Abstract
Bloch electrons lacking inversion symmetry exhibit orbital magnetic moments owing to the rotation around their center of mass; this moment induces a valley splitting in a magnetic field. For the graphene/hBN moiré superlattice, inversion symmetry is broken by the hBN. The superlattice potential generates a series of Dirac points (DPs) and van Hove singularities (vHSs) within an experimentally accessible low energy state, providing a platform to study orbital moments with respect to band structure. In this work, theoretical calculations and magnetothermoelectric measurements are combined to reveal the emergence of an orbital magnetic moment at vHSs in graphene/hBN moiré superlattices. The thermoelectric signal for the vHS at the low energy side of the holeside secondary DP exhibited significant magnetic fieldinduced valley splitting with an effective gfactor of approximately 130; splitting for other vHSs was negligible. This was attributed to the emergence of an orbital magnetic moment at the second vHS at the holeside.
Introduction
Berry curvature and orbital magnetic moment are two pseudovectors that are observed in the presence of spatial inversion asymmetry. These enable us to control valley contrasting phenomena in twodimensional materials^{1,2,3}; therefore, they are essential in the field of valleytronics. The presence of the Berry curvature induces electrons to have anomalous velocity perpendicular to an applied electric field, thus exhibiting the valley Hall effect. The valley Hall effect has been demonstrated at the gapped Dirac band such as gapped bilayer graphene and monolayer graphene/hBN moiré superlattice^{4,5,6}. In contrast to this, an orbital magnetic moment will induce an energy shift under the application of the perpendicular magnetic field; this is known as valley Zeeman splitting and has been demonstrated in gapped bilayer graphene^{7,8,9} as well as transition metal dichalcogenides^{10}. Both Berry curvature and orbital magnetic moment, in principle, strongly depends on the local structure of the band. Here we choose graphene/hexagonalboron nitride (hBN) as a model system to investigate the orbital magnetic moment in relation to the band structure.
Results
Band structure and orbital magnetic moment in graphene/hBN
A pristine monolayer graphene is an inversion symmetric crystal (Fig. 1a). At the corner of the hexagonal Brillouin zone, different bands are connected at the Dirac point (DP) located at the K and K′point, and the energies between different valleys (K and K′) are degenerated. When graphene is transferred on a hBN insulator through van der Waals force, two significant changes occur in the band structure. First, the interaction between graphene and hBN breaks the spatial inversion symmetry of graphene. Second, the lattice mismatch between two crystals generates a moiré pattern (Fig. 1b). Owing to the moiré superlattice potential, the band structure of graphene is completely reconstructed^{11,12,13}, as shown in Fig. 1c, d, and calculated at the Kpoint using an effective continuum theory^{12}. Herein, to emphasize the effect of inversion asymmetry, two different calculations are performed: one that holds spatial inversion symmetry (inversion symmetric model shown in Fig. 1c) and one that uses an inversion asymmetric (Fig. 1d) model (see Supplementary Note 1). The inversion asymmetric model was demonstrated to comprehensively match the experimental results on band structure as well as Landau quantization^{11,12,13,14,15,16}. The plot of energy with respect to the density of states (DOS) derived from Fig. 1d is shown in Fig. 1e. Because of the moiré potential, series of subDPs, such as secondaryDPs (SDPs) or tertiaryDPs (TDPs), are generated at both low and highenergy sides of the main DP^{11,12,13,17,18,19}. These result in local minima in the DOS (Fig. 1e). Moreover, saddle points between the DPs, known as van Hove singularities (vHSs), are represented as local maxima. The most striking effect of inversion asymmetry indicated in Fig. 1c, d is the opening of the band gap at several points^{11,12,13}, which is indicated by the red and bluedashed circles. First, a band gap is opened at the main DP (i.e., the Kpoint in the first band) and both electron and holeside SDPs (i.e., the Xpoint; bluedashed circles in Fig. 1d). This gapped DP has been reported to exhibit a finite Berry curvature^{4,5,6}.
In this study, we observed that there was another point (i.e., a second vHS in the holeside located at the Ypoint (reddashed circle in Fig. 1d)) that exhibited a pronounced effect of inversion asymmetry; this point has not been discussed to date. In the symmetric model (Fig. 1c), this point is a point of contact between the holeside of the second and third bands. Inversion asymmetry induces a significant gap opening; as a result, this point becomes the vHS point in the inversion asymmetric model. This is a unique property of this particular vHS because the influence of inversion symmetry on other vHSs from the first to the third bands is limited. The opening of a gap with inversion asymmetry naturally results in the generation of an orbital magnetic moment m(k) at this vHS. This is an analog of a spin angular moment of the electron. However, m(k) is orbital in nature, and it is responsible for the splitting between the K and K′ valley instead of the up and downspin. A general form of the m(k) is expressed as the following equation^{2,3,9}:
where \(\left. {\left {u\left( {\mathbf{k}} \right)} \right.} \right\rangle\) is the periodic part of the Bloch function, H(k) is the Bloch Hamiltonian, and ε(k) is the dispersion of band. We calculated m(k) for the band structure in Fig. 1d and obtained m(k) with respect to the wavevector of the first, second, and third bands in the holeside of the main DP, as depicted in Fig. 1f–h, respectively. m(k) is large for the gapped DPs, including both the main DP and SDPs^{3}. Moreover, the calculation shows that there is a noticeably large m(k) presented at the second vHS(h; i.e., the Ypoint in the second band). The second vHS(h) yields an m(k) value of 66 μ_{B}, corresponding to a valley gfactor of ~130. This value is noticeably large as an orbital magnetic moment of vHS, considering that m(k) is negligibly small at other vHSs such as Ypoint in the first band (see Supplementary Note 2 for the detail comparison).
Allelectrical magnetothermoelectric measurement
To verify the existence of orbital moment at the vHS, we performed magnetothermoelectric measurement, as schematically illustrated in Fig. 2a; the optical micrograph of the device is shown in Fig. 2b. Two different flakes of graphene encapsulated with hBN were placed on the SiO_{2}/dopedSi substrate. We created trench in the hBN to reduce capacitive coupling between the two graphene flakes (see Methods section and Supplementary Note 3). The graphene depicted on the right has a graphite gate (GrG), which can apply a local backgate voltage of V_{R} to control the carrier density n_{R}. The carrier density of the graphene on the left n_{L} is controlled using the backgate voltage V_{L}. The lattice of the graphene on the left is aligned with the underlying hBN substrate, to ensure a large period of moiré potential. In contrast, the period of the graphene on the right is small because of the significant rotational misalignment between graphene and hBN. As illustrated in Fig. 2c, the nonaligned graphene is used as a local heater by applying current to induce Joule heating. The heat radiated from the nonaligned graphene induces heating of the aligned graphene, thereby creating a temperature gradient perpendicular to the channel of the aligned graphene. In the presence of this temperature gradient and the magnetic field perpendicular to the plane B, thermoelectric voltage V_{ind} is generated along the channel direction of the aligned graphene, due to the Nernst effect^{20,21,22,23,24}. This voltage can be detected using the metal electrode connected at both ends of the graphene. The thermopower is significantly enhanced at the vHS as it is demonstrated by photoNernst measurement^{25}. Therefore, our allelectrical magnetothermoelectric measurement provides sensitive probes for vHSs.
The twoterminal resistance R of the aligned graphene is measured under a sweep of n_{L} at 2.0 K; the results are plotted in Fig. 2d. In addition to the resistance peak caused by the main DP at n_{L} = 0, other peaks from the SDP at the electron and the holeside at n_{L} = +2.20 and −2.25 × 10^{12} cm^{−2}, respectively, are visible. Although its origin is not clear at this moment, there is a slope background in R that increases towards the left in the figure; we infer this is related to the inhomogeneity of metal/graphene contact. Based on these n_{L} values, the period of moiré potential λ is determined using the relationship \(n_{{\mathrm{SDP}}} = 8/\left( {\sqrt 3 \lambda ^2} \right)\) where n_{SDP} represents the carrier density at the resistance peak position of the SDP^{14,15,16}. Using the obtained λ value of ~14 nm, we determined the lattice alignment angle between graphene and hBN as θ ~ 0°^{17}. During thermoelectric measurement, a constant power P = 1 mW is applied to the nonaligned graphene using a source meter, while maintaining its carrier density at charge neutrality n_{R} = 0 at B = 0 and quantum Hall filling factor ν = −6 such that n_{R} = 6eB/h in the magnetic field, where e depicts electron charge and h the Planck constant. This maintain the twoterminal resistance of heater graphene to be kept ~10 kΩ throughout the measurement from no magnetic field to high magnetic field (see Supplementary Note 4). Under this condition, the increase in the average temperature of the aligned graphene is estimated to be ~10 K. V_{ind} as a function of n_{L} is measured at different magnetic fields B, and the results are presented in Fig. 2e by the solid black lines. For B = 0 T, the V_{ind} signal is negligibly small. The small signal around n_{L} = +2.20 and −2.25 × 10^{12} cm^{−2} at B = 0 T can be attributed to a Seebeck effect of SDP, originating from an asymmetry of the contacts. On increasing B, initially, the V_{ind} signals that appeared at the main DP and the SDP had similar amplitudes. Thereafter, the V_{ind} signal at the DPs exhibits a peak in positive B and a dip in negative B. For comparison, the V_{ind} signal obtained from nonaligned graphene/hBN device is provided in Supplementary Note 5. In addition, a larger signal appears at a position away from the SDPs, as indicated by the arrows in Fig. 2e. Evidently, the signs of the values of V_{ind} at these locations differ from those of the DPs, such that the signal from the three DPs exhibit dips in negative B values, whereas the signals indicated by the arrows exhibit peaks. These results are compared to the calculated DOS derived from Fig. 1e plotted versus carrier density n, as depicted in Fig. 2f. Here, the carrier density was calculated by integrating DOS such that \(n\left( {E_F} \right) = {\int}_0^{E_F} {{\mathrm{DOS}}\left( E \right)dE}\) The unit of this axis is normalized using the total number of electron states divided by the area of a filled Bloch band n_{0}. Therefore, n/n_{0} = ±4 and 8 correspond to the SDP and TDP, respectively. A comparison of Fig. 2d–f indicates that additional large signals of V_{ind} are observed at three vHSs, namely the first vHS(e), first vHS(h), and second vHS(h). The first vHSs on both the electron and holeside had previously been investigated^{25}; here, we investigated the second vHS(h) which was observed^{26,27} but not discussed before. In addition to these, it is noteworthy that the V_{ind} signal largely increases around n_{L} = −4.65 × 10^{12} cm^{−2} in Fig. 2e due to the presence of TDP, the location of which can be seen from Fig. 2f; however, a detail investigation of this TDP is difficult on this device since the demonstration of carrier density higher than that of Fig. 2e is limited by the breakdown of SiO_{2} dielectric.
The V_{ind} versus n_{L} are measured in a magnetic field from B = + 0 to 2.23 T and results are shown in Fig. 2g. The magnetothermopower from the second vHS(h) and first vHS(h) exhibits a considerably different behavior. Their positions are depicted by a solid magenta circle and a blue circle, respectively, in Fig. 2g. The dipshape signal generated from the second vHS(h) exhibits significant splitting with increasing B (see also refs. ^{26,27}), whereas the V_{ind} signals from other vHSs, i.e., the first vHS(h) and the first vHS(e) (see Fig. 3a, b), did not exhibit any evident splitting with increasing B.
Valley splitting of the thermoelectric voltage at vHS
To investigate the origin of the magnetic fieldinduced splitting in V_{ind} at the second vHS(h), we present a comparison between the experimental values and the calculated DOS in aligned graphene/hBN obtained from the continuum model^{12}. In Fig. 3a, we depict the plot of V_{ind} as a function of n_{L} and B plotted for +B and −B. The signals arising from three vHSs—the first vHS(e), first vHS(h), and second vHS(h)—are extracted from the data, and plotted with green, blue, and magenta solid circles, respectively in Fig. 3b (details of the extraction procedure as well as the V_{ind} data in high magnetic field region are presented in Supplementary Note 6). We present a calculated DOS as a function of n/n_{0} and B in Fig. 3c–e. The calculated DOS as a function of energy E is also provided in Supplementary Note 7. Here, we plotted the total DOS, including the DOS for both K and K′ valleys (Fig. 3c), DOS for K valley (Fig. 3d), and DOS for K′ valley (Fig. 3e), separately. A good coincidence was observed between Fig. 3a, c; the Binduced splitting of vHSs as well as the Landau fan diagram of the DPs were completely reproduced by the calculation. By plotting the K and K′valley contributions to the DOS separately in Fig. 3d, e, we observed that the DOS peak at the second vHS(h) for the K(K′) valley changed to the larger (smaller) n/n_{0} value with the magnetic field. Furthermore, we plotted individual traces of Fig. 3d, e in Fig. 3f, for a comparison with the experimental data of V_{ind} shown in Fig. 3g (The data plotted in Fig. 3g is same data as Fig. 2g). These data also exhibited good agreement with each other. Therefore, we conclude that the magnetic field induced splitting of the second vHS(h) indicated by our results is caused by the large magnetic fieldinduced valley splitting.
Discussion
The significant valley splitting corresponding to a valley gfactor of ~130 extracted from Fig. 1g indicates the emergence of a large m(k) at the second vHS(h). To obtain a finite m(k) from Eq. 1, the matrix element for the inter subband coupling should be nonzero^{28} (see Supplementary Note 8 for a detailed discussion). This can only be achieved at a point of contact between the bands (i.e., DP) in the graphene/hBN under the inversion symmetric model. As seen from the comparison between Fig. 1c, d, the second vHS(h) in Fig. 1d is a point of contact between holeside second and third band at Ypoint under inversion symmetry; thus, it is a subDP. This subDP is gapped under the introduction of inversion asymmetry and becomes a vHS point. Therefore, the inter subband coupling at this vHS point is still nonzero, even after the opening of the gap. This hidden DP nature of the Ypoint at the holeside second and third band demonstrates the emergence of an orbital magnetic moment at the second vHS(h) in the graphene/hBN moiré superlattice. Our allelectrical magnetothermoelectric measurements unveil this unique band property of graphene/hBN moiré superlattice. Since there is a growing interest in twisted bilayer or twisted multilayer graphene^{29,30,31,32,33,34,35,36}, this method may also provide a sensitive probe to detect orbital magnetic moment with respect to different band structures of such new material systems.
Methods
Sample fabrication
A device was fabricated using sequential dry release transfer of an individual flake. Two different flakes of monolayer graphene were successively transferred on top of the hBN/graphite structure on the SiO_{2}/dopedSi substrate. Finally, both flakes of graphene were encapsulated with another hBN. The flake transfer was performed using a poly(propylene) carbonate (PPC)based dry release transfer method^{37,38}, and the thickness of hBN was chosen to be ~30–40 nm. Using ebeam (EB) lithography and reactive ion etching employing a mixture of CF_{4} and Ar gases, a trench was fabricated in the hBN located between the two graphene flakes. Thus, the connection through hBN between the two graphene flakes was eliminated. Finally, a contact pattern was created using an EB lithography with PMMA resist, and a 90nmthick Au/5nmthick Cr stack was deposited using EB evaporation. It should be noted that during device fabrication, the device was annealed at 350 °C under an Ar/3% H_{2} atmosphere after each flake transfer. This annealing caused the transferred graphene flakes to rotate within the plane toward the preferred orientation with respect to the adjacent hBN^{39,40}. In the device presented in Fig. 2a, b, the graphene on the left is almost perfectly aligned with hBN, to generate moiré superlattice potential; however, the graphene on the right was not aligned.
Experimental setup
A liquid Hecooled variable temperature cryostat equipped with a superconducting magnet was used for transport measurement. The differential resistance of graphene was measured by applying an AC current I_{ac} = 10 nA, with a frequency of 18 Hz; the AC voltage was measured using a lockin amplifier. A source measure unit (Keithley 2400) was used to apply constant power to the heater graphene.
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was supported by CREST, Japan Science and Technology Agency (JST) (grant number JPMJCR15F3), and JSPS KAKENHI (grant numbers JP19H02542, JP19H01820, JP20H00127, JP20H00354, and JP20H01840). J.A.C. and P.M. acknowledge the support of NYU Shanghai (StartUp Funds), NYUECNU Institute of Physics at NYU Shanghai, and New York University Global Seed Grants for Collaborative Research. J.A.C. acknowledges the support from the National Science Foundation of China (NSFC; Grant No. 11750110420). P.M. acknowledges the support from the Science and Technology Commission of Shanghai Municipality (STCSM; Grant No. 19ZR1436400).
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R.M. and T.M. conceived the experiment. K.K. fabricated the device. M.K. developed the theoretical aspects. K.K. and R.M. performed the measurements with the help of S.M. J.A.C. and P.M. performed the theoretical calculation and analyzed the results. T.T. and K.W. grew the hBN crystals. R.M. and T.M. wrote the paper with input from all authors. All authors contributed to extensive discussions of the results.
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Moriya, R., Kinoshita, K., Crosse, J.A. et al. Emergence of orbital angular moment at van Hove singularity in graphene/hBN moiré superlattice. Nat Commun 11, 5380 (2020). https://doi.org/10.1038/s4146702019043x
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DOI: https://doi.org/10.1038/s4146702019043x
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