New two-dimensional phase of tin chalcogenides: candidates for high-performance thermoelectric materials
Baojuan Dong, Zhenhai Wang, Nguyen T. Hung, Artem R. Oganov, Teng Yang, Riichiro Saito, Zhidong Zhang
NNew two-dimensional phase of tin chalcogenides: candidates for high-performance thermoelectricmaterials
Baojuan Dong,
1, 2, 3
Zhenhai Wang,
3, 4, 5, ∗ Nguyen T. Hung, Artem R.Oganov,
3, 7, 8
Teng Yang,
1, 2, †
Riichiro Saito, and Zhidong Zhang
1, 2 Shenyang National Laboratory for Materials Science, Institute of Metal Research,Chinese Academy of Sciences, Shenyang 110016, China University of Chinese Academy of Sciences, Beijing 100049, China Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, 3 Nobel St., Moscow 143026, Russia School of Telecommunication and Information Engineering,Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210003, China Emanuel Institute of Biochemical Physics RAS, 119334, 4 Kosigin St, Moscow, Russia Department of Physics, Tohoku University, Sendai 980-8578, Japan Moscow Institute of Physics and Technology, 9 Institutskiy Lane,Dolgoprudny City, Moscow Region 141700, Russia Federation International Center for Materials Discovery, Northwestern Polytechnical University, Xi’an, 710072, PR China (Dated: January 3, 2019)Tin-chalcogenides SnX (X = Te, Se and S) have been arousing research interest due to their thermoelectricphysical properties. The two-dimensional (2D) counterparts, which are expected to enhance the property, nev-ertheless, have not been fully explored because of many possible structures. Generating variable compositionof 2D Sn − x X x systems (X = Te, Se and S) has been performed using global searching method based on evo-lutionary algorithm combining with density functional calculations. A new hexagonal phase named by β (cid:48) -SnXis found by Universal Structure Predictor Evolutionary Xtallography (USPEX), and the structural stability hasbeen further checked by phonon dispersion calculation and the elasticity criteria. The β (cid:48) -SnTe is the most stableamong all possible 2D phases of SnTe including those experimentally available phases. Further, β (cid:48) phases ofSnSe and SnS are also found energetically close to the most stable phases. High thermoelectronic (TE) perfor-mance has been achieved in the β (cid:48) -SnX phases, which have dimensionless figure of merit (ZT) as high as ∼ ∼ ∼ × cm − . The high TE performance is resulted from ahigh power factor which is attributed to the quantum confinement of 2D materials and the band convergence nearFermi level, as well as low thermal conductivity mainly from both low elastic constants due to weak inter-Snbonding strength and strong lattice anharmonicity. PACS numbers: 81.05.Zx, 72.20.Pa, 71.20.Mq, 71.20.Nr, 72.20.-i, 73.63.-b
I. INTRODUCTION
Group IV-VI alloys have been intensively studied with itsmany physical properties including ferroelectricity [1], topo-logical insulator [2] and, in particular, thermoelectricity [3–5]. Thermoelectric (TE) materials, which directly convertwaste heat into electricity, have drawn an attention in thelast few decades. The conversion efficiency of TE materialscan be evaluated by the dimensionless figure of merit ZT (= σ S κ T , in which σ , S , κ and T represent the electrical con-ductivity, Seebeck coefficient, thermal conductivity and tem-perature, respectively). Among the group IV-VI alloys, tinand lead chalcogenides [3–6] have been attracting increas-ing interest in thermoelectric community with the structuraland electronic structural anisotropy and intrinsic lattice an-harmonicity [5, 6]. Lattice anharmonicity helps to suppressthermal conductivity, while anisotropy is related to the con-finement effect which has proved to be efficient in improvingthermoelectric performance according to Hicks-Dresselhaus ∗ [email protected] † [email protected] theory [7, 8] if the confinement length is smaller than thermalde Broglie length [9].With the development of exfoliation and synthesis method,many two-dimensional (2D) van der Waals materials in-cluding graphene, black phosphorene (BP), transition metaldichalcogenides (TMDs) and tin chalcogenide (SnX) has beensynthesized [10–13]. The exfoliated semiconducting mono-layer BP and SnX have shown much improved thermoelectricperformance ( ZT ∼ β (cid:48) phase and shown in a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n hole carrier flow a) b) electron flow n-typep-type n-type p-type electrical current -100 -50 0 50 100 n (10 cm -2 ) FIG. 1.
Promising thermoelectric properties of new structural phase of SnTe . (a) Schematics of β (cid:48) -SnTe for thermoelectrics, (b) temper-ature and carrier concentration dependent dimensionless figure of merit ZT . The values (0.96, 2.45, 3.81) on the left panel are the peak ZT values of β (cid:48) -SnTe at 300K, 600K, 900K, respectively. Fig. 1(a), has been found by using USPEX. The β (cid:48) -SnX havebeen checked to be thermodynamically stable. Owing to thelow lattice thermal conductivity κ l and high σ , as explainedin the following section, high thermoelectric performance isachieved in the β (cid:48) -SnX phases. For example, as seen fromFig. 1(b), ZT of β (cid:48) -SnTe at a carrier concentration around afew 10 cm − can be obtained up to 2.45 and 3.81 at 600 and900 K, respectively.The paper is organized as follows. We briefly introducethe computational methods in II. In III we shows the mainresults of the β (cid:48) -SnX, including the structural stability in III-A, the thermal transport properties in III-B, and thermoelectricproperties in III-C. Finally we draw a conclusion in IV. II. METHOD
The structure search of 2D tin chalcogenides is performedby USPEX [19–22] combined with Vienna ab initio simula-tion package (VASP) [23]. In our variable-composition US-PEX calculations, the thickness of 2D crystals is restricted inrange of 0-6 ˚A, the total number of atoms is set to be 2-12,while 80 layer groups are chosen for the symmetry in gen-eration of initial 2D structures. Total energy is calculatedwithin the framework of Projector Augmented Wave (PAW)method [24]. Generalized gradient approximation (GGA) [25]is used to treat the electronic exchange correlation interaction.More details on the parameters can be referred to the supple-mentary information [26].Electronic transport properties are calculated by solving thesemi-classical Boltzmann transport equation within the con-stant relaxation time approximation as implemented in theBoltZTraP package [27]. Since there is no experimental dataof electrical conductivity available for the new β ’-phases toevaluate the relaxation time τ , τ is estimated based on carriermobility µ , for example, τ = m ∗ µ e , m ∗ is the effective massof carrier, carrier mobility µ is calculated based on the de-formation potential theory [28–33]. The calculated τ at room temperature varies from a few tens to a few hundred femtosec-onds (10 − s), which has the same order of magnitude as thecalculated values in other 2D materials [34, 35]. More detailson how to get the carrier mobility µ and relaxation time τ canbe found in the supplementary information [26]. Although therelaxation time of electron depends on the Fermi energy, weadopted the constant relaxation approximation for simplicity.In order to calculate the relaxation time by first principles cal-culations, we should consider electron-phonon interaction andphonon-phonon interaction for estimating the relaxation timeof an electron and a phonon in the same level of approxima-tion, which should be a future problem. Phonon dispersion re-lation is calculated by Phonopy package [36]. The κ l is evalu-ated by phonon lifetime, which is self-consistently calculatedin the ShengBTE package [37]. The second-order harmonicinteratomic force constants (IFCs) are calculated within thePhonopy package, and the third-order anharmonic IFCs areevaluated by using 3 × × III. RESULTS AND DISCUSSIONS
To understand such a high thermoelectric performance inthe β (cid:48) -SnX, we investigate the structural, thermal and elec-tronic transport properties of the β (cid:48) -SnX systems as follows. A. Structure and stability of β (cid:48) -SnX First, we show the results of global search on 2D structuresof tin chalcogenides Sn − x X x (X = Te, Se, S). In Fig. 2, weshow the formation enthalpy ∆ H (defined in Eq.(1) in sup-plementary) of tin chalcogenide 2D systems as a function ofchalcogenide composition in the variable-composition convexhulls as predicted by USPEX. In the convex hull, the zero lineconnects two points at ∆ H = 0 for the most stable 2D ele-mentary structures of Sn and chalcogenide as predicted by a) b) c) d) Top view Side view zigzag a r m c ha i r XSn a P-3m1 thickness M K SnTeSnS SnS SnSe SnSe FIG. 2.
Convex hull of Sn-X (X = Se, S and Te) materials searched by USPEX and atomic structure of β (cid:48) -SnX . USPEX-predictedformation enthalpy ∆ H of 2D bi-element structures with different stoichiometries between (a) Sn and Se, (b) Sn and S, and (c) Sn and Te. Theblue and purple dots represent the stable structural phases experimentally observed, and the red dots represent the new β (cid:48) -phase of SnX. (d)Both top and side views of atomic structure for β (cid:48) -SnX, where light-blue and dark-yellow represent Sn and X (X = Te, Se, S), respectively.The unit cell is marked by the pink dash lines and the first Brillouin Zone is shown. Armchair, zigzag and thickness directions are indicatedby arrows.TABLE I. Structural and mechanical parameters for the β (cid:48) -SnX. Here a is the lattice constant, b Sn − Sn ( b Sn − X ) is the bond length for Sn andSn(X), and thickness for 2D SnX is the vertical distance between the two outermost X atoms in the unit of angstrom, which are shown in Fig.1. In-plane Young and shear module in the unit of Nm − are listed.a b Sn − Sn b Sn − X thickness Young’s modulus shear modulus Poisson ratio( ˚A) ( ˚A) ( ˚A) ( ˚A) ( Nm − ) ( Nm − )SnTe 4.34 3.36 2.97 5.39 47.15 18.42 0.28SnSe 4.09 3.37 2.76 5.26 45.77 17.17 0.33SnS 3.95 3.38 2.64 5.12 45.32 16.53 0.37 USPEX. 2D Sn − x X x is more stable than the reactant ma-terials only when ∆ H is below the zero line. Out of morethan 2600 structures generated, we show only those with ∆ Hlower than 0.5 eV/atom. The two most stable structures ofSn-Te as highlighted in blue and red dots in the convex hull inFig. 2(a) are the commonly observed puckered orthorhombicSnTe phase (Fig.1S(a) in the supplementary) [1] and the new β (cid:48) SnTe phase, respectively. ∆ H of the β (cid:48) SnTe is lower by 19meV/atom than the puckered orthorhombic SnTe phase. The β (cid:48) SnTe has actually been proposed to be a stable semicon-ductor by Sa [38] and Zhang et al. [39]. Here we substantiatestructural stability of the β (cid:48) phase in a convex hull with allpossible stoichiometries considered.The β (cid:48) phase of both Sn-Se and Sn-S has also been ob-tained close to the convex hulls, as shown in red dots inFig. 2(b,c). However, the β (cid:48) phase is less stable than thepuckered orthorhombic phase [40], with ∆ H slightly higherby 8 meV/atom for SnSe and 42 meV/atom for SnS. Addi- tionally, octahedral 1T phases of both SnSe and SnS , theformer of which has been synthesized by experiment [41], isfound to be stable in the convex hull, as shown in purple dotsin Fig. 2(b,c). In the current paper, we will focus only on the β (cid:48) phases.All the β (cid:48) phases have P ¯3 m (cid:93) β (cid:48) -SnX (X = Te, Se, S) shown in Fig. 2(d) and Table I, re-spectively. From Fig. 2(d), the β (cid:48) structure can be viewed as abuckled hexagonal lattice of Sn with two X atoms (one up andone down) at the center of hexagon. Or as shown in Fig.1S(b)in supplementary, it can be considered as two stacked β -SnTemonolayers, one of which takes a series of symmetry opera-tions (inversion + glide) to get the second layer, which makesthe β (cid:48) phase distinct from and more stable than AB-stacked β -bilayer in which a translation symmetry exists between two β monolayers. The relative stability of the β (cid:48) -SnTe phase overthe AB-stacked β -bilayer is analyzed in more details in sup- SnTe SnSe SnS a) b) c) zigzag a r m c ha i r FIG. 3.
Lattice thermal properties of β (cid:48) -SnX . (a) Phonon dispersion relation, (b) lattice thermal conductivity κ L , and (c) normalizedcumulative thermal conductivity κ c / κ L as a function of phonon frequency. plementary [26].To check the stability of the β (cid:48) -SnTe, we calculated thephonon dispersion relation of the β (cid:48) phase of SnX, as shownin Fig. 3(a). No imaginary phonon frequencies are found nearthe Γ point, showing that the β (cid:48) phases are dynamically sta-ble. And the stability is also checked by the elastic parametersin Table I from the standard criteria of elastic stability [42–44]. In fact, the necessary condition for stable 2D materials isthat all elastic constant C i j should be positive [45], which issatisfied in our β (cid:48) -SnX materials (more details in supplemen-tary [26]). B. Thermal transport properties of β (cid:48) -SnX Thermoelectric properties consist of thermal and electronictransport properties. We first study the thermal transport prop-erties of β (cid:48) -SnX. Since the thermal transport of lattice is re-lated to the mechanical properties, let us discuss the mechan-ical properties firstly. The Young’s modulus of the β (cid:48) -SnX(less than 50 N/m) as shown in Table I are much smallerthan other 2D materials like graphene ( ∼
345 N/m) and phos-phorene ( ∼ β (cid:48) -SnXare found less than 20 N/m. From the phonon dispersion inFig. 3(a), we can see anti-crossing of the phonon dispersionbetween low-frequency optical vibration modes with acousticphonon modes for the β (cid:48) -SnX. These optical phonon modescorrespond to two-fold degenerate in-plane shearing modesand out-of-plane breathing mode. Interestingly, their vibra-tional frequency are almost independent of materials at around50 cm − at Γ point, which is closely related to their simi-lar and low values of the shear modulus as given in Table I. In Fig. 3(b), the calculated κ L is plotted as a function of T for armchair and zigzag directions. For example, κ L of β (cid:48) -SnTe at 300 K along armchair direction is as low as 2.87 W m − K − . A typical T dependence of κ L ( κ L ∼ / T ) revealsthat the Umklapp process in the phonon scattering is essentialfor the temperature range that we studied. In Fig. 3(c), weshow the normalized κ L by cumulative thermal conductivity κ c as a function of frequency ω at room temperature. κ c is thevalue of κ L when only phonons with mean free paths belowa threshold are considered [37]. Over 90% of the κ L is con-tributed by phonon modes with frequency below 80 cm − forSnTe, in which the three acoustic modes and the three low-frequency optical modes contribute to κ L .The low lattice thermal conductivity of the β (cid:48) -SnX arisesnot only from low elastic constants due to weak Sn-Sn bond-ing strength, but also from strong lattice anharmonicity. In thelow-frequency region, Fig.S3(a) in supplementary shows thatthe anharmonic scattering dominates the phonon-phonon in-teractions (PPIs) by comparing the anharmonic three-phononscattering rates (ASRs) and isotropic scattering rates (ISRs).These ASRs are mainly contributed by phonon absorptionprocess (ASRs+). Among the three β (cid:48) -SnXs, the strongestASRs are found in SnS, which corresponds to the lowest κ L .It is worth noting that the highest value of ASRs are locatedbetween around 50 and 100 cm − , where acoustic and threelow-frequency optical modes are mixed to one another. Thuswe expect that the inter-band scattering between the acousticand optical modes are associated with the large ASRs. b)c) S ( m V / K ) SnTe SnSe SnS a) SnTe SnSe SnSSnTe SnSe SnSelectron hole
FIG. 4.
Electrical transport properties of β (cid:48) -SnX . (a) Electronic band structure. (b) Seebeck coefficients S, and (c) electrical conductivity σ of SnX for zigzag and armchair directions, as a function of carrier density for T = 300, 600 and 900 K.TABLE II. Carrier mobility at 300K and effective mass for SnX. The effective mass is in units of electron mass m (9.11 × − kg ). Themethod in the supplementary gives more details on calculating carrier mobility µ . [26]SnTe SnSe SnShole electron hole electron hole electroncarrier mobility µ zigzag 1364 1112 1275 853 1220 764( cm V − s − ) armchair 576 834 579 694 468 660effective mass m ∗ zigzag 0.213 0.144 0.228 0.169 0.363 0.218( m ) armchair 0.227 0.144 0.228 0.163 0.363 0.212 C. Thermoelectric properties of β (cid:48) -SnX Based on the calculated results, we will discuss the thermo-electric properties of β (cid:48) -SnX.
1. Seebeck coefficient
The calculated structural stability and low thermal conduc-tivity suggest that β (cid:48) -SnX can be considered suitable for ther-moelectric applications. To unveil its potential for energy con-version between heat and electricity, we look into the rele-vant electronic band structure and electrical properties, bothof which reinforce its capacity for such an application. InFig. 4(a), we show electronic band structures of β (cid:48) -SnXs. In- direct band gaps of β (cid:48) -SnXs exist near the zone center. Thevalue of energy gaps are around 1.0 eV, which are independentof chalcogenide atoms. It is noted that the conduction bandshave been upshifted to fit the band gap obtained by hybridfunctional calculations [47], which usually give a more reli-able band gap size. According to E g ∼
10 K B T opt rule [48],the optimal working temperature for thermoelectric applica-tions of such materials should be around 1000 K.Thermoelectric properties are closely related to electronicband structure. For all β (cid:48) -SnXs, we found the following fea-tures in the electronic bands: (1) Band dispersions of va-lence band maximum (VBM) and conduction band minimum(CBM) along both Γ K and Γ M directions are quite similar,which corresponds to a similar effective mass along both thezigzag and armchair directions as given in Table II. Accord-ing to Cutler et al. [49] and Snyder et al. [50], for a parabolic
SnTe SnSe SnSelectron hole SnSSnTe SnSe SnSelectron hole holeelectron electron hole A r m c ha i r Z i g z ag a)c) d)b) P F ( W K - m - ) P F ( W K - m - ) ZT ZT FIG. 5.
Thermoelectric performance for β (cid:48) -SnX . (a,c) Power factor (PF) and (b,d) Figure of merit ZT of SnX as a function of doping level n at different temperature. Here n is the electron (negative) or hole (positive) doping per unit surface area for 2D SnX. Blue, green and red colorrepresents 300K, 600K and 900K, and solid and dash line represents zigzag and armchair direction which is shown in Fig.2. The maximal PFand ZT at optimal doping level as a function of temperature and crystal direction are shown in (c) and (d). band within the energy-independent scattering approximation,the Seebeck coefficient takes the form of S = π k B m ∗ T eh ( π n ) / ,where m ∗ is the effective mass of the carrier and n is the car-rier concentration. From this formula, similar to the effec-tive mass m ∗ , one expects no directional dependence of theSeebeck coefficient, as shown in Fig. 4(b), in which dashedlines are completely overlapped by solid lines. (2) A usualquadratic dispersion relation appears for the carriers at theCBM, while a quartic band dispersion (E k ∼ k ) is found at theVBM, which usually brings about flat bands near Fermi level.Thus constant electronic density of states (DOS) appears nearCBM, while a van-Hove DOS singularity divergence appearsnear VBM [51], as shown in Fig.4S in supplementary. Sucha difference of DOS between VBM and CBM explains whythe effective mass of holes is larger than that of electrons, asis listed in Table II.In Fig. 4(b) and (c), we show Seebeck coefficient and elec-trical conductivity. For β (cid:48) -SnSe and β (cid:48) -SnS, the Seebeckcoefficient of hole carriers is larger than that of electron; while the electrical conductivity σ of hole is smaller thanthat of electron, which is expected for a parabolic band withinenergy-independent scattering approximation [50]. However,an opposite trend is found in β (cid:48) -SnTe that Seebeck coefficientof electron is higher than that of hole, which is due to theconvergence of conduction band minimum at Γ point with flatband edge at M point. Such type of band convergence is muchadvantageous for an enhancement of Seebeck coefficient [52].
2. Electrical conductivity
In Fig. 4(c), we plot the calculated electrical conductivity σ as a function of carrier concentration n for zigzag and arm-chair directions. Decent electrical conductivity σ as high as afew 10 S/m at room temperature is obtained for the three β (cid:48) -SnXs. Differences of σ for different materials along differentdirections can be understood from the carrier mobility µ andthe effective mass m ∗ as is listed in Table II. From Table II,we can point out that (1) the effective mass m ∗ depends noton crystal direction, but on carrier type, for example, m ∗ h > m ∗ e ; (2) carrier mobility µ along the zigzag direction is largerthan that along the armchair direction, due to a smaller defor-mation potential along the zigzag direction than the armchairdirection. All these features lead to a preference of zigzagover armchair direction and electron over hole carrier for op-timal electrical conductivity σ , which is indicated by the com-parison between solid (zigzag) and dashed (armchair) lines asis shown in Fig. 4(c). Moreover, σ decreases with increas-ing temperature, which is associated with the intrinsic phononscattering mechanism. The electrical thermal conductivity κ e is also calculated based on the Boltzmann transport theory, asgiven in Fig.5S, and fits the Wiedemann-Franz law in combi-nation with σ .
3. Power factor and figure of merit
With Seebeck coefficient, electrical conductivity and ther-mal conductivity available, we finally evaluate power factor(PF) and dimensionless figure of merit (ZT). In Fig. 5(a) and(b), we plot the dependence of carrier type, crystalline direc-tion and temperature for PF and ZT. The optimal PF and ZT are shown in Fig. 5(c,d). As seen from Fig. 5(a), PF remains ashigh as 0.01 WK − m − or above in a wide range of tempera-ture at carrier concentration from 10 to 10 cm − . Becauseof the high PF and relatively low κ , it is no surprise to ob-serve quite promising value of ZT in β (cid:48) -SnX. From Fig. 5(b),all ZT of β (cid:48) -SnX show above 1.0 at 900 K in the interesteddoping range ( | n | < × cm − ). ZT of β (cid:48) -SnTe can evengo above 2.0 at 600K, which makes the material very compet-itive against the present commercialized thermoelectric mate-rials. From Fig. 5(a, b), both PF and ZT are larger for holethan for electron in β (cid:48) -SnS and β (cid:48) -SnSe, mainly due to thesmaller Seebeck coefficient of electron than hole. As for β (cid:48) -SnTe, we get a better thermoelectric performance of electronthan hole, which is due to a large S and PF from the concept of‘band convergence’ [52, 53] at CBM concurrent with decentelectrical conductivity from the smaller effective mass of elec-tron than that of hole. In Ref. [52], the dependence of optimalPF opt on ∆ E for a generic systems with band convergence isgiven quantitatively within two-band model, with ∆ E definedas valley splitting energy. PF opt decreases exponentially withincreasing ∆ E within a few k B T. In our case, ∆ E is the en-ergy difference of the CBMs between the K and M points inFig. 4(a), ∆ E is 0.15, 0.36 and 0.28 eV for β (cid:48) SnTe, SnSe andSnS, respectively, which explains why a much bigger PF opt ofthe n-type β (cid:48) SnTe is obtained than that of the n-type β (cid:48) SnSeand SnS.In Fig. 5(c,d), we show the optimal values of PF and ZT fortwo types of carriers along armchair and zigzag directions at T= 300, 600 and 900 K. It is more clear to see in Fig. 5(a,b) thatn-type β (cid:48) -SnTe has a much better thermoelectric performancethan β (cid:48) -SnS and β (cid:48) -SnSe, while p-type β (cid:48) -SnX shows verydecent performance but no obvious difference of PF and ZT from SnS to SnTe.It should be pointed out that we expect some discrepancybetween theoretical and experimental values. For our estima- tion, there are following reasons for discrepancy: 1) the con-stant relaxation time approximation was used for electronictransport properties, where the real relaxation time may varywith the carrier concentration; 2) only isotopic and three-phonon scattering was considered here for κ L . The con-stant relaxation time approximation may overestimate the σ .However, κ L may also be overestimated without consider-ing enough scattering rates coming from the impurity, defect,grain boundary and dislocation and so on. Considering thatthe two parameters κ L and σ are both overestimated, the devi-ation of TE performance may be alleviated in part by the twoeffects. Therefore, our estimated TE performance may give areasonable agreement with the experimental values.Finally due to confinement effect for 2D system [7, 8], it isimportant to evaluate the PF enhancement factor f E [9], whichis defined as f E = ( L Λ ) D − , where L is the spatial confinementlength and Λ (= (cid:113) π ¯ h k B Tm ∗ ) is the so-called thermal de Brogliewavelength and D (= 1 or 2) is the dimension. Here we con-sider the PF enhancement from 3D to 2D. L is taken from theinterlayer distance in 3D counterparts. Values of Λ , L and F E for β (cid:48) -SnX are given in Table III in supplementary. Take n-type β (cid:48) -SnTe as an example, we find that Λ ∼ ∼ E ∼ β (cid:48) -SnX in 2D form is much enhanced upon its 3D counterpart.In summary, for the new β (cid:48) phase of SnX, the decent ther-moelectric properties beyond traditional thermoelectric mate-rials occur because of the following reasons: (1) The low di-mensional structure with high elastic and dynamic stability,which shows substantial enhancement of power factor uponthe bulk phase due to the larger thermal de Broglie length Λ . (2) The low shear modulus within the layer giving riseto an ultralow frequency of the shearing mode which can cou-ple very effectively with the acoustic phonon mode to greatlysuppress the lattice thermal conductivity. (3) The convergenceof electronic bands at the valence and conduction band edges.All the above factors appear concurrently and coherently tolead to the good thermoelectric performance. IV. CONCLUSION
In this paper, we found new-phase SnX ( β (cid:48) phase) whichis suitable for thermoelectric application by combining ab ini-tio density functional theory with genetic algorithm and semi-classical Boltzmann transport theory. The β (cid:48) phase is eitherthe most stable phase ( β (cid:48) -SnTe), or close to the most sta-ble phases (such as orthorhombic phase of SnSe and SnS)which are experimentally observed. Phonon dispersion re-lation calculation and elasticity criteria are used to confirmthe structural stability. A low lattice thermal conductivity isobtained for β (cid:48) -SnX, mainly because of hybridization acous-tic phonon modes with low-frequency inter-layer shearingvibration modes. A decent value of power factor ( ∼ − K − ) is also observed from our calculations, which isascribed to band convergence at CBM and quartic electronicband dispersion at VBM. A competitive dimensionless figureof merit can be obtainable in β (cid:48) -SnX within practical dop-ing of a few 10 cm − , in particular, ZT over 2.5 can bereached in β (cid:48) -SnTe at 900 K. Thermoelectric performance of β (cid:48) -SnX can be further optimized with transport along zigzagcrystalline direction. Our theoretical study deems to facilitatediscovering new phase for optimizing thermoelectric perfor-mance by experiment. V. ACKNOWLEDGEMENT
This work is supported by the National Key R&D Pro-gram of China (2017YFA0206301) and the Major Programof Aerospace Advanced Manufacturing Technology ResearchFoundation NSFC and CASC, China (No. U1537204).A.R. O. and B.J. D. thank the Russian Science Founda- tion (Grant 16-13-10459). R.S. acknowledges MEXT-JapanGrants Nos. JP25107005, JP25286005, JP15K21722 andJP18H01810. N.T.H. acknowledges JSPS KAKENHI GrantsNo. JP18J10151. Z.H. W. thanks the National Science Foun-dation of China (Grant No.11604159) and the Russian Scien-tific Foundation (Grant No.18-73-10135). Calculations wereperformed on XSEDE facilities and on the cluster of the Cen-ter for Functional Nanomaterials, Brookhaven National Lab-oratory, which is supported by the DOE-BES under contractno. DE-AC02-98CH10086. 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