Large bulk photovoltaic effect and second-harmonic generation in few-layer pentagonal semiconductors PdS2 and PdSe2

Vijay Kumar Gudelli; Guang-Yu Guo

doi:10.1088/1367-2630/ac231c

1. Introduction

Because of their extraordinary electronic and optical properties, two-dimensional (2D) materials such as graphene, atomically thin (few-layer) transition metal dichalcogenides (TMDCs) and black phosphorus, have attracted an enormous amount of interest in recent years, finding diverse applications in electronic, opto-electronic and nonlinear photonic devices with superior performances. Among them, group 6B TMDC semiconductors with chemical formula MX₂ (M = Mo, W; X = S, Se) and each layer made up of a 2D hexagonal array of M atoms sandwiched between the similar arrays of X atoms, constitute a particularly interesting family of 2D materials. In particular, they were found to exhibit an indirect to direct band gap transition when they were thinned down to a monolayer (ML) [1]. This makes the MX₂ MLs semiconductors with a direct band gap, thus becoming promising materials for, e.g. electro-optical devices with efficient light emission [1] and field effect transistors with high on-off ratios [2]. Furthermore, these hexagonal 2D MX₂ materials with an odd layer-number lack the spatial inversion symmetry, although their bulk crystals are centrosymmetric. This broken inversion symmetry makes them exhibit novel properties of fundamental and technological interests, especially second-order nonlinear optical (NLO) responses such as second-harmonic generation (SHG) [3–7] and bulk photovoltaic effect (BPVE) [8].

Stimulated by the recent fabrications of few-layer PdSe₂ via molecular beam epitaxy, chemical vapor deposition and mechanical exfoliation [9–11], palladium-based TMDC 2D materials have also attracted much attention in the past five years [12]. Indeed, these few-layer Pd-based TMDC materials were found to exhibit some desired properties for applications, such as tunable band gap, high carrier mobility, anisotropy, enhanced thermoelectric property and ultrahigh air stability [9, 12–16]. There are several distinct differences between group 6B TMDC and Pd-based TMDC 2D materials. In particular, compared with Mo and W atoms, Pd atoms have a nearly filled d-shell and thus there are stronger hybridization between Pd d-orbitals and chalcogen p-orbitals in Pd-based TMDC 2D materials. This result in stronger covalent bonding within each PdX₂ (X = S and Se) layer as well as stronger interlayer binding. The latter gives rise to layer-number dependent properties especially band gap size [9, 12, 16]. In contrast to groups 5B and 6B TMDCs, bulk PdS₂ and PdSe₂ crystallize in the orthorhombic layered structure (see figure 1(a)) with the centrosymmetric Pbca space group [17, 18]. Furthermore, in each PdX₂ layer, a Pd atom bonds with four chalcogen atoms, and Pd and chalcogen atoms form a buckled pentagonal layered structure (see figure 1(c)) [9, 14]. Depending on their thickness (i.e. layer-number), few-layer PdX₂ structures have different symmetries compared with their bulk crystals. Interestingly, 2D PdX₂ materials with an even layer-number crystallizes in a noncentrosymmetric structure with space group Pca2₁ and point group symmetry of C_2v (or mm2), while 2D PdX₂ materials with an odd layer-number form a centrosymmetric structure of space group P2₁/c [9, 12, 19]. Note that 2D group 6B TMDC structures with an odd-layer number are noncentrosymmetric, while that with an even layer-number have the inversion symmetry. Therefore, as for few-layer group 6B TMDC materials with an odd layer-number, 2D PdX₂ structures with an even layer-number are expected to show second-order NLO properties. Indeed, strong angle-dependent SHG signals in bilayer (BL) and four-layer (FL) PdSe₂ were observed recently [20], although the precise SHG susceptibility was not determined. Motivated by these exciting developments, in this work we perform a systematic theoretical study of the NLO responses of BL and FL PdX₂, based on first-principles density functional theory (DFT) calculations.

**Figure 1.** (a) Crystal structure of bulk PdX₂ (side view along the b-axis), where blue dashed lines (magenta dashed lines) represent the unit cell of BL PdX₂ (FL PdX₂). (b) Crystal structure of BL PdX₂ (side view along the a-axis). (c) Top view of the PdX₂ structures. Here black solid lines (red dashed lines) show the unit cell (the pentagon) of a PdX₂ layer. (d) 2D Brillouin zone (BZ) for BL and FL PdX₂ structures. The blue rectangle denotes the irreducible BZ wedge. The conduction band minimum is located at a general k-point on the Γ–M line (see the text).
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In this paper, we focus on three principal second-order NLO responses of few-layer PdX₂ materials, namely, SHG, linear electric-optical (LEO) effect and BPVE. The SHG, one of the best-known second-order NLO effects, converts the two photons of the same-frequency into a new photon with a doubled photon energy [3, 4]. Noncentrosymmetric materials with a large SHG susceptibility χ⁽²⁾(−2ω; ω, ω) have many applications in nonlinear photonic devices such as frequency conversion. The SHG is also a powerful probe of the symmetry of surfaces and interfaces [3]. Here we find that all PdX₂ structures possess large χ⁽²⁾ in the visible frequency spectrum and BL PdSe₂ has the largest χ⁽²⁾ value of 1.4 × 10³ pm V⁻¹ at 1.9 eV. The LEO effect refers to the linear refractive index change (Δn) with the applied electric field strength (E), Δn = n³ rE/2, where n is the refraction index and r is the LEO coefficient [4]. The LEO effect thus allows one to use an electrical signal to control the amplitude, phase or direction of a light beam in the NLO material, and leads to a widely used means for high-speed optical modulation and sensing devices (see, e.g. reference [21] and references therein). We find that the LEO coefficients of BL PdS₂ and BL PdSe₂ are significant and comparable to that of trilayer group 6B TMDC semiconductors [5]. The BPVE (also known as photogalvanic effect) refers to the generation of dc photocurrents in noncentrosymmetric materials [8]. In a nonmagnetic semiconductor, there are two main contributions to the BPVE, namely, the circular injection current and linear shift current [8, 22–24]. Materials having large BPVE are crucial for applications in photovoltaic solar cells and high sensitive photodetectors. Here we predict that the BPVE in the considered few-layer PdX₂ structures is generally strong, with a large shift current conductivity of up to 130 μA V⁻² and injection current susceptibility of up to 100 × 10⁸ A V⁻² s⁻¹ in the visible frequency range. These superior NLO responses of the BL and FL PdX₂ structures will make them valuable for technological applications in NLO and electro-optic devices such as light signal modulators, frequency converters, electro-optical switches, photovoltaics and photodetector applications.

2. Computational methods

Bulk PdX₂ crystallize in a layered orthorhombic structure with space group Pbca (see figure 1), as mentioned above. The experimental lattice constants are a = 5.460 Å, b = 5.541 Å and c = 7.531 Å for PdS₂ [17] and a = 5.7457 Å, b = 5.8679 Å and c = 7.6976 Å for PdSe₂ [18]. The bulk unit cell contains two X–Pd–X MLs stacked along the c-axis and each nearly squared inplane unit cell contains two chemical formulas (f.u.) (i.e. six atoms) (see figure 1). In each X–Pd–X layer, interestingly, a Pd atom bonds with four chalcogen atoms, and Pd and chalcogen atoms form a rare pentagonal structure (see figure 1(c)) [9, 14]. In the present calculations, a BL (FL) structure is constructed by cutting two (four) X–Pd–X layers out of the bulk crystal. The slab-superlattice approach is adopted with the separations of neighboring slabs being at least 15 Å. We notice that a number of the structural optimization calculations for bulk PdSe₂ using more than ten exchange-correlation functionals have been carried out [9, 11, 19, 25]. The discrepancies between the experimental and theoretical lattice constants are large, varying from 2% all the way up to 20% depending strongly on the exchange-correlation functional used [9, 25]. On the other hand, the experimental inplane lattice constants of BL PdSe₂ are only slightly larger than the corresponding lattice constants of bulk PdSe₂ [11]. Thus, we use the experimental bulk structural parameters in the present calculations. We believe that using the experimental structural parameters of atomically thin PdX₂ would not significantly change the calculated electronic and optical properties of BL and FL PdX₂ to be presented below.

The electronic structure calculations are performed using the accurate projector augmented wave method [26] as implemented in the Vienna ab-initio simulation package (VASP) [27, 28]. For the exchange-correlation potential, we adopt the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof parametrization [29]. A large plane-wave energy cutoff of 400 eV is used throughout the calculations. The valence configuration of Pd atom is taken as 4d⁹ 5s¹, S atom is 3s² 3p⁴ and for Se atom it is 4s² 4p⁴. A k-point mesh of 18 × 18 × 1 is used in the Brillouin zone (BZ) integrations for few-layer PdX₂ structures. All the calculations are performed within the scalar-relativistic projector augmented potentials, with the energy convergence up to 10⁻⁶ eV between the successive iterations.

All the linear and NLO properties of the 2D PdX₂ structures are calculated from the self-consistent electronic band structures within the linear response formalism with the independent-particle approximation (IPA). Specifically, we first calculate the imaginary part (ɛ''(ω)) of the dielectric function due to direct interband transitions by using the Fermi golden rule [30]. The obtained ɛ''(ω) is used to get the real part (ɛ'(ω)) of the dielectric function by a Kramer–Kronig transformation [30]. Given the complex dielectric function (ɛ' + iɛ''), all other linear optical properties can be calculated. For example, complex refraction index (n + iκ) = $\sqrt{{\varepsilon }^{\prime }+i{\varepsilon }^{{\prime\prime}}}$ and absorption coefficient α = ωɛ''/(nc) where n is the refraction index, κ is the extinction coefficient and c is the speed of light.

For the BPVE, the dc photocurrent density along the a-axis in a noncentrosymmetric material under the applied optical electric fields E_b and E_c may be written as [22, 23, 32]

$\begin{equation}{J}_{a}(0)=\sum\limits _{bc}{\sigma }_{abc}(0;\omega ,-\omega ){E}_{b}(\omega ){E}_{c}(-\omega ),\end{equation} \tag{ 1 }$

where the photocurrent conductivity σ_abc is a third-rank tensor [22]. For a nonmagnetic semiconductor, the dc photocurrent contains two main contributions, namely, the linear shift current [22] and also the circular injection current [32]. That is, ${\sigma }_{abc}={\sigma }_{abc}^{sh}+{\sigma }_{abc}^{inj}$ . Within the length gauge formalism, the shift current conductivity ${\sigma }_{abc}^{sh}$ can be written in terms of the interband position matrix element ${r}_{ij}^{a}$ and its momentum derivative ${r}_{ij;b}^{a}$ [22]. By replacing ${r}_{ij}^{a}$ with ${p}_{ij}^{a}/i{{\epsilon}}_{ij}$ where ${p}_{ij}^{a}=\langle \mathbf{k}\mathit{\text{j}}\vert {\hat{p}}_{a}\vert \mathbf{k}\mathit{\text{i}}\rangle$ is the a-component of the momentum matrix element, _ji = (_{k
j} − _{k
i}) and _{k
i} is the energy of the |k n⟩ band state, one may obtain [33]

$\begin{equation}{\sigma }_{abc}^{sh}(\omega )=\frac{\pi }{{\Omega}{N}_{k}}\sum\limits _{\mathbf{k}}\sum\limits _{i\in VB}\sum\limits _{j\in CB}\frac{1}{{{\epsilon}}_{ij}^{2}}\sum\limits _{l\ne i,j}\;\;\mathrm{I}\mathrm{m}\left\{\frac{{p}_{il}^{a}\langle {p}_{ji}^{b}{p}_{lj}^{c}\rangle }{2{{\epsilon}}_{il}}+\frac{{p}_{lj}^{a}\langle {p}_{ji}^{b}{p}_{il}^{c}\rangle }{2{{\epsilon}}_{jl}}\right\}\delta ({{\epsilon}}_{ji}-\omega ),\end{equation} \tag{ 2 }$

where $\langle {p}_{jl}^{b}{p}_{li}^{c}\rangle =\frac{1}{2}({p}_{jl}^{b}{p}_{li}^{c}+{p}_{li}^{b}{p}_{jl}^{c})$ , Ω is the unit cell volume and N_k is the number of k-points used. The injection current conductivity ${\sigma }_{abc}^{inj}=\tau {\eta }_{abc}$ where τ is the relaxation time of photoexcited carriers and η_abc is the injection current susceptibility, which can also be written in terms of ${r}_{ij}^{a}$ and ${r}_{ij;b}^{a}$ (see equation (31) in reference [32]). Again, by substituting ${r}_{ij}^{a}$ with ${p}_{ij}^{a}/i{{\epsilon}}_{ij}$ , one would get

$\begin{equation}{\eta }_{abc}^{inj}(\omega )=\frac{2\pi }{{\Omega}}\sum\limits _{\mathbf{k}}\;{w}_{\mathbf{k}}\sum\limits _{i\in VB}\sum\limits _{j\in CB}\frac{{{\Delta}}_{ji}^{a}\mathrm{I}\mathrm{m}[{p}_{ji}^{b}{p}_{ij}^{c}]}{{{\epsilon}}_{ij}^{2}}\delta ({{\epsilon}}_{ji}-\omega ),\end{equation} \tag{ 3 }$

where ${{\Delta}}_{ji}^{a}=({p}_{jj}^{a}-{p}_{ii}^{a})$ . Note that the atomic unit is adopted here and hence ℏ = m_e = 1.

The imaginary part [ ${\chi }_{(abc)}^{\prime \prime (2)}(-2\omega ,\omega ,\omega )$ ] of the SHG susceptibility is obtained from the self-consistent electronic band structure by using the expression already reported elsewhere [30, 34]. The real part [ ${\chi }_{(abc)}^{\prime (2)}(-2\omega ,\omega ,\omega )$ ] of the SHG susceptibility is then obtained from ${\chi }_{abc}^{\prime \prime (2)}$ by the Kramer–Kronig transformation [30, 34]. We also calculate the low-frequency LEO coefficients of the considered materials using the obtained static dielectric constants and SHG susceptibility. The LEO coefficients in the zero frequency limit are given by

$\begin{equation}{r}_{abc}(0)=-\frac{2}{{\varepsilon }_{a}(0){\varepsilon }_{b}(0)}\enspace \underset{\omega \to 0}{\mathrm{lim}}\enspace {\chi }_{abc}^{\prime \prime (2)}(-2\omega ,\omega ,\omega ).\end{equation} \tag{ 4 }$

To ensure the accuracy of the calculated optical properties we use denser k-point meshes of 110 × 110 × 1 for BL structures and 60 × 60 × 1 for FL structures. Furthermore, to ensure that '(ω) and χ'⁽²⁾ calculated via Kramer–Kronig transformation are reliable, at least 150 and 300 energy bands are included in the present optical calculations for BL and FL structures, respectively. The unit-cell volume is not well-defined for low-dimensional systems. Therefore, similar to the previous calculations [5, 30, 34], we use the effective unit-cell volume (i.e. a × b × nh, where n is the layer-number and h is the effective layer thickness) of the 2D material rather than the volume of the supercell which is arbitrary. Since each unit cell of bulk PdX₂ contains two layers, the effective layer thickness h is 3.766 Å for few-layer PdS₂ and 3.849 Å for few-layer PdSe₂.

Correct band gaps would be important for obtaining accurate optical properties. However, in general the GGA functional is known to underestimate the band gaps because some many-body effects especially the quasiparticle self-energy corrections are neglected. Therefore, we perform the band-structure calculations using the hybrid Heyd–Scuseria–Ernzerhof (HSE) functional, which is known to produce much improved band gaps for semiconductors [37, 38]. We then use the HSE band gaps and calculate all the optical properties with the well-known scissors-correction (SC) [35]. In the SC calculations, the conduction bands are uniformly up-shifted so that the band gap would match the HSE gap together with the renormalized transition matrix elements [35]. All the optical properties presented in this paper are obtained with this SC scheme.

3. Results and discussion

3.1. Electronic structure

The calculated GGA band structures of BL PdX₂ and FL PdX₂ are presented, respectively, in figure 2 and figure S1 of the supplementary information (SI) (https://stacks.iop.org/NJP/23/093028/mmedia). All the four structures are an indirect band gap semiconductor. The valence band maximum (VBM) is located at the Γ point and the conduction band minimum (CBM) is located on the M–Γ line in the 2D Brillouin zone (see figure 1(d)) for both BL and FL PdS₂ structures (figure 2(a) and figure S1(a)). In the PdSe₂ structures, the VBM is located in the Γ–X direction and the CBM is located on the M–Γ line (figure 2(b) and figure S1(b)). The present band structure of BL PdSe₂ agrees rather well with that reported in references [9, 11, 19]. The calculated band gaps of all the structures are listed in table 1, together with the available experimental band gaps for BL and FL PdSe₂ [9, 11]. Table 1 shows that the band gap decreases significantly as the S atoms are replaced by the Se atoms in PdX₂ and also as we move from the BL to FL PdX₂ structure, indicating the tunability of the band gap by chalcogen substitution and also by layer-number variation.

Table 1. Band gaps (in eV) of BL and FL PdX₂ calculated with GGA and HSE functional. The available experimental band gap values are in parentheses for comparison.

	PdS₂		PdSe₂
	GGA	HSE	GGA	HSE
BL	0.78	1.90	0.74	1.62 (1.15 [11])
△E = E_HSE − E_GGA		1.12		0.88
FL	0.40	1.47	0.27	1.11 (1.06 [9])
△E = E_HSE − E_GGA		1.07		0.84

The calculated total- and orbital-projected density of states (DOS) for BL PdX₂ and FL PdX₂ are presented in figure 3 and in figure S3 of the SI, respectively. Figure 3 shows that the upper valence band edge and lower conduction band edge are contributed almost equally by the Pd d orbitals and chalcogen (X) p orbitals (see figures 3(a) and (b)). This indicates a strong covalent bonding in the PdX₂ structures (see also [36]), being in rather strong contrast to group 6B TMDCs (e.g. MoS₂) which may be called charge-transfer semiconductors [5]. This is due to the nearly filled Pd d states in the PdX₂ structures while in MoS₂ the Mo d states are less than half-filled [5]. Furthermore, orbital-projected DOS spectra show that the contribution at the upper valence band edge comes predominantly from Pd ${d}_{{z}^{2}}$ and chalcogen p_z with minor contribution from p_x,y states while the contribution at the lower conduction band edge comes from Pd ${d}_{xy,{x}^{2}-{y}^{2}}$ and chalcogen p_x,y and p_z states. Thus, the optical transitions in PdX₂ take place from the valence states of the hybridized Pd d and chalcogen p states to the conduction band states of Pd d orbitals.

Table 1 shows that the GGA band gaps for BL and FL PdSe₂ are significantly smaller than the corresponding experimental values [9, 11], indicating that the GGA functional generally underestimates the band gaps of semiconductors, as mentioned in the preceding section. Therefore, we also perform the band-structure calculations using the HSE functional [37, 38] and the HSE band structures are presented in figure S2 in the SI. Although the dispersions of the HSE band structures are similar to that of the corresponding GGA band structures, the band gaps from the HSE functional are significantly larger than that of the GGA ones (table 1). Furthermore, the HSE band gaps of BL and FL PdSe₂ are in better agreement with the experimental values than the GGA band gaps (table 1). Therefore, all the optical properties presented in the following sections are calculated within the SC scheme [35] by using the HSE band gaps (table 1). Nevertheless, we notice that the experimental band gap for BL PdSe₂ is significantly smaller than that of the HSE calculation (table 1). This discrepancy could be due to the effect of the graphene substrate [11] because the environment can significantly affect the electronic properties of 2D materials [39, 40]. For example, the graphene substrate can introduce extra screening, thus reducing the band gap [39, 40]. In particular, putting a BN sheet on graphene would reduce the band gap of BN ML by as much as ∼1.0 eV (14%) [39].

3.2. Linear optical properties

The calculated dielectric functions of all the four PdX₂ structures are plotted in figure 4. As can be expected of a 2D material, there are huge differences between the out-of-plane and in-plane components of the dielectric functions (see figure 4). For example, the real part of the z-polarized dielectric function ${\varepsilon }_{zz}^{\prime }$ is only about half of that for the in-plane polarized dielectric function ${\varepsilon }_{xx}^{\prime }$ and ${\varepsilon }_{yy}^{\prime }$ below 3 eV. The imaginary part of the z-polarized dielectric function ɛ''_zz is about four times smaller than that of the in-plane dielectric functions ɛ''_xx and ɛ''_yy. There are also discernible differences between the two in-plane components of the dielectric functions, namely, ɛ_xx and ɛ_yy (see figure 4). In particular, the real dielectric constant of ${\varepsilon }_{xx}^{\prime }$ is slightly higher than ${\varepsilon }_{yy}^{\prime }$ in the energy range of 2.5–3.0 (2.0–2.5) eV and 4.3–5.6 (4.0–5.0) eV, whereas ${\varepsilon }_{yy}^{\prime }$ in the energy range of 3.0–4.3 (2.5–4.0) eV and also in 5.6–6.0 (5.0–5.7) eV is higher than ${\varepsilon }_{xx}^{\prime }$ for BL PdS₂ (PdSe₂) (see figures 4(a) and (c)). In FL PdS₂ (PdSe₂) structure, the real dielectric constant ${\varepsilon }_{xx}^{\prime }$ is higher than ${\varepsilon }_{yy}^{\prime }$ in the energy range of 2.2–2.7 (1.9–2.5) eV and 4.3–5.6 (4.0–5.6) eV, whereas ${\varepsilon }_{yy}^{\prime }$ in the energy range of 2.7–4.3 (2.5–4.0) eV and also in 5.6–6.00 (5.0–6.0) eV is slightly higher than ${\varepsilon }_{xx}^{\prime }$ (see figures 4(e) and (g)). Similar profile is found in the imaginary part of the dielectric constant, where ɛ''_xx is higher than ɛ''_yy in the energy range of 2.6–3.1 eV (2.5–3.4 eV) and 3.9–5.0 eV (5.0–6.0 eV) for BL (FL) PdS₂ and 2.2–2.9 eV (2.1–3.0 eV) and 4.5–5.4 eV (4.5–5.3 eV) for BL (FL) PdSe₂, whereas in the other energy windows ɛ''_yy is higher than ɛ''_xx for both BL (FL) PdS₂ and PdSe₂ structures, see figures 4(b) and (d) (figures 4(f) and (h)). It is worth mentioning that the large imaginary part of the dielectric function of all the PdX₂ structures spans over a wide range of the visible frequency range. This suggests that these 2D PdX₂ structures will be useful for opto-electronic applications such as high solar-absorption efficiency solar cells [41].

3.3. Shift and injection currents

As mentioned before, the point symmetry group of BL and FL PdX₂ is C_2v with the C₂ rotation axis along the y-axis. Therefore, there are only five inequivalent nonzero shift current tensor elements [42], namely, ${\sigma }_{xxy}^{sh}={\sigma }_{xyx}^{sh}$ , ${\sigma }_{yxx}^{sh}$ , ${\sigma }_{yyy}^{sh}$ , ${\sigma }_{yzz}^{sh}$ and ${\sigma }_{zyz}^{sh}={\sigma }_{zzy}^{sh}$ . Similarly, there are only two inequivalent nonzero injection current susceptibility elements, namely, η_xxy = −η_xyx and η_zzy = −η_zyz [42]. Since the photocurrent cannot flow along the out-of-plane direction (the z-axis), we will not consider the nonzero elements of ${\sigma }_{zyz}^{sh}={\sigma }_{zzy}^{sh}$ and η_zzy = −η_zyz in the rest of this paper.

The calculated four inequivalent nonzero shift current conductivity elements for BL and FL PdX₂ are plotted in figure 5. Figure 5 shows that in all the PdX₂ structures, the four shift current conductivity elements are zero below the band gap but increase rapidly above the band gap. Notably, the ${\sigma }_{xxy}^{sh}$ element in BL PdS₂ dominates the low photon energy range of 2.4 ∼ 3.2 eV with a prominent peak of height of 160 μA V⁻² at 2.9 eV. The ${\sigma }_{xxy}^{sh}$ of BL PdS₂ has a second peak with the reduced maximum of 65 μA V⁻² at 3.9 eV (see figure 5(a)). For BL PdSe₂, these two prominent peaks in the ${\sigma }_{xxy}^{sh}$ spectrum become comparable with the maximum values of ∼115 μA V⁻² and ∼126 μA V⁻² at 2.6 eV and 3.6 eV, respectively (see figure 5(b)). In both BL structures, the magnitudes of the ${\sigma }_{yxx}^{sh}$ and ${\sigma }_{yyy}^{sh}$ spectra are also pronounced in the photon energy range from the absorption edge to ∼5.0 eV. For example, the ${\sigma }_{yyy}^{sh}$ of BL PdS₂ has a negative peak at ∼3.9 eV with the maximum value of −90 μA V⁻². Figure 5 also indicates that the magnitude of the ${\sigma }_{yzz}^{sh}$ spectra from all the four structures is much smaller than all the other shift conductivity elements. This may be attributed to the fact that the absorptive part of the out-of-plane polarized dielectric function element ɛ_zz is much smaller than that of the in-plane polarized dielectric function elements ɛ_xx and ɛ_yy (see figure 4). In the FL structure, the maximum shift current conductivity is from ${\sigma }_{xxy}^{sh}$ ( ${\sigma }_{yyy}^{sh}$ ) of the order of ∼97 (∼90) μA V⁻² at a photon energy of 2.9 (2.5) eV for PdS₂ (PdSe₂) (see figures 5(c) and (d), respectively). Similar to the BL structures, FL PdX₂ also have contributions of ∼50 μA V⁻² from other elements such as ${\sigma }_{yyy}^{sh}$ for FL PdS₂ and ${\sigma }_{xxy}^{sh}$ , ${\sigma }_{yxx}^{sh}$ for FL PdSe₂ in the energy range of 3–4 eV (see figures 5(c) and (d)).

The calculated nonzero injection current susceptibility element η_xxy is displayed as a function of photon energy for both BL and FL PdX₂ in figure 6. Using the typical relaxation time τ = 0.04 ps for 2D PdX₂ materials [13], we obtain the injection photocurrent conductivity ${\sigma }_{xxy}^{inj}=\tau {\eta }_{xxy}$ , as shown in figure 6. For all the PdX₂ structures, both η_xxy and ${\sigma }_{xxy}^{inj}$ are zero below the absorption edge but they increase rapidly above the absorption edge. In BL PdS₂, the calculated injection conductivity ${\sigma }_{xxy}^{inj}$ has two negative prominent peaks with the maximum values of −240 and −360 μA V⁻² at 2.7 and 4.7 eV, respectively (see figure 6(b)). Similarly, in BL PdSe₂, we also find two pronounced peaks with larger maximum values of −400 and −580 μA V⁻² at 3.1 and 4.0 eV, respectively (see figure 6(d)). In the FL structures, the magnitudes of both η_xxy and ${\sigma }_{xxy}^{inj}$ spectra are smaller compared with that from the BL structures (figure 6). In particular, the magnitudes of the η_xxy and ${\sigma }_{xxy}^{inj}$ of FL PdSe₂ are generally less than half of that from BL PdSe₂ (see figures 6(c) and (d)). Nonetheless, FL PdS₂ and FL PdSe₂ do have a rather pronounced peak in the ${\sigma }_{xxy}^{inj}$ spectrum with the peak value being about −200 μA V⁻² at 3.1 and 2.8 eV, respectively.

**Figure 6.** (a) and (c) Injection current susceptibility (η_xxy) tensor and (b) and (d) injection conductivity (σ_xxy) of BL and FL PdS₂ [PdSe₂].
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Let us now compare the BPVE in the present structures with the well-known BPVE materials to access their application potentials in photovoltaic solar cells and opto-electronic devices. The theoretically predicted shift current conductivity for the archetypal ferroelectrics PbTiO₃ and BaTiO₃ [43], have a value within 10 μA V⁻² in the visible frequency range, which is in agreement with the earlier experiments [44]. These values are several times smaller than the present predictions for the 2D PdX₂ materials, as shown in figure 7(a). Recently, the shift current conductivity of some chiral materials was predicted to be rather large, being in the range of 20 ∼ 80 μA V⁻² in the visible frequency range [45]. Furthermore, ML group-IV monochalcogenides were found to exhibit large shift current conductivity of about 100 μA V⁻² in the visible frequency range. Nevertheless, these values are smaller or at best comparable to the present predictions for the 2D PdX₂ structures (see figure 7). Interestingly, the calculated injection current susceptibility of the present PdX₂ structures is two-orders larger than the experimental values of 1.5 × 10⁸ A V⁻² s⁻¹ and 4 × 10⁸ A V⁻² s⁻¹ for semiconductors CdSe and CdS [46], respectively, which are in the same order of magnitude with the theoretical predictions reported in reference [32]. Among the 2D materials, as figure 7(b) shows, the injection current susceptibility of ML group-IV chalcogenides was reported to be one-order larger compared to the present PdX₂ structures [51, 54].

**Figure 7.** Comparison of maximum NLO responses of the considered PdX₂ structures with other reported NLO materials within the visible frequency range (i.e. up to 4.0 eV): (a) shift current conductivity [43, 45–50], (b) injection current susceptibility [46, 49, 51] and (c) SHG susceptibility [5, 52, 53].
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Nevertheless, 2D PdX₂ structures are atomically thin with thickness being about two-orders of magnitude smaller than visible light wavelength. One may wonder whether such atomically thin films would absorb sufficient light and generate significant photocurrent. In figure 8, we display the absorbance spectra of 2D PdX₂ structures. It is clear from figure 8 that although atomically thin, BL and FL PdX₂ can absorb around 10% and 20% of incident light above the absorption edge, respectively. Now we evaluate the photocurrent due to the BPVE in 2D PdX₂ structures. Let us consider the normal incidence of linearly polarized light of intensity I_b(0). The shift current is then given by

$\begin{align}{J}_{abb}^{sh}& =w{\sigma }_{abb}^{sh}{\int }_{0}^{d}\vert {E}_{b}(z){\vert }^{2}\mathrm{d}z=w{\sigma }_{abb}^{sh}\frac{2{I}_{b}(0)}{c{\varepsilon }_{0}}{\int }_{0}^{d}{\text{e}}^{-{\alpha }_{bb}z}\mathrm{d}z& \\ & =w\enspace \mathrm{d}{\sigma }_{abb}^{sh}\frac{2{I}_{b}(0)}{c{\varepsilon }_{0}{\alpha }_{bb}d}(1-{\text{e}}^{-{\alpha }_{bb}d}),& \end{align} \tag{ 5 }$

where w is the width of the sample, d = nh is the effective film thickness, I_b(z) = cɛ₀|E_b(z)|²/2, ɛ₀ is the vacuum permittivity and c the speed of light. Here we choose the same experimental parameters w = 0.15 cm and I_b(0) = 0.5 mW cm⁻¹ as in the experiment on BaTiO₃ [44]. The calculated shift current spectra are plotted in figure 8. Strikingly, the shift currents due to the BPVE in the atomically thin PdX₂ semiconductors are comparable to the measured one in bulk ferroelectric BaTiO₃ [44] (see figure 2 in reference [43]). This clearly indicates that 2D PdX₂ semiconductors would find promising applications in, e.g. solar cells.

**Figure 8.** Absorbance and shift current ( ${J}_{abb}^{sh}$ ) of BL PdS₂ and PdSe₂ as well as FL PdS₂ and PdSe₂. Absorbance is given by absorption coefficient (α) times film thickness (d). Shift current is calculated by using the parameters for the experiment on BaTiO₃ [44], namely, incident light intensity I_bb(0) = 0.5 mW cm⁻² and sample width w = 0.15 cm.
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**Figure 8.** Absorbance and shift current ( ${J}_{abb}^{sh}$ ) of BL PdS₂ and PdSe₂ as well as FL PdS₂ and PdSe₂. Absorbance is given by absorption coefficient (α) times film thickness (d). Shift current is calculated by using the parameters for the experiment on BaTiO₃ [44], namely, incident light intensity I_bb(0) = 0.5 mW cm⁻² and sample width w = 0.15 cm.
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3.4. Second-harmonic generation

For a noncentrosymmetric material, the nonzero elements of its SHG susceptibility tensor are the same as that of its shift current conductivity tensor. Therefore, as for the shift current conductivity, the SHG susceptibility tensor of the BL and FL PdX₂ has five independent nonzero elements, i.e. ${\chi }_{xxy}^{(2)}={\chi }_{xyx}^{(2)}$ , ${\chi }_{yxx}^{(2)}$ , ${\chi }_{yyy}^{(2)}$ , ${\chi }_{yzz}^{(2)}$ and ${\chi }_{zyz}^{(2)}={\chi }_{zzy}^{(2)}$ [3, 4]. The real and imaginary parts as well as the absolute of these nonzero elements for BL PdX₂ and FL PdX₂ are presented in figures 9 and 10, respectively. Figures 9 and 10 show that the imaginary (absorptive) parts of the SHG susceptibility for both BL and FL PdX₂ are zero for photon energy being smaller than half of the band gaps but they increase rapidly above half of the band gaps. Furthermore, below half of the band gaps, the real (dispersive) parts of the SHG susceptibility are small and remain almost constant. These nonzero dispersive parts of the SHG susceptibility below half of the band gaps give rise to the low-frequency LEO effect in the PdX₂ structures, which will be discussed in the next section. As for the imaginary parts, they increase rapidly above half of the band gaps.

**Figure 9.** Real (upper panels) and imaginary (middle panels) parts as well as absolute (lower panels) of the SHG susceptibility of BL PdS₂ and BL PdSe₂.
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**Figure 10.** Real (upper panels) and imaginary (middle panels) parts as well as absolute (lower panels) of the SHG susceptibility of FL PdS₂ and FL PdSe₂.
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Among the nonzero SHG susceptibility elements, ${\chi }_{yxx}^{(2)}$ and ${\chi }_{yyy}^{(2)}$ generally exhibit larger magnitudes in the visible frequency range for all the PdX₂ structures (see figures 9(c) and (f) as well as figures 10(c) and (f)). In BL PdS₂, both ${\chi }_{yyy}^{(2)}$ and ${\chi }_{yxx}^{(2)}$ spectra show a pronounced peak with almost the same maximum value of ∼1.2 × 10³ pm V⁻¹ at slightly different photon energies of 3.1 and 3.3 eV, respectively (see figure 9(c)). They come from the negative peak in the real and imaginary parts of the ${\chi }_{yyy}^{(2)}$ and ${\chi }_{yxx}^{(2)}$ spectra at 3.1 and 3.3 eV, respectively (see figures 9(a) and (b)). The ${\chi }_{yyy}^{(2)}$ spectrum of BL PdS₂ exhibits the second maximum of 0.9 × 10³ pm V⁻¹ at 1.6 eV (see figure 9(c)). It originates from the negative peak near 1.6 eV in both real and imaginary parts of the ${\chi }_{yyy}^{(2)}$ spectrum (see figures 9(a) and (b)). Remarkably, the ${\chi }_{yyy}^{(2)}$ absolute spectrum of BL PdSe₂ has two gigantic peaks of heights of 1.1 and 1.4 × 10³ pm V⁻¹ at 1.4 and 1.9 eV, respectively (see figure 9(f)) and they stem from the negative and positive peaks at 1.4 and 1.9 eV in the real and imaginary parts of the spectrum (see figures 9(d) and (e)). The ${\chi }_{yxx}^{(2)}$ spectrum of BL PdSe₂ has a broad twin peak of magnitude of 1.1 × 10³ pm V⁻¹ centered at ∼1.9 eV (see figure 9(f)). Nonetheless, below the band gap, the magnitudes of the ${\chi }_{xxy}^{(2)}$ and ${\chi }_{yzz}^{(2)}$ spectra of the PdX₂ structures are generally comparable or even larger than that of the ${\chi }_{yyy}^{(2)}$ and ${\chi }_{yxx}^{(2)}$ spectra. For example, both BL PdS₂ and BL PdSe₂ have a rather broad peak of ∼0.8 × 10³ pm V⁻¹ at 1.5 and 1.3 eV, respectively (see figures 9(c) and (f)). All the ${\chi }_{abc}^{(2)}$ spectra from the FL PdX₂ structures are generally smaller than the corresponding spectra of the BL PdX₂ structures. Nonetheless, the magnitudes of the ${\chi }_{yyy}^{(2)}$ and ${\chi }_{yxx}^{(2)}$ spectra of FL PdSe₂ do peak at 1.2 and 1.7 eV with the large maximum values of 0.94 × 10³ and 1.2 × 10³ pm V⁻¹, respectively (see figure 10(f)).

As shown before (see, e.g. reference [5] and references therein), the prominent features in the SHG susceptibility are generally caused by either single (ω) and double (2ω) photon resonances or both. Thus, to help understand the origins of the prominent features in the calculated SHG spectra, we plot the absolute values of the imaginary parts of the nonzero SHG elements of BL PdX₂ and FL PdX₂ in figures 11 and 12, respectively, along with the absorptive parts of the corresponding dielectric functions ɛ''(ω) and ɛ''(2ω). Figures 11(a) and (b) (figures 12(a) and (b)) show that the prominent features in the |χ''⁽²⁾| spectra of BL (FL) PdS₂ in the energy range of 1.2–2.5 eV (1.1–2.2 eV) below the band gap look similar to the features in the ɛ''(2ω) spectra, indicating that they are due to double-photon resonances. Similarly the pronounced features in the $\vert {\chi }_{yyy}^{\prime \prime (2)}\vert$ and $\vert {\chi }_{yxx}^{\prime \prime (2)}\vert$ spectra above the band gap of 2.5 eV (2.2 eV), have a shape similar to that in the ɛ''(ω), suggesting that they are caused primarily by single-photon resonances. On the other hand, the $\vert {\chi }_{yyy}^{\prime \prime (2)}\vert$ and $\vert {\chi }_{yxx}^{\prime \prime (2)}\vert$ spectra above the band gap of 2.5 eV (2.2 eV) have a much reduced amplitude and also are rather oscillatory, being rather similar to the ɛ''(2ω) spectra in this regime. This indicates that they stem mainly from the double (2ω) photon resonances.

**Figure 11.** (a) and (c) Absolute value of the imaginary part of the SHG susceptibility and (b) and (d) imaginary dielectric constant of BL PdS₂ [PdSe₂].
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**Figure 12.** (a) and (c) Absolute value of the imaginary part of the SHG susceptibility and (b) and (d) imaginary dielectric constant of FL PdS₂ [PdSe₂].
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We now compare the calculated SHG susceptibility of BL and FL PdX₂ with that reported for other NLO materials. ML group 6B TMDC semiconductors such as MoS₂, are considered to be the most promising 2D NLO materials because of their direct band gaps and large SHG susceptibility (see reference [5] and references therein). Remarkably, figure 7(c) shows that the maximum values of the SHG susceptibility of BL PdX₂ are comparable or even larger than that of ML group 6B TMDCs [5]. Similarly, FL PdX₂ generally have the SHG susceptibility that are comparable or even larger than that of TL group 6B TMDCs [5]. Interestingly, ReS₂ is another rare TMDC that exhibits second-order NLO responses only when the number of layers is even. The measured SHG susceptibility of BL ReS₂ is large, being about 900 pm V⁻¹ at 0.8 eV [55]. Nevertheless, this is smaller than that of BL PdS₂ and BL PdSe₂.

3.5. Linear electro-optical coefficient

Here we estimate the LEO coefficients of BL and FL PdX₂ structures, based on the obtained SHG susceptibility at low-frequency limit and static dielectric constant. Note that the LEO coefficients we present here represent only the electronic contribution. There are other contributions to the LEO coefficient such as ionic and piezoelectric contributions, which are beyond of the scope of the present work. The calculated LEO coefficients r_abc(0) at zero frequency along with the static dielectric constants and SHG susceptibilities ${\chi }_{xyz}^{(2)}(0,0,0)$ are listed in table 2. It is clear from table 2 that BL PdS₂ and BL PdSe₂ have much larger LEO values than FL PdS₂ and FL PdSe₂. BL PdS₂ and BL PdSe₂ also exhibit a rather strong anisotropy in the LEO effect. Semiconductor GaAs was reported to have an LEO coefficient of r_xyz(0) = −1.5 pm V⁻¹ [57]. Recent calculations [5] predicted that the magnitudes of the LEO coefficients of ML group 6B TMDC semiconductors are about 1.5 pm V⁻¹, being close to that of GaAs. Table 2 indicates that the LEO coefficients for BL PdS₂ and BL PdSe₂ are many times smaller than that of GaAs [57] and also ML group 6B TMDC semiconductors [5]. Nevertheless, they are in the same order of magnitude as that of trilayer group 6B TMDC semiconductors [5].

Table 2. Calculated static dielectric constants (ɛ_x, ɛ_y and ɛ_z), second-order optical susceptibility χ⁽²⁾(0) in pm V⁻¹, and linear electro-optical coefficients r_abc in pm V⁻¹ of BL and FL PdX₂ structures.

	ɛ_x	ɛ_y	ɛ_z	xxy	yxx	yyy	yzz
BL-PdS₂	12.95	12.73	5.97
χ⁽²⁾(0) (pm V⁻¹)				−11	−8.80	−13	−7.04
r_abc (pm V⁻¹)				0.13	0.11	0.16	0.09
BL-PdSe₂	13.84	13.90	6.68
χ⁽²⁾(0) (pm V⁻¹)				−28	−22	−24	−18
r_abc (pm V⁻¹)				0.29	0.23	0.25	0.19
FL-PdS₂	12.09	11.92	5.40
χ⁽²⁾(0) (pm V⁻¹)				−6.75	1.39	−5.41	−0.42
r_abc (pm V⁻¹)				0.09	−0.02	0.08	0.01
FL-PdSe₂	14.84	14.64	8.46
χ⁽²⁾(0) (pm V⁻¹)				−0.96	−12	−4.90	−5.78
r_abc (pm V⁻¹)				0.01	0.11	0.05	0.05

4. Discussion and conclusions

It was recently pointed out [58] that the size of the band gap of a semiconductor is a principal factor that determines the strength of its second-order NLO responses. To understand other origins of the large NLO responses in the 2D PdX₂ structures compared with other NLO materials of similar band gaps, we calculate the deformation charge density, which is defined as the difference between the valence charge density and the superposition of the free atomic charge densities. The calculated deformation charge density distributions for the four 2D PdX₂ structures look very similar. Thus, here we focus only on the deformation charge density of BL PdSe₂, which is displayed in figure 13. Clearly, there is a significant buildup of the electron charge in the vicinity of the Pd–Se bond center by depleting the charge around the Pd atoms along the bond directions. This is caused by the strong directional covalent bonding in BL PdSe₂, and can lead to an enhanced optical responses due to large spatial overlap between the wavefunctions of the initial and final states, and high anisotropy which would result in large NLO response values [58–60].

**Figure 13.** Contour plot of the deformation charge density distribution of BL PdSe₂. The contour interval is 0.002 e Å⁻ ³. The electron accumulation is depicted by positive contours (blue solid lines), while the electron depletion is represented by negative contours (red dashed lines). The black dashed line indicates the 2D unit cell (see figure 1(c)).
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We notice that the NLO responses of FL PdX₂ are generally weaker than that of BL PdX₂ (see figures 5–7, 9 and 10). In particular, figure 6(c) shows that the injection current susceptibility of FL PdSe₂ is about half of that of BL PdSe₂. This can be roughly explained as follows. As mentioned earlier, 2D PdX₂ materials with an odd layer-number is centrosymmetric, i.e. they have no second-order NLO response. On the other hand, a 2D PdX₂ material with an even layer-number, which can be considered as a 2D PdX₂ material with an odd layer-number plus one PdX₂ ML, is noncentrosymmetric. In other words, in either BL or FL PdX₂, effectively, only one PdX₂ ML is second-order NLO active. Consequently, since the effective unit cell volume for FL PdX₂ is a factor of 2 larger than that of BL PdX₂, the magnitude of the NLO responses of FL PdX₂ would be only half of that for BL PdX₂ if both structures had a similar band gap and also interlayer interaction is negligible (see equations (2) and (3)). Nevertheless, FL PdX₂ generally has a smaller band gap than BL PdX₂. Also interlayer interaction is not negligibly small. Therefore, the strength of the second-order NLO responses of FL PdX₂ generally falls between half and one of that for BL PdX₂. This is quite similar to 2D group 6B TMDC semiconductors (MX₂) where the magnitude of SHG for TL MX₂ is about 1/3 of that of ML MX₂ [5].

In conclusion, we have systematically studied the second-order NLO responses of BL and FL PdX₂ structures using first-principles DFT calculations. First of all, we predict that the BPVE in the considered 2D PdX₂ structures is generally strong, with a large shift current conductivity of up to 130 μA V⁻² and injection current susceptibility of up to 100 × 10⁸ A V⁻² s⁻¹ in the visible frequency range. Indeed, the calculated shift currents in these atomically thin PdX₂ semiconductors (figure 8) are comparable to that observed in bulk ferroelectric BaTiO₃ [44]. Secondly, we find that all PdX₂ structures possess large SHG susceptibility χ⁽²⁾ in the visible frequency spectrum and BL PdSe₂ has the largest χ⁽²⁾ value of 1.4 × 10³ pm V⁻¹ at 1.9 eV. Thirdly, we find that the LEO coefficients of BL PdS₂ and BL PdSe₂ are significant. These superior NLO responses of the BL and FL PdX₂ structures will make them valuable for technological applications in NLO and electo-optic devices such as light signal modulators, frequency converters, electro-optical switches, photovoltaics and photodetector applications. Finally, the strong NLO responses of BL and FL structures of PdX₂ are attributed to strong intralayer directional covalent bonding and also 2D quantum confinement.

Acknowledgment

The authors acknowledge the support by the Ministry of Science and Technology and National Center for Theoretical Sciences, Taiwan. The authors also thank National Center for High-performance Computing, Taiwan for the computing time.

Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

Large bulk photovoltaic effect and second-harmonic generation in few-layer pentagonal semiconductors PdS₂ and PdSe₂

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Abstract

1. Introduction

2. Computational methods