Dependence of Persistent Gaps at Landau Level Crossings on Relative Spin

We report measurements of the quantum Hall state energy gap at avoided crossings between Landau levels originating from different conduction band valleys in AlAs quantum wells. These gaps exhibit an approximately linear dependence on the magnetic field over a wide range of fields and filling factors. More remarkably, we observe an unexpected dependence of the gap size on the relative spin orientation of the crossing levels, with parallel-spin crossings exhibiting larger gaps than antiparallel-spin crossings.

Figure 1

(a) ${\rho }_{xx}$ vs $B_{\perp }$ at $T=0.3\mathrm{K}$ as $n$ increases from 3.5 (top) to $6.6\times {10}^{11}{\mathrm{cm}}^{-2}$ (bottom). Traces are vertically offset for clarity. (b) ${\rho }_{xx}$ at integer $\nu$ extracted from the traces in (a). (c) Simulated energy level diagram (energy units are arbitrary) for the data in (a) and (b), showing reasonable agreement between the indicated LL crossings and the extrema in ${\rho }_{xx}$ at integer $\nu$. Solid (dotted) lines are majority (minority) spin LLs, and parallel and antiparallel LL crossings are shown by open and solid symbols, respectively. Our notation indicates the energy levels’ valley, LL index, and spin.

Figure 2

Energy gaps ($\Delta$) at LL crossings as derived from temperature-dependent activation measurements. Different symbols correspond to different $\nu$ as indicated, and open (solid) symbols are parallel- (antiparallel-) spin crossings. Black points are for sample $A$, blue for sample $B$, and red for a point derived from the data of Ref. 12. The lines are guides to the eye.

Figure 3

Energy gap ($\Delta$) at the $\nu =3$ LL crossing as sample $A$ is tilted through the first coincidence angle at fixed $n$ and $B_{\perp }$. The vertical dotted line indicates the coincidence angle, and schematic fan diagrams on either side of this coincidence are shown as insets.