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Ultrahigh thermoelectric performance of Janus {\alpha}-STe2 and {\alpha}-SeTe2 monolayers

Combined with first-principles calculations and semiclassical Boltzmann transport theory, Janus {\alpha}-STe2 and {\alpha}-SeTe2 monolayers are investigated systematically. Janus {\alpha}-STe2 and {\alpha}-SeTe2 monolayers are indirect semiconductors with band gaps of 1.20 and 0.96 eV. It is found they possess ultrahigh figure of merit (ZT) values of 3.9 and 4.4 at 500 K, much higher than that of the pristine {\alpha}-Te monolayer (2.8). The higher ZT originates from Janus structures reduce lattice thermal conductivities remarkably compared with pristine {\alpha}-Te monolayer. The much higher phonon anharmonicity in Janus monolayers leads to significant lower lattice thermal conductivity. It is also found electronic thermal conductivity can play an important role in thermoelectric efficiency for the materials with quite low lattice thermal conductivity. This work suggests the potential applications of Janus {\alpha}-STe2 and {\alpha}-SeTe2 monolayers as thermoelectric materials and highlights Janus structure as an effective way to enhance thermoelectric performance.


Introduction
During the past several decades, thermoelectric (TE) materials attract much attention as they can convert waste heat into useful electrical power directly. Generally, a dimensionless figure of merit ZT can be used to evaluate the conversion efficiency, which can be expressed as = 2 /( + ), where the , , T, and are Seebeck coefficient, electrical conductivity, absolute temperature, electronic thermal conductivity, and lattice thermal conductivity, respectively. Moreover, 2 is the power factor (PF). A high ZT value requires a high 2 and low thermal conductivity ( + ). Unfortunately, the parameters which determine the ZT value are usually interrelated. For instance, S and σ generally behave in an opposite trend, greatly complicating the optimizing of ZT values 1-3 . Usually, a TE material should possess a narrow band gap, due to excellent electronic transport properties 4,5 . However, it is reported recently that materials with wide band gaps can also have excellent TE performance due to high motilities and low lattice thermal conductivities 6-dimensionality can improve the ZT value of the TE materials, as the quantum confinement effect and the interface/surface scattering effect [9][10][11][12] .
Since the successful exfoliation of graphene in 2004 13,14 , a new field in the study of two-dimensional (2D) materials has been opened up.
Recently, motivated by the successful synthesis of tellurene (2D tellurium) 22 , the 2D materials of group VI attract much attention, and various 2D materials of Te and Se have been synthesized experimentally [22][23][24][25][26][27][28] . It is found the tellurene possesses high carrier mobility on the order of 10 3 cm 2 V -1 s - 1 22 , outstanding air stability 29 , and a high on/off ratio on the order of 10 6 24 , leading to potential applications in field-effect transistor (FET), photodetectors, and sensors. Extraordinary electronic transport properties have also been found for few-layer tellurene 24,30,31 . Furthermore, it also shows good potential applications for thermoelectric devices with high thermoelectric performance 11,32,33 .
Recently, Janus 2D materials become a hot issue due to the mirror asymmetry in the structure and resultant distinct properties. The Janus transition metal dichalcogenides (TMDs) have been investigated theoretically and experimentally [34][35][36][37] . Investigations suggest that the Janus monolayer has promising applications in various fields such as electronics, optoelectronics, photocatalysts, and gas sensing [38][39][40][41] . In addition, theoretical researches find Janus monolayers usually reduce obviously compared with the pristine monolayers, such as Janus MoSSe, ZrSSe, PtSSe, and SnSSe [42][43][44][45] . Thus, the Janus structure may possess higher ZT than the pristine material due to its low thermal conductivity 46 .
In this work, we have investigated the TE properties of Janus α-STe 2 and α-SeTe 2 monolayers based on first principles. For comparison, the pristine α-Te monolayer was investigated as well. The dynamic and thermal stability is confirmed by phonon dispersion and the ab initio molecular dynamics (AIMD) simulations. It is found Janus α-STe 2 and α-SeTe 2 monolayers possess ultra-high ZT values which are up to about 4.0 at 500 K, much higher than the maximum ZT value of about 2.8 for α-Te.
The ultra-high ZT originates from the good electronic transport properties and ultra-low thermal conductivity. Janus structures reduce the electronic transport properties as the breaking of inversion symmetry, but also impair significantly, leading to the enhancement of ZT. Furthermore, the importance of electronic thermal conductivity to TE efficiency is highlighted. The physical mechanisms of low are also investigated. It is found the much higher phonon anharmonicity results in the lower in Janus monolayers. And the broken inversion symmetry leads to the higher phonon anharmonicity in Janus monolayers hence lower .

Computational Methods
Based on the density functional theory (DFT), all the first-principles calculations were performed by the Vienna ab initio simulation package (VASP) 47,48 . The Perdew−Burke−Ernzerhof (PBE) generalized gradient approximation (GGA) was used as the exchange-correlation functional 49 .
For structure optimization, a plane-wave kinetic energy cutoff of 600 eV was chosen, which is much higher than the maximum recommended cutoff to ensure the good convergence of the calculations. During the optimization, a total energy convergence criterion of 10 −

Results and Discussions
The structure of the α-Te monolayer belongs to ̅ 3 1 (164) symmetry group, similar to 1T-MoS 2 . The Janus monolayers are built from the pristine α-Te by replacing Te atoms of the outer layer with S/Se atoms.
The optimized structures of three monolayers are shown in Fig. 1        The phonon dispersions along high-symmetry lines are investigated, as shown in Fig. 3. The corresponding projected phonon density of states (PhDOS) are also displayed together. There is no negative frequency, indicating the dynamical stability of the three monolayers. In fact, the stabilities of the Janus monolayers are also confirmed in previous works 59, 63 . There are three acoustic phonon branches and six optical phonon where h, , * , and are Planck constant, Boltzmann constant, effective mass, and electron charge, respectively. It is found | | is proportional to * , and inversely proportional to n at a given temperature.
From Table 3, it is found * of the hole is much larger than that of the electron, due to the smaller band curvature around the VBM than the CBM as shown in Fig. 2. Thus, | | of p-type doping are bigger than the ones of n-type doping for three monolayers. | | are close to each other among the three monolayers. For instance, at 500 K and 10 13 cm -2 of n, | | are 323, 331, and 312 μV/K for α-STe 2 , α-SeTe 2 , and α-Te monolayers, respectively.
Furthermore, | | at 500 K is the higher than 300 K, as it is proportional to T based on Eq. (1). The carrier relaxation time is required to obtain 55 . From the conventional Boltzmann transport theory, electrical conductivity is proportional to carrier relaxation time τ, which can be expressed as = * , where is the carrier mobility. By using the deformation potential (DP) theory, of 2D materials can be written as 66 : where ℏ and 2 are the reduced Planck constant and 2D elastic constant, respectively. is the average effective mass calculated by = √ * * . However, for in-plane isotropic materials here, * = * . It is found that increases with the increasing carrier concentration n, and decreases with temperature T. The curves of 300 K are higher than the ones of 500 K. On the whole, of n-type doping is higher than p-type doping at the same n and T. The dependencies of can be interpreted by a simple model = 67 . It correlates positively with n, contrary to the situation of S. Furthermore, is inversely proportional to * and T based on Eq.
(2). As * of the electron is smaller than that of the hole for each monolayer, of n-type doping is higher.
Then, based on the S and , the power factor PF as a function of carrier concentration n are calculated and shown in Fig. 4 , where L is Lorenz number 69 . The calculated of three monolayers at 300 and 500 K are displayed in Fig. 6. Since and are linearly related, show similar dependences on n and * . are very small when n is lower than 10 13 cm -2 , whereas they increase sharply and exceed 15 Wm -1 K -1 at 10 14 cm -2 . On the whole, are much larger for ntype doping than p-type doping when n is lower than 10 13 cm -2 , as * of the electron is lighter than the hole. Combined with lower PF, the ZT value of n-type doping should be much smaller than p-type doping. As the * of the hole in α-Te monolayer is the lightest among three monolayers, of the α-Te monolayer is the largest. For instance, are 1.30, 0.89, 1.82 Wm -1 K -1 for n-type doped α-STe 2 , α-SeTe 2 , and α-Te monolayers at 300 K, 5 × 10 12 cm -2 . It is also found are independent on T approximately. However, our discussion is based on the results of full solution, as our work focuses more on the theoretical research. The values are in agreement with previous works 33,59 . The are much lower than many 2D monolayers, such as graphene (3716 Wm -1 K -1 ) 74, 75 , silicene (28.3 Wm -1 K -1 ) 74, 75 , blue phosphorene (106.6 Wm -1 K -1 ) 74 , and MoS 2 monolayer (87.6 Wm -1 K -1 ) 76 .
The ultralow generally trend to high TE efficiency. It is also found are close to at 5 × 10 12 cm -2 and room temperature.  Furthermore, the normalized contribution of each mode to total at 300 K is also investigated, as listed in   75,81 . Overall, | | of α-Te is smaller than α-STe 2 and α-SeTe 2 , as shown in Fig. 7(d)-(f), especially for the TA phonon branch. Specifically, the maximum value of | | is only 2.6 for α-Te, while the ones can reach 20 for α-STe 2 and α-SeTe 2 . The phonon relaxation times ℎ of three monolayers are also exhibited in Fig. 7(g)-(i). ℎ of α-STe 2 and α-SeTe 2 are significantly lower than that of α-Te, resulting from the much higher | |. It is concluded as the breaking of the inversion symmetry with respect to the central plane of the structure, the Janus α-STe 2 and α-SeTe 2 introduce much stronger anharmonicity, leading to abnormally lower than α-Te 45 . Therefore, the much lower of α-SeTe 2 monolayer mainly leads to a much higher ZT value than α-Te. For the three monolayers studied here, higher PF doesn't always lead to higher ZT. For example, α-Te has the highest PF (0.040 Wm -1 K -2 at 500 K) and the lowest ZT, whereas α-SeTe 2 possesses the lowest PF (0.028 Wm -1 K -2 at 500 K) and the highest ZT.

Conclusions
In summary, the electronic, thermal transport properties and TE performances of Janus α-STe 2 and α-SeTe 2 monolayers are investigated using first-principles calculations combined with semi-classical Boltzmann theory. For comparison, the TE properties of pristine α-Te monolayer are also studied. The electronic band structures, effective masses, and carrier relaxation times are similar to each other. Janus structures have different VBM, larger band gaps, and reduced PF values, compared with the pristine α-Te. However, the breaking of the inversion symmetry greatly reduces the by enhancing the phonon anharmonicity in Janus α-STe 2 and α-SeTe 2 monolayers. Furthermore, Janus monolayers also possess smaller . Note here plays an important role in ZT as it can be comparable with . Thus, the ZT values of Janus α-STe 2 and α-SeTe 2 monolayers are much higher than the one of the pristine α-Te monolayer. Particularly, the ZT value of the Janus α-SeTe 2 monolayer is as high as about 4.4 at 500 K. Our work indicates the Janus α-SeTe 2 and α-STe 2 are promising thermoelectric materials. Furthermore, it also suggests that the Janus structure is an effective method to reduce and enhance ZT.