Hybrid nodal-chain semimetal with emergent flat band in MgCaN2

The distinct over-tiltings of band crossings in topological semimetals generate the type-I and type-II classification of Dirac/Weyl and nodal-line fermions, accompanied by exotic electronic and magnetic transport properties. In this work, we propose a concept of hybrid nodal-chain semimetal (NCSM), which is identified by linked type-I and type-II nodal rings (NRs) and hosts inevitable flat band. Using first-principles calculations and structure search technique, a new ternary nitride MgCaN2 is proposed as the first candidate to realize a novel 3D hybrid NCSM state. Remarkably, flat band is emergent along specific direction as a characteristic signature of such a hybrid nodal-chain, thus offering a platform to explore the interplay between topological states and flat bands. By analyzing the mirror Z2 topological invariant and developing a low-energy effective k⋅p model, we unveil the physical origin of the hybrid nodal-chain structure with multiple-mirror protected mechanism. Moreover, when the linked NRs are projected onto the (010) and (001) surfaces, considerable drumhead-like topological surface states can be illustrated with unique connection patterns. These results expand our understanding of NCSMs from significant band tilting effects and provide a new candidate on realizing hybrid nodal-chain fermion for further theoretical and experimental investigations.


Introduction
Topological semimetals, emerging as new topological quantum states, have recently attracted vast research interest in both communities of condensed matter physics and materials science [1][2][3][4][5]. Thereinto, the nodal line semimetals (NLSMs) with one-dimensional band degeneracies in the momentum space are of remarkable importance due to their intriguing electronic and transport properties [6][7][8]. In contrast to 0D nodal point and 2D nodal surface, such 1D nodal line embedded in the 3D Brillouin zone (BZ) is endowed with abundant geometrical and topological structures. Specifically, a single nodal line that subjects to certain crystal symmetry and material parameters may take the forms of an extended straight line, a closed ring [9], or a twisty curve [10,11], demonstrating the morphology of the diversification [12,13]. Importantly, for systems with various crystalline symmetries, multiple nodal loops may intersect or interwind with each other, evolving into diverse topological structures such as nodal-chain [14][15][16], nodal-link [17][18][19], nodal-knot [20], and nodal-net [21][22][23][24], providing abundant topological quantum states with exotic properties.
Among them, nodal-chain semimetal (NCSM) represents a typical topological structure of multiple NLSMs with distinct topological excitations to achieve non-trivial transport physics [14,25]. As shown in figure 1(a), two nodal rings (NRs) (represented by blue and green colors) lying on orthogonal planes can interlock together to form a 1D nodal-chain structure extending along the intersecting line, i.e. k y direction. When two sets of 1D nodal-chains couple together, a 2D nodal-chain structure is formed extending along k y and k z directions, as seen in figure 1(c). Such a nodal-chain fermion was initiatively reported by Bzdušek et al [14] in the non-centrosymmetric and non-symmorphic crystals with multiple glide-plane symmetries in the presence of spin-orbital coupling (SOC). Soon, it was generalized to the spinless systems with SU(2) symmetry without SOC by Takahashi et al [26]. In addition, NCSMs were also found to exist in certain centrosymmetric and symmorphic materials [27][28][29][30][31], such as HfC [15], and MgSrSi-type compounds [32].
Learned from the successfully classification of nodal-point semimetals, a NLSM can also be categorized into different subgroups based on the fold of degeneracy [33,34] (i.e. two-fold Weyl or four-fold Dirac NLs), the tilting effect [35][36][37][38][39][40][41] (i.e. type-I, II, or hybrid NLs) and the order of dispersion [42,43] (i.e. linear, quadratic, and cubic NLs). These NLSMs can achieve unconventional magnetic responses, such as a zero-field magnetic breakdown and anisotropy in cyclotron resonance [44,45], thereby inspiring interest on exploring systems with non-trivial band topology. Motivated by the prosperous of emergent NLSMs, we ask whether a universal classification could be extended into NCSMs, and what kind of related physical phenomena it will bring about. In fact, from the perspective of degeneracy, a four-fold degenerate 'Dirac nodal chain' was introduced in rhenium dioxide [46], in contrast to previous double-degenerated 'Weyl nodal chain' . Recently, a 'hybrid nodal-chain' that is composed of P·T (P: space inversion symmetry, T: time reversal symmetry)-protected NLs and mirror-protected NLs was predicted in an orthorhombic graphene network from the perspective of distinctive symmetry [47]. Albeit, the band tilting effect on NCSMs is until now rarely explored and the associated physical mechanism is still uncovered.
In this work, we aim at expanding the category of NCSMs in terms of the band tilting effect. As schematically shown in figure 1, when two linked nodal-rings belong to the same type-I or type-II case, it can be classified as type-I or type-II NCSM, keeping pace with the category of NLSM. Unusually, when two nodal-rings inside the NCSM host completely different dispersion types, e.g. one belongs type-I and the other belongs type-II, it gives birth to a new distinctive nodal-chain state, termed as hybrid NCSM. In the following, we identify by ab initio calculations and structure search method a new ternary nitride MgCaN 2 crystal as a novel hybrid NCSM, characterized by the linked type-I and type-II NRs around the Fermi level. The mirror Z 2 topological invariants and an effective k · p model were adopted to reveal the multi-mirror protection mechanism. Significantly, a flat-band is emergent as a characteristic signature of such NCSM along certain direction, which serves as a platform for exploring interplay between NCSM states and flat-band physics. Moreover, the topological surface states with unique connection patterns are demonstrated. In the final discussion part, we suggest the possible synthesis of MgCaN 2 from CaN 2 precursor by typical ion implantation strategy, and propose a family of ternary nitride XYN 2 (X,Y = Be, Mg, Ca, Sr Ba) as good candidates for studying distinct NCSM physics.

Computational methods
The crystal structure of ternary nitride MgCaN 2 was predicted using the crystal structure analysis by particle swarm optimization code [48,49], which has been widely used to search materials with excellent stability and intriguing physical properties. Structural optimization and electronic structure calculations were carried out by using Vienna ab-initio simulation package [50,51] with Perdew-Burke-Ernzerhof parameterized generalized gradient approximation (PBE-GGA) [52]. The energy cutoff of 520 eV and a k-point mesh of 12 × 12 × 8 are chosen. The ionic relaxations were performed until the force on each atom was less than 0.01 eV Å −1 and convergence criterion for the self-consistent electronic minimization loop was set to 10 −6 eV. Besides, the phonon spectrum was calculated with the PHONOPY package [53], and the thermal stability was evaluated by performing ab-initio molecular dynamics (AIMD) simulations based on an NVT ensemble with temperature controlled by a Nosé-Hoover bash [54,55]. To analyze the electronic topological properties, the tight-binding Hamiltonian was constructed through projecting the Bloch states into the maximally localized Wannier functions [56,57], and surface states were calculated using the iterative Greens function method as implemented in the WannierTools package [58].

Crystal structure and stability of MgCaN 2
The crystal structure of ternary nitride MgCaN 2 was unveiled by an extensively swarm-intelligence structure search method, which is further optimized based on DFT calculations. As displayed in figure 2(a), the identified MgCaN 2 compound crystallizes in a tetragonal lattice with space group of P4/mmm (No. 123). It includes spatial inversion (P), four-fold rotation (C 4z ) and three orthogonal mirror (M x,y,z ) symmetries. The Ca and N atoms are located at Wyckoff positions 1d (1/2, 1/2, 1/2) and 2g (0, 0, 3/8). They form a typical CaN 2 layer as observed in CaN 2 crystal [59]. The Mg atom intercalates between two layers of CaN 2 with Wyckoff position of 1a (0, 0, 0). The stability of MgCaN 2 is firstly investigated by computing the phonon spectrum, as shown in figure 2(b). The absence of imaginary-frequency phonon mode throughout the Brillouin zone confirms the dynamical stability of MgCaN 2 . Moreover, the low-frequency modes (∼0-6 THz) mainly derive from Ca and Mg atoms, while N atoms gain more phonon population in higher frequency modes as expected. In particular, there exist a branch of isolated highest-frequency mode, which is ascribed to the sturdy N-N covalent bond. Besides, the thermal stability of MgCaN 2 crystal is also confirmed by AIMD simulations at 300 K and 500 K (see figure S1 in supporting information (SI)).
The electron localization function of MgCaN 2 is demonstrated in figure 2(c). On the (100) plane, there is a dumbbell-shaped iso-value surrounding and inside N-N dimer [60], indicating strong ionic bonding of Ca-N and covalent bonding of N-N, respectively. While on (001) plane crossing the Mg atoms, the electrons tend to diffuse into the interstitial region, manifesting typical metallic bonding nature as that of free electron gas in alkali metal. In addition, Bader charge analysis gives that each Ca atom loses 1.38 electron (e) while each Mg atom only loses 0.96 e. All these electrons are transferred to coordinated N atoms. Overall, it reveals that the interactions between Ca-N and Mg-N are both ionic bonding, meanwhile the left electron of Mg + contributes to the metallic bond.

Hybrid NCSM state in MgCaN 2
The band structure and partial density of states (PDOSs) of MgCaN 2 without SOC at PBE-GGA level is displayed in figure 2(d). Given the light-element nature of Mg, Ca, and N atoms, the SOC effect on the electronic structure is rather weak and thus can be neglected (see figure S2 in SI). We focus on the lowest conduction band and highest valence band (marked by orange color), where several nontrivial band degeneracies emerge around the Fermi level. Based on the calculated PDOS, the low-energy electronic states in MgCaN 2 are mainly determined by N 2p states, with neglectable contribution of Mg 3s and Ca 4s states. First, the band along Γ-Z shows two-fold degeneracy, in which each point exhibits a quadratic dispersion relation in the k x -k y plane. This belongs to the quadratic nodal line fermion as proposed by Yu et al [42]. Second, various of seemingly isolated band-crossing points exist along Z-R-X, Γ-R, M-X-A paths, which are also checked by using a more advanced hybrid Heyd-Scuseria-Ernzerhof functional [61] (see figure S3 in SI). As displayed in figures 3(a)-(c), via careful nodal-point searching within the whole Brillouin zone, we reveal that these band crossings actually belong to three kinds of inequivalent NRs, labeled as NR1, NR2, NR3. The NR1 lies in the horizontal M z -invariant plane (ΓXM) encircling M point, the NR2 lies in the vertical M y -invariant plane (ΓZRX) centered around R point, and the NR3 lies in the vertical M x -invariant plane (ARXM) centered around X point. Further, 3D band-structures on corresponding mirror-invariant planes are calculated to analyze the band dispersion type of three NRs. As explicitly demonstrated in figures 3(d)-(f), the results show that NR1 and NR3 belong to type-I NRs, whereas NR2 shows type-II feature with over-tilted dispersion relation.
The unique connections for these NRs are unveiled. In figure 3(j), the type-I NR1 exteriorly contacts with the type-I NR3, to form a type-I nodal-chain periodically elongated along k y (XM) direction. More intriguingly, as seen in figure 3(k), the type-II NR2 connects with the type-I NR3, resulting in a hybrid  nodal-chain extended along k z (XR) direction. Sharing the same NR3, the type-I and hybrid nodal-chains naturally link together to generate a 2D nodal-chain located on k x = π plane. Further, considering the C 4z rotation symmetry of tetragonal lattice, two equivalent 2D nodal-chains on k x = π and k y = π planes have to link together, leading to the realization of a 3D hybrid nodal-chain structure across the entire BZ, as schematically displayed in figure 3(l) (see the calculated result in figure S4 in SI). Therefore, the distinct band tilting of the multiple linked NRs generates a novel hybrid NCSM state in MgCaN 2 crystal.

Symmetry analysis and topological invariant
To reveal the symmetric protection mechanism of the NRs and nodal chains in MgCaN 2 , we calculated the eigenvalues of M x,y,z for Bloch states on M x,y,z -invariant planes as shown in figures 3(g)-(i). The fact that two crossing bands around the Fermi level hosting opposite eigenvalues indicates these NRs are protected by corresponding mirror symmetries. In analogy to defining the Chern number for identifying an isolated Weyl point, one can use the quantized Berry phase to characterize the topological properties of a continuous nodal line in 3D BZ [10,62,63]. With the protection of mirror as well as spatial inversion symmetry the Berry phase along a closed loop is quantized to be zero or π module to 2π, which represents odd or even times of the nodal line passing through the chosen loop. As shown in figures 3(j) and (k), the Berry phase of nodal-chains is calculated for distinctive loops. We find that Berry phase equals to π for L 1,2,3 and 0 for L 4,5 paths, respectively. This result is consistent with the previous theoretical analysis because the L 1,2,3 encircles the NR one time while the L 4,5 encircles the NR two times.
To further reveal the underlying formation mechanism of NRs and nodal chains in MgCaN 2 , a Z 2 topological invariant derived from the knowledge of the eigenvalues of mirror operator is defined as, where the sum runs over four time-reversal invariant points (TRIPs) Γ m on each mirror-invariant plane. The quantity δ(Γ m ) = n∈occ ξ m n , in which ξ m n = ±1 is the eigenvalue of the nth band at Γ m . The product involves all the occupied bands. In analogy to the well-known parity criterion for topological insulators [64], here the Z 2 invariant ν = 0 or 1 indicates the existence of odd or even numbers of NR in associated mirror-invariant planes [6]. Moreover, the specific values of δΓ m can reflect the shapes and connection patterns of NRs, which is utilized to jude whether the nodal-chain appears or not.
Then we apply this criterion to the electronic structure of MgCaN 2 . With the eigenvalues of M x,y,z at each TRIP for all the occupied bands, the δΓ m is readily obtained and Z 2 invariant ν is further calculated, from which one can identify the existence of distinctive NRs and nodal chains. For instance, as for the M x -invariant plane (ARXM) in figure 3(c), the δ(R) = δ(A) = δ(M) = −δ(X) = +1 leads to ν = 1. Hence, there must be at least a nodal line inside this plane. More interesting, the opposite δ between X and R/A/M suggests that the band inversion should happen for odd times along arbitrary k-path connecting X and the other points, which directly leads to the emergence of a closed NR surrounding X point, namely NR3. Analogously, in figures 3(a) and (b) the associated topological indexes on ΓXM and ΓZRX ensure the appearance of NR1 centered at M point and NR2 centered at R point, respectively. Because the XM belongs to the intersection of ΓXM and ARXM planes, such that the band crossing point at XM belongs to NR1 and NR3, simultaneously. As a result, the X-centered NR1 and M-centered NR3 have to interlock with each other exactly at the nexus point [65], forming a 1D nodal-chain along XM direction. Similar analysis works for XR path as the intersection of ΓZRX and ARXM planes, giving anther 1D nodal-chain along XR direction. These results derived from topological indexes are qualitatively in agreement with results from first-principle calculations.

Effective k · p Hamiltonian and inevitable flat band
To obtain quantitatively understanding of these linked NRs, particularly the distinctive dispersion relations, we now construct the low-energy effective k· model for each NR on corresponding mirror-invariant plane. In general, a two-band k · p Hamiltonian around the Γ m in BZ can be written as: where f 0,x,y,z (q) are real functions, and q = (q x , q y , q z ) is the momentum vector respect to the Γ m . σ 0 is the identity matrix, and σ x,y,z denotes the Pauli matrices. Now, we consider the symmetry constrains of the Hamiltonian in equation (2). In the absence of SOC, the time-reversal symmetry can be taken as T = K, where K is the complex conjugation operator. As the requires the f y (q) and f 0,x,z (q) to be odd and even functions, respectively. As for the M i =x,y,z -invariant plane, based on the fact that two bands have opposite mirror eigenvalues, then M i can be written as M i = σ z , which further adds constraint f x,y (q) and f 0,z (q) to be odd and even functions, respectively. Hence, in the M i -invariant plane, up to the second order of q, the model Hamiltonian is with and Here Applying the replacements of q j by |q| cosθ and q k by |q| sinθ. The eigenvalues of Hamiltonian equation (3) are, with ϵ 0 = w 0 + m 0 , w = w j cos 2 θ + w k sin 2 θ, and m = m j cos 2 θ + m k sin 2 θ. Hence, the dispersion relation (type-I or type-II) of NRs in M i -invariant planes can be estimated by a direction-dependent parameter s(θ), which is defined as: The nonzero s(θ) can break the electron-hole symmetry, leading to the asymmetric nature of conduction and valence bands around the Γ m point. It allows us to roughly estimate the types of dispersion for the crossing points. Specifically, for s(θ) < 1, this case indicates along q(θ) direction the parabolic conduction and valence bands have opposite opening direction, thus leading to type-I band crossing point as shown in figures 4(a) and (b). In contrast, s(θ) > 1 indicates the conduction and valence bands have same opening direction, thus generating type-II band crossing points as shown in figure 4(d). It's worth mentioning that for s(θ) = 1, this special case implies a flat band along certain direction, generating unique type-III band crossing [66], as displayed in figure 4(c). The values of s(θ) for target NR1/NR2/NR3 are given in figure 4(e). We find that s(θ) << 1 for NR1 (the circle in green color) along any directions, demonstrating that NR1 is a typical type-I NR. For NR2 (blue color), the condition s(θ) > 1 is satisfied for most of θ aside from θ = 90 • or 270 • (along RX), where the s = 0.97 ≈ 1. Thus, NR2 can be approximatively taken as a type-II NR. As for NR3 (red color), one can find s(θ) < 1 always hosts for arbitrary direction, which indicates that NR3 belongs to type-I. Specially, the s(90 • ) = 0.98 ≈ 1, implies the emergence of flat band along XR direction, as confirmed by the DFT result in figure 3(i). Significantly, the rise of flat band is a dominant characteristic of hybrid NCSM. Since a hybrid nodal-chain state is formed by connected type-I NR centering on Γ 1 and type-II NR centering on Γ 2 , which require s ⩽ 1 and s ⩾ 1 along Γ 1 Γ 2 , simultaneously. As a result, the s = 1 is restricted along Γ 1 Γ 2 , which inevitably results in a flat band. Applying this criteria to MgCaN 2 , wherein the hybrid NCSM is composed by the type-I NR3 centering at X and the type-II NR2 centering at R. Thus, the Γ 1 and Γ 2 correspond to X and R points, respectively. This means the flat band will emerge along XR path, which corresponds to the directions of θ = 90 • or 270 • , respectively, as pointed out by the arrows in figure 4(e). This is exactly the origin of the flat band along the intersectional XR in MgCaN 2 . The emergent flat band in hybrid NCSM can create novel type-III Dirac node as the critical state between type-I and type-II Dirac nodes, which will enrich the classification of topological quantum phases and offer a feasible strategy in designing flat-band materials [67]. Moreover, the interplay between the nontrivial topology and flat band in hybrid NCSM may provide a promising route towards unconventional superconductivity and correlated electronic states [68,69].

Topological surface states
NCSMs usually possess distinctive surface states on different direction. In general, when a nodal-chain is projected to a certain plane, different NRs will project into either closed rings or line segments, which divide the projected surface into topologically distinctive regions. Therefore, as shown in figure 5, in order to comprehensively demonstrate the topological drumhead surface states and further reveal the underlying coupling effect between different NRs, we have computed the (001) and (010) surface states for MgCaN 2 by using the post-processing code of WannierTools [58]. For the (001) surface in figure 5(a), the NR1 projects into a closed ring (green circle) encircling M, while the NR3 projects into a line segment (red line), which connects NR1 with its periodic repeated unit. Besides, the NR2 is also projected into a line segment (blue line) connecting NR3 at X. Those projected NRs divide (001) surface into two separated regions: region-I, projection of NR1 which is a closed isolated region; region-II, outside projection area of NR1, which is an extended region that stretches across the entire BZ. To reveal the nontrivial surface state of NCSM, two typical paths are chosen: cut-1 and cut-2. The cut-2 traverses region-I and region-II, crossing the projected NR1 (green circle) and avoiding touching other line segments (blue/red lines). In contrast to cut-2, the cut-1 additionally crosses the line segments. From surface spectra along cut 1 in figure 5(b) and cut 2 in figure 5(c), we discover that the typical drumhead surface state [70] only emerges in the region-II, and would be pinned by the projection of NR2.
For the (010) surface in figure 5(d), the NR2 on k y = 0 plane projects into a closed ring (blue circle) encircling R, while the NR3 on k y = π plane projects into a closed ring (red circle) encircling Γ, which divides (010) surface into three separated regions of I, II and III. Meaningwhile, the equivalent NR2 from k x = 0 plane projects into a line segment (blue line) along ΓZ which connects projected NR3 (red ring), while the equivalent NR3 on k x = π plane projects into a line segment (red line) along XR which connects the projected NR2 (blue ring). Besides, the NR1 is also projected into a line segment (green line) along ΓX, which connects the projected NR3. The surface spectra along two typical paths are displayed, cut 3 in figure 5(e) and cut 4 in figure 5(f). For cut 3, which passes through region-I and -II, we find the drumhead surface state from the type-II Dirac point of NR2 passes through the region-II, and is pinned by the projected NR2 on ΓZ. For cut 4, which passes through region-II and -III, we discover that the drumhead surface state from the projected NR3 (red ring and line) only exists inside the region-II. Such rich and extended surface states of MgCaN 2 in the momentum space are feasibly measured by surface experimental techniques, such as scanning tunneling microscope or angle-resolved photoemission spectroscopy [71][72][73].

Discussions and conclusions
Though MgCaN 2 material is discovered from the structure search method, we propose a possible synthetic route of MgCaN 2 compound by the powerful technique of ion intercalation [74][75][76]. As an typical alkaline earth diazenides, CaN 2 crystallizes [59] in a tetragonal lattice with space group of I4/mmm (139), as shown in figure 6. It can be regarded as AB-stacking of CaN 2 layers along c direction. By interpolating Mg ion into the interstitial region between two adjacent CaN 2 layers, the interlayer distance is significantly enlarged in accompany with interlayer sliding, which leads to the formation of a new phase MgCaN 2 . With unique ionic-bond between Mg + and [CaN 2 ] − and covalent-bond within [CaN 2 ] − , ternary nitride MgCaN 2 belongs famous Zintl compounds, such as CaAl 2 Si 2 [77,78]. Moreover, a group of isoelectronic compounds with chemical formula XYN 2 (X, Y = Be, Mg, Ca, Sr Ba) are designed as illustrated in figures S5 and S6 on SI. These materials are revealed to host abundant topological semimetals states including NLSM and NCSM with distinctive dispersion types as summarized in table S1 in SI.
In summary, based on first-principles calculations, structure search method, and symmetry analysis, we have identified a new Zintl compound MgCaN 2 as a novel hybrid NCSM, with interlocked type-I and type-II NRs around the Fermi level. The calculated phonon spectrum and AIMD simulations have confirmed the dynamical and thermal stability of MgCaN 2 , demonstrating its promise for experimental synthesis. The underlying multiple-mirror protected mechanism of the hybrid nodal-chain structure is unveiled by analyzing the mirror Z 2 topological invariant and developing a k · p effective Hamiltonian. More importantly, we point out that flat band along a certain direction is emergent as a characteristic signature of such a hybrid nodal-chain, thus serving a platform to study the interplay between the topological semimetal states and flat bands. Furthermore, the considerable drumhead-like topological surface states are observed on the (010) and (001) surfaces with unique connection patterns, benefiting experimental detection of the nontrivial nodal-chain band topology. These results expand our understanding of NCSMs from new perspective of significant band tilting effects, and provide a promising candidate to realize hybrid nodal-chain state for further theoretical and experimental investigations.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).