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 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. PACS numbers: 72.80.Ey, 73.21.Fg, 73.43.-f, 73.43.Qt In multi-component, two-dimensional (2D) quantum Hall (QH) systems, the presence of discrete electronic degrees of freedom can significantly alter the electronelectron interaction. Recently, there has been increasing interest in the interplay between different discrete degrees of freedom in these systems. For example, the spin can influence the correlated � = 1 phase in a double layer system [1], or change the ground state symmetry of QH ferromagnetic phases at crossings of Landau levels (LLs) that originate from different confinement subbands in a wide GaAs quantum well (QW) [2]. The interplay between spin and conduction band valley has also been demonstrated, such as through the dependence of spin susceptibility on valley degeneracy in AlAs QWs [3] or the dependence of valley splitting gaps in SiGe/Si/SiGe QWs on the spins of the valley levels [4]. We studied crossings between the LLs of one valley with the lowest LL of another valley in 2D electrons in AlAs QWs. This is similar to the experiments of Ref. [2] where confinement subband crossings were studied, except we are able to vary the magnetic field position of the valley crossings, allowing us to study the field dependence of persistent gaps at those crossings. Also, the transfer of charge between different confinement subbands has a capacitive energy component due to charge redistribution along the confinement direction, whereas the valley degree of freedom has no spatial charge redistribution and, hence, no capacitive energy associated with it, yielding a somewhat simpler system. We observe an approximately linear dependence of the persistent gaps at crossings of different valley LLs on magnetic field and, surprisingly, a dependence on the relative spin of the levels, with larger gaps for the parallel spin case than for antiparallel. Bulk AlAs has an indirect band gap with conduction band minima at the six equivalent X-points of the Brillouin zone. The Fermi surface of electrons, therefore, consists of three (six half) prolate ellipsoids (or valleys), each with its major axis oriented parallel to the crystal axis along which the valley center is displaced. Electrons occupying these anisotropic valleys have a longitudinal band effective mass of ml = 1.04 and transverse masses of mt = 0.21, in units of the vacuum electron mass. When confined to AlAs QWs thicker than 55 u, as is the case for our samples, the strain associated with lattice mismatch between the AlAs QW and the GaAs substrate on which it is grown causes only the valleys oriented in the plane of the QW to be occupied up to the highest accessible densities. We will refer to these in-plane valleys as X and Y. The band effective mass, associated with the density of states and the cyclotron energy, for these valleys is given by mb = (mlmt) 1/2 = 0.47. We measured two samples (A and B) from a wafer containing a 110 u AlAs QW [5]. Each sample was patterned in a Hall bar geometry and fitted with metallic front and back gates to vary the charge density, n, from 1.5 to 8.2 x 10 11 cm 2 . Typical mobilities are near 26 m 2 /Vs. Ohmic contacts were made by depositing AuGeNi and alloying in a reducing environment. The samples were thinned to approximately 150 µm and glued to a piezoelectric actuator to allow for in situ application of a symmetrybreaking, in-plane strain that varies the energy splitting of the two occupied valleys, as described previously [6]. We employed standard low-current, low-frequency lockin techniques, and the samples were cooled in a pumped 3 He refrigerator with a base temperature, T, of 0.3 K that