Abstract
Topological nodal line semimetals, a novel quantum state of materials, possess topologically nontrivial valence and conduction bands that touch at a line near the Fermi level. The exotic band structure can lead to various novel properties, such as longrange Coulomb interaction and flat Landau levels. Recently, topological nodal lines have been observed in several bulk materials, such as PtSn_{4}, ZrSiS, TlTaSe_{2} and PbTaSe_{2}. However, in twodimensional materials, experimental research on nodal line fermions is still lacking. Here, we report the discovery of twodimensional Dirac nodal line fermions in monolayer Cu_{2}Si based on combined theoretical calculations and angleresolved photoemission spectroscopy measurements. The Dirac nodal lines in Cu_{2}Si form two concentric loops centred around the Γ point and are protected by mirror reflection symmetry. Our results establish Cu_{2}Si as a platform to study the novel physical properties in twodimensional Dirac materials and provide opportunities to realize highspeed lowdissipation devices.
Introduction
The discovery of topological insulators has ignited great research interest in the novel physical properties of topological materials in the past decade^{1, 2}. A characteristic feature of topological insulators is the existence of topologically nontrivial surface states that are protected by time reversal symmetry. Recently, tremendous research interest has moved from traditional topological insulators to topological semimetals which have vanishing densities of states at the Fermi level. The valence and conduction bands in topological semimetals can touch at either discrete points or extended lines, forming Dirac/Weyl semimetals^{3,4,5,6} and nodal line semimetals^{7,8,9,10,11,12,13,14,15,16,17,18}. The band degeneracy points or lines in topological semimetals are also protected by symmetries and are thus robust against external perturbations.
On the other hand, twodimensional materials have also attracted broad scientific interest because of their exotic properties and possible applications in highspeed nanodevices^{19}. Therefore, it is natural to ask the following question: do threedimensional topological semimetals have counterparts in twodimensional materials? The realization of twodimensional topological semimetals will provide new platforms for the design of novel quantum devices at the nanoscale. For Dirac semimetals, a twodimensional counterpart is graphene, when spin orbit coupling (SOC) is neglected^{4, 5}. Twodimensional nodal line fermions are predicated to exist in monolayer hexagonal lattices^{20}, honeycombkagome lattices^{21} and monolayer transition metalgroup VI compounds^{22}. The nodal lines in these materials are protected by (slide) mirror symmetry and require negligible SOC. However, the experimental realization of such structures in real materials is quite challenging; thus, it is necessary to search for new and realizable twodimensional materials that host robust nodal lines.
In this work, we study monolayer Cu_{2}Si, which is composed of a honeycomb Cu lattice and a triangular Si lattice. In the freestanding form, all Si and Cu atoms are coplanar^{23} and thus mirror reflection symmetry with respect to the xy plane (M_{z}) is naturally expected; this is important for the existence of twodimensional nodal lines. Importantly, the experimental synthesis of monolayer Cu_{2}Si is easy and has already been realized decades ago. One method to synthesize Cu_{2}Si is the direct growth of copper on Si(111)^{24,25,26}. However, monolayer Cu_{2}Si forms a quasiperiodic superstructure on Si(111) without precise long range periodicity. Alternatively, a commensurate 1 × 1 structure of monolayer Cu_{2}Si has been synthesized on a Cu(111) surface using chemical vapor deposition (CVD) methods^{27, 28}. As the 1 × 1 lattice of Cu_{2}Si approximately matches the \(\left( {\sqrt 3 \times \sqrt 3 } \right)\)R30° superlattice of Cu(111), large domains of ordered Cu_{2}Si can form on the Cu(111) surface. However, previous investigations of Cu_{2}Si have primarily concentrated on the structural and chemical properties, and a detailed investigation of its band structures is still lacking. Here, our comprehensive theoretical calculations show the existence of two Dirac nodal loops centred around the Γ point. The gapless nodal loops are protected by mirror reflection symmetry. These intriguing band structures have been directly observed by angleresolved photoemission spectroscopy (ARPES) measurements of Cu_{2}Si/Cu(111). Both nodal loops survive in the Cu_{2}Si/Cu(111) system because of the weak substrateoverlayer interaction.
Results
Firstprinciples calculations on freestanding Cu_{2}Si
The band structures of freestanding Cu_{2}Si without SOC are shown in Fig. 1b. Within 2 eV of the Fermi level, there are three bands: two holelike bands (labelled α and β) and one electronlike band (labelled γ). All three bands form closed contours on the Fermi surface: a hexagon, a hexagram, and a circle, respectively, as shown in Fig. 1e. Interestingly, we find that band γ crosses bands α and β linearly in all directions without opening of energy gaps, thus forming two concentric Dirac nodal loops centred around the Γ point (labelled NL1 and NL2). In Fig. 1f, we present the momentum distribution of gapless nodal points in the Brillouin zone, which shows a hexagon (NL1) and a hexagram (NL2), respectively.
It is natural to ask whether the two gapless Dirac nodal loops in Cu_{2}Si are symmetryprotected. To answer this question, we calculated the M_{z} parity of each band without SOC and find that the parity of band γ is opposite to that of bands α and β, as indicated by the plus and minus signs in Fig. 1b (see Supplementary Note 1 for orbital analysis). The opposing M_{z} parities indicate that band γ does not couple with bands α and β, and therefore both Dirac nodal loops remain gapless. This result provides strong evidence that the two gapless nodal loops are protected by mirror reflection symmetry.
After including SOC, each band is doubledegenerate and the two degenerate bands have opposing M_{z} parity. As a result, the mirror reflection symmetry cannot protect the nodal loops anymore. Along the preceding Dirac nodal loops, the bands with the same parity will couple with each other, resulting in the opening of band gaps. The calculated band structures with SOC clearly show that the nodal lines are fully gapped (Fig. 1c, g, h). However, the size of the SOC gap is quite small because of the weak intrinsic SOC strength in Cu_{2}Si. After artificially increasing the SOC strength, the size of the gap increases accordingly, as shown in Fig. 1d. These results show that mirror reflection symmetry only protects the Dirac nodal loops in the absence of SOC.
To further validate the role of mirror reflection symmetry in the protection of the gapless nodal loops, we introduce artificial perturbations to break the mirror reflection symmetry. The first method involves the introduction of buckling in the honeycomb Cu lattice while keeping the Si atoms unchanged, as schematically shown in Fig. 2a. This kind of buckling in a honeycomb lattice is similar to the intrinsic buckling in silicene and germanene, with neighbouring atoms buckled upwards and downwards, respectively^{29}. The second method to break the mirror reflection symmetry involves shifting the Si atoms downwards, as schematically shown in Fig. 2b. Our calculated band structures without SOC show that both nodal lines are gapped except the remaining gapless Dirac points along the ΓM and ΓK directions (Fig. 2c, d). These results confirm that the two gapless nodal loops in the absence of SOC are protected by the mirror reflection symmetry. The remaining gapless Dirac cones may be protected by other crystal symmetries, as discussed in the Supplementary Note 2.
ARPES measurements on Cu_{2}Si/Cu(111)
To directly confirm the intriguing nodal loop properties in Cu_{2}Si, we performed highresolution ARPES to measure its band structures. We synthesized monolayer Cu_{2}Si by directly evaporating atomic Si on Cu(111) in an ultrahigh vacuum. This sample preparation method is superior to the previously reported CVD methods^{27, 28}, as it can avoid exotic impurities introduced by the gases. The asprepared Cu_{2}Si sample is of high quality (see the Supplementary Note 3 for details) and is thus suitable for the highresolution ARPES measurements.
Monolayer Cu_{2}Si forms a \(\left( {\sqrt 3 \times \sqrt 3 } \right)\)R30° superstructure with respect to the Cu(111)1 × 1 lattice, in agreement with previous reports^{27, 28}. A schematic drawing of the Brillouin zones of Cu_{2}Si and Cu(111) is shown in Fig. 3a. In Fig. 3b–e, we show the evolution of constant energy contours (CECs) with binding energies. Several pockets centred at the Γ point can be seen: one hexagon, one hexagram, and one circle. Because of the matrix element effects, the ARPES intensities are anisotropic, resulting in a rhombic shape at the Fermi level. The hexagon, hexagram and circle become clearer at higher binding energies. These bands agree well with bands α, β, and γ from our calculations (Fig. 1e). With increasing binding energies, the sizes of the hexagon and hexagram increase, while the size of the circle decreases. In particular, the circle is larger than the hexagon and hexagram at the Fermi level and becomes smaller at binding energies higher than 1.0 eV, indicating the existence of two gapless or gapped nodal loops centred at the Γ point.
In Fig. 3f, we show the ARPES intensity plots measured along the ΓK direction. The band crossings, i.e., the nodal lines, are clearly observed at both sides of the Γ point (indicated by the black arrows), in agreement with the evolution of the CECs from Fig. 3b–e. The band crossing is located at deeper binding energy compared with the freestanding Cu_{2}Si (Fig. 1), which might originate from the electron doping of the metallic Cu(111) substrate. Furthermore, linear dispersion is observed near the nodal lines, in agreement with our theoretical calculation results. These results show that the quasiparticles in Cu_{2}Si are Dirac nodal line fermions. Because of the small separation of α and β bands along the ΓK direction, the two bands are not clearly resolved in Fig. 3f. According to our theoretical calculations, the two bands are well separated along the ΓM direction (Fig. 1). As ΓM is a mirror axis (M_{ σ }), one can unambiguously determine even/odd M_{ σ } parity of the bands based on the polarization dependence of the matrix element. The ARPES intensity plots measured along the ΓM direction are shown in Fig. 3g, h. We find that the α and γ bands are observed with p polarized light and the β band is observed with s polarized light, indicating that M_{ σ } parity of the α and γ bands is even, and that of the β band is odd. The result is fully consistent with our theoretical calculations. Besides the polarization of the incident photons, the spectral weight of the bands is also dependent on the photon energy. With 60eV photons, only the γ band is observable, while the spectral weight of the α and β bands is suppressed (Fig. 3k).
To further confirm the intriguing band structures in Cu_{2}Si, we show the ARPES intensity plots along a series of parallel momentum cuts in the ΓK direction, as indicated in Fig. 4a. From cut 1 to cut 10, band β first moves closer to band α until k _{ y } = 0 (cut 3), and then separates again. We note that cut 8 is measured along one edge of the hexagonal NL1, so one can see a relatively flat band near E _{B} = 1.0 eV (indicated by a black arrow). These results agree well with our calculated band structures and thus confirm the existence of two Dirac nodal loops in Cu_{2}Si.
Discussion
As Cu_{2}Si is only one atomic layer thick, all three bands are expected to have twodimensional characteristics, i.e., no k_{z} dispersion, which can be confirmed by ARPES measurements with different photon energies. In Fig. 3f, i, j, we present the ARPES intensity plots along the ΓK direction measured with three different photon energies, which shows no obvious change for all the three bands. The photon energyindependent behaviour indicates no k_{z} dispersion for the three bands, in agreement with their twodimensional characteristics. The intensity variance of these bands originates from the matrix element effects in the photoemission process. On the other hand, the bulk bands of Cu(111) are expected to be folded to the first Brillouin zone of Cu_{2}Si because of the Umklapp scattering of the \(\left( {\sqrt 3 \times \sqrt 3 } \right)\)R30° superlattice. However, we did not observe the folded bulk bands of Cu(111) within our experimental resolution. This result indicates a weak interaction between Cu_{2}Si and Cu(111) that results in the negligible intensity of the folded bands.
It should be noted that the mirror reflection symmetry in Cu_{2}Si is indeed broken when it is prepared on the Cu(111) surface. This is because one side of Cu_{2}Si is vacuum, while the other side is the Cu(111) substrate. The interaction with the substrate could open a band gap at the nodal lines because of the breaking of mirror reflection symmetry. However, we do not find a clear signature of gap opening in our ARPES measurements, which is another evidence for the weak interaction of Cu_{2}Si and the Cu(111) substrate. Indeed, our firstprinciples calculation results including the substrates show that the surface Cu_{2}Si layer remains approximately planar on Cu(111) (see Supplementary Note 4 for details), in agreement with previous theoretical and experimental results^{30, 31}. These results indicate that the mirror reflection symmetry is preserved to a large extent. As a result, no obvious gap opens at the Dirac nodal lines within our experimental resolution, in agreement with our calculated band structures including the substrates (Supplementary Fig. 5). As we have discussed in Fig. 1c, SOC can also lead to gap opening at the nodal lines, but the intrinsic SOC in Cu_{2}Si is too weak to induce detectable gaps.
Our theoretical and experimental results have unambiguously confirmed the existence of two concentric Dirac nodal loops in monolayer Cu_{2}Si. These nodal loops are protected by the mirror reflection symmetry and thus robust against symmetry conserving perturbations. These results not only extend the concept of Dirac nodal lines from three to twodimensional materials, but also provide a new platform to realize novel devices at the nanoscale. As copper has already been widely used for the preparation and subsequent transfer of graphene, we expect that monolayer Cu_{2}Si will be able to be transferred to insulating substrates, which is crucial for future transport measurements and device applications. On the other hand, although the breaking of mirror reflection symmetry will destroy the global Dirac nodal loops, our theoretical calculations have found the formation of gapless Dirac cones along the ΓM and ΓK directions (Fig. 2). This result suggests the possibility of tuning the Dirac states in Cu_{2}Si by controllable breaking of the mirror reflection symmetry, which might be realized by selecting appropriate substrates. Finally, we want to emphasize that the Dirac nodal lines in Cu_{2}Si are protected by the crystalline symmetry, instead of the band topology as with generic topological semimetals. The symmetry protected Dirac nodal lines can serve as a good platform for the topological phase transition among twodimensional Dirac nodal line, topological insulator, topological crystalline insulator and Chern insulator by inducing certain mass terms^{21, 22, 32}.
Methods
Sample preparation and ARPES measurements
The sample preparation and photoemission measurements were performed at BL9A (Proposal number 16AG005) and BL9B (Proposal Number 17AG011) of HiSOR at Hiroshima University and the VUVPhotoemission beamline of the Elettra synchrotron at Trieste. Singlecrystal Cu(111) was cleaned by repeated sputtering and annealing cycles. Monolayer Cu_{2}Si was prepared by directly evaporating Si on Cu(111) while keeping the substrate at approximately 500 K. The structure of Cu_{2}Si was confirmed by the appearance of sharp \(\left( {\sqrt 3 \times \sqrt 3 } \right)\)R30° lowenergy electron diffraction (LEED) patterns. The pressure during growth was less than 1 × 10^{−7} Pa. After preparation, the sample was directly transferred to the ARPES chamber without breaking the vacuum. During the ARPES measurements, the sample was kept at 40 K. The energy resolution was better than 20 meV. The pressure during measurements was below 2 × 10^{−9} Pa.
Firstprinciples calculations
Our firstprinciples calculations were carried out using VASP (Vienna abinitio simulation package)^{33} within the generalizedgradient approximation of the Perdew, Burke, and Ernzerhof^{34} exchangecorrelation potential. A cutoff energy of 450 eV and a kmesh of 27 × 27 × 1 were chosen for selfconsistentfield calculations. The lattice constant of 4.123 Å was obtained from the experimental value, and the thickness of vacuum was set to 18 Å, which is adequate to simulate twodimensional materials. The convergence criteria of total energy and the force of each atom were 0.001 eV and 0.01 eV Å^{−1}, respectively. The spinorbital effect was considered in part of our calculations.
Data availability
The data sets generated during and/or analyzed during the current study are available from the corresponding authors on reasonable request.
Additional information
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Acknowledgements
We thank Professor X. J. Zhou for providing the Igor macro to process the ARPES data. This work was supported by the Ministry of Education, Culture, Sports, Science and Technology of Japan (Photon and Quantum Basic Research Coordinated Development Program), the MOST of China (Grant Nos. 2014CB920903, 2016YFA0300600, 2013CBA01601, and 2016YFA0202300), the NSF of China (Grant Nos. 11574029, 11225418, 11674366, and 11674368), the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB07020100), and the US Department of Energy (DOE), Office of Science (OS), Office of Basic Energy Sciences, Division of Materials Science and Engineering (Grant No. DEFG0207ER46383).
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Authors
Contributions
B. Feng, L.C. and K.W. conceived the research. B. Feng, S.I., Y.F., S.W., S.K.M., P.S., P.M., M.A. and I.M. performed the ARPES measurements. B.Fu, S.K., C.C.L., O.S. and Y.Y. performed the theoretical calculations. P.C., K.W. and L.C. performed the STM experiments. All authors contributed to the discussion of the data. B. Feng wrote the manuscript with contributions from all authors.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Lan Chen or Yugui Yao or Iwao Matsuda.
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