Controllable Interface Junction, In-Plane Heterostructures Capable of

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Controlled Interface Junction, In-Plane Heterostructures Capable of Mechanically Mediated On-Demand Asymmetry of Thermal Transports Yuan Gao, and Baoxing Xu ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.7b11508 • Publication Date (Web): 12 Sep 2017 Downloaded from http://pubs.acs.org on September 12, 2017

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ACS Applied Materials & Interfaces

Controlled Interface Junction, In-Plane Heterostructures Capable of Mechanically Mediated On-Demand Asymmetry of Thermal Transports Yuan Gao and Baoxing Xu * Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904, USA

* Corresponding author: [email protected] ABSTRACT: Designing structures with thermal rectification performance that can be regulated by or adapted to mechanical deformation is in great demand in wearable electronics. Herein, using non-equilibrium molecular dynamics simulation, we present an in-plane graphene-boron nitride heterostructure with a controlled interface junction and demonstrate that its thermal transport ability is asymmetric when reversing the direction of heat flow. Such thermal rectification performance can be further regulated by applying an external tensile loading due to the mitigation of stress concentration, phonon resonance and phonon localization. The analyses on heat flow distribution, vibrational spectra and phonon participation suggest that the out-of-plane phonon modes dominate thermal rectification at a small tensile strain, while the mechanical stress plays a dominant role in regulation at a large tensile strain due to the weakened localization of out-of-plane phonon modes. The effect of tensile loading on the thermal rectification is demonstrated by selective interface junction-enabled heterostructures, and the results indicate that both asymmetry and direction of thermal transport can be controlled by introducing defects to the interface junction and/or applying mechanical tensile strain. These findings and models are expected to provide an immediate guidance for designing and manufacturing 2D material-based devices with mechanically tunable thermal management capabilities. KEYWORDS: thermal rectification, imperfect junction, stress concentration, heterostructure, phonon resonance 1

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1. INTRODUCTION In-plane heterostructures that comprise two different atomic monolayers with a coplanar heterojunction have exhibited many unique electronical,1 magnetic,2-4 thermal5-7 and mechanical properties8-9, beyond either of the individual component monolayer structures. Most of these novel properties are strongly dependent on heterojunctions, and the interface junctions are considered the essential element for the creation of next-generation, high-performance wearable devices with unprecedented functionalities. For example, the mechanical stress in polycrystalline graphene could be manipulated by tilting the junctions of grain boundaries to match the requirement for biological and electronic applications.10 Creating such selective boundary junctions in hexagonal-boron nitride would also introduce net charges and lead to a significant reduction in bandgap.11 When a temperature loading is applied to the coplanar heterostructures and reverses its loading direction, the magnitude of heat flow shows a notable difference, analogous to the current flow across a p-n junction electronic or optical-electronic diode.12-15 This thermal rectification performance also strongly relies on the in-plane heterojunctions,5, 16-18 and varies with both inherent atomic structures19-20 and external fields, such as mechanical loadings.21-23 For instance, the topological defects at the interface have been employed to improve the thermal transport across the interface in the graphene-boron nitride heterostructure by reducing phonon scattering compared to perfect bonding interfaces.6 A significant enhancement of thermal conductance in graphene-boron nitride heterostructures can also be achieved by applying a tensile strain, which improves the alignment of out-of-plane mode (ZA) phonon bands in both graphene and boron nitride.24 Further simulations show that the effect of tensile strain on thermal transport depends on the size of graphene-boron nitride heterostructures owing to the competition between in-plane phonons softening and flexural phonons stiffening.25 In addition, the optimization on graphene-boron nitride in-plane heterostructures shows that a negative differential thermal resistance can be realized by introducing a special edge geometry

26-27

effect and lattice vibration mismatch.5,

and is believed to be caused by phonon resonance 28

These studies have suggested great potential

applications of in-plane heterostructures such as thermal rectifiers,29-31 thermal transistors,32 2


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thermal diode circuits,33 devices in thermal memory26 and thermal signal processing.34 In these applications, thermal rectification performance will be key, in particular, its variation with the imperfection density at the interface, which is currently unclear.35-38 More importantly, motivated by both fundamental considerations in thermal engineering and the ever-growing demand in applications in areas of electronics and controllable thermal managements, design of deterministic structures whose thermal rectification performance can be dynamically controlled, in particular under an external field such as mechanical loading, needs to be achieved. In the present work, by controlling both of the topological imperfection density at the interface junctions in graphene/boron nitride (GBN) in-plane heterostructures and the external tensile loading, we demonstrate their regulatory roles in tuning the thermal rectification performance using non-equilibrium molecular dynamics (NEMD) simulation. The thermal rectification in the heterostructures is caused by phonon resonance and phonon localization. More importantly, further simulations show that the thermal rectification can be reduced by applying an external tensile strain. The competition and coordination mechanism of stress concentration and phonon resonance for the GBN in-plane heterostructures with topological junctions under a uniaxial tensile strain are elucidated to reveal the mechanism of thermal rectification. As for the demonstration of potential applications, we present two novel conceptual designs of the non-uniform imperfect junction-enabled GBN heterostructures and show that the on-demand path of heat transport can be well controlled by introducing imperfection interface and tensile strain. These models and the application demonstration are expected to provide an immediate guidance on the design of flexible and stretchable electric and thermal devices with controllable thermal management capabilities. 2. COMPUTATIONAL MODELING AND METHODOLOGY 2.1 Computational Modeling and Method. Fig. 1a presents the atomic model of GBN in-plane heterostructure with a size of 14.9 nm×6.0 nm. The interface is located in the middle and has a width of 0.5nm. SW-5577 defects that are formed by rotating two bonded carbon atoms 90 degrees about the midpoint of their pairwise bond,39 and often used in structures with interfaces,40-41 are selectively employed to achieve controlled interface junctions in GBN 3



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in-plane heterostructures, and the ratio of the number of implemented defect units to the maximum allowable number of defect units at the interface is defined as the interface junction parameter . The atomic structures of interface junctions with =0, 1/7, 3/7,

5/7 and 1 are highlighted in Fig. 1a and the GBN in-plane heterostructures with these

interface junctions will be investigated as representatives in this work. All molecular dynamics simulations were performed with LAMMPS.42 Non-periodic boundary condition was applied in all directions. The boundary atoms at both ends in the x-direction were fixed. The time step was set as 0.5 fs. The atomistic interactions were described by Tersoff force field.43 We should note that the Tersoff potential has been well-acknowledged in the studies of thermal transport in graphene-boron nitride systems6, 44 and proves to accurately reproduce the tensile properties of graphene and boron nitride,45-46 which are critical in our current study. The structure was first relaxed in canonical ensemble (NVT ensemble) with Nose-Hoover thermostat at 300 K for 1 ns. Next, a uniaxial quasistatic loading under the strain rate of 0.5 ns was applied to heterostructures by uniformly projecting the coordinates of atoms in the x-direction every 1000 time steps. The nominal tensile strain was calculated via =

, where and are the stretched and initial length

of the structure, respectively, and =14.9 nm. The nominal tensile stress was calculated via

= , where is the reactive force of the fixed boundary atoms, and = is the

cross-sectional area. =6.0 nm is the width of the structure. is the thickness of the GBN heterostructure and is taken as 0.335 nm.47 Afterward, to measure the heat transport of the GBN heterostructures, the atoms within 1.5 nm of both ends were selected as heat baths. In simulations, the two heat baths were maintained at 390 K and 210 K with the temperature difference of 180 K between them by the Nose-Hoover thermostat for 8.0 ns to reach a steady-state of thermal transport. We should note that a temperature difference of ~200 K is usually employed in numerical simulations to highlight the thermal rectification performance.48-49 In another 8.0 ns, the heat flow was extracted from the slope of the linear regression curve by fitting the cumulative energy change in the heat baths with simulation time (Fig. S1). 4


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2.2 Calculations of Vibrational Spectra. The vibrational spectra were calculated through +

&'% ∙' )

= ! " #$% &'

∙' )

* , where is angular frequency and , is atomic velocity

vector. The symbol ⋅ stands for dot product and the symbol < > denotes the average over atoms in specific groups. The atoms in both graphene and boron nitride domains, which are more than 1.6 nm (>1.5nm, the width of baths) away from the ends, were taken so as to avoid the effect of the boundary constraints and thermal reservoirs. The calculation of out-of-plane phonon spectra only involves the z component of velocity vectors, and the in-plane phonon spectra only takes into account the velocity of atoms in the x and y-directions. The overlap between spectra reflects the capability of phonon transport across the interface,6, 13 and can be determined through the mode matching theory

.=

4

4

! /0 $ /12 $ 3$ 4

! /0 $ 3$ ! /12 $ 3$

,50 where /

and 5 are the spectra of graphene and boron nitride domain, respectively. Furthermore, the out-of-plane and in-plane spectra overlaps can be obtained to uncover the roles of in-plane and out-of-plane phonons in thermal rectification, and are defined as .6 = 4

! /07 $ /127 $ 3$ 4 4 ! /07 $ 3$ ! /127 $ 3$

and .# =

4

! /08 $ /128 $ 3$ , 4 4 ! /08 $ 3$ ! /128 $ 3$

respectively, where /6 and

/# are the out-of-plane and in-plane phonon spectra of graphene, respectively, and 56 and 5# are the out-of-plane and in-plane phonon spectra of boron nitride, respectively.

2.3 Calculations of Atomic Heat Transfer Vector. The atomic heat flux vector is defined as

9: = ; "# ,# − # ,# , where =, ", , and are the atomic volume, energy, velocity vector

and stress tensor, respectively, and the subscript > refers to the ith atom. We should note that the stress tensor of an atom is calculated based on the atomic interaction and has the dimension of stress×volume with an energy unit. The heat flux vectors were obtained by

averaging data for 8.0 ns after a steady state of thermal transport was reached.

3. RESULTS AND DISCUSSION 3.1. Thermal rectification performance in GBN Heterostructure. When the heat flows from graphene to boron nitride region, Fig. 1b shows that the heat flow decreases at first at

a small , and then increases as further increases. In contrast, shows an opposite

variation with when the direction of heat flow reverses. The simulations on the individual 5



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graphene and boron nitride nanoribbons, which have the same dimensions as when they are in the heterostructures, are also performed here as references (Fig. S2). The results suggest that both of their intrinsic thermal conductivities show an overall decrease with the increase of tensile strain. Therefore, the nonlinear variations in GBN heterostructures in the presence of interfaces are expected to be caused by the competition between phonon softening, phonon resonance and stress concentration at the interface, which will be illuminated in Section 3.3. Additionally, with the same temperature difference between hot and cold reservoirs, the magnitude of heat flow from boron nitride to graphene domain is higher than that in the opposite direction, indicating a clear thermal rectification performance in GBN heterostructures, analogous to the current flow in an electric diode. More importantly, with the increase of tensile strain , the difference between in these two cases becomes smaller, suggesting the regulation on thermal rectification by an external mechanical strain. As an alternative proof, given the same magnitude of temperature difference between heat baths, a linear temperature profile is obtained in the graphene and boron nitride domains, but a clear temperature drop exists at the interface when heat transports in both directions, as shown in Fig. S3. A higher temperature drop across the interface is observed when heat transports from boron nitride to graphene domain, which further confirms the asymmetry of thermal transport in the GBN heterostructures. Note that the temperature drop near thermostats is caused by the local

thermal

conductivity

and

the

scattering.51-53

phonon-boundary

Since

the

phonon-boundary scattering is independent of temperature while the local thermal conductivity decreases with the increase of temperature,54 a higher temperature drop is obtained near the hot thermostat than near the cold thermostat. Besides, as increases, more defects will intensify the impediment to heat transport across the interface, leading to a decrease in , as shown in Fig. 1c. In particular, a clear decrease is observed when the heat flows from boron nitride to graphene domain. We further define the thermal rectification ratio via @ =

A12→0 A0→12 A0→12

to quantitatively

characterize the thermal rectification performance, where /→5 and 5→/ are the heat flow in the direction from graphene to boron nitride region and from boron nitride to graphene region, respectively. Fig. 1d shows that @ increases approximately by 100% at 6


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first before reaching a peak at = 5%, and decreases with a further increase of ,

weakening the asymmetry of thermal transport. Besides, at a small (7.5%), a smaller leads to a smaller @. The competing

effects of and on @ are further confirmed in Fig. 1e, and are expected to result from

the mechanisms of stress, phonon resonance and phonon localization near the interfaces. In addition, it is emphasized that a larger temperature difference between heat bathes or a smaller structure usually yields a higher thermal rectification ratio @. Nevertheless, the thermal rectification ratio @ will not be zero as long as a temperature difference exists.55 Fig.

S4 shows the effect of the temperature difference between heat bathes on the thermal D

rectification of the heterostructure with an imperfect junction ( = E ), and the thermal rectification still remains when the temperature difference is as low as 30 K, which is commonly employed in experiments.19, 56 3.2. Mechanical Stress and Thermal Transport Analyses. Fig. 2a presents the snapshots of von Mises stress distribution near the interfaces in the heterostructures. The von Mises stress

is

asσGH = IJ Kσ

− σJJ

J

+ KσJJ − σDD

J

+ KσDD − σ

defined J

J + 6σJJD + σJD + σ J ], where

σ represents stress, and subscripts 1, 2, 3 are coordinate directions (i.e. x, y and z accordingly). For the heterostructure with a perfect interface junction (=0), the stress gradient across the interface is very small due to the similar lattice structures of graphene and boron nitride. A clear stress concentration is observed when defects are introduced to the interfaces, and is more severe in the graphene domain because of its higher in-plane stiffness57. As the applied strain increases, the stress concentration becomes weaker. To quantitatively describe the contribution of interfacial stress, the stress concentration factor, O% , is extracted by taking the ratio of the averaged stress of carbon and boron nitride atoms at the interface (#P% ) to the far-field stress (QQ ) in their corresponding graphene and boron nitride domains, i.e. O% = #P% /QQ . The far-field stress (QQ ) in graphene and boron nitride

domains is determined at the location where the effect of interface can be neglected,58 and the details are given in Fig. S6a-c. Fig. 2b shows a higher O% in graphene than in boron nitride. 7



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Besides, a higher leads to a higher O% . O% in both graphene and boron nitride domains

decreases with the increase of , which is well consistent with von Mises stress distribution

in Fig. 2a. Usually, a higher O% will constrain vibrations of atoms and weaken their thermal

transport ability,6, 21 thus leading to a lower from graphene to boron nitride domain than

from boron nitride to graphene domain. Moreover, an enhanced resistance against the heat transport across the interface at selective junctions at a higher O% is consistent with a decreased at a larger in Fig. 1c.

Analogous to the thermal rectification ratio @, the stress concentration factor ratio R is also calculated to further characterize the effect of stress concentration on the thermal rectification, and is defined as R =

ST0 ST12 ST12

, where O%/ and O%5 are the stress

concentration factors in the graphene domain and boron nitride domain, respectively. Fig. 2c shows that R decreases monotonously with the increase of the tensile strain . In particular,

at >7.5%, this monotonous decrease agrees well with the decrease of @ in Fig. 1d,

indicating the dominant role of mechanical stress in the thermal rectification at a larger strain. As for comparison, the elevated @ , despite reduced R , at a small strain implies the out-of-plane phonon resonance and phonon localization play a more important role, which also agrees well with the thermal transport mechanism of graphene nanoribbons in tension.21 To further demonstrate the resistance effect of stress concentration on thermal transport at interface junctions, Fig. 2d gives the snapshots of atomic heat flux vectors in the D

heterostructure with = E at different tensile strains (see Methods). Note that the summation of the heat flux vector over the atoms between the heat baths multiplied by the cross-sectional area will be the total heat flow across the structures, which is consistent with a steady heat conduction. A strong congestion (green region) is observed near the defects at the interface and is caused by in-plane phonon localization (will be discussed in details in Section 3.4) and stress concentration. In contrast, the perfect interface has a lower stress, and a higher heat flux (red region) is observed. When the heat flows from boron nitride to graphene domain, the magnitude of average heat flux is higher than that along the opposite direction, echoing well with the thermal rectification. At elevated strains, the effect of 8


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interface becomes weak with a decreased congestion of heat flow, and the heat flow tends to be uniform. In addition, with more defects in the interfaces, as shown in Fig. S7a-e, the congestion of heat flow will be stronger because of the enhanced stress concentration. These snapshots indicate that the mechanical strain is capable of being used to tune thermal rectification performance in heterostructures by weakening the interface stress concentration. 3.3. Vibrational Spectrum and Mode Matching Analyses. Fig. 3a shows the out-of-plane D

vibrational spectra of graphene and boron nitride atoms in the heterostructure with = E, and a clear difference between them is observed. In particular, when the heat flows from graphene to boron nitride domain, the phonon modes in graphene distribute in the frequency ranges of 0-20 and 20-40 THz with a minimum magnitude at 20 THz. Considering the frequency range of intrinsic graphene out-of-plane phonon modes (0-30 THz),59 the phonon modes with a frequency higher than 30 THz are contributed by the atoms near the interface.7 In contrast, the spectrum of boron nitride is higher between 0 and 10 THz and between 15 and 30 THz. When the direction of heat flow reverses with the same temperature difference between heat bathes, since a higher temperature will result in a higher phonon population in vibrational spectrum,5, 60 the phonon peak magnitudes in vibrational spectra of both graphene and boron nitride domains change. To reveal the mechanism of thermal rectification, the popular out-of-plane vibrational spectra overlap .6 is extracted. When heat flows from boron nitride to graphene domain in the absence of mechanical tensile strain ( = 0), the

peak at 13 THz in graphene spectrum broadens, leading to a significant increase of the overlap .6 in comparison with the spectrum obtained from the case with opposite heat transport direction. The enhancement of .6 leads to an increased heat flow from boron

nitride to graphene domain, which is consistent with the calculations in Fig. 1b-e. When heat transports from graphene to boron nitride domain at a small tensile strain (5%), the spectra of both graphene and boron nitride are similar to the ones at = 0 and the overlap of their spectra .6 approximately remains with the same value (Fig. 3b). When heat transports from

boron nitride to graphene domain, the small peak at 8 THz in the out-of-plane spectrum of boron nitride is broadened in comparison with the one at = 0, leading to a slight increase in the overlap. As the tensile strain further increases to 10% (Fig. 3c), the overlap of .6 9



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shows an increase when the heat flows from graphene to boron nitride domain. When the heat flow reverses its direction, .6 shows a slight decrease but is still larger than that in the absence of tensile strain, indicating the mitigation effect of a larger strain on the phonon localization and phonon resonance. In comparison with the out-of-plane vibrational spectra and resulting spectra overlap, Fig. 3d shows the in-plane vibrational spectra of graphene and boron nitride, and no obvious difference is observed. Besides, the overlap .# is approximately the same before and after the switch of heat flow directions. This negligible difference in .# remains after applying 5%

and 10% tensile strain to heterostructures, as shown in Fig. 3e and f, which suggests that the thermal rectification should be contributed by the out-of-plane phonon resonance, and is independent of in-plane phonon resonance. We should note that once the heterostructure is stretched by the tensile loading, the resulting stress or strain in the graphene or boron nitride domain will constrain the lattice deformation and softens phonon modes, leading to lattice anharmonicity22. As a consequence, a shift of high frequency peak (~48 THz in the absence of tensile strain in Fig. 3d) to lower frequency (~46 THz at 5% and ~43 THz at 10% tensile strain in Fig. 3e and f), often referred to as red shift, is observed in the spectra of both graphene and boron nitride. In addition, at a higher strain, a larger red shift is observed in boron nitride than in graphene. These uneven shifts decrease the overlap of their spectra, and lead to a clear reduction in .# at 5% and 10% tensile strain, as shown in Fig. 3e and f.

Nevertheless, .# still remains approximately the same after reversing heat flow direction at the same tensile strain. Fig. 3g shows the variation of .6/→5 with the tensile strain . At a small tensile strain (ε ≤ 5%), the out-of-plane vibrational spectra overlap .6/→5 is low, indicating a weak phonon resonance and a very limited thermal transport ability across the interface. Meanwhile, the imperfections at the heterojunction will intrigue stress concentration and phonon scattering. However, at ε ≤ 5%, the effect of stress concentration is not significant in comparison with that of out-of-plane phonon resonance (Fig. S5). Besides, the imperfect junction mainly influences in-plane phonon modes (Fig. S11a and b). Therefore, the 10


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imperfection has little effect on heat flow from graphene to boron nitride domain /→5 at

≤ 5%, and the softened in-plane phonon modes (Fig. 3d-f) result in an overall decrease in thermal transport, leading to a minimum /→5 at ε = 5%, as shown in Fig. 1b. As the

tensile strain increases, .6/→5 increases, and the enhanced phonon resonance leads to a reduction in thermal resistance and an increase of /→5 (Fig. 1b). At the same time, the

larger with a higher stress concentration (Fig. 2a) results in /→5 being negatively

dependent on at ≥ 10% (Fig. 1b). On the other hand, when the heat flows from boron

nitride to graphene domain, .X5→/ increases at the beginning and slightly decreases after the tensile strain reaches 5% (Fig. 3g), leading to the initial increase in phonon resonance and thus an increase in 5→/ . As the tensile strain increases, the enhanced stress will lead to the decrease of heat flow 5→/ . Moreover, the heterojunction with a higher has more severe

stress concentration (Fig. 2b) and thus more rapid decrease of 5→/ is obtained (Fig. 1b). In

addition, with a relatively high .X5→/ , a small contribution of out-of-plane phonon localization to the thermal resistance is expected in comparison with that of stress concentration, and the effect of on 5→/ is more obvious when heat transfers from boron nitride to graphene. As a result, the magnitudes of heat flow in opposite directions show a maximum difference at = 5%, and therefore a maximum @ is obtained at = 5%,

which agrees well with results in Fig. 1d and e. Similar results of overlap for different are obtained in Fig. S8. Furthermore, we calculate Y6 =

Z[12→0 Z[0→12 Z[0→12

to characterize the effect of

out-of-plane phonon modes on thermal rectification, and its variation with is plotted in Fig. 3h. Y6 shows an increase initially and then decreases, which is in agreement with the

variation of @, indicating the dominant role of out-of-plane phonon modes resonance in

thermal rectification at a small tensile strain. Fig. 3i presents the effect of on Y6 . When

the tensile strain is small (< 5%), Y6 decreases with the increase of ; At a high tensile strain (=10%), Y6 increases with . For comparison, the overlap .# and the resulting Y#

for in-plane phonon modes are also calculated and plotted in Fig. S9. .# shows a

monotonous decrease with due to the increased in-plane phonon mode mismatch between 11



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the spectra of graphene and boron nitride domains, but remains approximately the same when the heat flow switches its direction. As a consequence, Y# keeps a small fluctuation around zero, further indicating that thermal rectification is independent of the in-plane phonon modes. 3.4. Phonon Localization and Energy Distribution. To further probe the effect of the stress concentration and the out-of-plane phonon localization on thermal rectification, we calculate ∗ the phonon participation ratio of each phonon mode via \] = ^ ∑#∑a `#a,] `#a,] J ,

61

where N is the number of atoms, d is the polarization of interest (d= x, y or z), `#a,] is the eigenvector component of ith atom in the e th phonon mode in d direction. At the

equilibrium position of atoms in both graphene and boron nitride domains, `#a,] can be obtained by calculating the Hessian matrix with lattice dynamics simulation.62 The calculation of \] indicates that the phonon participation ratio is independent of temperature and lies between O(1) for completely delocalized modes and O(1/N) for completely localized phonon modes.13 Here \] < 0.25 is taken as the criterion to ensure the delocalization of

most phonon modes. In the absence of tensile strain, Fig. 4a shows that most localized modes (i.e. \] < 0.25) emerge in the frequency range of 0-30 THz, which corresponds well to the appearance of significant out-of-plane phonon modes in Fig. 3a-c. In contrast, the majority of phonons in the pure graphene and boron nitride nanoribbons (Fig. S10 ) has a participation ratio \] larger than 0.25 in the range of 0-30 THz, which indicates a significant out-of-plane phonon localization. Moreover, an overall reduced phonon participation is found in the heterostructure, and is caused by the mismatch of the intrinsic phonon modes of graphene and boron nitride. With the increase of interface defects (i.e. larger ), lower phonon participation ratios are obtained in the high frequency range (>40 THz), which is dominated by in-plane phonon modes, indicating that the imperfect junctions will lead to in-plane phonon localization and reduction in thermal transport, in consistency with Fig. 1c. The similar observations are also obtained in Fig. S11a and b when the tensile strain is applied. Fig. 4b presents the effect of tensile strain on the participation ratio. In the high frequency range (>40 THz), a significant reduction of phonon participation ratio and red shift of phonon 12


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frequency are found as tensile strain increases, and could attribute to the lattice deformation, anharmonicity

22

and phonon localization of in-plane phonon modes. In the low frequency

range, where out-of-plane phonon modes are dominant, a slight increase of phonon participation ratio is observed, suggesting a weaker contribution of out-of-plane phonon localization to the thermal rectification at large strains. To understand the evolution of in-plane and out-of-plane phonons, we investigate the energy spatial distribution of both localized in-plane and out-of-plane phonons in the heterostructure. The energy of the localized phonons at a location is calculated via h = ∑$i +

J

ℏ k# ,61, 63 where ℏ is the reduced Planck constant, is the angular frequency,

i is the occupation number in the Bose-Einstein distribution and k# is the local vibrational

∗ density of states. More specifically, k# = ∑] ∑a `#a,] `#a,] l − ] , where l denotes

Dirac delta function, and ] is the angular frequency of the eth phonon mode. For in-plane

phonon modes, d is x and y, and for out-of-plane modes, only d=z is counted. Fig. 4c shows the energy distribution of localized out-of-plane phonon modes in the heterostructure D

with = E. A higher energy of localized phonons exists in the graphene region and it corresponds well with the wider range of frequencies in out-of-plane spectra in graphene (Fig. 3a-c). The concentration of localized phonon energy in graphene is more obvious when heat transfers from graphene to boron nitride domain, narrowing the propagation path of delocalized phonon modes and thus limiting the thermal conductance.13 When heat transfers from boron nitride to graphene, the difference between average localized phonon energy in the domains of the two materials becomes smaller, which leads to an increased phonon resonance. As a consequence, an enhanced thermal transport ability in the heterostructures is achieved, which further leads to the thermal rectification. At an external mechanical tensile strain, the energy of localized phonons becomes smaller in the entire structure, indicating its mitigation effect on out-of-plane phonon localization. To further reveal the regulatory role of tensile strain, the out-of-plane localized phonon energy along the y-axis is averaged and plotted with x-coordinates (i.e. the direction of heat transport) normalized by the length of structure m/ in Figure 4c. It shows that when heat travels from graphene to boron nitride, 13



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the localized phonon energy is maintained at a higher magnitude of plateau in the graphene domain than that in the boron nitride domain, associated with a clear drop near the interface; when the heat flow reverses its direction, both plateaus shrink to the middle with a smaller drop between them, indicating a reduced difference of localized phonon energy distribution in both graphene and boron nitride. This reduction will lead to an enhanced phonon resonance of these localized phonons, and thus improve the thermal transport ability. The “shrink” becomes the most obvious at 5% tensile strain, which is consistent with the optimized thermal rectification ratio at 5% tensile strain shown in Fig. 1d. Similar conclusions can be drawn from the out-of-plane phonon energy of heterostructures with =0 and =1, as shown in Fig S12a-b. Moreover, the reduced overall energy of out-of-plane localized phonon modes echoes with the increased phonon participation ratio at a large strain in the frequencies 0-30 THz in Fig. 4b and also is consistent with Y6 in Fig. 3h. Besides, this finding further confirms that the effect of out-of-plane phonon mode localization is weak at large tensile strain, and the thermal rectification is dominated by the stress concentration near the interfaces, which is in agreement with a monotonous decrease of both @ and R (>7.5%). This confirmation is further supported by the energy of localized phonon modes in heterostructures with = 0

and = 1, as shown in Fig. S12a-b. Similarly, we have also calculated the energy distribution of in-plane localized phonon modes and their variation along the x-axis, as shown in Fig.S13a-c. Strong in-plane phonon localizations are observed at interfaces where defects are introduced as well as at the boundaries, which corresponds well to the decreased phonon participation ratio in Fig. 4a and Fig. S11a and b. In addition, this significant in-plane localized phonon energy at the interface echoes well with the heat congestion in Fig. 2d, which further indicates the impediment of in-plane phonon localization to the thermal transport across the interface. The increase of will result in more severe phonon localization and leads to an enhanced resistance to heat transport. This enhancement becomes larger when the tensile strain increases, which is opposite to the effect of tensile strain on out-of-plane phonon mode, but consistent with a decreased participation ratio at a higher range of phonon frequencies (>40 THz) in Fig. 4b. 14


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3.5. Conceptual Heterostructure Systems with Controllable Thermal Transport Paths. To demonstrate the potential applications of mechanical tensile strain and selective junction interfaces in the thermal transport of heterostructures, we present two conceptual designs of heterostructure systems. Fig. 5a shows the atomic modeling of the first heterostructure system composed of alternatively arranged graphene and boron nitride sheets with parallel interfaces but different junction parameters =0, 1, referred to as thermal antiparallel shunt. In this system, both ends and middle regions were selected as heat baths to generate a temperature gradient. When the heat flows from the middle region with a high temperature (hot bath) to both ends with the same temperature (cold bath), Fig. 5b shows a clear difference in the heat flows in the two directions. Besides, compared to the same system with a uniform interface junction (=0) (Fig. S14a-c), the defects (=1) at the interface deteriorate the resistance to heat transport, and thus amplifies the asymmetry of heat transport in these two opposite directions. When a 10% tensile strain is applied to the heterostructure system, the asymmetry of heat transport in both directions is significantly reduced. When the hot and cold thermostats switch with the same temperature gradient, the asymmetric heat flow is also observed and will decrease with the increase of the applied tensile strain, as shown in Fig. S14d-i. Fig. 5c presents a second graphene-boron nitride heterostructure system with one interface. Its interface consists of equal junctions but with different parameters (=0, 1), referred to as thermal parallel shunt. When the heat flows from boron nitride to graphene domain, Fig. 5d shows that heat flow with greater magnitude is much easier to cross the interface with =0 than with =1. When the direction of the temperature gradient switches,

the asymmetry of heat flow across the interfaces with both =0 and =1 decreases. Besides, upon applying a tensile strain, the dependence of heat flow in both directions on the interface

parameter becomes weak. The preference of heat flow in these two heterostructure systems indicates that the asymmetric thermal transport can be tuned by interface junctions, and more importantly it can be reduced by an external mechanical strain. 4. CONCLUSIONS

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In summary, we systematically investigate the thermal rectification in graphene-boron nitride (GBN) in-plane heterostructures with controlled interface junctions and their response to an external mechanical tensile strain. Using non-equilibrium molecular dynamics (NEMD) simulation, we show that the asymmetric heat transport decreases with the introduction of junction defects. Upon applying a mechanical deformation to GBN heterostructures, it will increase at a small tensile strain, but decrease at a large tensile strain. This competing effect of heterojunction interfaces and mechanical tensile strain are probed through the stress analysis, vibrational spectra, phonon participation ratio, and heat flow distribution. At a small tensile strain, the thermal rectification is dominated by out-of-plane phonon mode resonance and localization, and with the increase of tensile strain, the mechanical stress concentration at the interface plays a dominant role. Based on the dependence of thermal transport across heterojunction on defects and tensile strain, we put forward two conceptual designs of controllable thermal transport heterostructure systems. By introducing selective junctions and a certain level of tensile strain, the thermal rectification performance can be well regulated. Besides, the heat can also be controlled to transfer along a desirable path through a careful design and selection of junction interfaces. These findings are expected to lay a foundation in the design and manufacturing of thermal functional devices such as thermal rectification devices with mechanically tunable thermal rectification, and also shed light on functional electric devices with controllable thermal management through mechanical deformation. ASSOCIATED CONTENTS Supporting Information Effect of temperature difference. Stress-strain curves. Temperature profile. Von Mises stress distribution and far field stress-strain curves. Heat flow distribution in heterostructures. Phonon spectra, mode matching factors, phonon participation ratio, and localized modes distribution plots of heterostructures. AUTHOR INFORMATION 16


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Corresponding Author *Email: [email protected] Notes The authors declare no competing financial interest. ACKNOWLEDGEMENTS This work is supported by the start-up funds at the University of Virginia. REFERNCES (1) Gong, Y.; Lin, J.; Wang, X.; Shi, G.; Lei, S.; Lin, Z.; Zou, X.; Ye, G.; Vajtai, R.; Yakobson, B. I.; Terrones, H.; Terrones, M.; Tay, Beng K.; Lou, J.; Pantelides, S. T.; Liu, Z.; Zhou, W.; Ajayan, P. M., Vertical and In-plane Heterostructures from WS2/MoS2 Monolayers. Nat. Mater. 2014, 13 (12), 1135-1142. (2) Wen, Y.-N.; Xia, M.-G.; Zhang, S.-L., Size Effect on the Magnetic and Electronic Properties of the Monolayer Lateral Hetero-junction WS2-MoS2 Nanoribbon. Appl. Surf. Sci. 2016, 371, 376-382. (3) Kan, D.; Aso, R.; Sato, R.; Haruta, M.; Kurata, H.; Shimakawa, Y., Tuning Magnetic Anisotropy by Interfacially Engineering the Oxygen Coordination Environment in a Transition Metal Oxide. Nat. Mater. 2016, 15 (4), 432-437. (4) Liu, H.-J.; Lin, J.-C.; Fang, Y.-W.; Wang, J.-C.; Huang, B.-C.; Gao, X.; Huang, R.; Dean, P. R.; Hatton, P. D.; Chin, Y.-Y.; Lin, H.-J.; Chen, C.-T.; Ikuhara, Y.; Chiu, Y.-P.; Chang, C.-S.; Duan, C.-G.; He, Q.; Chu, Y.-H., A Metal–Insulator Transition of the Buried MnO2 Monolayer in Complex Oxide Heterostructure. Adv. Mater. 2016, 28 (41), 9142-9151. (5) Chen, X.-K.; Xie, Z.-X.; Zhou, W.-X.; Tang, L.-M.; Chen, K.-Q., Thermal Rectification and Negative Differential Thermal Resistance Behaviors in Graphene/hexagonal Boron Nitride Heterojunction. Carbon 2016, 100, 492-500. (6) Liu, X.; Zhang, G.; Zhang, Y.-W., Topological Defects at the Graphene/h-BN Interface Abnormally Enhance Its Thermal Conductance. Nano Lett. 2016, 16 (8), 4954-4959. (7) Liu, B.; Baimova, J. A.; Reddy, C. D.; Dmitriev, S. V.; Law, W. K.; Feng, X. Q.; Zhou, K., Interface Thermal Conductance and Rectification in Hybrid Graphene/Silicene Monolayer. Carbon 2014, 79, 236-244. (8) Wei, W.; Dai, Y.; Huang, B., Straintronics in Two-Dimensional In-plane Heterostructures of Transition-Metal Dichalcogenides. Phys. Chem. Chem. Phys. 2017, 19 (1), 663-672. (9) Mortazavi, B.; Ostadhossein, A.; Rabczuk, T.; Van Duin, A. C., Mechanical Response of All-MoS2 Single-layer Heterostructures: A ReaxFF Investigation. Phys. Chem. Chem. Phys. 2016, 18 (34), 23695-23701. (10) Wei, Y.; Wu, J.; Yin, H.; Shi, X.; Yang, R.; Dresselhaus, M., The Nature of Strength Enhancement and Weakening by Pentagon–heptagon Defects in Graphene. Nat. Mater. 2012, 11 (9), 759-763. 17



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Figure 1. Thermal rectification in graphene-boron nitride (GBN) heterstructures with controlled interface junctions. (a) Atomic model of GBN heterostructures (left) and highlighted structures in interface junctions (right). (b) and (c) Comparison of heat flow in heterostructures along both directions as functions of tensile strain and junction parameter . (d) and (e) Variation of thermal rectification @ as tensile strain , and hereojunction parameter . 21



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Figure 2. Stress analysis and heat flow distribution in graphene-boron nitride (GBN) heterostructures. (a) Snapshots of von Mises stress distribution near the interface. (b) Variation of stress concentration O% in graphene and boron nitride domains as tensile strain , and hereojunction parameter . (c) Comparison of relative stress concentration R in different heterostructures as functions of tensile strain . (d) Heat flow distribution in heterostructure with = 3/7 at different tensile strains and flow directions.

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Figure 3. Vibrational spectra of graphene and boron nitride domain in graphene-boron nitride (GBN) heterostructures. (a), (b) and (c) Out-of-plane vibrational spectra 6 of graphene and boron nitride in the heterostructure with = 3/7 at the tensile strain of = 0, 5% and 10%. (d), (e) and (f) In-plane vibrational spectra # of graphene and boron nitride in the heterostructure with = 3/7 at the tensile strain of = 0, 5% and 10%. (g) Overlap of out-of-plane phonon spectra .6 in heterostructure with = 3/7 at different strains. (h) Relative overlap Y6 in heterostructure with = 3/7 at different strains. (i) Variation of relative overlap of out-of-plane phonon spectra Y6 as functions of junction parameter .

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Figure 4. Activity of phonon participation and localized phonon modes in graphene-boron nitride (GBN) heterostructures. (a) Phonon participation ratio in heterostructures with = 0, 3/7 and 1 in the absence of tensile strain = 0 . (b) Comparison of phonon participation ratio in heterostructure with = 3/7 at different tensile strain . (c) Energy distribution of localized out-of-plane phonon modes (top) and the averaged plots as y-axis with x-position normalized by the length l (bottom) in heterostructure with = 3/7 in both directions of heat flow at different tensile strains.

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Figure 5. Demonstration applications of controllable thermal transport paths in the heterostructure systems with selective interface junctions. (a) Atomic structure of GBN heterostructure system composed of alternatively arranged graphene and boron nitride sheets with parallel interfaces but different junction parameters (=0, 1), referred to as thermal antiparallel shunt. (b) Comparison of heat flux in the “thermal antiparallel shunt” system at 0% and 10% tensile strain. (c) Atomic structure of GBN heterostructure system with an equally shared interface but different junction parameters (=0, 1), referred to as thermal parallel shunt. (d) Comparison of heat flux in both directions in the “thermal parallel shunt” system at 0% and 10% tensile strain. 25



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TOC

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Controllable Interface Junction, In-Plane Heterostructures Capable of

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