Geometric Resonance of Four-Flux Composite Fermions

TL;DR

This study investigates four-flux composite fermions near filling factor 1/4 using geometric resonance, revealing unique symmetry properties and anisotropies that differ from two-flux CFs, and suggesting different effective masses on either side of 1/4.

Contribution

First direct experimental probing of the Fermi sea of four-flux composite fermions near ν=1/4, uncovering novel symmetry and anisotropy features.

Findings

Symmetric geometric resonance minima around ν=1/4 without in-plane magnetic field.

Asymmetry in minima positions when an in-plane magnetic field is applied.

Indication of different effective masses for four-flux CFs on either side of ν=1/4.

Abstract

Two-dimensional interacting electrons exposed to strong perpendicular magnetic fields generate emergent, exotic quasiparticles phenomenologically distinct from electrons. Specifically, electrons bind with an even number of flux quanta, and transform into composite fermions (CFs). Besides providing an intuitive explanation for the fractional quantum Hall states, CFs also possess Fermi-liquid-like properties, including a well-defined Fermi sea, at and near even-denominator Landau level filling factors such as $\nu=1/2$ or $1/4$. Here, we directly probe the Fermi sea of the rarely studied four-flux CFs near $\nu=1/4$ via geometric resonance experiments. The data reveal some unique characteristics. Unlike in the case of two-flux CFs, the magnetic field positions of the geometric resonance resistance minima for $\nu<1/4$ and $\nu>1/4$ are symmetric with respect to the position of $\nu=1/4$. However, when an in-plane magnetic field is applied, the minima positions become asymmetric, implying a mysterious asymmetry in the CF Fermi sea anisotropy for $\nu<1/4$ and $\nu>1/4$. This asymmetry, which is in stark contrast to the two-flux CFs, suggests that the four-flux CFs on the two sides of $\nu=1/4$ have very different effective masses, possibly because of the proximity of the Wigner crystal formation at small $\nu$.

Equations

$$k_{x,y}=\sqrt{\frac{n}{\pi }}{\left(1-\frac{2^{10}}{3^5{\pi }^6}\frac{e^2{B}_{||}^2}{{\hslash }^2}\frac{w^4{m}_z}{m_{||}}\right)}^{\pm 1/4}.$$

Full text

Ultra-low-disorder two-dimensional electron systems (2DESs) subjected to a perpendicular magnetic field ($B_{\perp}$) give rise to a plethora of quantum many-body phases of matter. Many of these phases can be understood based on composite fermions, quasiparticles comprised of an electron and an even number of flux quanta Jain.PRL.1989 ; Jain.2007 ; Halperin.PRB.1993 . Near Landau level (LL) filling factor $\nu=1/2$, e.g., an electron merges with two flux quanta to form a two-flux composite fermion (2CF). While the electron system is highly interacting and is in a high $B_{\perp}$, the 2CFs behave as essentially non-interacting particles and only feel an effective magnetic field $B^{*}=B-B_{\nu=1/2}$, where $B_{\nu=1/2}$ is the field at $\nu=1/2$. Importantly, these 2CFs occupy a Fermi sea at $\nu=1/2$ and can execute cyclotron motion near $\nu=1/2$ at small $B^{*}$, similar to their fermion counterparts near $B=0$ Halperin.PRB.1993 . With the application of a one-dimensional periodic perturbation to the 2DES, if the 2CFs can complete a cyclotron orbit ballistically, then they exhibit a geometric resonance (GR) when their orbit diameter equals the period of the perturbation. Such a resonance provides a direct and quantitative way to explore some of the fundamental properties of 2CFs Willett.PRL.1993 ; Kang.PRL.1993 ; Smet.PRL.1999 ; Kamburov.PRL.2013 ; Kamburov.PRB.2014 ; Kamburov.PRL.2014 . For example, recent GR measurements of 2CF Fermi sea revealed an unexpected asymmetry between the two sides of $\nu=1/2$ Kamburov.PRL.2014 . This asymmetry, and more generally the question of particle-hole symmetry, inspired renewed interest in the physics of a half-filled LL Barkeshli.PRB.2015 ; Kachru.PRB.2015 ; Son.PRX.2015 ; Balram.PRL.2015 ; Balram.PRB.2016 ; Wang2.PRB.2016 ; Wang3.PRB.2016 ; Mulligan.PRB.2016 ; Geraedts.Science.2016 ; Zucker.PRL.2016 ; Balram.PRB.2017 ; Wang.PRX.2017 ; Cheung.PRB.2017 ; Pan.NatPhys.2017 ; Geraedts.PRL.2018 ; Goldman.PRB.2018 ; Son.Annul.Rev.Cond.Mat.Phys.2018 ; Mitra.preprint . Notable among the new studies is the theory involving a Dirac fermion description Son.PRX.2015 ; Wang2.PRB.2016 ; Wang3.PRB.2016 ; Mulligan.PRB.2016 ; Geraedts.Science.2016 ; Zucker.PRL.2016 ; Wang.PRX.2017 ; Cheung.PRB.2017 ; Pan.NatPhys.2017 ; Geraedts.PRL.2018 ; Goldman.PRB.2018 ; Son.Annul.Rev.Cond.Mat.Phys.2018 ; Mitra.preprint .

Qualitatively similar to the case of $\nu=1/2$, at $\nu=1/4$ electrons merge with four flux quanta and form a four-flux CF (4CF) Fermi sea. Unlike $\nu=1/2$, there is no obvious particle-hole symmetry at $\nu=1/4$ footnote . This provides motivation for studies of 4CFs whose physics could be distinct from 2CFs. However, measurements of 4CFs are very scarce Leadley.PRL.1994 ; Yeh.PRL.1999 ; Pan.PRB.2000 , partly because they require very high magnetic fields, and also because of the proximity of $\nu=1/4$ to the Wigner crystal formation near $\nu=1/5$ Andrei.PRL.1988 ; Jiang.PRL.1990 ; Goldman.PRL.1990 . Therefore, many fundamental questions have remained unanswered: Do 4CFs have properties similar to the 2CFs? Do 4CFs show an asymmetry in the field positions of the GR minima similar to 2CFs Kamburov.PRL.2014 ? What happens to the 4CF Fermi sea when the Fermi sea for zero-field electrons is highly anisotropic? Our GR measurements reported here provide answers to these fundamental questions, and reveal surprises for 4CFs.

Our experimental platform is a molecular beam epitaxy grown 2DES, with density $n=1.78\times 10^{11}$ cm*-2* and low-temperature mobility $1.4\times 10^{7}$ cm${}^{2}/$Vs, confined to a modulation-doped, 40-nm-wide, GaAs quantum well SM . In our GR measurements, we impose a minute periodic density modulation, the estimated magnitude of which is about $0.5\%$ Mueedstripe.PRL.2016 . As illustrated in Fig. 1(a), this is achieved by fabricating a one-dimensional superlattice of period $a=240$ nm, consisting of stripes of negative electron-beam resist on the surface of a lithographically-defined Hall bar Kamburov.PRB.2014 ; Skuras.APL.1997 ; SM ; Endo.PRB.2000 ; Endo.PRB.2001 ; Endo.PRB.2005 ; Mueedstripe.PRL.2016 ; Davies.PRB.1994 ; Kamburov3/2.PRB.2014 ; Kamburov.PRL.2014 ; Mueed.PRB.2017 ; Mueed.PRL.2015 ; Mueed.PRL.2016 ; Shafayat5/2.PRL.2018 . Thanks to the piezoelectric effect in GaAs, the strain from this surface superlattice propagates to the 2DES which is 235 nm underneath the sample surface and leads to a small density modulation.

Weakly-interacting CFs subjected to an effective perpendicular magnetic field $B^{*}$ execute circular cyclotron motion with an orbit radius of $R_{c}^{*}=\hbar k_{F}^{*}/eB^{*}$, the size of which is determined by the magnitude of the Fermi wave vector of the CFs, $k_{F}^{*}$ footnote1 . If the CFs have a sufficiently long mean-free-path so they can complete a ballistic cyclotron orbit, then a GR occurs when the orbit diameter becomes commensurate with the period ($a$) of the modulation; see Fig. 1(a) for a schematic illustration. More quantitatively Smet.PRL.1999 ; Kamburov.PRL.2013 ; Kamburov.PRB.2014 ; Kamburov3/2.PRB.2014 ; Kamburov.PRL.2014 ; Mueed.PRB.2017 , when $2R_{c}^{*}/a=i+1/4$ ($i=1,2,3,...$), GRs manifest as minima in magneto-resistance at $B_{i}^{*}=2\hbar k_{F}^{*}/ea(i+1/4)$. Thus, $k_{F}^{*}$ can be deduced directly from the positions of $B_{i}^{*}$. Such direct measurement of $k_{F}^{*}$ not only provides a proof for the existence of a CF Fermi sea and a measure of its spin polarization but also enables one to quantitatively investigate how the anisotropy of the electron Fermi sea transfers to the CF Fermi sea Smet.PRL.1999 ; Kamburov.PRL.2013 ; Kamburov.PRB.2014 ; Kamburov3/2.PRB.2014 ; Kamburov.PRL.2014 ; Mueed.PRB.2017 ; Shafayat5/2.PRL.2018 ; Mueed.PRL.2015 ; Mueed.PRL.2016 . Here, we apply this technique in very high magnetic fields (using a 45 T hybrid magnet) to investigate the 4CFs near $\nu=1/4$.

We first show in Fig. 1(b) a representative magneto-resistance trace, exhibiting well-developed GR features flanking symmetrically a deep V-shaped minimum at $\nu=1/4$. Figure 1(c) zooms in around $\nu=1/4$. From the period of the modulation, $a=240$ nm, we determine the expected positions for the primary $i=1$ GR resistance minima according to $B_{i=1}^{*}=2\hbar k_{F}^{*}/ea(1+1/4)$ where $B_{i=1}^{*}=B_{i=1}-B_{\nu=1/4}$. We assume a fully spin-polarized CF sea and mark the expected positions for $B_{i=1}$ in Fig. 1(c) considering two possibilities: (i) black solid lines for $k_{F}^{*}=\sqrt{4\pi n}$, and (ii) orange dashed lines for $k_{F}^{*}$ changing according to the magnetic length, i.e., $k_{F}^{*}=\sqrt{4\pi n}\times\sqrt{B/B_{\nu=1/4}}$ Halperin.PRB.1993 ; Son.PRX.2015 ; Wang2.PRB.2016 ; Wang3.PRB.2016 ; Wang.PRX.2017 ; Cheung.PRB.2017 ; Son.Annul.Rev.Cond.Mat.Phys.2018 ; Mitra.preprint . The difference between the expected $B_{i=1}$ for the two assumptions is very small and cannot be resolved in our experiments. From Fig. 1(c), it is clear that the observed GR minima positions are in excellent agreement with the expected $B^{*}_{i=1}$, confirming that the 4CFs near $\nu=1/4$ are fully spin polarized footnote.n1 . More importantly, unlike the 2CF GRs flanking $\nu=1/2$ Kamburov.PRL.2014 , the GR features for 4CFs are quite symmetric around $\nu=1/4$. This is reasonable, considering that the minority carrier density, which was found experimentally in Ref. Kamburov.PRL.2014 to determine $k_{F}^{*}$ for 2CFs, is the same on the two sides of $\nu=1/4$ and is equal to $n$ footnote.n2 .

A fundamental question regarding emergent quasiparticles such as CFs in high magnetic fields is how an anisotropy in the Fermi sea of the electrons at zero field affects the CF Fermi sea Kamburov.PRL.2013 ; Kamburov.PRB.2014 ; Gokmen.Natphy.2010 ; Haldane.PRL.2011 ; Yang.PRB.2012 ; Wang.PRB.2012 ; Yang.PRB.2013 ; Balram2.PRB.2016 ; Jo.PRL.2017 ; Ippoliti1.PRB.2017 ; Ippoliti2.PRB.2017 ; Ippoliti3.PRB.2017 ; Lee.Preprint.2018 . To address this question, we apply an in-plane magnetic field ($B_{||}$) which, through its coupling to the out-of-plane motion of the electrons in a quasi-2D system, severely distorts the Fermi sea of the low-field electrons Kamburov.PRB.2012 ; Kamburov.PRB.2013 ; Mueed.PRL.Fermi2015 . The application of $B_{||}$ shrinks the real-space cyclotron orbit diameter in the in-plane direction perpendicular to $B_{||}$, thereby shrinking the Fermi sea in the direction of $B_{||}$.

The subsequent anisotropy of the CF cyclotron orbit can be determined in a straightforward manner via measuring the positions of the CF GR minima along the two perpendicular arms of the L-shaped Hall bar [inset of Fig. 1(b)]. Since the reciprocal-space ($k$-space) orbits are expected to be a scaled version of the real-space trajectories, rotated by $90^{o}$ AM.BOOK , our GR measurements then directly probe the Fermi sea shape. In our experiments, we tilt the sample so that $B_{||}$ is always along $[110]$, with $\theta$ denoting the angle between the field direction and the normal to the 2D plane [Fig. 1(b) inset].

As seen in Fig. 2, the application of $B_{||}$ affects the positions of the 4CF GR minima. Traces for the two arms of the Hall bar along $[110]$ and $[\mathrm{bar}{1}10]$ are shown in Figs. 2(a) and 2(b). In both panels, the vertical dotted lines mark the expected positions of the $i=1$ CF GR minima for spin-polarized 4CFs with a circular Fermi sea, i.e., $B_{i=1}^{*}=2\hbar\sqrt{4\pi n}/ea(1+1/4)$. These lines match the observed positions of the resistance minima for the bottom traces of Fig. 2, which were taken at $\theta=0^{o}$. When we increase $\theta$ and thereby $B_{||}$, for the [110] arm [Fig. 2(a)], the positions of the two GR minima shift away from $\nu=1/4$ to larger values of $|B_{\perp}^{*}|$. In contrast, the GR minima for the $[\mathrm{bar}{1}10]$ arm [Fig. 2(b)] move towards smaller $|B_{\perp}^{*}|$. Using the field positions of the GR minima along the $[110]$ and $[\mathrm{bar}{1}10]$ directions, we directly extract the magnitude of the Fermi wavevector $\bm{k}_{F}^{*}$ along $[\mathrm{bar}{1}10]$ and $[110]$, respectively; we use the expression: $k_{F}^{*}=B_{i=1}^{*}ea(1+1/4)/2\hbar$.

The most surprising finding of our study emerges in Fig. 3 where we show the deduced $k_{F}^{*}$, normalized to $k_{F0}^{*}$, the value of $k_{F}^{*}$ at $B_{||}=0$. We observe a remarkable difference in the deduced $k_{F}^{*}$ for $\nu>1/4$ and $\nu<1/4$. For both cases, with increasing $B_{||}$, $k_{F}^{*}$ along $[\mathrm{bar}{1}10]$ increases while along $[110]$ it decreases. However, for $\nu<1/4$, the change is much slower compared to $\nu>1/4$. This is different from the 2CF Fermi sea at $\nu=1/2$ where both sides show similar anisotropy with increasing $B_{||}$ Kamburov.PRL.2013 ; Kamburov.PRB.2014 . The difference is particularly puzzling considering that 2CFs and 4CFs form in the same LL.

Before discussing the asymmetry observed in Fig. 3 data, we emphasize that our measured Fermi sea anisotropy for 4CFs is qualitatively different from the electron Fermi sea anisotropy at $B_{\perp}=0$. The comparison is summarized in Fig. 4 where we show the Fermi contours of the electrons (top panels), calculated self-consistently based on the $8\times 8$ Kane Hamiltonian Winkler.2003 ; Kamburov.PRB.2013 ; Mueed.PRL.Fermi2015 , and the 4CF Fermi contours deduced from our measurements (bottom panels). For electrons, the Fermi sea becomes severely distorted with increasing $B_{||}$ and even splits into two tear-shaped seas, signaling the formation of a bilayer system, as confirmed in experiments Kamburov.PRB.2013 ; Mueed.PRL.Fermi2015 . In stark contrast, the 4CFs Fermi sea is much less anisotropic and remains connected even at the highest $B_{||}=30$ T. This is similar to what was seen for 2CFs except that our measured Fermi sea anisotropy for 4CFs is even smaller than for 2CFs for the same quantum well width Kamburov.PRB.2014 . At $B_{||}=25$ T, e.g., $k_{F}^{*}/k_{F0}=1.9$ for the 2CFs Kamburov.PRB.2014 while the 4CFs exhibit $k_{F}^{*}/k_{F0}=1.3$ and $1.6$ for $\nu<1/4$ and $\nu>1/4$, respectively.

For an understanding of the qualitative difference between the Fermi sea anisotropies of electrons and 4CFs, we use a simple model, inspired by Fermi liquid theory. Developed in Ref. Kamburov.PRB.2014 to explain 2CF data, this model takes into account the coupling of $B_{||}$ to the out-of-plane (orbital) motion of the quasi-2D charged particles confined to a quantum well of width $w$, and provides an estimate for the Fermi sea anisotropy Stern.PRL.1968 . In the limit of a small anisotropy where this model is valid, it yields an elliptical Fermi contour with minor and major Fermi wave vectors:

[TABLE]

Here $B_{||}$ is along the $x$ direction, and $m_{||}$ and $m_{z}$ are the particles’ effective mass in the 2D plane and out-of-plane, respectively. It is reasonable to expect that the physics of CFs characterizes the in-plane dynamics of the quasiparticles in our experiments. According to Fermi liquid theory, $m_{||}$ should then be approximately the effective mass of CFs that contains electron-electron interaction and is about unity Jain.2007 ; Halperin.PRB.1993 ; Du.PRL.1993 ; Manoharan.PRL.1994 . (All effective masses are in units of the free electron mass.) On the other hand, the quantized perpendicular motion of the quasiparticles giving rise to the formation of electric subbands should reflect the band dynamics which is characterized by the band mass of electrons in GaAs, $m_{z}=0.067$. The approximate validity of this simple model for the 2CFs was demonstrated in Ref. Kamburov.PRB.2014 where the much smaller measured 2CF Fermi sea anisotropy compared to that of the zero-field electrons and its dependence on $w$ was explained.

In Fig. 3(b) we show the predictions of Eq. (1) (orange curves) using different values of $m_{||}$, with $m_{z}$ fixed at 0.067. The curve with $m_{||}=2.4$ fits the $\nu<1/4$ data reasonably well. For $\nu>1/4$, none of the curves fit the experimental data well footnote3 , but a comparison with the data suggests that $m_{||}$ is smaller than 2.4, namely that there is an asymmetry between $m_{||}$ for $\nu<1/4$ and $\nu>1/4$. It is noteworthy that a qualitatively similar asymmetry in 4CF mass ($m^{*}$) was also deduced from measuring the temperature dependence of the strengths of fractional quantum Hall states near $\nu=1/4$ Pan.PRB.2000 . For $\nu>1/4$, the measured $m^{*}$ for 4CFs was found to be consistent with the value expected based on the 2CF mass (after scaling with $m^{*}\propto\sqrt{B_{\nu}}$ to take into account that $m^{*}$ is proportional to the Coulomb energy Jain.2007 ; Halperin.PRB.1993 ; Du.PRL.1993 ; Manoharan.PRL.1994 ). However, for $\nu<1/4$, a much larger $m^{*}$ was deduced and was attributed to the formation of the pinned, magnetic-field-induced Wigner crystal (WC) which manifests as an insulating phase near $\nu=1/5$ Jiang.PRL.1990 ; Goldman.PRL.1990 .

For our sample, the expected $m^{*}$ for 4CFs near $\nu=1/4$ based on the 2CF $m^{*}$, and using the $m^{*}\propto\sqrt{B_{\nu}}$ scaling, is $\simeq$ 1.4 footnote2 . The data of Fig. 3(b) suggest that $m^{*}$ for $\nu<1/4$ is larger than this value, qualitatively consistent with the data of Ref. Pan.PRB.2000 . While the proximity to the WC formation as suggested in Ref. Pan.PRB.2000 and confirmed by numerical calculations Zu.PRB.1999 might be a possible explanation for the larger 4CF $m^{*}$ on the $\nu<1/4$ side in our sample also, we would like to emphasize an important point. The $m^{*}$ measured in Ref. Pan.PRB.2000 and calculated in Ref. Zu.PRB.1999 were in the range $0.237>\nu>0.222$, relatively close to the insulating phase that sets in at $\nu=0.21$ Jiang.PRL.1990 ; Goldman.PRL.1990 ; Pan.PRB.2000 . In contrast, we observe GR resistance minima in the range $0.246>\nu>0.242$, reasonably far from 0.21, and very close to $\nu=1/4$. It would therefore be surprising if the WC formation would affect the 4CFs so significantly SM . We hope that our data would stimulate theoretical work for a quantitative understanding of 4CF properties and, in particular, the strong asymmetry we observe for the 4CFs Fermi sea anisotropy on the two sides of $\nu=1/4$.

Note added, recently we came to know of J. Wang’s proposal of a flux-attached Dirac fermion theory where the Berry phase of composite fermion Fermi liquid at $\nu=1/4$ is computed Wang.PRL.2019 . This theory perhaps can provide an explanation of the asymmetry that we observe for the 4CFs.