Abstract
Motivated by the recent experimental synthesis of twodimensional semiconducting film PdSe_{2}, we investigate the electronic and thermal transport properties of PdSe_{2} monolayer by using the density functional theory and semiclassical Boltzmann transport equation. The calculated results reveal anisotropic transport properties. Low lattice thermal conductivity about 3â€‰Wm^{âˆ’1} K ^{âˆ’1} (300K) along the x direction is obtained, and the dimensionless thermoelectric figure of merit can reach 1.1 along the x direction for ptype doping at room temperature, indicating the promising thermoelectric performance of monolayer PdSe_{2}.
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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 decade^{1}. The conversion efficiency of a thermoelectric material is quantified by the dimensionless thermoelectric figure of merit (ZT), which is defined as ZTâ€‰=â€‰S^{2}ÏƒT/(Îº_{ e }â€‰+â€‰Îº_{ l }), where S is the Seebeck coefficient, Ïƒ the electrical conductivity, T the absolute temperature, Îº_{ e } and Îº_{ l } the electronic and lattice thermal conductivities, respectively. Obviously, higher power factor (PFâ€‰=â€‰S^{2}Ïƒ) and lower thermal conductivity are beneficial for improving the thermoelectric performance. The allscale electronic and atomistic structural engineering techniques have been used to enhance ZT values to 2 within a temperature range of 700â€‰~â€‰900â€‰K^{2,3,4,5}. 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 923â€‰K^{6}, although this value falls quickly for lower temperatures.
Since the discovery of graphene in 2004^{7,8}, many 2D structures of inorganic layered materials, such as black phosphorus^{9,10,11} and hBN^{12,13} etc., have been experimentally realized during the last decade. It has been proposed that lowdimensional 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 states^{14,15,16}. And even in high dimensional materials, one can make use of the effective low dimensionality of the electron band to increase the thermoelectric performance^{17,18,19}. 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 properties^{20,21,22}. However, the ZT values of 2H MoSe_{2}, MoS_{2} and WSe_{2} monolayers are about 0.1 at 1200â€‰K^{23}, 0.11 at 500â€‰K^{24} and 0.7 at high temperature^{23}, respectively. It was confirmed that such a low ZT is mainly caused by a high lattice thermal conductivity Îº_{ l }. While those with CdI _{2} type typically represented by Mâ€‰=â€‰Ti, Zr, Hf, etc. have much lower lattice thermal conductivities. For example, the Îº_{ l } values of monolayer ZrSe_{2} and HfSe_{2} are 1.2 and 1.8â€‰Wm^{âˆ’1} K^{âˆ’1}â€‰^{25}, respectively at 300â€‰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 theoretically^{26,27,28,29,30}. Importantly, the monolayer PdSe_{2} 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 airstable. The experimental results by Oyedele et al. demonstrated that fewlayer PdSe_{2} displayed tunable ambipolar charge carrier conduction with a high electron apparent fieldeffect mobility of ~158â€‰cm^{2} V^{âˆ’1} s^{âˆ’1}. In addition, the puckered 2D PdSe_{2} flakes exhibit a widely tunable band gap that varies from metallic (bulk) to ~1.3â€‰eV (monolayer). Motivated by this, we expand our knowledge on the thermoelectric properties on the monolayer PdSe_{2} in this work. And to the best of our knowledge, there is no utter investigation in the thermoelectric properties of the monolayer PdSe_{2}. In this paper, we investigate PdSe_{2} 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 PdSe_{2} is an indirect semiconductor, with a bandgap value of 1.38â€‰eV, which is in good agreement with ref.^{28}. Based on the electronic and phononic properties, we study the thermoelectric properties of monolayer PdSe_{2}. We obtain the Seebeck coefficients for monolayer PdSe_{2} and a maximum ptype figure of merit, 1.1, along the x direction at the optimal doping (300â€‰K). We also find anisotropic characters in electrical conductivity and thermal conductivity which are derived from the asymmetric structure of the monolayer PdSe_{2} in plane.
Results and Discussions
Geometric structure
In our calculations, the monolayer structure is obtained from the experimental bulk structure PdSe_{2} with aâ€‰=â€‰5.75â€‰Ã…, bâ€‰=â€‰5.87â€‰Ã…, and câ€‰=â€‰7.69â€‰Ã…^{31}. The monolayer PdSe_{2} is cut through the (0 0 1) plane of the PdSe_{2} crystal, and a vacuum slab about 21â€‰Ã… is added in the direction perpendicular to the nanosheet plane (z direction). As shown by the side view and projected top view of the PdSe_{2} monolayer in Fig.Â 1(a) and (b), each Pd atom binds to four Se atoms in the same layer, two neighboring Se atoms can form a covalent SeSe bond^{32} 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 pca2_{1} evolving from bulk to monolayer, which has been found in experiments^{27}. The unit cell of monolayer PdSe_{2} is displayed in Fig.Â 1(c) and the optimized lattice parameters of monolayer PdSe_{2} are aâ€‰=â€‰5.7538â€‰Ã… and bâ€‰=â€‰5.9257â€‰Ã…, which are in good agreetment with the previous reports^{26,27}.
In order to verify the stability of the monolayer PdSe_{2}, we perform phonon dispersion calculations^{33}. As represented in Fig.Â 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 reports^{28,31}.
Electronic transport properties
Experimental and theoretical studies have demonstrated that monolayer PdSe_{2} exhibits high mobility and Seebeck coefficient^{26,27}, which are beneficial for the thermoelectric transport. Now we first turn to the investigation of electronic transport properties. Based on the abovedetermined configuration, we calculate the electronic band structure with the Brillouin zone path along Î“â€‰âˆ’â€‰Xâ€‰âˆ’â€‰Mâ€‰âˆ’â€‰Yâ€‰âˆ’â€‰Î“ as shown in Fig.Â 1(c). Computed via the TBmBJGGA potential with spinorbit coupling (SOC) included, the PdSe_{2} monolayer is semiconducting with an indirect band gap of 1.38â€‰eV, which is in general agreement with the previous reports^{26,34}, as depicted in Fig.Â 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 Î“ and X (0.5, 0, 0) points. The projected density of states reveals that the dstates of the transition metal atoms and pstates of the selenium atoms contribute most to the states at both VBM and CBM.
The effective mass m^{*} near the Fermi energy is an important parameter for the thermoelectric transport^{35}, which can be extracted from the highprecise energy band calculation via the equation
where Ñ› is the reduced Plankâ€™s constant, E(k_{ Î± }) is the band index Î± 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 TableÂ 1, the effective mass along Î“X and Î“Y are 0.30(e), âˆ’0.25(h) and 0.12(e), âˆ’0.16(h), respectively. Obviously, in the m_{ e } unit of free electron mass, the effective masses along Î“X are significantly larger than that along Î“Y direction and even in the same direction there are slightly differences between holes and electrons, indicating the anisotropic electronic properties of monolayer PdSe_{2}. 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 PdSe_{2}, we investigate its carrier mobilities on the basis of BardeenShockley deformation potential (DP) theory in 2D materials^{36,37}. Note that the DP theory has been successfully performed to present the carrier mobility of many 2D structures^{38,39,40,41}. 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 (Î¼) of 2D structure can be expressed as
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). C_{2D} is the inplane 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, l_{0} and S_{0} 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 TableÂ 1. The inplane effective elastic modulus is 1.92 (x direction) and 1.17 (y direction) eV/Ã…^{2} much lower than those of MoS_{2} (7.99â€‰eV/Ã…^{2})^{39} and PdS_{2} (3.62â€‰eV/Ã…^{2} in the x direction and 5.11â€‰eV/Ã…^{2} in the y direction)^{30}, indicating that PdSe_{2} is much softer than MoS_{2} and PdS_{2} monolayer. As have been investigated in previous works, such large flexible deformation may improve the electronic properties via the compression (tensile) strain^{29,42,43,44}. By fitting the band edgestrain curves, we find that the deformation potentia_{ l }s (E _{ l }) of holes are rather small, namely âˆ’2.61 (x direction) and âˆ’2.89 (y direction), compared with the values of electrons of âˆ’8.49 (x direction) and âˆ’9.11 (y direction) cm^{2} V^{âˆ’1} s^{âˆ’1}, respectively. Deformation potential constants describe the scattering caused by electronacoustic phonon interactions. Thus, small of deformation potential constants may lead to large carrier mobilities. Then, based on the EquationÂ 2, the acoustic phononlimited carrier mobilities have been estimated. As shown in TableÂ 1, the mobilities of electrons are 159.92 and 211.59â€‰cm^{2} V^{âˆ’1} s^{âˆ’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 PdSe_{2} monolayer are larger than those of the MoS_{2}^{39} and PdS_{2}^{30}, indicating that the monolayer PdSe_{2} would be a quite promising material for electronic and thermoelectric applications.
Now we are in a position to evaluate the electronic transport coefficients such as Seebeck coefficient S and electrical conductivity Ïƒ, based on the CRTA Boltzmann theory. The left (right) panels of Fig.Â 4 show the transport coefficients along the x and y directions as a function of the electron (hole) concentration at Tâ€‰=â€‰300 K. It is clear that the Ïƒ in Fig.Â 4(a,b) increases with the increasing carrier concentration while the magnitude of S in Fig.Â 4(c,d) decreases with doping. The electrical conductivity Ïƒ of monolayer PdSe_{2} exhibits remarkable anisotropic behaviors with (Ïƒ_{ y }/Ïƒ_{ x })â€‰~2.3 for ntype doping and (Ïƒ_{ x }/Ïƒ_{ y })â€‰~2.4 for ptype at 1.1â€‰Ã—â€‰10^{13}â€‰cm^{âˆ’2} concentration. The calculated Seebeck coefficients along the x and y directions as a function of carrier concentration are shown in Fig.Â 4(c) and (d) for n and ptype doping, respectively. We find a larger asymmetry of the Seebeck coefficient for ptype doping than for ntype doping, which is in good agreement with the recent report^{26}. This anisotropy in the thermopower values in the two different directions might enable to design transverse thermoelctric device^{45}. It is important to note that the Seebeck coefficients for both n and ptype doped monolayer PdSe_{2} are substantially high at room temperature, reaching a peak value of 660 Î¼V/K at an electron concentration around 1.25â€‰Ã—â€‰10^{11}â€‰cm^{âˆ’2} and with an average value in the range of 300â€“340â€‰Î¼V/K. These values of S for monolayer PdSe_{2} compare favorably with those reported for some other 2D materials^{30,39}. FigureÂ 4(e) and (f) depict the power factor (PF) S^{2}Ïƒ at room temperature along the x and y directions for n and pdoped PdSe_{2} monolayer, respectively. The results reflect significant anisotropy in the power factor with the PF_{ x }/PF_{ y }â€‰~1.9 for ptype doping and (PF)_{ y }/(PF)_{ x }â€‰~2 for ntype doping at concentration around 1.1â€‰Ã—â€‰10^{13}â€‰cm^{âˆ’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.
Phononic transport
FigureÂ 2 shows the phonon dispersion relations of monolayer PdSe_{2} at its equilibrium volume along the high symmetric Î“â€‰âˆ’â€‰Yâ€‰âˆ’â€‰Mâ€‰âˆ’â€‰Xâ€‰âˆ’â€‰Î“ directions. It is noteworthy that the phonon spectrums of monolayer PdSe_{2} is very distinct from the MoS_{2} type monolayer. The maximum frequency of the acoustic mode markedly drop to rather low value of 3.7â€‰THz, while for monolayers of MoSe_{2} and WSe_{2} it is 5.4â€‰THz and 4.8â€‰THz, respectively, and even higher for monolayer MoS_{2} with the value of 7.5â€‰THz. Such low frequency suggests the low group velocity of acoustic modes in monolayer PdSe_{2}. As acoustic modes contribute mostly to the lattice thermal conductivity Îº_{ l }, lower Îº_{ l } in this PdSe_{2} monolayer is expected.
Now we turn to the computation of lattice thermal conductivity Îº_{ l }. As mentioned above, we estimate Îº_{ 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.Â 5, Îº_{ l } decreases following a T^{âˆ’1} dependence with the increasing temperature, suggesting that Umklapp phonon scattering dominates threephonon interactions^{46}. From the calculations, the obtained lattice thermal conductivity of monolayer PdSe_{2} is 3.7 (1.4) and 7.2 (2.7)â€‰Wm^{âˆ’1} K^{âˆ’1} at 300â€‰K (800â€‰K) along the x and y directions, respectively, which are much lower than MoS_{2}^{47} and GX_{2} monolayers^{48}. 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.
Dimensionless figure of merit ZT
The electronic thermal conductivity Îº_{ e } of monolayer PdSe_{2} is calculated via the WiedemannFranz law Îº_{ e }â€‰=â€‰Lâ€‰ÏƒT. Within the relaxation time approximation, the Seebeck coefficient can be calculated independently of the relaxation time Ï„, but evaluation of the electrical conductivity requires knowledge of Ï„. Here we take into account only the intrinsic scattering mechanism, namely, the interaction of electrons with acoustic phonons. Then the relaxation time Ï„ can be evaluated from the equation Ï„â€‰=â€‰Î¼m*/e, here the carrier mobility Î¼ and effective mass m* have been calculated in subsection of Electronic transport properties, as listed in TableÂ 1.
Combining the electronic and thermal transport properties, we now evaluate the thermoelectric performance of the PdSe_{2} monolayer. FigureÂ 6 shows the figure of merit ZT value for both n and p doped PdSe_{2} 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 ntype doped monolayer PdSe_{2} are rather small and almost isotropic with the maximum value of 0.13 with the corresponding concentration 3â€‰Ã—â€‰10^{13}â€‰cm^{âˆ’2}. However, for ptype doped monolayer PdSe_{2}, 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 6.5â€‰Ã—â€‰10^{12}â€‰cm^{âˆ’2} and 0.5 along the y direction at the carrier concentration of 2â€‰Ã—â€‰10^{13}â€‰cm^{âˆ’2}, respectively. Therefore, heavily doped ptype PdSe_{2} 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 PdSe_{2} monolayer is very soft near point Î“, 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.
Conclusion
In summary, by means of firstprinciples calculation, the geometrical structure, mechanical, electronic and thermal transport properties of monolayer PdSe_{2} are systematically investigated. In contrast to TMCs, monolayer PdSe_{2} has strong anisotropic mechanical, electronic and thermal transport properties, leading to anisotropic thermoelectric properties. We find that PdSe_{2} is a semiconductor with an indirect band gap of 1.38â€‰eV and a hole mobility as high as 1929â€‰cm ^{2} V^{âˆ’1} s^{âˆ’1}. The inplane effective elastic modulus are rather low, suggesting the flexible mechanical properties in this structure. Furthermore, monolayer PdSe_{2} has a low lattice thermal conductivity about 3â€‰Wm^{âˆ’1} K^{âˆ’1} along the x direction at room temperature. Combining its high Seebeck coefficient and markedly low thermal conductivity, monolayer PdSe_{2} shows an optimum ZT value of 1.1 (300K) at optimal doping. Therefore, our results indicate monolayer PdSe_{2} is a material with promising thermoelectric performance.
Computational Methods
The initial structure of monolayer PdSe_{2} is optimized through DFT with the planewave based Vienna abinitio simulation package (VASP)^{49,50}, using the projector augmented wave (PAW) method. For the exchangecorrelation functional, we have used the PerdewBurkeErnzerhof version of the generalized gradient approximation (GGA)^{51}. A planewave cutoff energy of 400â€‰eV and an energy convergence criterion of 10^{âˆ’7}â€‰eV are adopted throughout calculations. The spinorbit coupling (SOC) is not considered in the structure relaxation. For ionic relaxation calculations, a 11â€‰Ã—â€‰11â€‰Ã—â€‰1 MonkhorstPack kmeshes^{52} are used and the structure is considered to be stable when the HellmannFeynman forces are smaller than 0.001â€‰eV/Ã…. For the slab model, a 21â€‰Ã… 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 allelectron fullpotential WIEN2k code^{53} using recently implemented Tran and Blahaâ€™s modified BeckeJohnson (TBmBJ)^{54} exchange potential plus generalised gradient approximation (GGA) with the SOC included. The TBmBJGGA 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 approximations^{55}. The number of plane waves in a Fourier expansion of potential in the interstitial region was restricted to R_{ MT }â€‰Ã—â€‰K_{ max }â€‰=â€‰8. The muffin tin radii for Se and Pd are 2.1 and 2.2 a.u., respectively. We used 19â€‰Ã—â€‰19â€‰Ã—â€‰1 kpoint MonkhorstPack mesh for electronic band structure calculations.
Based on the selfconsistent converged electronic structure calculations, we have employed the eigenenergies on a very dense nonshifted 8000 kpoint 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 relaxationtime approximations (CRTA) as implemented in the BoltzTraP software^{56}, which neglects the weak energy dependence of relaxation time but retains some temperature and doping dependence^{57}. 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 axes^{58,59,60,61}. A comprehensive description of the Boltzmann transport theory in the relaxation time approximation can be found elsewhere^{23}. A brief summary of formalism used in this work is provided below^{62}. The energy projected transport distribution (TD) tensor is defined as
where group velocity \({v}_{\alpha }(i,k)=\frac{1}{\hslash }\frac{\partial {\varepsilon }_{i,k}}{\partial {k}_{\alpha }}\), N is the number of kpoints sampled, Ï„_{i, k} is the band index i and wave vector k dependent relaxation time, Î± and Î² 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 Î¼, can be written as
where Î© is the volume of unit cell and f_{0} is the FermiDirac distribution function. Thus, by using the CRTA, Ï„ is exactly cancelled out in EquationÂ 5. From the above calculations we can obtain the Seebeck coefficient S and the electrical conductivity over relaxation time (Ïƒ/Ï„) as well. The electronic thermal conductivity k_{ e } is calculated using the WiedemannFranz law, k_{ e }â€‰=â€‰Lâ€‰ÏƒT, where L is the Lorenz number. In our calculations we use Lâ€‰=â€‰2.4â€‰Ã—â€‰10^{âˆ’8}â€‰J^{2} K^{âˆ’2} C^{âˆ’2}â€‰^{63}.
To confirm the dynamic stability of the PdSe_{2} monolayer, we have calculated the phonon spectrum using a finite displacement method implemented the Phonopy code interfaced with the VASP code^{50,64}. At the same time the secondorder harmonic IFCs of monolayer PdSe_{2} and third order anharmonic IFCs were calculated using a 4â€‰Ã—â€‰4â€‰Ã—â€‰1 supercell and a 3â€‰Ã—â€‰3â€‰Ã—â€‰1 supercell with Î“ point, respectively. Based on an adaptive smearing approach to the conservation of energy^{65} and with an iterative solution method^{66}, we then solved the phonon Boltzmann transport equation using ShengBTE^{67}.
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Acknowledgements
The authors gratefully acknowledge the financial supports from the National Natural Science Foundation of China (Grant No. 51401031).
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D.Q. and P.Y. conceived and designed the research. D.Q. carried out the calculations and analyzed the calculated results with the helps from P.Y., G.Q.D., X.J.G. and H.Y.S. discussed the related calculated results. All authors reviewed the manuscript.
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Qin, D., Yan, P., Ding, G. et al. Monolayer PdSe_{2}: A promising twodimensional thermoelectric material. Sci Rep 8, 2764 (2018). https://doi.org/10.1038/s41598018209189
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DOI: https://doi.org/10.1038/s41598018209189
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