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Lattice Mismatch Dominant Yet Mechanically Tunable Thermal Conductivity in Bilayer Heterostructures Yuan Gao, Qingchang Liu, and Baoxing Xu ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.6b01674 • Publication Date (Web): 19 Apr 2016 Downloaded from http://pubs.acs.org on April 21, 2016
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Lattice Mismatch Dominant Yet Mechanically Tunable Thermal Conductivity in Bilayer Heterostructures Yuan Gao1, Qingchang Liu1, and Baoxing Xu1, 2* 1
Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA 2
Institute for Nanoscale and Quantum Scientific and Technological Advanced Research, University of Virginia, Charlottesville, VA 22904, USA
*Corresponding author:
[email protected] (BX)
ABSTRACT: Heterostructures that are assembled by interfacing two-dimensional (2D) materials offer a unique platform for the emerging devices with unprecedented functions. The attractive functions in heterostructures that are usually absent and beyond the single layer 2D materials are largely affected by the inherent lattice mismatch between layers. Using nonequilibrium molecular dynamics simulations, we show that the phonon thermal transport in the graphene-MoS2 bilayer heterostructure is reduced by the lattice mismatch, and the reduction can be mitigated well by an external tension, weakening the effect of inherent mismatch-induced strain on thermal conductivity. Mechanical analysis in each layered component indicates that the external tension will alleviate the lattice mismatch-induced deformation. The phonon spectra are also softened by the applied tension with a significant shift of frequency from high to low modes. A universal theory is proposed to quantitatively predict the role of the lattice mismatch in thermal conductivity of various bilayer heterostructures, and shows good agreement with simulations. Keywords: Heterostructures; 2D materials; Lattice mismatch; Thermal conductivity; External tension
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Two-dimensional (2D) materials continue to attract much attention for the unique mechanical,1-3 electronic
4-9
and thermal properties.10-12 Accompanying with the emergence of more 2D
materials such as hexagonal boron nitride (hBN),13 molybdenum disulfide (MoS2),14 black phosphorus (BP)
15
and tungsten disulfide (WS2)
16
since the discovery of graphene,17
heterostructures that are constructed by interfacing two or more layers of two-dimensional (2D) materials through interlayer van der Waals interactions, also referred to as van der Waals heterostructures, are designed to harness drawbacks of the single layer 2D materials, and to achieve unprecedented properties beyond that of individual components.18-24 For example, the intrinsic weak absorption characteristics in graphene can be enhanced by assembling together with 2D molybdenum disulfide (MoS2) materials that possess a strong optical absorption and visible-range bandgap.25,
26
The assembled graphene-MoS2 heterostructure has demonstrated
great applications in photon harvesting and photocurrent generation,8 photoresponsive memory 27
and tunable electrical semiconductor diode.28 In the graphene-hBN heterostructure, the
presence of hBN layer minimizes the carrier scattering, leading to an order of magnitude higher in electronic quality of graphene than that of a single isolated graphene sheet.29 The hBN layer can also serve as a protective cover for a long exposure of graphene to ambient conditions without deterioration.30 A similar protection to MoS2 from radiation damage by graphene layer has also been found recently in heterostructures.31, 32 These outstanding properties in heterostructures that are distinct from its components are usually attributed to the lattice mismatch at the interfaces. Besides, for either electrical current or thermal transport in heterostructures, thermal property is very critical, where the mismatchinduced strain plays a significant role in affecting thermal transport. In principle, the lattice mismatch alters vibrations of atoms and couplings due to the constraints among layers and also leads to spontaneous strain energy in the interface associated with deformation of the planar layers,33-37 which, in turn, is expected to yield changes of electrical and thermal properties of its 2D material components. The mismatch-induced strain has been measured in heterostructures such as hBN-graphene,38 graphene-MoS2
33, 39
and MoS2-WS2.34 However, a quantitative
relationship between thermal properties such as thermal conductivity and lattice mismatch is lacking to date. More importantly, the development of the ability to control and manipulate the thermal transport in heterostructures is of great interest for fundamental physics and potential applications in thermal devices with tunable thermal functionalities. 2 ACS Paragon Plus Environment
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In the present work, we investigate the thermal conductivity in parallel to the interface of graphene-MoS2 bilayer heterostructures by using nonequilibrium molecular dynamics (NEMD) simulations, and probe the influence of the lattice mismatch. It is found that the lattice mismatch leads to reduction of phonon thermal transport due to the out-of-plane deformation of graphene layer. Moreover, we demonstrate that the out-of-plane displacement can be alleviated by applying an external tensile strain in the heat flow direction, weakening the influence of lattice mismatch. The regulation of thermal conductivity via the applied strain is confirmed by both mechanical deformation and phonon spectra of the individual graphene and MoS2 layer in the heterostructure. Further, a unified theoretical model is proposed to understand the effect of lattice mismatch, and is verified by performing extensive simulations on the popular bilayer heterostructures of graphene-MoS2, graphene-hBN, and hBN-MoS2, graphene-black phosphorus, hBN-black phosphorus. RESULTS AND DISCUSSION Mitigation of lattice mismatch-induced deformation by tension. Figure 1a illustrates the computational model of a bilayer van der Waals heterostructure that consists of one pristine graphene layer and one pristine MoS2 layer. The lattice mismatch ratio between them is defined as = ( − )/, where =0.312 nm and =0.249 nm are the first lattice constant of MoS2 and graphene,33 respectively, and =0.253. The heat transfer is applied in parallel to the interface of heterostructures, and the external tensile strain is applied to the same direction as the heat flow. Computations on the bilayer graphene and MoS2 structures are performed individually for comparisons. The details of model and simulation methods can be found in the method section. Figure 1b shows the nominal stress-strain curves of the graphene-MoS2 heterostructure, bilayer graphene and bilayer MoS2. For the bilayer graphene, the stress increases linearly at a small strain, followed by a nonlinear variation till to a sudden drop that corresponds to the failure of carbon-carbon bonds, which is similar to that of a single layer graphene under uniaxial tension.40 As for the bilayer MoS2, the stress increases almost linearly until the failure of the structure occurs, which is analogous to that of a single layer MoS2.33, 41 At the same strain, the stress of the graphene-MoS2 heterostructure is higher than that of bilayer MoS2 but lower than that of bilayer graphene. The failure strain is almost the same as the bilayer MoS2,
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demonstrating that the deformation of heterostructure is dominated by the MoS2 layer with weaker stretchability.33 To reveal the effect of the applied tensile strain on lattice mismatch-induced layer deformation, the variation in the positions of carbon and molybdenum disulfide atoms in graphene-MoS2 heterostructure at equilibrium in comparison to their individual bilayer structures are extracted. Figure 1c shows the out-of-plane displacement for both graphene and MoS2 layer components, . An obvious out-of-plane deformation in the graphene sheet is observed, and the periodic pattern is attributed to the non-centrosymmetric atomistic structure of MoS2. Besides, in contrast to the planar carbon ring structure in graphene layer, since the distribution of molybdenum and sulfur atoms in the MoS2 layer show a trigonal structure with a relatively high thickness, the lattice mismatch leads to a weak influence,42 and a very small displacement to MoS2 layer is observed. When an external strain is applied, it will stretch the carbon-carbon bonds and constrain the out-of-plane deformation, leading to a decreased out-of-plane displacement. The similar reduction to the deformation on MoS2 layer is also observed. Different from the obvious out-of-plane displacement, the lattice mismatch causes a relatively small displacement to the in-plane displacement, , in graphene layer owing to the high in-plane stiffness of graphene (Figure S1).1 Note that the in-plane displacement of MoS2 is greater than that of graphene due to inherent 3D atomistic structure that facilitates the in-plane displacement. Besides, in comparison to the obvious mitigation to out-of-plane deformation by the applied tension, the effect of tensile strain on the in-plane displacement is small. Thermal conductivity. Figure 2a shows the thermal conductivity of graphene-MoS2 heterostructure, , bilayer graphene, and bilayer MoS2, . In the absence of the applied tensile strain, , and are 100.79, 8.50 and 34.05 W/mK, respectively, consistent with previous reports.42-45 As increases, they all decrease monotonously, and is in between and , which remains with the increase of the strain , similar to that observation in the stress-strain curve in Figure 1b. To understand the effect of the lattice mismatch, we calculate the thermal conductivity of individual graphene and MoS2 layers in the heterostructure by monitoring the heat flow in each layer (see method section for computational settings). Their variations with the applied strain are plotted in Figure 2b. It shows that the thermal conductivity of graphene in heterostructure 4 ACS Paragon Plus Environment
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is lower than that of bilayer graphene structure (Figure 2a); while the thermal conductivity of MoS2 in heterostructure is close to that of bilayer MoS2 structure (Figure 2a), implying that the thermal conductivity of MoS2 is barely affected by interaction in its cross-plane direction, consistent with the fact that thermal conductivity of MoS2 is independent of layer numbers.46 Similar to and , both and decrease monotonously with the increase of . Consider the equivalence of the total heat flow, we have = + , where is the heat flow through the entire heterostructure, and and are the heat flow passing through individual graphene and MoS2 layer, respectively. Since the temperature gradient is approximately the same (Figure S2b), based on Fourier’s Law, we have = ( + )/ , where and are the thickness of graphene and MoS2 layer in heterostructure, respectively and are approximately equal to the thickness of a single graphene and MoS2 layer.33, 44 is the thickness of heterostructure, and = + .44 Taking =0.335 nm, and =0.609 nm,33 we can calculate , as shown in Figure 2b (dash line), and they show good agreement with MD results that are extracted from graphene-MoS2 heterostructure as an entire structure (Figure 1a). To clarify the effect of the lattice mismatch in heterostructure and the tensile strain on , the thermal conductivity of individual graphene and MoS2 layer in heterostructrue is normalized by their corresponding bilayer structures, and is plotted in Figure 2c and d, respectively. For the graphene layer, / is 0.788 (< 1.0) at =0, indicating that is reduced by the presence of MoS2 layer that causes the out-of-plane displacement intrigued by the lattice mismatch (Figure 1c). With the increase of , / increases, implying that the restriction on thermal transport by the lattice mismatch is weakened by the tensile strain. The alleviated effect agrees well with the decreased out-of-plane mismatch displacement under a tensile strain (Figure 1c). In contrast to an increased / with the applied strain in graphene, / remains to be near 1, demonstrating that the thermal transport across MoS2 is barely affected by the lattice mismatch, independent of the applied tensile strain, which is also in good agreement with the very small out-of-plane and in-plane displacement in MoS2 layer (Figure 1c and Figure S1). Phonon spectrum. To further probe the atomic mechanism of heat conduction and the effect of tensile strain and mismatch-induced displacement in the graphene-MoS2 heterostructure, the 5 ACS Paragon Plus Environment
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most popular phonon frequency spectrum ( ) is investigated. Figure 3a shows the total phonon spectrum of graphene in graphene-MoS2 heterostructure without and with tensile strain. Two main peaks are observed. One is located within high frequency range at around 53 THz, and is dominated by the in-plane phonon mode (Figure S3a), the other is located within low frequency range at about 17 THz, and is dominated by the out-of-plane phonon mode (Figure S3b). As the tensile strain increases, the high frequency mode shifts to a low one, which indicates a softened phonon transport and lattice anharmonicity.47, 48 The low frequency mode remains without an obvious shift, suggesting the dominant role of in-plane phonon mode in graphene layer. The shift of high frequency phonon model that leads to softening of phonon mode impedes phonon thermal transport and reduces thermal conductivity (Figure 2a). Different from spectrum in graphene, Figure 3b show that peaks of phonon spectrum in MoS2 layer only occur at relatively low frequency. Further decoupling analysis of phonon spectrum (Figure S3c and d) indicates that both in-plane and out-of-plane phonon spectra play a significant role on thermal transports in response to the applied tensile strain. This joint contribution is attributed to the trigonal atomistic structure of MoS2 layer. Similar to the frequency shift phenomena observed in graphene, as the tensile strain increases, the peaks at such as 12 THz, 10 THz, and 9 THz shift to lower frequencies, indicating similar softening phenomena in MoS2 layer with that of graphene on phonon modes and lattice anharmonicity, thus leading to a decrease of thermal conductivity in tension (Figure 1a and b). To investigate influence of the lattice mismatch on phonon activity, Figure 3c shows the in-plane phonon spectrum, ( ), of graphene in graphene-MoS2 heterostructure and bilayer graphene under no tensile strain. The comparison shows an obvious decrease in the peak of phonon spectrum of graphene near high frequency modes in the heterostructure. The reduction is expected to result from the enhanced phonon scattering and impedance to phonon thermal transport. As the tensile strain increases to 15% as shown in Figure 3d (to 5% as shown in Figure S4a), the peak difference of phonon spectra from graphene in heterostructure and bilayer graphene decreases, suggesting that the tensile strain mitigates the out-of-plane displacement and attenuates the effect of out-of-plane displacement on in-plane phonon activity. However, since the out-of-plane displacement does not change the mechanical confinement of atoms in out-ofplane direction, the out-of-plane phonon activity is expected not to vary and the in-plane phonon spectrum at low frequency shows almost the same in both heterostructure and bilayer graphene, 6 ACS Paragon Plus Environment
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as shown in Figure 5a-c, consistent with independent in-plane phonon activity of out-of-plane deformation in graphene.49 The high frequency phonons associated with out-of-plane deformation dominates thermal transport in graphene, and the effect of the lattice mismatch on thermal conductivity is reduced, consistent with Figure 2c. In contrast to the significant changes of phonon models for graphene in the bilayer graphene and heterostructure, Figure 3e shows that there is almost no difference between the in-plane phonon spectra of MoS2 in heterostructure and bilayer MoS2. Besides, as the tensile strain increases (Figure 3f and Figure S4b), no change is found. The similar no difference is also found in the out-of-plane phonon spectra (Figure S5d-f). The nearly same phonon activity indicates that the lattice mismatch does not affect the thermal conductivity of MoS2, indicating good agreement with the approximate constant / with the increase of the applied strain in Figure 2d. Unified Theory and Verification. From the classical lattice thermal transport theory, we know = , where , and are the specific heat, phonon group velocity and phonon mean free path, respectively. Since no obvious softening or stiffing on modes with frequency shifts is observed in phonon spectra of graphene in heterostructure in comparison to that in the bilayer graphene (Figure 3c and d, Figure S4a), and are considered not to change.50, 51 Based on Matthiessen rule on the phonon mean free path ,52 for the phonon mean free path in bilayer graphene, , we have
!"#
=
$%&
%$+
$%&'
, where ()() is the phonon-phonon scattering
length and ()* is the phonon-boundary scattering length and represents the phonon scattering at the interface.53 For the graphene in heterostructure, the mean free path can be written "#&+,&"-
=
!"#
+
.+/.012%
, where 3 4356) reflects the scattering length associated with the
lattice mismatch. When a tensile strain is applied, it reduces the out-of-plane displacement and attenuates the limitation of phonon mean free path. 3 43567) can be approximately estimated by
3 43567) = 8 (1 + :) , and it corresponds to a longer length than ()* , which agrees with the dominant role of phonon boundary scattering.54, 55 When there is no lattice mismatch, =0, 3 43567) is infinite and will not affect the phonon mean free path, and reduces to ;
