Evidence for a fractionally quantized Hall state with anisotropic longitudinal transport

Abstract

At high magnetic fields, where the Fermi level lies in the N=0 lowest Landau level (LL), a clean two-dimensional electron system (2DES) shows numerous incompressible liquid phases which exhibit the fractional quantum Hall effect (FQHE; ref. 1). These liquid phases do not break rotational symmetry, exhibiting resistivities which are isotropic in the plane. In contrast, at lower fields, when the Fermi level lies in the N≥2 third and several higher LLs, the 2DES exhibits a distinctly different class of collective states. In particular, near half-filling of these high LLs the 2DES exhibits a strongly anisotropic longitudinal resistance at low temperatures^2,3. These ‘stripe’ phases, which do not exhibit the quantized Hall effect, resemble nematic liquid crystals, possessing broken rotational symmetry and orientational order^4,5,6,7,8. Here we report a surprising new observation: an electronic configuration in the N=1 LL, the resistivity tensor of which simultaneously exhibits a robust fractionally quantized Hall plateau and a strongly anisotropic longitudinal resistance resembling that of the stripe phases.

Main

The sample (details described in Methods) we employ is a square shaped GaAs/AlGaAs heterostructure with edges parallel to the 〈110〉 and  crystal directions, henceforth referred to as the and directions, respectively. Substantial in-plane magnetic fields B_∥ may be added to the field  perpendicular to the 2DES plane by tilting the sample at low temperatures. For the present studies, B_∥ lies along the , or 〈110〉, direction. Figure 1 shows the longitudinal and Hall resistances at T≈15 mK with the magnetic field perpendicular to the 2DES plane (tilt angle θ=0). For the field range shown, the Fermi level lies in the lower spin branch of the N=1LL where the filling fraction runs from ν=2 to ν=3. Deep minima in the longitudinal resistances and associated plateaux in R_{x y} and R_{y x} clearly signal the presence of FQHE states at ν=7/3, 5/2 and 8/3. Whereas the ν=7/3=2+1/3 and 8/3=2+2/3states may be kin to the well-known ν=1/3and 2/3 FQHE states in the N=0 lowest LL (ref. 1; alternatives do exist; see ref. 9), the ν=5/2 state¹⁰ is thought to be an example of the non-abelian Moore–Read paired composite fermion state¹¹. In addition to these and a few weaker FQHE states, the four known re-entrant integer quantized Hall states¹², in varying stages of development, are also evident in Fig. 1. These insulating phases are poorly understood but may be related to the ‘bubble’ phases found in the flanks of the N≥2 LLs and which also exhibit re-entrant integer Hall quantization^2,3,4,5,6. As the data in Fig. 1 make clear, the longitudinal and Hall resistances are very similar for current flow along 〈110〉 and . (The small differences between R_{x x} and R_{y y} for are probably due to extrinsic sample-dependent effects of no relevance here.) Unlike the situation in the N≥2 LLs, no anisotropic phases have been found in 2D electron systems in the N=1 LL, at least in the absence of an external symmetry breaking field such as an in-plane magnetic field B_∥. (Anisotropy in the N=1 LL has been observed in 2D hole systems^13,14.)

Figure 1: Hall and longitudinal resistances at T≈15 mKversus magnetic field in the N=1 Landau level.

ν=7/3, 5/2, and 8/3 FQHE states are indicated by cross-hairs in a and arrows in b. R_{x y} and R_{x x} are the Hall and longitudinal resistances, respectively, for mean current flow along the 〈110〉 direction; for R_{y x} and R_{y y} mean current flow is along . Sample is perpendicular to the magnetic field (θ=0°). Green arrows in a indicate locations of re-entrant integer quantized Hall states.

Tilting the 2DES relative to the magnetic field direction has a profound influence on the various collective phases found in the N=1 LL. In agreement with prior studies^12,15,16,17, we find that the ν=5/2 FQHE state and the re-entrant integer quantized Hall states are suppressed by an in-plane magnetic field component, B_∥. In addition to the destruction of these quantized Hall states, the general trend of the longitudinal resistance throughout the N=1 LL is to become increasingly anisotropic as B_∥ is initially applied, with R_{x x} (for which the mean current direction lies along B_∥) growing significantly larger than R_{y y} (refs 18, 19).

Figure 2 shows the temperature dependence of the longitudinal resistances R_{x x} and R_{y y} at ν=7/3 for various tilt angles θ. As expected, at θ=0 (Fig. 2a) we find R_{x x} and R_{y y} are very nearly equal at all temperatures. Below about T=100 mK the ν=7/3FQHE begins to develop, with R_{x x} and R_{y y} drop** rapidly towards zero in unison as the temperature falls. This temperature dependence is well-approximated by simple thermal activation, R∼exp(−Δ/2T), with Δ≈225 mK.

Figure 2: R_{x x} and R_{y y} versus temperature at ν=7/3 for θ=0°, 19°, 44° and 76°.

R_{x x} is shown as red dots and R_{y y} as blue triangles.

Tilting the sample to just θ=19° (Fig. 2b), where B_∥=0.97 T, creates a substantial anisotropy in the longitudinal resistance, with R_{x x} exceeding R_{y y}. The anisotropy is present at relatively high temperatures (T∼300 mK), well above where both resistances begin to fall sharply as the FQHE develops. Increasing the tilt angle to θ=44° (Fig. 2c), where B_∥=2.72 T, enhances both the anisotropy and the temperature below which the resistances begin their FQHE-induced fall. This latter effect reflects the tilt-induced increase of the ν=7/3 FQHE energy gap noted previously in similar samples^{20, crystal directions, henceforth referred to as the and directions, respectively. InSn ohmic contacts are positioned at the corners and side midpoints of the sample. Longitudinal resistance (R_{x x} and R_{y y}) measurements are performed by driving an a.c. current (typically 2 nA at 13 Hz) between midpoint contacts on opposite sides of the sample and detecting the resulting voltage difference between corner contacts on one side of the mean current axis. For the Hall resistances (R_{x y} and R_{y x}), the voltage difference between the two midpoint contacts on opposite sides of the current axis is recorded. The sample is mounted on a rotating platform (allowing for the application of an in-plane magnetic field component) thermally linked to the mixing chamber of a dilution refrigerator.}