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

TL;DR

This study reveals that the 1T'' phase of MoSe2 exhibits high mobility and thermoelectric power factor, making it a promising candidate for thermoelectric applications based on first-principles calculations.

Contribution

It systematically investigates the electronic and transport properties of 1T'' phase transition-metal dichalcogenides, highlighting MoSe2's superior thermoelectric performance.

Findings

MoSe2 has a hole mobility of 690 cm^2/Vs at room temperature.

MoSe2's Seebeck coefficient reaches 300 μV/K at room temperature.

MoSe2 exhibits a thermoelectric power factor of around 6×10^-3 W/mK^2 at low to intermediate temperatures.

Abstract

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.

Tables

Table 1. Table 1: Equilibrium lattice constant ( a 0 subscript 𝑎 0 a_{0} ), bandgap (E gap ) and carrier effective mass (m ∗ ) of MoS 2 , MoSe 2 and MoTe 2 .

	MoS2	MoSe2	MoTe2
$a_0$ (Å)	6.44	6.68	7.11
Egap (eV)	0.10	0.12	0.16
m${}_h{}^{}{}_{}{}^{\ast }$ (m0)	0.227	0.535	0.707
m${}_e{}^{}{}_{}{}^{\ast }$ (m0)	0.303	0.937	0.872
	WS2	WSe2	WTe2
$a_0$ (Å)	6.50	6.71	7.13
Egap (eV)	metal	metal	0.03
m${}_h{}^{}{}_{}{}^{\ast }$ (m0)	-	-	0.566
m${}_e{}^{}{}_{}{}^{\ast }$ (m0)	-	-	0.437

Equations

$$\frac{1}{{\tau }_{n\mathbf{k}}^0}=$$

$$\times [\left(1-f_{m\mathbf{k}+\mathbf{q}}^0+n_{\mathbf{q}v}\right)\delta \left({\epsilon }_{n\mathbf{k}}-{\epsilon }_{m\mathbf{k}+\mathbf{q}}-\hslash {\omega }_{\mathbf{q}v}\right).$$

$$+\left(f_{m\mathbf{k}+\mathbf{q}}^0+n_{\mathbf{q}v}\right)\delta \left({\epsilon }_{n\mathbf{k}}-{\epsilon }_{m\mathbf{k}+\mathbf{q}}+\hslash {\omega }_{\mathbf{q}\nu }\right)],$$

$${\mu }_{\mathrm{e}}=\frac{-e}{n_{\mathrm{e}}\Omega }\sum _{n\in \mathrm{CB}}\int \frac{d\mathbf{k}}{{\Omega }_{\mathrm{BZ}}}\frac{\partial f_{n\mathrm{k}}^0}{\partial {\epsilon }_{n\mathrm{k}}}{v}_{n\mathbf{k}}{v}_{n\mathbf{k}}{\tau }_{n\mathbf{k}}^0,$$

$$\sigma =e^2\int \Xi (\epsilon )\left(-\frac{\partial f^0}{\partial \epsilon }\right)\mathrm{d}\epsilon ,$$

$$S=\frac{e}{\sigma }\int \Xi (\epsilon )\left(-\frac{\partial f^0}{\partial \epsilon }\right)\frac{\epsilon -\mu }{T}\mathrm{d}\epsilon ,$$

Full text

I Introduction

Since the discovery of 2D materials, transition metal dichalcogenides (TMDCs) Wang2012 ; Feng2012 ; Conley2013 ; Castellanos-Gomez2013 ; Chhowalla2013 are particularly interesting due to the semiconducting characteristics with strong stability and large flexibility. The unique structural, mechanical, optical, electrical, and thermal properties Fiori2014 ; Bernardi2017 make they have potential applications in photovoltaics Britnell2013 , transistors Fang2012 ; Liu2013 ; Yoon2011 ; Mak2013 ; Qiu2015 , Optoelectronic Huo2014 , photodetector and molecular sensing Wang2012 . The usual crystallographic form of monolayer TMDCs is the hexagonal 1H phase, in which TMDCs have the high on-off ratio (108) with carrier mobility of 200 cm2/Vs at room temperature Radisavljevic2011 ; Levi2013 . The exciton energy and strong spin-valley coupling of 1H phase also provide novel platform for intriguing nanoelectronic devices Xu2014 ; Mak2014 ; Cui2015 ; Roy2013 ; Qi2015 . TMDCs also exhibit other metastable trigonal polymorphic forms with different degrees of structural distortion (1T, 1$T^{\prime}$, 1$T^{\prime\prime}$, and 1$T^{\prime\prime\prime}$) Song2015 ; Zhang2018 ; Linghu2019 ; Zhao2018 ; Kan2014 ; Calandra2013 ; Bruyer2016 ; Zhuang2017 ; Pal2017 ; Zhou2018 ; Singh2015 . Thereinto 1T is the primary structure and adopts the octahedral coordination with point group D3d. In the octahedral crystal field, 4d orbitals of Mo atoms are split into the eg orbitals (d${}_{x^{2}-y^{2}}$, d${}_{z^{2}}$) over t2g orbitals (dxy, dxz, dyz), and the partially filled t2g orbitals induce metallic conductivity Chhowalla2013 . Due to the Peierls instability, the distortion of octahedral [MoS6] in 1T phase can result in other metastable polymorphs with lower symmetry Chou2015 ; Eda2012 , where Mo-Mo associations take place, such as the dimerization (1$T^{\prime}$) Yu2018 and trimerization (1$T^{\prime\prime}$ and 1$T^{\prime\prime\prime}$) Fang2018 ; Shirodkar2014 ; Shang2018 . The spontaneous symmetry breaking of structural distortion lift the degeneracy of electronic states to lower the energy. In these metastable structures, the higher conductivity of metallic 1T phase make them as excellent electrocatalysts for hydrogen evolution, rechargeable batteries and supercapacitors Acerce2015 ; Voiry2013 ; Lukowski2013 . Many novel physical properties are also revealed in the distorted structures. For example, the strong spin-orbital coupling (SOC) make 1$T^{\prime}$ TMDCs to be large-gap quantum spin hall insulators Qian2014 ; Choe2016 ; Tang2017 . The nontrivial geometry with the trimerization of Mo atoms in 1$T^{\prime\prime\prime}$ phase lead to the ferroelectricity with high carrier mobility simultaneously Bruyer2016 .

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 Cutler1969 , low-dimensional materials TMDCs have natural advantage in thermoelectric (TE) applications, an important and meaningful crossing field of physics, materials and energy Bell2008 ; Heremans2013 ; Dresselhaus2007 ; Zhao2014 . Therefore, more recently people have paid attention to TMDCs in the prospect of thermoelectricity Huang2013 ; Fan2014 ; Huang2014 ; Babaei2014 ; Wang2017 ; Wu2014 ; Yoshida2016 ; TWang2016 . The efficiency of TE materials depends on their dimensionless figure-of-merit ZT defined as ZT = $\sigma$S2T/$\kappa$. S is the Seebeck coefficient, $\sigma$ is the electrical conductivity, T is the absolute temperature, and $\kappa$ 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 $\sigma$, but opposes a large S. Hence, power factor ($\sigma$S2) 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 WSe2 single crystals and obtain an power factor of $\sim$ 4 10*-3* W/mK2  Yoshida2016 . Another experiment report a power factor of MoS2 as large as 8.5 10*-3* W/mK2 at room temperature Hippalgaonkar2017 , exfoliated samples by the scotch-tape method. Moveover, it is found that the Kondo effect can improve the power factor of MoS2 Wu2019 to much quite high value of 50 10*-3* W/mK2. 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 heavily-doped small-bandgap semiconductors Gascoin2005 ; Lee2012 ; Goldsmid2014 , 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 1$T^{\prime\prime}$ phase of transition-metal dichalcogenides with small bandgap, such as 0.1 eV in 1$T^{\prime\prime}$-MoS2 Zhuang2017 , 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 electron-phonon coupling. The lower carrier effective mass and the weakest electron-phonon scattering make 1$T^{\prime\prime}$ MoSe2 has high mobility of 690 cm2/Vs at room temperature. Moreover, duo to the advantages of small bandgap and suitable carrier effective mass on the enhancement of Seebeck coefficient (300 $\mu$V/K), we obtain that MoSe2 has high value around 6 10*-3* W/mK2 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 $|n\mathbf{k}\rangle$ ($\varepsilon_{n\mathbf{k}}$), via emission or absorption of phonons ($\omega_{\mathbf{q}\nu}$), the relaxation time $\tau_{n\mathbf{k}}^{0}$ is associated with the imaginary part of the Fan-Migdal electron self-energy Giustino2017 , defined by Ponce2018

[TABLE]

where $\Omega_{\rm BZ}$ 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 $g_{mnv}(\mathbf{k},\mathbf{q})$ are the probability amplitude for scattering from an initial electronic state $|n\mathbf{k}\rangle$ into a final state $|m\mathbf{k}+\mathbf{q}\rangle$ via a phonon $|\mathbf{q}\nu\rangle$, as obtained from density-functional perturbation theory (DFPT) Baroni2001 ; Giustino2017 ; Ponce2018 .

In the self-energy relaxation time approximation (SERTA) Ponce2018 , the electron carrier mobility takes the simple form

[TABLE]

where $v_{n\mathbf{k}}$ is the group velocity of electronic state $|n\mathbf{k}\rangle$ and $\Omega$ is the volume of the crystalline unit cell. Based on the relaxation time $\tau_{n\mathbf{k}}^{0}$, the TE transport ($\sigma$ and S) as a function of the chemical potential $\mu$ and of the temperature T is the following expressions Huang2018 ; Pizzi2014 :

[TABLE]

[TABLE]

where $\Xi(\varepsilon)$ is the transport distribution function, defined as $\Xi(\varepsilon)=\sum_{n,\mathbf{k}}v_{n\mathbf{k}}v_{n\mathbf{k}}\tau^{0}_{n\mathbf{k}}\delta\left(\varepsilon-\varepsilon_{n\mathbf{k}}\right)/\Omega$.

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 Giannozzi2017 . The exchange and correlation energy was in the form of Perdew-Burke-Ernzerhof (PBE) Perdew1996 . Due to the existence of heavy transition metal element, the fully relativistic SOC was included in all calculations. By requiring convergence of results, the kinetic-energy cutoff of 40 Ry and the Monkhorst-Pack k-mesh of 16$\times$16$\times$1 were used in the calculations dealing with the electronic ground-state properties. The phonon spectra were calculated on a 4$\times$4$\times$1 q grid using DFPT. In order to obtain the stable structure, the atomic positions were relaxed fully with the energy convergence criteria of 10*-5* Ry and the force convergence criteria of 10*-4* Ry/a.u. In the monolayer structure, a vacuum layer with 15 Å was set to avoid the interactions between the adjacent atomic layers. Within the EPW code Ponce2016 of QUANTUM ESPRESSO in conjunction with the WANNIER90 Mostofi2008 ; Mostofi2014 , electron-phonon coupling was calculated on a 40$\times$40$\times$1 q grid with dense k points of 160$\times$160$\times$1 by the Wannier-Fourier interpolation technique of maximally localized Wannier functions Giustino2007 ; Marzari2012 .

III Results

According to the sample preparation in the present experiment Song2015 ; Zhang2018 ; Linghu2019 ; Zhao2018 ; Kan2014 ; Calandra2013 ; Bruyer2016 ; Zhuang2017 ; Pal2017 ; Zhou2018 ; Singh2015 , 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 1$T^{\prime}$ (space group P21/m) and trimeric structure 1$T^{\prime\prime}$ (space group P3) and 1$T^{\prime\prime\prime}$ (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 (1$T^{\prime}$) and trimerization (1$T^{\prime\prime}$) of nearest-neighboring transition metal atoms Linghu2019 , respectively. And the K3 distortion Shirodkar2014 , a small rotary polymerization of three nearest-neighboring Mo atoms, leads to a lower symmetry cell tripled $T^{\prime\prime\prime}$ structure. A case study of MoS2, the total energy difference relative to the 1H phase shows that 1T and 1$T^{\prime}$ phases have the highest and lowest total energy in the metastable phases, respectively, when two trimeric structures have similar total energy. In the 1$T^{\prime\prime}$ phase, it is found that interatomic distance (2.77 Å) of three Mo atoms in $2a\times 2a$ superstructure is much shorter than that in 1T phase (3.22 Å), marked by Mo3 for simplicity. Other one Mo atom (marked by Mo1) has little deviation relative to the corresponding Mo atom in 1T phase. And the equilibrium lattice constant ($a_{0}$=6.44 Å) of MoS2 agrees well with the previous studies Zhuang2017 ; Linghu2019 . 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 Mo2+ is almost identical to W2+, thus the change of $a_{0}$ induced by the W element is much smaller, as summarized in Tab. 1.

Because the small bandgap Zhao2018 of MoS2 in 1$T^{\prime\prime}$ phase is advantageous to enhance the thermoelectricity, here we mainly study the 1$T^{\prime\prime}$-phase MX2 (M=Mo, W; X=S, Se, Te). As shown in Fig. 2, The band structures indicate that three MoX2 all have direct bandgap at K point, when only WTe2 in WX2 is semiconductor with very small indirect-bandgap of 0.03 eV [Tab. 1]. For the case of valence band, there is a second energy maximum ($\Gamma_{v}$) at $\Gamma$ point for all MX2, closing to valence band maximum (VBM) with small energy difference. And the second energy minima ($\Gamma_{c}$) of conduction band at $\Gamma$ point exists only in MoS2 and WS2. Moreover, the stronger SOC of heavy transition metal atom also generate the larger spin splitting at both conduction band minimum (CBM) and VBM. Now we analyze the source for the bandgap in 1$T^{\prime\prime}$ phase according to the projected band structure of MoS2 [Fig. 3]. CBM and VBM around the direct bandgap are mainly composed of Mo atomic orbitals, when there are only S atomic orbitals (px and py) at $\Gamma_{c}$ [Fig. 3(a)]. And the short interatomic distance of trimeric Mo3 results in the short Mo-S bonding length as well as the large energy difference between the bonding states and antibonding states of Mo3-4d orbitals. Therefore the distribution of Mo3-4d orbitals are far away from CBM and VBM, which are contribution from the Mo1-d orbitals, as shown in Fig. 3(b). According to the comparation between the distorted octahedral [Mo1S6] and Oh-[MoS6] in 1T phase [Fig. 3(d)], it is found that the angles between para-position Mo-S bonds $\theta$ is 175*∘* of 1$T^{\prime\prime}$ different from the 180*∘* of 1T phase and the six Mo-S bond lengths of 1$T^{\prime\prime}$ don’t have the same value. These small distortions can break the double degeneration of eg 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 Mo1-d${}_{z^{2}}$ orbitals and weakens coupling between the S-p and Mo1-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 $\Gamma_{c}$. Hence, the bigger bandgap exists in the cases of heavier chalcogens [Tab. 1]. However, the space between W atomic layer and X atomic layer is smaller than that in MoX2, so the effect of W atom contrary to that of heavy chalcogens and make WX2 have very small bandgap even be metal.

In order to ensure the stability of 1$T^{\prime\prime}$ phase, we also calculate the phonon spectra. As shown in the bottom half of Fig. 2, only WSe2 has the large imaginary frequency and other systems all have dynamics stabilities. Because the $\Gamma$ point has symmetry of C3v (3m) point group in 1$T^{\prime\prime}$ phase, 33 optical phonon modes can be decomposed by three irreducible representations: A1 (8 modes), A2 (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 MoS2 and 235.8 cm*-1* of WTe2. 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 MX2 in $T^{\prime\prime}$ phase, we investigate the carrier doping and temperature dependences of mobility of MoX2 (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 MoX2 [Tab. 1] and increase with atomic number of chalcogens. by contrast, WTe2 has a heavier m${}^{*}_{h}$ than m${}^{*}_{e}$, because VBM locate at $\Gamma$ point, differ from the K point for MoX2. Hence the next study keystone is hole-carrier transport properties and the doping range is set as 0.02 $\sim$ 20$\times$1012 cm*-2*. To facilitate analysis of relative contribution of phonons with different frequencies to electron-phonon scattering, we calculate the transport spectral function $\alpha^{2}_{tr}$F($\omega$) Allen1978 ; Xu2013 , obtained by the phonon self-energy with doping in semiconductor. Figure 4(a) plots the $\alpha^{2}_{tr}$F($\omega$) of MoX2 with n${}^{h}_{\rm 2D}$=2$\times$1012 cm*-2*. It can be seen that the peak intensities of $\alpha^{2}_{tr}$F($\omega$) in MoS2 are higher than those in other two cases and MoSe2 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, MoS2 and MoTe2 have strong electron-phonon coupling, which is almost absence from MoSe2. And in the latter, the peak value of MoSe2 is much smaller than that in MoS2 or MoTe2. From the above, MoSe2 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 MoS2, MoSe2 and MoTe2 are 42, 690, and 176 cm2/Vs at the low carrier concentration, respectively. It is noteworthy that the mobility of 1$T^{\prime\prime}$-MoSe2 is much higher than that of 1H-phase TMDCs in experiments and predicted calculations Radisavljevic2011 ; Levi2013 ; Kaasbjerg2012 ; Li2013 ; Li2015 ; Ge2014 ; Ong2013 ; Wang2016 ; Sohier2018 . 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${}_{2}<$ MoS2. Thereby they result in the much higher hole carrier mobility of MoSe2 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 MoX2, 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 MoSe2 with the increase of temperature [Fig. 4(c)]. And the temperature dependent hole carrier mobilities of MoX2 (X=S, Se, Te) are fitted to be proportional to T*-2.0*, T*-1.9* and T*-1.9*, respectively.

Based the relaxation time of electron-phonon scattering, we calculate the electrical conductivity $\sigma$ and Seebeck coefficient S of MoSe2 according to Eqs.(3) and (4). The proportionality between $\sigma$ and n2D*$\mu$ can derive the ascending curve of $\sigma$ with carrier concentration and low $\sigma$ 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). 1$T^{\prime\prime}$ MoSe2 has a large Seebeck coefficient, and the maximum value of S, 422 $\mu$V/K of 100 K to 205 $\mu$V/K of 500 K, shifts to high doping concentration and decrease as temperature increases, similar to the previous results of H-phase TMDCs Kumar2015 ; Gandi2014 . At room temperature, S can reach up to 300 $\mu$V/K when n${}^{h}_{\rm 2D}$=1$\times 10^{12}$ cm*-2*, catching up to and even surpassing the experimental values of many two materials GShi2015 ; Oh2016 ; Ng2019 ; Guo2018 ; Zeng2018 ; Hippalgaonkar2017 ; Yoshida2016 ; Pu2016 ; Saito2016 . In Mott formula Cutler1969 , 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 GShi2015 , 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 ($\sigma$S2) has a large value in the middle and low temperature zone (100 K$\sim$500 K). The highest value of 10.2$\times$10*-3* W/mK2 with n${}^{h}_{\rm 2D}$=2$\times 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 $\sim$6.0 10*-3* W/mK2, well above the present the experimental measurements of intrinsic power factor in the vast majority of TMDCs and some classic TE materials, such as SnSe, Bi2Te3 and PbTe Ng2019 ; Heremans2008 ; Oh2016 ; Zhao2016 . It indicates the 1$T^{\prime\prime}$-phase MoSe2 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 monolayer thickness and impact the interaction between Mo and Se atoms Chang2013 , we also expect and study the effect of small compressive strain ( $\epsilon$=($a-a_{0}$)/$a_{0}\times$100%$\leq$-2.0%) on the transport property of MoSe2, in order to enhance the bandgap as well as the temperature range of thermoelectric application. In the electronic band structure with $\epsilon$=-2.0%, the bandgap increases to 0.19 eV and the hole carrier effective mass m${}^{*}_{h}$ is light to 0.488 m0 [Fig. 6]. But the energy difference between $\Gamma_{v}$ and VBM almost disappear, which can enhance the intervalley scattering of $\Gamma$ and K, assisted by the K-vector phonons, similar to the intervalley scattering in 1H MoS2 Li2013 ; Ge2014 . Consequently, after the introduction of small compressive strain, the hole carrier mobility drops as well as the the decrease of power factor (1$\sim$3 10*-3* W/mK2) [Fig. 6]. And the peak values of power factor all locate at the range of high concentration ($\geq 2$ 1012 cm*-2* ).

IV Conclusion

In summary, by using the first-principles calculations with Boltzmann transport theory, we studies systematically the metastable monolayer 1$T^{\prime\prime}$ phase MX2, 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 [MoX6]. And the extremely weak electron-phonon coupling of MoSe2 gives rise to its hole carrier mobility as high as 690 cm2/Vs at 300K. Moreover, combining the Seebeck coefficient around 300 $\mu$V/K, it is obtained that the TE power factor of MoSe2 keeps above 6 10*-3*W/mK2 in the large range of temperature (100K$\sim$500K). Our results illustrate the outstanding potential 1$T^{\prime\prime}$ MoSe2 on TE materials.

Acknowledgements.

This work was supported by the NSFC (Grants No.11747054), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.2018M631760), the Project of Heibei Educational Department, China (No. ZD2018015 and QN2018012), and the Advanced Postdoctoral Programs of Hebei Province (No.B2017003004).