Semiconductor Two-Dimensional PdQ2 (Q=S, Se) Monolayer: Strain Modulating Electronic Band Gaps and SQ E ciencies


 We studied the physical, electronic transport and optical properties of a unique pentagonal PdQ2 (Q= S, Se) monolayers. The dynamic stability of 2D - wrinkle 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.

Gronvold et al., studied the PdS2/PtS2 and concluded that noble metals (Pd and Pt) would be a layered structure with sulphur atoms [15,16]. Afterwards, in 2015 Wang et al., theoretically reported that 2D PdS2 is semiconductor with moderate indirect band gap up to ~ 1.60 eV, which is very significant in fabrication of nanodevices [17]. Moreover, Cheng et al., [18] have developed ultrafast Yb doped fiber laser using few layers of PdS2 and indicated that layered PdS2 can be a favourable candidate for photonics application. Some work have also reported on the layered PdSe2 via method of mechanical exfoliated, atomic-resolution scanning transmission electron microscopy (AR-STEM) and chemical vapor deposition (CVD) [14,[19][20][21][22][23][24]. For Example, Akinola et al., [24] had fabricated few layers puckered pentagonal PdSe2 by AR-STEM method and demonstrated that, PdSe2 exhibits good ambipolar semiconducting nature with room temperature electron-apparent field-effect mobility (up to ~ 158 cm 2 V -1 s -1 ) and its stability is remain same up to 60 days. It has been noted that having high mobility (electron/hole) and band gap between 1.2 ~ 1.9 eV of TMD, it makes them suitable for the FETs (field effect transistors) and CMOS applications, respectively [25][26][27][28].
Recently, Weiting and their research group [29] had successfully synthesized 2D PdSe2 nanosheets on 300 nm SiO2/Si substract and claimed that it is really promising material to make infrared photodetector due to its high-photoresponsivity (~ 660 A.W -1 under 914 nm laser) in harsh condition, which may be benefitted in military field for night-time detection because of its ability to work at night as well as in bad whether conditions. Beside these, few other reports theoretical are also available on the 2D pentagonal type PdQ2 monolayers [17,[30][31][32][33][34]. Deng et al., [31] investigated the strain applied mechanical, electronic and optical properties of PdS2, PdP2 and PtSe2. Except this, none of works have been reported on strain applied mechanical, electronic, optical and electronic transport properties of 2D penta-PdQ2 (Q= S, Se). Further, the lower band gap of 2D-PdQ2 monolayers has also motivated us to investigate absorbance of solar radiation on nanosheet. Hence, we also studied the strain dependent solar cell power conversion efficiency % (PCE) of penta-PdQ2 (Q= S, Se) monolayers.
In this work, we have investigated the geometric, mechanical, electronic transport and optical properties of penta-PdQ2 using density functional theory (DFT). After the optimization of structures, we analysed the lattice parameters (Å), band gap (eV) and effective mass (m*). Then after we applied biaxial strain ɛ (%) and obtained its bulk modulus B (GPa).
The ultimate tensile strength (UTHS) is also found under the tensile strain (+ɛ%) for PdQ2 (Q= S, Se). We also studied the influence of loading on the band edges and obtained carrier mobility (µ2D) along biaxial strain ɛ (%). We have adopted Shockley-Queisser (SQ) method to study Solar cell efficiency η (%) and calculated the maximum power density ( ). The optical properties including the dielectric constants, absorption co-efficient, refractive index, and reflectivity of penta-PdQ2 (Q= S, Se) monolayers are also reported in this paper. Herein, section 2 contains the methedology of computations, section 3 includes the results and disccusion part of the work on penta-PdQ2 (Q= S, Se) monolayers.

Computational Methods
The structural, electronic transport and optical properties of 2D wrinkled PdQ2 (Q= S, Se) monolayers were performed with in the SIESTA code [35]. The exchange-correlation functional approach was used in term of Perdew-Burke-Ernzerhof (PBE) type pseudopotential [36]. The basic unit cell of 2D-PdQ2 (Q= S, Se) contains two Pd atoms and four Q (Q=S, Se) atoms as shown in Figure 1. An energy cut-off of 450 and 300 Ry for PdS2 and PdSe2 monolayers were adopted, respectively. The doubled zeta plus (DZP) basis set was used with an energy of 0.02 Ry to expand the Kohn-Sham orbital. The Гcentred mesh of 20×20×1 and 15×15×1 k-points were sampled for PdS2 and PdSe2 monolayers, respectively under the Monkhorst-pack scheme in the 2D Brillouin zone [37]. The force 0.01 eV/Å was kept to relaxing the unstrained and strained structures of penta-PdQ2 (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 (EF) is shifted at the zero energy. The parameters of carrier mobility of penta-PdQ2 (Q=S, Se) monolayers were found such as i.e., effective mass (m*), stiffness constants (C2D) and deformation energy (E1) approximations using the following formula proposed by the Bardeen and Shockley [39], µ2D here, e is elementary charge of an electron, ђ is the reduced Planck's constant, C2D is the in-plane stiffness constant and for 2D system it is defined as 2 = Where, T (E, Vbias) is the transmission probability of an electron incident at an energy E through the device under the bias voltage Vbias, and are the electrostatic potentials of left and right electrodes at a particular voltage bias and G0 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.

Stability and Mechanical Properties
First, the geometric structures of pristine 2D penta-PdQ2 (Q = S, Se) were systemitically relaxed and lattice structures of penta-PdQ2 were obtained. It is built up with two Pd and four Q (S, Se) atoms in the unitcell as shown in Figure 1(a, b). The differential charge density in Figure 1 (c, 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-Q1, Pd-Q2 and Q1-Q2, thickness t (Å) and cohesive energy Ecoh (eV/atom) of PdQ2 monolayers.    --(N/m) were investigated under uniaxial and biaxial strain on the penta-PdQ2 monolayers. The range of applied strain (%) is taken in interval of -5% ≤ ≤ +5% in the step of 1%. The bulk modulus computed by applying biaxial load (x and y) on the penta-PdQ2 monolayer with following relation, [44,45] Where, S0 is unstrained cross-sectional area of the unit cell. Es and A are the total strain energy and area of respected applied strain, respectively. Figure S2 [44,46], that indicate that penta-PdQ2 has stiffer and better resistance to deformation compared to antimony (Sb) and arsenene (As) monolayers. Afterwards, we have calculated the in-plane for penta-PdQ2. To obtained Young modulus from first-principles calculation, the following formula was employed [44,45], Here, S0 is unstrained cross-sectional area of the unicell. Next, 2 ɛ 2 shows the second derivative of strain energy (eV) with applied load. Here, the compressive and tensile strains are applied to the penta-PdQ2 in the longitudinal or transverse direction. The Young moduli obtained by using the curves shown in Figure S3 (a, b)  Further, we have also explored the mechanical stability of penta-PdQ2 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-PdQ2 monolayers are shown in Figure   2 (a, 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-PdS2, the calculated values of UTSR along x (y) directions is 16% (18%) at stress of 3.44 GPa and 3.43 GPa, respectively (see Figure 2(a)). Whereas, in penta-PdSe2 (see Figure 2(b)) it is observed at 17% (19%) with stress value of 3.46 GPa (3.40) GPa. The present values of UTSR for penta-PdQ2 are higher than the bluephosphorene monolayer (~16%) [51]. Thus, the UTSR values suggest that the penta-PdQ2 monolayer is relatively more flexible.

Electronic Properties
The band structure, total density of states (TDOS) and partial density of states (PDOS) of penta-PdQ2 monolayers along high symmetry points of Brillouin zone (BZ) have also investigated. At PBE level, from Figure 3 (a, b) we can see that the penta-PdS2 (PdSe2) 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-PdS2 is Eg PBE = 0.94 eV, while band gap for penta-PdSe2 is Eg PBE = 1.26 eV as seen in Figure 3 (a, b).We also did calculations using GGA+U, LDA, LDA+U and HSE06 approximations for PdS2 (PdSe2) and obtained indirect band gaps are 1.05 (1.26) eV, 0.92 (1.31) eV, 0.94 (1.30) and 0.92 eV, respectively. Next, having heavier element (Pd) in 2D penta-PdQ2 material, the spin-orbit coupling (SOC) effect may play a crucial role and could influence in the band gap (Eg) of material. Thus, we have also checked the SOC effect on the penta-PdQ2 monolayer and results are shown in Figure S4 (a, b) Figure 3 (a, 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-PdQ2 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 S5 (a, b), (ESI † ) for PdS2 and PdSe2 monolayers. we have also investigated PDOS (States/eV) as displayed in Figure 3 (c, d), to further look into partial contribution of atoms in penta-PdQ2 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.

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 (-ε%) and tensile (+ε%) strain on penta-PdQ2 (Q = S, Se) monolayers and traced influence on the band edges of CBM and VBM in the band structure, where system is anisotropic (a not equal to b). The strength of strain along biaxial direction is given by [52], ε = [(S-S0)/S0] × 100 % ; here, S0 is the unstrained lattice constant and S is the strained lattice constant of the monolayer. The response of applied strain to band gaps of penta-PdQ2 is shown in Figure 4. As stated before, the fundamental indirect band gap of penta-PdS2 and penta-PdSe2 is 0.96 eV and 1.26 eV, respectively. The Figure 4 reflects that the band gap of penta-PdQ2 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-PdS2 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-PdS2 is ~ 1.35 eV at ε = -5% and for penta-PdSe2 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-PdS2 (PdSe2) 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-PdQ2 behaviour arises due to crossing one band of VBM at the Fermi level ( Figure S6 (a, b), ESI † ). In contrary, up on the critical tensile (+ε%) strain, it occurred due to touching one band of CBM at the Fermi level ( Figure S7 (a, b), ESI † ). Additionally, we have also examined the phonon spectrum for each applied strain on penta-PdQ2 and computed its phonon frequency as summarized in Table S1, ESI † . Thus, our outcomes accomplish those electronic properties of studied 2D penta-PdQ2 (Q= S, Se) materials are tuneable by applying biaxial strain, which establish that the penta-PdQ2 is a promising candidate for designing flexible nanodevices. Table 2. The carrier effective mass |m*| (me, the mass of free electrons), deformation potential constant E1 (eV), stiffness constant C2D (J m -2 ), carrier mobility (μ) for electron (e) and hole (h) and carrier mobility ratio (R) along biaxial direction of penta-PdQ2 monolayers at 300 K.
Owing to tunability of penta-PdQ2 monolayers, we inspired to calculate the carrier mobility of penta-PdQ2 monolayers. For that the Bardeen and Shockley formalism [39] had been applied, we have calculated the effective mass (m*), deformation potential E1 (eV) and stiffness constant C2D (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 [53], Next, deformation potential E1 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 MoS2, WS2 and phosphorene. The needed ingredients for carrier mobility (µ) are summarized in Table 2 (0.14 me), respectively. Our computed deformation potential E1 (eV) for penta-PdS2 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-PdSe2, the value of E1 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 C2D for penta-PdQ2 has anisotropic behaviour. The in-plane stiffness constant obtained for penta-PdS2 and penta-PdSe2 are 121.76 J m -2 and 94.62 J m -2 , respectively. The stiffness constant C2D of penta-PdS2 is larger than the penta-PdSe2, indicating that PdSe2 monolayer is softer than the penta-PdS2 monolayer. As shown in Table   2, the achieved electron mobility of penta-PdS2 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- and R is 3.43. However, it is observed that the hole mobility of penta-PdSe2 is higher than the 2D phosphorene [54] (i.e., 640-700 cm 2 V -1 s -1 ) and BN nanosheet (i.e., 500 cm 2 V -1 s -1 ) [55], indicating that penta-PdSe2 would be a promising material for modelling electronic applications.

Current-Voltage (I-V) Characteristics
To understand the current sensitivity of the penta-PdQ2 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 electrons are driven from the cold to the hot lead through central region.
Here, we have taken 10 primitive cells, among them the lead had built up with fourunit cell (24 atoms) and central region consist of 12-unit cells (72 atoms). The I-V characteristic of both penta-PdS2 and penta-PdSe2 monolayers are shown in Figure 6 (a, 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-PdS2 and PdSe2 monolayers, respectively. Apparently, it can be seen form Figure 6 (a, b) penta-PdS2 is more current-sensitive than the penta-PdSe2 monolayer. Although, in the case of penta-PdS2, 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) [56] in the bias range 2.0-3.5 V (see Figure 6(a)). While in case of penta-PdSe2, the NDC effect is seen in range of 1.0-4.5 V (see Figure 6

SQ Efficiency
We also focused on the power conversion efficiency (PCE) of penta-PdQ2 (Q = S, Se) using Shokley-Queisser (SQ) limit that gives the percentage of power converted to electrical energy [58,59]. For this we required several parameters such as short circuit current density ( ), open circuit voltage ( ), maximum power density ( ) and efficiency (%). The theoretical short circuit current density is calculated using following relation [60,61], where is the elementary charge, ( ) represents the absorbance of layer [60] and ( ) is the photon flux density as specified in the AM1.5G spectrum [62], Also, the theoretical reverse saturation current 0 is defined as 0 = ∫ ( ) ( ) ∞ 0 [63]; Here ( , ) is the black body spectrum and the radiative recombination fraction, which is supposed to be unity [58]. Thus, the total current density can be obtained using the formula [63], where is the Boltzmann's constant, is the voltage over the absorber of the 2D devices. By the relation of = , the maximum power density can be evaluated from the maxima of the − curve, as presented in Figure 7(a, b). Eventually, with help of above-mentioned parameters, now we have obtained solar cell efficiency η (%) by formula [59], = where (1000 2 ⁄ ) is total incident power density from solar irradiation of AM1.5G and is maximum power density. of strain taken) at T = 300 K (a) for penta-PdS2 and (b) for penta-PdSe2.
As mentioned in the SQ limit theory, the electronic band gap (Eg) 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 [64].
As seen from the Table 3, the remarkable conversion efficiency for pristine penta-PdS2 and penta-PdSe2 is observed and are 30.79 and 33.54%, respectively. Also, the efficiency of penta-PdSe2 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 Figure 8 (a, b). Therefore, we have also checked the conversion efficiency under compressive (-%) and tensile strain (+ %) for both the monolayers and presented in Figure 8 (a, b). It is noted that, when we have exerted compressive strain (-%) on the penta-PdQ2 the efficiency at certain strain is higher as compared to the pristine (0%), that of 33.93% (at -5% of penta-PdS2) and -33.94% (at -2% of penta-PdSe2). On other hand, under tensile strain the maximum efficiency arises that of 25.72% at +1% of penta-PdS2 and 33.63% at +1% of penta-PdSe2, which is lower than with respective each of pristine (0%). This suggests that, by applying biaxial strain on the penta-PdQ2 the related SQ efficiency could be tuned. Also, the fabrications of penta-PdQ2 solar cell have potential to improve photovoltaic performance compared to traditionally use Si-based solar cells.

Optical Properties
In order to evaluate light absorption ability of penta-PdQ2,  shaded by the light-pink and light-green colour, respectively. Figure 9 (a, b) shows the graphical image of real ( ') and imaginary ( ") components of penta-PdQ2, that simply gives the idea about the electronic polarizability of the material from the Clausis -Mossotti [66] relation. It can be seen from the Figure 9 (a, b), the static dielectric constant ( ∥ (0), ⊥ (0)) of penta-PdS2 is (3.56, 2.62), while static dielectric constant of penta-PdSe2 is 4.0 ( ∥ (0)) and 2.94 ( ⊥ (0)). This means that its (0) of parallel optical vector have more dominant polarizability compared to the perpendicular of (0). The optical peculiarity is strongly connected to the imaginary part of the dielectric function "(ω), which reflects the transition between the energy bands near the EF levels and that would be govern the linear response of the material to light under small wave vector as depicted in Figure 9 (c, d). Therefore, the imaginary part "(ω) in case of penta-PdS2, the one major peak occurred at 1.98 eV in the visible (VIS) region for both ⟂ " ( ) and ǁ " ( ) case as plotted in Figure 9 (c). While, in case of penta-PdSe2, it is clearly seen (Figure 9 (d)) 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 ⟂ " ( ) and ǁ " ( ) are similar on the photon energy spectrum for both penta-PdQ2 monolayers.
The optical absorption spectra of the penta-PdQ2 were recorded by absorption coefficient ( ) vs photon energy (eV) as shown in Figure 9 (e, f). For penta-PdQ2, 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-PdS2, 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 ( ) of penta-PdS2 for parallel and perpendicular polarization is 3.65 x 10 5 and 4.04 x 10 5 , respectively (See Figure   9 (e)). While, in case of penta-PdSe2, the first peak is located at 2.2 eV in the IR region and kick off in visible region at highest absorption coefficient Ι ( ) of 4.12×10 5 . However, with perpendicular polarization the highest coefficient found is about 5.28×10 5 in the UV region (See Figure 9 (f)). Thus, the penta-PdQ2 monolayers have shown prominent absorption in VIS and UV region, which suggest a good prospect of penta-PdQ2 monolayers as UV filters and UV photodetectors.
Next, the computed refractive index ( ) and reflectance ( ) of penta-PdQ2 have the same evolution curve as plotted in Figure S8 (a, b), ESI † and S9 (a, b), ESI † . In penta-PdS2, we observed static refractive index as ( ∥ (0), ⊥ (0)) = (1.90, 1.63) (see Figure   S8 (a), ESI † ). With the parallel polarization, the ( ) 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 ( ) is also shown in Figure S9 (a), ESI † . The static reflectance ( ∥ (0), ⊥ (0)) of penta-PdS2 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-PdSe2, the static refractive index ( ∥ (0), ⊥ (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-PdSe2 monolayer by the Zhao et al [67] group. The first and highest peak of ( ) is trapped in the VIS region at energy of 1.68 eV, at that energy the ∥ (ω) is 2.65. The intense peak of ⊥ (ω) is also located at ħω = 1.68 eV in the VIS region with ⊥ (ω) of 1.68. The static reflectance ( ∥ (0), ⊥ (0)) of penta-PdSe2 in VIS region is about to (11%, 7%) as seen in Figure S9 (b), ESI † . Also, the highest peak of ⊥ ( ) is occurred in the VIS region with the reflectance of 26%, that reflectance is in good agreement with Zhao et al [67] reported results of penta-PdSe2 monolayer. Overall, the prominent peaks appeared in the ( ) and ( ) 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-PdQ2 (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-PdS2 and penta-PdSe2, respectively.
From mechanical stability point of view, penta-PdS2 withstands up to 16% (18%) in x (y) directions, while penta-PdSe2 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 PdS2) and 3.0 V (for PdSe2), hence this feature leads us to conclude that penta-PdQ2 monolayers will be consider in future as promising material for NDC-based nanodevices. More importantly, the SQ efficiencies for pristine PdS2 and PdSe2 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-PdQ2