Abstract
Engineering thermal transport in two dimensional materials, alloys and heterostructures is critical for the design of nextgeneration flexible optoelectronic and energy harvesting devices. Direct experimental characterization of lattice thermal conductivity in these ultrathin systems is challenging and the impact of dopant atoms and heterophase interfaces, introduced unintentionally during synthesis or as part of deliberate material design, on thermal transport properties is not understood. Here, we use nonequilibrium molecular dynamics simulations to calculate lattice thermal conductivity of \({\mathrm {(MoW)Se_2}}\) monolayer crystals including \({\mathrm {Mo}}_{1x}{\mathrm {W}}_x{\mathrm {Se_2}}\) alloys with substitutional point defects, periodic \({\mathrm {MoSe_2}\mathrm {WSe_2}}\) heterostructures with characteristic length scales and scalefree fractal \({\mathrm {MoSe_2}}{\mathrm {WSe_2}}\) heterostructures. Each of these features has a distinct effect on phonon propagation in the crystal, which can be used to design fractal and periodic alloy structures with highly tunable thermal conductivities. This control over lattice thermal conductivity will enable applications ranging from thermal barriers to thermoelectrics.
Introduction
Two dimensional semiconductors are an important class of functional nanomaterials with promising electronic and mechanical properties for optoelectronic and thermoelectric applications. Monolayer transition metal dichalcogenides of composition \({\mathrm {AB_2}}\) (A = Mo/W and B = S/Se/Te) have recently attracted a lot of attention for optoelectronic properties arising from their favorable electronic band gaps in the range of 1.0–2.0 eV, high chargecarrier mobilities and large on/off ratios^{1,2,3,4}. Thermal engineering of these monolayered materials remains a challenge for the design of devices based on twodimensional materials. For instance, materials for thermal barrier coatings and thermoelectric energy generation require tight control over phonon transport over a wide range of frequencies to achieve minimal thermal conductivities^{5}, whereas materials for optoelectronic devices, where thermal dissipation is key, have opposing design requirements^{6}. Extensive efforts have been made to develop monolayered materials for thermoelectric applications, where a low lattice thermal conductivity is essential for achieving a high figure of merit^{7,8,9}. While several twodimensional and layered materials have been characterized experimentally and computationally for their thermal transport properties^{10,11,12,13,14}, a systematic understanding of the role of point and extended defects and interfaces on controlling thermal conductivity in twodimensional semiconducting systems like transition metal dichalcogenides is lacking.
However, several previous experimental and theoretical investigations have attempted to modulate lattice thermal transport in these material systems by a combination of alloying, interfacial and microstructural engineering and phase patterning. Alloying modifies thermal transport in materials by affecting one or more of the following material parameters—crystal structure, atomic mass^{15}, interatomic bonding and anharmonicity^{16,17} and is effective in scattering highfrequency phonons^{5}. Formation of interfaces and superlattice structures in nanomaterials are very promising for controlling phonon scattering, particularly for low frequency phonons over 1–2 THz^{18,19,20,21,22}. Scaleinvariant fractal patterning, which results in features of multiple sizes, are widely pursued to affect phonons over a wide range of frequencies and mean free paths^{23}. These panoscopic techniques for hierarchicaldesign have been applied to identify electroncrystal and phononglass materials with excellent thermoelectric properties^{24}.
In this study, we use nonequilibrium molecular dynamics simulations to compute lattice thermal conductivity of monolayer \({\mathrm {(MoW)Se_2}}\) systems, including \({\mathrm {Mo}}_{1x}{\mathrm {W}}_x{\mathrm {Se_2}}\) alloys and fractal heterostructures and periodic superlattices constructed out of two transition metal dichalcogenides, \({\mathrm {MoSe_2}}\) and \({\mathrm {WSe_2}}\), suitable for ultrathin electronic applications. This distribution of point defects, heterophase interfaces and a range of feature sizes allows us to explore the influence of each of these features on phonon scattering and identify guidelines for design of twodimensional material structures with tunable thermal transport properties.
Methods
Nonequilibrium molecular dynamics simulations for computing thermal conductivity of \({\mathrm {(MoW)Se_2}}\) layers
Lattice Thermal conductivity (\(\kappa _{lat}\)) of suspended monolayer crystals is computed using the socalled ‘direct’ method of nonequilibrium molecular dynamics simulations (Fig. 1a). This nonperturbative approach for the calculation of \(\kappa _{lat}\) for a heterogeneous system, is consistent with values extracted from classical equilibrium MD (EMD) simulations using Green–Kubo techniques^{25}, but does not suffer from deficiencies in the commonly adopted relaxation time approximation solutions to the Boltzmann Transport Equation, which are known to severely underpredict the thermal conductivity of several 2D materials including transition metal dichalcogenides^{26,27}. To compute the \(\kappa _{lat}\) for thermal transport along the x direction in a \({\mathrm {(MoW)Se_2}}\) monolayer of dimensions \(2L \times L\), a predefined flux of thermal energy, \({\dot{Q}}\), is added to the atoms in a 100 Åstrip at \(x=\frac{L}{2}\) (‘Hot’ end) and an identical heat flux is removed from the system at \(x=\frac{3L}{2}\) (‘Cold’ end). Periodic boundary conditions along the x and ydirections, ensure an equal magnitude of thermal flux in the x and \(x\) directions from the ‘Hot’ to the ‘Cold’ ends. The thermal conductivity of the system can then be obtained directly from the steadystate temperature gradient using the Fourier law of heat conduction (Eq. 1).
where \(\kappa _{\mathrm {lattice}}\) is the thermal conductivity of the monolayer, \(\nabla T\) is the temperature gradient established between the heat source and heat sink due to the imposed heat flux, \({\dot{Q}}\). L and t are the effective width and thickness of the suspended monolayer. Thermal conductivity is calculated for four classes of \({\mathrm {(MoW)Se_2}}\) systems containing different barriers to phonon propagation, namely, pure \({\mathrm {MoSe_2}}\) and \({\mathrm {WSe_2}}\) crystals with no point defects or interfaces, \({\mathrm {Mo}}_{1x}{\mathrm {W}}_x{\mathrm {Se_2}}\) substitutional alloys (Fig. 1b), selfsimilar fractal \({\mathrm {MoSe_2}}/{\mathrm {WSe_2}}\) heterostructures (Fig. 1c), and periodic \({\mathrm {MoSe_2}}/{\mathrm {WSe_2}}\) superlattices with a characteristic length scale, l (Fig. 1d). The random \({\mathrm {Mo}}_{1x}{\mathrm {W}}_x{\mathrm {Se_2}}\) alloy is constructed by replacing x fraction of cation sites chosen at random in the \({\mathrm {MoSe_2}}\) lattice with \({\mathrm {W}}\) atoms. Such a random alloy configuration is consistent with real TMDC alloys synthesized by scalable techniques like chemical vapor deposition (CVD)^{28,29}. Periodic superlattices are constructed as a lattice of square \({\mathrm {WSe_2}}\) patches of size l in the \({\mathrm {MoSe_2}}\) matrix separated by heterophase interfaces along the zigzag and armchair directions. Selfsimilar fractal structures are constructed by substitutionally alloying \({\mathrm {W}}\) atoms in the cation sublattice of the \({\mathrm {MoSe_2}}\) crystal in the form of a Sierpinski carpet. Results from these deterministic fractals are expected to hold even for random fractal structures of the same fractal dimension such as amorphous twodimensional alloys^{30}. Both periodic superlattices and fractal heterostructures are constructed with atomicallysharp interfaces with no atomic mixing that can scatter shortwavelength phonons^{31,32}. Such epitaxial interfaces between isoelectronic materials is preferable for optoelectronic applications, since diffuse interfaces, grain boundaries, inclusions and pores can also detrimentally affect electrical transport^{5}. Figure 1c represents a representative fractal structures containing four levels of selfsimilarity. The choice of selfsimilarity level also dictates the overall stoichiometry of the fractal structure. All fractal structures are constructed such that the size of the smallest feature is larger than approximately 4 nm, reflecting the limits of current patterning technologies^{33}.
The average lattice strain in either the alloys or the heterostructures is less than − 0.075%, reflecting the nearidentical inplane lattice constants of \({\mathrm {MoSe_2}}\) and \({\mathrm {WSe_2}} \, (a({\mathrm {MoSe_2}}) = 3.289\,\AA\) and \(a({\mathrm {WSe_2}}) = 3.286\)Å)^{34,35}. Therefore, point defects and interfacial scattering results mainly from changes in the bonding interactions and atomic masses and the potential effect of longrange disorder and strain on the measured thermal transport is negligible. Details about the molecular dynamics simulations, including development of suitable empirical forcefields and workflow are given in Section I and II of the Supporting Information.
Results
Thermal transport in \({\mathrm {Mo}}_{1x}{\mathrm {W}}_x{\mathrm {Se_2}}\) alloys
Substitutional doping of \({\mathrm {MoSe_2}}\) by \({\mathrm {W}}\) atoms has a significant effect on the lattice thermal conductivity. Figure 2a shows the computed lattice thermal conductivity of the monolayer \({\mathrm {Mo}}_{1x}{\mathrm {W}}_x{\mathrm {Se_2}}\) alloy as a function of substitutional doping. Even moderate doping (\(x < 5\%\)) leads to greater than \(70\%\) reduction in lattice thermal conductivity relative to undoped crystals. Similar results were observed in various materials^{36,37,38,39,40,41}. Classical molecular dynamics simulations exclude electronic structure effects such as chargetransfer and charge carrier–phonon interactions, therefore the large reduction in \(\kappa _{\mathrm {lattice}}\) is attributable primarily to increased rate of point defect scattering that originates from both the mass difference and interatomic coupling force differences resulting in greater phonon localization and reduced meanfree paths^{42,43,44}. However, there is no noticeable change in other phonon characteristics such as phonon frequencies, group velocities and phonon density of states at low frequencies.
To quantify the phonon localization effect, we computed the phonon participation ratio \(P_{\lambda }\) for the unalloyed and defectfree \({\mathrm {MoSe_2}}\) single crystal and the 3.7% Wdoped \({\mathrm {MoSe_2}}\) alloy (Fig. 2b). The phonon participation ratio, \(P_{\lambda }\), measures the spatial localization of a phonon mode, \(\lambda\) and it is defined as^{45,46}
where N is the total number of atoms and \(\varepsilon _{i\alpha ,\lambda }\) is the \(\alpha ^{th}\) cartesian component of the eigenmode \(\lambda\) for the ith atom. \(P_{\lambda }\) is a dimensionless quantity ranging from 1/N to 1, with \(\approx 1\) denoting the propagating mode and \(\approx 0\) denoting the localized mode.
We observe that the degree of localization is enhanced for all phonons of finite frequency in doped \({\mathrm {MoSe_2}}\) crystal, as shown by the lower values of \(P_{\lambda }\) in doped\({\mathrm {MoSe_2}}\) as compared to that in dopantfree \({\mathrm {MoSe_2}}\) single crystal samples. This behavior is consistent with Anderson’s theory of localization of waves in disordered twodimensional media driven by interference between multiple wave scattering^{47} as well as experimental observations in other twodimensional materials^{48}. It can also be seen that substitutional point defects lead to a large suppression in thermal transport by highfrequency, low meanfreepath phonons, with participation ratios below 0.3. Similarly, long wavelength acoustic phonons (\(\omega < 4 \, \hbox {THz}\)), which dominate thermal transport in undoped \({\mathrm {MoSe_2}}\) undergo relatively less scattering, with participation ratios between 0.2 and 0.85, resulting in a finite thermal conductivity even at high doping level. Further, it is noticeable that thermal conductivity of the alloy remains constant and relatively insensitive to \({\mathrm {W}}\) content beyond approximately \(20\%\) alloying. This low and compositionindependent thermal conductivity implies that substitutional alloys are not suitable for thermal design applications.
Fractal \({\mathrm {MoSe_2}}{\mathrm {WSe_2}}\) heterostructures
There exist several empirical models to describe transport processes (electrical, thermal and mass) in porous, selfsimilar and fractal media^{49,50,51}. However, they provide a description of macroscopic properties of the system only in terms of the bulk properties of the individual phases, excluding any interfacial effects. The most common model for transport through irregular, porous and selfsimilar media is Archie’s law^{52}. This empirical relation, given by \({\dot{Q}} \propto \phi ^m/a\) relates flux (thermal or mass) through the medium, \({\dot{Q}}\) to the phase fraction, \(\phi\) and via the empirical exponent m which takes a value between 1.3–2.5 and tortuosity of the thermal path, a^{53}. An alternative model by Miller suggests that^{51}, \(\sigma _{max} = 1  \left( \frac{1}{12G}\right) c\), where c is the concentration of the \({\mathrm {WSe_2}}\) phase (assumed to be of zero conductivity) and \(\sigma _{max}\) is the thermal conductivity of the pure \({\mathrm {MoSe_2}}\) phase and G is some geometric parameter equal to 0.27 for square parches. Extending this thought, we can show that in a fractal of order n, the effective matrix around the largest central particle is a fractal of order \(n1\). Therefore, we can write \(\sigma _{n} = \sigma _{n1}* \left( 1  \frac{1}{12G}\right)\). This assumption is also common in more complex models for thermal transport in regular fractal systems. However, none of these models can accurately capture the gradual, nearlinear variation of \(\kappa _{lattice}\) with \({\mathrm {WSe_2}}\) phase fraction, shown in Fig. 3a, because they do not consider the role of the \({\mathrm {MoSe_2}}{\mathrm {WSe_2}}\) interfacial scattering of phonons, which is the dominant scattering mechanism in these systems and the thermal boundary resistance of the \({\mathrm {MoSe_2}}{\mathrm {WSe_2}}\) interface, as described by the acoustic mismatch model^{54}. Further, thermal transport in the resulting \({\mathrm {MoSe_2}}\) and \({\mathrm {WSe_2}}\) nano domains will also demonstrate significant size effects within the Casimir regime (i.e. smallest feature size < phonon mean free path). Therefore Archie’s law and other previously determined models cannot be applied, contrary to the results of Ref.^{53}.
In these selfsimilar structures, the reduction in thermal conductivity is caused by phonon scattering at \({\mathrm {MoSe_2}}/{\mathrm {WSe_2}}\) heterointerfaces. To understand this scattering process, we compute the timeaveraged heat flux on each atom in NEMD simulations using the expression
where e, \(v_i\), and \(S_{ij}\) are the energy, velocity vector, and local stress tensor at each atom^{55,56}. Figure 3b shows the computed peratom flux through the fractalpatterned \({\mathrm {MoSe_2}}/{\mathrm {WSe_2}}\) heterostructure. It is noticeable that the \({\mathrm {MoSe_2}}/{\mathrm {WSe_2}}\) interfaces are the primary source of phonon scattering and that the majority of the thermal flux flows through regions of the fractal structure that contain no \({\mathrm {MoSe_2}}/{\mathrm {WSe_2}}\) interfaces in the xdirection. The figure also shows that the majority of the thermal boundary resistance is concentrated at the interfaces closest to the hot or the cold end, consistent with observations from the Si–Ge system^{57}. The greater scattering due to heterostructures is also noticeable in the polar plot of atomic heat flux vectors Fig. 3c, which shows that thermal transport in the heterostructured crystal has a greater scatter away from the \(+x\) and \(x\) directions compared to heat flow in pure \({\mathrm {MoSe_2}}\).
Thermal transport in periodic superlattices
In order to understand if the inherent lack of periodicity in the fractal structure affects phonon propagation, we also compute the thermal conductivity of periodic \({\mathrm {MoSe_2}}{\mathrm {WSe_2}}\) superlattices with square patches of \({\mathrm {WSe_2}}\) patches embedded in a \({\mathrm {MoSe_2}}\) matrix (Fig. 1d). Specifically, we choose heterostructures of composition 29% \({\mathrm {WSe_2}}\), equal to that in a level 3 fractal heterostructure, for our simulations. At this constant composition, we can vary the periodicity of \({\mathrm {WSe_2}}\) patches to construct periodic heterostructures of different interfacial densities.
Figure 3d shows the nearlinear decrease in the computed thermal conductivity of the three periodic heterostructures as a function of interfacial density, as seen in other semiconducting systems like Si–Ge^{22,58}. It can be observed that the computed \(\kappa _{\mathrm {lattice}}\) for the thirdlevel fractal falls in line with the trend predicted by the periodic heterostructures. This linear and inverse dependence of thermal resistance with interfacial density (and not by their relative orientations and arrangement) in indicates that thermal transport in \({\mathrm {MoSe_2}}/{\mathrm {WSe_2}}\) heterostructures is dominated by conduction of incoherent phonons. The presence of interfaces and anharmonicity of the interatomic interactions lead to decoherence of phonons and their resulting particlelike behavior^{59}. Coherent phonons, which can traverse periodic heterostructure, but not nonperiodic fractal ones^{46}, contribute negligibly to the calculated thermal conductivity.
Design of heterostructures for tuning lattice thermal transport
This understanding of phonon scattering by point defects (like vacancies and dopant atoms) and heterostructure interfaces provides useful design guidelines for the construction of low thermal conductivity structure. Figure 4a shows one such heterostructure which attempts to maximize both the interfacial density as well as the concentration of dopant atoms in the \({\mathrm {WSe_2}}\) patches and the \({\mathrm {MoSe_2}}\) matrix. This ‘doped fractal’ structure was observed to have a thermal conductivity of only 15 W/mK, which is lower than that of either the 3%doped \({\mathrm {Mo}}_{1x}{\mathrm {W}}_x{\mathrm {Se_2}}\) alloy or the thirdlevel fractal \({\mathrm {MoSe_2}}{\mathrm {WSe_2}}\) heterostructure used to construct the ‘doped’ fractal structure (Fig. 4b). This behavior can be explained using Matthiessen’s rule of independent scattering events, where the overall scattering rate is a sum of individual scattering rates^{41}. These simulations show that careful control over doping and heterostructure construction can be used to controllably modify thermal conductivity of \({\mathrm {(MoW)Se_2}}\) monolayer single crystals.
Discussion
We have performed nonequilibrium molecular dynamics simulations using a specifically parameterized forcefield to compare the thermal conductivity of suspended \({\mathrm {Mo}}_{1x}{\mathrm {W}}_x{\mathrm {Se_2}}\) alloys with periodic and fractalpatterned \({\mathrm {MoSe_2}}{\mathrm {WSe_2}}\) heterostructures to identify the dependence of lattice thermal conductivity on dopant concentrations and interfacial densities. We show that even low dopant concentrations (\(< 5\%\) doping) can strongly localize highfrequency phonons in the \({\mathrm {(MoW)Se_2}}\) crystal leading to a large (\(> 70\%\)) reduction in the lattice thermal conductivity. Further, this low value of \(\kappa _{\mathrm {lattice}}\) is largely insensitive to dopant concentration and therefore alloying alone is not a viable strategy for controlling thermal conductivity. On the other hand, thermal transport in both periodic and fractal patterned heterostructures is dominated by incoherent phonon conduction and varies gradually and monotonically with the density of \({\mathrm {MoSe_2}}{\mathrm {WSe_2}}\) interfaces. Thermal conductivity can be controllably tuned by constructing doped fractal heterostructures where both scattering mechanisms operate.
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Acknowledgements
This work was supported as a part of the Computational Materials Sciences Program funded by the U.S Department of Energy, Office of Science, Basic Energy Sciences, under Award Number DESC0014607. All Simulations were performed at the Center for High Performance Computing of the University of Southern California.
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P.V., R.K.K. and A.N. conceived the simulations. A.K. and N.B. performed simulations and data analysis. All authors wrote and reviewed the manuscript.
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Krishnamoorthy, A., Baradwaj, N., Nakano, A. et al. Lattice thermal transport in twodimensional alloys and fractal heterostructures. Sci Rep 11, 1656 (2021). https://doi.org/10.1038/s41598021810554
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DOI: https://doi.org/10.1038/s41598021810554
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