II-a. Structural motif and phase stability
Figure 1(a) summaries the design of layered compounds based on the -MPd5- slab. Without layer separation, the -MPd5- slab 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 local 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 layer, which results in a body-centered tetragonal (tI16) structure in SG I4/mmm (139). 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.
As plotted in Fig. 1(b) for the projected density of states (PDOS) of NM InPd5I2, 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 4d 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 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 shown as the two splitting DOS peaks at 0 and + 2 eV. This interaction gives a sizable magnetic moment on Cr of 2.80 \(\:{\mu\:}_{B}\) 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 DSM, whose topological band structure will be detailed later.
The GS convex hull energy (Eh) for all the MPd5I2 compounds studied are plotted in Fig. 1(c) for PBEsol39 and also with vdW exchange functional of optB86b40. 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. 1(c), 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 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, which shows they are all thermodynamically stable as 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. 1(c). 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 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 prefers 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, Fig. 2(a) shows the calculated volume per atom is underestimated when compared to available experimental data. This underestimation of volume with PBEsol is 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 Fig.S1 of SM.
As shown in Fig. 2, for 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, 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 random alloy or solid solution with Pd. Because CoPd3 and NiPd3 both are only slightly above the GS hull at 0.06 eV/atom, they can be used as good representatives for the binary solid solution. We also include them in calculating the stability of the MPd5I2. Among the calculated MPd5I2, the NM compounds are more stable than the magnetic ones. Given 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 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 because the solid solution of MPd3, it is also possible to form 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.
II-b. 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 1L structures. Bulk band structures of other MPd5I2 can be found in Fig.S2 of SM. Figure 3(a) 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. 3(c). The highest valence band (N) and band below (N–2) according to simple filling are shown in red and blue, respectively. 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 the middle of \(\:{\Gamma\:}\)-Z. From the triple degeneracy point to the Z point, a doubly degenerated nodal line segment appears, also shown in Fig. 3(d) 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 as protected by the diagonal mirror symmetries.
With SOC, as plotted in Fig. 3(b), 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 four-fold rotational symmetry. Because of the time-reversal symmetry (TRS) and inversion symmetry, each band is still doubly degenerated. The BDP is zoomed in Fig. 3(e) 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), also shown as the red dots in Fig. 3(c). The band structure of TiPd5I2 along the Z-S1 direction is zoomed in Fig. 3(f) to show the small SOC-induced gap.
Interestingly for the highest valence band (red), it is also gapped from below by the next valence band (blue). The lower branch of the orbital degenerated band N–2 along \(\:{\Gamma\:}\)-Z forms a band inversion region around the Z point with the top valence band N. Wilson loop calculations show this N–2 band manifold hosts a strong TI (STI) state with the Fu-Kane41 topological index of (1;001). The Wilson loop with Wannier charge centers (WCC) on the kz=0.5 plane in Fig. 3(g) shows the non-trivial Z2 number with the odd number of crossings by the dashed line with WCC. Calculations with more recent r2SCAN + rVV1042 XC functional show similar band features (see Fig.S3 in SM). For mBJ43 functional, the BDP still remains, despite most of the valence bands being pushed lower and conduction bands higher in energy. But the band inversion at the Z point is lifted between band N–2 and N for mBJ functional giving no STI. The existence of this band inversion for STI or not in TiPd5I2 will be the features need to be verified in experiment.
To demonstrate the non-trivial band structure of TiPd5I2 with a BDP above and a STI below the highest valence band, we have calculated the (001) surface spectral functions from the Wannier functions. On (001) surface, the projections of the BDP at ± kz onto the same \(\:{\Gamma\:}\) point are isolated in energy and have no overlap with other bulk band projection because of the clean FS. There are topological surface states (TSS) stemming from the BDP projection as seen in Fig. 3(h) 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. 3(j) 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. 3(i) at EF+0.045 eV. Thus, bulk TiPd5I2 is a DSM with a clean FS surface and also possibly hosts a STI below the highest valence band.
When the vdW layered TiPd5I2 is exfoliated down to 1L, the band structure is plotted in Fig. 4(a). 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 in Fig. 4(b) indicates it is 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. 4(c) shows it is topologically trivial. The edge spectral functions are plotted in Fig. 4(d) and (e) for the different TiPd- and PdI-terminations, respectively. The topological edge states connect the gapped valence with conduction band projections and form TRS-protected edge Dirac points (EDP) at the \(\:\stackrel{-}{{\Gamma\:}}\) 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. So 1L TiPd5I2 is a tetragonal QSH insulator. Calculations with r2SCAN + rVV10 XC functional show similar band inversion feature at the \(\:{\Gamma\:}\) point for QSH (see Fig.S3 in SM). In contrast, HSE0644 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 one with 100 meV at \(\:{\Gamma\:}\) point (see Fig.S3(d)). Although not a QSH in HSE06, the 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.
Next for InPd5I2 as an example from group III MPd5I2, its bulk band structure with SOC is plotted in Fig. 5(a). Because of the TRS and inversion symmetry, each band is doubly degenerated. But because of the odd number of electrons, the highest valence band (red in Fig. 5(a)) is only half-filled, indicated by EF sitting right in the middle of the band width. But it is still meaningful to discuss the topological features of the band manifolds below and above the half-filled top valence band, although the FS is finite. Between the highest valence and lowest conduction band, there is only one crossing along the \(\:{\Gamma\:}\)-Z direction as protected by the 4-fold rotational symmetry, similar to TiPd5I2. The BDP is zoomed in Fig. 5(b) shown by the projection on I pz and Pd dxz and dyz orbitals with the 2-dimensional irreducible representations of \(\:{\Gamma\:}\)9 and \(\:{\Gamma\:}\)6. The BDP are 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 it hosts a STI state. The Wilson loop with WCC at the kz=0.5 plane is plotted in Fig. 5(c). To confirm the STI with surface spectral function on (001), the surface Dirac cone is shown clearly in Fig. 5(d) with the SDP right at the EF and inside the projected bulk gap. The spin-texture of the surface states at the EF is shown in Fig. 5(e) confirming the spin-momentum locking topological feature without overlapping with bulk band projection. In contrast, the BDP projection at EF+0.10 eV on (001) surface in Fig. 5(d) is buried inside the other bulk band projection and shows no TSS, unlike those in TiPd5I2 with a clean FS. Thus, group III MPd5I2 host both a BDP above and a STI below the half-filled highest valence band. For InPd5I2, both the SDP and BDP projection appear at the \(\:\stackrel{-}{{\Gamma\:}}\) point on (001), and they are also within an energy window of 0.1 eV with the SDP being right at the EF and BDP just above the EF.
The band structure of 1L InPd5I2 is plotted in Fig. 5(f). Again, due to the odd number of electrons, the top valence band is half-filled with EF sitting in the middle of the band width. But the valence band is continuously gapped from both below and above with band inversion, so topological properties can be calculated. The Wilson loop calculation of the band manifolds in Fig. 5(g) and (h) show the odd number of crossings of WCC confirming it hosts two QSH states. The edge spectral functions are plotted in Fig. 5(i) and (j) for two different terminations, which are rather similar. The TRS-protected EDP at EF+0.3 eV is for the upper QSH and the EDP at EF is for the lower QSH. Calculations with r2SCAN + rVV10, mBJ and HSE06 XC functionals all show similar band inversion and topological features (see Fig.S4 in SM), which are much less affected than TiPd5I2, because InPd5I2 bands are more metallic from the half-filled top valence band than TiPd5I2. So 1L InPd5I2 hosts two QSH states in a tetragonal structure despite being a metal. Together with the QSH insulator with a small indirect band gap or a narrow-gap semiconductor in 1L 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.
II-c. Magneto-anisotropy of CrPd 5 I 2
Among the magnetic 3d TM MPd5I2 compounds, the most interesting one is CrPd5I2 with the largest MAE and easy-axis along the c-axis. First without SOC, the spin DOS of bulk and 1L CrPd5I2 are plotted in Fig. 6(a) and (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 hybridization of Cr-3d with Pd-4d to form bonding and anti-bonding just above EF, the exchange splitting gives a sizable magnetic momentum of 2.8 \(\:{\mu\:}_{B}\) on Cr. The 1L spin DOS is similar and have narrower and sharper peaks due to the less band dispersion from the reduced interlayer interactions than bulk. 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. 6(c) and (d) for the \(\:\varDelta\:\)MAE = ± 0.03 meV/f.u, respectively, the positive MAE contribution (favoring the c-axis) in Fig. 6(c) is mostly around the \(\:{\Gamma\:}\) and X points. In contrast, the negative MAE contribution (favoring in-plane) in Fig. 6(d) is mostly from the Z, S point and also half way between \(\:{\Gamma\:}\) and X point. Going to the 1L CrPd5I2, the whole band width is reduced, but most of the hybridization peaks remain the same, which shows that the 1L can retain the chemical stability. The magnetic moment does not change much, the MAE is still quite high at 2.47 meV/f.u.
With SOC, the band structures of FM bulk and 1L CrPd5I2 are plotted in Fig. 6(e) and (f), respectively. The band double degeneracies are all lifted. The top valence band is shown in red and there are many bands crossing the EF and a more complicated FS than the NM TiPd5I2 and InPd5I2. These many crossings form 2-fold degenerated Weyl nodal lines as plotted in Fig. 6(g) and (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 1L CrPd5I2, there are three Weyl nodal loops, one around the X point and two around the M points. 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 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. With such a large 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.