Introduction

In nodal loop semimetals, band degeneracies form a closed loop in the three-dimensional Brillouin zone1. In these topological materials, topological protection is linked to additional discrete symmetries of the crystal2. Ab initio calculations paved the way in the search for the physical realization of nodal loop semimetals3. In many of these calculations, it was highlighted, that spin-orbit coupling in most cases splits up the nodal structure into discrete Weyl points. Compelling evidence for the existence of a nodal loop in ZrSiS was found in quantum oscillation measurements4. The experimental isolation of the stoichiometric compound Ca\(_3\)P\(_2\) is also considered an important step towards the technological exploitation of nodal semimetals5. Spin-orbit coupling is expected to be negligible in this system, which is composed of relatively lightweight elements. Furthermore, first-principles calculations of the bulk spectrum revealed a large nodal loop precisely at the Fermi level. Drumhead states on the surfaces of nodal semimetals, are defined by the surface projection of the nodal structure of the bulk. These topological boundary states have small kinetic energy and are thus susceptible to interactions, making them an ideal platform for emergent surface magnetism6. Spin-polarized surface states in SnTaS\(_2\) samples were for instance recently reported7. Interaction-induced magnetic properties have already been studied theoretically8,9 and observed experimentally10,11 in rhombohedral graphite, a material also exhibiting a helical nodal line in its bulk. In two dimensions an analogous magnetization of localized states pinned at the zig-zag termination of graphene ribbons was observed12 experimentally and studied theoretically13,14.

Surface and interface magnetism has already significantly impacted consumer electronics, with the giant magnetoresistance effect underpinning many data storage solutions15. Spintronics architectures incorporating magnetic boundaries promise smaller, faster, and more energy-efficient classical and quantum devices16. Understanding and exploiting the new avenue for surface and interface magnetism provided by the drumhead surface states of nodal line semimetals is thus desired.

In this work, we investigate surface reconstruction and surface magnetism in the stoichiometric nodal loop semimetal Ca\(_3\)P\(_2\). This material, first isolated by Xie and coworkers5, is a form of Ca\(_3\)P\(_2\) arranged in a hexagonal crystal structure similar to Mn\(_5\)Si\(_3\), however according to X-ray diffraction data all Ca sites have only a partial occupancy of 90%. Thus this arrangement yields a stable stoichiometry balancing the constituent Ca\(_{2+}\) and P\(_{3-}\) ionic cores. We applied a unified first principles approach, based on the SIESTA method17. This allowed us to perform geometrical relaxation of the considered slab structures, to obtain electronic spectra for systems with partial occupancy and to extract information regarding the localization and magnetization properties of the electronic states near the Fermi energy in the bulk and on the surface of the sample. Our results show how surface reconstruction and band bending may interfere with surface magnetism.

Results

In this section, we detail all our findings regarding the surface states of Ca\(_3\)P\(_2\). First, we present the structural and electronic bulk properties of the system obtained from our first principles approach. Second, we discuss structural properties and surface reconstruction of the two considered slab geometries. We then explore the spectral properties of the surface states. Finally, we turn to the discussion of surface magnetism in the stoichiometric system and beyond.

Properties of the bulk

The bulk crystal structure of Ca\(_3\)P\(_2\) is depicted in Fig. 1. The bulk is spawned by lattice vectors \(\textbf{a}_1=a(1,0,0)\), \(\textbf{a}_2=a(-1/2,\sqrt{3}/2,0)\) and \(\textbf{a}_3=c(0,0,1)\). The bulk structure consists of three distinct types of layers. Two layers, denoted by blue and orange colour contain three Ca and three P sites. We shall refer to these as common layers. A third type of layer, the spacing layer, contains two Ca sites and is situated between the common layers. A unit cell has two spacing layers and two common layers. In a given common layer both Ca and P sites form interlocked distorted triangular lattices. This is highlighted for the P sites in Fig. 1. where all nearest neighbour P sites are connected by solid (shorter bonds) or dashed (longer bonds) lines. In the bulk, the constituent layers are equally spaced, that is \(\Delta =c/4\) and the layers do not exhibit any buckling, that is \(\delta =0\). We note that in the case of slab calculations discussed below, both spacing and buckling are appreciably changed near the surface. We find that buckling is introduced in the common layer in such a way that all Ca sites are shifted together with respect to all P sites, this shift is denoted by \(\delta\). Structural relaxation of the bulk crystal yielded the parameters presented in Table 1. These are in general in good agreement with those reported in Ref. 5 although a was found slightly larger and c slightly lower in our calculations.

Fig. 1
figure 1

Real space structure of Ca\(_3\)P\(_2\). Top (a) and side (b) view of the unit cell. Lattice vectors \(\textbf{a}_1\), \(\textbf{a}_2\) and \(\textbf{a}_3\) spanning the unit cell are denoted by arrows, Ca sites are marked by disks while P sites by squares. Blue and orange colors refer to common layer 1. and 2. respectively. Spacing layers are marked by green.

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Table 1 Summary of optimized bulk lattice parameters.
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The electronic spectrum depicted in Fig. 2a,c. shows a clear signature of nodal line semimetals, namely a continuous line of band degeneracy in the plane spawned by \(\textbf{b}_1\) and \(\textbf{b}_2\) reciprocal vectors at the Fermi level. We again note a good qualitative agreement between our results and those reported in Ref. 5. However, we remark that the spectrum reported previously in Ref. 18 differs somewhat around the A-L-H plane, specifically in our case states of the conduction band approach the Fermi energy closer at the A point. This discrepancy can be attributed to the slightly different implementations of density functional theory (DFT). In our case, localized pseudo atomic orbital basis was used in conjunction with a pseudopotential approach, while those calculations presented in Ref. 18 used an all electron code with plane wave basis. In Fig. 2b we show the contributions of different species to the density of states. Orbitals corresponding to P atoms dominate the filled bands close to the Fermi energy, while states associted with Ca sites have higher contribution above the Fermi energy.

Fig. 2
figure 2

Electronic band structure of bulk Ca\(_3\)P\(_2\) along high symmetry points of the three-dimensional Brillouin zone (a). Green disks marking band degeneracies in the M-\(\Gamma\)-K plane. Subfigure (b) shows partial density of states corresponding to Ca and P sites and total density of states. Subfigure (c) depicts the Brillouin zone spawned by reciprocal lattice vectors \(\textbf{b}_1\),\(\textbf{b}_2\) and \(\textbf{b}_3\) as well as high symmetry points. The green curve, calculated as the degeneracy of the bulk bands, indicates the nodal loop.

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Slab calculations

In the following, we present the results of our ab initio calculations performed for slabs of a finite thickness. All presented results were obtained for slabs of 6 unit cell thickness in the \(\textbf{a}_3\) direction as our structural and magnetic calculations showed that the middle of these structures already approximated the bulk well. First, we discuss structural properties then we turn our attention to the electronic spectrum and magnetic properties.

Structural surface reconstruction

We considered two symmetric slab geometries differing in the type of the very first and last layer. We shall refer to slabs as capped if the first layer is a common layer, built from Ca and P atoms, while we shall call slabs terminated if the first layer is a spacing layer containing only Ca atoms.

We performed geometry optimization on both types of slabs. The resulting arrangements are presented in Fig. 3. For the capped configuration depicted in subfigure (a) one can observe that P atoms move towards the bulk and Ca atoms slightly elevate out from the surface. This introduces a buckling \(\delta\) in the order of 0.4Å on the surface. One can observe a considerable lateral displacement as well in the case of a capped slab. For terminated slabs shown in Fig. 3b Ca atoms, in layer 1. move slightly away from the system. On the next layer, P atoms also move away from the bulk. That is in this layer the atoms behave oppositely compared to how they did in the first common layer of the capped slab. Curiously the sample does not exhibit any substantial relaxation in the lateral direction. The observed scale of the buckling \(\delta\) is similar to that seen for the capped slab. Surface reconstruction, as it is expected, diminishes as we move towards the bulk. We note that for capped slabs after the first unit cell, i. e. after four layers the geometry of the bulk is recovered. In the case of terminated slabs, this recovery is somewhat slower, in particular, Ca atoms are still further than 0.1Å compared to their unrelaxed position in layer 6, which is in the second unit cell.

Fig. 3
figure 3

Structure of the surface of a slab before and after relaxation in the capped (a) and in the terminated (b) configuration.

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Electronic spectra of surface states

For slab geometries the periodicity of the system is two-dimensional. In this case, the high symmetry points of the bulk Brillouin zone collapse to the high symmetry points of a two-dimensional Brillouin zone. In particular, \(\Gamma\) and A are mapped to \(\Gamma\), L and M are mapped to M and H and K to K. In the following, we denote the wave vectors of this two-dimensional Brillouin zone with \(\vec {k}\) as as opposed to the three-dimensional notation \(\textbf{k}\).

The different geometrical arrangements of the two considered slab types result in a markedly different electronic structure for the two cases. First, we discuss capped slabs. In Fig. 4. the layer resolved spectral function, \(A_z(\vec {k},E)\) for capped slabs is shown for common layers. We note that spacing layers have a negligible spectral weight in the energy window shown thus we do not plot the spectral function localized on them. One can observe clear signatures of band bending in the presented spectral data. The majority of the spectral weight shifts towards lower energies as one approaches the bulk. On the first layer states with relatively low kinetic energy appear, however, they span the whole Brillouin zone contrary to what one expects from drumhead states. These states are also just under the Fermi level. At higher energies 0.5 eV above \(E_F\) on the first layer and slightly below in the layers closer to the bulk a relatively flat band also appears.

Fig. 4
figure 4

Layer resolved spectral function \(A_z(\vec {k},E)\) for a capped slab at with stoichiometric Ca occupations.

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Slabs in the terminated geometry exhibit less band bending, and the electronic states are lowered in energy compared to the capped arrangement, as is evident in Fig. 5. In this geometry, the first layer is a spacing layer with a considerable spectral weight localized on it, albeit all of this weight is above the Fermi level and forms a graphene-like band structure. As it was in the case for capped slab spacing layers further inside the system have negligible spectral weight in the shown energy window. In the common layers close to the surface the spectral function is prominent in a band situated close to the Fermi level. Comparing the features of this band one notes similarities to the band structure observed at 0.5 eV above the Fermi level in the capped system. This spectral feature might be best attributed to the drumhead surface states, however as in the case of the capped slab, the flat spectral features around the Fermi energy extend further in the Brillouin zone.

Fig. 5
figure 5

Layer resolved spectral function \(A_z(\vec {k},E)\) for a terminated slab with stoichiometric Ca occupations.

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As we discussed above, both slab configurations exhibit flat spectral features in the vicinity of the Fermi level, thus a natural expectation is that interactions drastically impact the system for instance by germinating a finite magnetic moment close to the surface of the slabs. Contrary to these expectations the magnetization of the system remains rather low in both cases. The largest moments in both cases are localized on common layers close to the surface but are only on the order of \(10^{-4}\mu _B\).

In an experimental setting, such as Ref. 5 the Ca site occupancy can differ from the stoichiometric \(90\%\). We now discuss magnetic spectral functions for \(88.8\%\) occupancy depicted in Figs. 6 and 7. In these cases an appreciable magnetic moment \(\approx 0.1 \mu _B\) will be localized on atoms near the surface as it is shown in Fig. 8 as well. The magnetic spectral functions depicted in Figs. 6 and 7. clearly show that at this occupancy electronic states in the two spin channels are split on common layers close to the surface, while towards the bulk this splitting is reduced. We note that the total spectral function, not shown here, in this case differs only slightly compared to those at the stoichiometric occupancy.

Fig. 6
figure 6

Layer resolved magnetic spectral function \(A_{\sigma ,z}(\vec {k},E)\) for a capped slab at \(88.8\%\) Ca occupancy.

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Fig. 7
figure 7

Layer resolved spin spectral function \(A_{\sigma ,z}(\vec {k},E)\) for a terminated slab at \(88.8\%\) Ca occupancy.

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Magnetism below the stoichiometric limit

Above we established that at slightly diminished Ca occupancy electronic states of opposing spin channels are energetically split. Below we give further analysis of this state from the perspective of the magnetic moments localized at atomic sites close to the surface.

In Fig. 8. magnetic moments m as a function of the z coordinate of atoms in the slab are presented for capped (a) and terminated (b) configurations respectively. In the capped configuration the largest magnetic moment is located on the surface, that is on layer 1., while no appreciable magnetism can be observed beyond the first unit cell. Magnetic moments on the surface for the capped configuration are strictly localized on the P atoms. In the case of a terminated slab, the very first layer is not magnetized. Layers 2. and 4. carry most of the magnetic moment in the system. In this case spacing layers around them also develop a small magnetic moment. Magnetic moments in the terminated configurations, as opposed to the capped configurations, are associated with the Ca atoms. As for the capped configuration, we observe no magnetization in the bulk for the terminated configuration.

Fig. 8
figure 8

Layer dependent magnetization m as a function of height z for capped and terminated configurations \(88.8\%\) Ca concentration.

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The magnetic moments discussed above were obtained in a collinear reference frame calculation. By fitting the energetics of infinitesimal spin rotations to the electronic structure we extracted effective Heisenberg exchange interactions14. For both types of slabs we calculated interactions only between sites which exhibited a larger magnetic moment than 0.05 \(\mu _B\), that is in the case of capped slabs P atoms on layers 1. and 3. and for terminated slabs Ca sites on layers 2. and 4. were taken into account. The obtained interactions are depicted in Fig. 9. Again the results show marked differences between capped and terminated slabs. In the case of capped slabs, the strongest interaction is ferromagnetic between P atoms with the smallest lateral displacement but in a different layer. Inspection of further neighbour terms reveals predominantly ferromagnetic couplings. We thus conclude that for capped slabs the surface layer shows a ferromagnetic arrangement. A more complicated magnetic texture can be inferred in the terminated slab. Coupling between nearest neighbours on layer 4. is the strongest coupling with an antiferromagnetic nature. Similarly, antiferromagnetic coupling can be observed for nearest neighbours on layer 2. This observation suggests that in terminated slabs localized moments will not prefer a co-linear arrangement, but rather a 120\(^{\circ }\) Neél configuration is expected19.

Fig. 9
figure 9

Effective Heisenberg couplings \(J_{ij}\) between magnetized layers in the capped (a) and terminated (b) configuration for \(88.8\%\) Ca concentration. In subfigure (a) the P atoms of layers 1. and 3. are denoted by orange and blue circles. In subfigure (b) the Ca atoms of layers 2. and 4. are denoted by squares. For both cases, the thickness of the connecting line is proportional to the interaction strength normalized to the strongest interaction. Red colors refer to ferromagnetic coupling while blue colors signal antiferromagnetic coupling. Black crosses mark sites inside the unit cell which were taken as the origin for the calculated exchange interactions.

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Discussion

In this manuscript, we report a concise ab initio study of surface reconstruction and surface localized magnetization in the nodal loop semimetal Ca\(_3\)P\(_2\). Our calculations go beyond previous studies18 for slabs of Ca\(_3\)P\(_2\) where a tight-binding model was fitted to the bulk band structure and the obtained slab spectrum was calculated in the absence of surface reconstruction. We investigated two types of slabs with different surface termination. We show that in the stoichiometric limit even though states with reduced kinetic energy are localized close to the surface, the interplay of geometrical reconstruction and band bending pushes these states just below the Fermi level impeding the emergence of surface magnetism. Even though the bulk exhibited a pronounced nodal loop, it is hard to identify the drumhead surface states in the slab calculations because of other non-dispersive bands present in the spectrum. Deviating slightly from the stoichiometric limit the surface is shown to be magnetic with a magnetic texture depending strongly on surface configuration. As an alternative to adjusting the occupancy of Ca sites, in an experimental setup where slabs of Ca\(_3\)P\(_2\) are investigated, a similar emergence of magnetism could be achieved by controlling the Fermi level of the system with a gate voltage.

The results presented here underscore the significance of surface reconstruction in altering the magnetic properties of nodal loop semimetals, emphasizing that even minimal deviations in atomic arrangements can have profound effects on surface magnetism. Future research focusing on fine-tuning these surface characteristics may pave the way for practical applications where the unique properties of nodal loop semimetals could be harnessed to engineer novel magnetic and electronic functionalities.

Methods

In all our ab initio calculations we used the SIESTA approach to DFT.17 Bulk calculations were performed with a mesh of \(21\times 21\times 21\) k-points while slab calculations were done on a \(21\times 21\) mesh. A real space grid cutoff of 1000 Ry was employed. The exchange and correlation energies were approximated by the use of the GGA-PBE functional20. Lattice relaxation was performed with a maximum force tolerance of 0.008 eV/Å. All relaxed structures were obtained by relaxing the system with fully occupied Ca sites and without spin polarization. To treat partial occupancy of the Ca sites we employed the virtual crystal approximation (VCA) in the spirit of previous works5,18 as it is implemented in the SIESTA package21. All our calculations employing VCA used a mixture of Ca and Ar atoms. We justify this choice by noting that in the experiments first reporting the isolation of Ca\(_3\)P\(_2\) samples were prepared in Ar atmosphere5. We note that VCA is a rough approximation compared to the more sophisticated Coherent Potential Approximation (CPA)22,23. To the best of our knowledge, no existing codes combine CPA with geometry relaxation, limiting our ability to fully capture the complexities of systems with partial occupancy. Future advances integrating CPA with geometry relaxation would improve accuracy in such studies. As the constituent elements were rather light in all of our calculations we neglected spin-orbit coupling. Spin-polarized calculations were carried out in a colinear reference frame. Magnetic moments were obtained by standard Mulliken charge analysis provided by SIESTA17. To find the ground state spin configurations e. g. those depicted in Fig. 8. we considered three initial spin configurations and let the electronic system converge. The three configurations considered were the ferromagnetic (all atoms polarized), the paramagnetic (no atoms polarized), and the opposing surface antiferromagnetic (common layers on the two opposing surfaces polarized in the opposite direction). In all cases, the ferromagnetic initial configuration arrived at the lowest total energy yielding a pattern reported here in Fig. 8.

The bandstructure, the partial and total density of states as well as the layer- and spin-resolved spectral function were calculated with the help of the sisl post-processing package24. For a given wave number \(\vec {k}\) SIESTA yields the overlap matrix \(S(\vec {k})\) and spin channel dependent Hamiltonian \(H^s(\vec {k})\). Solution of the generalized eigenvalue problem \(H^s(\vec {k})\varphi ^s_n(\vec {k})=E^s_n(\vec {k})S(\vec {k})\varphi ^s_n(\vec {k})\) in turn delivers the spectrum \(E^s_n(\vec {k})\) and eigenvectors \(\varphi ^s_n(\vec {k})\) in the basis of pseudo atomic orbitals. With these, the spin-resolved spectral function is given by

$$\begin{aligned} A_{s,z}(\vec {k},E)=-\frac{1}{\pi }\textrm{Im}\left( \sum _{l\in z} \frac{[\varphi ^s_n(\vec {k})]^*_l \left[ S(\vec {k})\varphi ^s_n(\vec {k}) \right] _l }{E+\textrm{i}\eta -E^s_n(\vec {k})} \right) , \end{aligned}$$

where the index l runs over all orbitals of atoms associated to layer z, and \(\eta\) is small positive number. The total and magnetic spectral functions are simply given by \(A_{z}(\vec {k},E)=A_{\uparrow ,z}(\vec {k},E)+A_{\downarrow ,z}(\vec {k},E)\) and \(A_{\sigma ,z}(\vec {k},E)=A_{\uparrow ,z}(\vec {k},E)-A_{\downarrow ,z}(\vec {k},E)\).

We obtained the magnetic exchange interactions by evaluating the Lichtenstein formula25 adapted to the case of non-orthogonal basis functions used by the SIESTA package14,26. The isotropic Heisenberg interaction \(J_{ij}\) between two atomic sites i and j is given by

$$\begin{aligned} J_{ij}=\frac{1}{2\pi }\int _{-\infty }^{E_F}\textrm{d}E\textrm{Im}\textrm{Tr}_l\left[ (H^\uparrow _{ii}-H^\downarrow _{ii}) G^\uparrow _{ij}(E) (H^\uparrow _{jj}-H^\downarrow _{jj}) G^\uparrow _{ji}(E) \right] , \end{aligned}$$

where \(\textrm{Tr}_l\) denotes trace over orbitals, \((H^\uparrow _{ii}-H^\downarrow _{ii})\) contains the matrix elements of the exchange field on-site i and \(G^s_{ij}(E)\) are the site and spin-dependent matrix elements of the Green’s function in real space. These parameters allow one to build a classical spin model defined by the Hamiltonian \(\mathcal {H}=-\frac{1}{2}\sum _{i\ne j}J_{ij}\textbf{e}_i\cdot \textbf{e}_j\), where the \(\textbf{e}_i\) are unit vectors are aligned with the magnetization of site i.