Manifestation of the exchange enhancement of valley splitting in the quantum Hall effect regime

We report a “dip” effect in the Hall resistance, $R_{xy}$, of a Si metal-oxide-semiconductor field-effect transistor in the quantum Hall effect regime. With increasing magnetic field, the Hall resistance moves from the plateau at Landau filling factor $\nu =6$ directly to the plateau at $\nu =4$, skipping the plateau at $\nu =5$. However, when the filling factor approaches $\nu =5$, the Hall resistance sharply “dives” to the value $1∕5(h∕e^2)$ characteristic of the $\nu =5$ plateau, and then returns to $1∕4(h∕e^2)$. This is interpreted as a manifestation of the oscillating exchange enhancement of the valley splitting when the Fermi level is in the middle between two adjacent valley-split Landau bands with the asymmetric position of the extended states.

Figure 1

Dependence of the Hall resistance ${\rho }_{xy}$ on the perpendicular magnetic field $B$ for different gate voltages $V_g$. The electron density is as indicated.

Figure 2

Dependence of the dimensionless resistivity ${\rho }_{xy}∕(h∕e^2)$ on the filling factor $\nu =nh∕eB$ for different gate voltages $V_g$. The inset shows the enhanced view of the “dip” effect.

Figure 3

Longitudinal resistivity in dimensionless units ${\rho }_{xx}∕(h∕2e^2)$ as a function of filling factor $\nu =nh∕eB$ for different $V_g$.

Figure 4

Comparison of QHE in $\mathrm{Si}∕\mathrm{SiGe}$ ($n=8.94\times {10}^{15}{\mathrm{m}}^{-2}$, solid line) and in Si-MOSFET with similar electron concentration ($n=8.80\times {10}^{15}{\mathrm{m}}^{-2}$, dashed line).

Figure 5

Schematic sketch of the splitting enhancement between adjacent valley-split LB when the filling factor passes over $\nu =5$. Delocalized states, shown as shaded bands, are displaced from the center of LB. Fractional numbers correspond to the plateau resistance in units $h∕e^2$. Magnetic field increases from $a$ to $c$.

Figure 6

Temperature dependence of the “dip” amplitude at $\nu =5$.