Design and predict tetragonal van der Waals layered quantum materials of MPd5I2 (M=Ga, In and 3d transition metals)

Introduction
Design and discovery of novel 2D van der Waals (vdW) layered topological and magnetic materials containing 3d transition metals (TM) are essential for advancing both fundamental science and applied technologies. For example, understanding and manipulating itinerant 3d TM magnetism in intermetallics are crucial to understand many emergent states, including superconductivity1,2,3,4,5. At the same time, rare-earth-free permanent magnets with 3d TM having large coercivity and saturation moment are sought after for renewable energy technologies6,7,8,9,10,11. When combining these two features in the same material, it is rare to find families of magnetic 2D vdW materials with only a handful examples such as CrX3 (X = I, Br, Cl)12,13,14,15,16,17, VI318, Cr2Si2Te619, Cr2Ge2Te620,21, Fe3GeTe222, and MnBi2nTe3n+1(MBT)23. The MBT systems are particularly interesting because most of them are intrinsic antiferromagnetic (AF) topological insulators (TI) that can be exfoliated to a few layers. The discovery of these magnetic 2D vdW materials have galvanized extensive research activities to study their unique properties.
The recent discovery24,25,26,27,28,29 of magnetic materials with the structural motif of -MPd5– and -MPt5– slabs in the tetragonal anti-CeCo5In structure has attracted wide attention for incorporating 3d TM magnetism of Cr, Mn, and Fe into slabs of Pd and Pt with large spin-orbit coupling (SOC). Ferromagnetic (FM) CrPt5P28 and MnPd5P29 have shown large magneto-anisotropy energy (MAE) for possible rare-earth-free permanent magnet applications, however, the easy axis is in-plane. These 1-5-1 compounds with separation of three-atomic-layered -MPd5– and -MPt5– slabs by a single layer of P or As are not exfoliable due to the strong bonding of P or As to Pd or Pt on both sides. But it hints at a new way to design 2D vdW materials by inserting more anion layers to separate the well-structured atomic metal slabs. On the other hand, non-magnetic (NM) 2D vdW materials that can be exfoliated to a single layer are in high demand to realize a range of emergent quantum states from superconductivity30 to fractional quantum anomalous Hall effect31,32,33 by twisting the few-layer systems, as well as charge density wave (CDW) induced quantum spin Hall (QSH) effect34 by gating. These new tetragonal vdW-layered materials predicted here can potentially provide a platform for the twisted few-layer systems with a tetragonal lattice35 besides the twisted hexagonal and orthorhombic lattices.
Here we report the design and prediction of 2D vdW-layered I-separated -MPd5– slabs in the body-centered tetragonal crystal structure of MPd5I2 with (M= Ga, In, and 3d TMs) by first-principles calculations. The phase stability calculations based on density functional theory36,37 (DFT) show that the NM ones are on the ground state (GS) hull and thermodynamically stable, similar to the already existing AlPd5I238, and a few magnetic ones are close to the GS hull and thus metastable. These materials have either topologically non-trivial band structures or interesting magnetic properties. In the two band manifolds near the Fermi energy (EF) separated by the top valence band from above and below, respectively, the NM compounds host a pair of bulk Dirac points (BDPs) just above and likely an additional strong TI state below the top valence band. While TiPd5I2 is a Dirac semimetal (DSM) with a relatively clean Fermi surface (FS), InPd5I2 has a half-filled top valence band with the surface Dirac point (SDP) from the strong TI state appearing at the EF. This combined feature results in the surface states with both the BDP projection and SDP appearing at the Γ point in a narrow energy window of 0.1–0.2 eV around the EF in TiPd5I2 and InPd5I2. The single layer (1 L) TiPd5I2 is a QSH insulator with an indirect global band gap of 30 meV or a narrow-gap semiconductor with a small direct gap of 100 meV depending on the DFT exchange-correlation (XC) functional. The 1 L InPd5I2 hosts two QSH states just above and below the half-filled top valence band with the band inversions involving pieces of flat bands. Among the magnetic ones, CrPd5I2 stands out for having the highest MAE of 2.88 meV/f.u. with the FM easy axis along the c-axis giving the desirable out-of-plane magnetic anisotropy needed for a rare-earth-free permanent magnet. The large MAE remains for the 1 L FM CrPd5I2. From the many band crossings around EF in both bulk and 1 L CrPd5I2, there are Weyl nodal lines protected by the horizontal mirror plane as the FM magnetic moment is along the c-axis.
Methods
DFT36,37 calculations have been performed with different XC functionals using a plane-wave basis set and projector augmented wave method39, as implemented in the Vienna ab initio simulation package40,41 (VASP). Besides PBEsol42, for vdW interaction, we have used vdW density functional (vdW-DF) of optB86b43 and the most recent r2SCAN + rVV1044. Band structures have been calculated with SOC and the results have also been checked with modified Becke-Johnson45 (mBJ) and HSE0646 exchange functionals. We have used a kinetic energy cutoff of 400 eV, Γ-centered Monkhorst-Pack47 with a k-point density of 0.025 1/Å and a Gaussian smearing of 0.05 eV. The ionic positions and unit cell vectors are fully relaxed, with the remaining absolute force on each atom being less than 1 × 10–2 eV/Å. For the 1 L structures, ionic relaxation is allowed in all directions, while the lattice vectors are only relaxed along the in-plane directions (x–y) with a 20 Å vacuum inserted along the out-of-plane (z) direction. In magnetic systems, calculations are initialized with a magnetic moment of 5 μB for transition metals and 0 μB for other elements in both FM and AF configurations. MAE calculations are performed by changing the global spin quantization axis from the z to x direction. Phase stability analysis is conducted using the convex hull algorithm48,49, which is implemented in the Pymatgen package50,51. The high throughput calculations on electronic structure and thermodynamics have been carried out in the workflow of high throughput electronic structure pakage (HTESP)52. To calculate Wilson loop and surface spectral functions, maximally localized Wannier functions (MLWF)53,54 and the tight-binding model have been constructed to reproduce closely the band structure within EF±1 eV by using Group III sp, TM sd, and I p orbitals. The surface spectral functions have been calculated with the surface Green’s function methods55,56 as implemented in WannierTools57. The phonon band dispersions are calculated with the small displacement method as implemented in Phonopy58.
Results
Structural motif and phase stability
Figure 1a summarizes the design of layered compounds based on the -MPd5– slabs. Without layer separation, the -MPd5– slabs with shared plane boundary is in the Cu3Au-type structure in space group (SG) Pm-3m (221) with M being surrounded by 12 nearest-neighbor Pd in a locally close-packed face-centered cubic (FCC) structure. As shown by the successful synthesis of MPd5P and MPd5As compounds27,29, an anionic layer can be inserted between the -MPd5– slabs to make new structures in the SG P4/mmm (123). With P/As in the nominal valence of 3–, it is possible to have two I to replace P, and more importantly, it becomes vdW layered, as shown by the existence of AlPd5I238, the only reported compound in this structure so far. It replaces P with two I and shifts the -MPd5– slabs in-plane for every other slab, which results in a body-centered tetragonal (tI16) structure in SG I4/mmm (139) with a large distance of 3.89 Å between the I atoms of neighboring slabs. A very recent experimental study59 on single crystal AlPd5I2 has shown that, indeed it can be exfoliated to a single AlPd5I2 slab with interesting properties. Here we design and predict new vdW-layered compounds in this structure by replacing Al with other group III elements and also 3d TMs via high throughput DFT calculations.

a Crystal structures of MPd5 motif in the three-atomic-layer slab with increasing distance between slabs as in MPd3 of Pm-3m (221) in Cu3Au-type, MPd5P of P4/mmm (123) and MPd5I2 of I4/mmm (139). The atomic species are shown in different colors and labeled accordingly. b Projected density of states (PDOS) on atomic orbitals of non-magnetic InPd5I2, TiPd5I2, and ferromagnetic (FM) CrPd5I2. c Convex hull energy (Eh) of MPd5I2 and MPd5I (M=Al, Ga, In, Ti, V, Cr, Mn, Fe, Co, and Ni) calculated in density functional theory (DFT) with PBEsol and optB86b exchange-correlation functionals. The ground state magnetic configurations with easy axis are drawn with the listed moment size on the magnetic ions (from V to Ni) and magneto-anisotropy energy (MAE). FM and antiferromagnetic (AF) are shown in red and blue shaded squares, respectively.
As plotted in Fig. 1b for the projected density of states (PDOS) of NM InPd5I2, NM TiPd5I2, and FM CrPd5I2, most of the I 5p orbitals hybridize with the bottom of Pd 4d orbitals from –6 to –4 eV to form the p-d bonding states. The p-d anti-bonding states are pushed to above 1 eV as empty states to gain more cohesion for the ternary compounds with additional bonding hybridization between Pd 5s (not shown) and I 5p orbitals in the same low-energy range. In contrast to I 5p, the p orbitals of group III elements at M site, for example In, mostly hybridizes with Pd 4 d states at a higher energy range (–4 to –2 eV), reflecting their electron positive character. But the states near the EF in InPd5I2 are dominated by Pd 4d and I 5p orbitals. Next, moving to 3d TM, Ti 3d orbitals hybridize extensively with Pd 4d orbitals giving the broader and lower Pd 4d-derived bands than those in InPd5I2 below the EF. There is also a large empty anti-bonding DOS peak just above the EF due to the 3d-4d hybridization. Then, for Cr with two more 3d electrons, these empty states get partially filled and induce a large exchange interaction, as shown by the two splitting DOS peaks at 0 and +2 eV. This interaction gives a sizable magnetic moment of 2.80 ({mu }_{B}) on Cr to prefer FM, and importantly, the easy axis is along the c-axis with a large MAE of 2.88 meV/f.u. The PDOS for other magnetic 3d TM MPd5I2 are similar in terms of band hybridizations. The NM TiPd5I2 is an interesting case with the right number of valence electrons that the anti-bonding states are almost completely empty, giving a minimum DOS at EF to form a semimetal, whose topological band structure will be detailed later.
The GS convex hull energy (Eh) for all the MPd5I2 compounds studied are plotted in Fig. 1c for PBEsol42 and also with the vdW exchange functional of optB86b43. To confirm that the introduction of I prefers two anion layers instead of one, besides the MPd5I2 in the I4/mmm structure, we have also calculated the hypothetical MPd5I in the P4/mmm structure. As shown in Fig. 1c, Eh for MPd5I are all above 0.10 eV/atom, much higher than MPd5I2, confirming the qualitative argument that the broken metallic interactions in separating -MPd5– slabs need to be compensated by a strong ionic interaction with enough anionic valence. For the group III elements, Al, Ga, and In, the Eh of MPd5I2 are all zero for PBEsol and slightly below 0.01 eV/atom for optB86b, showing they are all thermodynamically stable, because AlPd5I2 has already been found in experiment38, although optB86b gives a small positive Eh of 0.006 eV/atom. InPd5I2 has the smallest Eh, thus the most stable among the group III compounds. For the 3d TMs, first TiPd5I2 is quite stable on the GS hull even for optB86b and we found it is a DSM with a clean FS. Next for V and Cr, Eh becomes positive, but smaller than 0.10 eV/atom. Then Eh decreases for Mn and Fe at the middle of 3d TM series, and increases again for the late 3d Co and Ni, which are the least stable among the 3d TM MPd5I2. Overall, optB86b gives a higher Eh than PBEsol, showing a systematic shift between the different XC functionals. But the trends for the variation in Eh across the whole series for both MPd5I2 and MPd5I are the same for different XC functionals, showing the results are well converged.
The magnetic properties across the 3d TM series for MPd5I2 are quite interesting and tabulated at the top of Fig. 1c. Except for Ti being NM, the magnetic moment size increases first, starting with V, reaching the maximum of 3.92 ({mu }_{B}) for Mn before decreases at the end of the series for Co and Ni. With the gradual filling of the 3d orbitals, V, Cr, and Mn prefer FM, while Fe, Co, and Ni prefer AF. Importantly, both V and Cr prefer an easy axis along the c-axis, with Cr having the largest MAE of 2.88 meV/f.u., much higher than the 0.30 meV/f.u. for VPd5I2 and the rest. In contrast, FM Mn prefers the in-plane easy-axis, although with the largest moment. Then, for AF, first Fe moments prefer in-plane and then Co and Ni prefer out-of-plane directions.
To study phase stability and construct GS hull, all the existing binary and ternary compounds, together with the elemental ones in the ternary phase diagrams, have been computed, and their stability are calculated via different possible reaction paths. Although PBEsol gives AlPd5I2 on the GS hull, agreeing with experiment in Fig. 1c, Fig. 2a shows that the PBEsol-calculated volume per atom is underestimated when compared to the available experimental data. Then as shown in Fig. 2b, this underestimation of volume can be improved by using optB86b exchange functional. Also, to explicitly include vdW interaction for the 1-5-2 compounds, we have chosen optB86b vdW exchange functional for the phase stability plots in Fig. 2, as well as the band structure and magnetic property calculations. The GS hulls with PBEsol are similar and can be found in Supplementary Fig. 1.

a Volume per atom of the fully relaxed elemental, binary and ternary compounds in PBEsol are compared to available experimental data with the mean absolute percentage errors (MAPE) listed. b Same comparison for optB86b. c–l The calculated phase stability and structural energies of MPd5I2 (M=Al, Ga, In, Ti, V, Cr, Mn, Fe, Co, and Ni) in optB86b. The compounds on the ground state hull are labeled as green dots. The MPd5I2 are indicated with arrows and their respective hull energies are listed in parenthesis.
As shown in Fig. 2, for the Pd-I binary, there is only one stable line compound of PdI2. For M-I binaries, there are many stable line compounds for Ga, In, Ti, and V. For the rest, there is only one stable binary M-I compound, including CrI3 for Cr-I. For M-Pd binaries, Al, Ga, In, Ti, V, and Mn have many stable line compounds. Interestingly, for the -MPd5– motif in MPd3 or the Cu3Au-type, this binary line compound structure exists for In, Ti, V, Cr, Mn, and Fe with Pd. But for Ni and Co, they form a random alloy or solid solution with Pd. Because both CoPd3 and NiPd3 are only slightly above the GS hull at 0.06 eV/atom, they can be used as good representatives for the binary solid solution, and we include them in the phase stability calculation of MPd5I2. Among the calculated MPd5I2, the NM compounds are more stable than the magnetic ones. Given that AlPd5I2 with the Eh of 0.006 eV/atom in optB86b has already been synthesized, we predict the existence of GaPd5I2, InPd5I2, and TiPd5I2 because they are on the GS hull. For magnetic ones, Mn and Fe have the smallest Eh above the GS hull and then followed by V and Cr. Considering the approximations used in DFT calculations, we predict these four magnetic ternaries are metastable, also because the binary MPd3 line compounds with the -MPd5– motif are stable and found in experiments. Lastly, for Co and Ni, they have the largest Eh above GS hull even with PBEsol, and due to the solid solution of MPd3, it is also possible to form a solid solution for the ternary compounds. We predict these two are possible ternaries but with solid solution tendency, which needs to be further studied in the future.
From an experimental viewpoint, these MPd5I2 compounds are much more challenging to synthesize than MPd5P and MPd5As because of the higher vapor pressure or lower sublimation temperature of I than P and As. We have also studied the dynamical stability of these predicted compounds by calculating the phonon band dispersion. For example, the phonon band dispersions of both bulk and 1 L structures for TiPd5I2, InPd5I2, and CrPd5I2 are plotted in Supplementary Fig. 2. The absence of imaginary phonon modes shows that these predicted compounds in bulk and 1 L structures are dynamically stable.
Topological features of non-magnetic 1-5-2 compounds
For the band structures of NM MPd5I2, we chose TiPd5I2 and InPd5I2 to present the topological band features of both the bulk and 1 L structures. Bulk band structures of other MPd5I2 can be found in Supplementary Fig. 3. Figure 3a plots the band structure of bulk TiPd5I2 without SOC with the body-centered tetragonal Brillouin zone (BZ) and high symmetry k-points shown in Fig. 3c. The highest valence band (N) and band below (N–2) according to simple filling are shown in red and blue, respectively, with N being the number of valence electrons. Above the highest valence band, there is a sizable gap in most of the BZ, except for around the Z point in the (Gamma)-Z, Z-S1, and Y1-Z directions. Along the (Gamma)-Z direction, band N and N–2 are degenerate, giving triple degeneracy (or six-fold including spin) at the crossing point between the highest valence and lowest conduction band in middle of the (Gamma)-Z direction. From the triple degeneracy point to the Z point, a doubly degenerated nodal line segment appears, as also shown in Fig. 3d by plotting the zero-gap k-points in the whole BZ. The crossings along the Z-S1 and Y1-Z directions are parts of the nodal line loops around the Z point on the (110) and (1–10) planes, which are protected by the diagonal mirror symmetries.

a Band structure of TiPd5I2 without spin-orbit coupling (SOC) with the highest valence band (N) and band below (N-2) shown in red and blue, respectively. Green shade stands for I pz orbital projection. b Band structure of TiPd5I2 with SOC highlighting the top two valence bands and the same orbital projection as in (a). c Bulk Brillouin zone (BZ) with high symmetry k-points and those on (001) surface BZ are labeled. The bulk Dirac points (BDP) along (varGamma)-Z are indicated by red dots. d Nodal lines in TiPd5I2 without SOC between the highest valence and lowest conduction bands. e Bands zoomed in along (varGamma)-Z around the BDP between the highest valence and lowest conduction band. The green (gray) shade is for I pz (Pd dxz and dyz) orbital. f Bands zoomed in along the (Z)–({S}_{1}) direction with a small gap opening. The orange (cyan) shade is for Pd dyz (Ti dxz) orbital. g Wilson loop of Wannier charge centers (WCC) on the kz = 0.5 plane between band N–2 and N. h Surface spectral function along (bar{X})–(bar{varGamma })–(bar{M}) on (001). i (001) 2D surface Fermi surface at Fermi energy (EF) + 0.045 eV and j EF-0.185 eV with spin-texture shown in orange arrows.
With SOC, as plotted in Fig. 3b, the orbital degeneracy for the top two valence bands along the (Gamma)-Z direction is lifted, and the nodal loops are all gapped out, except for the crossing between the highest valence and lowest conduction band along the (Gamma)-Z, forming a BDP as protected by the fourfold rotational symmetry. Because of the time-reversal symmetry (TRS) and inversion symmetry, each band is still doubly degenerated. The BDP is zoomed in Fig. 3e along the (Gamma)-Z direction showing the zero gap and the switching between I pz and Pd dxz/dyz orbitals with the 2-dimensional irreducible representations of (Gamma)9 and (Gamma)6. The BDPs are at the momentum energy of (0, 0, ±0.1499 Å–1; EF + 0.0236 eV), as also shown by the red dots in Fig. 3c. The band dispersion of TiPd5I2 is zoomed along the Z-S1 direction in Fig. 3f to show the small SOC-induced gap.
Additionally, for the highest valence band N (red), it is also gapped from below by the next valence band N–2 (blue). The lower branch of the band N–2 along the (Gamma)-Z direction forms a band inversion region around the Z point with the top valence band N. Wilson loop calculation shows that this N–2 band manifold hosts a strong TI (STI) state with the Fu-Kane60 topological index of (1;001). The non-trivial Z2 number is shown by the Wilson loop of Wannier charge centers (WCC) on the kz = 0.5 plane in Fig. 3g with the odd number of crossings by the dashed line with WCC. Calculations with the more recent r2SCAN + rVV1044 XC functional give similar band features (see Supplementary Fig. 4) for both DSM and STI in TiPd5I2. For mBJ45 functional, the BDP still remains and predicts a DSM, despite most of the valence bands being pushed lower and conduction bands higher in energy. But the extra band inversion at the Z point is lifted between bands N–2 and N. The existence of this extra band inversion for STI or not in TiPd5I2 will be the features that need to be verified in the experiment.
To demonstrate the non-trivial band structure of TiPd5I2 with both a BDP above and an STI below the highest valence band, we have calculated the (001) surface spectral functions using the Wannier functions. On (001) surface, the projections of the BDP at ±kz onto the same (bar{Gamma }) point stand out nicely around the EF because of no overlap with other bulk band projection for the clean FS. There are topological surface states (TSS) stemming from the BDP projection, as seen in Fig. 3h at EF + 0.03 eV. Below that at EF–0.2 eV is the SDP from the STI of the N–2 band manifold also shown clearly even though on top of the other bulk band projections. The spin-texture of the surface Dirac cone is plotted at EF–0.185 eV in Fig. 3j, confirming the spin-momentum locking of the surface Dirac cone. The spin-momentum locking of the TSS stemming from the BDP projection is shown in Fig. 3i at EF + 0.045 eV. Thus, bulk TiPd5I2 is a DSM with a clean FS and also possibly hosts an STI just below the highest valence band.
When the vdW-layered TiPd5I2 is exfoliated down to 1 L, the band structure is plotted in Fig. 4a. A large band gap exists in most of the BZ, while a small gap appears along the (Gamma)-X direction, which is projected from the Z-S1 direction from the bulk band structure. Overall there is an indirect global band gap of 30 meV between the valence band maximum at X and conduction band minimum at (Gamma) point. The Wilson loop calculation of the highest valence band manifold (red) in Fig. 4b indicates a QSH with an odd number of crossings of WCC. In contrast, for the N–2 band manifold (blue), the even number of crossings in the Wilson loop in Fig. 4c shows it is topologically trivial. The edge spectral functions are plotted in Fig. 4d, e for the different TiPd- and PdI-terminations, respectively. The topological edge states connect the gapped valence with conduction band projections to form a TRS-protected edge Dirac points (EDP) at the (bar{Gamma }) point. While the EDP on the TiPd-termination is inside the QSH gap, that on PdI-termination is merged into the valence band projection at EF–0.06 eV. With EF cutting through the QSH gap, the spin-momentum locked edge states are unavoidable from TRS topological protection. Calculations with r2SCAN + rVV10 XC functional give a similar band inversion feature at the (Gamma) point for QSH (see Supplementary Fig. 4). In contrast, HSE0646 functional pushes the valence band lower and conduction bands higher in energy, and lift the band inversion and changes the indirect band gap to a direct gap of just 100 meV at the (Gamma) point (see Supplementary Fig. 4d). Such a small gap size can be potentially tuned to close and reopen by strain to induce the band inversion for a topological phase transition to realize a critical 2D Dirac point at the (Gamma) point and then a QSH.

a Band structure of 1 L TiPd5I2 with the highest valence band (N) and band below (N-2) shown in red and blue, respectively. b Wilson loop of Wannier charge center (WCC) between band N and N + 2. c Wilson loop between band N–2 and N. d Edge spectral functions along (bar{X})–(bar{varGamma })–(bar{X}) on TiPd- and e PdI-terminations.
Next for InPd5I2 as an example from group III MPd5I2, its bulk band structure with SOC is plotted in Fig. 5a. Due to TRS and inversion symmetry, each band is doubly degenerated. Because of the odd number of electrons, the highest valence band (red in Fig. 5a) is only half-filled, indicated by EF sitting right in the middle of the bandwidth. But it is still meaningful to discuss the topological features of the band manifolds above and below this half-filled top valence band with a finite FS. Similarly to TiPd5I2, between the highest valence and lowest conduction band of InPd5I2, there is only one crossing along the (Gamma)-Z direction as protected by the fourfold rotational symmetry for a pair of BDP. The BDP is zoomed in Fig. 5b shown by the projection on I pz and Pd dxz/dyz orbitals with the two-dimensional irreducible representations of (Gamma)9 and (Gamma)6. The BDP is at the momentum energy of (0, 0, ±0.0406 Å–1; EF + 0.0955 eV). In contrast, between the highest and the next valence band (blue), there is no band crossing. For such gapped band manifolds, the Fu-Kane topological index has been calculated as (1;001) showing a STI state with the Wilson loop of WCC at the kz = 0.5 plane being plotted in Fig. 5c. To confirm the STI with surface spectral function on (001), the surface Dirac cone is shown clearly in Fig. 5d with the SDP right at the EF inside the projected bulk gap. The spin-texture of the surface states at the EF is plotted in Fig. 5e, confirming the spin-momentum locking topological feature without the overlap with bulk band projection. In contrast, the BDP projection at EF + 0.10 eV on (001) surface in Fig. 5d is buried inside the other bulk band projection and shows no TSS, unlike those in TiPd5I2 with a clean FS. Thus, group III MPd5I2 hosts a BDP above and a STI below the half-filled highest valence band. For InPd5I2, both the SDP and BDP projection appear at the (bar{Gamma }) point on (001), and they are also within an energy window of 0.1 eV, with the SDP being right at the EF and the BDP just above the EF.

a Band structure of bulk InPd5I2 with the highest valence band (N) and band below (N-2) shown in red and blue, respectively. b Bands zoomed in along (varGamma)–Z around the Dirac point (DP) between the highest valence and lowest conduction band. The green (gray) shade is for I pz (Pd dxz and dyz) orbital. c Wilson loop of Wannier charge center (WCC) on the kz = 0.5 plane between band N–2 and N. d Surface spectral function along (bar{X})–(bar{varGamma })–(bar{M}) on (001). e (001) 2D surface Fermi surface at Fermi energy (EF) with spin-texture shown in orange arrows. f Band structure of 1 L InPd5I2 with the highest valence band (N) and band below (N-2) shown in red and blue, respectively. g Wilson loop of WCC between band N and N + 2. h Wilson loop between band N–2 and N. i Edge spectral functions along (bar{X})–(bar{varGamma })–(bar{X}) on InPd- and j PdI-terminations.
The band structure of 1 L InPd5I2 is plotted in Fig. 5f. Again, due to the odd number of electrons, the top valence band is half-filled with EF sitting in the middle of the bandwidth. But the valence band is continuously gapped from both below and above with band inversions, so topological properties can be calculated. The Wilson loop calculations of the band manifolds in Fig. 5g, h show the odd number of crossings of WCC, confirming it hosts two QSH states. The edge spectral functions are plotted in Fig. 5i, j for two different terminations, which are rather similar. The TRS-protected EDP at EF + 0.35 eV is for the upper QSH, and the EDP at EF is for the lower QSH. The topological edge states of the upper QSH around the (bar{Gamma }) point stand out, which is accessible, when the EF can usually be tuned by doping and gating. Calculations with r2SCAN + rVV10, mBJ, and HSE06 XC functionals all give the similar two band inversions and topological features (see Supplementary Fig. 5), which are much less affected than TiPd5I2, because InPd5I2 bands are more metallic from the half-filled top valence band than TiPd5I2. So 1 L InPd5I2 hosts two QSH states in a tetragonal structure despite being a metal. Additionally, the band inversions for the two QSH states around the EF in Fig. 5f involve pieces of flat bands from the orbital-decorated square lattice as recently proposed59. Together with the QSH insulator with a small indirect band gap or a narrow-gap semiconductor in 1 L TiPd5I2, the few-layer tetragonal systems of these exfoliable 1-5-2 compounds will be an interesting playground for emergent quantum states in future studies.
Magneto-anisotropy of CrPd5I2
Among the magnetic 3d TM MPd5I2 compounds, CrPd5I2 has the largest MAE and also the easy-axis is along the c-axis. First, without SOC, the spin DOS of the bulk and 1 L CrPd5I2 are plotted in Fig. 6a, b, respectively. While the spin down (majority) forms a pseudo gap near the EF, the spin up (minority) has a local DOS maximum at EF. Via the hybridization between Cr-3d and Pd-4d orbitals to form bonding and anti-bonding states just above EF, the exchange splitting interaction gives a sizable magnetic momentum of 2.8 ({mu }_{B}) on the Cr atom. The 1 L spin DOS in Fig. 6b is similar and has narrower and sharper peaks than the bulk in Fig. 6a due to the smaller band dispersion from the reduced interlayer interactions. To analyze the origin of the large MAE in CrPd5I2, we have calculated the k-point-resolved MAE over the entire BZ by fixing the magnetic charge density but rotating the magnetic axis from [001] to [100] with SOC. As shown in Fig. 6c, d for the (Delta)MAE = ± 0.03 meV/f.u., respectively, the positive MAE contribution (favoring the c-axis) in Fig. 6c is mostly around the (Gamma) and X points. In contrast, the negative MAE contribution (favoring in-plane) in Fig. 6d is mostly from the Z, S point and also half way between the (Gamma) and X points. Going to the 1 L CrPd5I2, the whole bandwidth is reduced, but most of the hybridization peaks remain in the same positions, which shows that the 1 L can retain the chemical stability. The magnetic moment does not change much, the MAE is still as high as 2.47 meV/f.u.

a, b Up (minority) and down (majority) density of state (DOS) for bulk and 1 L CrPd5I2 without spin-orbit coupling (SOC). c, d k-point resolved magneto-anisotropy energy (MAE) isosurface at ±0.03 meV/f.u. for bulk CrPd5I2 with switching the magnetic axis from [001] to [100] with SOC. e, f Band structure with SOC for bulk and 1 L CrPd5I2 with the highest valence band (N) shown in red. g, h Weyl nodal lines of bulk and 1 L CrPd5I2 between band N and N + 1.
The band structures of FM bulk and 1 L CrPd5I2 with SOC, are plotted in Fig. 6e, f, respectively. The band double degeneracies are all lifted with the top valence band shown in red. There are many bands crossing the EF, indicating a more complicated FS than the NM TiPd5I2 and InPd5I2. These many crossings form twofold degenerated Weyl nodal lines as plotted in Fig. 6g, h. For the FM bulk CrPd5I2, besides the main Weyl nodal loops on the kz = ±0.5 plane, there are also loops around the X points. For the FM 1 L CrPd5I2, there are three Weyl nodal loops, one around the X point and two around the M point. These Weyl nodal lines are within EF ± 0.2 eV.
The high MAE in CrPd5I2 reflects the unique structural motif of the -MPd5– slab, where each moment-bearing 3d TM atom is surrounded by Pd with much larger SOC strength. The distance among the 3d TM atoms is much larger than that in elemental solids. The magnetic coupling among the 3d TM atoms are through Pd with a larger SOC and itself is near the Stoner magnetic instability. Such a combination gives a range of magnetic configurations in MPd5I2. With the gradual filling of the 3d orbitals. V, Cr and Mn prefer FM, while Fe, Co and Ni prefer AF. Importantly, both V and Cr prefer easy axis along the c-axis with Cr having the largest MAE of 2.88 meV/f.u. In contrast, FM Mn prefers the in-plane easy axis, although with a larger moment. Then, for AF, first Fe prefers in-plane, and then Co and Ni prefer out-of-plane. Among the FM ones, with the largest MAE and easy-axis being out-of-plane, CrPd5I2 can give a large coercivity field, which is attractive for developing rare-earth-free permanent magnets.
Discussion
Using high throughput density functional theory calculations, we have explored the phase stability, topological and magnetic properties of MPd5I2 compounds (M=Ga, In and 3d TM), a family of -MPd5– slabs separated by two layers of I to design and predict new vdW-layered quantum materials with tetragonal structure. After confirming the existing AlPd5I2 is on the ground state (GS) hull, we find non-magnetic (NM) compounds with M=Ga, In, and Ti are also on the GS hull and thermodynamically stable. For the magnetic ones with 3d TM, we find V, Cr, Mn, and Fe are not far above the GS hull and metastable, given the existence of the binary structures with -MPd5– slab in the cubic MPd3 structure. For Co and Ni, the hull energy is the largest and also these MPd3 form random alloys. Using TiPd5I2 and InPd5I2 as examples, we show that the NM MPd5I2 host a pair of bulk Dirac points for the band manifolds just above the highest valence band and also likely an extra strong topological insulator (TI) state from below, respectively. While TiPd5I2 is a Dirac semimetal with a mostly clean Fermi surface, InPd5I2 has a half-filled top valence band with the surface Dirac point from the strong TI appearing at the Fermi energy (EF). This combination gives the (001) surface hosting both a surface Dirac point and a bulk Dirac projection just 0.1–0.2 eV separation at the (bar{Gamma }) point. From different exchange-correlation functionals, the 1 L TiPd5I2 is either a quantum spin Hall (QSH) insulator with an indirect global band gap of 30 meV or a narrow-gap semiconductor with a direct gap of 100 meV, which potentially can be tuned for a topological phase transition. In contrast, the 1 L InPd5I2 is always a metal from half band-filling and hosts two QSH states with pieces of the flat bands near EF. For the magnetic MPd5I2 with 3d TMs, the preferred magnetic ground state changes with the gradual filling of the 3d orbitals. V, Cr, and Mn prefer a ferromagnetic (FM) ground state with less or at half-filling, while Fe, Co, and Ni prefer an antiferromagnetic configuration with more than half-filling of the 3d orbitals. Interestingly both VPd5I2 and CrPd5I2 have their FM moment easy axis along the out-of-plane c-axis, a desirable feature to develop rare-earth-free permanent magnets. CrPd5I2 has a large MAE of 2.88 meV/f.u. Thus, our calculations predict that a group of MPd5I2 are synthesizable tetragonal vdW-layered quantum materials with either non-trivial topological features or highly tunable magnetic properties.
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