Quantum Hall effect in graphene: Interface makes a difference

Quantum Hall edge (QHE) states hold promises for constructing quantum excitations such as Majorana fermion. However, it often requires extreme conditions, holding it back from practical applications. We show an unusually robust QHE by coupling graphene to an interfacial long wavelength charge order.
Quantum Hall effect in graphene: Interface makes a difference
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Significance

Quantum Hall effect (QHE) is one of the very first topological electronic states found in history. It can be realized in a number of solid-state systems with the transversal conductance quantized by e2/h, with e and h being the elementary charge and the Planck constant, respectively [1–5]. This peculiar behaviour has been crucial for such as the implementation of quantum-based resistance standards with extremely high precision and reproducibility [6] and the construction of non-Abelian quantum states [7]. Among the few known systems manifesting QHE, graphene received special attention for its distinct band structure and the gate tunable nature.

It is known that QHE is often much influenced by the interplay between the host two-dimensional electron gases and the substrate, sometimes predicted to exhibit exotic topological states [8-9]. Yet the understanding of the underlying physics and the controllable engineering of this paradigm of interaction remain largely unexplored. Here, we demonstrate a hybrid system of graphene/CrOCl, in which an unusually robust QHE phase was observed thanks to the peculiar gate tunable interfacial charge coupling. 

Design and Approach

We dry-transferred graphene monolayer atop an anti-ferromagnetic insulator CrOCl equipped with dual gates, with the CrOCl and hBN as bottom and top dielectric, respectively (Fig. 1a). At low temperatures, the CrOCl is insulating enough, and a surface state emerges in CrOCl when subjected to a finite vertical electric field. At certain gate ranges, the bottom of surface band in CrOCl can be pulled down and matches the Fermi level of graphene, leading to charge transfer between graphene and the surface state in CrOCl via tunneling. 

Once the surface states are filled, the system becomes a three-capacitor model, instead of the two-capacitor one in conventional dual gated graphene devices. A simplified electro-static model of such a scenario is illustrated in Fig. 1b-c. As a consequence, dual gate mapping of the channel resistance of the system exhibits a largely distorted charge neutrality point (CNP) at 3 K, as shown in Fig. 1d. Notably, the bent CNP is much more resistive than the conventional CNP.

The above phenomena become even more pronounced when a finite perpendicular magnetic field is applied. The well-known ‘linear’ like stripes of Landau levels (LLs) are driven into a ‘cascade’ like pattern, separated by a clear phase boundary, as shown in Fig. 2. We call the conventional LLs phase as Phase-i, and the distorted cascade-like LLs phase as Phase-ii. 

Figure 1. (a) Optical image of a typical hBN/graphene/CrOCl device. (b)-(c) Electro-static model of the system, and (d) Dual-gate map of the channel resistance at 3 K.

Our theoretical considerations suggest that, due to e-e interactions, the electrons filled in the surface state (from the Cr-3d orbital, which is about 0.7 nm below graphene) can spontaneously form a long wavelength order, i.e., it undergoes a Wigner crystallization. Indeed, such surface state driven by vertical electrical fields is predicted to be a host of electronic crystals in many 2D systems, as calculated in a separate work [10]. Notice that the long wavelength charge order on 2D materials have been reported elsewhere via scanning tunneling microscopy [11].

We further considered the graphene subjected to such long wavelength electron charge super potential, using the continuum-model-based unrestricted Hartree-Fock method. A band reconstruction is revealed, which takes place once the system is driven into Phase-ii by gate voltages. And the Dirac cone of graphene becomes drastically sharpened, with a much enhanced Fermi velocity (Fig. 2). Our calculations also show that a gap opening is seen at the CNP in Phase-ii, which is in agreement with the experimental observations. 

Figure 2. (a) At certain vertical electrical fields, the bottom of surface band of CrOCl is pulled down to meet the Fermi level of graphene, and charge transfer start to take place via tunneling. (b) Dual-gate map of channel resistance of a typical device at B=5 T and T=1.5 K. A schematic picture of the band structure reconstruction in the conventional phase (Phase-i) and the strong interfacial coupling phase (Phase-ii) is illustrated.

Robust QHE Phase

Color maps at 3 K are shown in different planes in the B-Deff-ntot coordinates in Fig. 3a, with a line cut of Rxx and Rxy as a function of ntot, at a fixed  Deff=0.35 V/nm plotted in Fig. 3b. Here, Deff = (CtgVtg - CbgVbg)/2ϵ0, and ntot = (CtgVtg + CbgVbg)/e, with Ctg and Cbg being the top and bottom gate capacitances per area, respectively, and Vtg and Vbg are the top and bottom gate voltages, respectively. For simplicity, the exact numbers in all axes are omitted, with only ranges of them given in the graph. In such a data-cube presentation, all surprising experimental findings are rather straightforward to see, including: 1). the crossover from Fan- to Cascade-like Landau quantization; 2). the parabolic B-Deff relation in LLs; and 3). the very surprising zero field limit of the quantized plateau.

Figure 3. (a) Landau quantization of the current system in a parameter space of magnetic field, electrical field, and effective carrier doping. Data obtained at 3 K. (b) Line profile of longitudinal and transverse resistance at B= 14 T, and D= 0.35 V/nm.

It is noticed that in such an interfacial-coupling QHE phase (Phase-ii), the transverse resistance can be quantized in a very small magnetic field, as indicated in the right-most panel in Fig. 3a. And this behaviour can prevail up to liquid nitrogen temperature pretty robust compared to conventional QHE phases. For example, at T = 77 K, the graphene/CrOCl hybrid system can reach quantization of ν=±2 plateau in Phase-ii at as low as 350 mT. While the same quantization in conventional graphene at 77 K often requires a magnetic field over 10 T (Fig. 4). This is a leap forward, from liquid He temperature to liquid nitrogen temperature, for future constructions of quantum excitation states such as Majorana fermions (when the QHE is coupled to superconductivity) a route toward quantum computation. 

Figure 4. Among several recent systems, our work shows the lowest magnetic field (0.35 T) required for quantized edge states at 77 K.

The following video summarize the physical process of QHE phase engineered by the band structure reconstruction of graphene, which is driven by the enhanced e-e interaction from an interfacial long wavelength charge order:

Perspectives

Our findings suggest that the paradigm of charge transfer can play key roles in the engineering of quantum electronic states, when the e-e interactions are taking effects. To date, a number of effective band structure engineering have been manifested in 2D material, using the tuning knobs of high pressure, proximity effects, and twisted-moiré (sometimes with symmetry breaking that yield non-trivial topological states). The interfacial long wavelength charge order (presumably gate tunable) is yet the first time serving as a quantum superlattice to effectively drive the band reconstruction of the Dirac fermions. And, according to theory, such a quantum superlattice may be a universal phenomenon in many layered materials [10, 12], and enriched physical phenomena are yet to be discovered.

For more details, please see the original version of the manuscript in Nature Nanotechnology:

https://www.nature.com/articles/s41565-022-01248-4

References

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[2] von Klitzing, K. Quantum Hall effect: discovery and application. Annu. Rev. Condens. Matter Phys. 8, 13–30 (2017).

[3] von Klitzing, K. et al. 40 years of the quantum Hall effect. Nat. Rev. Phys 2, 397–401 (2020).

[4] Laughlin, R. B. Quantized Hall conductivity in two dimensions. Phys. Rev. B 23, 5632(R) (1981).

[5] Avron, J. E., Osadchy, D. & Seiler, R. A topological look at the quantum Hall effect. Phys. Today 56, 38 (2003).

[6] Ribeiro-Palau, R. et al. Quantum Hall resistance standard in graphene devices under relaxed experimental conditions. Nat. Nanotechnol. 10, 965–971 (2015).

[7] Michelsen, A. B. et al., Supercurrent-enabled Andreev reflection in a chiral quantum Hall edge state. arXiv. 2203.13384 (2022).

[8] Qiao, Z. et al. Quantum anomalous Hall effect in graphene proximity coupled to an antiferromagnetic insulator. Phys. Rev. Lett. 112, 116404 (2014).

[9] Zhang, J., Zhao, B., Yao, Y. & Yang, Z. Quantum anomalous Hall effect in graphene-based heterostructure. Sci. Rep. 5, 10629 (2015).

[10] Lu, X. et al. Synergistic interplay between Dirac fermions and long-wavelength order parameters in graphene-insulator heterostructures. arXiv:2206.05659 (2022)

[11] Wang, Z. et al. Imaging the quantum melting of Wigner crystal with electronic quasicrystal order. arXiv:2209.02191 (2022).

[12] Tseng, C. C. et al. Gate-tunable proximity effects in graphene on layered magnetic insulators. Nano Lett. https://doi.org/10.1021/acs.nanolett.2c02931 (2022).

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