Composite fermions in graphene fractional quantum Hall state at half filling: evidence for Dirac composite fermions
Composite fermions in fractional quantum Hall (FQH) systems are believed to form a Fermi sea of weakly interacting particles at half filling . Recently, it was proposed (D. T. Son, Phys. Rev. X 5, 031027 (2015)) that these composite fermions are Dirac particles. In our work, we demonstrate experimentally that composite fermions found in monolayer graphene are Dirac particles at half filling. Our experiments have addressed FQH states in high-mobility, suspended graphene Corbino disks in the vicinity of . We find strong temperature dependence of conductivity away from half filling, which is consistent with the expected electron-electron interaction induced gaps in the FQH state. At half filling, however, the temperature dependence of conductivity becomes quite weak as expected for a Fermi sea of composite fermions and we find only logarithmic dependence of on . The sign of this quantum correction coincides with weak antilocalization of composite fermions, which reveals the relativistic Dirac nature of composite fermions in graphene.
The fractional quantum Hall (FQH) state is a many body phenomenon where fractionally charged elementary excitations lead to quantization of the Hall conductance at fractional filling factor at carrier density and magnetic field Tsui et al. (1982). The generation of these incompressible liquid states requires a large Coulomb interaction energy compared with the disorder potential, putting strict requirements on temperature, the quality of the two-dimensional electron gas (2-DEG) and the strength of the magnetic field. Owing to reduced screening in atomically thin graphene, the electrons in graphene interact with larger Coulomb interaction energy than electrons in semiconductor heterostructures, providing an extraordinary setting for studies of FQH states and their description in terms of composite fermions Jain (1989, 2007, 2015).
The composite fermion theory Jain (1989) and Composite Fermion Chern-Simon (CFCS) theory have been very successful in outlining a unified picture of fractional quantum Hall effect. Lopez and Fradkin Lopez and Fradkin (1991) showed that the problem of interacting electrons moving in 2D in the presence of an external magnetic field is equivalent to a fermion system, described by a Chern-Simon gauge field, where electrons are bound to even number of vortex lines. Fluctuations in the gauge field were soon realized to have strong influence on the quantum correction of the composite fermion conductivity Kalmeyer and Zhang (1992). Subsequently, a Fermi liquid type of theory was proposed for half-filled Landau level Halperin et al. (1993) where various observables in the low-temperature limit are described in terms of Fermi liquid parameters Simon (1998), involving most notably the effective mass for composite fermions, which is expected to have a strong enhancement near half filling.
There is extensive experimental evidence in favor of weakly interacting Fermi sea of composite fermions effectively in a zero magnetic field at . Transport anomalies in the lowest Landau level of two-dimensional electrons at half filling were observed by Jiang et al. Jiang et al. (1989). Distinct features related with compressibility in surface-acoustic-wave propagation on high-quality AlGaAs/GaAs heterostructures were observed by Willett et al. Willett et al. (1990, 1993); Willett (1998). Furthermore, resonances at fields where the classical cyclotron orbit becomes commensurate with a superlattice have been found Kang et al. (1993); Deng et al. (2016). Strong enhancement of composite fermion mass near half filling has been observed in experiments on similar 2D electron gas heterostructures Manoharan et al. (1994); Du et al. (1994). Logarithmic temperature dependence of conductivity at half-integer filling factors has been observed, which has been interpreted to point towards residual interactions between composite fermions Kang et al. (1995); Rokhinson et al. (1995); Rokhinson and Goldman (1997).
A particle-hole symmetric model for the Fermi liquid ground state of a half-filled Landau level has recently been reinvestigated Son (2015), with the conclusion that composite fermions in a 2-DEG should, in fact, be Dirac particles. This conclusion has separately been verified by extensive numerical renormalization group analysis by Geraedts et al. Geraedts et al. (2016). As Dirac particles, composite fermions may pick up a Berry phase when traversing around a closed loop, which has significant implications on their backscattering dynamics. Reduced backscattering will lead to weak antilocalization (WAL) which has been observed for Dirac particles in graphene at small magnetic fields Tikhonenko et al. (2009). Similar behavior can be anticipated also for composite fermions in graphene acting as Dirac particles.
In this work, we investigate quantum corrections to conductivity for a half-filled quantum Hall state in monolayer graphene sample which is formed of a suspended Corbino disk with high mobility. We find that composite fermions in graphene are Dirac particles as evidenced by the observed weak (anti) localization behavior at half filling. Our results display typical logarithmic temperature corrections for two-dimensional weak localization (WL), but we find an opposite sign for these quantum corrections with respect to previous observations in 2D electron gas experiments Rokhinson et al. (1995); Rokhinson and Goldman (1997).
Suspended graphene Corbino disks provide a clear-cut setting to probe composite fermion physics in graphene at high fields. Recently, Corbino geometry was used in graphene experiments which focused on studies of magneto-conductance in the quantum Hall regime Zhao et al. (2012); Peters et al. (2014). These previous measurements on graphene Corbino disks have failed to show any fractional states, perhaps due to strong charge inhomogeneity induced by the substrate. Our experiments at mK temperatures, however, display a multitude of FQH states (up to 11 states) in the lowest Landau level which are clearly visible in magneto- and transconductance measurements Kumar et al. (). In our devices, conductance is governed by bulk properties instead of the chiral edge states of regular quantum Hall bars. Consequently, our current-annealed, suspended graphene Corbino disks form an excellent platform to investigate the dynamics of the free Fermi sea of composite fermions at half filling.
Interaction and weak localization corrections to conductivity in graphene are influenced by the pseudospin introduced by the two atom basis of the hexagonal lattice. The pseudospin leads to an additional Berry phase of in the backscattering wave interference, resulting in reduced back scattering or, in other words, antilocalization Suzuura and Ando (2002); Khveshchenko (2006); McCann et al. (2006). Real time reversal symmetry of graphene requires intervalley scattering which makes its weak localization phenomena intriguing. In fact, low field magnetoresistance in graphene may be positive or negative, depending on the strength of intervalley and dephasing scattering, as well as the scattering strength for TRS (time reversal symmetry) breaking in a single valley McCann et al. (2006); Tikhonenko et al. (2009); Couto et al. (2014).
A charge carrier is combined with two vortices in the composite fermion picture. The underlying valley structure will be reflected on the graphene composite fermions. Hence, we expect similar scattering processes among graphene composite fermions as for graphene Dirac particles at small fields. Consequently, magnetoresistance at can be either positive or negative depending on the rates of intervalley, dephasing and single-valley TRS breaking scattering for composite fermions denoted as , , and , respectively. Even though fluctuations due to Chern-Simon fields will enhance the dephasing rate of composite carriers, we expect the tendency towards weak antilocalization to remain, which reflects the presence of a Berry phase due to the Dirac nature of these composite carriers. Our results do confirm this expectation and display weak antilocalization behavior for graphene composite fermions.
I Samples and basic experimental results
The samples considered here are denoted by EV2 ( = 1900 nm, = 750 nm) and EV3 ( = 1600 nm, = 400 nm), where and denote outer and inner radii of the Corbino disks. Our sample fabrication is explained in detail in Ref. Kumar et al., . We exfoliated graphene using a heat-assisted exfoliation technique to maximize the size of the exfoliated flakes Huang et al. (2015). The contacts were deposited by developing further the ideas presented in Ref. Tombros et al., 2011. Strongly doped silicon Si++ substrate with 285 nm of thermally grown SiO was used as a global back gate. Annealing of samples on LOR was typically performed at a bias voltage of 1.60.1 V which is comparable with our HF etched, rectangular two-lead samples Laitinen et al. (2014). Larger Corbino disks can require higher annealing voltages, as was the case with sample EV3 where V was used.
Our suspended graphene structure is illustrated in Fig. 1a: the Corbino disk is supported only by the inner and outer Au/Cr leads. The conductivity was calculated from the measured conductance using . The field-effect mobility was determined using equation after subtracting out the contact resistance from and the measured minimum conductivity at the Dirac point ( V). Both samples had cm/Vs. The residual charge density was identified by looking for a cross-over between constant and power law behavior in vs. traces. Contact doping Laitinen et al. (2016) is estimated to correspond to a charge density of under the contact metal. Details concerning determination of basic parameters are given in Ref. Kumar et al., .
In zero magnetic field, the conductance of graphene in Corbino geometry at equals to according to conformal mapping theory Rycerz et al. (2009). After subtraction of the contact resistance, our measured conductivity is in line with the above theoretical value due to evanescent modes. Also the measured gate voltage dependence of in the unipolar regime was found to agree with theoretical formula, which at the same time gave an estimate = 410 for the contact resistance. The high quality of our samples is also manifested in the observability of broken symmetry states () down to 0.6 Tesla.
Our measurements down to 20 mK were performed on a BlueFors LD-400 dilution refrigerator. The measurement lines were twisted pair phosphor-bronze wires supplemented by three stage filters with a nominal cut-off given by Ohms and nF. However, due high impedance of the quantum Hall samples the actual cutoff is determined by the sample resistance. For magneto-conductance measurement in Fig. 1b, we used an AC peak-to-peak current excitation of nA at Hz. The other magnetoconductance measurements were conducted using DC with sufficient pause times, and thus no signal suppression by cut-off is present. Due to symmetry reasons of the Corbino geometry, the azimuthal electric field is zero. Consequently, our experiment measures directly the longitudinal conductivity of our sample.
Ii Magnetoconductance around 1/2 filling
Our data on electronic conductivity vs. gate-swept filling factor is displayed in Fig. 2 for several temperatures in the range K measured at T and 9 T. Half filling regime is characterized by the weakest temperature dependence of , which is in accordance with the absence of any energy gap in this regime. In both frames, only linear background variation of around is observed at K, which indicates washing out of quantum corrections of conductance due to strong dephasing.
At 5 T (see Fig. 2 a), we find a change in effective magnetoconductance defined as , which varies between positive and negative values with around at low temperatures. We interpret this as varying sign of as coming from a competition between WL/WAL behavior and the appearance of gaps of the fractional states at . This complicated localization behavior is assigned to fluctuations in Chern-Simon fields and the particular band structure of graphene influencing its weak localization properties McCann et al. (2006). By enhancing the applied magnetic field, the effective magnetoconductance in sample EV2 acquires a negative sign around at fields of T, as illustrated by the data in Fig. 2 b (in sample EV3 at T). The half-filling data at 9 T displays and , which form the fundamental findings before conversion to the composite fermion description.
The temperature dependence in Fig. 2 can also be employed to determine energy gaps around , for which we find at 5T K/T and K/T with K for particles and holes, respectively. Using the slopes , we obtain for the effective masses and , respectively, where denotes the electron mass.
The effective field seen by the composite fermions at half filling is . To compare the magnetoconductivity in Fig. 2 with phenomena at small fields, we note that there may be either positive or negative magnetoresistance in graphene McCann et al. (2006). The sign of magnetoresistance has been demonstrated to depend fundamentally on parameters and ; when both of these values are , weak antilocalization is preferred and Tikhonenko et al. (2009). In Ref. Tikhonenko et al., 2009, small ratios were achieved at elevated temperatures K. In our suspended sample, we were able to observe WAL even at 20 mK, and the range of weak antilocalization magnetoresistance was found to be within mT. This makes field-swept investigations of weak localization at challenging in our graphene samples, primarily since we have a charge inhomogeneity corresponding to variation . Nevertheless, we see clear quantum corrections in conductivity near half filling and these corrections depend logarithmically on temperature.
Iii Quantum corrections
Fig. 3a displays temperature dependence of conductivity for sample EV2 at . The data indicate increase in with temperature where ; here with the reference temperature taken as K. Altogether, the observed quantum correction is quite large and it amounts to % of over the measured range.
We assume that particle-hole symmetry is valid in our 2D graphene electron gas experiment Kivelson et al. (1997); Nagaosa and Fukuyama (1998); Son (2015). Then, the connection between electronic conductivity in units and the composite fermion conductivity is given by
Eq. 1 has been employed to extract the diagonal conductivity of composite fermions .
The inversion of Eq. 1 yields two solutions for . We retain only the metallic solution which reads as , where we have used the fact for the particle-hole symmetric model and that for our data. The resulting conductivity for composite fermions is displayed in Fig. 3b. Again, logarithmic dependence is obtained but now with , which has a different sign when compared to previous observations on 2D electron gas systems with composite fermions at Kang et al. (1995); Rokhinson et al. (1995); Rokhinson and Goldman (1997). One possible explanation for this different behavior is that Chern-Simon field fluctuations suppress very strongly weak localization contributions in regular 2D electron gas, leaving only interaction corrections to conductivity, while weak antilocalization effects still survive in graphene. In fact, the weak antilocalization properties of Dirac particles in graphene have their particular characteristics due to the specific valley structure of graphene McCann et al. (2006).
In the presence of impurity scattering, the composite fermion motion will be cut off by a finite transport mean free path . For , the random phase approximation result is where indicates the number of vortices connected to each electron Jain (2007). By fitting this formula to our data using , we obtain , which yields nm for the mean free path of composite fermions at T. This indicates clearly diffusive motion as . However, the impurity concentration in our sample does not coincide with this mean free path, and we conclude that this scattering length is set by the Chern-Simon field fluctuations.
The combined effect of the diffusive motion of composite fermions and the gauge field interaction has been studied by Khveshchenko Khveshchenko (1996, 1997) and by Mirlin and Wölfle Mirlin and Wölfle (1997). According to Khveshchenko Khveshchenko (1997), interference between disorder scattering and gauge fields leads to magnetoconductivity with effective :
where is the cyclotron frequency at effective field , denotes the transport scattering time and is related to the sign of quantum corrections to conductivity, i.e. and for WAL and WL, respectively. Hence, weak antilocalization ( in Fig. 3b) is tied with negative magnetoconductivity, which is in contrast to our observed positive sign. One possible explanation for our positive magnetoconductivity in comparison with this model can be that Chern-Simon fluctuations display field dependence and they are reduced when going off from half filling, resulting in an increased mean free path . Consequently, the magnetoconductance becomes positive around but the temperature dependence at constant mean free path retains the antilocalization behavior.
Charge puddles and impurities create a disorder potential in suspended graphene, which can support localized FQH states Kumar et al. (). The localized states can carry current across the sample via coupling to adjacent localized states across the gapped FQH regions. However, since the energy gaps around appear smaller than their width , there may be percolating paths around half filling, which guarantee that the nature of charge transport will not change sharply around . There is the possibility though that these percolation paths vary with magnetic field, which leads to sample dependent magnetoconductivity Mirlin et al. (1998). We have cooled our samples a few times and found that the variation of magnetoconductivity at half filling remains approximately within , but every time the sign of the quantum corrections have been the same at fields T.
Our suspended graphene samples contain always built-in non-uniform strain, which is seen as frozen ripples at room temperature. This strain is modified by applied gate voltage, which can induce additional rippling around the perimeter if the graphene sheet is able to slide against the metallic contact. Variation in strain will lead to locally varying pseudomagnetic fields that will lead to TRS-breaking scattering within one cone (i.e. shorten ). According to Ref. Couto et al., 2014, non-uniform strain is the main contributor to both elastic scattering time and , and within a factor of 2-3. This scenario explains the observation of WAL also in our sample at low magnetic fields.
Using a similar line of arguments as above, Chern-Simon fluctuations governing will dominate both scattering rates and at at high fields. Typically, it is argued that weak localization effects cannot be observed in the presence of Chern-Simon field fluctuations. This conclusion derives from the infrared divergence in the density of states of low frequency excitations at small wave vectors Simon (1998). In our sample, however, there is a rather high infrared cutoff due to small geometrical size and low carrier density. By taking and , we obtain for the cut-off . Consequently, the Chern-Simon fields do not kill weak localization effects of composite fermions in our graphene sample, but rather just lower the dephasing time (and the ratios and ) which favors weak antilocalization behavior among graphene Dirac particles Tikhonenko et al. (2009).
We have investigated fractional quantum Hall states in suspended graphene Corbino disk around half filling. We find weak logarithmic temperature dependence of conductivity, the sign of which indicates weak antilocalization behavior of graphene composite fermions. These observations with nearly zero effective field acting on composite fermions can be understood making a parallel with graphene Dirac particles at small magnetic fields and assuming an enhanced infra-red cut-off for Chern-Simon fluctuations in a graphene sample of small carrier density and geometrical size.
We thank C. Flindt, A. Harju, Y. Meir, T. Ojanen, S. Paraoanu, and E. Sonin for fruitful discussions. This work has been supported in part by the EU Framework Programme (H2020 Graphene Flagship) and the European Research Council (grant no. 670743), and by the Academy of Finland (projects no. 250280 LTQ CoE and 286098).
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