Two-dimensional pentagonal material Penta-PdPSe: A first-principle study

The atomic structure of the PdPSe monolayer in the diverse views is represented in Figs. 1(a). In the PdPSe, the lattice with a pentagonal primitive unit cell is determined by eight atoms (2Pd, 2 P, and 2 Se), and the primitive unit cell is shown as a red square. The pentagonal lattice of PdPSe belongs to the $Pbca$ (60) space group with a unit cell lattice constants of 5.86 Å (a) and 5.79 Å (b). The bond lengths are determined to be: 2.85 Å ($d_{1}$), 2.48 Å ($d_{2}$), 2.31Å ($d_{3}$) and 2.20 Å ($d_{4}$). While the bond angles of $\theta_{2}$), $\theta_{2}$ and $\theta_{3}$ are calculated 114.86, 103.83 and 112.19 ^∘,respectively. The PdPSe monolayer is not a planar structure. The buckling is determined as 1.46 Å. The structural parameters are summarized in Table I.

The cohesive energy per atom, the energy obtained by adjusting the atoms in a crystalline compared to the gas state, that is quantifying the stability of materials, was computed applying the following equation:

where $E_{Pd}$, $E_{P}$ and $E_{Se}$ represent the energies of isolated single Pd, P and Se atoms; $E_{tot}$ shows the total energy of the PdPSe. Additionally, n_Pd, n_P, and n_Se determines the number of Pd, P, and Se atoms in the primitive unit cell, respectively. Our results show that the cohesive energies are -0.18 eV/atom. The negative cohesive energy obtained for this layer suggests that the free-standing state of the nanostructure remains likely to be stable.

The difference charge density showed in Fig. 1(b) is specified as:

where $\rho_{tot}$, $\rho_{Pd}$, $\rho_{P}$ and $\rho_{Se}$ depicts the charge densities of the isolated atoms and PdPSe monolayer, respectively. The charge depletion and accumulation are illustrated by coloring schemes with yellow and blue regions, correspondingly. C and N atoms are negatively charged and are enclosed by the positively charged B atoms. Based on the charge transfer ($\Delta\rho$) analysis, the Se atoms gain as 0.18 from the nearby Pd and P atoms. The electrostatic potential of the PdPSe monolayer is flat in the vacuum area (Fig. 1(c)). Applying the formula $\Phi=E_{vacuum}-E_{F}$, the work function of the PdPSe was defined as 5.90 eV.

The phonon dispersion spectrum through the whole BZ is shown in Figs. 2(a). The phonon spectrum represents an in-plane transverse (TA) and a longitudinal (LA) optical mode, with a linear dispersion, and an out-of-plane flexure mode (ZA) with quadratic dispersion in the long-wavelength limit. The quadratic dispersion results in the large density of modes for ZA flexural phonons, which carry the most heat comparison LA and TA phonon modes. We may expect the other way around because the flexural acoustic ZA phonons have vanishing group velocities for a wave vector near zero. The selection rule for three-phonon scattering can also explain this unexpected result, which strongly restricts the phase space for this scattering [52]. Flexural modes play a vital role in contributing to the thermal conductivity of graphene and structures with flexural modes, both as carriers [52] and as scatterers [53]. Flexural modes are perceived in the eminent 2DMs such as graphene[52], hexagonal boron nitride (h-BN)[54], and arsenene[55].

The phonon dispersion spectrum is free from any imaginary frequency, discovering the dynamical stability of the PdPSe monolayer. In addition, a remarkable gap between the high-frequency and low-frequency optical branches is present. The splitting between the longitudinal and the transversal optical phonon frequencies (LO-TO splitting) presented at the $\Gamma$ point predicts the effects of the dipole-dipole interactions and the high-frequency dielectric constants [56].

We analyze the thermal stability of the PdPSe monolayer by the ab initio molecular dynamics (AIMD) trajectories at 600 K. The AIMD computational results for the PdPSe monolayer at 600 K is illustrated in Fig. 2(b). The inset in the panel of Fig. 2(b) demonstrates the optimized structure after five ps of simulation. Analysis of the AIMD trajectories shows that the 2D structure could stay intact at 600 K, with very stable temperature and energy profiles, proving the thermal stability of the PdPSe monolayer which is beneficial for the experimental synthesis of monolayer PdPSe.

Next, we investigate the mechanical stability and properties of the PdPSe. The linear elastic constants are calculated using the harmonic approximation. We find that the elastic parameters agree with the twelve Born criteria [61]. We calculate the bulk (B), shear (S) and Young’s moduli (Y) of PdPSe monolayer as 21.43 GPa, 15.36 GPa, and 37.19, respectively. The Poisson’s ratio ($\nu)$=0.21) suggests that the PdPSe monolayer is a brittle structure as it is less than 0.33 [62].

The electronic band structure show that the PdPSe monolayer is an indirect bandgap semiconductor with a forbidden gap, E_g, of 1.26 eV and 2.07 eV within the PBE and HSE06 functional, respectively (see Fig. 3). The HSE06 hybrid functional does not alter the type of indirect bandgap in the PdPSe monolayer. Notably, the band gap of PdPSe is similar to that of MoS₂ and WS₂ monolayers [tmd_bg]. The valance band minimum (VBM) and conduction band maximum (CBM) are located at the $\Gamma$ point and S point, respectively. The VBM and the CBM are almost isotropic around the $\Gamma$ point. Based on the partial density of state (PDOS) analysis, as depicted in Fig. 3(b), the VBM are composed mainly of the Pd-$d_{z^{2}}$-$d_{xy}$-$d_{x^{2}-y^{2}}$, P-$s,p_{x}$ and Se-$p_{x,z}$ orbitals states, while the CBM are originated from $d_{xz,yz}$, $p_{x,z}$ and $p_{x,y,z}$ of Pd, P and Se atoms of PdPSe, respectively.

The electron localization function (ELF) along the (0 0 1) plane held the monolayer as depicted in Fig. 3(d). The red (blue) color means high (low) electron density in the ELF. ELF takes a value between 0 and 0.5, where ELF = 0.5 corresponds to the perfect localization. Electron localization occurs on the Se atoms. The maximum ELF value between Pd and P/Se atoms is approximately 0.50, representing a large proportion of the covalent bond in the PdPSe monolayer.

We now investigate the optical properties of the PdPSe monolayer. The dielectric function (the real and imaginary parts) is calculated using the RPA method over the HSE06. Figs. 4(a,b) shows the real and imaginary parts of the dielectric function which describe the interaction between the electromagnetic field and monolayer PdPSe monolayer. The static value of the dielectric constant is the value of the real part at the zero energy which is found to be 5.73 and 5.63 for the xx and yy components, respectively. The dielectric function in the zz direction is very small as compared to the corresponding function in xx and yy components due to the monolayered nature of PdPSe. The dielectric function is anisotropic along the xx, yy, and zz directions. The first plasma frequencies appear at $\sim$ 3.01 eV and 2.97 eV along xx and yy direction, respectively. Other plasma frequencies appear at 8.95 eV and 9.10 eV along xx and yy direction, respectively. The imaginary part of the dielectric function represents the sum of all transitions from the valence bands to the conduction bands. It starts to increase above $\sim$ 2.1 eV which is related to the band gap value. The imaginary part increases from 2.1 eV to 4 eV which are in the visible and the ultraviolet ranges of light. Figure 4(c) shows the absorption spectrum for PdPSe monolayer. The onset of significant absorption occurs at $\sim$ 2.0 eV which is related to the onset of the imaginary part and the band gap of PdPSe. The absorption spectrum suggests that the PdPSe has the ability to absorb the visible and ultraviolet light range. Fig. 4(d) illustrates the band alignment of the oxidation and reduction potentials for water splitting with respect to the band edges of PdPSe monolayer suggesting that it is a potential candidate for photocatalytic water splitting. Most importantly, PdPSe monolayer exhibits a suitable band gap $\sim$ 2.07 eV and the band edges CBE and VBE must be higher (more negative) and lower (more positive) than the hydrogen reduction potential of H₊/H₂ and the water oxidation potential of H₂O/O₂, respectively, for efficient water splitting to take place. The CBE is computed from the relation $E_{CBE}=X-0.5E_{gap}-4.5$ eV [64, 65, 66], then the valence band edge is calculated from $E_{VBE}=E_{CBE}+E_{gap}$, where $X$ is the geometric mean of the electro-negativity of the ingredient atoms [67] and $4.5$ eV is the free energy of the electron (with respect to the vacuum level).

The calculated $S$, $\sigma$, $\kappa_{e}$ and power factor ($PF=S^{2}\sigma/\begin{tabular}[]{c}$\tau$\end{tabular}$) for both $p$- and $n$-type for monolayer PdPSe are shown in Fig. 5. The $S$ linearly decreases (increases) with increasing carrier concentration (temperature) and which is agreement with previous work [69]. At 300 K, the $S$ for the hole (electron) doping of 10¹¹ are 702 (668) and 669 (692) $\mu$VK^-1 along the $x$- and $y$-direction, respectively. The $S$ does not varry significantly with both doping and transport directions. In general, $\sigma/\tau$ and $\kappa_{e}/\tau$ increase with an increasing carrier concentration. The $\sigma/\tau$ for the holes along the $x$-direction is lower than that of the electrons due to the flat valence band while that of both electron and hole are similar along the $y$-direction. The $PF$ is related to the interplay between $S$ and $\sigma/\tau$. As shown in Figs. 5(g) and (f), the $PF$ increases with an increasing carrier concentration initially to reach a peak before decreases with higher carrier concentration. The maximum $PF$ values of 7.9 and 5.2 are achieved for $n$-type along the $x$-direction and $p$-type along $y$-direction, respectively. Such high values of $PF$ demonstrating that PdPSe is a promising material for thermoelectricity.