Observation of an intrinsic bandgap and Landau level renormalization in graphene/boron-nitride heterostructures

Abstract

Van der Waals heterostructures formed by assembling different two-dimensional atomic crystals into stacks can lead to many new phenomena and device functionalities. In particular, graphene/boron-nitride heterostructures have emerged as a very promising system for band engineering of graphene. However, the intrinsic value and origin of the bandgap in such heterostructures remain unresolved. Here we report the observation of an intrinsic bandgap in epitaxial graphene/boron-nitride heterostructures with zero crystallographic alignment angle. Magneto-optical spectroscopy provides a direct probe of the Landau level transitions in this system and reveals a bandgap of ~38 meV (440 K). Moreover, the Landau level transitions are characterized by effective Fermi velocities with a critical dependence on specific transitions and magnetic field. These findings highlight the important role of many-body interactions in determining the fundamental properties of graphene heterostructures.

Introduction

Heterostructures consisting of two-dimensional (2D) layers of graphene, hexagonal boron-nitride (h-BN), MoS₂ and so on coupled by van der Waals interactions^1,2 exhibit many intriguing physical properties² and new device functionalities that are not achievable by individual constituting materials^3,4,5,6,7. In particular, graphene/h-BN heterostructures have shown great potentials for band structure engineering of graphene^{8,9,10,11,12,13,14,15,16,17,18,19,20} including inducing a bandgap^{11,13,15,16,17,18,19,20} (Fig. 1a), which is of great fundamental^21,22,23,24 and technological²⁵ interest. The coupling between graphene and h-BN results in a periodic moiré superlattice potential due to a 1.8% lattice mismatch⁸, which gives rise to superlattice minibands and new Dirac points near the edges of the superlattice Brillouin zone^{8,9,10,11,12,14}. Furthermore, the local sublattice symmetry of graphene is broken owing to different local potentials produced by boron and nitrogen atoms^17,18,19 (Fig. 1b), inducing a local bandgap^18,19. Although this effect varies spatially and is predicted to nearly disappear after spatial averaging¹⁸, transport studies showed signatures of a global bandgap in these heterostructures^11,13. It is suggested that many-body interactions may be responsible for the observed bandgap^15,16, but the issue remains unresolved experimentally.

Figure 1: Graphene/h-BN heterostructures.

(a) Energy spectrum of pristine graphene (left) and gapped graphene (right). (b) Schematic of the moire pattern in graphene on h-BN with zero crystallographic rotation angle and an exaggerated lattice mismatch of 11% (carbon, grey; boron, blue; and nitrogen, red). The lattice alignments in different regions lead to different local sublattice symmetry breaking in graphene. (c) Atomic force microscopy image of a monolayer graphene sample grown on h-BN and treated by hydrogen plasma etching, with bare BN shown in dark colour. The inset shows the observed moiré pattern with a periodicity of 15±1 nm. (d) Schematic of the magneto-optical measurements.

Here we report the observation of a finite bandgap in epitaxially grown graphene/h-BN heterostructures with zero crystallographic rotation angle¹² (Fig. 1c; see Methods section) by probing their Landau levels (LL) using magneto-optical spectroscopy. The inter-band (inter-LL) transition peaks measured by optical spectroscopy are determined by peaks in the joint density of states between two bands (LLs), which are not limited by disorder^26,27 such as impurities and defects, and therefore enable the measurement of the intrinsic bandgap. On the other hand, disorder may lead to a reduced mobility gap (or thermal activation gap) measured by other techniques compared with the intrinsic gap^26,27.

At zero magnetic field, the energy dispersion of graphene with a bandgap Δ is^11,28 , where v_F is the Fermi velocity and p is the momentum (Fig. 1a). In a magnetic field, B, the electronic spectrum of pristine graphene is quantized into LLs described by refs 11, 28:

where e is the elementary charge, ℏ is Planck’s constant divided by 2π, the integer n is LL index, and sgn(n) is the sign function. The LLs for gapped graphene have the form²⁸:

which features two zeroth LLs labelled as n=+0 and n=−0 with energies of E_±0 =±Δ/2. Here δ is the Kronecker delta function. Therefore, the bandgap of graphene can be explored by probing its LL energy spectrum.

Our study provides a direct spectroscopic determination of the bandgap in epitaxial graphene/h-BN heterostructures from optical measurements of LL transitions. We observe an intrinsic bandgap of ~38 meV (440 K) in this system, which is comparable to the gap value found in transport studies^11,13. Moreover, we find different values of effective Fermi velocity for different LL transitions, indicating LL renormalization by interaction effects. These findings have broad implications for the fundamental understanding of graphene heterostructures and their potential applications.

Results

Transmission spectra in magnetic field

Infrared (IR) transmission spectra T(B) were measured in magnetic field applied perpendicular to the samples as shown in Fig. 1d (see Methods section). Figure 2 depicts the T(B)/T(B₀) spectra for a representative sample, where B₀=0 T. Data for more samples are shown in Supplementary Fig. 1. Three dip features denoted as T₁, T₂ and T₃ are observed, all of which systematically shift to higher energies with increasing magnetic field. The zero-field transmission spectrum T(B₀) of either pristine or gapped graphene shows a step-like feature without any sharp resonances in the energy range explored here, so the observed dip features in the T(B)/T(B₀) spectra are corresponding to transmission minima in T(B) and thus absorption peaks in magnetic fields (Supplementary Fig. 2 and Supplementary Note 1).

Figure 2: Magneto-transmission ratio spectra of graphene/h-BN.

(a) Colour rendition of the −ln[T(B)/T(B₀)] spectra as a function of magnetic field and energy for sample 1, where B₀=0 T. (b) Schematic of LLs of gapped graphene. The arrows indicate transitions observed in this study. (c,d) Several representative T(B)/T(B₀) spectra for sample 1; dashed lines are guides to eyes. For clarity, the data in c and d are displaced from one another by 0.06 and 0.02, respectively.

The effective bulk mobility of our samples estimated from the widths of the resonances in the optical spectra is higher than 50,000 cm² V⁻¹ s⁻¹ (Supplementary Fig. 3 and Supplementary Note 2). Our optical data also indicate that the Fermi energy for our samples is in the range of E_F <19 meV (Supplementary Note 3).

Observed LL transitions

The energies (E) of all observed features in graphene/h-BN exhibit an approximate linear dependence on (or equivalently, E² has an approximate linear dependence on B) in our spectral range as shown in Fig. 3. However, they all show non-zero energy intercepts at zero magnetic field under linear extrapolations, in stark contrast to the LL transitions of pristine graphene described by equation (1), which converge to zero energy at zero field^29,30. Similar behaviours were observed in all five samples we have measured (Supplementary Fig. 1). Within the non-interacting single-particle picture, the finite zero-field extrapolation values of all observed absorption energies suggest that the LLs of our graphene/h-BN samples are described by equation (2). From the selection rule³¹ for allowed optical transitions from LL_n to LL_n’, Δn=|n|−|n′|=±1, and a quantitative comparison with equation (2), we assign feature T₁ to transitions of LL_-1→LL₊₀ and LL_-0→LL₊₁ (Fig. 2b), with an energy given by:

Figure 3: LL transition energies of graphene/h-BN.

(a) All observed transitions shown in a E²–B plot. Symbols: data for sample 1 and 2. Solid lines: best fits to the data for sample 1 using equations (3)–(5), , and parameters discussed in the text. Δ=38 meV and are used for the fit to T₃ transition shown here. (b) The low energy part of a to highlight the extraction of the gap. Dashed line: a guide for eye showing linear extrapolation of the data. The error bars in both panels, δ(E²) , are calculated as δ(E²) = 2Eδ(E) , where δ(E) is the uncertainty in determining the energy of each LL transition from the T(B)/T(B₀) spectra.

Fitting the T₁ feature based on equation (3) from a least squares fit yields a bandgap Δ≈38±4 meV and an effective Fermi velocity (Supplementary Table 1 and Supplementary Note 4). We emphasize that the bandgap explored here is at the main Dirac point of graphene instead of the secondary Dirac points at the edges of graphene/BN superlattice Brillouin zone^8,9,10,11,12. The T₁ transition energies in Fig. 2 are well below the energy (~200 meV) of the secondary Dirac points^9,10,11,12, so the LLs in this energy region are not significantly affected by the strong band structure modifications at the boundary of the superlattice Brillouin zone, which is supported by the observed linear dependence of the LL transition energy.

The observed T₂ and T₃ features have higher energies compared with T₁ features and can be assigned as (Fig. 2b): T₂, LL₋₂→LL₊₁ and LL₋₁→LL₊₂; T₃, LL₋₃→LL₊₂ and LL₋₂→LL₊₃. Their energies are described by:

The energies of T₂ transition show a deviation from linear dependence above 4 T (or 220 meV in energy) as shown in Fig. 3, with E² deviating from a linear B-dependence, which is perhaps due to the effect of moiré superlattice or many-body interactions. Therefore, we focus on the low field (<4 T) region where the T₂ transition exhibits an overall linear dependence, which most likely arises from the intrinsic behaviours of gapped graphene alone. We observed a splitting of the T₂ transition near 169 meV owing to the coupling to the IR active phonon of h-BN, which nonetheless does not affect the main conclusions of our analysis because this effect only occurs in a very narrow field and energy range. Based on equation (4), we find that the T₂ transition in low field is consistent with a bandgap similar to that extracted from the T₁ transition, Δ≈38±4 meV, and an effective Fermi velocity (Supplementary Fig. 4 and Supplementary Note 4). The T₃ transition is discussed in details below and in Supplementary Fig. 5 and Supplementary Note 4.

Comparison with pristine and gapped graphene

We stress that our data on graphene/h-BN cannot be explained by many-body effects of pristine graphene (Supplementary Fig. 6 and Supplementary Note 5). One prominent feature of interaction effects in pristine graphene is that the effective Fermi velocity varies for different LL transitions, so that the energy ratios $E_{{\text{T}}_{\text{2}}}/[(\sqrt{2}+1)E_{{\text{T}}_{\text{1}}}]$ and $E_{{\text{T}}_{\text{3}}}/[(\sqrt{3}+\sqrt{2})E_{{\text{T}}_{\text{1}}}]$ are higher than one, as demonstrated by previous IR studies²⁹ and our data on graphene on SiO₂. However, the data for graphene/h-BN exhibit an entirely different behaviour (Fig. 4a) compared with interaction effects in pristine graphene. Instead, the energy ratios of different LL transitions for graphene/h-BN are consistent with the behaviours of gapped graphene. A theoretical result of $E_{{\text{T}}_{\text{2}}}/[(\sqrt{2}+1)E_{{\text{T}}_{\text{1}}}]$ based on equations (3) and (4) is shown in Fig. 4a, with Δ≈38 meV, and , which agrees very well with the experimental results.

Figure 4: Many-body effects on LL transitions for pristine and gapped graphene.

(a) Energy ratios of different LL transitions for graphene on SiO₂ (blue colour) and graphene/h-BN sample 1 (red colour) shown in a common vertical scale. Open symbols: $E_{{\text{T}}_{\text{2}}}/[(\sqrt{2}+1)E_{{\text{T}}_{\text{1}}}]$. Solid symbols: $E_{{\text{T}}_{\text{3}}}/[(\sqrt{3}+\sqrt{2})E_{{\text{T}}_{\text{1}}}]$. Red solid line: theoretical result of $E_{{\text{T}}_{\text{2}}}/[(\sqrt{2}+1)E_{{\text{T}}_{\text{1}}}]$ based on equations (3) and (4) for gapped graphene. The ratios for graphene on SiO₂ are >1, which is a signature of interaction effects in pristine graphene (Supplementary Note 5). On the other hand, the ratios for graphene/h-BN exhibit an entirely different behaviour, which is consistent with gapped graphene. The error bars of energy ratios are calculated using standard formulas for propagation of uncertainty for division based on the uncertainty in determining the energy of each LL transition from the T(B)/T(B₀) spectra. (b) Fermi velocity ratios of different LL transitions for graphene/h-BN with a constant . For T₂ transition, a constant Fermi velocity is extracted from the data. These ratios are distinct from the value expected from the single-particle picture, which is indicative of interaction effects in gapped graphene. The error bars indicate the range of Fermi velocity values that could fit the data in Fig. 3 (Supplementary Note 4).

Discussion

Previous transport measurements on graphene/h-BN heterostructures indicated a gap of ~300 K at 0.4° crystallographic rotational angle (θ)¹¹. A recent study¹³ reported the existence of large domains of graphene with the same lattice constant as h-BN separated by domain walls with concentrated strain for small θ, and a gap of 360 K was found for θ=0°. Our optical study provides a direct spectroscopic determination of the bandgap with similar magnitude (~440 K) in epitaxial graphene/h-BN heterostructures. This bandgap value is larger than those found in theories within the single-particle picture^18,19,20, which suggests the relevance of many-body interactions in generating the gap^15,16. It was argued that the gap at the Dirac point is greatly enhanced by interaction effects due to coupling to a constant sublattice-asymmetric superlattice potential¹⁵, which is not affected by the spatial variations shown in Fig. 1b. The intrinsic gap value for graphene/h-BN obtained in our study provides a critical input for the basic understanding of the gap in this system.

Our study further reveals the crucial role of many-body interactions in renormalizing LL transitions³² in graphene/h-BN heterostructures. Specifically, the effective Fermi velocity associated with the LL transitions varies with particular transitions as well as the magnetic field. We find that the T₃ transition cannot be consistently fitted by equation (5) using a constant v_F, so we use an effective field-dependent parameter (B) to describe this transition (Supplementary Note 4). Figure 4b depicts the Fermi velocity ratios and , both of which are higher than one and therefore very different from the constant v_F for all transitions expected from single-particle pictures. Intriguingly, shows a systematic increase in low magnetic fields. For T₂ transitions (consider LL₋₁→LL₊₂, for example), it shows even at 1 T field with $E_{{\text{T}}_{\text{2}}}$=115 meV (Supplementary Fig. 1c), which corresponds to E_LL+2 ~66 meV and E_LL−1 ~49 meV. According to theoretical studies³³, the band structure of graphene/BN at such low energy scales are quite linear and not strongly modified by the superlattice Dirac points (~200 meV; refs 9, 10, 11, 12), so the value extracted from data at low magnetic field (thus low energy) is little affected by the superlattice Dirac points. Similar argument can be made for T₁ and T₃ transitions at low magnetic fields and low energy. Note that the LL transitions at high field and high energy (for instance, T₂ transition above 4 T field) may be affected by the band structure modification due to the superlattice Dirac points^8,33, but our discussions here are only focused on the low field regime shown in Fig. 4b. Our results in Fig. 4b indicate LL renormalization due to many-body interactions in magnetic field. Theoretical studies^34,35,36,37 showed that interaction effects of electron-hole excitations between LLs, such as direct Coulomb interactions between the excited electrons and holes and the exchange self-energy of electrons and holes between LLs, can significantly renormalize the inter-LL transition energy. The observation shown in Fig. 4b in gapped graphene is qualitatively similar to the results from many-body theories^34,35,36,37 as well as experimental studies²⁹ on pristine graphene, so our results strongly suggest contributions of many-body effects to the inter-LL transitions^34,35,36,37. Further theoretical investigations are required to quantitatively understand these interactions in gapped graphene, with many open questions yet to be addressed such as the role of superlattice potential¹⁵ and bond distortion²⁰ in graphene/h-BN heterostructures.

Multi-valley (band extrema in momentum space) Dirac systems such as gapped graphene, silicene and 2D transition metal dichalcogenides are described by the same Dirac Hamiltonian and share several essential properties such as valley-dependent orbital magnetic moment and Berry curvature^23,38, which are intimately related to their unconventional valley-dependent LL structures^38,39,40. In this context, the strong LL renormalization observed here has broad implications for fundamental studies of many novel phenomena related to LLs in these Dirac materials, such as magnetic control of valley degree of freedom³⁸ and valley-spin polarized magneto-optical response^39,40.

In summary, we have observed a bandgap of ~38 meV (440 K) in graphene/h-BN heterostructures with zero crystallographic rotation angle using magneto-optical spectroscopy. The intrinsic gap value reported here is important for fundamental understanding of the bandgap and many-body interaction effects in this system. Our demonstration of a finite bandgap in epitaxial graphene/h-BN heterostructures can also lead to novel applications in electronics and optoelectronics.

Methods

Sample preparation and characterization

h-BN was mechanically exfoliated onto double-side-polished SiO₂/Si substrates with 300 nm SiO₂. Graphene was epitaxially grown on h-BN by remote plasma enhanced chemical vapour deposition¹². Some multilayer grains can be found on monolayer graphene in as-grown samples, so hydrogen plasma etching technique¹² was applied to reduce these additional grains. The resulting sample is continuous monolayer graphene with minor etched hexagonal pitches in plane, as shown in Fig. 1c. The moiré pattern (Fig. 1c) due to lattice mismatch shows a periodicity of 15±1 nm as measured by atomic force microscopy, indicating zero crystallographic alignment angle between graphene and h-BN¹². This moiré pattern is observed over the entire areas of all samples, establishing these epitaxial samples as single-crystalline and single-domain graphene heterostructures. The samples studied in this work have typical lateral sizes of ~100 μm. The observed sharp LL transitions indicate that the optical absorption of our samples is little affected by defects or grain boundaries. The effective mobility of our samples estimated from the widths of the resonances in the optical spectra is higher than 50,000 cm² V⁻¹ s⁻¹ (Supplementary Fig. 3 and Supplementary Note 2). The absence of the LL₋₁→LL₋₀ and LL₊₀→LL₊₁ transitions in our optical data indicates that the Fermi energy is within the gap for our samples, namely E_F <19 meV (Supplementary Note 3).

Magneto-transmission measurements

The measurements were performed at ~4.5 K in a superconducting or resistive magnet in the Faraday geometry (magnetic field perpendicular to the sample surface). IR light from a Fourier transform spectrometer is delivered to the sample using a copper light pipe, and the light transmitted through the sample is detected by a composite Si bolometer. The focus of the IR light on the sample is ~0.5–1 mm. To reduce the stray light around our small samples, an aluminium aperture of ~200 μm diameter was placed around the sample. We report data at energies above 60 meV corresponding to wavelengths shorter than 20 μm, which ensures that the wavelength is significantly smaller than the sizes of the samples, and therefore a macroscopic description of the data using optical constants is applicable.