Strain-Modulated Electronic Structure and Infrared Light Adsorption in Palladium Diselenide Monolayer

Two-dimensional (2D) transition-metal dichalcogenides (TMDs) exhibit intriguing properties for both fundamental research and potential application in fields ranging from electronic devices to catalysis. Based on first-principles calculations, we proposed a stable form of palladium diselenide (PdSe2) monolayer that can be synthesized by selenizing Pd(111) surface. It has a moderate band gap of about 1.10 eV, a small in-plane stiffness, and electron mobility larger than that of monolayer black phosphorus by more than one order. Additionally, tensile strain can modulate the band gap of PdSe2 monolayer and consequently enhance the infrared light adsorption ability. These interesting properties are quite promising for application in electronic and optoelectronic devices.

plane sandwiched by two Se planes. The thickness of the freestanding PdSe 2 monolayer measured from the distance between the two Se planes is 2.631 Å. The Pd-Se bond length is about 2.528 Å. The optimized lattice constant of the hexagonal lattice is 3.739 Å, quite close to that of the PtSe 2 monolayer, 3.7 Å 9 . Notably, a 3 × 3 PdSe 2 monolayer (11.217 Å) matches well with the 4 × 4 Pd(111) surface (11.186 Å), which implies high plausibility of growing PdSe 2 monolayer on Pd(111) surface via selenization. This allows us to build a PdSe 2 /Pd heterostructure by placing a 3 × 3 PdSe 2 on Pd(111) surface without considering the small lattice mismatch (~0.3%). Compared with freestanding monolayer, the buckling of the supported PdSe 2 increases to 2.710 Å and the bond length vary from 2.51 Å to 2.61 Å with an average of 2.55 Å, due to the substrate effect. The distance between PdSe 2 monolayer and substrate is about 2.16 Å, and the binding energy between them is − 1.04 eV per PdSe 2 unit. The PdSe 2 -Pd(111) interaction is much stronger than the van der Waals (VDW) interaction in graphite (~− 36 meV/atom), making the exfoliation of PdSe 2 from the Pd substrate difficult.
It is noteworthy that the lattice structure of hexagonal PtSe 2 monolayer differs significantly from that cut from orthorhombic PdSe 2 crystal 11 . The latter case has orthorhombic symmetry with four-coordinated Pt atoms and two-coordinated Se atoms, and is slightly stable than the hexagonal structure by about 26 meV/atom. However, the hexagonal PdSe 2 monolayer matches well with the Pd(111) surface in both symmetry and lattice constant along with a negative binding energy (− 1.04 eV per PdSe 2 unit) and thus has high plausibility on the Pd(111) surface as the surface is selenized 9 .
To confirm the dynamic stability of the PdSe 2 monolayer, we calculated the phonon spectrum using a finite displacement method implemented the Phonopy code interfaced the VASP code 13,14 . It was found that the phonon spectrum is free from imaginary frequency modes, as shown in Fig. 1(b), suggesting the dynamic stability of the PdSe 2 monolayer. Additionally, there is a small energy gap between the acoustic and optical branches in the phonon spectrum. Such an energy gap can protect the vibration of acoustic modes from being interrupted by optical phonon [15][16][17] , which leads to the resonators with higher quality factor than the graphene resonators 18 . Mechanical properties. Strain is not evitable as a monolayer is grown on a substrate. We therefore investigated the mechanical properties PdSe 2 monolayer from first-principles. To obtain the elastic modulus in harmonic range, we employed a rectangular supercell and applied strains along x-and y-direction, as shown in Fig. 1(a). The energy (E s ) dependence of strains is illustrated in Fig. 1(c). The data is fitted to a two-dimensional  , where ε x and ε y represent the strains applied along x-and y-direction, respectively. The total energy of system at the equilibrium state was set to zero. It was found that the two parameters a 1 and a 2 are almost identical due to the isotropy in the honeycomb symmetry 19 . The poisson's ratio (ν) and in-plane stiffness (C) can be obtained as υ = a a , where A 0 is the equilibrium area of the structure. The calculated poisson's ratio (0.29) is close to those of MoS 2 (0.25) and silicene (0.30) but larger than that of graphene (0.16). The in-plane stiffness of PdSe 2 monolayer, 54 J/m 2 , is much smaller than those of MoS 2 (123 J/m 2 ) and graphene (335 J/m 2 ) 19,21 , suggesting that PdSe 2 monolayer is softer than both graphene and MoS 2 monolayer.
The softness of PdSe 2 monolayer was also confirmed by the energetic and structural evolution as a function of biaxial strain. Figure 1(d) gives the energy increase of PdSe 2 monolayer as the biaxial strain varies from − 9% (compression) up to 35%. It is interesting to see that energy varies continuously without abrupt decrease, indicating that no Pd-Se bonds are broken in this region. As the tensile strain exceeds 35%, energy decreases drastically and irreversible structure modification along with Pd-Se bond breakage takes place. The two energy local minima at ε = 0 and 20.4% correspond to the equilibrium state and a metal-stable state of PdSe 2 monolayer. The later configuration has the Pd-Se bond length of 2.639 Å, slightly longer than that of the equilibrium state by about 4.4%, but the thickness is greatly reduced to 0.918 Å compared to the equilibrium state value 2.631 Å. The metastable state represents a low-buckled configuration of PdSe 2 monolayer under high tensile strain. Similar results have also been reported for silicene, but the critical tensile strain in PdSe 2 monolayer is much larger than that in silicene 21 .
Electronic properties. We then calculated the electronic structure of PdSe 2 monolayer from first-principles.
We employed the PBE functional and Heyd-Scuseria-Ernzerhof (HSE) screened Coulomb hybrid density functional 22 in the framework of DFT. The band structure and electronic density of states (DOS) near the Fermi level are illustrated in Fig. 2(a). Both functionals gave the semiconducting nature of PdSe 2 monolayer with an indirect band gap, similar to the case of orthorhombic PdSe 2 11 . The valence band maximum (VBM) resides at the Г (0, 0, 0) point while the conduction band minimum (CBM) locates in between Г (0, 0, 0) and M (0.5, 0, 0). Six valleys are therefore found in the whole Brillouin zone as shown in Fig. 2(b), which gives opportunities to valley devices. The PBE functional underestimated the band gap value (0.7 eV) compared with HSE functional (1.10 eV). The band gap of PdSe 2 monolayer is slightly smaller than that of PtSe 2 monolayer (1.2 eV) 9 and orthorhombic PdSe 2 (1.43 eV) 11 and suitable for adsorbing infrared light. From the orbital-resolved electron density of states shown in Fig. 2(a), we can see that the VBM is mainly contributed by the 4p orbits of Se atoms, while the CBM comes mainly from the 4d orbits of Pd and 4p orbits of Se. This is consistent with the features of the Kohn-Sham wavefunctions of the VBM and CBM plotted in Fig. 2(d). Considering Pd is a heave element, we also took the spin-orbit coupling (SOC) effect into account in electronic structure calculations. It was found that the energy degeneracy at some highly-symmetric points in BZ is lifted due to SOC (Fig. S2 in Supplementary  Information)  per Se atom transfer from upper Se layer to down Se layer, while the electron transfer from PdSe 2 monolayer to substrate is almost negligible (~0.007 |e| per PdSe 2 unit). This slightly reduces the workfunction of PdSe 2 to 5.20 eV compared with that of freestanding PdSe 2 monolayer (5.40 eV). We also calculated the electronic structures of PdSe 2 multilayers built by stacking PdSe 2 monolayers with different patterns. For the PdSe 2 bilayer, AA stacking pattern is energetically more favorable than AB pattern (Supplementary Information). The interlayer interaction greatly reduces the band gap of 2D PdSe 2 . The band gap of the PdSe 2 bilayer with AB pattern is only 0.33 eV (PBE), while the energetically preferred AA pattern becomes metallic.
We also calculated the electron mobility of PdSe 2 monolayer using the phonon-limited scattering model 23,24 . In the method, the electron mobility can be evaluated using the expression [24][25][26] where e is the electron charge, ℏ is Planck's constant divided by 2π , k B is Boltzmann's constant and T is the temperature. m e is effective mass of electron in the transport direction (either m x or m y along the x and y direction, respectively) and m a is the averaged effective mass of that along x and y direction determined by where E is the total energy and S 0 is the lattice area at equilibrium for 2D system. The electron mobilities along x-and y-direction are 9800 cm 2 v −1 s −1 and 42000 cm 2 v −1 s −1 at 300 K, respectively. These values are much higher than that in monolayer black phosphorus by more than one order of magnitudes. Such high electron mobility is quite promising for improving the photocatalytic activity of PdSe 2 monolayer, because the photo-generated electrons can be easily transferred 9 .
The electronic structure modification of PdSe 2 monolayer in response to tensile strain was then investigated. With the increase of tensile strain, the band gap of PdSe 2 monolayer decreases and comes to close as the tensile is larger than 14%. More interestingly, compressive strain also reduces the band gap of PdSe 2 monolayer. To reveal the origins of the strain-induced band gap modulation, we plotted the band structures of PdSe 2 monolayer under difference strains in Fig. 3(a). From this figure, we can see that under a tensile strain, the CBM move downwards while the VBM varies slightly, reducing the band gap. This may be related to the enhancement of the Se_4p and Pd_4d interaction due to the shorten distance between the Se plane and Pd plane. Under a compressive strain, however, the VBM is sensitive to strain and shifts upwards, leading to band gap reduction. Besides, it is interesting to see that tensile strain decreases the energy of the G points (illustrated by green arrow in Fig. 3(a)), and lifts the energies of the N and N' points (illustrated by red arrow in Fig. 3(a)). As the tensile strain is larger than 4%, the energy difference between the N and N' points becomes very small, making the PdSe 2 a quasi-direct band semiconductor.
Optical adsorption. The strain-induced band gap reduction affects the optoelectronic properties of PdSe 2 monolayer. We calculated the imaginary part of complex dielectric function using the expression:  2 ; where ′  and ″ is the real and imaginary part of complex dielectric function. The optical absorption coefficient I(ω ) calculated using HSE functional is plotted in Fig. 3(b). At equilibrium state, the adsorption peaks locate in the visible light and ultraviolet light region. A small compression has little affect on the adsorption properties. When a tensile strain is applied to PdSe 2 monolayer, adsorption peaks come to appear in the low energy region, due to the band gap reduction. Meanwhile, the adsorption to visible light is partially screened. For example, under the tensile strains larger than 12%, the adsorption coefficients are almost zero in the energy region from 1.0-2.0 eV. These fascinating properties are quite promising for infrared detectors where the screening of visible light adsorption is needed.

Conclusion
In summary, based on first-principles calculations, we proposed a stable form of palladium diselenide (PdSe 2 ) monolayer that may be synthesized by selenizing Pd(111) surface. The PdSe 2 monolayer possesses a moderate band gap of about 1.10 eV and high electron motilities of about 9800-42000 cm 2 v −1 s −1 , suitable for the electronic and optoelectronic devices working at infrared light region. The band gap of PdSe 2 monolayer can be modulated by applying tensile strains. With the increase of tensile strain, the band gap decreases and comes to close as the tensile strain is larger than 14%. Consequently, the light adsorption ability of the PdSe 2 monolayer in infrared light region is greatly enhanced. These findings are quite promising for application in electronic and optoelectronic devices.

Methods
Our first-principles calculations were performed within density-functional theory (DFT) using the Vienna ab initio simulation package known as the VASP code [27][28][29] . The projector augmented wave method (PAW) 30,31 was used to describe the electronic-ion interaction. The energy cutoff of the plane waves was set to 450 eV with an energy precision of 10 −5 eV. The electron exchange-correlation function was treated using a generalized gradient approximation (GGA) in the form proposed by Perdew, Burke, and Ernzerhof (PBE) 32 . The Monkhorst-Pack k-point meshes 33 for the Brillouin zone (BZ) sampling used in structural optimization and electronic structure calculations are 8 × 8 × 1 and 15 × 15 × 1, respectively. The primitive cell contains one Pd atom and two Se atoms which is periodically repeated along the x-and y-directions. A vacuum region up to 15 Å was applied along the z-direction to exclude the interaction between adjacent images. Both atomic positions and lattice vectors were fully optimized using the conjugate gradient (CG) algorithm until the maximum atomic forces were less than 0.0005 eV/Å. The lattice constants of the orthorhombic PdSe 2 crystal (a = 5.752 Å, b = 5.926 Å) obtained from our calculations agree well with the experimental data (a = 5.746 Å, b = 5.868 Å) 12 , confirming the validity of this strategy.