Monolayer PdSe2: A promising two-dimensional thermoelectric material

Motivated by the recent experimental synthesis of two-dimensional semiconducting film PdSe2, we investigate the electronic and thermal transport properties of PdSe2 monolayer by using the density functional theory and semiclassical Boltzmann transport equation. The calculated results reveal anisotropic transport properties. Low lattice thermal conductivity about 3 Wm−1 K −1 (300K) along the x direction is obtained, and the dimensionless thermoelectric figure of merit can reach 1.1 along the x direction for p-type doping at room temperature, indicating the promising thermoelectric performance of monolayer PdSe2.

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 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 PdSe 2 . We obtain the Seebeck coefficients for monolayer PdSe 2 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 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 a = 5.75 Å, b = 5.87 Å, and c = 7.69 Å 31 . The monolayer PdSe 2 is cut through the (0 0 1) plane of the PdSe 2 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 PdSe 2 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 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. 1(c) and the optimized  lattice parameters of monolayer PdSe 2 are a = 5.7538 Å and b = 5.9257 Å, 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. 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 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 PdSe 2 monolayer is semiconducting with an indirect band gap of 1.38 eV, which is in general agreement with the previous reports 26,34 , 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.
The effective mass m * near the Fermi energy is an important parameter for the thermoelectric transport 35 , which can be extracted from the high-precise energy band calculation via the equation 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 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 Bardeen-Shockley 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 (μ) of 2D structure can be expressed as    , 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/Å 2 much lower than those of MoS 2 (7.99 eV/Å 2 ) 39 and PdS 2 (3.62 eV/Å 2 in the x direction and 5.11 eV/Å 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 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) cm 2 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 cm 2 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 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 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 PdSe 2 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 × 10 13 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 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 p-type doped monolayer PdSe 2 are substantially high at room temperature, reaching a peak value of 660 μV/K at an electron concentration around 1.25 × 10 11 cm −2 and with an average value in the range of 300-340 μV/K. These values of S for monolayer PdSe 2 compare favorably with those reported for some other 2D materials 30,39 . Figure 4(e) and (f) depict the power factor (PF) S 2 σ at room temperature along the x and y directions for n-and p-doped PdSe 2 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 × 10 13 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.
Phononic transport. Figure 2 shows the phonon dispersion relations of monolayer PdSe 2 at its equilibrium volume along the high symmetric Γ − Y − M − X − Γ 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 3.7 THz, while for monolayers of MoSe 2 and WSe 2 it is 5.4 THz and 4.8 THz, respectively, and even higher for monolayer MoS 2 with the value of 7.5 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 κ l , lower κ l in this PdSe 2 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 interactions 46 . From the calculations, the obtained lattice thermal conductivity of monolayer PdSe 2 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 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 κ e of monolayer PdSe 2 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 PdSe 2 monolayer. Figure 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 n-type doped monolayer PdSe 2 are rather small and almost isotropic with the maximum value of 0.13 with the corresponding concentration 3 × 10 13 cm −2 . However, for p-type 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 6.5 × 10 12 cm −2 and 0.5 along the y direction at the carrier concentration of 2 × 10 13 cm −2 , respectively. Therefore, heavily doped p-type 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 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.

Conclusion
In summary, by means of first-principles 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 1.38 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 PdSe 2 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 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 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-meshes 52 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 code 53 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 approximations 55 . 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 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 , N is the number of k-points sampled, τ i, 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 where Ω is the volume of unit cell and f 0 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 J 2 K −2 C −2 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 second-order harmonic IFCs of monolayer PdSe 2 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 energy 65 and with an iterative solution method 66 , we then solved the phonon Boltzmann transport equation using ShengBTE 67 .