Probing spin-polarized currents in the quantum Hall regime

An experiment to probe spin-polarized currents in the quantum Hall regime is suggested that takes advantage of the large Zeeman-splitting in the paramagnetic diluted magnetic semiconductor zinc manganese selenide $\left({\mathrm{Zn}}_{1-x}{\mathrm{Mn}}_x\mathrm{Se}\right)$. In the proposed experiment spin-polarized electrons are injected by $\mathrm{ZnMnSe}$-contacts into a gallium arsenide (GaAs) two-dimensional electron gas (2DEG) arranged in a Hall bar geometry. We calculated the resulting Hall resistance for this experimental setup within the framework of the Landauer–Büttiker formalism. These calculations predict for 100% spin injection through the $\mathrm{ZnMnSe}$-contacts a Hall resistance twice as high as in the case of no spin-polarized injection of charge carriers into a 2DEG for filling factor $\nu =2$.We also investigated the influence of the equilibration of the spin-polarized electrons within the 2DEG on the Hall resistance. In addition, in our model we expect no coupling between the contact and the 2DEG for odd filling factors of the 2DEG for 100% spin injection, because of the opposite sign of the g-factors of $\mathrm{ZnMnSe}$ and $\mathrm{GaAs}$.

Figure 1

Hall bar geometry of a 2DEG (filling factor $\nu =2$) with $\mathrm{ZnMnSe}$-contacts, which are simulated in the Landauer–Büttiker formalism by regulators for the minority spin-type to be injected into or extracted from the 2DEG with some probability $ϵ$. These regulators simulate a spin imbalance between the edge channels provided by the $\mathrm{ZnMnSe}$-contacts. The actual location of these regulators has no physical meaning. The spin imbalance regulators represent a theoretical concept to simulate the spin aligning effect of the $\mathrm{ZnMnSe}$-contacts in the Landauer–Büttiker formalism.

Figure 2

Hall resistance $R_{xy}$ (left axis) and longitudinal resistance $R_{xx}$ (right axis) normalized to the Klitzing resistance $R_K=h∕e^2$ as a function of $ϵ$ for filling factor $\nu =2$, neglecting spin equilibration.The Hall resistance $R_{xy}$ decays with increasing injection probability $ϵ$ for the minority spin-type from $R_K$ to $R_K∕2$, reflecting the decreasing imbalance of the two edge channels for $ϵ\to 1$. The nonvanishing longitudinal resistance $R_{xx}$ can be explained by backscattering of electrons (see text).

Figure 3

(a) Normalized Hall resistance $R_{xy}$ and (b) normalized longitudinal resistance $R_{xx}$ as a function of $ϵ$ and the distance $l$ between the $\mathrm{ZnMnSe}$-contacts of a 2DEG with filling factor $\nu =2$ for an equilibration length of $l_{eq}=200\mu m$. The Hall resistance $R_{xy}$ decays exponentially with increasing distance of the $\mathrm{ZnMnSe}$-contacts, reflecting the equilibration of the two edge channels. The longitudinal resistance $R_{xx}$ vanishes for $l=0$ and approaches zero for infinite contact distances $l$. The nonvanishing longitudinal resistance for a finite contact distance $l$ can be explained by the backscattering of electrons (see text).

Figure 4

Hall resistance $R_{xy}$ as a function of the applied magnetic field $B$ for different distances $l$ between the $\mathrm{ZnMnSe}$-contacts for an equilibration length of $l_{eq}=200\mu m$. With increasing distance $l$ of the $\mathrm{ZnMnSe}$-contacts the Hall resistance $R_{xy}$ decays exponentially to approach the value of the Hall resistance for non-spin-polarized contacts (thin solid line). The reason that the Hall resistance for odd filling factors does not decay to the value for the case of non-spin-polarized contacts is the opposite sign of the g-factor in $\mathrm{ZnMnSe}$ and $\mathrm{GaAs}$ and the fact that the presented model does not include inter Landau level scattering (see text).