High Thermoelectric Performance in Two-Dimensional Tellurium: An Ab Initio Study

TL;DR

This study investigates the thermoelectric properties of monolayer $ ext{α}$-Te, revealing a high ZT value of 0.83, indicating its potential as an efficient thermoelectric material.

Contribution

It provides the first detailed analysis of the thermoelectric performance of the more stable $ ext{α}$-Te phase using ab initio and Boltzmann transport methods.

Findings

Maximum ZT of 0.83 achieved under optimal hole doping.

Monolayer α-Te exhibits promising thermoelectric efficiency.

Potential for α-Te to compete with other 2D thermoelectric materials.

Abstract

In 2016, bulk tellurium was experimentally observed as a remarkable thermoelectric material. Recently, two-dimensional (2D) tellurium, called tellurene, has been synthesized and has exhibited unexpected electronic properties compared with the 2D MoS$_2$. They have also been fabricated into air-stable and high efficient field-effect transistors. There are two stable 2D tellurene phases. One ($\beta$-Te) has been confirmed with an ultralow lattice thermal conductivity ($\kappa_L$). However, the study of the transport properties of the other more stable phase, $\alpha$-Te, is still lacking. Here, we report the thermoelectric performance and phonon properties of $\alpha$-Te using Boltzmann transport theory and first principle calculations. A maximum ZT value of 0.83 is achieved under reasonable hole concentration, suggesting that the monolayer $\alpha$-Te is a potential competitor in the thermoelectric field.

Equations

$$\text{Kn}=$$

$$\times [-\frac{\partial f(\mu ,T,{\epsilon }_i)}{\partial {\epsilon }_i}],$$

$$\sigma =e^2\text{K0},$$

$$S=\frac{1}{eT}\text{K1}{\text{K0}}^{-1},$$

$$\text{ke}=\frac{1}{T}({\text{K}}_2-{\text{K}}_1^2{\text{K}}_0^{-1}).$$

$${\kappa }_{\alpha \beta }=\frac{1}{V}\sum _{\lambda }{C}_{\lambda }{\upsilon }_{\lambda \alpha }{\upsilon }_{\lambda \beta }{\tau }_{\lambda },$$

$${\mu }_{2D}=\frac{e{\hslash }^3{C}_{2D}}{k_{B}T{m}^{\ast }{m}_d{E}_i^2}$$

$$\frac{1}{{\tau }_{\lambda }^B}=\frac{{\upsilon }_{\lambda }}{L},$$

Full text

1 Introduction

Recently, 2D materials have triggered a large number of interest due to the striking physical properties related to the low dimensionality 1, 2, 3. Graphene 1 in group-IV, borophene in group-III 4, 5 and black phosphorous in group-V 6, 7 have been extensively investigated experimentally and theoretically. However, researches related to group-VI monolayer materials are still lacking to date.

Recently, Zhu et al. 8 firstly have proposed three novel types of monolayer tellurene, namely $\alpha$-, $\beta$- and $\gamma$-Te. Afterwards, tellurene has been synthesized and attracted lots of interest 8, 9, 10, 11, 12, 13, 14, 15. These findings indicate that tellurene possesses unusually physical properties, especially for applications in electronic and thermoelectric devices. Furthermore, Wang et al. 10 pointed out that tellurene can be obtained in a very large scale and can be fabricated into air-stable and high-performance field-effect transistors, which makes it different from other 2D materials. Furthermore, tellurene has much larger carrier mobility than that of 2D MoS2 8. Few layer tellurene also has extraordinarily electronic transport properties 10, 14, 15.

$\beta$-Te has the intrinsic $\kappa_{L}$ of 2.16 and 4.08 W/mK along both directions (300 K) 12 and also shows significant anisotropy, indicating good potential in application of thermoelectric devices with ZT = 0.8 16. Besides, in 2016, bulk tellurium was experimentally observed as a remarkable thermoelectric material 17. Energetically, monolayer $\alpha$-Te is the most stable structure compared with the metastable $\beta$- and $\gamma$-Te phases 8. As the most stable phase, $\alpha$-Te is also a promising one to be synthesized experimentally and has applications in practical nanoelectronics.

Generally, we use ZT to assess the degree of thermoelectric material by $ZT={S^{2}\sigma T/(\kappa_{e}+\kappa_{L})}$, where S means Seebeck coefficient, $\sigma$ is the electric conductivity, T is the absolute temperature, $\kappa_{e}$ and $\kappa_{L}$ are the electronic and lattice thermal conductivity, respectively. Ideally, one would like to enhance the numerator and weaken the denominator at the same time. Fortunately, one can enhance the scattering of phonons by boundary, defect, isotopic effect, resonant bonding 18, rattle-like scattering 19, lone electron pairs 20, making composite structures 21 and even forming solid solutions 22. Unfortunately, $S$ and $\sigma$ unusually interweave and behave in an opposite trend, which induces the complexity to efficiently realize the waste heat recovery 21.

Interestingly, low dimensional such as 2D materials have the potential to break above relation between S and $\sigma$ due to the quantum confinement effect pointed by Dresselhaus et al. 3. There are three phases of atomically thin tellurium, namely $\alpha$-, $\beta$- and $\gamma$-Te. The first two are energetically, dynamically and mechanically stable. Furthermore, $\alpha$-Te is more stable than $\beta$-Te 8 that has an unusually low $\kappa_{L}$ 12 and superior thermoelectric property 16. However, relevant works about $\kappa_{L}$ and thermoelectric properties of the more stable $\alpha$-Te phase is still lacking. In this letter, we focus on the more stable phase, $\alpha$-Te, and would like to explore its novel electronic and thermal properties. We find a largest ZT (0.83) is achieved under reasonable hole concentration at 700 K for $\alpha$-Te.

2 Computational methods

We use vasp code and PBE functional 23, 24, 25. The cutoff energy is 500 eV. In order to obtain a more accurate electronic band structure and density of states, we use HSE06 approach 26. The van der Waals (vdW) correction proposed by Grimme 27 was taken into consideration in our calculations. The criterion of convergence energy and force are 10*-6* eV and 1 meV/Å. We use 17$\times$17$\times$1 k-mesh in the Brillouin zone.

In the framework of Boltzmann transport theory (BTE), the electrical transport properties, such as $\sigma$, S and $\kappa{{}_{e}}$ can be expressed as 28, 29:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\tau_{i}(\textbf{k})$ and $\varepsilon_{i}$ are the electronic relaxation time and energy eigenvalue. ke can be obtained based on the tensor Kn.

The in-plane $\kappa_{L}$ under relaxation time approximation can be obtained 12:

[TABLE]

where the meaning of each parameter are explained elsewhere 12. We use Phonopy 30 and ShengBTE 31 to deal with the harmonic and anharmonic force constants. We use the equation (4) to calculate the electronic thermal conductivity ke since sometimes the results from the Wiedemann-Franz law is unreasonable, such as in metallic VO2 32, clean graphene near the charge neutral point 33, and single-electron transistor 34. As a matter of fact, the Wiedemann–Franz law is only suitable for the system where the scattering of electrons in the material is dominanted by the elastic collision (“good metal”) 35. Therefore, we use the definition to calculate ke. It should be noted that an effective thickness should be defined in order to calculate the electronic and thermal properties for 2D materials. The thickness of $\alpha$-Te is 7.74 Å, whose definition is clearly defined in elsewhere 36, 12.

3 Results and discussion

The optimized structure of $\alpha$-Te is shown in Figure 1a and 1b with lattice constant 4.238 Å obtained by PBE+D2 functional. The $\alpha$-Te possesses P-3M1 (164) symmetry group belonging to trigonal system and isotropic pattern in 2D plane, which is quite different from the anisotropic bulk Te 37 and 2D $\beta$-Te 8, 12. From the top and side views, $\alpha$-Te and sandwiched 1T-MoS2 look alike in many ways. Intermediate Te atom is octahedrally coordinated to six neighboring Te atoms and upper and lower Te layers form the A-B like stacking. The coordination numbers in outer and centered Te are 3 and 6, which is the characteristic of a multivalency formation of Te atom located in the near bottom of the periodic table. The bond length in $\alpha$-Te is all 3.04 Å, which is larger than those of 2.77 Å and 3.03 Å (two type bonds) in monolayer $\beta$-Te 12.

Bulk Te was reported a direct semiconductor (Eg = 0.25 eV) 37 and it also has recently been experimentally observed as a remarkable thermoelectric material 17. Figure 1c and 1d show the electronic band structure and density of states (DOS) of $\alpha$-Te, indicating a near-direct band gap material (Eg = 1.11 eV) at HSE06 level. Our result is very consistent with the previous theoretical work 8.

A good thermoelectric material not only needs a minimum thermal conductivity, but also requires a simultaneously maximum power factor (PF). However, Seebeck S and $\sigma$ are usually inter-weaved. A large S requires a large carrier effective mass decided by 38 $S=\frac{8\pi^{2}k_{B}^{2}T}{3eh^{2}}m^{*}(\frac{\pi}{3n})^{2/3}$ in 3D semiconducting materials, but $\sigma$ is inversely proportional to the carrier effective mass due to $\mu=\frac{e\tau}{m^{*}}$.In 1993, Hicks and Dresselhaus proposed a seminal idea that sharpen the energy dependence of the DOS in 1D and 2D systems to alleviate the coupling between S and $\sigma$, consequently enhancing the PF 39, 40. Mahan and Sofo further generalized it to a refined sentence 41. Afterwards, these guiding principles triggered two interesting band structure shapes. One is “pudding-mold” 42, 43 and the other one is “Mexican-hat-shape” 44. Recently, a backward thinking emerges, which introduce quasi-one-dimensional electronic band dispersions in 2D or 3D materials to increase the thermoelectric performance 45, 46. A relatively flat band means a large DOS in a narrow energy region near the Fermi level. The dispersive band leads to a high carrier velocity, also indicates a small $m^{*}$ and therefore a high $\mu$ 47, 48, 49, 50. Interestingly, The valence bands near the Fermi level of $\alpha$-Te shown in Figure 1c has some hole pockets and relatively flat bands, which is the characteristic of a good thermoelectric property and is similar to renowned PbTe1-xSex thermoelectrics 51.

Based on the band structure, we could evaluate the S, $\sigma$, and $\tau$ of $\alpha$-Te. Since the melting point of tellurium is 723 K, we select three typical temperatures (300, 500 and 700 K) in the whole calculations. Figure 2a shows the Seebeck coefficients S in the n- and p-type of $\alpha$-Te as functions of carrier concentration. Overall, the absolute values of S decrease for both n- and p-type when increasing the carrier concentration, which reflects the inverse proportion between S and carrier concentration n. Moreover, the absolute value of S of hole doping is bigger than the electron doping in $\alpha$-Te. For instance, the value of $|\textit{S}|$ for n-type is 242.9 $\mu$V/K at 1012 cm*-2* carrier concentration at 500 K temperature, only around half of p-type 529.6 $\mu$V/K at same condition. The maximum value S of p-type $\alpha$-Te is around 700 $\mu$V/K at room temperature, which is the double of that 350 $\mu$V/K in bulk tellurium 17. This is the physical reason that the ZT of $\alpha$-Te are comparable with bulk tellurium, although the $\kappa_{L}$ in $\alpha$-Te (9.85 W/mK) is around 7 times of that in bulk tellurium (1.5 W/mK) 17. In this sense, if one further increases the phonon scattering using some defects and isotope effect in $\alpha$-Te, the $\kappa_{L}$ will decrease but with the same level of Seebeck coefficient S. Then the ZT of $\alpha$-Te will be significantly enhanced and surpass the bulk tellurium 17.

The difference between electron and hole doping in Seebeck S originates from the electronic band structure and DOS in Figure 1c and 1d. As we discussed above, S is primarily regulated by the effect mass m∗ and the magnitude of DOS. A more smooth valence band (hole-doping) compared with a conduction band (electron-doping) corresponds to a larger carrier m∗, indicating a large S since S is proportional to the m∗. Furthermore, DOS of p-type is obviously much larger than n-type of $\alpha$-Te around the Fermi level shown in Figure 1d. These evidences verify that the superior performance of the hole doping compared with the electron doping in $\alpha$-Te.

According to the BTE, the electronic conductivity $\sigma$ is dependent and is proportional to the relaxation time $\tau$. Therefore, a reasonable relaxation time $\tau$ should be chosen. As a matter of fact, $\tau$ in materials is a function of temperature and carrier concentration. So far, experimental measurement may be the only effective way to solve this issue. For theoretical calculations, $\tau$ is difficult to obtain with high precision. Specifically, there are mainly three methods, such as using a designated $\tau$ based on revelent experiments 52, 53, electro-phonon coupling implemented in EPW 54. Another approach is based on the well-known model: $\tau=\frac{m^{*}\mu}{e}$ in which $\mu$ is the carrier mobility and in 2D materials using deformation potential theory acoustic phonon limited $\mu$ could be expressed as 55:

[TABLE]

where md is the average effective mass dominated by md = $\sqrt{m_{x}^{*}m_{y}^{*}}$. Ei can be expressed as $E{{}_{i}}=\Delta E{{}_{i}}/(\Delta l/l{{}_{0}})$ in which $\Delta E{{}_{i}}$ is the ith band under small compression and expansion compared with the energy of unstrained system. Here, it supposes that the energy of core electrons, considered as the energy reference, do not change under the small deformation. This treating process is reasonable and have also been verified by the previous researches 55, 45. The detailed calculations is shown in the Supporting Information. Our calculated effective masses $m^{*}$ are 0.107 $m_{e}$ for the conduction band minimum and 0.164 $m_{e}$ for the valence band maximum and the carrier mobilities are 2.086$\times$103 cm2V*-1* s*-1* for electrons and 1.736$\times$103 cm2V*-1* s*-1* for holes. Our result is very consistent with the previous report 8. Therefore, based on $\tau=\frac{m^{*}\mu}{e}$, we can obtain the relaxation time of 0.127 ps for electrons and 0.163 ps for holes in $\alpha$-Te.

Contrary to the Seebeck S, electronic conductivity $\sigma$ increases with increasing carrier concentration. Figure 2b shows a larger $\sigma$ of electron doping compared with hole doping at the same carrier concentration due to the smaller effective mass (0.107 me for electrons $<$ 0.164 me for holes). Furthermore, $\sigma$ of two type $\alpha$-Te are insensitive to temperature, which is similar to that of $\beta$-Te 16.

A high PF needs a large S and a large $\sigma$ simultaneously. One should compromise between decreasing function of S and increasing function of $\sigma$. The calculated power factor ($\textit{S}^{2}\sigma$) that can demonstrate the thermoelectric performance of $\alpha$-Te is shown in Figure 2c. The PF for both types first increase, climb the summit at moderate carrier concentrations, and then decrease when increasing the concentration. The p-type PF is significantly larger than n-type of $\alpha$-Te, though they exhibit similar trend dependent on the carrier concentration. For instance, the maximum PF of holes and electrons are 74.6 mW/mK2 at 6.12$\times$1013 concentration and 21.2 mW/mK2 at 2.86$\times$1013 concentration at 500 K. The former one is around 2.5 times larger than that of the latter one and is 2 times larger than 2D SnSe 56.

More interestingly, the maximum PF of bulk tellurium at room temperature is around 1.3 mW/mK2 17, which is around a tenth of that in $\alpha$-Te (12 mW/mK2) at the similar condition shown in Figure 2c. The physical reason behind it is the very small effective mass $m^{*}$ of $\alpha$-Te. The $m^{*}$ of standard 2H-MoS2 is 0.47 $m_{e}$ for electrons and 0.58 $m_{e}$ for holes 8. However, in $\alpha$-Te, the $m^{*}$ is 0.107 me for electron and 0.164 $m_{e}$ for holes. Obviously, the effective mass of carriers in $\alpha$-Te is quite smaller than 2H-MoS2. A smaller $m^{*}$ will result in a larger electronic conductivity qualitatively decided by the simple model: $\sigma=\frac{ne^{2}\tau}{m^{*}}$. Moreover, Liu et al. 14 pointed out that lone-pair electrons in bulk Te will significantly enhance the interchain electronic transport. Similarly, the coordination number in $\alpha$-Te is 3 for outer Te atoms, which is only half of valence electrons in Te atoms (group-VI). Therefore, each outside Te atom exists 3 lone-pair electrons. These lone-pair electrons will enhance the hopping term between electrons and contribute to the superior electronic transport properties in $\alpha$-Te 8, 14.

An optimal ZT also needs a minimum thermal conductivity ($\kappa_{e}+\kappa_{L}$). For electronic thermal conductivity, ke has been discussed in the computational methods. Next, we would like to explore the phonon transport properties of $\alpha$-Te, as phonon dominates the thermal transport in semiconductors. The calculated phonon dispersions are shown in Figure 3a. Firstly, it shows that all phonon branches are free from negative and this result confirms the dynamical stability of $\alpha$-Te. Moreover, there is no phonon gap between acoustic and optical branches, indicating a strong optical-acoustic phonon scattering which will suppress the $\kappa_{L}$ 57 of materials.

A high-order ($\geq$ 3) scattering of phonons leads to a limited $\kappa_{L}$ 58, 59. Figure 3b shows the intrinsic $\kappa_{L}$ computed with both non-iterative and iterative method from 100 K to 700 K. We note that both methods give similar trends of $\kappa_{L}$ in the whole temperature range. However, the iterative method always gives values of $\kappa_{L}$ higher than the non-iterative method at the same temperature. We choose the values of iterative method in the following discussion. We find $\kappa_{L}$ of $\alpha$-Te can be well described by the T-1 curve. This phenomenon is also found in bulk Te 17 and $\beta$-Te 12. The value of intrinsic $\kappa_{L}$ is 9.85 W m*-1K-1* (300 K), which is twice as much as that of $\beta$-Te at the same temperature 12. Furthermore, $\kappa_{L}$ of $\alpha$-Te is also larger than bulk Te (2.77 W/mK) 12, which is consistent with the familiar trend ($\kappa_{2D}$ $>$ $\kappa_{3D}$) 60, 61, 12.

The normalized contribution of ZA, TA, LA and optical branches varying with temperature are plotted in Figure 3c, in which each mode varies small at different temperatures especially beyond 200 K, and can be regarded as independent of temperature, similar with the $\beta$-Te 12. At room temperature, three acoustic branches contribute about 90% to the total intrinsic $\kappa_{L}$, while the summation of all optical phonons is only 10%. Furthermore, the proportions of contributions to total $\kappa_{L}$ are about 25%, 47%, and 17% for LA, TA, and ZA phonons, in which ZA mode is the smallest one among three acoustic branches in $\alpha$-Te, quite different from graphene where the proportion of ZA mode is about 75% 62.

It was reported that there is a symmetry selection rule in flat graphene that only even number of ZA modes can be involved in scattering processes 63, 64. This rule originates from the reflection symmetry about c axis in the structure. The thermal resistance roots in the high order of phonon scattering. Due to the selection rule, ZA mode contributes mainly to the $\kappa$ in graphene. However, the lattice does not have the reflection symmetry about c axis in $\alpha$-Te. As a result the selection rule is broken, the importance of ZA mode in thermal transport of $\alpha$-Te is much smaller than graphene.

Moreover, it is worthwhile to explore the contribution of phonons with different frequencies to the total $\kappa_{L}$. Here the frequency-resolved $\kappa_{L}$ for $\alpha$-Te at 300 K is displayed in Figure 3d. It can be found that the main contribution originates from acoustic phonons with frequencies lower than 2 THz, while the contribution of optical phonons is very small. This is consistent with Figure 3c. Furthermore, the acoustic phonons with very low frequency (lower than 1 THz) contribute most to $\kappa_{L}$. In the following, we will find the behind reason is that phonons around $\Gamma$ point usually possess large phonon relaxation time.

All samples of material have finite size in practical experiment and device application, thus the additional boundary scattering will reduce $\kappa_{L}$, especially at nanoscale or at low temperatures. Usually, in nanostructures, $\kappa_{L}$ related to boundary scattering can be evaluated as 61, 12:

[TABLE]

where L is the size of material. This formula represents the situation for a completely diffusive boundary scattering of phonons. Cumulative $\kappa_{L}$ ($\wedge$ $<$ $\wedge_{max}$) can also be used to study the size effect, whose definition has been defined elsewhere 31. Figure 3e shows the size dependence of $\kappa_{L}$ for $\alpha$-Te with completely diffusive boundary scattering at room temperature and the cumulative $\kappa_{L}$ is displayed in Figure 3f. In Figure 3e, $\kappa_{L}$ becomes lower while the sample size getting smaller, as the boundary scattering becomes larger. For a specifical example, when the sample size is 1 $\mu$m, $\kappa_{L}$ can be suppressed to 7.56 W m*-1K-1*, about 78% of value for the infinitely large system. When the sample size is 10 $\mu$m, $\kappa_{L}$ reaches 91% of the infinitely large system. It implies the size effect always keeps till about 10 $\mu$ m, such as 2D WTe2 65.

The size effect can also be estimated from the cumulative $\kappa_{L}$ as a function of MFP shown in Figure 3f. We can find that the MFP range of phonons that contribute significantly to $\kappa_{L}$ is about 10 nm to 10 $\mu$m, which is consistent with our previous result in Figure 3e. In order to obtain the representative MFP, we fit cumulative $\kappa_{L}$ using a single parametric function: $\kappa(\wedge\leq\wedge_{max})=\kappa_{max}/(1+\wedge_{0}/\wedge_{max})$, where $\kappa_{max}$ is $\kappa_{L}$ of infinite size, $\wedge_{max}$ is the maximal MFP. The fitting parameter $\wedge_{0}$ can be interpreted as the typical MFP with which the phonons contribute significantly to $\kappa_{L}$. The fitted curve is shown in Figure 3f by dashed line. The parameter $\wedge_{0}$ is 94 nm, which could be interpreted as the representative MFP of $\alpha$-Te. This implies the $\kappa_{L}$ of $\alpha$-Te may decline sharply with the sample size of hundreds of nm. This property is propitious to the electronic and thermoelectric materials based on $\alpha$-Te in the nanoscale 10.

Group velocity of phonon has great effects on thermal transport of materials, as it is proportional to $\upsilon^{2}$ based on Eq. (3). The calculated $\upsilon^{2}$ with different frequencies are shown in Figure 4a. LA branch possesses the highest $\upsilon$ among three acoustic branches, specifically more than two times of that for ZA and TA phonons. The optical phonons around 3.7 THz has the largest $\upsilon$ but they only contribute 10% to total $\kappa_{L}$. Here $\tau$ is also displayed in Figure 4b. $\tau$ around $\Gamma$ point has the magnitude of 103-104 ps but $\tau$ of most phonons with frequency more than 5 THz is about 0.1-10 ps. That is the reason why phonons with very low frequency dominate the thermal transport in $\alpha$-Te (Figure 3d). Furthermore, $\tau$ of TA branch is the largest of all, about 104 ps around the $\Gamma$ point. That is the reason why TA mode contributes most to $\kappa_{L}$ even more than LA phonons (Figure 3c). It can also explain that optical phonons contribute little to $\kappa_{L}$ since most of them have ultralow $\tau$ of only 1 to 10 ps.

Grünerisen parameter provides crucial information on the anharmonic interactions of phonons. $\gamma$ in $\alpha$-Te for whole frequency spectrum is displayed in Figure 4c. $\gamma$ of TA and LA modes are in the range of 0-2, while those of optical modes are 1-6. Comparing with $\beta$-Te, $\gamma$ of $\alpha$-Te are smaller, indicating a lower anharmonicity and a higher $\kappa_{L}$ 12. Interestingly, it can be found $\gamma$ of acoustic and optical phonon modes have suddenly dropped around 2-2.5 THz. Since there is no acoustic-optical phonon gap, the large acoustic-optical phonon scattering would lead to a giant $\gamma$ and is also the source of ultralow $\kappa_{L}$ in $\beta$-Te 12. In the three-phonon scattering process, all phonon modes must be satisfied the conservation of energy and momentum simultaneously 66. The quantitative probability of all available three-phonon scattering in $\alpha$-Te are shown in Figure 4d. This P3 phase space is one order smaller than of $\beta$-Te, indicating a weaker anharmonicity of $\alpha$-Te compared with $\beta$-Te 12.

Based on the above results, intrinsic figure of merit ZT can be evaluated and exhibited in Figure 5. The climax ZT at 700 K are 0.57 and 0.83 for electron and hole doping $\alpha$-Te, which is as high as $\beta$-Te 16 and has the same order of bulk Te 17. Room temperature ZT of n-type $\alpha$-Te is 0.54. Our calculation suggests that monolayer $\alpha$-Te is a promising thermoelectric material. Furthermore, it can be promoted further via suppressing $\kappa_{L}$ such as grain boundary, isotope effect and vacancy 67.

4 Conclusion

In summary, we have systematically studied the extraordinarily high thermoelectric performance of monolayer $\alpha$-Te by the ab initio calculations. To explain the high ZT, we consider the electronic and thermal transport properties. A large and asymmetric electronic density of states induce relatively flat bands, leading to a large Seebeck coefficient S. A small effective carrier mass $m^{*}$ of CBM (VBM) will result in a larger electronic conductivity $\sigma$. Combining both, the PF in $\alpha$-Te (12 mW/mK2) exhibits ten times as large as that in bulk tellurium (1.3 mW/mK2) 17 at the similar condition. For lattice thermal conductivity $\kappa_{L}$, $\alpha$-Te consists of heavy atomic mass, which breeds low harmonic properties, such as low group velocities. For anharmonicity, strong phonon anharmonic scattering rates including acoustic-optical phonon scattering bring about a same order of magnitude $\kappa_{L}$ of $\alpha$-Te compared with the bulk Te and $\beta$-Te. A superior electronic properties and a suitable $\kappa_{L}$ combine together leading to the ultrahigh ZT in $\alpha$-Te. A maximum ZT of 0.83 is achieved with reasonable hole concentration at 700 K. Our results indicate that monolayer $\alpha$-Te is a promising thermoelectric material. We hope that $\alpha$-Te, more stable than $\beta$-Te, would be experimentally observed with high ZT in the near future.

5 Supporting Information

The Supporting Information is available free of charge on the ACS Publications website via the Internet at https://pubs.acs.org/journal/aamick.

Electronic band structure and effective mass around conduction band minimum and valence band maximum of two-dimensional tellurium.

**Author Information

**

**Corresponding Author

∗**E-mail: liugang8105@gmail.com

∗E-mail: Xonics@tongji.edu.cn

**ORCID

**Zhibin Gao: 0000-0002-6843-381X

Jie Ren: 0000-0003-2806-7226

**Notes

**The authors declare no competing financial interest.

{acknowledgement}

We thank our member Yi Wang for helpful discussions. This work is supported by the National Natural Science Foundation of China (No. 11775159), the Natural Science Foundation of Shanghai (No. 18ZR1442800), the National Youth 1000 Talents Program in China, and the Opening Project of Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology. Computational resources have been provided by the Tongji University.