Introduction

Thermoelectric materials, which enable a direct conversion between heat and electricity via either Seebeck or Peltier effect, have attracted much attention as a sustainable energy resource in the last decade1. The conversion efficiency of a thermoelectric material is quantified by the dimensionless thermoelectric figure of merit (ZT), which is defined as ZT = S2σT/(κ e  + κ l ), where S is the Seebeck coefficient, σ the electrical conductivity, T the absolute temperature, κ e and κ l the electronic and lattice thermal conductivities, respectively. Obviously, higher power factor (PF = S2σ) and lower thermal conductivity are beneficial for improving the thermoelectric performance. The all-scale electronic and atomistic structural engineering techniques have been used to enhance ZT values to 2 within a temperature range of 700 ~ 900 K2,3,4,5. Another promising simple structures exhibit intrinsically low thermal conductances without requiring sophisticate structural engineering such as SnSe crystal and with ZT value of 2.6 at 923 K6, although this value falls quickly for lower temperatures.

Since the discovery of graphene in 20047,8, many 2D structures of inorganic layered materials, such as black phosphorus9,10,11 and h-BN12,13 etc., have been experimentally realized during the last decade. It has been proposed that low-dimensional materials could have better thermoelectric performance than their bulk due to the diverse scattering mechanism for phonons and intrinsic energy dependence of their electronic density of states14,15,16. And even in high dimensional materials, one can make use of the effective low dimensionality of the electron band to increase the thermoelectric performance17,18,19. Recently, the class of transition metal dichalcogenide (TMD) with one layer of transition metal sandwiched between two layers of chalcogen atoms have been a subject of extensive studies due to their fantastic electronic properties20,21,22. However, the ZT values of 2H- MoSe2, MoS2 and WSe2 monolayers are about 0.1 at 1200 K23, 0.11 at 500 K24 and 0.7 at high temperature23, respectively. It was confirmed that such a low ZT is mainly caused by a high lattice thermal conductivity κ l . While those with CdI 2 type typically represented by M = Ti, Zr, Hf, etc. have much lower lattice thermal conductivities. For example, the κ l values of monolayer ZrSe2 and HfSe2 are 1.2 and 1.8 Wm−1 K−1 25, respectively at 300 K, leading to optimum ZT values of 0.87 and 0.95, respectively.

Most recently, another class of layered materials formed by noble metals, such as Pt and Pd, with S and Se atoms have been investigated both experimentally and theoretically26,27,28,29,30. Importantly, the monolayer PdSe2 has very recently been exfoliated from bulk crystals by Akinola D. Oyedele et al.28, which is a pentagonal 2D layered noble transition metal dichalcogenide with a puckered morphology that is air-stable. The experimental results by Oyedele et al. demonstrated that few-layer PdSe2 displayed tunable ambipolar charge carrier conduction with a high electron apparent field-effect mobility of ~158 cm2 V−1 s−1. In addition, the puckered 2D PdSe2 flakes exhibit a widely tunable band gap that varies from metallic (bulk) to ~1.3 eV (monolayer). Motivated by this, we expand our knowledge on the thermoelectric properties on the monolayer PdSe2 in this work. And to the best of our knowledge, there is no utter investigation in the thermoelectric properties of the monolayer PdSe2. In this paper, we investigate PdSe2 monolayer with the configuration of the above experiment, performing electronic structure, and phononic transport calculations based on density functional theory (DFT) and Boltzmann transport theory. The results show that monolayer PdSe2 is an indirect semiconductor, with a band-gap value of 1.38 eV, which is in good agreement with ref.28. Based on the electronic and phononic properties, we study the thermoelectric properties of monolayer PdSe2. We obtain the Seebeck coefficients for monolayer PdSe2 and a maximum p-type figure of merit, 1.1, along the x direction at the optimal doping (300 K). We also find anisotropic characters in electrical conductivity and thermal conductivity which are derived from the asymmetric structure of the monolayer PdSe2 in plane.

Results and Discussions

Geometric structure

In our calculations, the monolayer structure is obtained from the experimental bulk structure PdSe2 with a = 5.75 Å, b = 5.87 Å, and c = 7.69 Å31. The monolayer PdSe2 is cut through the (0 0 1) plane of the PdSe2 crystal, and a vacuum slab about 21 Å is added in the direction perpendicular to the nano-sheet plane (z direction). As shown by the side view and projected top view of the PdSe2 monolayer in Fig. 1(a) and (b), each Pd atom binds to four Se atoms in the same layer, two neighboring Se atoms can form a covalent Se-Se bond32 and two Pd atoms and three S atoms can form a wrinkled pentagon, which is rather rare in known materials. In addition, we note that the space group has changed from pbca to pca21 evolving from bulk to monolayer, which has been found in experiments27. The unit cell of monolayer PdSe2 is displayed in Fig. 1(c) and the optimized lattice parameters of monolayer PdSe2 are a = 5.7538 Å and b = 5.9257 Å, which are in good agreetment with the previous reports26,27.

Figure 1
figure 1

(a) and (b) are the top and side views of PdSe2 monolayer, respectively. (c) The unit cell and corresponding Brillouin zone path with the high-symmetry points at Γ(0, 0, 0), X(0.5, 0, 0), M(0.5, 0.5, 0) and Y(0, 0.5, 0). The lattice parameters are denoted as a and b, which are along the x and the y directions, respectively. Cyan: Pd atom. Yellow:Se atom.

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In order to verify the stability of the monolayer PdSe2, we perform phonon dispersion calculations33. As represented in Fig. 2, there are no soft modes in the calculated phonon dispersions, indicating the dynamical stability of this structure. This is also consistent with the previous reports28,31.

Figure 2
figure 2

Calculated phonon dispersion spetrum of monolayer PdSe2. Here the band structure is along the high-symmetry points at Γ(0, 0, 0), X(0.5, 0, 0), M(0.5, 0.5, 0), Y(0, 0.5, 0) and Γ(0, 0, 0).

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Electronic transport properties

Experimental and theoretical studies have demonstrated that monolayer PdSe2 exhibits high mobility and Seebeck coefficient26,27, which are beneficial for the thermoelectric transport. Now we first turn to the investigation of electronic transport properties. Based on the above-determined configuration, we calculate the electronic band structure with the Brillouin zone path along Γ − X − M − Y − Γ as shown in Fig. 1(c). Computed via the TB-mBJ-GGA potential with spin-orbit coupling (SOC) included, the PdSe2 monolayer is semiconducting with an indirect band gap of 1.38 eV, which is in general agreement with the previous reports26,34, as depicted in Fig. 3. The conduction band minimum (CBM) locates at the M (0.5, 0.5, 0) points, while the valence band maximum (VBM) locates in the interval between Γ and X (0.5, 0, 0) points. The projected density of states reveals that the d-states of the transition metal atoms and p-states of the selenium atoms contribute most to the states at both VBM and CBM.

Figure 3
figure 3

Calculated electronic band structure (left) and density of states (right) of monolayer PdSe2 with indirect band gap of 1.38 eV. Here the band structure is along the high-symmetry points at Γ(0, 0, 0), X(0.5, 0, 0), M(0.5, 0.5, 0), Y(0, 0, 0.5) and Γ(0, 0, 0). The solid arrows indicate the lowest energy transitions between the valence band maximum (VBM) and conduction band minimum (CBM).

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The effective mass m* near the Fermi energy is an important parameter for the thermoelectric transport35, which can be extracted from the high-precise energy band calculation via the equation

$$\frac{1}{{m}_{\alpha }^{\ast }}=\frac{1}{\hslash }\frac{{\partial }^{2}E({k}_{a})}{{\partial }^{2}{k}_{\alpha }}$$
(1)

where ћ is the reduced Plank’s constant, E(k α ) is the band index α and wave vector k dependent energy. Thus, on the basis of the electronic band calculations, we can obtain the effective m* of electrons and holes in the x and y directions. As listed in Table 1, the effective mass along Γ-X and Γ-Y are 0.30(e), −0.25(h) and 0.12(e), −0.16(h), respectively. Obviously, in the m e unit of free electron mass, the effective masses along Γ-X are significantly larger than that along Γ-Y direction and even in the same direction there are slightly differences between holes and electrons, indicating the anisotropic electronic properties of monolayer PdSe2. Besides the band gap and effective mass, carrier mobility is another important factor for semiconducting materials in electronic transport properties. Therefore, in order to obtain more information on the transport properties of monolayer PdSe2, we investigate its carrier mobilities on the basis of Bardeen-Shockley deformation potential (DP) theory in 2D materials36,37. Note that the DP theory has been successfully performed to present the carrier mobility of many 2D structures38,39,40,41. Although the results may be less accurate, it can still reflect the basic and general thermoelectric performance of materials. According to the DP theory, the carrier mobility (μ) of 2D structure can be expressed as

$$\mu =\frac{e{\hslash }^{3}{C}_{2D}}{{k}_{B}T{m}^{\ast }{m}_{d}{E}_{l}^{2}}$$
(2)

where k B is the Boltzmann constant, T is the temperature, m d is the average effective mass defined as \({m}_{d}=\sqrt{{m}_{x}^{\ast }{m}_{y}^{\ast }}\) (\({m}_{x}^{\ast }\) and \({m}_{y}^{\ast }\) are the effective mass along the x and y directions, respectively). C2D is the in-plane effective elastic modulus for 2D system defined as \({C}_{2D}={\frac{1}{{S}_{0}}\frac{{\partial }^{2}E}{\partial {(l/{l}_{0})}^{2}}|}_{l={l}_{0}}\), where E and l are the total energy and lattice constant after deformation, l0 and S0 are the lattice constant and cell area at equilibrium for 2D system. E l is the deformation potential constant determined by \({E}_{l}={\frac{\partial {E}_{edge}}{\partial (l/{l}_{0})}|}_{l={l}_{0}}\), where E edge is the energy value of CBM (for electrons) and VBM (for holes). All the results are summarized in Table 1. The in-plane effective elastic modulus is 1.92 (x direction) and 1.17 (y direction) eV/Å2 much lower than those of MoS2 (7.99 eV/Å2)39 and PdS2 (3.62 eV/Å2 in the x direction and 5.11 eV/Å2 in the y direction)30, indicating that PdSe2 is much softer than MoS2 and PdS2 monolayer. As have been investigated in previous works, such large flexible deformation may improve the electronic properties via the compression (tensile) strain29,42,43,44. By fitting the band edge-strain curves, we find that the deformation potentia l s (E l ) of holes are rather small, namely −2.61 (x direction) and −2.89 (y direction), compared with the values of electrons of −8.49 (x direction) and −9.11 (y direction) cm2 V−1 s−1, respectively. Deformation potential constants describe the scattering caused by electron-acoustic phonon interactions. Thus, small of deformation potential constants may lead to large carrier mobilities. Then, based on the Equation 2, the acoustic phonon-limited carrier mobilities have been estimated. As shown in Table 1, the mobilities of electrons are 159.92 and 211.59 cm2 V−1 s−1 in the x and y directions, respectively. Whereas the mobilities of holes are 1928.99 (x) and 1498.03 (y), which are much larger than those of electrons mainly due to the rather small E l . However, the mobilities of both holes and electrons for the PdSe2 monolayer are larger than those of the MoS239 and PdS230, indicating that the monolayer PdSe2 would be a quite promising material for electronic and thermoelectric applications.

Table 1 The computed effective mass (m*), average effective mass (m d ), elastic modulus C2D, DP constant E l , carrier mobility (μ), relaxation time (τ) of electrons and holes along the x and y directions for the PdSe2 monolayer at 300 K.
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Now we are in a position to evaluate the electronic transport coefficients such as Seebeck coefficient S and electrical conductivity σ, based on the CRTA Boltzmann theory. The left (right) panels of Fig. 4 show the transport coefficients along the x and y directions as a function of the electron (hole) concentration at T = 300 K. It is clear that the σ in Fig. 4(a,b) increases with the increasing carrier concentration while the magnitude of S in Fig. 4(c,d) decreases with doping. The electrical conductivity σ of monolayer PdSe2 exhibits remarkable anisotropic behaviors with (σ y /σ x ) ~2.3 for n-type doping and (σ x /σ y ) ~2.4 for p-type at 1.1 × 1013 cm−2 concentration. The calculated Seebeck coefficients along the x and y directions as a function of carrier concentration are shown in Fig. 4(c) and (d) for n- and p-type doping, respectively. We find a larger asymmetry of the Seebeck coefficient for p-type doping than for n-type doping, which is in good agreement with the recent report26. This anisotropy in the thermopower values in the two different directions might enable to design transverse thermoelctric device45. It is important to note that the Seebeck coefficients for both n- and p-type doped monolayer PdSe2 are substantially high at room temperature, reaching a peak value of 660 μV/K at an electron concentration around 1.25 × 1011 cm−2 and with an average value in the range of 300–340 μV/K. These values of S for monolayer PdSe2 compare favorably with those reported for some other 2D materials30,39. Figure 4(e) and (f) depict the power factor (PF) S2σ at room temperature along the x and y directions for n- and p-doped PdSe2 monolayer, respectively. The results reflect significant anisotropy in the power factor with the PF x /PF y  ~1.9 for p-type doping and (PF) y /(PF) x  ~2 for n-type doping at concentration around 1.1 × 1013 cm−2. The anisotropy in power factor arises from the large anistotropy of the conductivity and Seebeck coefficient for p and n types, as described above.

Figure 4
figure 4

Calculated thermoelectric transport coefficients σ (a,b), S (c,d), and S2σ (e,f) vs carrier concentration for n-(left panels) and p-type (right panels) doped PdSe2 along the x (black lines) and the y (red lines) directions at room temperature.

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Phononic transport

Figure 2 shows the phonon dispersion relations of monolayer PdSe2 at its equilibrium volume along the high symmetric Γ − Y − M − X − Γ directions. It is noteworthy that the phonon spectrums of monolayer PdSe2 is very distinct from the MoS2 type monolayer. The maximum frequency of the acoustic mode markedly drop to rather low value of 3.7 THz, while for monolayers of MoSe2 and WSe2 it is 5.4 THz and 4.8 THz, respectively, and even higher for monolayer MoS2 with the value of 7.5 THz. Such low frequency suggests the low group velocity of acoustic modes in monolayer PdSe2. As acoustic modes contribute mostly to the lattice thermal conductivity κ l , lower κ l in this PdSe2 monolayer is expected.

Now we turn to the computation of lattice thermal conductivity κ l . As mentioned above, we estimate κ l by means of the phonon Boltzmann transport equation and DFT as implemented in VASP and ShengBTE code. As presented by the fitted lines in Fig. 5, κ l decreases following a T−1 dependence with the increasing temperature, suggesting that Umklapp phonon scattering dominates three-phonon interactions46. From the calculations, the obtained lattice thermal conductivity of monolayer PdSe2 is 3.7 (1.4) and 7.2 (2.7) Wm−1 K−1 at 300 K (800 K) along the x and y directions, respectively, which are much lower than MoS247 and GX2 monolayers48. It is obvious that the lattice thermal conductivity exhibits large directional anisotropy which may be due to differences in group velocity, anharmonicity and scattering phase space along the different directions.

Figure 5
figure 5

Calculated lattice thermal conductivity of monolayer PdSe2 along the x (red solid line) and the y (black solid line) directions from 300 K to 800 K with the interval of 100 K.

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Dimensionless figure of merit ZT

The electronic thermal conductivity κ e of monolayer PdSe2 is calculated via the Wiedemann-Franz law κ e  = L σT. Within the relaxation time approximation, the Seebeck coefficient can be calculated independently of the relaxation time τ, but evaluation of the electrical conductivity requires knowledge of τ. Here we take into account only the intrinsic scattering mechanism, namely, the interaction of electrons with acoustic phonons. Then the relaxation time τ can be evaluated from the equation τ = μm*/e, here the carrier mobility μ and effective mass m* have been calculated in subsection of Electronic transport properties, as listed in Table 1.

Combining the electronic and thermal transport properties, we now evaluate the thermoelectric performance of the PdSe2 monolayer. Figure 6 shows the figure of merit ZT value for both n and p doped PdSe2 monolayer along the x and y directions as a function of the carrier concentration at room temperature. We can see that the ZT values of n-type doped monolayer PdSe2 are rather small and almost isotropic with the maximum value of 0.13 with the corresponding concentration 3 × 1013 cm−2. However, for p-type doped monolayer PdSe2, ZT values exhibit the strong anisotropic property, with the value along the x direction being much larger than that along the y direction. The largest ZT value of 1.1 can be obtained in the x direction at the carrier concentration of 6.5 × 1012 cm−2 and 0.5 along the y direction at the carrier concentration of 2 × 1013 cm−2, respectively. Therefore, heavily doped p-type PdSe2 may offer excellent thermoelectric performance for applications such as powergeneration. It is worthwhile to note that we have not considered the thermoelectric performance at higher temperature since the ZA mode of PdSe2 monolayer is very soft near point Γ, hence, it may be difficult to remain stable at high temperature. Usually, the thermoelectric performance at room temperature is the most importantly information we need for it is better to discover thermoelectric materials working under room temperature.

Figure 6
figure 6

Calculated thermoelectric figure of merit (ZT) of monolayer PdSe2 with n-type doping (left plane) and p-type doping (right plane) at 300 K along the x (black solid lines) and y (red solid lines) directions.

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Conclusion

In summary, by means of first-principles calculation, the geometrical structure, mechanical, electronic and thermal transport properties of monolayer PdSe2 are systematically investigated. In contrast to TMCs, monolayer PdSe2 has strong anisotropic mechanical, electronic and thermal transport properties, leading to anisotropic thermoelectric properties. We find that PdSe2 is a semiconductor with an indirect band gap of 1.38 eV and a hole mobility as high as 1929 cm 2 V−1 s−1. The in-plane effective elastic modulus are rather low, suggesting the flexible mechanical properties in this structure. Furthermore, monolayer PdSe2 has a low lattice thermal conductivity about 3 Wm−1 K−1 along the x direction at room temperature. Combining its high Seebeck coefficient and markedly low thermal conductivity, monolayer PdSe2 shows an optimum ZT value of 1.1 (300K) at optimal doping. Therefore, our results indicate monolayer PdSe2 is a material with promising thermoelectric performance.

Computational Methods

The initial structure of monolayer PdSe2 is optimized through DFT with the plane-wave based Vienna ab-initio simulation package (VASP)49,50, using the projector augmented wave (PAW) method. For the exchange-correlation functional, we have used the Perdew-Burke-Ernzerhof version of the generalized gradient approximation (GGA)51. A plane-wave cutoff energy of 400 eV and an energy convergence criterion of 10−7 eV are adopted throughout calculations. The spin-orbit coupling (SOC) is not considered in the structure relaxation. For ionic relaxation calculations, a 11 × 11 × 1 Monkhorst-Pack k-meshes52 are used and the structure is considered to be stable when the Hellmann-Feynman forces are smaller than 0.001 eV/Å. For the slab model, a 21 Å thick vacuum layer was used to avoid the interactions between adjacent monolayers.

After determining the equilibrium structure, we have performed electronic structure calculations employing the all-electron full-potential WIEN2k code53 using recently implemented Tran and Blaha’s modified Becke-Johnson (TB-mBJ)54 exchange potential plus generalised gradient approximation (GGA) with the SOC included. The TB-mBJGGA potential for electronic properties and band gap with higher accuracy and less computational effort as compared to hybrid functional and GW overcomes the shortcoming of underestimation of energy gap in both LDA and GGA approximations55. The number of plane waves in a Fourier expansion of potential in the interstitial region was restricted to R MT  × K max  = 8. The muffin tin radii for Se and Pd are 2.1 and 2.2 a.u., respectively. We used 19 × 19 × 1 k-point Monkhorst-Pack mesh for electronic band structure calculations.

Based on the self-consistent converged electronic structure calculations, we have employed the eigenenergies on a very dense nonshifted 8000 k-point mesh in the full Brillouin zone (BZ). Thermoelectrical transport properties were calculated by solving the Boltzmann transport equations within the rigid band (RBA) and constant relaxation-time approximations (CRTA) as implemented in the BoltzTraP software56, which neglects the weak energy dependence of relaxation time but retains some temperature and doping dependence57. This CRTA approach has been tested earlier and found to work quite well in calculating the Seebeck coefficient in a variety of thermoelectric materials even for materials with highly anisotropic crystal axes58,59,60,61. A comprehensive description of the Boltzmann transport theory in the relaxation time approximation can be found elsewhere23. A brief summary of formalism used in this work is provided below62. The energy projected transport distribution (TD) tensor is defined as

$${\sigma }_{\alpha \beta }(\varepsilon )=\frac{{e}^{2}}{N}\sum _{i,k}{\tau }_{i,k}{v}_{\alpha }(i,k){v}_{\beta }(i,k)\delta (\varepsilon -{\varepsilon }_{i,k})$$
(3)

where group velocity \({v}_{\alpha }(i,k)=\frac{1}{\hslash }\frac{\partial {\varepsilon }_{i,k}}{\partial {k}_{\alpha }}\), N is the number of k-points sampled, τi, k is the band index i and wave vector k dependent relaxation time, α and β are the Cartesian indices, and e is the electron charge. Then the electrical conductivity and Seebeck coefficient as a function of temperation T and chemical potential μ, can be written as

$${\sigma }_{\alpha \beta }(T,\mu )=\frac{1}{{\rm{\Omega }}}\int {\sigma }_{\alpha \beta }(\varepsilon )[-\frac{\partial {f}_{0}(T,\varepsilon ,\mu )}{\partial \varepsilon }]d\varepsilon $$
(4)
$${S}_{\alpha \beta }(T,\mu )=\frac{1}{{e}^{2}T{\rm{\Omega }}}{\sigma }_{\alpha \beta }(T,\mu )\int {\sigma }_{\alpha \beta }(\varepsilon )(\varepsilon -\mu )\times [-\frac{\partial {f}_{0}(T,\varepsilon ,\mu )}{\partial \varepsilon }]d\varepsilon $$
(5)

where Ω is the volume of unit cell and f0 is the Fermi-Dirac distribution function. Thus, by using the CRTA, τ is exactly cancelled out in Equation 5. From the above calculations we can obtain the Seebeck coefficient S and the electrical conductivity over relaxation time (σ/τ) as well. The electronic thermal conductivity k e is calculated using the Wiedemann-Franz law, k e  = L σT, where L is the Lorenz number. In our calculations we use L = 2.4 × 10−8 J2 K−2 C−2 63.

To confirm the dynamic stability of the PdSe2 monolayer, we have calculated the phonon spectrum using a finite displacement method implemented the Phonopy code interfaced with the VASP code50,64. At the same time the second-order harmonic IFCs of monolayer PdSe2 and third order anharmonic IFCs were calculated using a 4 × 4 × 1 supercell and a 3 × 3 × 1 supercell with Γ point, respectively. Based on an adaptive smearing approach to the conservation of energy65 and with an iterative solution method66, we then solved the phonon Boltzmann transport equation using ShengBTE67.