Valley-Tunable Even-Denominator Fractional Quantum Hall State in the Lowest Landau Level of an Anisotropic System

Fractional quantum Hall states (FQHSs) at even-denominator Landau level filling factors ($\nu$) are of prime interest as they are predicted to host exotic, topological states of matter. We report here the observation of a FQHS at $\nu =1/2$ in a two-dimensional electron system of exceptionally high quality, confined to a wide AlAs quantum well, where the electrons can occupy multiple conduction-band valleys with an anisotropic effective mass. The anisotropy and multivalley degree of freedom offer an unprecedented tunability of the $\nu =1/2$ FQHS as we can control both the valley occupancy via the application of in-plane strain, and the ratio between the strengths of the short- and long-range Coulomb interaction by tilting the sample in the magnetic field to change the electron charge distribution. Thanks to this tunability, we observe phase transitions from a compressible Fermi liquid to an incompressible FQHS and then to an insulating phase as a function of tilt angle. We find that this evolution and the energy gap of the $\nu =1/2$ FQHS depend strongly on valley occupancy.

Figure 1

Observation of FQHS at $\nu =1/2$ in an $X$-valley occupied AlAs 2DES when subjected to a tilted magnetic field; $\theta$ denotes the angle between $B_{\perp }$ and total $B$.

Figure 2

Sample description and valley tuning. (a) First Brillouin zone of bulk AlAs, showing anisotropic conduction band valleys. (b) Sample geometry, showing the orientation of the two occupied valleys ($X$ and $Y$) and the measured resistances ($R_{[100]}$ and $R_{[010]}$). Electrical contacts to the sample are denoted by 1–8. For $R_{[100]}$, we pass current from contact 8 to 2 and measure the voltage between contacts 6 and 4. For $R_{[010]}$, the current is passed from 8 to 6 and we measure the voltage between 2 and 4. (c) Experimental setup for applying in-plane strain ($ϵ$). (d) Resistance of the sample at $B=0$ and $T\simeq 0.03\mathrm{K}$, measured as a function of $ϵ$. (e) Direction of tilted magnetic field with respect to the crystallographic directions.

Figure 3

Tilt evolution of magnetoresistance data near $\nu =1/2$ for different valley occupancies, all traces taken at $T\simeq 0.03\mathrm{K}$, and the tilt angles are indicated in each panel. The direction of $B_{||}$ is along [100], i.e., along $R_{[100]}$. (a) and (b) contain data for cases when the electrons occupy only $X$ or only $Y$ valley, respectively. In both (a) and (b), with increasing $\theta$, the 2DES at $\nu =1/2$ undergoes transitions from a compressible phase to an incompressible FQHS, and then finally to an insulating phase. However, the transitions happen at relatively smaller $\theta$ for case (a) where the electrons occupy the $X$ valley. Insets in each panel show the calculated charge distribution for the corresponding $B_{||}$ that is experienced by the electrons at $\nu =1/2$ [52].

Figure 4

Strength of the $\nu =1/2$ FQHS, as defined in the inset to (a), plotted against $B_{||}$ for different valley occupancies. (a), (b) Data for the hard and easy axis directions (see text). (c),(d) Extraction of the pseudo activation gap (${}^{1/2}\mathrm{\Delta }$) of the $\nu =1/2$ FQHS from $T$ dependence of $R_{[100]}$ (black) and $R_{[010]}$ (red) for the two valleys.