Lung-I. Huangab,
Yanfei Yangac,
Randolph E. Elmquista,
Shun-Tsung Lo*d,
Fan-Hung Liud and
Chi-Te Liang*bde
aNational Institute of Standards and Technology (NIST), Gaithersburg, MD 20899, USA
bDepartment of Physics, National Taiwan University, Taipei 106, Taiwan. E-mail: ctliang@phys.ntu.edu.tw
cDepartment of Physics, Georgetown University, Washington, DC 20057, USA
dGraduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan. E-mail: shuntsunglo@mail.ncku.edu.tw
eGeballe Laboratory for Advanced Materials (GLAM), Stanford University, Stanford, CA 94305, USA
First published on 22nd July 2016
We report on magneto-transport measurements on low-density, large-area monolayer epitaxial graphene devices grown on SiC. We observe temperature (T)-independent crossing points in the longitudinal resistivity ρxx, which are signatures of the insulator-quantum Hall (I-QH) transition, in all three devices. Upon converting the raw data into longitudinal and Hall conductivities σxx and σxy, in the most disordered device, we observed a T-driven flow diagram approximated by the semi-circle law as well as the T-independent point in σxy near e2/h. We discuss our experimental results in the context of the evolution of the zero-energy Landau level at low magnetic fields B. We also compare the observed strongly insulating behaviour with metallic behaviour and the absence of the I-QH transition in graphene on SiO2 prepared by mechanical exfoliation.
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Although in most cases, transport in graphene on SiO2 prepared by mechanical exfoliation shows metallic behavior or a very weak T dependence,1,2 insulating behavior in the sense that the resistivity decreases with increasing T can appear in suspended graphene on SiO2,6 and in graphene on hexagonal boron nitride (h-BN) when sublattice symmetry is broken.7 It is also known that h-BN can substantially increase the mobility of graphene device and the induced sublattice symmetry breaking allows the observation of Zeeman spin degeneracy lifting of the LLs in the presence of a magnetic field.8,9 Interestingly, recent experiments show very low conductivity near the charge neutrality point for monolayer graphene on boron nitride with a suspended top gate7 and for monolayer epitaxial graphene (EG) with a point-like constriction caused by bilayer patches.10 Such important results on monolayer graphene suggest further studies are required and may be related to the possible splitting of the zeroth LL11 at low B. Moreover, insulating behavior and a temperature-independent point in the measured resistivity are observed in a disordered monolayer EG device.12 Here, we address the two aforementioned fundamental issues: the fate of the zero-energy LL at low fields and the insulating behavior in disordered graphene. In the most disordered EG device, we observe a well-defined T-independent point in the measured Hall conductivity σxy and the appearance of a semicircle relation in the T-driven flow diagram.13 Such results are in sharp contrast to the theoretical understanding of the zero-energy LL which is believed to be B-independent. Moreover, our data provide a thorough understanding of the low-field insulator-quantum Hall (I-QH) transition in disordered EG as well as the metallic-like behavior in graphene on SiO2.
Large-area EG devices are suitable for studies of QH transitions and insulating behavior since the long-range effects of increasing disorder may be hidden by local or size-dependent phenomena for small samples.16 Moreover, in EG grown on SiC,17,18 EF can be pinned to the localized states19 such that the ν = 2 QH plateau extends from a low field (∼1 T) to exceptionally high values (30 T),20 making EG an ideal system for studying an isolated low-field QH transition, although no such high-field transition has been reported. A possible reason for this is the reservoir model responsible for the long ν = 2 QH plateau19 so that one does not observe the high-field insulating state. Measurements on large-area (0.6 mm × 0.1 mm) devices were made in a perpendicular magnetic field up to 9 T in a variable-temperature cryostat using standard low-frequency lock-in techniques.
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Fig. 1 AFM images taken on (a) EG1, (b) EG2, and (c) EG3. Left: surface and right: phase measurements. |
The longitudinal and Hall resistivities (ρxx and ρxy) for EG1, EG2, and EG3 at various T are plotted in Fig. 2a–c. The low-T resistivity of EG1 is nearly two times lower than that of EG2. However, the mobility of EG1 is lower than that of EG2. Therefore it is not possible to tell whether the level of disorder in EG1 or in EG2 is higher. Nevertheless, since both the mobility and zero-field conductivity of EG3 is the lowest among the three devices, we believe that EG3 is the most disordered sample. We can immediately see the T-independent points in ρxx at crossing fields Bc in all three samples. For B < Bc, the device behaves as an insulator in the sense that ρxx decreases with increasing T.21 For B > Bc, the device shows QH-like behavior and ρxx increases with increasing T.21–24 Our results show characteristics of the insulator to ν = 2 QH transition observed in disordered 2D systems.20–22 Like other disordered 2D systems, localization and interaction effects are observed in our devices (see ESI†).
To further study the observed I-QH transition,12 we plot σxx and σxy for EG1, EG2 and EG3 in Fig. 3a–c. Interestingly, a clear T-independent crossing point in σxy develops near e2/h for EG2 and EG3. In the scaling theory of the QH effect, values of σxy that are half multiples of e2/h (per spin) behave as unstable points under renormalization.21 Therefore the observed crossing point near e2/h suggests a delocalization/localization process occurs when the zeroth LL passes upwards through EF when B is decreased.22
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Fig. 3 The directly converted conductivities, σxx and σxy, at different T for (a) EG1, (b) EG2, and (c) EG3. The vertical arrows indicate the temperature increase. The temperature points are the same as those given in the caption of Fig. 2 for each sample. |
A T-driven flow diagram in the (σxy, σxx) plane can be used to study the physics of localization processes in 2D systems.25,26 A field-induced transition involves a transition between two fixed points in this diagram, with a sudden increase and a similar decrease in σxx once the LL is emptied or filled. It has been experimentally verified that this transition traces out a semicircle25 in the (σxy, σxx) plane and for systems with a single conduction channel the semicircle represents a critical boundary for the QH state. The semicircle is centered at (0, e2/h) and follows (σxx)2 + (σxy − e2/h)2 = (e2/h)2, where the transition to the ν = 2 QH state occurs.
Fig. 4a and b show that samples EG1 and EG2 develop robust ν = 2 QH characteristics to the right of the semicircle (σxy > e2/h) at fields B ≈ 1 T, and approach the limiting point of the QH state at (2e2/h, 0). Conductivity data is given in Fig. 4 for all three EG samples with arrows showing T-driven flow superimposed at a series of fixed B. For a given sample, results at constant magnetic field strength that follow a vertical T-driven flow line corresponds to a critical field denoted by Bcσ identified as a crossing point of constant conductivity σxy. Similar curved arrows show how flow divides along the critical boundary of the QH state, shown by a dotted semicircle, starting at an unstable point indicated by a black dot. EG1 avoids the critical boundary with high conductivity (σxx ≈ 4e2/h) at low fields, and the vertical flow line occurs at σxy < e2/h. Vertical T-driven flow arrows in Fig. 4b and c show that the crossing magnetic field Bcσ occurs close to σxy = e2/h for both EG2 and EG3, while the magnitude of σxx decreases from σxx ≈ 2e2/h to σxx ≈ e2/h. Thus, we can characterize the T-driven flow for increasing disorder strength in our samples by vertical flow along σxy = e2/h, the line that points toward the center of the ν = 2 QH semicircle. Elsewhere the flow diverges from verticality especially near the semi-circle boundary, as clearly seen for sample EG3, where flow lines become nearly tangent to the semicircle.
Based on the floating up picture,27,28 Kivelson, Lee, and Zhang have proposed the global phase diagram (GPD) which describes possible phase transitions in a 2D system.29 When the spin degeneracy is considered, for a strongly disordered 2D system in which the spin-splitting is not well-resolved, the only I-QH transition is the 0–2 transition, where the numbers 0 and 2 correspond to the insulating phase and the ν = 2 QH state. This 0–2 transition and the 2–0 transition, from the QH state to the insulating regime, are equivalent within the GPD framework.29 The establishment of the semicircle relation for the 0–2 transition requires that the lowest extended band continuously floats up above EF with smaller B.22,23,27–29 Experimental evidence for the floating-up of the extended states in GaAs has been claimed.30 On the other hand, at low magnetic fields, extended states can float up and then merge in a Si 2D system.31 It was pointed out that chaotic potentials and possible oscillation of the boundary between the metallic and insulating phases32 can make the observation of the pronounced floating-up of the extended states not realizable.33
The semicircle-like flow lines obtained on EG3 appears to be in line with the levitation of the zeroth LL in disordered graphene, linking the observed insulating behavior in EG3 at low fields to the zeroth Landau band floating up above.30 However, without the ability to tune the carrier density to trace the crossing point in σxy in our case, we cannot confirm the floating-up of the extended states at low B. The semicircle law does not provide a good explanation for the transition in the cleaner devices EG1 and EG2. The possible origin is that their weak disorder prohibits the observation of the levitating Landau band. Moreover we found that the slope of σxy at Bcσ scales with temperature following T−κ with κ = 0.21 and 0.36 for EG2 and EG3, respectively (see Fig. S7 in the ESI†). At such low-field transitions, the Zeeman splitting plays a minor role, preserving the spin degeneracy. Therefore the increase in κ can be attributed to the breaking of sublattice symmetry in the presence of potential fluctuations, which may split the zeroth Landau band.34 With the strongest disorder in EG3, the semicircle relation between σxy and σxx becomes apparent, linking out results with possibly B-dependent zeroth Landau band due to disorder.
For EG3, the T-independent crossing point in ρxx occurs at the filling factor νc = nh/(eBc) of 0.6, which is in agreement with the recently reported value for the high-field levitation of the zeroth Landau band.35 However for EG1 and EG2, it corresponds to νc = 16 and νc = 7, respectively, which is much higher than that for EG3. These values deviate from the prediction of plateau-to-plateau transition between the ν = 6 and ν = 2 QH state, suggesting that the transition in weakly disordered EG1 and EG2 does not result from the N = 1 Landau band passing through the Fermi energy with magnetic field. In addition, at zero magnetic field, we have estimated the width Γ ≈ ħ/τ of Landau level broadening due to disorder. The results are 23 meV, 24 meV, 76 meV for EG1, EG2, and EG3, respectively. However the Fermi energy lies at EF = 49 meV, 35 meV, and 28 meV for EG1, EG2, and EG3. Interestingly, for EG3, EF is smaller than the estimated Γ. This finding infers a narrowing of the zeroth Landau band, which is robust against some sorts of disorder36 such that we can still observe the ν = 2 quantum Hall character in highly disordered EG3. It is worthwhile noting that the finite size effect and the charge transfer from the buffer layer/SiC interface (which partially determines the carrier density in a QH state) to the graphene sheet20 would modify the transitions. Moreover, since we have observed logarithmic temperature dependent Hall slope in all the devices due to interaction effects (see Fig. S4a in the ESI†), electron–electron interactions which are not considered within the global phase diagram may be regarded as perturbation/modification to the original floating-up picture.
It was shown that graphene-substrate-induced sublattice symmetry breaking coupled with charge disorder in epitaxial graphene layer can substantially modify the transport properties of graphene.37 We note that in graphene on h-BN, strongly insulating behavior solely due to graphene-substrate related sublattice symmetry breaking is observed.6 Interestingly, such an insulating phase makes a direct transition to the ν = 0 state at an extremely low field (B ∼ 0.1 T) without an intermediate transition to the ν = 2 QH state,5 in sharp contrast to our experiments. In our case, the mobility of EG3 is 20 times lower than that of the graphene on h-BN. The stronger disorder and the fact that our device is not exactly at the Dirac point should inhibit the formation of the ν = 0 state as supported by no sign of the ν = 0 plateau in σxy. Therefore although insulating behavior can be observed in both graphene on h-BN and disordered EG, we observe a transition from the insulating phase to the ν = 2 QH state as well as the semi-circle-like T-driven flow diagram, in line with floating up of the N = 0 electron LL due to stronger disorder compared with that of Amet et al.7 Our results, together with the pioneering work of Amet et al. suggest that sublattice symmetry breaking plays an important role in the observed insulating behavior in graphene subject to the environment effect. The strength of disorder, however, determines the allowable transition between the insulating state and the ν = 2 QH state or the ν = 0 state.
The unique B-dependent carrier density in epitaxial graphene grown on SiC, which can be ascribed to the reservoir model, has a pronounced effect on the QH transition. We would like to point out that though such an effect is believed to be responsible for the extremely long ν = 2 quantum Hall plateau, it should not significantly affect the low-field I-QH transition observed in our devices. The reason for this is that a good crossing point requires fixed carrier density in the system as previously observed in conventional semiconductor-based systems in which the carrier density is B-independent.21–23 Moreover, in all the theoretical studies on the I-QH transition, the carrier density is assumed to be constant, independent of both temperature and magnetic field.27–29 Therefore the reservoir model describing charge transfer between epitaxial graphene and the SiC substrate as a function of B (ref. 16) should not play an important role in the observed low-field I-QH transition in the work of Pallecchi et al.12 as well as in our devices.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra07859a |
This journal is © The Royal Society of Chemistry 2016 |