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# Ferromagnetic fractional quantum Hall states in a valley degenerate two-dimensional electron system 

Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 

Medini Padmanabhan, T. Gokmen, and M. Shayegan 

(Dated: November 2, 2018) 

We study a two-dimensional electron system where the electrons occupy two conduction band val- leys with anisotropic Fermi contours and strain-tunable occupation. We observe persistent quantum Hall states at filling factors ν = 1 / 3 and 5/3 even at zero strain when the two valleys are degenerate. This is reminiscent of the quantum Hall ferromagnet formed at ν = 1 in the same system at zero strain. In the absence of a theory for a system with anisotropic valleys, we compare the energy gaps measured at ν = 1 / 3 and 5/3 to the available theory developed for single-valley, two-spin systems, and find that the gaps and their rates of rise with strain are much smaller than predicted. 

In a single-electron picture, there should be no quan- tum Hall effect (QHE) at odd-integer Landau level (LL) filling factors ( ν ) in a two-dimensional electron system (2DES) with a vanishing Lande g -factor. In an interact- ing 2DES, however, a QH state exists at ν = 1 even in the limit g = 0 [1, 2]. The ground state is a ferromagnet, and it is the large Coulomb (exchange) energy cost of a spin flip that leads to a sizable energy gap separating this ground state from its excitations. These charged ex- citations, called skyrmions, have a nontrivial, long-range spin texture with a slow rotation of the spin as a function of distance to minimize the Coulomb energy cost [1–5]. 

A question naturally arises as to whether there are similar ferromagnetic ground states in the fractional quantum Hall effect (FQHE) regime. The presence of such a state with finite energy gap at g = 0 along with skyrmionic excitations was theoretically predicted for ν = 1 / 3 [1, 6]. Experimentally, it is challenging to tune the g -factor to zero while keeping the quality of the 2DES sufficiently high to exhibit FQHE. There has been only one study [7] to date probing the state at ν = 1 / 3 under the condition of g → 0, which was achieved via ap- plying hydrostatic pressure. The data revealed no FQH resistance minimum at ν = 1 / 3 when g ≃ 0 [8], but from measurements at non-zero values of g , a finite but small energy gap was deduced for the ν = 1 / 3 FQH state. 

Here we report magnetotransport measurements for 2DESs confined to AlAs quantum wells where the elec- trons occupy two conduction band minima (valleys). In this system, the energy splitting between the two valleys can be controlled in situ via applying in-plane strain. Considering the valley degree of freedom as an isospin, the condition of zero valley splitting is identical to the limit of g = 0 in a two-spin system. Under this condi- tion, a ferromagnetic ground state and skyrmionic exci- tations have already been reported at ν = 1 [9–11]. Here we report the observation of a well-developed QH resis- tance minimum at ν = 1 / 3 as we tune the valley splitting through zero. We also observe a pronounced resistance minimum at ν = 5 / 3 which is the particle-hole conju- gate state of ν = 1 / 3 in our two-component (two-valley) system. These observations strongly suggest the pres- 

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ence of ferromagnetic FQH states at these fillings. From the temperature ( T ) dependence of the resistance min- ima, we deduce energy gaps for these states as a function of applied strain. The gaps increase initially and then saturate, as expected qualitatively. Quantitatively, how- ever, both the gaps and their dependencies on strain are much weaker than the predictions of theories which were developed for the single-valley, two-spin systems. 

Our samples are AlAs quantum wells grown by molecu- lar beam epitaxy [10]. Contacts were made with GeAuNi alloy and Hall bars were defined. Metallic front and back gates were deposited on the samples, thus allowing us to control the 2D electron density, n . We report results for three samples from two wafers. The quantum well in sample A is 12 nm wide while samples B and C have well 

