Large thermoelectric power factor of high-mobility 1T'' phase of transition-metal dichalcogenides

The experimental studies about monolayer transition metal dichalcogenides in the recent year reveal this kind of compounds have many metastable phases with unique physical properties, not just 1H phases. Here, we focus on the 1T'' phase and systematically investigate the electronic structures and transport properties of MX2 (M=Mo, W; X=S, Se, Te) using the first-principles calculations with Boltzmann transport theory. And among them, only three molybdenum compounds has small direct bandgap at K point, which derive from the distortion of octahedral-coordination [MoS6]. For these three cases, hole carrier mobility of MoSe2 is estimated as 690 cm^2/Vs at room temperature, far more high than that of other two MoX2. For the reason, the combination of the modest carrier effective mass and weak electron-phonon coupling lead to the outstanding transport performance of MoSe2. The Seebeck coefficient of MoSe2 is also evaluated as high as 300 10^-6 V/K at room temperature. Due to the temperature dependent mobility of T^-1.9 and higher Seebeck coefficient at low temperature, it is found that MoSe2 has a large thermoelectric power factor around 6 10^-3 W/mK^2 in the low to intermediate temperature range. The present results suggests 1T'' MoSe2 maybe a excellent candidate for thermoelectric material.

In addition, due to the proportional relation between Seebeck coefficient and the energy derivative of the electronic density of states around Fermi level in the Mott formula 45 , low-dimensional materials TMDCs have natural advantage in thermoelectric (TE) applications, an important and meaningful crossing field of physics, materials and energy 46-49 . Therefore, more recently people have paid attention to TMDCs in the prospect of thermoelectricity [50][51][52][53][54][55][56][57] . The efficiency of TE materials depends on their dimensionless figure-of-merit ZT defined as ZT = σS 2 T/κ. S is the Seebeck coefficient, σ is the electrical conductivity, T is the absolute temperature, and κ is the total thermal conductivity and characterizes the heat leakage. Reaching high ZT has remained demanding because of the complicated relation between these individual parameters, especially the electrical conductivity and Seebeck coefficient. In general, the competition appears between these two properties, a small carrier effective mass favors high σ, but opposes a large S. Hence, power factor (σS 2 ) is often used to represent the electron energy conversion capability in TE materials. Recently, by using electric double-layer technique (EDLT) with the gate dielectrics of ionic liquids, researchers measure the ultrathin WSe 2 single crystals and obtain an power factor of ∼ 4 10 −3 W/mK 2 55 . Another experiment report a power factor of MoS 2 as large as 8.5 10 −3 W/mK 2 at room temperature 58 , exfoliated samples by the scotch-tape method. Moveover, it is found that the Kondo effect can improve the power factor of MoS 2 59 to much quite high value of 50 10 −3 W/mK 2 . While the power factor in other TE experiment about TMDCs is much lower than that in the above experiments. The main reasons is that low electrical conductivity limits the power factor for TE applications.
As is well-known, prevalent TE materials are heavilydoped small-bandgap semiconductors [60][61][62] , which can hold the balance between high Seebeck coefficient of semiconductor and high electrical conductivity of metals. Therefore, in the present work, we focus on the 1T phase of transitionmetal dichalcogenides with small bandgap, such as 0.1 eV in arXiv:1907.12037v2 [cond-mat.mtrl-sci] 31 Jul 2019 1T -MoS 2 29 , and explains the origin of small bandgap from the structure distortion. Since carrier doping at high concentration of EDLT has been successfully used to improve the performance of TMDCs, this work also systematically explore the dependents of electronic transport for a large range of carrier-doping concentrations by considering the electronphonon coupling. The lower carrier effective mass and the weakest electron-phonon scattering make 1T MoSe 2 has high mobility of 690 cm 2 /Vs at room temperature. Moreover, duo to the advantages of small bandgap and suitable carrier effective mass on the enhancement of Seebeck coefficient (300 µV/K), we obtain that MoSe 2 has high value around 6 10 −3 W/mK 2 in a larger temperature range.

II. METHODS
In the diffusive transport regime, electronic transport of a material can be calculated based on the Boltzmann transport equation (BTE). In the consideration of electron-phonon scattering in and out of the state |nk (ε nk ), via emission or absorption of phonons (ω qν ), the relaxation time τ 0 nk is associated with the imaginary part of the Fan-Migdal electron selfenergy 63 , defined by 64 where Ω BZ is the volume of the first Brillouin zone, f and n are the Fermi-Dirac and Bose-Einstein distribution functions, respectively. In Eq.(1), The electron-phonon matrix elements g mnv (k, q) are the probability amplitude for scattering from an initial electronic state |nk into a final state |mk + q via a phonon |qν , as obtained from density-functional perturbation theory (DFPT) [63][64][65] .
In the self-energy relaxation time approximation (SERTA) 64 , the electron carrier mobility takes the simple form where v nk is the group velocity of electronic state |nk and Ω is the volume of the crystalline unit cell. Based on the relaxation time τ 0 nk , the TE transport (σ and S) as a function of the chemical potential µ and of the temperature T is the following expressions 66,67 : where Ξ(ε) is the transport distribution function, defined as Ξ(ε) = n,k v nk v nk τ 0 nk δ (ε − ε nk ) /Ω. Technical details of the calculations are as follows. All calculations in this work were carried out in the framework of density-functional theory (DFT) as implemented in the QUANTUM ESPRESSO package 68 . The exchange and correlation energy was in the form of Perdew-Burke-Ernzerhof (PBE) 69 . Due to the existence of heavy transition metal element, the fully relativistic SOC was included in all calculations. By requiring convergence of results, the kineticenergy cutoff of 40 Ry and the Monkhorst-Pack k-mesh of 16×16×1 were used in the calculations dealing with the electronic ground-state properties. The phonon spectra were calculated on a 4×4×1 q grid using DFPT. In order to obtain the stable structure, the atomic positions were relaxed fully with the energy convergence criteria of 10 −5 Ry and the force convergence criteria of 10 −4 Ry/a.u. In the monolayer structure, a vacuum layer with 15Å was set to avoid the interactions between the adjacent atomic layers. Within the EPW code 70 of QUANTUM ESPRESSO in conjunction with the WAN-NIER90 71,72 , electron-phonon coupling was calculated on a 40×40×1 q grid with dense k points of 160×160×1 by the Wannier-Fourier interpolation technique of maximally localized Wannier functions 73,74 .

III. RESULTS
According to the sample preparation in the present experiment [22][23][24][25][26][27][28][29][30][31][32] , there are mainly three distorted phases from 1T phase (space group P-3m1). They all have lower symmetry than 1T and can be classified into two cases: dimeric structure 1T (space group P21/m) and trimeric structure 1T (space group P3) and 1T (space group P31m), as show in Fig. 1. The Peiels distortions of the prototypical 1T phase in the one direction and two directions along lattice vectors lead to the dimerization (1T ) and trimerization (1T ) of nearestneighboring transition metal atoms 25 , respectively. And the K 3 distortion 37 , a small rotary polymerization of three nearestneighboring Mo atoms, leads to a lower symmetry cell tripled T structure. A case study of MoS 2 , the total energy difference relative to the 1H phase shows that 1T and 1T phases have the highest and lowest total energy in the metastable phases, respectively, when two trimeric structures have similar total energy. In the 1T phase, it is found that interatomic distance (2.77Å) of three Mo atoms in 2a × 2a superstructure is much shorter than that in 1T phase (3.22Å), marked by Mo 3 for simplicity. Other one Mo atom (marked by Mo 1 ) has little deviation relative to the corresponding Mo atom in 1T phase. And the equilibrium lattice constant (a 0 =6.44Å) of MoS 2 agrees well with the previous studies 25,29 . The heavy chalcogens elongate a 0 significantly, accompanied by the slight bigger space between X atomic layer and Mo atomic layer, because of the increase of ionic radius with the atomic number of chalcogens. However, the ionic radius of Mo 2+ is almost identical to W 2+ , thus the change of a 0 induced by the W element is much smaller, as summarized in Tab. I.
Because the small bandgap 26 of MoS 2 in 1T phase is advantageous to enhance the thermoelectricity, here we mainly study the 1T -phase MX 2 (M=Mo, W; X=S, Se, Te). As shown in Fig. 2 Fig. 3(a)]. And the short interatomic distance of trimeric Mo 3 results in the short Mo-S bonding length as well as the large energy difference between the bonding states and antibonding states of Mo 3 -4d orbitals. Therefore the distribution of Mo 3 -4d orbitals are far away from CBM and VBM, which are contribution from the Mo 1 -d orbitals, as shown in Fig. 3 Fig. 3(d)], it is found that the angles between para-position Mo-S bonds θ is 175 • of 1T different from the 180 • of 1T phase and the six Mo-S bond lengths of 1T don't have the same value. These small distortions can break the double degeneration of e g orbitals (d x 2 −y 2 , d z 2 ) and produce the small bandgap [ Fig. 3(c)]. For the case of heavy chalcogens, X atom tinily moves backward the Mo atom, which strengthens the coupling between the X-p and Mo 1 -d z 2 orbitals and weakens coupling between the S-p and Mo 1 -d x 2 −y 2 orbitals. These modulations of couplings lead to the higher bonding state of d x 2 −y 2 (CBM), the lower bonding state of d z 2 (VBM) and the rise of Γ c . Hence, the bigger bandgap exists in the cases of heavier chalcogens [Tab. I]. However, the space between W atomic layer and X atomic layer is smaller than that in MoX 2 , so the effect of W atom contrary to that of heavy chalcogens and make WX 2 have very small bandgap even be metal. In order to ensure the stability of 1T phase, we also calculate the phonon spectra. As shown in the bottom half of Fig. 2, only WSe 2 has the large imaginary frequency and other systems all have dynamics stabilities. Because the Γ point has symmetry of C 3v (3m) point group in 1T phase, 33 optical phonon modes can be decomposed by three irreducible representations: A 1 (8 modes), A 2 (3 modes) and E (11 double degenerate modes). With the increase of atomic mass, the highest phonon frequency obviously decrease, such as 448.6 cm −1 of MoS 2 and 235.8 cm −1 of WTe 2 . And the greater proportion of chalcogens also give rise to the more obvious changes of phonon frequency with the different chalcogens. In addition, the small mass ratio of M and X atoms can close the frequency gap between acoustic phonons and optic phonons, as shown in Fig. 2.
Basing on the stable semiconductor with suitable bandgap of MX 2 in T phase, we investigate the carrier doping and temperature dependences of mobility of MoX 2 (X=S, Se, Te) with the consideration of electron-phonon scattering. Firstly, we estimate the carrier effective mass of hole (m * h ) and electron (m * e ) on the basis of band structures and find that m * h is lighter than m * e of MoX 2 [Tab. I] and increase with atomic number of chalcogens. by contrast, WTe 2 has a heavier m * h than m * e , because VBM locate at Γ point, differ from the K point for MoX 2 . Hence the next study keystone is holecarrier transport properties and the doping range is set as 0.02 ∼ 20×10 12 cm −2 . To facilitate analysis of relative contribution of phonons with different frequencies to electronphonon scattering, we calculate the transport spectral function α 2 tr F(ω) 75,76 , obtained by the phonon self-energy with doping in semiconductor. Figure 4(a) plots the α 2 tr F(ω) of MoX 2 with n h 2D =2×10 12 cm −2 . It can be seen that the peak intensities of α 2 tr F(ω) in MoS 2 are higher than those in other two cases and MoSe 2 has the lowest value in the whole spectrum space. Of particular note is the low frequency region around 40 cm −1 and intermediate frequency region around 200 cm −1 . In the former, MoS 2 and MoTe 2 have strong electron-phonon coupling, which is almost absence from MoSe 2 . And in the latter, the peak value of MoSe 2 is much smaller than that in MoS 2 or MoTe 2 . From the above, MoSe 2 has the weakest electron-phonon coupling, to the benefit of high-performance carrier transport. As show in Fig. 4(b), the room-temperature hole carrier mobilities of MoS 2 , MoSe 2 and MoTe 2 are 42, 690, and 176 cm 2 /Vs at the low carrier concentration, respectively. It is noteworthy that the mobility of 1T -MoSe 2 is much higher than that of 1H-phase TMDCs in experiments and predicted calculations 15,16,[77][78][79][80][81][82][83] . Here are two important factors need to be considered. One is the hole carrier effective mass, proportional to the atomic mass of chalcogens. Other one is the electron-phonon coupling cause the scattering, whose the order of from weakest to strongest intensity is MoSe 2 < MoTe 2 < MoS 2 . Thereby they result in the much higher hole carrier mobility of MoSe 2 than other two cases. The down trend of the mobility on the carrier concentration also derive from the strong electron-phonon coupling of high density of electronic states at high concentration. As a contrast, the electron carrier mobilities of MoX 2 , plotted in the inset of Fig. 4(b), are lower than hole carrier by reason of heavy carrier effective mass. Furthermore, phonon concentration has positive correlation relationship with temperature, thus the high temperature causes increased electron-phonon scattering, as the declining mobility of MoSe 2 with the increase of temperature [ Fig. 4(c)]. And the temperature depen-  ) and (4). The proportionality between σ and n 2D *µ can derive the ascending curve of σ with carrier concentration and low σ at high temperature, as shown in Fig. 5(a). The hole-doping Seebeck coefficients as functions of carrier concentration at different temperatures are also plotted in Fig. 5(b). 1T MoSe 2 has a large Seebeck coefficient, and the maximum value of S, 422 µV/K of 100 K to 205 µV/K of 500 K, shifts to high doping concentration and decrease as temperature increases, similar to the previous results of H-phase TMDCs 84,85 . At room temperature, S can reach up to 300 µV/K when n h 2D =1×10 12 cm −2 , catching up to and even surpassing the experimental values of many two materials 55,58,[86][87][88][89][90][91][92] . In Mott formula 45 , hole-doping S of semiconductor is inversely proportional to doping concentration ( in direct proportion to the chemical potential). However, the small bandgap easily causes the bipolar effect at low doping concentration 86 , which make S has proportional with doping concentration and the sign reversal of S with the increasing negative contribution of thermally excited electrons. As shown in Fig. 5(c), the power factor (σS 2 ) has a large value in the middle and low temperature zone (100 K∼500 K). The highest value of 10.2×10 −3 W/mK 2 with n h 2D =2×10 12 cm −2 @200 K. And it is more important that over a large temperature range, the maximal power factor of different temperatures can stay around ∼6.0 10 −3 W/mK 2 , well above the present the experimental measurements of intrinsic power factor in the vast majority of TMDCs and some classic TE materials, such as SnSe, Bi 2 Te 3 and PbTe [91][92][93][94] . It indicates the 1T -phase MoSe 2 as a high-performance candidate TE materials in the low to intermediate temperature range. Inspired by the similar effect of heavy chalcogens and compressive strain, which both lead to the increase of mono-layer thickness and impact the interaction between Mo and Se atoms 95 , we also expect and study the effect of small compressive strain ( =(a − a 0 )/a 0 ×100%≤-2.0%) on the transport property of MoSe 2 , in order to enhance the bandgap as well as the temperature range of thermoelectric application. In the electronic band structure with =-2.0%, the bandgap increases to 0.19 eV and the hole carrier effective mass m * h is light to 0.488 m 0 [Fig. 6]. But the energy difference between Γ v and VBM almost disappear, which can enhance the intervalley scattering of Γ and K, assisted by the K-vector phonons, similar to the intervalley scattering in 1H MoS 2 78,80 . Consequently, after the introduction of small compressive strain, the hole carrier mobility drops as well as the the decrease of power factor (1∼3 10 −3 W/mK 2 ) [ Fig. 6]. And the peak values of power factor all locate at the range of high concentration (≥ 2 10 12 cm −2 ).

IV. CONCLUSION
In summary, by using the first-principles calculations with Boltzmann transport theory, we studies systematically the metastable monolayer 1T phase MX 2 , including electronic structure, electron-phonon coupling, carrier mobility and TE power factor. The small direct bandgap at K point of three molybdenum compounds is attributed to the distorted octahedral coordination of [MoX 6 ]. And the extremely weak electron-phonon coupling of MoSe 2 gives rise to its hole carrier mobility as high as 690 cm 2 /Vs at 300K. Moreover, combining the Seebeck coefficient around 300 µV/K, it is obtained that the TE power factor of MoSe 2 keeps above 6 10 −3 W/mK 2 in the large range of temperature (100K∼500K). Our results illustrate the outstanding potential 1T MoSe 2 on TE materials.