Mechanical and electronic properties of van der Waals layered hcp PdH2.

Mechanical and electronic properties of palladium dihydrides (PdH2) as a function of pressure were studied by ab initio calculations based on density functional theory (DFT). The ab initio random structure searching technique was employed for screening potential PdH2 crystal structures under high pressure. A hexagonal close packed (hcp) phase of PdH2 with space group P63mc was reported. The structure geometry and elastic constants were calculated as a function of pressure. It was found that H atoms are in the interstitial position of Pd atoms layer at 0 GPa. There is an electronic topology transition of hcp PdH2 at 15 GPa. When pressure exceeds above 15 GPa, one hydrogen atom occupies the tetrahedral site and another hydrogen atom locates in the interstitial position. When the c/a ratio is between 1.765 to 1.875, the hcp PdH2 is mechanically stable, and the Pd-H2b bond is the major factor that limits the mechanical stability. The elastic constant C44 is the first one that cannot satisfy the mechanical stability criteria under pressure. The anisotropy parameters are far from 1(one) shows that the hcp PdH2 is a highly anisotropic structure. The electronic structure study indicates that the bonding force between Pd and H atoms along the z-axis direction increases with the increasing pressure. Also, the phonon dispersion study shows that PdH2 is dynamic stability under pressure. The results suggest that hcp PdH2 can be metastable in van der Waals layered structure.

of H concentration from 0 to 0.76 indicating. It has been reported by Greenwood and Earnshaw that the metallic conductivity reduces as hydrogen is absorbed of PdH x reduces with the increase of x, until at around PdH 0.5 the solid becomes a semiconductor 14 . The H concentration x in PdH x depends on the temperature and pressure. By heating the Pd film to 600 °C in a hydrogen atmosphere at pressure 0.1 GPa, a high H concentration phase PdH 1.33 was obtained 15 . PdH 2 has also attracted the attention of researchers because of the increase of H concentration compared with monohydride 16 . Experiments show that PdH x can be formed on the top of multi-wall carbon nanotubes 17 . For 0 < x < 1, it is stable in face-centered cubic (fcc) structure. While, for x = 2 it has a hexagonal close-packed (hcp) structure. Palladium was studied in a high pressure hydrogen atmosphere up to 20 GPa. However, the PdH 2 was not observed by X-ray diffraction until the pressure up to 20 GPa 18 .
Ab initio studies have been performed for the stability of PdH 2 structure. It was found that the PdH 2 is octahedrally-centered and H 2 dimer is inside the fcc PdH 2 . The H 2 dimer and octahedron have the same axis. There is repulsive interaction between the H atoms in the octahedral cage, which causes two atoms move to the tetrahedral sites 19 . The chemical bonding properties were studied by analyzing the electronic structure and partial density of states (PDOS). The overlap of Pd 4d-H 1s states is the most important for stabilizing the dihydride. However, the antibonding Pd 4d-H 1s states set in lower energy because of the downshift of the central group of 4d bands. As a consequence, the destabilization of dihydride was observed as compared to PdH 20 . The formation enthalpy calculations show that the PdH 2 in fcc structure is unstable and this will decompose into fcc PdH and H 2 21,22 .
In the present work, the random search study was performed to predict the structure of PdH 2 at ambient as well as at the high pressures. An energy minimum structure is confirmed. It is an hcp structure, and the space group is P6 3 mc. The structural parameters of hcp and fcc PdH 2 are discussed as a function of pressure. The elastic constants of hcp PdH 2 as a function of pressure are obtained through ab initio calculations using the stress-strain method. The electronic structure of hcp PdH 2 is also analyzed by band structure and DOS projected on atoms and orbitals. From the obtained results, the mechanical and electronic properties of the PdH 2 are analyzed.

Results and Discussion
crystal structure. A global energy minimum phase of PdH 2 structure was found by ab initio random structure searching technique as shown in Fig. 1. It is an hcp structure with the space group P6 3 mc. In this structure, Pd atoms are positioned at 2b (1/3 2/3 z) and H atoms occupy the 2a (0 0 z) and 2b (1/3 2/3 z) site. The atomic symmetry of both Pd and H atoms is 3m. As shown in Fig. 1(a), at 0 GPa, Both of H atoms 2a sites (H 2a ) and 2b sites (H 2b ) are three-coordinated with Pd atoms. As the pressure increases above 15 GPa (see in Fig. 1(b)), the coordination number of H 2b atoms with Pd atoms increased to four.
For comparison, the fcc PdH 2 with the space group Fm3m and F43m 21 were considered. In the Fm3m structure, Pd and H atoms are located at the Wyckoff positions 4a (0 0 0) and 8c (1/4 1/4 1/4). The atomic symmetries of Pd and H atoms are 43m and m3m, respectively. The 8c H (H 8c ) atoms occupy two tetrahedral (T) sites. In the F43m structure, Pd atoms located at the Wyckoff positions 4a (0 0 0) and the H atoms occupy the 4b (1/2 1/2 1/2) and 4c (1/4 1/4 1/4) site. The atomic symmetries of Pd and H atoms are 43m. The 4b H (H 4b ) atoms and 4c H (H 4c ) atoms occupy octahedral (O) sites and T sites, respectively. The enthalpy differences ∆H for P6 3 mc and F43m PdH 2 relative to Fm3m PdH 2 are obtained. Figure 2 shows the enthalpy difference ∆H including zero-point energy and zero-point energy as a function of pressure. The ΔH takes positive values for F43m structure at all studied pressure indicating that the F43m structure is unstable than the Fm3m structure. At the pressure lower than 3 GPa or higher than 95 GPa, ΔH of P6 3 mc structure takes negative values. It suggests that the hcp structure is more stable than the F43m structure in these pressure ranges. Phonon dispersion in the whole Brillouin zone of hcp PdH 2 at 0 GPa is shown in Fig. 2(a). The inexistence of imaginary frequencies indicates the dynamic stability of the phases at 0 GPa.
The zero-point energy (ZPE) of the system is defined as the free energy of the system at 0 K. To improve accuracy, the zero point vibration energy was corrected in the energy calculations. The zero-point energy monotonous increase with the increase of pressure, except for the P6 3 mc structure at 15 GPa as shown in Fig. 2(b). The reason for the discontinuity of zero energy is that the hcp PdH 2 undergoes an isostructural phase transition when the pressure is about 15 GPa. The detailed discussion is carried out in the later discussion on structural parameters. The results show that the P6 3 mc PdH 2 is, in fact, a different phase under pressure. For comparison, we have included the hcp P6 3 mc and fcc Fm3m structures in the subsequent calculations of the structural parameters.   As is shown in Fig. 3(c), the c/a ratio decreases dramatically from 2.718 to 1.875 with the increase of pressure from 0 GPa to 15 GPa. As the PdH 2 is further compressed, the decrease of c/a ratio becomes slower with the increase of pressure. This indicates that the interlayer bonding force between Pd and H atoms is significantly enhanced when the pressure reaches above 15 GPa. The c/a ratio almost tends to an ideal value of 1.613 as the pressure exceeds 100 GPa. The c/a ratio changes with the pressure, indicating that the hcp PdH 2 is highly anisotropic. The lattice parameter a of both hcp and fcc PdH 2 decreases with the increase of pressure, except a slight increase at around 15 GPa. When pressure is lower than 15 GPa, the c decreases rapidly compare to a with the increase of pressure. With further increase in pressure, the decrease of c becomes slower.
Pressure affects the distance between the two H atoms. As is shown in Fig. 3(d), the H-H distance decreases with the increase of pressure. The hcp H 2a -H 2b distance is always less than fcc H 8c -H 8c distance. The H 2a -H 2b distance is closer to H 8c -H 8c distance at a pressure below 15 GPa than the pressure above 15 GPa.
Discontinuous change of bond length, bond angle, c/a, lattice parameter a, and the crystal symmetry remains unchanged at 15 GPa, indicating that there is an isostructural phase transition of hcp PdH 2 . For hcp PdH 2 , a = 2.973 Å and c/a = 2.718 at 0 GPa and a = 3.034 Å and c/a = 1.875 at 15 GPa.

Mechanical properties.
Elastic constants are the quantities to characterize the elasticity of materials which determine the response of materials to external forces. Elastic constants C ij of hcp PdH 2 as a function of pressure are listed in Table 1. It shows that all of the C ij increases monotonously with the increase of pressure. The magnitude of all the C 11 , C 12 , C 13 and C 33 are greater than the magnitude of the applied pressure except for the C 44 . The smaller increasing rate in the value of C 44 leads to that the shear deformation increase faster than before with the increase of pressure. It implies that the pressure reduces the stability of the structure.
The obtained bulk modulus B, shear modulus G, B/G ratio, Young's modulus E and Poisson's ratio ν of PdH 2 as a function of pressure are listed in Table 2. The B, G and E increase with the increase of pressure. All the B/G values are larger than 1.75 showing a ductile character that does not change with external pressure. The ν varies from 0.40 to 0.44. All of the ν is larger than 0.25, indicating that the PdH 2 is an ionic bonding material.
The mechanical stability for hcp PdH 2 as shown in Fig. 4. When the pressure range is 13 to 29 GPa, the C 44 − P > 0 is fulfilled. It means that the shear strain in (100) plane does not cause mechanical instability under the corresponding pressure range. The C 11 − |C 12 | − 2P convert to negative when pressure reaches around 57 GPa. For hcp structure, the shear elastic constants C 66 = C 11 − C 12 , hence the shear strain in (001) plane does not cause mechanical instability when the pressure is below 57 GPa. The C 11 − P > 0 and C 33 − P > 0 are fulfilled (see Table 1), indicating that one axial strain along the [100] and [001] directions does not lead to mechanical instability. The coupling strain along the [100] and [120] directions will lead to mechanical instability because of the C 11 − |C 12 | − 2P < 0 when the pressure is above 57 GPa. The (C 33 − P) (C 11 + C 12 ) − 2(C 13 + P) 2 > 0 is fulfilled when the pressure lower than 43 GPa. It indicates that the coupling strain along the [100], [120] and [001] directions does not cause mechanical instability when the pressure is in this pressure range. All the mechanical stability criteria are all fulfilled in the pressure range of 13 to 29 GPa. While the enthalpy differences for hcp PdH 2 relative to fcc PdH 2 are positive at this pressure range which implies that the hcp PdH 2 can be a metastable structure in the pressure range of 13 to 29 GPa. It means that the hcp PdH 2 is metastable when the c/a is in the range of 1.765 to P  [110] direction is related to C 11 and C 11 − C 12 . Therefore, the Pd-H 2b bond is the major factor that limits the mechanical stability of hcp PdH 2 .
Elastic anisotropic is also a fundamental parameter regarding mechanical properties. In this work, the c/a value changes with pressure. It means that the structure is always varying with the applied pressure. When the c/a value is in 1.765-1.875, the PdH 2 is mechanically stable. Here, we have discussed the anisotropy parameters of PdH 2 with c/a 1.827. For an isotropic medium, ΔP = ΔS 1 = ΔS 2 = 1. PdH 2 is anisotropic. For PdH 2 , the compressional anisotropy ΔP = 0.60. The PdH 2 is more easily compressed in the [001] direction than the [100] direction. The shear anisotropy ΔS 1 and ΔS 2 are 1.43 and 0.57, respectively. Due to the small C 44 , the shear anisotropy is large. It indicates that the largest shear deformation occurs in {100} plane and the slip is most likely to occur between planes parallel to {001} plane. All the anisotropy parameters are far away from 1(one), which means that the hcp PdH 2 is highly anisotropic. The results indicate that the Pd-H bonds are stronger in the layer which is parallel to the {001} plane than between the layers. electronic structure. The fat band along the high-symmetry directions of the BZ for PdH 2 at different pressure are shown in Fig. 5. The d-band on Pd atom has the main contribution to the electronic structure. The flat bands near the Fermi level suggest that the hcp PdH 2 is potential superconducting material. The band moves to lower energy with the increase of pressure. It also shows that two type-I Dirac point appears at K and H high-symmetry points indicating that the hcp PdH 2 is a topological-like structure. When the pressure increase from 0 GPa to 15 GPa, the gap between Dirc point appears. As the pressure increases from 0 GPa to 15 GPa, the flat band of H 2b and H 2a along Γ to A point move from −1.00 to 0.93. The anti-bonding was formed at 15 GPa. This sudden change in electronic structure at 15 GPa is due to the electronic topological transition of Fermi surface morphology. There are flat bands along with the high-symmetry directions [100] and [110] in the BZ. With the increase of pressure, the flat bands move to high energy because the partially occupied Pd 4d states were excited to a higher energy state. The flat bands move to the Fermi level at 40 GPa. When the pressure is higher than 40 GPa, the flat bands move across the Fermi level.
The density of states (DOS) projected on atoms and orbitals for PdH 2 at different pressure is shown in Fig. 6. This figure shows a pressure-induced metal to semimetal transition in the PdH 2 . It is seen that the Pd 4d states  Applied pressure reduces the distance between two layers of PdH 2 . As is shown in Fig. 6 (15 GPa), the interaction between Pd and H due to the two peaks at around the Fermi level. The Pd d z 2 states produce the main contribution to the DOS at around Fermi level. Above the Fermi level, the first peak shows that the H 2b 1s states contribute more to DOS than the H 2a 1s states. The first peak below the Fermi level shows that the H 2a 1s and H 2b 1s states almost have the same contribution to the DOS. The Pd d z 2 states interact with the H 1s states lead to an increase in interlayer bonding force between Pd and H. The peaks around the Fermi level move towards the Fermi level as the increase of pressure. As can be seen in Fig. 6 (40 GPa), the peak is on the Fermi level. This peak is derived from the flat zone along with the high-symmetry directions [100] and [110] in the Brillouin zone. It suggests that the hardness of the material is strengthened.
As the PdH 2 is further compressed, as in Fig. 6 (70 GPa), the peak moves above the Fermi level. The DOS has a minimum value at the Fermi level. The results show a phase transition from metal to semimetal in the PdH 2 under pressure.
The DOS of both H 2a and H 2b 1s states are flat at around Fermi level at 0 GPa. However, they are not flat as the PdH 2 is compressed. When the pressure was applied, some small peaks for H 1s states can be found at around Fermi level because of the Pd 4d states interact with other states. Hybridization of Pd d z 2 states and H 1s states increases the bonding force between layers of PdH 2 . These results indicate that the PdH 2 is more stable under pressure than at 0 GPa.

conclusions
In conclusion, a palladium hydride (PdH 2 ) with high hydrogen concentration is reported. The mechanical and electronic properties of PdH 2 were investigated by the ab initio calculations. We have reported elastic parameters such as bulk modulus, shear modulus, Young's modulus, Poisson's ratio within Hill approximation. It has been found that the hcp PdH 2 is mechanically stable when the c/a ratio is between 1.765 to 1.875. The Pd-H 2b bond is the major factor that limits the mechanical stability of this structure. The analysis of mechanical stability and anisotropy also shows that the interatomic forces of hcp PdH 2 are weaker between the layers which is parallel to the {001} plane than in the layer. Therefore, the cause of PdH 2 structural instability is the slip between {001} planes. Our study shows that the PdH 2 is dynamic stable at 0 GPa and the interatomic forces between layers increases with the increase of pressure. The results also show that there is an electronic topology transition of hcp PdH 2 at 15GPa. Our results suggest that PdH 2 can be stabilized in a metastable form. www.nature.com/scientificreports www.nature.com/scientificreports/ Methods. The structural and elastic constants of PdH 2 have been studied by performing first principles calculations. The calculations were achieved based on the density functional theory (DFT) 23 . The ab initio random structure searching technique 24 was used to find potential PdH 2 crystal structures at ambient and high pressure. The random search study was performed at 0, 50, and 100 GPa, with 1, 2, 3, and 4 PdH 2 units per simulation cell. The ab initio random structure searching technique generated unit cells of random shapes with reasonable volumes by calculating the ground state structure as well as determining the positions of PdH 2 formula in the cells. A plan-wave basis-set energy cutoff of 260 eV and an initial Brillouin Zone (BZ) sampling grid of 2π × 0.07 Å −1 were found to be sufficient for the initial searches. The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) 25 parameterization for the exchange-correlation functional and ultrasoft pseudopotential 26 were used for the structure searches by CASTEP code 27 .
The stable structure of PdH 2 was studied by ab initio lattice dynamics with a supercell approach, as implemented in the Vienna ab initio Simulations Package (VASP) code 28 and the phonopy package 29 . The structure was done using the projector augument-wave (PAW) 30 method and GGA was used to describe the electronic exchange-correlation effects. The calculation used 2 × 2 × 2 supercells, with consisting of 48 atoms, for P6 3 mc PdH 2 by using a plane-wave basis-set energy cutoff of 700 eV and sampling the BZ with fixed k-mesh 13 × 13 × 4.
The geometry relaxation and elastic constants calculations were performed using the VASP code. The exchange-correlation functional was described within the GGA of PBE. The outer electron configuration is 4p 6 5s 1 4d 9 for Pd. The plane-waves kinetic energy cut-off was set to be 900 eV. The "High" precision setting was used to avoid wrap around errors in the calculations. The vdW-DF2 31 functional was used to include the van der Waals interactions in the PBE functional. In a weakly bonded layered system, this function can calculate accurately equilibrium spacing and binding energy compare to the vdW-DF function 32 . The van der Waals (vdW) forces include attraction and repulsions between atoms, molecules, and surfaces, as well as other intermolecular forces. They differ from covalent and ionic bonding in that they are caused by correlations in the fluctuating polarizations of nearby particles. For the sparse systems, including soft matter, van der Waals complexes, and layered materials, which have interparticle separations, the vdW forces are important for the interactions of nonlocal and long-ranged. For the structural optimization and self-consistent calculations, the energy and forces convergence criterion of the electronic self-consistency were chosen as 10 −8 eV and 10 −3 eV/Å per atom. The k-meshes were generated automatically to divide the BZ in each direction. A k-mesh 17 × 17 × 17 was generated by Gamma method for hcp PdH 2 and a 22 × 22 × 22 was generated by Monkhorst-Pack method for fcc PdH 2 . These settings ensure a high convergence of 1 meV per unit cell in the total energy and accurate values of the forces in the atoms. For calculations of the hcp PdH 2 electronic band and DOS, the k-mesh was increased to 27 × 27 × 27.
Ab initio calculations were performed for enthalpies as a function of pressure. The enthalpy H at 0 K can be obtained by using the expression: where E 0 is the total energy, E ZPE is the zero-point energy, P is the hydrostatic pressure and V is the volume.
For the hcp PdH 2 , the elastic constants were calculated by using VASP software based on the strain-stress approach 33 . The number of k-mesh was increased to 27 × 27 × 27 for elastic constants calculations. A 0.01 Å positive and negative displacement was applied for each atom. For an hcp crystal, there are five independent elastic constants, C 11 , C 12 , C 13 , C 33 and C 44 . According to Hooke's law, the stress-strain relationship can be written as: where C 66 = 1/2 ( − C C 11 12 ). The elastic tensor is determined by performing six finite distortions of the lattice and deriving the elastic constants from the strain-stress relation.
The bulk modulus and shear modulus are used to describe the material's response to compress and shear stress. Here, we use the Hill approximations 34 , which are an average of the Voigt 35 and Reuss 36 elastic constants, to calculate the bulk modulus B and shear modulus G. The Voigt and Reuss approximations, labeled with subscripts V and G, are defined by: where S ij represents the elements of elastic compliance matrix which is equal to the reciprocal of elastic constant matrix. Then, the bulk modulus B and shear modulus G were obtained by: The G represents the resistance to plastic deformation and the B represents the resistance to fracture. The value of B/G is used to characterize the ductility of the material 37 . The material behaves in a ductile nature if B/G > 1.75, otherwise a brittle nature.
Young's modulus Y describes the response to linear stress. Poisson's ratio ν measures the phenomenon of deformation perpendicular to the loading direction when the material is compressed or stretched. The bond sorting can use the value of the ν. The ν is very small than 0.25 for a covalently bonded compound. While, for a typical ionic compound, the ν is nearly 0.25 or more. Young's modulus and Poisson's ratio 38  A crystalline structure is mechanical stable, if elastic energy is always positive. Elastic constants are used to determine the mechanical stability. At 0 GPa, the mechanical stability criteria for hcp structure are featured as 39 : where P is external pressure. The mechanical stability criteria show the response of the material to axial and tangential strain. The C 44 − P > 0 is related to shear strain in (100) face. The C 11 − C 12 − 2P ≡ C 66 − P > 0 is related to shear strain in (001) face. The C 11 − P > 0 and C 33 − P > 0 are related to the axial strain along the [100] and [001] direction, respectively. The C 11 − |C 12 | − 2P > 0 is related to the coupling strain along the [100] and [120] directions, which is same as the strain along the [110] direction. The (C 33 − P) (C 11 − P) − (C 13 + P) 2 > 0 is related to the coupling strain along the [100] and [001] directions. The (C 33 − P) (C 11 + C 12 ) − 2(C 13 + P) 2 > 0 is related to the coupling strain along the [100], [120] and [001] directions.
Elastic anisotropic is a fundamental parameter for mechanical properties. For an hcp crystal, the elastic anisotropic is described by the following formulas 41 : where ΔP is anisotropy for compressional wave, ΔS 1 and ΔS 2 are anisotropy for shear wave, polarized perpendicular to the basal plane and polarized in the basal plane, respectively. These three parameters characterize the anisotropy of the three main acoustic modes. The acoustic anisotropy in turn indicates the anisotropy of the elastic constants.