# Particle-Hole Symmetry Breaking of ν = 1/2 Composite Fermions

The physics of an interacting two-dimensional electron system (2DES) at low temperatures and strong perpendicular magnetic field is elegantly described by the composite fermion (CF) formalism in which an even number of flux quanta is attached to each electron in high magnetic field. A fundamental property of CFs at half-filled Landau levels is that they occupy a Fermi sea and therefore possess a Fermi contour. We have recently probed this Fermi contour in a number of commensurability experiments performed on samples with a unidirectional periodic potential modulation in the 2D plane. In these experiments, when the quasi-classical CF cyclotron orbit diameter matches an integer multiple of the period, a resistance minimum is expected. Examples of such oscillations can be seen in the image to the right.

We recently observed a pronounced asymmetry in the magnetic field positions of the commensurability resistance minima of fully spin-polarized CFs with respect to the field at ν = 1/2 (see figure to the left). The asymmetry is seen across a wide range of 2D densities and modulation periods. We can explain the asymmetry quantitatively if we assume that the CFs are fully spin-polarized and their density is equal to the density of the minority carriers in the lowest, spin-resolved Landau level (LL), namely the density of electrons when ν < 1/2 and of holes when ν > 1/2. Our results provide direct evidence that CFs are formed by pairing up of the minority carriers in the lowest spin-resolved LL with flux quanta. They further indicate a breaking of the particle-hole symmetry for spin-polarized CFs near ν = 1/2.

# Ferromagnetic Wigner Crystal of Two-flux Composite Fermions

The re-entrant integer quantum Hall effect (RIQHE) near ν = 4/5 was recently observed in very clean 2D electron systems, and was interpreted as the particle-hole counterpart of the Wigner crystal observed at low fillings near ν = 1/5. The RIQHE is only seen above a critical density (n_{c}) which is larger for narrower quantum wells, and its emergence is accompanied by an intriguing disappearance and reappearance of the ν = 4/5 fractional quantum Hall state (FQHS) at n_{c}. In this study, we demonstrate that the transition of the ν = 4/5 FQHS is consistent with a spin-polarization transition of two-flux composite Fermions (^{2}CFs). Therefore the RIQHE is a ferromagnetic Wigner crystal of ^{2}CFs.

The figure shows the longitudinal (R_{xx}) and Hall (R_{xy}) magneto-resistances measured in a 65-nm-wide GaAs quantum well at a fixed density n = 1.0 x 10^{11} cm^{-2} and different tilting angles θ. At θ = 0°, a strong FQHS is seen at ν = 4/5. The R_{xx} minimum at ν = 4/5 disappears at θ ≈ 30° and reappears at higher θ, signaling the destruction and resurrection of the FQHS. Two minima in R_{xy} on the sides of ν = 4/5 develop at θ > 30°. As θ is further increased, the R_{xy} minimum at ν > 4/5 deepens and an R_{xx} minimum starts to appear at the same ν (see vertical arrows in the figure). At the highest tilting angles θ > 40°, these minima merge into the wide R_{xy} plateau and the R_{xx} minimum near ν = 1. This evolution, and in particular the transition of the ν = 4/5 FQHS and the subsequent development of the RIQHE with increasing θ is very similar to what was seen as a function of increasing electron density.

Our extensive study shows that the ν = 4/5 FQHS transition signals the full spin polarization of the ^{2}CFs. Such a spin transition is only expected if we assume that the ^{2}CFs interact with each other to form ^{2}CF-FQHSs at fractional ^{2}CFs filling factors ν^{CF} = ν⁄(1−2ν). Theoretical calculations predict a spin transition of the ^{2}CF-FQHS at ν^{CF} = 4/3 (ν = 4/11), when the Zeeman energy (E_{Z}) in units of the Coulomb energy (e^{2}/4πεl_{B}) is ≈ 0.025. In our experiments, the ν = 4/5 FQHS, which corresponds to a ^{2}CF-FQHS at ν^{CF} = -4/3, exhibits a transition at E_{Z}/(4πεl_{B}) ≈ 0.02, close to the theoretical prediction. Moreover, we find two spin transitions of the ν = 5/7 FQHS (ν^{CF} = -5/3), also consistent with this interacting ^{2}CF picture.

# Evidence of a Warped Fermi Contour for Composite Fermions

Composite fermions (CFs), quasi-particles formed by attaching an even number of flux quanta to each carrier in high perpendicular magnetic fields (B), capture many phenomena exhibited by an interacting system of two-dimensional (2D) carriers such as the fractional quantum Hall effect. The flux attachment cancels out the external B at a half-filled Landau level, enabling CFs to occupy a Fermi sea and possess a Fermi contour, similar to their B = 0 carrier counterparts. Because the CFs are primarily a manifestation of interaction, one might argue that their physical properties should retain no memory of the B = 0 particles, including their energy band structure. Recently we found tantalizing evidence through commensurability measurements that hole-flux CFs confined to a wide GaAs quantum well (QW) exhibit a warped Fermi contour qualitatively similar to their B = 0 hole counterparts.

In commensurability experiments, a resistance minimum is observed in the longitudinal resistance (R_{xx}) when the quasi-classical CF cyclotron orbit diameter becomes commensurate with the period of the modulation. The field position of the minimum provides a direct measure of the Fermi wave vector (k_{F}). As illustrated in (a) in the above figure, the Fermi contour of 2D holes confined to a wide GaAs QW is expected to be significantly warped. Our low-field commensurability measurements on 2D holes in a 35-nm QW indeed demonstrate such warping, with k_{F} along [110] being about 20% larger than k_{F} expected if the hole Fermi contour were circular (see (b)); this is consistent with the average warping (~25%) calculated for the two spin species whose contours are split because of the spin-orbit interaction. Our data show that the CFs, too, exhibit a warped Fermi contour! As clearly seen in (c), the positions of the observed R_{xx} minima (vertical arrows) are measurably farther from where we would expect the minima if the CF Fermi contour were circular (red tick marks). From the observed positions, we deduce a warping of about 10%. Our results demonstrate that the warping of the hole Fermi contour is partly transferred to the hole-flux CFs. Complementing the recent measurements which have shown that the CF Fermi contours can be distorted via the application of a parallel magnetic field, our data illustrate that the band structure of the B = 0 particles can directly affect the CF Fermi contour.

# ν = 1/2 Fractional Quantum Hall Effect in Tilted Magnetic Fields

The fractional quantum Hall state (FQHS) at Landau level filling factor ν = 1/2 has been observed in 2D electron and hole systems confined to wide GaAs quantum wells (QWs). In wide QWs, at densities low enough so that the charge distribution is single-layer-like, the ground state at ν = 1/2 is compressible. As the density is increased and the charge distribution becomes bilayer-like, a FQHS at ν = 1/2 appears. At higher densities, a correlated, bilayer insulating state emerges and the ν = 1/2 FQHS disappears.

We performed the first systematic study of the evolution of the correlated states at ν = 1/2 in a wide QW in the presence of parallel magnetic field (B_{||}). The figure shows the magnetoresistance traces, for 2D electrons confined to a 65-nm-wide GaAs QW at a fixed density of 1.4 x 10^{11} cm^{-2}, taken at different tilting angles (θ). The state at ν = 1/2 is compressible at θ = 0° but a FQHS appears as soon as the sample is tilted. The FQHS becomes stronger with tilting up to θ = 35° and then is destroyed at higher θ by an insulating phase. This evolution is qualitatively similar to the one observed at θ = 0° as the density is increased. In our presentation, we explain this similarity by demonstrating how the high B_{||} couples to the out-of-plane motion of the 2D electrons confined to a wide QW and renders the system increasingly bilayer-like as B_{||} is increased. Based on data similar to those shown here but taken at different fixed densities, we also present a density vs B_{||} phase-diagram for the three different phases (compressible, FQHS, insulating) of the 2D system.

# Even-denominator Fractional Quantum Hall Effect

A strong perpendicular magnetic field B applied to a two-dimensional (2D) electron system quantizes the electron energies into discrete Landau levels (LLs). At very low temperatures, and if disorder is low, electron-electron interaction leads to new phenomena. An example is the fractional quantum Hall effect (FQHE), the condensation of 2D electrons into many-body incompressible states which are stable predominantly at odd-denominator fractional LL filling factors ν.

Here we describe unexpected phenomena in 2D hole systems (2DHSs) confined to GaAs quantum wells (QWs). In our study, we observe an unusual crossing of the two lowest-energy LLs. This crossing leads to a weakening or disappearance of the commonly seen odd-denominator FQHE states in the filling range 1/3 ? ν ? 2/3. But, surprisingly, a new FQHE state at the even-denominator filling ν = 1/2 comes to life at the crossing. Our results attest to the rich many-body physics of the 2DHSs.

The figure shows the longitudinal (Rxx) and Hall (Rxy) resistance traces for a 2DHS confined to a 30-nm-wide QW for several densities, ranging from p = 1.20 to 1.72 (in units of 10^{11} cm^{-2}). At the lowest p (top trace), R_{xx} minima are observed at numerous odd-denominator fillings such as ν = 2/3, 2/5, 3/5, 3/7, 4/7, etc. As p increases, starting at ≈ 1.32, an R_{xx} minimum develops at ν= 1/2, and quickly deepens and turns into a zero-resistance plateau centered at ν= 1/2 for p = 1.47. Concomitantly, the R_{xy} trace exhibits a Hall plateau quantized at 2h/e^{2}, signaling the formation of a strong FQHE state at ν = 1/2. At a slightly higher density, p = 1.59, the ν = 1/2 Rxx minimum becomes weak. The strong ν = 1/2 minimum returns again at higher p and the ν = 1/2 FQHE persists up to the highest densities we can achieve in this sample.

The behavior of R_{xx} near ν = 1/2, and in particular the strengths of the nearby FQHE states, provide clear demonstration of an unusual crossing of the two lowest-energy hole energy levels in a large perpendicular magnetic field. Surprisingly, the ν = 1/2 FQHE is very strong at this crossing and becomes weaker on either side of the crossing, while all the odd-denominator FQHE states, on the other hand, are strong except at the crossing. We tentatively interpret this FQHE as a two-component 331 state. A detailed understanding of its origin and properties await future research.

# Berry's phase in a 2D hole system under stain

We study the Aharonov-Bohm effect in a 2D hole system under stain. Due to strong spin-orbit coupling, the signature of spin Berry phase can be observed in the resistance oscillations across the ring. We investigate the effect of strain on the Berry phase of spins that traverse the mesoscopic ring structure.

Aharonov-Bohm effect is a manifestation of wave-like nature of charged particles that traverse two different paths, resulting in oscillations of resistance as a function of magnetic flux that are enclosed by the mesoscopic ring structure. The resistance oscillations can be understood as the result of two interfering coherent electron waves that acquire phase difference while travelling two different paths. Interestingly, in a system with strong spin-orbit coupling, the charge wave function is also coupled with the spin degree of freedom. This spin related phase factor produces the additional beating structure in AB oscillation signals. It was found that the beating pattern was associated with the geometrical phase of spin, or spin Berry phase. In this project, we use strain to manipulate the spin geometric phase in the AB oscillations. By employing piezo-driven uniaxial strain we can tune the strength of spin-orbit coupling, which results in the spin-splitting by the effective in-plane field. We have demonstrated previously that strain can effectively modify the spin-split bands in the 2D hole system. The use of strain for spin manipulation in nanostructures possesses a great potential for future spin interference and quantum information devices.