Letter pubs.acs.org/NanoLett
Topological Defects at the Graphene/h‑BN interface Abnormally Enhance Its Thermal Conductance Xiangjun Liu, Gang Zhang,* and Yong-Wei Zhang Institute of High Performance Computing, A*STAR, Singapore, 138632 S Supporting Information *
ABSTRACT: Low thermal conductance across interface is often the limiting factor in managing heat in many advanced device applications. The most commonly used approach to enhance the thermal conductance is to reduce/ eliminate the interfacial structural defects. Using a graphene/h-BN (Gr/h-BN) interface, we show surprisingly that topological defects are able to enhance the thermal conductance across the interface. It is found that the phonon transmission across the Gr/h-BN interface with 5|7 defects is higher than that of the pristine interface, which is in strong contrast to the common notion that interface defects promote phonon scattering. By analyzing the strain distribution and phonon vibrational spectra, we find that this abnormal enhancement in interfacial thermal conductance originates from the localization of the stress fields arising from misfit dislocations and their out-of-plane deformations at the interface. In the presence of the defects, the overall mismatch strain is reduced. In addition, the out-of-plane deformations screen the long-ranged dislocation strain fields, resulting in the stress fields to be localized only at the cores of the defects. This abnormal mechanism provides a new dimension to enhance the interfacial thermal conductance in two-dimensional heterostructures. KEYWORDS: Thermal management, interfacial thermal conductance, graphene, h-BN
C
nanoelectronic devices, the interfacial thermal resistance in general plays a critical role in dictating the overall heat transport. As a structurally continuous sheet formed by electrically conducting graphene and insulating hexagonal boron nitride (hBN), the graphene/h-BN (Gr/h-BN) in-plane heterostructure has remarkable advantages for use in nanoscale integrated circuits and electronic devices. Meanwhile, the rapid development in materials synthesis techniques has made many novel two-dimensional (2D) structures possible. For example, recently, 2D Gr/h-BN in-plane heterostructures were successfully synthesized.17 In particular, it is now possible to grow atomically sharp interfaces between graphene and h-BN domains using combined atmospheric pressure chemical vapor deposition (CVD) and reactive ion etching.18,19 Furthermore, the presence of topological 5|7 defects at the interface of graphene/h-BN heterostructure was also observed experimentally.20 These fascinating advances in the synthesis of 2D heterostructures have laid a promising foundation for realizing such interfaces. Recently, the phonon properties of both graphene and h-BN have been studied, and the stable interfacial geometry has been demonstrated.17 Thus, the Gr/hBN interface has become an ideal candidate for studying the fundamental mechanism of interfacial thermal conduc-
ontrolling, especially, reducing interfacial thermal resistance, has long been the focus in the study of nanoscale thermal management. Typically, engineering interfacial structures is a commonly used method.1−5 For example, for an interface with different atomic masses, a 20%−30% enhancement in the total thermal conductance can be achieved by inserting an interfacial layer to mediate the vibrational mismatch at the interface.1 However, this method was shown to highly depend on the mass ratio of the two materials. For instance, the maximum increase of 30% can be achieved at a mass ratio of 3, but it converges to a value only less than 10% when the ratio is higher than 5.1 Moreover, the extremely thin interlayer between the two mass-mismatched materials is obviously difficult to fabricate.5 Graphene has been found to have many fascinating properties, such as ultrahigh mechanical stiffness, strength and elasticity, and superior electrical and thermal conductivity.6 The ultrahigh thermal conductivity of graphene raises the hope for its applications in thermal management in nanoelectronic devices.7−11 Recently, the interfacial thermal resistances between graphene and various materials, such as graphene/ Cu,12 graphene/4H-SiC,13 graphene/Si,14 graphene/SiO2,15 and graphene/multilayer graphene,16 have been explored, and various effects on their interfacial thermal conductance, including temperature, system size, and interface roughness, have also been examined.12−16 These studies have shown that although graphene exhibits an ultrahigh thermal conductivity, due to the presence of a large number of interfaces in © 2016 American Chemical Society
Received: April 15, 2016 Revised: July 1, 2016 Published: July 7, 2016 4954
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Figure 1. Heterostructure with different interfaces between graphene and h-BN. (a) B−C connected coherent interface, NB−C; (b) N−C connected coherent interface, BN−C; (c) B−C connected incoherent interface with 5|7 topological defects, NB−C5|7; (d) N−C connected incoherent interface with 5|7 topological defects, BN−C5|7. B, N, C atoms are shown in purple, blue, and gray color, respectively.
tance.21−25 Although the interfacial thermal conductance of the Gr/h-BN interface was reported,21−25 how to increase the thermal conductance via interface engineering is still an open question. In addition, there is a lattice mismatch along the Gr/ h-BN interface. In particular, for a long interface topological defects may be energetically favorably introduced into the interface. Then, how those topological defects affect the interfacial thermal conductance is another open question. In this work, using Gr/h-BN interface as an example, we explore the effect of interfacial topological defects on the interfacial thermal conductance. Surprisingly, we find that the topological defects are able to enhance the interfacial conductance, which is in strong contrast to the commonly held notion that interface defects reduce phonon transmission. Based on our molecular dynamics calculation results and lattice dynamics analysis, we reveal a new mechanism that is responsible for the abnormal enhancement in the interfacial thermal conductance. The findings elucidated here provide a new design route for engineering interfacial thermal conductance. Theory and Computational Methods. Schematics of the atomistic configurations of 2D Gr/h-BN in-plane heterostructures are shown in Figure 1. Since both experimental26,27 and theoretical26 studies have shown that zigzag linking edges were preferably formed in the Gr/h-BN heterostructures, here only the zigzag-oriented interfaces are considered. The heterostructures without any defects at the interface, named coherent interface, are shown in Figure 1a,b. If B atoms are bonded with C atoms at the interface, the heterostructure is named as NB−C; on the other hand, if N atoms are bonded with C atoms at the interface, the heterostructure is named as BN−C. In accordance to the experimental observation,20 5|7 defects are considered at the Gr/h-BN interface, which is named as incoherent interface, for example, NB−C5|7 and BN− C5|7, as shown in Figure 1c,d. The lattice constants of graphene and h-BN are aC = 2.46 Å and aBN = 2.52 Å, giving rise to a lattice mismatch strain of χ = (aBN − aC)/aC = 2.44%. In all the simulated systems, the lengths of graphene and h-BN are LC = 106.52 Å and LBN = 109.12 Å, respectively. The width of graphene is W C = 22a C . In the coherent Gr/h-BN heterostructures, the width of h-BN is WBN = 22aBN to form a coherent interface with graphene. While in the incoherent
Gr/h-BN heterostructures, WBN = 20aBN is chosen to form an incoherent interface with graphene, and the number of 5|7 defects per unit length is about 0.37 nm−1. Here we set the xaxis to be the direction of the heat current, and the y-axis to be parallel to the Gr/h-BN interface. To mimic the Gr/h-BN heterostructure as a ribbon structure, free boundary condition is applied along the y- and z-directions. Molecular dynamics (MD) simulations are employed to study the thermal transport across the Gr/h-BN interfaces using the LAMMPS package.28 In all MD simulations performed here, the Tersoff potential29,30 is used to describe the covalent interactions between C, B, and N atoms with optimized parameters for thermal properties.31−33 The simulations are carried out with a time step of 0.25 fs throughout. The velocity Verlet algorithm is employed to integrate Newton’s equations of atomic motion. First, the system was equilibrated at a constant temperature of T = 300 K for 200 ps using Nosé−Hoover temperature thermostat34 (NVT ensemble). After the constant temperature relaxation, we continued to relax the system with NVE (constant volume and no thermostat) ensemble for 200 ps. During this stage, the total energy and temperature of the system were monitored. We found that the total energy was conserved and the temperature of the entire system remained constant with fluctuations around 300 K, which indicated that the system had reached equilibrium. Next, we computed the interfacial thermal conductance of the system using nonequilibrium molecular dynamics method. To establish a temperature gradient along the longitudinal xdirection and make heat energy transfer from graphene to hBN, the atoms close to the right end (of graphene) and the left end (of h-BN) were placed into hot and cold Nosé−Hoover reservoirs with temperatures set to be TH = 310 K and TC = 290 K, respectively. The simulations were then performed long enough (1.25 ns) to allow the system to reach the nonequilibrium steady state, where the temperature gradient was well established, and the heat flux going through the system became time-independent. The total heat flux J in the longitudinal direction was calculated by using28 4955
DOI: 10.1021/acs.nanolett.6b01565 Nano Lett. 2016, 16, 4954−4959
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Nano Letters ⎡N 1⎢ ∑ εivi + 1 J= V ⎢⎣ i 2 +
1 6
N
∑ ijk ; i ≠ j ≠ k
K−1 m−2, respectively. Both of them are three order higher than the out-of-plane ITC between graphene and h-BN (7.4 × 106 W K−1 m−2),35 and one order higher than that of chemically bonded graphene−metal interfaces (2.5 × 108 W K−1 m−2)36 predicted by first-principles calculations, indicating that in-plane hybrid Gr/h-BN material is efficient for heat transport. These values are close to those predicted from atomistic Green’s function calculations,23 which are about 3.52 × 109 W K−1 m−2. The ITC of BN−C is 48% higher than that of NB−C, which is due to the difference in bonding strength across the boundary.21−25 It is known that although both C−B and C− N interactions are covalent in nature, the strength of C−N bond is higher than that of C−B bond, explaining the advantage of the C−N interface for phonon transport. From lattice dynamic point of view, the key factor that determines the phonon transport across two joining materials is the overlapping of phonon density of states (PDOS) between them.37−40 Because phonon energy is essentially the energy of atomic vibrations, the atomic vibrational analyses for C, B, and N atoms are carried out in the frequency domain. The PDOS of these three atoms in the frequency domain can be calculated by taking the Fourier transform of the velocity autocorrelation functions of atoms belonging to the different domains of the system
N
∑ (Fij·vi)rij ij ; i ≠ j
⎤ (Fijk ·vi)(rij + rik)⎥ ⎥⎦
(1)
where, εi and vi are the energy and velocity associated with atom i, respectively. Vector rij denotes the interatomic distance between atoms i and j, and Fij and Fijk denote the two-body and three-body force, respectively. V is the volume of the studied system. After the system reached the nonequilibrium steady state, a time averaging of temperature and heat flux was performed for an additional 20 ns. Results and Discussion. A typical temperature profile at steady state in the NB−C heterostructure is shown in Figure 2.
D(ω) =
∫0
τ
Γ(t )exp(−iωt )dt
(3)
where ω is the frequency, D(ω) is the PDOS at frequency ω, and Γ(t) = ⟨v(t)v(0)⟩/⟨v(0)v(0)⟩ is the velocity autocorrelation function. v(t) is the atom velocity, ⟨···⟩ denotes time and atom number-averaged velocity autocorrelation function, and τ = 15 ps is the time duration for computing the velocity autocorrelation function and its corresponding discrete Fourier transform in a series of short runs. As shown in Figure 3, Figure 2. Temperature distribution along the heat flux (x) direction in the NB−C heterostructure. The inset is the temperature profile of the NB−C heterostructure.
It is clear that there is a temperature jump δT across the interface, indicating the existence of interfacial thermal resistance between graphene and h-BN. We calculated the value of interfacial thermal conductance (ITC) using J λ= (2) δT where λ is the ITC, and J is the heat flux across the interface. It is worth mentioning that we used the linear fitting and extrapolation to determine δT. From Figure 2 and eq 2, we obtained the values of ITC across the Gr/h-BN interfaces at room temperature, which are shown in Table 1. It is seen that the interfacial bonding configuration has significant effects on the ITC. The heterostructure with the N−C bonded interface has a higher ITC than that with the B−C bonded interface. The ITCs of NB−C and BN−C are 4.35 × 109 and 6.42 × 109 W
Figure 3. Phonon density of states: C atoms in graphene, B and N atoms in h-BN. The temperature is at 300 K.
Table 1. Interfacial Thermal Conductance, Temperature Jump, and Heat Flux in Gr/h-BN Heterostructures with Different Interfaces ITC (× 109 W K−1 m−2) temperature jump (K) heat flux (× 109 W m−2)
NB−C
BN−C
NB−C5|7
BN−C5|7
4.35 ± 0.13 6.79 ± 0.15 29.51 ± 0.56
6.42 ± 0.22 5.37 ± 0.09 34.48 ± 0.70
5.16 ± 0.18 6.63 ± 0.17 34.22 ± 0.69
7.09 ± 0.15 5.45 ± 0.14 38.61 ± 0.93
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the interface with tensile stress being on the graphene side and compressive stress being on the h-BN side. The amplitude of stress decays rapidly with the distance away from the interface. When phonons travel across the interface region, the rapidly changed strain field induces strong phonon scattering, resulting in a large interfacial thermal resistance. However, at the interface with topological defects, the stress distribution becomes nonuniform. As shown in Figure 4d−f, the stress mainly concentrates at the positions of 5|7 defects, which is in strong contrast to the Gr/h-BN coherent interface (Figure 4b,c), in which the stress is uniform along the interface. The formation of misfit 5|7 defects at the interface is an effective mechanism to accommodate large lattice mismatch strain. The underlying reason is that the misfit dislocations relieve a fraction of mismatch strain at a cost of forming 5|7 defects at the interface. Importantly, it was demonstrated that 5| 7 defects at the interface are able to make the Gr/h-BN heterostructure structurally more stable and energetically more favorable.43 After energy relaxation, the interface exhibits a severe out-of-plane deformation in the vicinity of 5|7 misfit dislocation cores, as shown in Figure 4d. The deformation makes a sharp curvature on the Gr/h-BN plane surface, but the curvature decays quickly and much of the Gr/h-BN remains flat. This is consistent with previous studies that showed that besides 5|7 misfit dislocation, the out-of-plane deformations can further substantially reduce the dislocation formation energy in a 2D material.42,44,45 As shown in Figure 4e,f, the 5|7 dislocations and localized out-of-plane deformations help relieve large strains at the interface. As a result, the stress is only localized around the defects. We have also calculated the cross-sectional distribution of the heat flux density along the Gr/h-BN interface using eq 1, and the results are shown in Figure 5. In the calculations, the cross-
phonon vibrational spectrum of N atom has a larger overlapping domain with that of C atom in the frequency range between 23 and 33 THz in comparison with that of B atom, indicating that N atom has more phonon modes to transport thermal energy to C atom, leading to a higher ITC at the N−C bonded interface. One striking feature observed here is that the 5|7 defects at the interface surprisingly enhance the ITC of the Gr/h-BN heterostructures. The ITC of NB−C5|7 is 5.16 × 109 W K−1 m−2, which is 18.6% higher than that of NB−C. The ITC of BN−C5|7 is 10.4% higher than that of BN−C. We have also examined various influential factors, including the width, length, boundary condition, and the spacing and density of the defects, on this abnormal enhancement. Remarkably, the enhancement of ITC caused by the topological defects at the interface persists in all these different cases (see our detailed simulation results in Supporting Information). However, it is well-known that the 5|7 defects generally lead to a significant reduction of thermal conductivity of 2D materials (e.g., graphene, h-BN) due to the enhanced local phonon scattering by these defects.40−42 For example, it was shown that with the similar number of 5|7 defects per unit length (0.398 nm−1), the thermal conductivity of 5|7-defected graphene decreases by more than 25%.40 As shown in Table 1, for the Gr/h-BN heterostructures with and without 5|7 defects, the temperature jump δT at the interfaces are almost the same, which is about 6.7 and 5.4 K for B−C and N−C bonded interfaces, respectively. Thus, the difference of ITCs between the Gr/hBN heterostructures with and without 5|7 defects is mainly due to the difference in heat flux across the boundary. Because both N−C and B−C interfaces show a similar defect effect on thermal conductance, in the following, without loss of generality, we use BN−C interface to explore the underlying mechanism. At the defect-free Gr/h-BN coherent interface, as shown in Figure 4a−c, there is a uniform mismatch stress along
Figure 5. Cross-sectional heat flux density distribution in BN−C and BN−C5|7 heterostructures.
section of the Gr/h-BN heterostructure is uniformly divided into 11 columns. It is seen that for defect-free Gr/h-BN heterostructure, the local heat flux at the edge is more than 25% lower than that at the center. It is known that there are edge scattering-induced localized phonon modes in the low frequency region in nanoribbons and nanowires.46,47 Hence, the localized phonons are responsible for the reduction of the local heat flux at the edges. For the BN−C heterostructure with
Figure 4. Stress distribution in BN−C and BN−C5|7 heterostructures. (a) BN−C heterostructure after relaxation. (b) Stress σx in the xdirection of BN−C; (c) stress σy in the y-direction of BN−C; (d) BN−C5|7 heterostructure after relaxation. (e) Stress σx in the xdirection of BN−C5|7; (f) stress σy in the y-direction of BN- C5|7. 4957
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Nano Letters coherent interface, the heat flux density in the sections from P2 to P10 is almost uniformly distributed. However, in the BN− C5|7 heterostructure with incoherent interface the heat flux density is higher than that in the BN−C heterostructure. Hence, we can deduce that there are two competing factors on phonon transport, (1) ITC enhancement arising from the stress relaxation along the Gr/h-BN boundary due to the introduction of the topological defects, which gives rise to the increase in interfacial thermal conductance, and (2) ITC reduction arising from the topological defects themselves, which gives rise to the local phonon scattering. As shown in Figure 5, although the local heat flux density at most part of the Gr/h-BN boundary increases upon introducing 5|7 defects, the local heat flux at the defect positions reduces with respect to the pristine interface. However, the heat flux reduction at the 5|7 defects is offset by the large increase in heat flux at positions without defects. Therefore, the total heat current through the incoherent Gr/hBN interface is higher than that through the coherent one. It is known that the PDOS overlap of the two materials forming the interface gives a reasonable measure for the phonon transmission across the interface as ITC and heat current correlate strongly with the overlap of the PDOS.48,49 To further understand the effect of 5|7 defects on the interfacial thermal conductance, we calculated the PDOS of C and N atoms on both sides of the interfaces, and the calculation results are shown in Figure 6. For the coherent Gr/h-BN interface, at
frequency regime can have considerable effect on ITC. For the incoherent interface with 5|7 defects, due to the atomic scale of the defects the long-wavelength phonons are unable to sense the defects and thus propagate through as if they were traveling across the coherent interface. As a result, the atomic defects have negligible influence on the long-wavelength phonons. Hence, the high frequency phonons play an important role: The topological defects are able to soften the abruptly changed stress field at the interface and induce more overlap in PDOS on the two sides, as shown in Figure 6b. To quantify the overlap of PDOS, we calculated the value of overlap (S) as48 ∞
S=
∫0 DC(ω)DN(ω)dω ∞
∞
∫0 DC(ω)dω ∫0 DN(ω)dω
(4)
It is found that SBN−C5|7 is about 0.034, which is larger than SBN−C (∼0.02). These results are consistent with our MD simulation results that ITC at BN−C5|7 is higher than that at BN−C. Therefore, phonons, which originally cannot propagate across the interface, are able to transmit via new channels, in the 45−50 THz frequency range. For phonon transport in a homogeneous material, atomic defects are able to lead to a reduction in thermal conductivity via phonon backscattering. For an individual interface, however, atomic defects suppress the abrupt change in the stress field, leading to an increase of phonon transmittance. It is worth mentioning that in addition to the in-plane heterostructure such as the graphene/h-BN junctions studied here, there exist other possible heterostructures, such as the graphene/h-BN cross-plane configurations and one-dimensional (1D) nanotube junctions.52−55 In the hybrid BN-carbon nanotubes,52 stress mismatch is also present at the interface, which is similar to the graphene/h-BN heterostructure considered here. Therefore, introduction of topological defects at the interface is also expected to enhance the ITC at junctions of these hybrid BN-carbon nanotubes. 4. Conclusions. We have investigated the interfacial thermal conductance of 2D graphene/h-BN heterostructures by using nonequilibrium molecular dynamics simulations. Four kinds of interfacial bonding configurations (two coherent and two incoherent) have been considered. It is found that the graphene/h-BN heterostructures have a remarkably high interfacial thermal conductance, which is one order higher than that of chemically bonded metal−graphene interfaces. Because of the stronger covalent bond, ITC of the N−C bonded interface is over 48% higher than that of the B−C bonded interface. We also find surprisingly that when 5|7 defects are introduced at the interface, the ITC is enhanced by more than 10% compared with the coherent counterpart. On the basis of the stress distribution and phonon vibrational spectrum analysis, we find that this abnormal enhancement in ITC originates from the localization of the stress field by misfit dislocations and the out-of-plane deformations at the interface. In the presence of 5|7 defects, the overall mismatch strain is reduced. In addition, the out-of-plane deformations screen the long-ranged dislocation strain field, resulting in the stress field to be localized only at the cores of the 5|7 dislocations. Our findings here provide a new means for engineering interfacial thermal conductance in graphene/h-BN and other 2D heterostructures.
Figure 6. Phonon density of states for carbon and nitrogen atoms at the interfaces in (a) BN−C and (b) BN−C5|7 heterostructures.
low frequencies there is significant PDOS overlap, indicating that phonon transmission is high. However, at high frequencies, the PDOS peak of graphene is different from that of h-BN, which is the main origin for the interfacial thermal resistance at the boundary of these two dissimilar materials.23 For ITC between two dissimilar solids, in addition to the contribution from phonons in the low frequency range below the optical modes, phonons modes that are above the cutoff frequency of the heavier material can also contribute to ITC, indicating that inelastic processes that occur at the contact interface provide additional transport channels.50,51 Therefore, phonons in high 4958
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b01565. The effects of various influential factors on enhancement of interfacial thermal conductance. The detailed stress distribution. (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported in part by a grant from the Science and Engineering Research Council (152-70-00017). The authors gratefully acknowledge the financial support from the Agency for Science, Technology and Research (A*STAR), Singapore and the use of computing resources at the A*STAR Computational Resource Centre, Singapore.
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DOI: 10.1021/acs.nanolett.6b01565 Nano Lett. 2016, 16, 4954−4959
