Strain modulating electronic band gaps and SQ efficiencies of semiconductor 2D PdQ2 (Q = S, Se) monolayer

We studied the physical, electronic transport and optical properties of a unique pentagonal PdQ2 (Q = S, Se) monolayers. The dynamic stability of 2Dwrinkle like-PdQ2 is proven by positive phonon frequencies in the phonon dispersion curve. The optimized structural parameters of wrinkled pentagonal PdQ2 are in good agreement with the available experimental results. The ultimate tensile strength (UTHS) was calculated and found that, penta-PdS2 monolayer can withstand up to 16% (18%) strain along x (y) direction with 3.44 GPa (3.43 GPa). While, penta-PdSe2 monolayer can withstand up to 17% (19%) strain along x (y) dirrection with 3.46 GPa (3.40 GPa). It is found that, the penta-PdQ2 monolayers has the semiconducting behavior with indirect band gap of 0.94 and 1.26 eV for 2D-PdS2 and 2D-PdSe2, respectively. More interestingly, at room temperacture, the hole mobilty (electron mobility) obtained for 2D-PdS2 and PdSe2 are 67.43 (258.06) cm2 V−1 s−1 and 1518.81 (442.49) cm2 V−1 s−1, respectively. In addition, I-V characteristics of PdSe2 monolayer show strong negative differential conductance (NDC) region near the 3.57 V. The Shockly-Queisser (SQ) effeciency prameters of PdQ2 monolayers are also explored and the highest SQ efficeinciy obtained for PdS2 is 33.93% at −5% strain and for PdSe2 is 33.94% at −2% strain. The penta-PdQ2 exhibits high optical absorption intensity in the UV region, up to 4.04 × 105 (for PdS2) and 5.28 × 105 (for PdSe2), which is suitable for applications in optoelectronic devices. Thus, the ultrathin PdQ2 monolayers could be potential material for next-generation solar-cell applications and high performance nanodevices.

. (a) Top-view of penta-PdQ 2 (Q = S, Se) monolayer structure. Red line indicats the unitcell of the system. (b) Side views from z-direction of the atomic structure of penta-PdQ 2 . The navy blue and light-green circles represent the Pd and Q (S, Se) atoms, respectively. The deformation charge density plot (c) for penta-PdS 2 (d) for penta-PdSe 2 ; where red and green colour indicate the charge accumulation (electron excess) and charge deplation (electron loss). The iso-level is 0.063 e/Å 3 for both monolayers. www.nature.com/scientificreports/ force 0.01 eV/Å was kept to relaxing the unstrained and strained structures of penta-PdQ 2 (Q = S, Se) monolayers. The phonon dispersive curves were calculated using the density functional perturbation theory (DFPT) formalism 38 . In all calculation, the Fermi level (E F ) is shifted at the zero energy. The parameters of carrier mobility of penta-PdQ 2 (Q = S, Se) monolayers were found such as i.e., effective mass (m*), stiffness constants (C 2D ) and deformation energy (E 1 ) approximations using the following formula proposed by the Bardeen and Shockley 39 , 3k B T|m * | 2 (E1) 2 ; here, e is elementary charge of an electron, ђ is the reduced Planck's constant, C 2D is the inplane stiffness constant and for 2D system it is defined as where S 0 is cell area and a 0 is the lattice constant at equilibrium, E and a are the total energy and lattice constant of the monolayer after deformation, k B is the Boltzmann constant, T is the temperature (300 K). Also, m* is the effective mass in the transport direction and E 1 is the deformation potential (DP) constant denoting the shifting of each band edges due to the applied biaxial strain ε xy defined as E 1 = ∂E edge ∂( a a 0 ) ; where E edge is energy value of VBM (for holes) and CBM (for electrons). The electronic transport properties of PdQ 2 monolayers are calculated by utilizing the non-equilibrium Green's function (NEGF) in the form of I-V response by TRANSIESTA module of SIESTA code 40,41 . The electric current through the scattering region is calculated by exploring the Landauer-Buttiker formalism, where T (E, V bias ) is the transmission probability of an electron incident at an energy E through the device under the bias voltage V bias , µ L and µ R are the electrostatic potentials of left and right electrodes at a particular voltage bias and G 0 is the unit of quantum conductance. The optical properties were investigated including the dielectric function and absorbance coefficient by DFT within the random phase approximation (RPA) 42 . The dielectric function is given by ɛ (ω) = ɛ' (ω) + i ε'' (ω). Here, ε'(ω) and ε''(ω) are the real and imaginary parts of the complex dielectric function 43 , respectively.

Results and discussion
Stability and mechanical properties. First, the geometric structures of pristine 2D penta-PdQ 2 (Q = S, Se) were systemitically relaxed and lattice structures of penta-PdQ 2 were obtained. It is built up with two Pd and four Q (S, Se) atoms in the unitcell as shown in Fig. 1a,b. The differential charge density in Fig. 1c,d shows an accumulation and depletion of charges between Pd and Q (S, Se) atoms. In Table 1, the relaxed structural parameters are summarized with ground state properties such as, lattice parameter (Å), distance d (Å) between Pd-Q 1 , Pd-Q 2 and Q 1 -Q 2 , thickness t (Å) and cohesive energy E coh (eV/atom) of PdQ 2 monolayers.
As shown, in Table 1, the obtained lattice parameters a (b) of PdS 2 and PdSe 2 monolayers are 5.62 (5.42) Å and 5.94 (5.68) Å, respectively. The atomic distance between Pd-S 1 (Se 1 ), Pd-S 2 (Se 2 ), and S 1 (Se 1 )-S 2 (Se 2 ) atoms are 2.35 (2.45) Å, 2.34 (2.42) Å and 2.14 (2.42) Å, respectively and shows good agreement with previously reported results 17,29,[32][33][34] . To evaluate stability of resulted structures, the cohesive energy is calculated using the formula: E coh = Σ n X E X -E PdQ2 /N. where E PdQ2 denotes total energy of PdQ 2 (Q = S, Se) unit cell, N is total atom in unit cell and n X and E X are number of atoms of each element and atomic energies of each atom in the unit cell, respectively. It is found that cohesive energies for PdS 2 and PdSe 2 monolayer are −4.70 and −4.30 eV/atom, respectively. Although this is larger than that for the 2D-antimony (−4.03), 2D-arsenene (−2.96), 2D-MoS 2 (−4.11), 2D-MoSe 2 (−3.94) and 2D-black phosphorous (−3.48). This is evidence that penta-PdQ 2 monolayers have strong bonding in its ring-network. In addition, to confirm the kinetic stability of penta-PdQ 2 (Q = S, Se) structures, we calculated the phonon dispersion curves along high symmetric points for both PdS 2 and PdSe 2 monolayers and results are as shown in Figure S1 (a, b), ESI † . The highest phonon frequencies extended is up to 2275 cm -1 and 2193 cm -1 for PdS 2 and PdSe 2 monolayers, respectively. Evidently, no negative acoustics branch is appeared in the first Brillouin zone, itself proves the dynamical stability of both the monolayers at room temperature.
Next, we focused on mechanical properties of penta-PdQ 2 and evaluated Young's modulus Y (N/m) and bulk modulus B (N/m). The Young's moduli Y (N/m) and bulk moduli B (N/m) were investigated under uniaxial and biaxial strain on the penta-PdQ 2 monolayers. The range of applied strain ε(%) is taken in interval of −5 % ≤ ε ≤ +5 % in the step of 1%. The bulk modulus computed by applying biaxial load (x and y) on the penta-PdQ 2 monolayer with following relation 44,45 , www.nature.com/scientificreports/ where S 0 is unstrained cross-sectional area of the unit cell. E s and A are the total strain energy and area of respected applied strain, respectively. Figure S2 (a, b), ESI † shows the strain energy vs area curve for pristine and loaded structures of PdS 2 and PdSe 2 monolayers, respectively. This curve is fitted by the polynomial curve fitting and with help of Eq. (1), we have obtained bulk modulus B (N/m) as shown in Figure S2a 44,46 , that indicate that penta-PdQ 2 has stiffer and better resistance to deformation compared to antimony (Sb) and arsenene (As) monolayers. Afterwards, we have calculated the in-plane Young's modulus Y (N/m) in longitudinal (along x-axis) or transverse (along y-axis) direction for penta-PdQ 2 . To obtained Young's modulus from first-principles calculation, the following formula was employed 44,45 , Here, S 0 is unstrained cross-sectional area of the unicell. Next, ∂ 2 E s ∂ε 2 shows the second derivative of strain energy (eV) with applied load. Here, the compressive and tensile strains are applied to the penta-PdQ 2 in the longitudinal or transverse direction. The Young's moduli obtained by using the curves shown in Figure S3a,b, ESI † . The calculated Young's modulus Y (N/m) along x (y) direction are 74.13 (42.40) N/m and 65.26 (28.62) N/m for penta-PdS 2 and penta-PdSe 2 , respectively. The results indicating that, Young's modulus decreases as we switched S to Se, and this is due to the increment in the Pd-S and Pd-Se bond strength. Xiong et al. 47 had calculated orientation-dependent Young's modulus of 2D orthorhombic MX 2 (M = Ni, Pd; X = S, Se, Te) and reported values are ~ 50 (37) N/m for PdS 2 (PdSe 2 ). From the present investigation of Young's modulus of penta-PdQ 2 , we suggest that the 2D penta-PdQ 2 can be preferred candidates for flexible devices because of having ultra-low values of Young's modulus, compared to that of 2D graphene (340 N/m) and 2D MoS 2 (125 N/m) 48,49 .
Further, we have also explored the mechanical stability of penta-PdQ 2 monolayers and examined that up to which values of strain ( % ) the monolayers can withstand? It is worth to find curve of stress-strain relation called ultimate tensile strength (UTSH) curve 50 . The UTSH is representing the maximum stress value (GPa) that a monolayer can withstand prior to the fractured structure set. This can be calculated by the components of stress tensor with respect applied strain. The computed UTSH for penta-PdQ 2 monolayers are shown in Fig. 2a,b. The stress value, where the slope of the stress-strain curve becomes zero indicates the value of UTSH, and the strain at this value represents the magnitude of ultimate tensile strain (UTSR). In case of penta-PdS 2 , the calculated values of UTSR along x (y) directions is 16% (18% ) at stress of 3.44 GPa and 3.43 GPa, respectively (see Fig. 2a). Whereas, in penta-PdSe 2 (see Fig. 2b) it is observed at 17% (19% ) with stress value of 3.46 GPa (3.40) GPa. The present values of UTSR for penta-PdQ 2 are higher than the blue-phosphorene monolayer (~ 16%) 51 . Thus, the UTSR values suggest that the penta-PdQ 2 monolayer is relatively more flexible.
Electronic properties. The band structure, total density of states (TDOS) and partial density of states (PDOS) of penta-PdQ 2 monolayers along high symmetry points of Brillouin zone (BZ) have also investigated. At PBE level, from Fig. 3a,b we can see that the penta-PdS 2 (PdSe 2 ) is an indirect band gap with valence band maximum (VBM) and conduction band minimum (CBM) tracing at S-point (M-point) and Γ-point, respectively. The obtained indirect band gap of penta-PdS 2 is E g PBE = 0.94 eV, while band gap for penta-PdSe 2 is E g PBE = 1.26 eV as seen in Fig. 3a,b, both the values are consistent with previously reported results 30,31 . As DFT-PBE functional may underestimate the bandgaps of semiconductors, we also used GGA + U, LDA, LDA + U and HSE06 52 approximations to ensure calculation reliability and the obtained indirect electronic bandgap of PdS 2 (PdSe 2 ) are   53 . Next, having heavier element (Pd) in 2D penta-PdQ 2 material, the spin-orbit coupling (SOC) effect may play a crucial role and could influence in the band gap (E g ) of material. Thus, we have also checked the SOC effect on the penta-PdQ 2 monolayer and results are shown in Figure S4a,b, ESI † . Using PBE + SOC functional calculation, the obtained indirect band gap of penta-PdS 2 is E g PBE+SOC = 1.16 eV, whereas band gap of penta-PdSe 2 is E g PBE+SOC = 1.51 eV. Recently, the Yang et al. 32 , have reported electronic properties of penta-PdS 2 from PBE (PBE + SOC) calculation and reported indirect band gap of 1.08 eV (1.13 eV), which is comparable with our results within of 12.96% (11.5% ) deviation. Moreover, a research group of Sun et al. 34 , have reported a band gap of penta-PdSe 2 with PBE level theory and mentioned indirect band gap of 1.38 eV, which is higher by 8.6% than our obtained band gap (~ 1.26 eV). Furthermore, with SOC effect, Qin et al. 33 , has computed indirect band gap of penta-PdSe 2 monolayer via TB-mBJ-GGA potential and reported band gap 1.38 eV. The deviation of present band gap of penta-PdSe 2 with PBE + SOC calculation is 9 % . Further, the VBM and CBM are substantially coincide with each other regardless of PBE or HSE06 (with or without SOC) level calculations. Also, the HSE06 level calculations are really time expensive and need heavy computing power, as a results of that, we only used PBE functional level band structures for further calculations by applying strain. The nature of CBM and VBM are vary with applying strain on the PdQ 2 monolayer to reach up the optimal value of 1.3 eV for solar cell applications, which is discussed in SQ Efficiency section. As presented in Fig. 3a,b, it is noticed that the TDOS (States/eV) are mostly ascend due to 'd' orbitals of Pd atom and 'p' orbitals of Q (S, Se) atom in the penta-PdQ 2 monolayer, which are also consistent to the past reported results 32,34 . Additionally, the TDOS difference from PBE to PBE + SOC calculations are shown in Figure S5a,b, (ESI † ) for PdS 2 and PdSe 2 monolayers. we have also investigated PDOS (States/eV) as displayed in Fig. 3 (c, d), to further investigate partial contribution of atoms in penta-PdQ 2 and we conclude that the conduction bands (CBs) and valence bands (VBs) are highly occupied by the 4d-Pd states and 3p-Q (Q = S, Se) states, respectively. Transport properties. Carrier mobility. We further focused on the band alignment in terms of strain effect to drive the electronic properties. Simultaneously, we have also calculated carrier mobility (µ) of monolayer, which is one of essential factor for a high-performance device like FET. For this, we have exerted compressive  Fig. 4. As stated before, the fundamental indirect band gap of penta-PdS 2 and penta-PdSe 2 is 0.96 eV and 1.26 eV, respectively. The Fig. 4 reflects that the band gap of penta-PdQ 2 is gradually decreases almost linearly in presence of tensile strain (+ ε% ) compared to fundamental band gap (0% ). Interestingly, upon increases compressive strain (−ε% ) band gap of penta-PdS 2 increases up to -5% compressive strain and then decreases gradually with respect to 0 % . It is noteworthy that, the maximum value of band gap of penta-PdS 2 is ~ 1.35 eV at ε = −5% and for penta-PdSe 2 is ~ 1.36 eV at ε = −1 % . Remarkably, the resistive phase transition is seen at higher compressive (−ε% ) and tensile strain (+ ε% ) in both cases. The semiconducting nature of penta-PdS 2 (PdSe 2 ) is transform into semi-metallic (~ 0 eV) by applying the biaxial compressive and tensile strain of −13% (−13% ) and + 7% (+ 9% ), respectively. Under the critical compressive strain (−ε% ), the resistive phase change (semiconductor to semi-metallic) in penta-PdQ 2 behaviour arises due to crossing one band of VBM at the Fermi level (Fig.  S6a,b), ESI † ). In contrary, up on the critical tensile (+ ε% ) strain, it occurred due to touching one band of CBM at the Fermi level (Fig. S7a,b, ESI  † ). Additionally, we have also examined the phonon spectrum for each applied strain on penta-PdQ 2 and computed its phonon frequency as summarized in Table S1, ESI † . Thus, our outcomes accomplish those electronic properties of studied 2D penta-PdQ 2 (Q = S, Se) materials are tuneable by applying biaxial strain, which establish that the penta-PdQ 2 is a promising candidate for designing flexible nanodevices.
Owing to tunability of penta-PdQ 2 monolayers, we inspired to calculate the carrier mobility of penta-PdQ 2 monolayers. For that the Bardeen and Shockley formalism 39 had been applied, we have calculated the effective mass (m*), deformation potential E 1 (eV) and stiffness constant C 2D (J m -2 ) to check the flow of charge carrier along biaxial strain. The carrier mobility depends on the most influenced factor on the carrier effective mass m* of electron (e) and hole (h), that is directly derived from the electronic band structure as 55 , Next, deformation potential E 1 is another vital parameter to effect carrier mobility and is obtained from the deformation theory (DP) 39 , which has been also successfully used method in previous reports for 2D MoS 2 , WS 2 and phosphorene. The needed ingredients for carrier mobility (µ) are summarized in Table 2. Along biaxial strain, the obtained effective mass (m*) of electron and hole for penta-PdS 2 (PdSe 2 ) are 0.39 m e (0.58 m e ) and 0.72 m e (0.14 m e ), respectively. Our computed deformation potential E 1 (eV) for penta-PdS 2 is −6.44 eV for e and 6.84 eV for h, that is consistent with ones reported by Wang et al. 17 , along x or y direction. While in case of penta-PdSe 2 , the value of E 1 is −2.91 eV for e and 6.14 eV for h that is also in accordance with the results of Qin et al., 33 for x or y direction. Whereas the in-plane stiffness constant C 2D for penta-PdQ 2 has anisotropic behaviour. The inplane stiffness constant obtained for penta-PdS 2 and penta-PdSe 2 are 121.76 J m -2 and 94.62 J m -2 , respectively. The stiffness constant C 2D of penta-PdS 2 is larger than the penta-PdSe 2 , indicating that PdSe 2 monolayer is softer than the penta-PdS 2 monolayer. As shown in Table 2, the achieved electron mobility of penta-PdS 2 monolayer is 258 cm 2 V −1 s −1 , which is quite higher than the hole mobility ~ 67 cm 2 V −1 s −1 . Also, the carrier mobility ratio R comes out to be 0.26. Moreover, Wang et al. 17 has reported the electron mobilities (cm 2 V −1 s −1 ) of penta-PdS 2 as 40.97 (x) and 169.11 (y), while hole mobilities (cm 2 V −1 s −1 ) as 339.25 (x) and 91.73 (y). On other hand, if we compare carrier transport along biaxial strain of penta-PdSe 2 the hole mobility is 1518 cm 2 V −1 s −1 that is greater than the electron mobility ~ 442.49 cm 2 V −1 s −1 and R is 3.43. However, it is observed that the hole mobility  www.nature.com/scientificreports/ of penta-PdSe 2 is higher than the 2D phosphorene 56 (i.e., 640-700 cm 2 V −1 s −1 ) and BN nanosheet (i.e., 500 cm 2 V −1 s −1 ) 57 , indicating that penta-PdSe 2 would be a promising material for modelling electronic applications.

Current-voltage (I-V) characteristics.
To understand the current sensitivity of the penta-PdQ 2 monolayers, we have examined the I-V characteristics, based on the equivalent transport theory 41 . Figure 5 represents the schematic view of two-terminal device, where we have considered LE (left electrode), RE (right electrode) and scattering region of same material. Here, we have taken 10 primitive cells, among them the lead had built up with four-unit cell (24 atoms) and central region consist of 12-unit cells (72 atoms). The I-V characteristic of both penta-PdS 2 and penta-PdSe 2 monolayers are shown in Fig. 6a,b. It is seen that, no significant current is observed till ~ 1.0 V and 1.5 V then it starts to increase with increasing applied voltages due to semiconducting nature of penta-PdS 2 and PdSe 2 monolayers, respectively. Apparently, it can be seen form Fig. 6a,b penta-PdS 2 is more current-sensitive than the penta-PdSe 2 monolayer. Although, in the case of penta-PdS 2 , the first peak in current is seen at 2.5 V with 14 µA and second current peak is located at 4.0 V with current of 13.2 µA. This is indicating a pronounced negative differential conductance effect (NDC) 58 in the bias range 2.0-3.5 V (see Fig. 6a). While in case of penta-PdSe 2 , the NDC effect is seen in range of 1.0-4.5 V (see Fig. 6b). More precisely, the first NDC effect is occurs at the Table 2. The carrier effective mass |m*| (m e , the mass of free electrons), deformation potential constant E 1 (eV), stiffness constant C 2D (J m −2 ), carrier mobility (μ) for electron (e) and hole (h) and carrier mobility ratio (R) along biaxial direction of penta-PdQ 2 monolayers at 300 K.

System
Carrier type m*/m e E 1 (eV) C 2D (J m −2 ) μ (cm 2 v −1 s −1 ) R = μ h /μ e   SQ efficiency. We also focused on the power conversion efficiency (PCE) of penta-PdQ 2 (Q = S, Se) using Shokley-Queisser (SQ) limit that gives the percentage of power converted to electrical energy 60,61 . For this we required several parameters such as short circuit current density ( J sc ), open circuit voltage ( V oc ), maximum power density ( P max ) and efficiency η(%) . The theoretical short circuit current density J sc is calculated using following relation 62,63 , where e is the elementary charge, A(E) represents the absorbance of layer 62 and I sun (E) is the photon flux density as specified in the AM1.5G spectrum 64 , Also, the theoretical reverse saturation current J 0 is defined as 65 ; Here I bb (E, T) is the black body spectrum and the radiative recombination fraction, which is supposed to be unity 60 . Thus, the total current density can be obtained using the formula 65 , where k is the Boltzmann's constant, V is the voltage over the absorber of the 2D devices. By the relation of P = JV , the maximum power density P max can be evaluated from the maxima of the J − V curve, as presented in Fig. 7a,b. Eventually, with help of above-mentioned parameters, now we have obtained solar cell efficiency η(%) by formula 59 , where P in 1000 W/m 2 is total incident power density from solar irradiation of AM1.5G and P max is maximum power density. As mentioned in the SQ limit theory, the electronic band gap (E g ) plays a pivotal role in the calculation of the material efficiency. Moreover, the maximum possible SQ efficiency limit is specified of 33.70% in circumstance of semiconductor with 1.34 eV optimum band gap 66 . As seen from the Table 3, the remarkable conversion efficiency for pristine penta-PdS 2 and penta-PdSe 2 is observed and are 30.79 and 33.54% , respectively. Also, the efficiency of penta-PdSe 2 is larger than the efficiency of benchmark GaAs (~ 31.4% ) solar cell with single junction. Next, the band gap of both monolayers could be well tuned from 0.34 eV ~ 1.35 eV under influence of biaxial strain ( % ) as shown in Fig. 8a,b. Therefore, we have also checked the conversion efficiency under compressive (−ε% ) and tensile strain (+ ε% ) for both the monolayers and presented in Fig. 8a,b. It is noted that, when we have exerted compressive strain (−ε% ) on the penta-PdQ 2 the efficiency at certain strain is higher as compared to the pristine (0 % ), that of 33.93 % (at −5 % of penta-PdS 2 ) and −33.94 % (at −2 % of penta-PdSe 2 ). On other hand, under tensile strain the maximum efficiency arises that of 25.72 % at + 1 % of penta-PdS 2 and 33.63 % at + 1 % of penta-PdSe 2 , which is lower than with respective each of pristine (0 % ). This suggests that, by applying biaxial strain on the Figure 7. The calculated J − V characteristic and power density curves of (a) for penta-PdS 2 and (b) for penta-PdSe 2 for pristine and compressive (-ε% ) or tensile (+ ε% )) loaded strain (maximum efficiency η ( ε% ) of strain taken) at T = 300 K. . www.nature.com/scientificreports/ penta-PdQ 2 the related SQ efficiency could be tuned. Also, the fabrications of penta-PdQ 2 solar cell have potential to improve photovoltaic performance compared to traditionally use Si-based solar cells.

Scientific Reports
Optical properties. In order to evaluate light absorption ability of penta-PdQ 2 , we have calculated the real ( ε ') and imaginary ( ε ") dielectric functions, absorption coefficient I(ω) , refractive index n(ω) , and reflectivity R(ω) of penta-PdQ 2 monolayers with parallel (E c) and perpendicular (E ⊥ c) electric vector (E). The computation is carried out using the Kramers-Kroning (KK) relationship 67 Figure 9a,b shows the graphical image of real ( ε ') and imaginary ( ε ") components of penta-PdQ 2 , that simply gives the idea   www.nature.com/scientificreports/ about the electronic polarizability of the material from the Clausis-Mossotti 68 relation. It can be seen from the Fig. 9a,b, the static dielectric constant ( ε (0), ε ⊥ (0)) of penta-PdS 2 is (3.56, 2.62), while static dielectric constant of penta-PdSe 2 is 4.0 ( ε (0)) and 2.94 ( ε ⊥ (0)). This means that its ε(0) of parallel optical vector have more dominant polarizability compared to the perpendicular of ε(0). The optical peculiarity is strongly connected to the imaginary part of the dielectric function ε"(ω), which reflects the transition between the energy bands near the E F levels and that would be govern the linear response of the material to light under small wave vector as depicted in Fig. 9 (c, d). Therefore, the imaginary part ε"(ω) in case of penta-PdS 2 , the one major peak occurred at 1.98 eV in the visible (VIS) region for both ε ⊥ (ω) and ε || (ω) case as plotted in Fig. 9c. While, in case of penta-PdSe 2 , it is clearly seen (Fig. 9d) that there are two intense peaks appeared in the visible region (VIS) at ħω = 2.04 eV and 2.76 eV due to the electronic transition of '4p' states of Se atom and '4d' states of Pd atom between conduction bands (CBs) and valence bands (VBs) of the electronic band structure. Apparently, the peaks tendency of ε ⊥ (ω) and ε || (ω) are similar on the photon energy spectrum for both penta-PdQ 2 monolayers.  www.nature.com/scientificreports/ The optical absorption spectra of the penta-PdQ 2 were recorded by absorption coefficient I(ω) vs photon energy (eV) as shown in Fig. 9e,f. For penta-PdQ 2 , the first absorption peak orientation is in the ~ 1.5-5 eV and second broad peak's orientation is cover the UV region up to photon energy of 16 eV. For penta-PdS 2 , the low energy excitonic peak occur around 2.1 eV in infrared region (IR) and that peak rise towards the visible region at 3.78 eV for both the polarization. The maximum absorption coefficient I(ω) of penta-PdS 2 for parallel and perpendicular polarization is 3.65 × 10 5 and 4.04 × 10 5 , respectively (See Fig. 9e). While, in case of penta-PdSe 2 , the first peak is located at 2.2 eV in the IR region and kick off in visible region at highest absorption coefficient I(ω) of 4.12 × 10 5 . However, with perpendicular polarization the highest coefficient found is about 5.28 × 10 5 in the UV region (See Fig. 9f). Thus, the penta-PdQ 2 monolayers have shown prominent absorption in VIS and UV region, which suggest a good prospect of penta-PdQ 2 monolayers as UV filters and UV photodetectors.
Next, the computed refractive index n(ω) and reflectance R(ω) of penta-PdQ 2 have the same evolution curve as plotted in Figure S8 (a, b), ESI † and S9 (a, b), ESI † . In penta-PdS 2 , we observed static refractive index as ( n (0), n ⊥ (0)) = (1.90, 1.63) (see Figure S8 (a), ESI † ). With the parallel polarization, the n(ω) is rising in the VIS region up to ħω = 1.78 eV, then going down up to energy of 2.32 eV. Similarly, in case of perpendicular polarization it has also gone down in the VIS region with 1.23 of n (ω) by the same spectrum photon energy. Moreover, the reflectivity R(ω) is also shown in Figure S9 (a), ESI † . The static reflectance ( R � (0), R ⊥ (0) ) of penta-PdS 2 in VIS region is about to (10% , 5 % ) and the most elevated peak of reflectance is found to be in the VIS region with reflectance of 31% as seen in Figure S9 (a, b), ESI † . While, for penta-PdSe 2, the static refractive index ( n (0), n ⊥ (0)) is (2.0, 1.71) as shown in Figure S8 (b), ESI † , which is also in good agreement with previous reported work on penta-PdSe 2 monolayer by the Zhao et al. 69 group. The first and highest peak of n(ω) is trapped in the VIS region at energy of 1.68 eV, at that energy the n (ω) is 2.65. The intense peak of n ⊥ (ω) is also located at ħω = 1.68 eV in the VIS region with n ⊥ (ω) of 1.68. The static reflectance ( R � (0), R ⊥ (0) ) of penta-PdSe 2 in VIS region is about to (11% , 7 % ) as seen in Figure S9 (b), ESI † . Also, the highest peak of R ⊥ (ω) is occurred in the VIS region with the reflectance of 26% , that reflectance is in good agreement with Zhao et al. 69 reported results of penta-PdSe 2 monolayer. Overall, the prominent peaks appeared in the n(ω) and R(ω) in the VIS region, that means most of incident light energy is reflected and refracted in VIS region and only small amount of incident energy is absorbed by the material.

Conclusions
In the summary of the work, the first principles method has been employed to study the band structure, bulk modulus, Young's modulus, transport properties such as carrier mobility and I-V curve, and optical properties of penta-PdQ 2 (Q = S, Se) monolayers. The positive phonon spectrum gives the ground state dynamical stability of both the materials. The obtained electronic band gap is 0.94 eV and 1.26 eV for penta-PdS 2 and penta-PdSe 2 , respectively. From mechanical stability point of view, penta-PdS 2 withstands up to 16% (18% ) in x (y) directions, while penta-PdSe 2 is up to 17% (19% ), which undoubtedly implies that each monolayer is flexible up to that strain. Interestingly, the I-V curve shows the NDC effect beyond the bias voltage of 2.5 V (for PdS 2 ) and 3.0 V (for PdSe 2 ), hence this feature leads us to conclude that penta-PdQ 2 monolayers will be consider in future as promising material for NDC-based nanodevices. More importantly, the SQ efficiencies for pristine PdS 2 and PdSe 2 were 30.79% and 33.54% , respectively, that could be essential utilized in the solar cell application. The computed optical properties reveal that the absorption range of penta-PdQ 2 is very broad in UV regions. The maximum absorption coefficient I(ω) of penta-PdS 2 and penta-PdSe 2 are found in the UV region are 4.04 × 10 5 and 5.28 × 10 5 , respectively. Collectively, due to unique electronic, mechanical, transport and optical properties of penta-PdQ 2 monolayers, the 2D materials have application prospect in the arena of semiconducting nanodevices.