A theoretical prediction of super high-performance thermoelectric materials based on MoS2/WS2 hybrid nanoribbons

Modern society is hungry for electrical power. To improve the efficiency of energy harvesting from heat, extensive efforts seek high-performance thermoelectric materials that possess large differences between electronic and thermal conductance. Here we report a super high-performance material of consisting of MoS2/WS2 hybrid nanoribbons discovered from a theoretical investigation using nonequilibrium Green’s function methods combined with first-principles calculations and molecular dynamics simulations. The hybrid nanoribbons show higher efficiency of energy conversion than the MoS2 and WS2 nanoribbons due to the fact that the MoS2/WS2 interface reduces lattice thermal conductivity more than the electron transport. By tuning the number of the MoS2/WS2 interfaces, a figure of merit ZT as high as 5.5 is achieved at a temperature of 600 K. Our results imply that the MoS2/WS2 hybrid nanoribbons have promising applications in thermal energy harvesting.

Environmental pollution and energy shortages are two big concerns in modern society. Thermoelectric materials, which can convert waste heat in the environment to electricity, are expected to be helpful in resolving these two issues [1][2][3][4] . The energy conversion efficiencies of thermoelectric materials are measured by the so-called figure of merit ZT which is defined as σ = / ZT S T k 2 , where S is the Seebeck coefficient, σ is the electronic conductance, and k ( = + k k k e p ) is the total thermal conductance including contributions of electrons (k e ) and phonons (k p ). Therefore, a high-performance thermoelectric material should have high electron conductance and low thermal conductance, i.e., electron crystals and thermal glasses. However, the ZT values of most bulk materials are very small (much less than 1.0) because their electronic and thermal properties always have the same trends 5-8 . There are extensive studies to search for high ZT materials. It is reported that the ZT values of some materials are improved significantly after nanocrystallization due to drastic changes of electronic and thermal properties [9][10][11][12][13][14][15][16] . For example, theoretical calculations proved that the ZT values of quasi-one nanowires have a larger increase than those of bulk and two-dimensional structures 2 nanostructured bismuth antimony telluride showed experimentally higher ZT values than the bulk because of a sharp reduction in k p 6 . Beyond the intrinsic improvement, the ZT values of nanostructures can be further enhanced by various modifications, such as hybridization [10][11][12] , doping 13,14 , absorption 15,16 , etc. Previous theoretical studies indicated that hybrid nanostructures, such as SiGe alloys and hybrid BN/graphene nanoribbons, possess higher thermoelectric properties than single nanostructures 11,12 . Even if the thermoelectric performances of nanostructures are much better than those of bulk, most of them still cannot meet the requirements for real world applications. As such, the search for high-performance thermoelectric materials for energy harvesting applications has become a main focus in the thermoelectric field.
Recently, the thermoelectric properties of single-layer or few-layer transition metal dichalcogenides (TMD) MX 2 (M = Mo, W, while X = S, Se etc.) have attracted attention [17][18][19][20][21] . MoS 2 and WS 2 are two typical TMDs, which are considered as excellent electronic materials because of direct band gaps and high carrier mobility. Electron transistors based on MoS 2 and WS 2 have been reported and show high electronic performance, while the thermal conductivities of the two nanosheets are relatively low [22][23][24][25][26] . Therefore, MoS 2 and WS 2 monolayer should have high ZT values, which have been proven by previous theoretical studies 18,19,21 . Meanwhile, some interesting MoS 2 /WS 2 hybrid nanostructures have been synthesized experimentally and studied theoretically [27][28][29][30] . However, the hybrid MoS 2 /WS 2 nanoribbons have not been synthesized yet to the best of the authors' knowledge. Their thermoelectric properties are still unknown.
In this paper, thermoelectric transport in MoS 2 /WS 2 hybrid nanoribbons is studied using nonequilibrium Green's function (NEGF) methods combined with first-principles and molecular dynamics methods [31][32][33][34][35] . The hybrid nanoribbons show high-performance thermoelectric properties compared to pure MoS 2 and WS 2 nanoribbons. Furthermore, the ZT values can be enhanced by modulating the number of interfaces in the structures, which approach 5.5 at 600 K and 4.0 at 300 K, respectively. The variations of the Seebeck coefficient, electronic, and thermal conductances are analyzed to interpret the enhanced thermoelectric properties. The super high ZT values indicate that the MoS 2 /WS 2 hybrid nanoribbons are ideal high-performance thermoelectric materials with high energy conversion efficiencies.

Results and Discussion
The MoS 2 /WS 2 hybrid nanostructures, as shown in Fig. 1, can be divided into three parts, one central scattering region and two (left and right) leads. The central region consists of finite periodic structures. Each period has one finite MoS 2 nanoribbon and one finite WS 2 nanoribbon. The two leads are semi-infinite MoS 2 or WS 2 nanoribbons. The length and width of the central region are labeled as L and W, respectively, and the length of a period of the central region is labeled as L′. Therefore, = × ′ L N L , where N is the periodic number of the central scattering region.
A tight-binding (TB) Hamiltonian is used to describe the electronic properties of the hybrid structures, where the three terms represent the Hamiltonians of the MoS 2 nanoribbons, WS 2 nanoribbons, and their interactions, respectively. , t 1 2 and ε , 1 2 are the third-order hopping integrals for the nearest-neighbor atoms and site energies, which can be obtained from the GGA parameters in ref. 36 (1), we can use the NEGF method to calculate the electronic transport properties, including electronic conductance σ, Seebeck coefficient S, and electronic thermal conductance k e . To verify the reliability of the TB parameters, we compare the electronic density of states (DOS) calculated by NEGF and density functional based tight binding (DFTB) methods 37,38 . The results indicate that the TB parameters can present the same energy gaps to those of the DFTB method, and the DOS profiles based on the TB parameters are also approximately similar to those of the DFTB method (see Fig. S1 in the Supplementary Information). For thermal transport properties, the phonon thermal conductance of the nanoribbons can be calculated using a harmonic approximation. One just needs to substitute the Hamiltonian matrix H by the mass-weighed force constant matrix K. Please see the Methods section for more details.
We calculate ZT as a function of chemical potential µ at T = 300 K for a MoS 2 /WS 2 hybrid nanoribbon (L = 8.37 nm and W = 2.34 nm) where the central region only consists of one MoS 2 /WS 2 interface, as shown in Fig. 2(a). The results for pure MoS 2 and WS 2 nanoribbons with the same size are also shown for comparison. It is found that the maximum values of ZT (Max ZT) for the three structures all appear at µ < 0, moreover, the Max ZT of the MoS 2 /WS 2 nanoribbon is larger than those of the two pure structures. The Max ZT of the hybrid nanoribbon is 2.3, while values for WS 2 and MoS 2 are 1.6 and 1.5, respectively. Therefore, after hybridization, ZT values of the pure nanoribbons are increased approximately by 1.5 times at room temperature.
The effect of temperature T on the Max ZT of the three structures is illustrated in Fig. 2(b). All of the ZT curves display the same trend-an increase at low temperature to the maximum followed by reduction at high temperature. With the variation of T, the highest ZT of the MoS 2 /WS 2 hybrid nanoribbon appears at T = 600 K with a value of 3.5 while the maximum values for the pure MoS 2 and WS 2 nanoribbons are only 2.3 (at 700 K) and 1.8 (at  To scrutinize the mechanism of the increment of ZT in hybrid structures over pure ones, we calculate their electronic and thermal properties, such as σ, S, k e , and T p . Equations (2-4) demonstrate that σ, S, and k e are functions of chemical potential µ and temperature T, while k p is only a function of temperature T. Figure 3(a-c) show σ, S, and k e as a function of µ for the three structures at T = 300 K. Seen from Fig. 3(a), the σ curve of the WS 2 nanoribbon is somewhat similar to that of the MoS 2 nanoribbon, except the threshold values of the latter are smaller than that of the former, which is mainly because the band gap of the WS 2 nanoribbon is smaller than that of the MoS 2 nanoribbon. For the MoS 2 /WS 2 hybrid nanoribbon, its energy gap should be equal to the wider one, i.e., equal to that of MoS 2 nanoribbon. The results of DFTB calculation further confirm this point (see Fig. S1 in the Supplementary Information). Therefore, the σ curve of the hybrid nanoribbon has the same threshold as that of the MoS 2 nanoribbon. Meanwhile, the interface scattering to the electrons is weak due to the similarity of the electronic properties of MoS 2 and WS 2 , and thus σ values of the hybrid nanoribbon are just a little smaller than those of the pure nanoribbons. The maximum S usually depends linearly on the band gap 39,40 , therefore, one can see from Fig. 3(b) that the S values of hybrid and MoS 2 nanoribbons are larger than that of WS 2 nanoribbons. The electronic thermal conductance k e shown in Fig. 3(c) almost coincides with the behavior of electron conductance σ, because both of them are attributed to the contribution of electrons. Figure 3(d) presents another ZT factor, the phononic thermal conductance k p as a function of T. The MoS 2 nanoribbon exhibits a higher k p than the WS 2 nanoribbon while the k p of the MoS 2 /WS 2 hybrid nanoribbon is the lowest, which is only about 0.5 times that of WS 2 and MoS 2 nanoribbons. From these results, one can find that the improved ZT value of the hybrid nanoribbon is mainly originated from the sharp decrease of phononic thermal conductance k p while the effect of the interface on σ, S, and k e is relatively small.
In order to understand the drop of k p in the hybrid structure, we compare the phonon transmissions ω ( ) T p in the MoS 2 , WS 2 , and hybrid nanoribbons, as shown in Fig. 4(a). The comparison reveals that two reasons cause the drop of k p in the hybrid structure. One is the shrinkage of the spectrum ranges where have effective phonon transmission. For example, the gaps between acoustical and optical phonons for MoS 2 and WS 2 nanoribbons are 24 and 73 − cm 1 , respectively, while that for hybrid nanoribbon is 87 − cm 1 . The other is the reduction of the transmission coefficients T p . One can find that, as ω > − cm 80 1 , T p of all phonons in the hybrid structure are smaller than those in the pure nanoribbons. The two aspects demonstrate that the interface in the hybrid nanoribbon vastly weaken the phonon transport. The LDOS in the hybrid structures in Fig. 4(b) further illustrates this point. There are many localized phonons at the interface, which acts as a potential barrier blocking the phonons from left to right. It is noted that the phonon localization at the nanoribbon edges is small although the edges are unpassivated. We also compare thermal conductances of nanoribbons with different types of edges, such as unpassivated, S-half passivated 41,42 and periodic edges (see Fig. S2(c) in the Supplementary Information). The results indicate that the effect of dangling bonds, phonon localization, and scattering at the edges on the thermal transport is very weak.
To analyze the effect of temperature T on the Max ZT of the MoS 2 /WS 2 hybrid nanoribbon, we calculate the variations of µ, σ S 2 , σ S T 2 , and k (k e ) corresponding to the Max ZT under temperature T, as shown in Fig. 5. It is seen from Fig. 5(a) that the chemical potential µ gradually shifts to the Fermi level with the increase of T. The variation of thermopower σ S 2 in Fig. 5(b) is irregular due to the different chemical potential µ and the small amplitude of variation, indicating the effect of temperature T on the thermopower is weak. This is further proved by the nearly linear relation between σ S T 2 and T in Fig. 5(c). Therefore, the effect of temperature T on the Max ZT is determined by the relation of thermal conductance k and T. The total thermal conductance k is composed of electronic thermal conductance k e and phononic thermal conductance k p . These three quantities as a function  Fig. 2(b). The σ S 2 is normalized by the σ of temperature are plotted in Fig. 5(d) for comparison. The total thermal conductance k increases slowly with the temperature as T < 600 K and then increases rapidly as T > 600 K, which is induced by the sharp increase of k e after T > 600 K. The variation of k e with T is related to the shift of chemical potential µ and the Fermi distribution. As a result, the highest ZT value appears at T = 600 K. Because the MoS 2 /WS 2 interface reduces the phononic thermal conductance significantly with small penalty on other components, we can further increase the ZT by introducing more interfaces in the system. This could be achieved by enlarging the spatial frequencies of interfaces, which is equivalent to period number N while the total length is fixed. Figure 6(a) shows Max ZT values for MoS 2 /WS 2 hybrid nanoribbons whose central scattering region includes N periods (N units of MoS 2 /WS 2 ). The length L of the hybrid nanoribbons is fixed at 8.37 nm, and thus the increase in periodic number N represents the corresponding decrease of the periodic length ′ L . One can find that the Max ZT increases with N as N < 6, and then decreases as N further increases. The highest ZT exists at N = 6, and the values at T = 100, 300, and 600 K are 2.0, 4.0, and 5.5, respectively, which are 2 ~ 3 times those of pure WS 2 and MoS 2 nanoribbons. Obviously, the multi-periodic hybrid structures further enhance the thermoelectric properties.
To examine the effect of periodic number N on ZT as well as its components, we study the normalized σ, S, σ S 2 , and k for multi-periodic hybrid nanoribbons at T = 300 K. They vary as a function of the periodic number N, as shown in Fig. 6(b). The normalized σ and k decrease as N increases and N < 4. This is because the number of the interfaces in the central scattering region increases with the periodic number N, which strengthens the scattering of electrons and phonons and thus σ and k decrease. As discussed above, the effect of the interfaces on the phonon transport is larger than that on the electron transport, and thus the decrease speed of k is faster. As N > 4, the central region gradually becomes a super lattice, therefore σ and k decrease slowly at first and then increase with N. On the other hand, the variation of S is inversely proportional to that of σ and k. It increases with N as N < 4 and then decreases slowly. The variation of σ S 2 with N is small, indicating that the improved ZT values for the multi-periodic structures are still originated from the sharp decrease of k.
As mentioned above, in-plane lateral hybrid MoS 2 /WS 2 heterojunctions have been synthesized by ambient-pressure chemical vapor deposition (CVD) and single-step vapor phase growth 27,28 . Therefore, the hybrid MoS 2 /WS 2 nanoribbons with one interface (N = 1) in Fig. 1 can be obtained by cutting these two-dimensional heterojunctions 43 . Although the hybrid MoS 2 /WS 2 heterojunctions with multiple interfaces have not been reported up to date, these heterojunctions are expected to be synthesized in the near future, by following the similar synthetic process of periodic BN/graphene heterostructures created by lithography patterning and sequential CVD growth steps 44 . Then, the hybrid MoS 2 /WS 2 nanoribbons with periodic interfaces can also be obtained. The high-performance thermoelectric properties of these hybrid nanoribbons make them have promising applications in thermal energy harvesting.

Summary
In summary, we study thermoelectric properties of MoS 2 /WS 2 hybrid nanoribbons, by using the NEGF method combined with first-principles and molecular dynamics methods. The hybrid nanoribbons exhibit super high thermoelectric properties. With the drastic reduction of k p and little change in the electronic properties, the maximum ZT of the hybrid nanoribbon with one interface is increased to 1.5 ~ 2 times that of pure nanoribbons. Moreover, the ZT values can be further increased by modulating the interface number. The highest ZT value of hybrid nanoribbons can approach 4.0 at 300 K and 5.5 at 600 K. Therefore, MoS 2 /WS 2 hybrid nanoribbons are very promising materials for high-performance thermoelectric devices.

Methods
In this section we have outlined some key steps in applying the NEGF approach for the electronic and phononic transport. We can use the NEGF method to calculate the electronic transmission coefficient Fermi-Dirac distribution function at the chemical potential µ and temperature T. Thermal transport properties of the hybrid nanoribbons can be calculated as follow. At first, the Stillinger-Weber (SW) potential 47 parameters used to describe the interatomic interactions in the hybrid nanoribbons can be obtained by the software GULP based on molecular dynamics scheme 48,49 . These parameters can fit well the phonon dispersions calculated by the first-principles method. Then, the force constants matrices K of the hybrid structures can be also obtained by GULP according to the potentials. In the GULP, the force constant is