arrow notations are the same as in Fig. 3, with the addition that horizontal arrows signify spin directions parallel to $\mathit{k}$.

Figure 5

Conduction- and valence-band dispersions for directions parallel (a) and perpendicular (b) to the in-plane magnetization direction [10] at $T⪡T_c$, for a semimetallic structure $(d=73\mathrm{\AA })$ with a “hybridization gap.” The arrow notations are the same as in Fig. 4, except that even the strongly mixed hole spin states are indicated with horizontal arrows for clarity. Electronlike portions of the subbands are indicated by the solid and dashed curves, whereas holelike regions are shown as points (small or large).

Figure 6

Conduction- and valence-band dispersions at $T>T_c$ (a) and $T⪡T_c$ with magnetization along the growth axis (b), for a semimetallic structure $(d=67\mathrm{\AA })$. The keys for the curve types and arrow directions are the same as in Fig. 5, with the addition that diagonal arrows represent states that are spin-polarized mostly along the growth axis.