Abstract
Topological superconductors are an intriguing and elusive quantum phase, characterized by topologically protected gapless surface/edge states residing in a bulk superconducting gap, which hosts Majorana fermions. Unfortunately, all currently known topological superconductors have a very low transition temperature, limiting experimental measurements of Majorana fermions. Here we discover the existence of a topological Dirac–nodalline state in a wellknown conventional hightemperature superconductor, MgB_{2}. Firstprinciples calculations show that the Dirac–nodalline structure exhibits a unique onedimensional dispersive Dirac–nodal line, protected by both spatialinversion and timereversal symmetry, which connects the electron and hole Dirac states. Most importantly, we show that the topological superconducting phase can be realized with a conventional swave superconducting gap, evidenced by the topological edge mode of the MgB_{2} thin films showing chiral edge states. Our discovery may enable the experimental measurement of Majorana fermions at high temperature.
Introduction
Superconducting and topological states are among the most fascinating quantum phenomena in nature. The entanglement of these two states in a solid material into a topological superconducting state will give rise to even more exotic quantum phenomena, such as Majorana fermions. Recently, much effort has been devoted to searching for topological superconductors (TSCs). The first way to realize a TSC phase is by the proximity effect via formation of a heterojunction between a topological material and a superconductor (SC).^{1,2,3} Cooper pairs can tunnel into a topological surface state (TSS), forming a localized state that hosts Majorana bound states at magnetic vortices,^{1,2,3} or into a spinpolarized TSS, leading to halfinteger quantized conductance.^{4} Second, TSCs can be made by realizing superconductivity in a topological material^{5,6,7,8,9,10,11} or conversely by identifying the topological phase in a superconductor.^{12,13,14,15} Broadly speaking, it is preferred to work with one single material, because interfacing two materials may suffer from the interface reaction and lattice mismatch between a SC and a topological material. Regardless of which approach to create a TSC, however, a common challenge is that all the known TSCs to date have a very low transition temperature. For example, up to now, a few superconducting/topological heterostructures are realized with low critical temperature (T_{c}) ~ 4 K.^{3} The superconducting transition temperature induced by doping and/or pressurizing a few topological insulators has a T_{c} in the range of 4 ~ 9 K,^{5,6,7,8,9,10,11} while the T_{c} for some materials where topological and superconducting phases coexist is <7 K.^{12,13,14} Hightemperature twodimensional (2D) SC FeSe on a SrTiO_{3} substrate has recently been shown to also host a 2D topological insulating phase with hole doping, whereas its superconducting phases require electron doping.^{15}
Here, we discover the existence of a topological phase in a conventional SC of MgB_{2} with a T_{c} of ~40 K, the highest transition temperature known for a bulk BCS SC. Based on firstprinciples calculations, we demonstrate a topological Dirac nodal line (DNL)^{16,17,18} structure in MgB_{2}, exhibiting a unique combination of topological and superconducting properties. The characteristic 1Ddispersive DNL is shown to be protected by both spatialinversion and timereversal symmetry, which connects the electron and hole Dirac states. A topological surface band of the (010) surface of MgB_{2} shows a highly anisotropic band dispersion, crossing the Fermi level within the superconducting gap. The essential physics of the DNL structure in MgB_{2} is further analyzed by effective tightbinding (TB) models, and the effects of superconducting transition on thin films are also studied. Most importantly, Majorana edge mode, having protected chiral bands crossing zero energy, can be realized.
Results and discussion
MgB_{2} has the AlB_{2}type centrosymmetric crystal structure with the space group P6/mmm (191). As shown in Fig. 1a, it is a layered structure with alternating closepacked trigonal layers of Mg, while B layers form a primitive honeycomb lattice like graphene. The optimized lattice constants are a = 3.074 Å and c = 3.518 Å, which agree well with the experimental^{19,20} and other theoretical results.^{21,22} The first Brillouin zone (BZ) of bulk MgB_{2} and the projected surface BZ of the (010) plane are shown in Fig. 1b.
To reveal the electronic and topological properties of MgB_{2}, its electronic band structure is calculated with or without spin–orbit coupling (SOC) (see Supplementary Information). Figure 1c, d shows the dominant bonding character of the B sheet and the bulk band structure of MgB_{2} with SOC, respectively. The band structure exhibits linear dispersions at both K and H point. The resulting band structure can be easily understood in terms of the B sublattice. The bonding character of the bands in Fig. 1d is indicated by a color gradient. Dirac bands are derived from B p_{z} states (πbonding) like the graphene Dirac state and other three bands below Dirac bands are from B p_{x,y} states (σbonding).^{21} Mg s states are pushed up by the B p_{z} orbitals and donate fully their electrons to the Bderived conduction bands. All bands from B s and p orbitals are highly dispersive, the bands from σbonding are more localized in a 2D B sheet (no dispersion along the z direction), while the Dirac bands are quite isotropic with high dispersion along the z direction. Substantial k_{z} dispersion of the p_{z} bands produces a Fermi surface that is approximately mirror reflected with respect to a plane between the k_{z} = 0 [in units of 2π/c] and k_{z} = 0.5 planes, with one pocket (electonlike) coming from the antibonding and the other (holelike) from the bonding p_{z} band.
Remarkably, the Dirac bands are also dispersive along the K–H directions, so that we have a dispersive Dirac–nodalline structure (Fig. 1e, f). Figure 1e shows the band structure along the highsymmetry points that are located in the same plane as k_{z} is varied. At k_{z} = 0, there is a holedoped Dirac band with a Diracpoint energy of 1.87 eV, while at k_{z} = 0.5, there is an electrondoped Dirac band with a Diracpoint energy of −1.91 eV. As k_{z} increases, the holedoped Dirac state changes to the electrondoped Dirac state continuously, and the critical point is located at k_{zc} = 0.218, where the carrier changes sign. Usually, SOC may open up gaps at the band crossing points; inversion and/or timereversal symmetry is insufficient to protect the band crossings. But additional symmetry, like nonsymmorphic symmetry, can protect nodal points or lines. For MgB_{2}, the Dirac nodal line is protected by crystal symmetry and SOC may introduce a small gap, leading to a topological insulating phase. However, the SOC strength of B is negligibly small (~μeV), even weaker than C and N. Consequently, the topological nodal line of MgB_{2} survives by a combination of inversion, timereversal, and crystal symmetries, along with a weak SOC of B. We note that MgB_{2} also has a nodal chain structure^{23} among the conduction or valence bands (see Supplementary Information), but these states are located far from the Fermi level.
We define two independent topological Z_{2} indices denoted by ζ_{1} and ζ_{2},^{24} one on a closed loop wrapping around the nodal line and the other on a cylinder enclosing the whole line, respectively (Fig. S2). The expression of ζ_{1} is described by the Berry phase, \(\phi = {\oint}_C d\vec k \cdot \vec A(\vec k)\), as a line integral along a closed path
where \(\vec A(\vec k)\) is the Berry connection, \(\vec A(\vec k) = i\langle n\nabla n\rangle\). The simplest Hamiltonian near the K(K′) point for the Dirac nodal line of MgB_{2} is
where σ_{i} are Pauli matrices. For any path in the (k_{x}, k_{y})plane that goes around the Dirac line, the index ζ_{1} is 1, which means the Dirac lines are stable against perturbations. For the topological invariant of the Dirac nodal line, one can calculate the index ζ_{2} using the flow of Wannier charge centers on a set of loops covering the enclosing manifold.^{25,26} For the DNL in MgB_{2}, the index ζ_{2} is 0. But the nodal lines cannot shrink to a point, instead, they appear in pairs and can only be annihilated in pairs.^{27} Furthermore, the parities of energy states can be used to assign the Z_{2} topological invariants in topological Dirac nodal line semimetals. We found that MgB_{2} is characterized by weak Z_{2} indices^{28} (see Table S1).
To further reveal the topological nature of the Dirac–nodalline–semimetal (DNLS) state in MgB_{2}, we also calculate the surface states. The existence of TSSs and Fermi arcs is one of the most important signatures of DNLS. Figure 2 shows the calculated surfacestate spectrum of Mgterminated (010) MgB_{2} surface using an iterative Green’s function method. We calculated the band dispersions perpendicular to the \(\tilde \Gamma  {\tilde{\mathrm Z}}\) direction with four representative cuts at different k_{sy} values in the surface BZ, whose momentum locations are indicated in Fig. 1b. As the bulk BZ is projected onto the (010) surface BZ, one can expect that the nodal line is located at k_{sx} = 2/3 along the symmetry line between \(\tilde \Gamma \left( {{\tilde{\mathrm Z}}} \right)\) and \({\tilde{\mathrm X}}\left( { {\tilde{\mathrm U}}} \right)\). The TSSs of DNLS connect two gapless Dirac points, which are the surface projections of the nodal points in the nodal line (Fig. 2a). Interestingly, as k_{sy} values increase, the type of bulk Dirac band is changed from the holedoped to electrondoped. Accordingly, the TSSs connecting two Dirac points are also changed from hole to electron type. Moreover, these TSSs are quite anisotropic: the surface band dispersion is almost flat along the k_{sx} direction, but highly dispersive along the k_{sy} direction (Fig. 2a–c). The Bterminated surface states are shown in Fig. S3.
We now discuss the Fermi surface. The evolution of the Fermi surface is also obtained from the Green’s function method for different values of Fermi energy (E_{F}) (Fig. 2d). The Fermi surface is composed of one Fermi arc, touching at two singularity points, where the surface projections of bulk Dirac points locate. The touching points which are indicated by red dots are also the jointed points between hole (h) and electron (e) pockets. As E_{F} increases, the touching point is shifted from k_{sy} = 0.5 [in units of 2π/c] to k_{sy} = 0. Notably, we can observe the Fermi arc as long as E_{F} is between −1.91 eV and 1.87 eV. Such a large energy window for observing the Fermi arc is a unique and a useful feature of dispersive DNL in MgB_{2}. Interestingly, beyond the electronic DNL state in MgB_{2}, phononic Weyl nodal lines and their nontrivial phononic arc states were also predicted in MgB_{2}.^{29}
The essential physics of the DNL structure in MgB_{2} is further characterized by an effective TB model using B p_{z} orbital (see Supplementary Information), which gives a quite good description with densityfunctional theory (DFT) calculations (Fig. 3a). Given the DNL structure of MgB_{2}, interesting features of quantum oscillation (the Shubnikov–de Hass or the de Haas–van Alphen effect) can be observed, to manifest the predicted nontrivial Berry phase.^{17} In graphene, such nonzero Berry phase has been confirmed.^{30,31} Similarly, the effective model of MgB_{2} exhibits a pseudospin vortex texture arising from the banddegeneracy point (Fig. 3b). We note that the pseudospin is actually independent of k_{z}. One can clearly see the two inequivalent BZ corners (K and K′) having different topologies and being characterized with opposite Berry phases. To measure the nontrivial Berry phase via Landaufantype analysis, the electron orbits should enclose the vortex points in a magnetic field. Figure 3c shows the calculated Fermi surface at different energy levels. For an undoped system, the electron and hole extremum orbits in a magnetic field along the z direction at E_{F} enclose the Γ point, giving rise to a zero Berry phase. However, for a doped system, the Fermi surface topology changes for different doping levels. At E_{F} = +1 eV (E_{F} = −1.2 eV), the hole (electron) extremum orbit encloses the vortex point to enable a nonzero Berry phase (Fig. 3b, c). There are three different (energy) ranges of doping characterized with different Berry phases. If E_{F} locates at 0.29 eV < E_{F} < 3.25 eV (−3.48 eV < E_{F} < −1.17 eV), the hole (electron) extremum orbit encloses the vortex point for a nonzero Berry phase. For −1.17 eV < E_{F} < 0.29 eV, the electron (hole) extremum orbit excludes the vortex point having the zero Berry phase. Thus, to observe the nonzero Berry phase, electron or hole doping is needed. It is known that for MgB_{2}, the Mg atoms can be substituted by Al to form Mg_{1−x}Al_{x}B_{2}, and B can be substituted by C to form Mg(B_{1−y}C_{y})_{2},^{32,33,34} to achieve electron doping. Using the rigid band and virtual crystal approximation, we have calculated the alloy band structures, which confirm an upward shifting of E_{F} due to electron doping (Fig. S4). The nontrivial Berry phase can be measured with a doping concentration of x > 0.17 or y > 0.07. Moreover, the biaxial compressive strain also induces electron doping in MgB_{2} (Fig. S5).
On the one hand, our discovery of a topological state in MgB_{2} might not appear surprising from the theoretical point of view, because of its similarity (the B hexagonal plane) to graphene/graphite. On the other hand, it is quite surprising from the experimental perspective, considering the fact that MgB_{2} has been studied for over decades, but none of the experiments have detected any topological signature. This is because there are some fundamental differences between MgB_{2} and graphene/graphitebased systems as we have revealed here. In particular, we show that in order to detect the topological signatures in MgB_{2}, one has to do experiments differently from before, e.g., by measuring angleresolved photoemission spectroscopy on the (001) instead of commonly used (111) surface and magnetoresistance in the doped sample instead of the intrinsic one. However, our most important discovery is possibly the topological superconducting state in MgB_{2} as we discuss below.
We now consider the superconducting effect on DNLS. MgB_{2} is a conventional BCS superconductor with two superconducting gaps, Δ_{σ} and Δ_{π}, which arise from the σ and π bands of the B electrons, respectively. The magnitude of the energy gap ranges from 1.5 to 3.5 meV for the π band and 5.5 to 8 meV for the σ band.^{35,36,37,38,39,40,41} The strong electron–phonon coupling is known in MgB_{2}, as reflected by strong electronpair formation of the σbonding states. Because the charge distribution of the σbonding states is not symmetrical with respect to the inplane positions of boron atoms, the σbonding states couple very strongly to the inplane vibration of boron atoms.^{37} On the other hand, the πbonding states, which are related with DNLs, form weaker pairs. However, this pairing is enhanced by the coupling to the σbonding states. Due to the coupling between π and σ states, π and σ gaps vanish at the same transition temperature T_{c} ~ 39 K, although their values are greatly different at low temperatures. Since the DNL in MgB_{2} originates from the π band, we first consider the superconducting gap for the π band. But the promising TSC usually requires a spintriplet (oddparity) pairing state. For the bulk DNL with superconductivity, there is no spectral density in the swave superconducting gap (Fig. S6). In general, it is hard to realize the TSC phase using a bulk state.
To realize the TSC state, we now focused on the MgB_{2} thin films. It was predicted that the MgB_{2} thin films are mechanically stable and could be grown owing to the selfdoping effect.^{42} Furthermore, fewmonolayer MgB_{2} has already been synthesized experimentally on some substrates.^{43,44,45,46} Figure 4 shows the band structure of MgB_{2} (001) thin films. The dispersive DNL structure is projected onto a point and presented a formation of multiple Dirac states. These apparent Dirac states are also present in the surface bands for the two types of surface. The π states of the boron layer at the Bterminated (Mgterminated) surface are indicated by red (blue) color. The Dirac state of the Bterminated surface shows hole type, while the Mgterminated surface shows electron type. One can see that the Dirac state of the Bterminated surface behaves independently. Moreover, the swave and multiplegap superconductivity in MgB_{2} thin films is retained up to a high critical temperature of 20–50 K.^{47,48} Taking advantage of these unique features of thin films, we can realize a 2D TSC state.
Conventional swave superconductivity has been utilized to generate TSCs via proximity to some materials.^{49} The Dirac state in the boron layer of the Bterminated surface can be considered as the effective 2D hexagonal lattice model. To realize Majorana fermions with the swave superconducting gap, we consider a hexagonal lattice TSC based on a model of boron layer with Rashba SOC and exchange field. The corresponding TB Hamiltonian is given by
where \(c_{i\alpha }^\dagger (c_{ia})\) is the creation (annihilation) operator on site i with spin α, and σ are the Pauli matrices. The 〈i, j〉 represents the nearestneighboring (NN) sites. The first term is the NN hopping term in the boron layer. The second term is the Rashba SOC arising from a perpendicular electric field to the B layer adjacent to a substrate, with λ_{R} and d_{i,j} representing the coupling strength and a unit vector from site j to site i, respectively. The V_{z} (Δ_{π}) in the third (fourth) term corresponds to the exchange field (superconducting gap of the π band). The μ in the last term is chemical potential. We can transform the Hamiltonian of Eq. (3) to the Bogoliubov–de Gennes (BdG) Hamiltonian H_{BdG} in the momentum space (see Supplementary Information).
If neglecting the Rashba and exchange field term, the system is topologically trivial with a swave superconducting gap, which is the same as the bulk DNL state. As expected, the superconducting gap mixes the Dirac states, resulting in the disappearance of gapless edge modes. When the Rashba and exchange field are turned on, spinup and spindown bands are split and mixed. The 2D BdG Hamiltonian with a broken timereversal symmetry belongs to the topological class D,^{50} which is characterized by an integer number. The Chern number C_{1} can be calculated by^{51}
where \(f_{xy}(\vec k)\) is the Berry curvature
Here, \(u_{m\vec k}\left( {E_{m\vec k}} \right)\) is the mth eigenvector (eigenvalue) of H_{BdG}; f_{m} is Fermi occupation factor. Interestingly, we found that the boron layer of MgB_{2} thin film is topologically nontrivial in the case of V_{z} ≥ Δ_{π} with the Chern number C_{1} = 4. We note that this criterion does not depend on the value of the Rashba coupling. Figure 5a shows the calculated energy spectrum of BdG Hamiltonian near the crossing points with different V_{z} values. We have chosen the pairing Δ_{π} as 3 meV, similar to the experimental swave gap in MgB_{2}. For a pristine hexagonal boron layer, the Dirac bands appear at the \(\overline {\mathrm{K}}\) and \(\overline {{\mathrm{K}}^\prime }\) points, respectively. If the exchange and Rashba SOC interactions are considered, the trivial swave superconducting gap near the Dirac point turns into a topological gap. As the magnetic exchange field V_{z} increases, the topological gap increases. Further, we study the topological phase diagram as shown in Fig. 5b. The bandgap (E_{g}) is calculated in the (k_{x}, k_{y})plane. The phase boundary between the normal SC and TSC is determined by the dashed curves, i.e., V_{z} = Δ_{π}. To visualize the formation of TSSs within the superconducting gap, we calculated the Majorana edge states with a semiinfinite boron sheet. Figure 5c, d shows the energy spectra of the zigzag edge near the \(\overline {\mathrm{K}}\) and \(\overline {{\mathrm{K}}^\prime }\) points. We found that there exist two zeroenergy states in each valley with the same propagation direction (v_{F} < 0), which is in agreement with C_{1} = 4. This indicates that these zeroenergy states are topologically protected Majorana edge states. An armchair edge also exhibits four protected bands crossing zero energy (Fig. S8).
Finally, we address the experimental feasibility of the TSC from a quantum anomalous Hall (QAH) phase of graphene having a similar structural system with the boron layer. Graphene on the magnetic insulator, such as BiFeO_{3},^{52} RbMnCl_{3},^{53} and Cr_{2}Ge_{2}Te_{6},^{54} can have an exchange field (V_{z} ~ 70–240 meV) and Rashba SOC (λ_{R } ~ 1–4 meV), realizing the QAH state. We suggest that MgB_{2} thin films be epitaxially grown on these magnetic substrates, and the exchange field and Rashba SOC can be induced in the boron layer adjacent to the substrate. Since the MgB_{2} thin films naturally have swave superconductivity, the MgB_{2} thin films are expected to become TSC with the induced exchange field (V_{z} ≥ Δ_{π}) and Rashba SOC.
In summary, based on firstprinciples calculations and model analysis, an intriguing inversion and timereversal symmetry protected Dirac nodal line state is revealed in a hightemperature superconductor MgB_{2}. Our finding provokes an exciting opportunity to study a topological superconducting phase in an unprecedented high temperature and may offer a promising material platform to building novel quantum and spintronics devices. It will stimulate future studies of topological phases in a broader range of superconducting materials, such as a honeycomb lattice layered structure.
Methods
We performed firstprinciples calculations within the framework of densityfunctional theory (DFT) using the Perdew–Burke–Ernzerhoftype generalized gradient approximation (GGA) for the exchangecorrelation functional, as implemented in the Vienna ab initio simulation package.^{55,56} All the calculations are carried out using the kinetic energy cutoff of 500 eV on a 12 × 12 × 12 Monkhorst–Pack kpoint mesh. All structures are fully optimized until the residual forces are less than 0.01 eV/Å. The SOC is included in the selfconsistent electronic structure calculation. We construct Wannier representations by projecting the Bloch states from the firstprinciples calculation of bulk materials onto Mg s and B s, p orbitals.^{57} Based on Wannier representations, we further calculate the surface density of states and Fermi surface using the surface Green’s function method for the (010) surface of a semiinfinite system.^{58,59}
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
References
 1.
Fu, L. & Kane, C. L. Superconducting proximity effect and Majorana fermions at the surface of a topological insulator. Phys. Rev. Lett. 100, 096407 (2008).
 2.
Akhmerov, A. R., Nilsson, J. & Beenakker, C. W. J. Electrically detected interferometry of Majorana fermions in a topological insulator. Phys. Rev. Lett. 102, 216404 (2009).
 3.
Wang, M.X. et al. The coexistence of superconductivity and topological order in the Bi_{2}. Se_{3} thin films. Science 336, 52–55 (2012).
 4.
Stern, A. NonAbelian states of matter. Nature 464, 187–193 (2010).
 5.
Hor, Y. S. et al. Superconductivity in Cu_{x}Bi_{2}Se_{3} and its implications for pairing in the undoped topological insulator. Phys. Rev. Lett. 104, 057001 (2010).
 6.
Wray, L. A. et al. Observation of topological oer in a superconducting doped topological insulator. Nat. Phys. 6, 855 (2010).
 7.
Sasaki, S. et al. Topological superconductivity in Cu_{x}Bi_{2}Se_{3}. Phys. Rev. Lett. 107, 217001 (2011).
 8.
Sato, T. et al. Fermiology of the strongly spinorbit coupled superconductor Sn_{1−x}In_{x}Te: implications for topological superconductivity. Phys. Rev. Lett. 110, 206804 (2013).
 9.
Zhang, J. L. et al. Pressureinduced superconductivity in topological parent compound Bi_{2}Te_{3}. PNAS 108, 24–28 (2011).
 10.
Kirshenbaum, K. et al. Pressureinduced unconventional superconducting phase in the topological insulator Bi_{2}Se_{3}. Phys. Rev. Lett. 111, 001 (2013).
 11.
Zhao, L. et al. Emergent surface superconductivity in the topological insulator Sb_{2}Te_{3}. Nat. Commun. 6, 8279 (2015).
 12.
Sakano, M. et al. Topologically protected sface states in a centrosymmetric superconductor β–PdBi_{2}. Nat. Commun. 6, 8595 (2015).
 13.
Guan, S.Y. et al. Superconducting topological sface states in the noncentrosymmetric bulk superconductor PbTaSe_{2}. Sci. Adv. 2, e1600894 (2016).
 14.
Lv, Y.F. et al. Experimental signature opological superconductivity and Majorana zero modes on β–PdBi_{2} thin films. Sci. Bull. 62, 852–856 (2017).
 15.
Wang, Z. F. et al. Topological edge states hightemperature superconductor FeSe/SrTiO_{3} (001) film. Nat. Mater. 15, 968–973 (2016).
 16.
Burkov, A. A. & Hook, M. D., & Balents, L. Topological nodal semimetals. Phys. Rev. B 84, 235126 (2011).
 17.
Kim, Y., Wieder, B. J., Kane, C. L. & Rappe, A. M. Dirac line nodes in inversionsymmetric crystals. Phys. Rev. Lett. 115, 806 (2015).
 18.
Yu, R., Weng, H., Fang, Z., Dai, X. & Hu, X. Topological nodeline semimetal and Dirac semimetal state in antiperovskite Cu_{3}PdN. Phys. Rev. Lett. 115, 036807 (2015).
 19.
Jones, M. E. & Marsh, R. E. The preparation and structure of magnesium boride, MgB_{2}. J. Am. Chem. Soc. 76, 1434–1436 (1954).
 20.
Nagamatsu, J., Nakagawa, N., Muranaka, T., Zenitani, Y. & Akimitsu, J. Superconductivity at 39K in magnesium diboride. Nature 410, 63–64 (2001).
 21.
Kortus, J., Mazin, I. I., Belashchenko, K., Antropov, V. P. & Boyer, L. L. Superconductivity of metallic boron in MgB_{2}. Phys. Rev. Lett. 86, 4656 (2001).
 22.
Liu, A. Y., Mazin, I. I. & Kortus, J. Beyond Eashberg superconductivity in MgB_{2}: anharmonicity, twophonon scattering, and multiple gaps. Phys. Rev. Lett. 87, 087005 (2001).
 23.
Bzdušek, T. et al. Nodalchain metals. Nature 538, 75–78 (2016).
 24.
Fang, C., Weng, H., Dai, X. & Fang, Z. Topological nodal line semimetals. Chin. Phys. B 25, 117106 (2016).
 25.
Bzdušek, T. & Sigrist, M. Robust doubly crged nodal lines and nodal surfaces in centrosymmetric systems. Phys. Rev. B 96, 155105 (2017).
 26.
Fang, C., Chen, Y., Kee, H.Y. & Fu, L. Topological nodal line semimetals with and without spinorbital coupling. Phys. Rev. B 92, 081201 (2015).
 27.
Hyart, T., Ojajärvi, R. & Heikkilä, T. T. Two topologically distinct diracline semimetal phases and topological phase transitions in rhombohedrally stacked honeycomb lattices. J. Low. Temp. Phys. 191, 35–48 (2018).
 28.
Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).
 29.
Xie, Q. et al. Phononic weyl nodal straight lines in hightemperature superconductor MgB_{2}. arXiv:1801.04048 (2018).
 30.
Novoselov, K. S. et al. Twodimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).
 31.
Zhang, Y., Tan, Y.W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005).
 32.
de la Peña, O., Aguayo, A. & de Coss, R. Effects of Al doping on the structural and electronic properties of Mg_{1−x}Al_{x}B_{2}. Phys. Rev. 66, 012511 (2002).
 33.
Wilke, R. H. T. et al. Systematic effects of carbon doping on the superconducting properties of Mg(B_{1−x}C_{x})_{2}. Phys. Rev. Lett. 92, 217003 (2004).
 34.
Kortus, J., Dolgov, O. V., Kremer, R. K. & Golubov, A. A. Band filling and interband scattering effects in MgB_{2}: carbon versus aluminum doping. Phys. Rev. Lett. 94, 027002 (2005).
 35.
Szabó, P. et al. Evidence for two superconducting energy gaps in MgB_{2} by pointcontact spectroscopy. Phys. Rev. Lett. 87, 137005 (2001).
 36.
Giubileo, F. et al. Twogap state density in MgB_{2}: a true bulk property or a proximity effect? Phys. Rev. Lett. 87, 177008 (2001).
 37.
Choi, H. J., Roundy, D., Sun, H., Cohen, M. L. & Louie, S. G. The origin of the anomalous superconducting properties of MgB_{2}. Nature 418, 758–760 (2002).
 38.
Eskildsen, M. R. et al. Vortex imaging in the π band of magnesium diboride. Phys. Rev. Lett. 89, 187003 (2002).
 39.
Souma, S. et al. The origin of multiple superconducting gaps in MgB_{2}. Nature 423, 65–67 (2003).
 40.
Xi, X. X. Twoband superconductor magnesium diboride. Rep. Prog. Phys. 71, 116501 (2008).
 41.
Iavarone, M. et al. Twoband superconductivity in MgB_{2}. Phys. Rev. Lett. 89, 187002 (2002).
 42.
Tang, H. & IsmailBeigi, S. Selfdoping in boron sheets from first principles: a route to structural design of metal boride nanostructures. Phys. Rev. B 80, 134113 (2009).
 43.
Cepek, C. et al. Epitaxial growth of MgB_{2} (0001) thin films on magnesium singlecrystals. Appl. Phys. Lett. 85, 976 (2004).
 44.
Petaccia, L. et al. Characterization of highquality MgB_{2} (0001) epitaxial films on Mg(0001). New J. Phys. 8, 12 (2006).
 45.
Bekaert, J. et al. Free surfaces recast superconductivity in fewmonolayer MgB_{2}: combined firstprinciples and ARPES demonstratn. Sci. Rep. 7, 14458 (2017).
 46.
Cheng, S.H. et al. Fabrication and characterization of superconducting MgB_{2} thin film on graphene. AIP Adv. 8, 075015 (2018).
 47.
Morshedloo, T., Roknabadi, M. R. & Behdani, M. Firstprinciples study of the superconductivity in MgB_{2} bulk and in its bilayer thin film based on electronphonon coupling. Phys. C. 509, 1–4 (2015).
 48.
Bekaert, J., Aperis, A., Partoens, B., Oppeneer, P. M. & Milošević, M. V. Evolution of multigap superconductivity in the atomically thin limit: Strainenhanced threegap superconductivity in monolayer MgB_{2}. Phys. Rev. B 96, 094510 (2017).
 49.
Lutchyn, R. M., Sau, J. D. & Das Sarma, S. Majorana fermions and a topological phase transition in semiconductorsuperconductor heterostructures. Phys. Rev. Lett. 105, 077001 (2010).
 50.
Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).
 51.
Ghosh, P., Sau, J. D., Tewari, S. & Das Sarma, S. NonAbelian topological order in noncentrosymmetric superconductors with broken timereversal symmetry. Phys. Rev. B 82, 184525 (2010).
 52.
Qiao, Z. et al. Quantum anomalous Hall effect in graphene proximity coupled to an antiferromagnetic insulator. Phys. Rev. Lett. 112, 116404 (2014).
 53.
Zhang, J., Zhao, B., Yao, Y. & Yang, Z. Quantum anomalous Hall effect in graphenebased heterostructure. Sci. Rep. 5, 10629 (2015).
 54.
Zhang, J., Zhao, B., Yao, Y. & Yang, Z. Robust quantum anomalous Hall effect in graphenebased van der Waals heterostructures. Phys. Rev. B 92, 165418 (2015).
 55.
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169 (1996).
 56.
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
 57.
Mostofi, A. A. et al. wannier90: A tool for obtaining maximallylocalised Wannier functions. Comput. Phys. Commun. 178, 685 (2008).
 58.
Sancho, M. P. L., Sancho, J. M. L. & Rubio, J. Quick iterative scheme for the calculation of transfer matrices: application to Mo (100). J. Phys. F. 14, 1205 (1984).
 59.
Wu, Q., Zhang, S., Song, H.F., Troyer, M. & Soluyanov, A. A. WannierTools: an opensource software package for novel topological materials. Comput. Phys. Commun. 224, 405 (2018).
Acknowledgements
We would like to thank Tomáš Bzdušek, QuanSheng Wu, and Alexey A. Soluyanov for helpful discussions. K.H.J., H.H., and F.L. acknowledge financial support from DOEBES (No. DEFG0204ER46148). We also thank the Supercomputing Center at NERSC and CHPC at the University of Utah for providing the computing resources.
Author information
Affiliations
Contributions
K.J. and F.L. designed the research. K.J. performed theoretical calculation, H.H., J.M., Z.L., and L.L. discussed the results, and K.J. and F.L. prepared the paper.
Corresponding author
Correspondence to Feng Liu.
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Received
Accepted
Published
DOI
Further reading

Topological nodal line state in superconducting NaAlSi compound
Journal of Materials Chemistry C (2019)

Straight nodal lines and waterslide surface states observed in acoustic metacrystals
Physical Review B (2019)

Robust topological nodal lines in halide carbides
Physical Chemistry Chemical Physics (2019)