On the Generalized Thermal Conductance Characterizations of Mixed

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On the Generalized Thermal Conductance Characterizations of Mixed 1D-2D van der Waals Heterostructures and Their Implication for Pressure Sensors Yuan Gao, and Baoxing Xu ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.8b03752 • Publication Date (Web): 03 Apr 2018 Downloaded from http://pubs.acs.org on April 3, 2018

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ACS Applied Materials & Interfaces

On the Generalized Thermal Conductance Characterizations of Mixed 1D-2D van der Waals Heterostructures and Their Implication for Pressure Sensors Yuan Gao and Baoxing Xu * Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA

* Corresponding author: [email protected] Abstract: The emergence of ever-growing two-dimensional (2D) materials has made revolutionary innovations on van der Waals (vdW) heterostructural designs by integrating them with other low-dimensional materials to achieve unprecedented and/or multiple functionalities that are beyond individual components. Guided by full-scale molecular dynamics simulations, we present a mixed-dimensional heterostructure by vertically stacking 1D and 2D materials through non-covalent vdW interactions and demonstrate that the thermal conductance can be generalized into a unified model by incorporating their mechanical properties and geometric features. Simulation analyses further reveal the strong dependence of thermal conductance on the location and magnitude of an external pressure loading applied to the local vdW heterojunctions. The underlying thermal transport mechanism is uncovered through the elucidation of the mechanical deformation, curvature morphology and density of atomic interactions

at

the

heterojunctions.

A

proof-of-conceptual

design

of

such

a

heterostructure-enabled pressure sensor is explored by utilizing the unique response of thermal transport to mechanical deformation at heterojunctions. These designs and models are expected to broaden the applications and functionalities of mixed-dimensional heterostructures and will also offer an alternative strategy to leverage thermal transport mechanisms in the design of high-performance vdW heterostructure-enabled sensors.

Keywords ： Mixed-dimensional heterostructure, Thermal conductance, Mechanical deformation, Pressure sensor 1

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1. Introduction Heterostructures composed of distinct layered nanomaterials through non-bonded van der Waals interactions, often referred to as van der Waals (vdW) heterostructures, have been largely inspired by the successful isolation of graphene, and are considered to open a new era for exploring fundamental novel properties that yield design of devices with new functionalities beyond the individual layer components1-4. As a consequence, various heterostructures have been designed and synthesized by stacking two-dimensional (2D) materials in a well-defined order to achieve on-demand properties, for example, a low composite Young’s modulus in MoS2-WS2 heterostructures,5 gate-tunable diode performance in MoS2-WSe2 heterostructures6-7 and black phosphorus-MoS2 heterostructures8, controlled transistor behavior in graphene-hBN-MoS2 heterostructures9, and strain controllable bandgap10 and thermal properties11 in various bilayer heterostructures. In theory, the non-bonded vdW interactions are not limited to 2D layered stacking heterostructures, and any dimensional materials that possess a dangling-bond-free surface could interact with another one via vdW forces and form vdW heterostructures, in particular, vdW heterostructures with combinations of different low-dimensional nanomaterials.12-15

In comparison with 2D layered heterostructures, mixed-dimensional heterostructures such as 2D-1D and 2D-0D heterostructures introduce a local interfacial disorder at heterojunctions, benefitting the exploration of spatial-dependent properties and large-area, multipixel functional devices4, 16-19. In addition, the break of the dimensional barrier will allow inheriting properties from both 1D and 2D materials and introduces unprecedented properties. For example, 1D-2D heterostructures have exhibited a low power consumption because of their small functional “contact” region.17 A remarkable tunability in the rectification ratio, as high as five orders of magnitude, has been achieved in 1D carbon nanotubes (CNTs)-2D MoS2 p-n diode.20 Combining 2D graphene with 1D CNTs to construct a 1D-2D heterostructure has been utilized to enhance the light absorption in photodetector.21 A similar heterostructure that is built by sandwiching 2D MoS2 with a vertical arrangement of two 1D CNTs and forms a point contact between them has also been used in a field-effect transistor and it exhibits a superior on/off ratio and sensitive light detecting performance.17 2


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Among the unique properties of mixed 1D-2D heterostructures, thermal transport properties, which play an important role in thermal management of various electronic and thermal devices22-23, are very critical for maintaining optimal functionality of these devices. Unlike the thermal transport across 2D layered stacking heterostructures, the thermal transport in 1D-2D vdW heterostructures strongly relies on the density of interatomic interactions because of the small area of heterojunctions,24 and it is also different from the phonon resonance mechanism in co-planar heterostructures with a line-like interface.25 Besides, the small local junction areas will enable the deterministic arrangements in vdW 1D-2D heterostructures such as the construction of point contact by crossing alignments of 1D materials in vdW 1D-2D-1D heterostructures17, which will facilitate the design of functional heterostructures with high-density multipixel capability such as enhanced sensing performance and local accuracy via sensor array.

In the present study, we construct a 1D-2D heterostructures and investigate their thermal properties using full-scale reverse non-equilibrium molecular dynamics simulations (RNEMD). RNEMD simulations have been widely employed to study thermal properties of low-dimensional materials such as CNTs and graphene whose thermal transports are dominated by phonon mechanism.26-28 We find that the thermal conductance across the mixed-dimensional heterojunction of the 1D-2D heterostructures can be well predicted through a unified theoretical model. This model has taken into account the mechanical, material and geometric features of both 1D and 2D materials and layer numbers of 2D materials and is validated by considering several typical 2D materials including graphene, hexagonal boron nitride (hBN), black phosphorus (BP) and molybdenum disulfide (MoS2). The effect of an external pressure applied to the heterojunction on thermal transport is further studied, and the results show that the sensitivity of thermal conductance is independent of the local pressure. The underlying thermal transport is uncovered from the understanding of both mechanical deformation and the density of atomic interaction at the heterojunctions. A pressure sensor enabled by such 1D-2D heterostructures is designed and demonstrates a new strategy of pressure sensing by leveraging the unified thermal transport mechanism at the local heterojunctions. 3



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2. Computational Modeling and Methodology 2.1. Computational Modeling and Method. Figure 1a depicts the computational modeling of the 1D-2D heterostructure with 2D materials -graphene sandwiched by two 1D carbon

nanotubes (CNTs). The dimension of 2D layer is taken as =14.51 nm in length and

=14.77 nm in width while the number of inserted layer can be variable; the length of both CNTs are the same and taken as =19.70 nm, whereas their diameters will vary from

0.678 to 3.254 nm. To generate the temperature gradient across the 2D layers, the hot and cold reservoirs are set up at two CNTs, guiding the heat flow across top and down heterojunctions between CNT and graphene in the z-direction. All MD simulations were performed with LAMMPS.29 In the computational modeling of 1D-2D heterostructures, the carbon interactions in CNTs and graphene layers were modelled by AIREBO potential30. The atomic interaction in boron nitride, black phosphorus and molybdenum disulfide were described by Tersoff,31 Stillinger-Weber32 and REBO33 potentials, respectively. The non-bonded van der Waals interactions were modeled by Lennard-Jones (LJ) potential34-35 . The cutoff distance in LJ potential was taken as =1 nm and is widely used in the studies of thermal transport.24,

36-37

A non-periodic boundary

condition was applied in all directions. The boundary atoms in x-direction of the lower CNT were fixed in all directions to prevent its random motion, and the boundary atoms in x-direction of 2D layers and the upper CNT were fixed in x and y-direction but free in z-direction to reproduce the natural and stable geometry of the van der Waals junctions. These boundary settings also help reach stable heterojunctions during simulations and no drifting or shifting is observed, as shown in Figure S1. The time step was set as 0.5 fs. 2.2. Calculation of thermal conductance. The heterostructures were first relaxed in the canonical (NVT) ensemble for 1 ns with a system temperature of 300 K controlled by Nosé-Hoover thermostat. To measure the thermal conductance, the atoms within 3 nm from the end of upper and lower CNTs were selected as hot and cold reservoirs, respectively. In micro-canonical (NVE) ensemble, at each time step, a constant amount of kinetic energy was 4


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added to/subtracted from the atoms in hot/cold reservoirs to introduce a constant heat flow of 10-60 nW for different systems from top to bottom of the heterostructures in z-direction. After 0.5 ns, the temperature of each part of the system reached a stable state and became time independent (Figure S2). As a consequence, a temperature difference of ~100 K between hot end and cold end were created (Figure S3). This temperature difference is considered to minimize both non-linear effect and statistical uncertainties,38 and has been applied to investigate thermal transport across different junctions in computations.39-40 Afterwards, the temperature data in next 2 ns were taken to calculate the thermal conductance via =

,

where is the heat flow, and and are the average temperature of

the upper and lower CNTs, respectively. 2.3. Applying an external pressure on heterostructures. To introduce external pressure to the local heterojunction, an external yet equal force in the negative z-direction was applied to each atom in the region of upper CNT, marked in the schematics. The projection of the region in x-y plane is in a square shape with the side length equal to the diameter of the upper CNT . Once the heterostructures arrive at the new equilibrium subjected to the local pressure,

the temperature difference will be created through the upper and lower CNTs to study the effect of pressure on thermal conductance.

3. Results and Discussion 3.1. Generalized Model of Thermal Conductance. Due to the curvature difference of 1D CNTs and 2D graphene and orientation setup of these two CNTs, a slight local deformation in the 2D graphene at the heterojunction is observed at the equilibrium. Generally, this local mechanical deformation depends on the bending stiffness of CNTs and 2D materials.41 A smaller bending stiffness of 2D layers or a large bending stiffness of CNTs will lead to a stronger atomic interaction in the heterojunction and thus a larger deformation in 2D materials. Figure 1b shows the thermal conductance of the heterostructures as a function

of the relative bending stiffness / , where and are the bending stiffness

of CNTs and 2D layers, respectively. A higher / will lead to a larger deformation 5



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of 2D layer at heterojunctions, and results in a higher . This enhanced holds true for

various materials including monolayer graphene, hBN, BP, MoS2 and multilayer graphene, suggesting that the out-of-plane local deformation promote the thermal transport across the heterojunctions associated with vdW interaction. We should note that the bending stiffness of

multilayer graphene cannot be calculated by multiplying the layer number with the

bending stiffness of monolayer due to the sliding between each layer, and is obtained separately from the square-power law confirmed in experiments.42 Besides, the correlation

between and / for all monolayer 2D materials follow the same linear behavior

in the log-log coordinate system, but will change when multiple layer 2D materials are employed.

In essence, the thermal transport based on the vdW interactions in heterostructures with multiple layers will rely on the number of pairwise atomic interactions near the heterojunctions.22, 40, 43-44 Therefore, in addition to the mechanical deformation, the thermal conductance of heterostructures will also change with the lattice structures of 2D materials, layer number of 2D materials and the spatial distance between CNT and 2D layers. To consider all these factors, we here define a dimensionless coefficient , =

!"#

%$∙(

'

%$∙∙ )

* , where is the diameter of CNTs and + is the lattice constant of 2D layers.45-47

/+ reflects the atomistic alignment between CNTs and 2D materials, and a larger /+

will lead to a stronger atomic interaction. is the effective distance of atomic interactions

between the CNT and the center of the 2D layers and can be estimated as = 0.5(0 + 2), where 0 and 2 are the vdW thickness of CNTs

48

and 2D layers,45, 49-51 respectively. *

is the layer number of 2D materials, and the 2nd order reflects powered atomistic interactions in both upper and lower junctions. Tables S1 and S2 in the supporting material summarize these parameters from literatures. Figure 1c presents the thermal conductance of

heterostructures for all employed monolayer 2D materials (i.e. graphene, hBN, BP, MoS2)

and multiple layered graphene. The variation of with for all these materials and layers

can be featured as a unified linear function in the log-log coordination system. As references,

Figure S4 shows the variation of with the diameter of CNTs for monolayer 2D 6


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materials and multiple layers of graphene, and confirms the size dependence of CNTs on thermal transport across vdW heterojunctions in CNTs and 2D layers. The unified thermal transport behavior of 1D-2D heterostructures indicates a one-to-one correspondence between and .

3.2. Gaussian Curvature in 2D Layers. The out-of-plane mechanical deformation of 2D materials at the heterojunctions mainly results from the curvature difference of 1D CNT and 2D materials and also is associated with small bending stiffness of 2D materials in comparison with CNTs. This local mechanical deformation will allow conformal atomic interactions between CNTs and 2D materials. As a consequence, the effective “contact” interactive region between CNTs and 2D materials increases, leading to stronger atomic interactions and thus enhancing the thermal conductance. Mechanically, this local deformation and geometric feature near the heterojunctions can be indicated by the non-zero absolute value of Gaussian curvature in 2D layers. For heterostructure systems with different CNT diameters, 2D layer materials and 2D layer numbers, Figure 2 shows comparisons of their out-of-plane deformation ( ) and the absolute value of spatial Gaussian curvature

distribution of 2D layers (|g|). For the monolayer 2D materials in heterostructures, the lager CNT diameter or lower bending stiffness of 2D materials leads to a larger absolute Gaussian curvature, being consistent with a higher thermal conductance in Figure 1b. By contrast, for multiple layered 2D materials in heterostructures, a larger layer number of 2D materials results in smaller mechanical deformation because of larger overall bending stiffness. The local maximum |6| at the heterojunction decreases to approximately zero as the number of

layer increases to five but leads to larger than that expected in Figure 1b because of the

absence of material and geometric features in / . For the layer number of 2D materials (*>1), the out-of-plane deformation and Gaussian curvature |6| in profiles were

obtained by averaging the out-of-plane deformation in each layer at each location in the x-y-plane. The calculations on 2D material BP, effects of CNT diameter and layer numbers have further confirmed these results, as given in Figures S5, S6 and S7. The higher Gaussian

curvature |6| at both edges and heterojunction enhances the atomic interactions and 7



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promotes the thermal transport, which agrees with an enhanced thermal conductance in Figure 1b and Figure S4. 3.3. Density of Atomic Interactions. The thermal transport across the vdW heterojunction is essentially contributed by each atom pair via vdW atomic interactions, and the theory of atomic interaction density can be employed to probe the underlying mechanism. The

effective pairwise contribution 7 of atom pairs, which comprise of the 8 th atom in the upper

CNT and the jth atom in the 2D layer at upper interface and the 9 th atom in the 2D layer and the th atom in the lower CNT at lower interface, separated by the distance of :; and D /EF ? − > D /EF ? , D ≤ EF ≤

EF >

0, G

7>:; ? =

)

M < D 1, and 7(

0, O

distance that corresponds to the minimum of the potential and is equal to 2P 0, σ is the

distance parameter in L-J potential and is the cut-off distance beyond which the pairwise contribution can be neglected and is set as 1 nm in all calculations.24, 36 When the distance

between an atom pair is smaller than that corresponding to the minimum of potential, the interaction of the atom pair is considered to be strong enough and the contribution n is equal to 1. On the other hand, the interaction is considered to be negligible when is larger than the cut-off distance , and the contribution n is hence 0. When falls in between the

cut-off distance and the distance at the minimum potential, the contribution can be scaled by using the LJ potential that was employed to describe the non-bonded vdW interactions among atoms in the modeling. Accordingly, the effective contribution of cumulated pairwise

interactions at upper and lower interface can be obtained via *RSS_ = ∑E ∑F 7(EF ) and

*RSS_V = ∑M ∑ 7(M ), respectively, which represent the effective number of atom pairs formed by atoms in both CNTs and 2D layers.24 Figure 3b shows *RSS_ and *RSS_V as

functions of CNT diameter for different 2D monolayers. We should note that both *RSS_ 8


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and *RSS_V are obtained from the deformed heterojunctions and thus can capture all features of mechanical, material and geometric behaviors. Both *RSS_ and *RSS_V at

two interfaces increase with the increase of the diameter of CNTs, which indicates that larger

CNTs introduce larger deformation at heterojunctions and more atom interactions, corresponding with the findings in mechanical deformation in Figure 2a and b. Besides, given the same diameter of CNTs, 2D monolayer materials with a lower bending stiffness in the heterostructure show a higher *RSS_ and *RSS_V , indicating that a smaller bending

stiffness of 2D layers will lead to a stronger interaction, which also agrees well with a larger absolute Gaussian curvature in Figure 2a, c and d. To incorporate *RSS_ and *RSS_V with the thermal conductance of heterostructures,

we further define a contribution factor via W = *RSS_ *RSS_V , which reflects the total

number of atomic routes for heat transfer across these top and bottom heterojunctions. For heterostructures consisting of multilayer 2D materials, our calculations (Figure S8) show the

temperature gradient between individual 2D layer is negligible, and the calculation of W can

be obtained by considering the total thickness of 2D layer. Figure 3c shows a relationship

between W and and a linear variation is observed in the log-log plot, independent of 2D materials and layer numbers. Figure 3d further shows the variation of with W in a log-log

plot, and a linear relationship is also observed. Both one-to-one correspondences of W with and with W suggest that should reflect the atomic interactions at the heterojunctions

and can be used to determine the thermal conductance, which is consistent with Figure 1c. By contrast, the phonon resonance theory, which is usually employed in the study of thermal transport across bonded interfaces is also calculated and given in Figure S9. No obvious difference is observed in both phonon density state and overlap coefficient, which indicates that the thermal transport across vdW heterojunctions in layered stacking heterostructures cannot be accurately captured. In comparison with bonded atomic interaction, phonon frequencies of the intralayer phonon modes in vdW heterostructures are usually an order of magnitude higher than those of interlayer modes, suppressing the contribution of intralayer

phonon modes to thermal conductance across vdW heterojunction52-54. Instead, the 9



contribution factor W that represents the density of atomic interactions can be utilized for

understanding the underlying thermal transport mechanism in vdW heterostructures.

3.4. 1D-2D Heterostructures for Pressure Sensors. Given the thermal conductance of 1D-2D heterostructure is closely associated with density of atomic interactions at vdW heterojunctions, the heterojunctions are expected to be highly sensitive to an external loading such as pressure that could lead to mechanical deformation of 2D materials and density of atomic interactions, holding great potential for designing a pressure sensor. Figure 4a

illustrates the 1D CNTs-2D graphene heterostructure subjected to a pressure X on the

heterojunction. We should mention that, although the pressure is on the order of GPa because of the nanosized contact area (~1 nm2) at heterojunctions between CNTs and 2D materials,

the applied force was on the order of ~1 nN, which is comparable to experiments.55-56 As X

increases, an approximately linear monotonous increase of thermal conductance is obtained, as shown in Figure S10. Taking the slope of a -X curve as the sensitivity of the

thermal transport of the heterostructure to an external pressure, referred to as Y, Figure 4b shows that Y is independent of the layer numbers of graphene sandwiched between two

CNTs. Mechanically, an external pressure will decrease the equilibrium distance between CNT and 2D layers, resulting in a stronger atomic interaction and larger W, and thus an

enhanced . Therefore, the minimum force/pressure detected by such designed sensor

depends on the capability that can alter the density of atomic interactions at heterojunctions and is associated with the equilibrium distance between CNT and 2D layers. Besides, the weaker vdW interaction and/or small thickness of 2D materials, the easier deformation that can be tuned, and the higher sensitivity can be obtained. Figure 4c shows an obvious

enhancement of the relative overall contribution factor (W − WZ )/WZ at a larger X, where WZ

corresponds to X=0. This linear monotonous increase of (W − WZ )/WZ also agrees with that of ( − Z )/Z in Figure S10. In addition, when an external pressure is applied to the heterojunction, the resulting distribution of stress change in the loading direction (z-direction)

can be monitored and is given in Figure S11. By comparing with that in the absence of the

pressure loading, Figure 4d demonstrates a clear increase in the monolayer graphene at X=2

GPa. More importantly, the position of the pressure loading can be accurately located, 10


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suggesting that both the magnitude and location of the applied pressure could be determined through the measured thermal properties of 1D-2D heterostructures. With the same mechanism, to further demonstrate the application of the thermal properties of 1D-2D heterostructures in pressure sensing, a pressure sensor consisting of an array of heterojunctions though multiple CNTs can be designed for pressure mapping. Figure 5a shows the illustrative schematics of the pressure sensor. A single layer graphene with size of 29.27 nm in length and 31.78 nm in width is sandwiched between three pairs of CNTs with same length of 36.03 nm to construct an array of nine heterojunctions, and the diameters of

these three CNTs are ) =1.085 nm, =1.356 nm and [ =1.898 nm, respectively. A

pressure with the same magnitude X) =2 GPa is applied to three different heterojunctions via

the top CNTs, as illustrated in Figure 5a. Similar to that Figure 4d, the resultant stress variation in graphene in comparison without the external pressure (Figure S12) can be

obtained and is shown in Figure 5b. Although the heterojunction area associated with different diameters of CNTs, the stress distribution in these three positions where the external applied is applied shows a good similarity, in particular, the maximum stress. As references, the pressure change at each heterojunctions is the same as shown in Figure S13. To reflect the thermal conductance through each heterojunction, we define a local contribution factor at each location in the 2D layer W ( = *RSS__E *RSS_V_E , where 8 represents the ith atom in

the 2D layer. The distribution plots of *RSS__E , *RSS_V_E and W ( in the graphene with

and without the loadings is given in Figure S14-S15. Further, define the relative local contribution factor by considering the geometric size of upper CNTs and 2D layer, W\ =

]^ ]^__ ]^__

∙ ' $` , where W (_Z represents the local contribution factor in the absence (

of external applied loadings. Figure 5c shows the summation of variation of thermal

conductance ∑ W \ at each heterojunction. A clear difference at the regions with and without

external pressure can be identified. More importantly, these three regions with the external

pressure show the same magnitude of variation of thermal conductance, being consistent with the same magnitude of applied pressure (X) =2GPa), demonstrating the success of sensing the

external pressure through the measurement of thermal conductance at local heterojunctions. In addition, when this pressure X) is applied to different locations, it can also be mapped 11



accurately through the thermal conductance at corresponding locations, as shown in Figure S16.

To further examine the functionality and sensitivity of the pressure sensor, three different

pressures P =1.5 GPa, P[ =1 GPa and Pb = 0.67 GPa but in the same locations with that X)

are applied, as illustrated in Figure 5d. Figure 5e shows the stress change in graphene by

comparing the stress distribution with and without external pressures (Figure S12a and c). A lower external pressure results in a lower stress increase in corresponding loading locations.

Besides, Figure 5f shows that the summation of variation of thermal conductance ∑ W \ in

these three regions are different. A lower pressure corresponds to a lower ∑ W \ , which is also

consistent with the magnitude of applied pressures. Therefore, the 1D-2D heterostructures array can be used to identify both magnitude and location of the applied pressures by extracting the local thermal properties and successfully demonstrates the performance of

pressure sensing. We should note that the small increase of stress in Figure 5b and e and local contribution factor in Figure 5c and f in the heterojunction without external pressures is led by the overall displacement of CNTs and can be neglected in comparison with those in heterojunctions with pressure applied.

4. Conclusions We have constructed a mixed-dimensional van der Waals (vdW) heterostructure by sandwiching 2D layered nanomaterials into the 1D nanomaterials and systematically investigated their thermal conductance using non-equilibrium molecular dynamics simulations and theoretical analyses. The results show that the thermal conductance will not only depend on local mechanical deformation of 2D materials at the heterojunction associated with bending stiffness, but will also rely on spatial equilibrium distance at heterojunction, and lattice structures and layer number of 2D materials, and can be described through a unified model by integrating both contributions. Both mechanical deformation of 2D materials and the density of atomic interaction at the heterojunctions are studied to reveal the underlying unique thermal transport mechanism and are in good agreement with the generalized unique 12


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model. Besides, when an external pressure is applied to the heterojunctions, the thermal conductance shows a monotonous variation and the variation sensitivity is independent of layer numbers. By utilizing the one-to-one correspondence between thermal conductance of heterostructures and the local pressure applied at heterojunctions, we put forward a conceptual design of a pressure sensor enabled by 1D-2D heterostructure and successfully demonstrate its capability of sensing to external pressures with high accuracy. These designs and findings are expected to lay a foundation for understanding thermal transport, mechanical deformation and their couplings in mixed-dimensional vertical heterostructures, and they are also expected to provide an immediate application guidance for designing mechanical sensor such as pressure sensor by using mix-dimensional heterostructures underpinned by thermal transport mechanisms.

Associated Content Supporting Information Thermal conductance of heterostructures as functions of CNT diameters and external pressure. Out-of-plane profiles with Gaussian curvature distribution. Temperature distribution in the heterostructures with multilayer graphene. Phonon spectra and phonon resonance analyses. Stress distribution in 2D layer in the heterostructures with external pressure loadings. Stress distribution, effective contribution factor distribution and local contribution factor distribution of sensor arrays with external pressure loadings. Demonstration of sensor array with another loading condition.

Author Information Corresponding Author *E-mail: [email protected]

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Figure 1. Computational model of 1D-2D van der Waals (vdW) heterostructures and their thermal conductance. (a) Molecular modeling of the 1D-2D heterostructure with a monolayer 2D materials (left: perspective view; right: side view). The 2D layer is sandwiched between the same two orthotropic 1D carbon nanotubes (CNTs), forming a 1D-2D heterostructure through non-covalent vdW interactions. Hot and cold reservoirs are assigned to the two ends of upper and lower nanotubes, generating a heat flow across the heterojunctions. (b) Thermal conductance of heterostructures as a function of relative bending stiffness between 1D and 2D materials / for monolayer and multiple layer 2D materials. G: graphene, hBN: hexagonal boron nitride, BP: black phosphorus (BP), and MoS2: molybdenum disulfide. (c) Thermal conductance of heterostructures as a function of the proposed dimensionless coefficient (=

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Figure 2. Out-of-plane mechanical deformation (do) and Gaussian curvature (g) distribution in 2D layers in heterostructures. (a) Monolayer graphene (G) with CNTs of 1.085 nm in diameter. (b) Monolayer graphene (G) with CNTs of 2.170 nm in diameter. (c) Monolayer boron nitride (BN) with CNTs of 1.085 nm in diameter. (d) Monolayer molybdenum disulfide (MoS2) with CNTs of 1.085 nm in diameter. (e) Trilayer graphene (G) with CNTs of 2.170 nm in diameter. (f) Pentalayer graphene (G) with CNTs of 2.170 nm in diameter.

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Figure 3. Density of atomic interactions in heterojunctions. (a) Schematic illustration of atom pairs at two heterojunctions in 1D-2D heterostructure. 8, c, 9 and illustrate the atomic positions in the upper CNT, 2D layer, and lower CNT at the heterojunctions,

respectively and form two atom pairs (i, j) and (k, l), and :; and

On the Generalized Thermal Conductance Characterizations of Mixed

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