Physical Properties and Photovoltaic Application of Semiconducting Pd₂Se₃ Monolayer

Abstract

Palladium selenides have attracted considerable attention because of their intriguing properties and wide applications. Motivated by the successful synthesis of Pd₂Se₃ monolayer (Lin et al., Phys. Rev. Lett., 2017, 119, 016101), here we systematically study its physical properties and device applications using state-of-the-art first principles calculations. We demonstrate that the Pd₂Se₃ monolayer has a desirable quasi-direct band gap (1.39 eV) for light absorption, a high electron mobility (140.4 cm²V⁻¹s⁻¹) and strong optical absorption (~10⁵ cm⁻¹) in the visible solar spectrum, showing a great potential for absorber material in ultrathin photovoltaic devices. Furthermore, its bandgap can be tuned by applying biaxial strain, changing from indirect to direct. Equally important, replacing Se with S results in a stable Pd₂S₃ monolayer that can form a type-II heterostructure with the Pd₂Se₃ monolayer by vertically stacking them together. The power conversion efficiency (PCE) of the heterostructure-based solar cell reaches 20%, higher than that of MoS₂/MoSe₂ solar cell. Our study would motivate experimental efforts in achieving Pd₂Se₃ monolayer-based heterostructures for new efficient photovoltaic devices.

1. Introduction

Two-dimensional (2D) transition metal chalcogenides (TMCs), including semiconducting MoS₂ [1], MoSe₂ [2], WS₂ [3], WSe₂ [4], ReS₂ [5], PtS₂ [6], PdSe₂ [7,8], and metallic VS₂ [9] and NbS₂ [10] are of current interest because of their extraordinary properties and practical applications in catalysis [11], electronics [12,13,14], optoelectronics [15,16] and valleytronics [17,18]. Among them, the layered PdSe₂ has attracted special attention due to its unique atomic configuration and electronic properties [8,19,20]. Whereas previous studies mainly focused on the PdSe₂ monolayer that has the same structural form as a single layer of the bulk PdSe₂ [7,21,22]. Very recently, Lin et al. reported the successful exfoliation of a new monolayer phase with a stoichiometry of Pd₂Se₃ [23], and found that Se vacancies in the pristine PdSe₂ reduce the distance between the layers, melding the two layers into one, thus, resulting in the formation of the Pd₂Se₃ monolayer. Due to its structural novelty, subsequent efforts have been made to further explore this new material, including its electronic and optical properties [24] and thermoelectric performance [25], as well as theoretical calculations and experimental synthesis of the lateral junctions between a PdSe₂ bilayer and the Pd₂Se₃ monolayer [26].

We noticed that in Reference [24] the results were obtained from standard density functional theory (DFT) calculations (the Perdew-Burke-Ernzerhof (PBE) functional [27] for the generalized gradient approximation (GGA)), which is well-known to underestimate the electronic band gap of semiconductors. However, the accurate description of electronic structure is important for further investigation of electronic and optical properties. To overcome this limitation, various theoretical approaches have been developed. Among them, the hybrid functional that combines standard DFT with Hartree-Fock (HF) calculations has been widely used for calculating the band gaps, because it predicts more reliable physical properties and keeps a good compromise with computational efficiency. Therefore, we use the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [28,29] to study the electronic, transport and optical properties of the newly synthesized Pd₂Se₃ monolayer. We show that this monolayer possesses a desirable bandgap for light harvesting, offering better opportunity for photovoltaic applications. Moreover, its electronic structure can be effectively tuned by applying biaxial strain, and indirect to direct bandgap transition occurs with a small critical strain of 2%. In addition, a stable Pd₂S₃ monolayer can be formed by substituting Se with S, which can be used to construct a type-II heterostructure with the Pd₂Se₃ monolayer. The heterostructure-based solar cell can reach a high power conversion efficiency (PCE) of 20%. These fascinating properties make the Pd₂Se₃ monolayer a promising candidate for future applications in nanoscale electronics and photonics.

2. Computational Methods

Within the framework of DFT, our first-principles calculations are performed using the projector augmented wave (PAW) method [30] as implemented in the Vienna Ab initio Simulation Package (VASP) [31]. The Perdew-Burke-Ernzerhof (PBE) functional [27] with the generalized gradient approximation (GGA) is used to treat the electron exchange-correlation interactions in crystal structure calculations, while the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [28,29], which includes the Hartree-Fock exchange energy and the Coulomb screening effect, is used to calculate the electronic and optical properties. A kinetic energy cutoff of 350 eV is set for the plane wave basis. The convergence criteria are 10⁻⁵ eV and 10⁻³ eV/Å for total energy and atomic force components, respectively. The Brillouin zone is represented by k points with a grid density of 2π × 0.02 Å⁻¹ in the reciprocal space using the Monkhorst-Pack scheme [32]. An adequate vacuum space (~20 Å) in the direction perpendicular to the sheet is used to minimize the interlayer interactions under the periodic boundary condition. Spin-orbit coupling (SOC) interactions are not included since our calculation shows that the SOC has negligible effect on electronic structure of the monolayer (see Figure S2). Phonon dispersion and density of states (DOS) are calculated using the finite displacement method [33] as implemented in the Phonopy code [34].

In the calculation of carrier mobility (µ), we consider the perfect crystal of the monolayer without defects and impurities. In addition, carrier mobility is a function of temperature. We set the temperature to be 300 K in our calculation, since most devices work at room temperature. In this situation, the dominant source of electron scattering is from acoustic phonons and the carrier mobility can be obtained using deformation potential theory proposed by Bardeen and Shockley [35], which has been successfully employed in many 2D materials [36,37,38,39]. Using effective mass approximation, the analytical expression of carrier mobility in 2D materials can be written as

$$\mu =\frac{e{\hslash }^{3}C}{k_{\mathrm{B}}T{m}^{*}{m}_d{E}_1^2}$$

(1)

C is the elastic modulus of the 2D sheet, T is the temperature, which is taken to be 300 K in our calculations, m* = ħ²[∂²E(k)/∂k²]⁻¹ is the effective mass of the band edge carrier along the transport direction and m_d is the average effective mass determined by $m_d=\sqrt{m_x^{*}{m}_y^{*}}$. E₁ is the DP constant defined as the energy shift of the band edge with respect to lattice dilation and compression, and k_B and ħ are Boltzmann and reduced Planck constants, respectively.

The optical absorption coefficient (α) can be expressed as [40,41,42]

$$\alpha (\omega )=\sqrt{2}\omega {\left[\sqrt{{\epsilon }_1^2(\omega )+{\epsilon }_2^2(\omega )}-{\epsilon }_1(\omega )\right]}^{1/2}$$

(2)

where ε₁(ω) and ε₂(ω) are the real and imaginary parts of the frequency-dependent dielectric functions which are obtained using the time-dependent Hartree-Fock approach (TDHF) based on the HSE06 hybrid functional calculations [43]. The model, developed by Scharber et al. for organic solar cells [44] and exciton-based 2D solar cells [45,46,47,48,49], is used to calculate the maximum PCE in the limit of 100% external quantum efficiency (EQE), which can be written as

$$\eta =\frac{{\beta }_{FF}{V}_{oc}{J}_{sc}}{P_{solar}}=\frac{0.65(E_g^d-\mathsf{\Delta }{E}_c-0.3){\displaystyle {\int }_{E_g^d}^{\infty }\frac{P(\hslash \varpi )}{\hslash \varpi }d(\hslash \varpi )}}{{\displaystyle {\int }_0^{\infty }P(\hslash \varpi )d(\hslash \varpi )}}$$

(3)

Here, the fill factor ${\beta }_{FF}$, which is the ratio of maximum power output to the product of the open-circuit voltage (V_oc) and the short-circuit current (J_sc), is estimated to be 0.65 in this model. V_oc (in eV) is estimated by the term ($E_g^d-\mathsf{\Delta }{E}_c-0.3$), where $E_g^d$ is the bandgap of the donor and $\mathsf{\Delta }{E}_c$ is the conduction band (CB) offset between donor and acceptor. J_sc is obtained by ${\displaystyle {\int }_{E_g^d}^{\infty }\frac{P\left(\hslash \varpi \right)}{\hslash \varpi }d\left(\hslash \varpi \right)}$, and the total incident solar power per unit area P_solar is equal to ${\displaystyle {\int }_0^{\infty }P(\hslash \varpi )d(\hslash \varpi )}$. Here, ħ and ϖ are reduced Planck constants and photon frequency, and P(ℏϖ) is the air mass (AM) 1.5 solar energy flux (expressed in W m⁻² eV⁻¹) at the photon energy (ℏϖ).

3. Results and Discussion

3.1. Geometric Structure of Pd₂Se₃

Figure 1a,b shows the optimized monolayer structures of Pd₂Se₃ and PdSe₂ respectively (the structural details are listed in Table S1, Supplementary Materials). For simplicity, we refer these two monolayer structures as Pd₂Se₃ and PdSe₂ in the following discussions unless stated otherwise. There are similarities as well as differences between the two structures. On one hand, they both consist of a layer of metal Pd atoms sandwiched between the two layers of chalcogen Se atoms, and each Pd atom binds to four Se atoms forming the square-planar (PdSe₄) structural units. On the other hand, Pd₂Se₃ indeed distinguishes itself from PdSe₂ in the following characteristics: (1) Pd₂Se₃ possesses Pmmn symmetry (point group D_2h) with four Pd and six Se atoms in one unit cell. While the symmetry of PdSe₂ is P2₁/c (point group C_2h) and each unit cell contains two Pd and four Se atoms. The different crystal symmetries result in distinct resonance in the Raman spectroscopy, which can serve as an efficient and straightforward clue for experimentalists to confirm the formation of Pd₂Se₃. The details about the calculated Raman spectra of Pd₂Se₃ and PdSe₂ are presented in the Supporting Information (Figure S1). (2) There are two chemically nonequivalent Se in Pd₂Se₃, marked in orange (Se2) and yellow (Se1) respectively. The two neighboring Se2 atoms form a covalent Se-Se bond while each Se1 atom is unpaired and binds to four neighboring Pd atoms. Whereas in PdSe₂, all Se atoms are in dimers and form the Se-Se bonds. (3) In Pd₂Se₃, the Se-Se dumbbells are parallel to the Pd layer, while in PdSe₂, they cross the Pd layer. (4) Pd₂Se₃ and PdSe₂ have different charge-balanced formulas, written as (Pd²⁺)₂(Se²⁻)(Se₂²⁻) and (Pd²⁺)(Se₂²⁻) respectively, due to the different chemical environments of Se atoms in the two structures. Since the properties of materials are essentially determined by their geometric structures, one can expect that Pd₂Se₃ would possess some new and different properties from those of PdSe₂.

3.2. Electronic Properties of Pd₂Se₃

We then investigated the electronic properties of Pd₂Se₃ by calculating its electronic band structure and density of states (DOS) using the hybrid HSE06 functional. Figure 2a shows the calculated band structure around the Fermi level and corresponding total and partial DOS. The bandgap size of Pd₂Se₃ is 1.39 eV, close to the optimum value (~1.3 eV) for solar cell materials [50,51,52]. Although Pd₂Se₃ is an indirect bandgap semiconductor with the valence band maximum (VBM) located at the Γ point and the conduction band minimum (CBM) located on the Y-M path, Pd₂Se₃ can be considered as a quasi-direct bandgap semiconductor because of the existence of the sub-CBM at the Γ point (CB2) that is only marginally higher in energy than the true CBM (the energy difference is less than 50 meV). The weakly indirect bandgap is desirable for photovoltaic applications since it can simultaneously increase optical absorbance and photocarrier lifetimes [41,53,54]. To assess the effect of SOC interaction, we computed the band structure of Pd₂Se₃ at the level of HSE06+SOC. The results in Figure S2 reveal that the SOC in Pd₂Se₃ is weak and has negligible effect on the bandgap of this structure. Hereafter, we do not include the SOC interaction and just use the HSE06 scheme for calculations in this study.

An analysis of the partial DOS in Figure 2a indicates that the electronic states of valance and conduction bands mainly originate from Se 4p and Pd 4d orbitals. In addition, the overlap of the orbital-projected DOS implies strong hybridization, that is, the formation of covalent bonds between Se 4p and Pd 4d orbitals. By calculating wave functions for the VBM and CBM, we visualized their electronic states showing distinct antibonding features for both of them (see Figure 2b). However, to gain a better understanding for the covalent bonding in this 2D structure, the electronic bands not only limited to near the Fermi level but also in a large energy range should be taken into account.

Figure 2c displays the band structure and partial DOS including all occupied and sufficient unoccupied states of Pd₂Se₃. Combining crystal field theory and crystal structure chemistry analysis, we can clearly identify the electronic states in the energy range from −17 to 4 eV. From partial DOS, the bands in the energy range of −17 ~ −12.5 eV are primarily from Se 4s orbitals. According to the different bonding states, they can be classified into three groups. The bottom and upper subsets correspond to the bonding and antibonding states dominated by the formation of Se-Se bonds, and the middle part is the nonbonding state of the unpaired Se 4s orbitals. When the energy goes up, there occurs Pd 4d orbitals. In Pd₂Se₃, the Pd atom is coordinated in a nearly perfect square-planar geometry, and its 4d orbitals split into four energy levels, i.e., e_g (d_xz/d_yz), a_1g (d_z2), b_2g (d_xy), and b_1g (d_x2-y2) from low to high energy. These d orbitals overlapping with Se 4p orbitals constitutes the bands in the energy range of −7.5 ~ 4 eV. In Figure 2d, we present the schematic drawing of DOS and energy level diagram of Pd₂Se₃ to explain details about how Pd 4d orbitals interact with Se 4p orbitals. It shows that a_1g, b_2g, and b_1g orbitals hybridize with Se 4p orbitals leading to lower energy bonding states and higher energy antibonding states, whereas the nonbonding states in the energy range from −4 to −1.7 eV stemming mainly from e_g orbitals. More importantly, the bandgap that separates occupied and unoccupied states lies in the antibonding region and amounts to the splitting energy between and states, consistent with the results of wave functions for the VBM and CBM in Figure 2b. The systematic and deep exploration of electronic structure of Pd₂Se­₃ is crucial for understanding its properties and origins of intriguing physical phenomena.

3.3. Strain Engineering of Electronic Band Structure of Pd₂Se₃

From above electronic structure analysis, it is clear that Pd₂Se₃ is a covalent semiconductor, and a connection between elastic strain and its electronic structure is expected. This is because elastic strain generally weakens the covalent interaction as the bonds lengthen, exerting efficient modulation on the band energies and bandgap. For this reason, we applied a biaxial tensile strain to Pd₂Se₃ and study its effect on the electronic bands of Pd₂Se₃.

Figure 3a shows the evolution of band structure with biaxial strain varying from 0% to 9%. It indicates that both direct and indirect bandgaps increase, and a transition from indirect bandgap to direct bandgap occurs when the biaxial strain is applied. To acquire a more accurate energy profile, we present the strain-dependent bandgaps in Figure 3b, which clearly shows the increasing trend of bandgaps and the bandgap transition from indirect to direct at the critical strain of 2%.

The strain-dependent bandgap of Pd₂Se₃ can be understood by analyzing its electronic structure in Figure 2, which shows that the valence and conduction bands both originate from antibonding states. Application of a tensile strain increases the Pd–Se bond length thus decreases the amount of orbital overlap, leading to the stabilization of valence and conduction bands and reducing them in energy. This is consistent with our results, as shown in Figure 3c, which displays the strain-dependent energy levels for the VBM and CBM. However, since the biaxial tensile strain not only enlarges the bond length but also distorts the square-planar ligand field (see Figure S3 for details), the valence band responds more strongly to strains than the conduction band, resulting in the increase of bandgap in the imposed strain filed.

Additionally, we also examined the structural stability under biaxial strains. The phonon dispersion in Figure S4 demonstrates that the structure remains stable under the strain of 9%. The large strain tolerance and an electronic structure that has a continuous response in the imposed strain field indicate the great potential of Pd₂Se₃ in future flexible electronics.

3.4. Transport Properties of Pd₂Se₃

We also investigated the transport properties of Pd₂Se₃ by calculating its room-temperature carrier mobilities as summarized in

Table 1. Calculated deformation potential constant (E₁), elastic modulus (C), effective mass (m^*), and mobility (μ) for electron and hole in the x and y directions for Pd₂Se₃ and PdSe₂ monolayers at 300 K.

	Carrier Type	E1 (eV)	C (N/m)	m* (me)	$\mathit{\mu }$ (cm2V−1s−1)
Pd2Se3	electron (x)	3.756	33.02	0.762	101.9
	electron (y)	3.785	32.93	0.543	140.4
	hole (x)	2.870	33.02	9.029	7.3
	hole (y)	12.082	32.93	0.187	19.9
PdSe2	electron (x)	9.542	32.45	0.429	43.36
	electron (y)	9.982	55.05	0.390	73.99
	hole (x)	3.352	32.45	0.656	97.94
	hole (y)	3.074	55.05	1.401	92.54

. One can see that the mobilities for both electrons and holes are slightly anisotropic along the x and y directions, due to the structural anisotropy of Pd₂Se₃. Meanwhile, the electron mobility along the y direction is estimated to be 140.4 cm²V⁻¹s⁻¹, significantly higher than that of hole. When compared with PdSe₂, whose carrier mobilities are calculated at the same theoretical level and listed in Table 1, Pd₂Se₃ possesses higher electron mobility and lower hole mobility, showing strong asymmetry in electron and hole transport. Although the carrier mobilities of Pd₂Se₃ is lower than the theoretical predicted carrier mobilities of some other 2D materials [36,38,55], it is still commendable if realized in practice [20].

3.5. Optical Properties of Pd₂Se₃

Attracted by the suitable bandgap and intriguing electronic properties of Pd₂Se₃, we further explored its light-harvesting performance by calculating the dielectric functions based on the hybrid HSE06 functional. Figure 4a shows the real (ε₁) and imaginary (ε₂) parts of the frequency-dependent complex dielectric functions of Pd₂Se₃. With the dielectric functions, we derive its optical absorption coefficient (α), as shown in Figure 4b. For comparison, the absorption spectra of PdSe₂ was also calculated. We notice that, for both Pd₂Se₃ and PdSe₂, the overall absorption coefficients are close to the order of 10⁵ cm⁻¹ and only show little difference along the x and y directions, which are considerably desirable for optical absorption. Moreover, as shown in Figure 4b, the absorption coefficient of Pd₂Se₃ is slightly larger than that of PdSe₂ in nearly the entire of the energy range, indicating the improved light-harvesting performance of Pd₂Se₃ as compared with PdSe₂. Furthermore, we also investigated the biaxial strain influence on the optical performance of Pd₂Se₃. The calculated strain-dependent optical absorption spectra are presented in Figure 4c. It shows that the strain slightly affects the optical absorption of Pd₂Se₃, which is favorable for applications in flexible systems since it guarantees steady performance of devices under stretching.

3.6. Extension and Photovoltaic Application of Pd₂Se₃

Moreover, we further explored the feasibility of other Pd₂X₃ monolayer phases, with X to be S and Te, respectively. Bulk PdS₂ has the same geometrical structure as that of bulk PdSe₂, thus the Pd₂S₃ monolayer might be experimentally synthesized following the same synthetic method as that of the Pd₂Se₃ monolayer. However, bulk PdTe₂ prefers a 1T configuration, indicating that the Pd₂Te₃ monolayer might be inaccessible. To confirm our assumption, we calculated the phonon dispersions of the two structures. No imaginary mode exists in the phonon spectra of Pd₂S₃ (see Figure 5a), indicating its dynamical stability of the monolayer. Whereas the phonon spectra of the Pd₂Te₃ monolayer shows imaginary frequency near the Γ point (see Figure S5), demonstrating its structural instability. We then calculated the electronic band structure of the stable Pd₂S₃ monolayer (Figure 5a), verifying the feature of a semiconductor with an indirect bandgap of 1.48 eV.

Since 2D TMCs can be vertically stacked layer-by-layer forming the van der Waals heterostructures which can efficiently modulate properties of materials for applications in nanoscale electronic and photovoltaic devices, here we propose a van der Waals heterostructure composed of the Pd₂Se₃ and Pd₂S₃ monolayers (see Figure 5b) and study its interesting properties. A key indicator for heterostructures is the band alignment that defines the type of heterostructures. Thus, we calculated the band alignment of the Pd₂Se₃ and Pd₂S₃ monolayers, as shown in Figure 5c. One can see that the Pd₂S₃/Pd₂Se₃ heterostructure has a type-II (ladder) band alignment, which allows more efficient electron-hole separation for lighting harvesting. Such type-II heterostructure can be used as active materials in excitonic solar cells (XSCs) [46,47,48,49]. For the Pd₂S₃/Pd₂Se₃ heterostructure, the Pd₂Se₃ monolayer is the donor and the Pd­₂S₃ monolayer serves as the acceptor. With the approximation that the HSE06 bandgap equals optical bandgap and using the model developed by Scharber et al. [44], we obtained the upper limit of the PCE, reaching as high as 20% (Figure 5d). For comparison, we also calculated the band alignments for the PdS₂/PdSe₂ and MoS₂/MoSe₂ heterostructures, and find that they both are type-II heterostructures with predicted PCEs to be 14% and 12% respectively. The high PCE of the Pd₂S₃/Pd₂Se₃ heterostructure renders it a promising candidate in flexible optoelectronic and photovoltaic devices.

4. Conclusions

In summary, on the basis of DFT calculations, we systematically studied the properties and potential applications of the recently synthesized Pd₂Se₃ monolayer by focusing on its geometric structure, electronic band structure, and optical adsorption. Comparing with the previously reported PdSe₂ monolayer, we found that the Pd₂Se₃ monolayer has the following merits: (1) A suitable quasi-direct bandgap (1.39 eV) for light absorption, (2) a higher electron mobility (140.4 cm²V⁻¹s⁻¹) and (3) a stronger optical absorption (~ 10⁵ cm⁻¹) in the visible solar spectrum, showing promise of Pd₂Se₃ as an absorber material for future ultrathin photovoltaic devices. In addition, the Pd₂Se₃ monolayer combining with the stable Pd₂S₃ monolayer can form a type-II heterostructure, and the heterostructure solar cell system can achieve a 20% PCE. These findings would encourage experimentalists to devote more effort in developing Pd₂Se₃-based devices with high performance.