Phonons, Localization, and Thermal Conductivity of Diamond

Letter pubs.acs.org/NanoLett

Phonons, Localization, and Thermal Conductivity of Diamond Nanothreads and Amorphous Graphene Taishan Zhu† and Elif Ertekin*,†,‡ †

Department of Mechanical Science and Engineering, University of Illinois at Urbana−Champaign, 1206 W Green Street, Urbana Illinois 61801, United States ‡ International Institute for Carbon Neutral Energy Research (WPI-I2CNER), Kyushu University, 744 Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan S Supporting Information *

ABSTRACT: Recently, the domains of low-dimensional (low-D) materials and disordered materials have been brought together by the demonstration of several new low-D, disordered systems. The thermal transport properties of these systems are not well-understood. Using amorphous graphene and glassy diamond nanothreads as prototype systems, we establish how structural disorder affects vibrational energy transport in low-dimensional, but disordered, materials. Modal localization analysis, molecular dynamics simulations, and a generalized model together demonstrate that the thermal transport properties of these materials exhibit both similarities and differences from disordered 3D materials. In analogy with 3D, the low-D disordered systems exhibit both propagating and diffusive vibrational modes. In contrast to 3D, however, the diffuson contribution to thermal transport in these low-D systems is shown to be negligible, which may be a result of inherent differences in the nature of random walks in lower dimensions. Despite the lack of diffusons, the suppression of thermal conductivity due to disorder in low-D systems is shown to be mild or comparable to 3D. The mild suppression originates from the presence of lowfrequency vibrational modes that maintain a well-defined polarization and help preserve the thermal conductivity in the presence of disorder. KEYWORDS: Phonons, vibrational energy transport, disorder, graphene, diamond nanothreads

D

of materials such as nanocrystalline SiGe,6 nanoporous silicon7 and germanium,8 and functionalized graphene motivate their use as advanced thermoelectrics.9,10 When considering vibrational energy transport, there are several reasons why disorder in low-dimensional (low-D) materials may introduce effects that are distinct from three dimensions (3D). On one hand, the effects of any perturbation are typically more dramatic in lowD, so localization of transporting modes may be more enhanced. On the other hand, low-D materials are featured by long wavelength vibrational modes with long mean free paths, giving rise to large thermal conductivities that have been suggested by some to even diverge with increasing system size.11,12 How these modes are affected as disorder is gradually introduced into a low-D material is not known. Also, 2D materials exhibit anomalous flexural vibrational modes (e.g., the ZA modes of graphene), which have parabolic dispersion, and their localization behavior may be different from that of conventional linear modes. While the nature of localization in 3D disordered materials is associated with the presence of diffusive modes that transport heat in a random-walk

isorder in the atomic configuration of a material refers to a lack of regular patterns or predictability in the atomic positions. One consequence of disorder, if the degree of randomness is sufficiently large, is localization (sometimes called Anderson localization or strong localization), which refers to the absence of waves in a disordered medium.1,2 Although the effect was originally introduced to suggest the possibility of electron localization within a semiconductor arising from the presence of defects or impurities, the phenomenon is more general and applies to electromagnetic waves, lattice waves, quantum waves, spin waves, and others. The breakdown of the wave picture, and the resulting localization of the modes, can have dramatic effects on electronic, thermal, magnetic, and other properties of materials. This work in particular is concerned with the effect of structural disorder on phonon localization and vibrational energy transport in low-dimensional systems. Establishing how heat transport in low-dimensional systems is affected by disorder is important for both fundamental and technological reasons. For example, poly- and nanocrystalline silicon can play key roles in integrated circuits;3 polycrystalline yttria-stabilized zirconia has applications in thermal barrier coatings,4 and nanocrystalline diamond may be useful for thermal management in microelectronics.5 Exotic applications © 2016 American Chemical Society

Received: February 8, 2016 Revised: June 17, 2016 Published: July 7, 2016 4763

DOI: 10.1021/acs.nanolett.6b00557 Nano Lett. 2016, 16, 4763−4772

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Figure 1. (a) Structural model of a diamond nanothread with 20% Stone−Wales defect density and (b) radial distribution functions for diamond nanothreads with varying Stone−Wales defect densities. (c) Structural model of amorphous graphene, which maintains an sp2 configuration. (d) Radial distribution functions of 2D pristine and amorphous graphene. The structural models for both disordered systems are generated using the approaches described in the text.

manner,13,14 it is unknown whether such modes exist in low-D and what their nature may be. In the past few years the previously disparate fields of lowdimensional and disordered materials have been bridged by several developments. One is the emergence of the class of onedimensional diamond nanothreads (DNTs), which were recently synthesized in the laboratory for the first time.15 These nanothreads, illustrated in Figure 1a,b, are comprised of a hydrogenated (3,0) nanotube backbone of which the bonding exhibits an sp3 configuration. Additionally a random selection of bonds undergo a 90° rotation, introducing a distribution of Stone−Wales defects at high density along the tube. These defects introduce a 5|7|7|5 set of rings in the otherwise hexagonal lattice and are shown in green in Figure 1a. Another relevant development is the successful demonstration of “amorphous graphene”, which is a two-dimensional sheet that maintains the sp2 bond order of pristine graphene, but contains a disordered distribution of rings of different sizes varying from 4 to 8 atoms.16 This material results from electron irradiation of a graphene layer, under which graphene undergoes a transformation via step-by-step nucleation and growth of a twodimensional amorphous layer composed of sp2-hybridized carbon atoms. Figure 1c,d shows a picture of such a system. Relatedly, nanoporous graphene has also recently been synthesized and proposed for use in water desalination.17 In this work we use both glassy diamond nanothreads and amorphous graphene as examples to better understand how disorder affects vibrational transport in low-D.

To establish the features of vibrational transport in these lowD, disordered materials, we invoke vibrational mode analysis, molecular dynamics simulations, and a generalization of the Debye/Peierls18−20 and Allen/Feldman approach.13,14 Using vibrational mode analysis, we demonstrate that the lattice modes of disordered graphene and disordered diamond nanothreads share several features in common with lattice modes of disordered 3D materials, but also exhibit some differences. In analogy with 3D, we find the existence of Anderson localized modes as well as both propagating and diffusive modes. In contrast to 3D, however, we predict that the diffusive modes contribute negligibly to vibrational energy transport across the full temperature range. These diffusive modes are largely responsible for thermal transport in amorphous 3D materials at most temperatures of interest (T > 50−200 K, depending on the material), so their absence in low-D might be expected to result in a greater suppression of the thermal conductivity. By contrast, however, we find that the degree of suppression is similar to or even milder than in 3D despite the lack of contributions from diffusive modes. Our analysis shows that the lower frequency modes of amorphous graphene maintain a well-defined polarization, with atoms vibrating in the out-of-plane direction, despite the disorder. Relatedly, the low-frequency modes of the disordered 1D diamond nanothreads exhibit polarizations reminiscent of longitudinal and twist modes of the ordered 1D systems. The resilience of these low frequency modes to disorder, combined with their large relative contribution to the total thermal 4764


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Figure 2. Participation ratio of vibrational modes in (a) 1D diamond nanothreads and (b) 2D amorphous graphene, compared to their crystalline counterparts. For both systems, participation ratios are systematically reduced due to the structural disorder, which indicates the presence of delocalized and localized modes. Visual comparisons between modes of different frequency are presented in (c) for diamond nanothreads (backbone) and (d) for amorphous graphene, where the arrows denote the vibration direction of each atom.

and thus generally underestimates thermal conductivity. For instance, the thermal conductivity of a 200 nm long (10,10) carbon nanotube according to nonequilibrium molecular dynamics from AIREBO can be ∼3.6 times lower than that from optimized Tersoff.27 Detailed comparisons of the potentials are available in the literature.26−28 To generate diamond nanothreads, C−C bonds from a pristine, hydrogenated (3,0) carbon nanotube are selected at random to undergo a Stone−Wales transformation to create a 5|7|7|5 set of rings in the hexagonal network29 (see Figure 1). The radial distribution functions g(r) corresponding to systems with Stone−Wales defect densities of 5%, 20%, 50%, and 75% are shown in Figure 1a and are in good agreement with previously reported results.15 For the following analysis we used 8 nm long DNT segments and considered defect densities of both 20% and 50%. The sample with a 20% defect density corresponds to four random bond rotations in the 8 nm segment; this defect density exhibits a radial distribution function that best matches experiment15 and also is the same as that used in ref 29. For comparison purposes, we also considered a higher defect density of 50%, corresponding to 10 Stone−Wales transformations in the 8 nm sample. In all cases, the defects are randomly distributed along the nanothread. The amorphous graphene sample has dimensions of 1.25 × 2.16 nm2 and is generated following a procedure outlined in detail elsewhere.30 Pristine samples of the same size are first melted into 2D carbon gases at T = 4500 K, then quenched to the target temperature in 1 ns, which is followed by a Nose-Hoover thermostatting for 0.5 ns. The resulting

conductivity in low-D systems, enables a larger degree of the thermal conductivity to be maintained as disorder is introduced, especially at low temperatures. This effect is captured both by our molecular dynamics simulations as well as a generalized Debye/Peierls and Allen/Feldman model.21 To begin our analysis, we generated atomic-scale models for both disordered systems by following well-established procedures from the literature. We use the optimized Tersoff potential22 in the case of 2D systems (graphene and amorphous graphene) and the AIREBO23 potential in the case of 1D systems (the hydrogenated (3,0) carbon nanotubes and diamond nanothreads) to describe the interatomic interactions. We selected the optimized Tersoff potential for 2D systems to facilitate comparison of our results to those typically presented in the literature for graphene and related materials.24,25 This potential was optimized to better describe acoustic phonon group velocities and thermal properties of sp2 carbon materials and is expected to describe our pristine and amorphous graphene samples well. The diamond nanothreads require a potential that includes covalent C−H interactions and can accommodate sp3 character, and thus we selected the adaptive intermolecular reactive bond order (AIREBO) potential. This also allows comparison of our results for the 1D systems to other ultrathin nanotubes recently reported in the literature.26 It should be noted that, due to the separate choices of empirical interaction potentials, the individual results presented for 1D and 2D in this work cannot be directly compared. Although the two potentials are related, the AIREBO potential in comparison to the optimized Tersoff potential suppresses group velocities 4765


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Figure 3. Polarization spheres for (a) nanothreads and (b) graphene. The first polarization sphere in each row is for the pristine configuration. The polarization spheres for the disordered analogues are shown for varying ranges of frequencies. The size of subspheres denotes the number of atoms vibrating in each direction. In the amorphous nanothreads, the lowest frequency modes maintain well-defined longitudinal and twist-like polarizations. In amorphous graphene the low frequency modes maintain a well-defined polarization indicative of flexural vibrations. In both disordered systems the loss of polarization and increasing diffuson character is evident as the frequency increases.

where N denotes the number of atoms in the sample. In our samples, N = 1920 for diamond nanothreads, and N = 1600 for amorphous graphene. One measure of the degree of spatial localization of a vibrational mode can be obtained from its participation ratio (PR), which is a measure of the fraction of atoms that participate in the vibration.14,33,34 The participation ratio for mode λ, pλ, is given by the expression

amorphous graphene buckles naturally to relieve residual stresses, as shown in Figure 1d, where the color map denotes the buckling height. We note that these samples maintain the sp2 connectivity of the graphene lattice and thus are different from those generated by introducing vacancies,31 in which dangling carbon bonds are also present. Our localization and thermal conductivity results for these low-D systems are compared to corresponding features of 3D amorphous materials. This includes both amorphous silicon and amorphous carbon. While the former is the most extensively characterized amorphous 3D system with well-established experimental and numerical results available,13,14,32 the latter, although less extensively characterized, may form a more fair comparison since our low-D systems are also carbon based. Accordingly, we have also carried out analysis of 3D amorphous sp3 carbon, details of which are provided in the Supporting Information. We performed a real-space vibrational mode analysis on our low-D samples. Assuming that the vibrational modes can be represented by the simple form uiα , λ = (1/ mi )εiα , λ exp(iωλt ), the eigenfrequencies and eigenvectors were obtained by solution of the lattice dynamical equations ωλ2εiα , λ =

1 = N ∑ (∑ εi*α , λεiα , λ)2 pλ α i

The participation ratios for perfect crystal modes are generally higher than for the disordered systems. For instance, for a typical phonon with a well-defined wave vector, all or most atoms participate in the vibrations, and the participation ratio is unity, or close. For a fully delocalized mode it can be as low as 6(1/N ). We quantified the modal PRs for all modes for glassy diamond nanothreads and amorphous graphene, and they are illustrated in Figure 2a,b in comparison to the ordered and crystalline counterparts. We show here the PRs for the diamond nanothread with 20% defect density. The PRs for the system with 50% defect density are similar and available in the Supporting Information. The crystalline counterparts are a pristine graphene sample of the same size and a hydrogenated (3,0) carbon nanotube of the same length but without Stone− Wales defects. For the crystalline materials, the PR varies from 0.6 to 1 for the ordered nanothreads, and it is uniformly equal to one for graphene according to Figure 2. The presence of structural disorder introduces modifications to the modal PRs for diamond nanothreads and for amorphous graphene. For most frequencies the PRs are reduced to approximately 0.4−0.6 for both glassy systems. Only for frequencies >40 THz do the PRs drop below 0.4, which is close to the Debye vibrational cutoff frequency (∼45 THz). Both the presence of lowfrequency modes with reduced PRs (in the range 0.4−0.6 for us) and the more dramatic drop at higher frequencies are

∑ Φiα ,jβ εjβ ,λ jβ

(1)

with matrix elements Φiα , jβ =

1 ∂ 2V mimj ∂uiαujβ

(3)

(2)

where V is the potential energy. In these expressions, λ is a particular vibrational mode, uiα is the displacement of atom i in the Cartesian direction α, mi is the mass of the atom, and ω is the modal eigenfrequency and ε iα the corresponding eigenvector component. The normal modes of the disordered systems are obtained by diagonalizing the 3N × 3N matrix Φ, 4766


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vibrations both in the direction along and normal to the tube axis. For the disordered diamond nanothreads with 20% defect density (labeled “amorphous”), Figure 3a also shows the polarization spheres. The polarization spheres corresponding to the system with 50% defect density are shown in the Supporting Information and exhibit similar characteristics. Remarkably, despite the disorder the lowest frequency modes with ω < 0.45 THz possess a well-defined polarization along the êz direction and thus exhibit characteristics similar in nature to longitudinal phonons. When modes with frequency ω < 3.2 THz are plotted together, several new modes containing atoms vibrating in or near the xy plane appear, suggesting vibrational characteristics similar to twist and/or transverse modes. However, as the plotted frequency range increases, the atomic vibrations become more randomized and uniformly distributed over the surface of the sphere, suggesting the usual loss of wave character for the higher frequency members. Figure 3b shows the corresponding polarization spheres for the 2D systems, crystalline and amorphous graphene. Now the êz direction indicates the out-of-plane direction. For the crystalline material, the polarization sphere (Figure 3b, leftmost) contains large dots at ±êz, corresponding to the flexural modes. The equatorial ring in the xy plane corresponds to the longitudinal and transverse modes, all of which contain atoms oscillating in plane. The ring is uniform, reflecting the isotropic nature of graphene, with vibrations for all atoms across all modes uniformly distributed in all directions. For amorphous graphene, we again show several spheres for specified vibrational frequency ranges, to again illustrate the evolution of the modal nature. Interestingly, the lowest frequency members (ω < 0.8 THz) mainly contain atoms with reasonably well-defined polarization vibrating along or close to ±êz. This suggests a character similar to the flexural modes of graphene and remarkably is more or less maintained for modes even up to frequencies of ω < 2.2 THz. As the plotted vibrational frequency range increases, the direction of atomic vibrations again become more randomized and uniformly distributed over the surface of the sphere. This is again consistent with the usual loss of wave character in disordered or amorphous systems, particularly for the highest frequency members. The third metric that we use to characterize the modes in our disordered samples is the modal diffusivity. In 1993 and in the years following, Allen and Feldman developed a harmonic theory of heat transport in disordered solids featured by delocalized and unpolarized vibrations.13,14 In their formalism, carriers in disordered materials are classified as extendons and locons. Locons are high frequency, highly localized carriers which contribute negligibly to thermal conductivity. Extendons are more delocalized carriers that do contribute, and they are further classified into two subgroups. While the lowest frequency members, propagons, retain a propagating character and transfer energy in a manner reminiscent of phonons, the higher frequency dif f usons are somewhat more localized modes that transfer heat in a diffusive manner. Diffusons are nominally decoupled harmonic oscillators that become effectively coupled by the presence of a temperature gradient via the heat current operator in order to permit heat flow.13 All together, the Allen− Feldman thermal conductivity can be expressed as

common features of disorder-induced localization not only limited to vibrational modes, but waves in general (electronic, spin, etc.) in disordered media. Examples of both localized and delocalized vibrational modes for amorphous graphene and diamond nanothreads are shown in Figure 2c,d, where distinct spatial spans can be observed in each case. From these figures we observe that characteristically the modes with PR around 0.4−0.6 remain reasonably delocalized.13,14 Additional insight into the character of the vibrational modes of the amorphous samples can be obtained by considering the modal polarization spheres.13,14,34 As a material becomes more disordered, the directions of the atomic vibrations of a given mode λ become less wavelike, well-defined, and more randomized. This loss of polarization and loss of the definition of the wave vector can be captured by the polarization sphere, which by convention is a plot of the normalized direction of vibration of all atoms i of the eigenvector εiα,λ for mode λ. Formally the polarization sphere shows eiα,λ, the projection of the modal eigenvectors εiα,λ onto unit spheres for all atoms i: eiα , λ =

εiα , λ ∑α (εi*α , λεiα , λ)

(4)

In a pristine crystalline material, the normal modes are all phonon modes possessing well-defined polarization. For example, for a ZA mode in crystalline graphene, all atoms oscillate in parallel along the êz direction, so each atom i has a unit polarization vector component of ±1 in the êz direction and 0 in the other directions. The polarization sphere would consist of dots marked only at ± êz, one dot for every atom. By contrast in an amorphous system for a mode that has lost all polarization, the associated atomic displacements would be entirely uncorrelated and distributed uniformly over the sphere. It is noteworthy that the polarization sphere shows not the direction of the wave vector, but rather the direction of atomic vibrations of a given mode directly. For example, in our hydrogenated (3,0) carbon nanotubes oriented along the z-axis, a TA mode with a wave vector along the tube axis (parallel to êz) would have atomic vibrations confined to the xy plane, and thus appear as dots on the equatorial xy plane of the polarization sphere. Figure 3 shows polarization spheres for glassy diamond nanothreads and for amorphous graphene, again in comparison to the crystalline analogue materials. Note that, different from convention, we simultaneously show the polarization sphere for all modes in a specified frequency range, rather than a single isolated mode, in order to reveal frequency-dependent localization trends. The size of the dots plotted on the spheres indicate the total number of atoms, summed over all modes in the frequency range, oscillating in that particular direction. While the PRs for the disordered systems in Figure 2 illustrate that most of the modes remain reasonably delocalized, the polarization characteristics reveal some surprising features. In Figure 3a, the leftmost panel shows the vibrational directions of the atoms for all modes for the ordered, hydrogenated (3,0) carbon nanotube oriented along the êz direction. The sphere shows the expected symmetries for the pristine system. For example, the six large spheres in the xy plane correspond to the twist modes TW and exhibit hexagonal symmetry about êz. The large spheres at ±êz correspond to longitudinal modes, and the equatorial ring in the xy plane corresponds to transverse modes. Finally, the three evenly spaced equatorial rings passing through the ±êz poles are modes that are helical in nature, containing

κAF = 4767

1 V

∑ cλDλ λ

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Figure 4. Spectral diffusivity for (a) 1D nanothreads and (b) 2D amorphous graphene. Two transitions can be observed for both systems: the Ioffe− Regel edges (∼0.45 THz for nanothreads and ∼0.6 THz for amorphous graphene), which separate diffusons from propagons, and the mobility edges (∼47.5 THz for 1D nanothreads and ∼55.8 THz for 2D graphene), which separate locons from extendons.

where V is the volume, cλ is the heat capacity of mode λ, and Dλ its diffusivity35 Dλ =

πV 2 3ℏ2ωλ2

modes are typically not populated at most temperatures of interest. The milder transition in Figure 4 between propagons and diffusons occurs at frequencies of around 0.45 THz and 0.6 THz, respectively, for diamond nanothreads and for amorphous graphene. By contrast, the corresponding transition for 3D amorphous carbon occurs at around 0.6 THz (see Supporting Information); for 3D amorphous silica and amorphous silicon the corresponding values have been reported to be 0.25 THz32 and 1.1 THz,13,14 1.8 THz,32 respectively. For the 1D nanothreads, the transition in Figure 4 is consistent with the polarization spheres in Figure 3a, which indicate a quick change in polarization from longitudinal to transverse and twist-like across the boundary. For 2D graphene, the transition at 0.6 THz is smoother, and we find that it does not correspond to an observable sharp change in the polarization spheres in Figure 3b. For instance, the spheres show that several modes with frequencies as large as ∼2.2 THz maintain atomic vibrations largely aligned with the out-of-plane direction. Some examples of these vibrational modes are plotted in Figure 4. The participation ratios, polarization spheres, and diffusivity analysis suggests that the effects of disorder on vibrational transport in low-dimensional carbon systems exhibit some similarities and some differences from the conventional wisdom about the effects of disorder in three dimensions. On one hand the trends in the participation ratios are similar, and the diffusivity analysis suggests the presence of propagons, diffusons, and locons. The modal diffusivity plots for diamond nanothreads, amorphous graphene, and amorphous 3D carbon all exhibit apparent propagon/diffuson boundaries in the freqeuncy range around 0.4 to 0.6 THz. On the other hand, the polarization spheres suggest that, in amorphous graphene, the low-frequency vibrational modes exhibit a polarization reminiscent of graphene flexural modes which persists to large frequencies. In diamond nanothreads the lowest frequency modes are reminiscent of longitudinal modes (as well as some transverse and twist modes), and maintain their polarization to frequencies around 0.45 THz. Having scrutinized the atomic scale character of the vibrations and their localization, we now explore the consequences of these features on the thermal conductivity of the materials. For this, we have implemented two independent approaches. The first invokes equilibrium molecular dynamics (EMD) simulations and the Green−Kubo

∑ |Sλ ,μ|2 δ(ωλ − ωμ) λ≠μ

(6)

where Sλ,μ is the heat current operator in the harmonic approximation.13 Typically when the modal diffusivities in disordered systems are plotted as a function of frequency, two transitions are observed. There is a sudden drop at high frequency and a milder transition at lower frequency from a power-law scaling regime to a constant plateau. The sudden drop at high frequency is the mobility edge, associated with a transition from diffusons to locons (strong localization). The mild transition at low frequency represents the boundary between propagons and diffusons and is often called the Iof fe− Regel transition. We performed a modal diffusivity analysis for the vibrational modes of the diamond nanothreads and amorphous graphene, the results of which are shown in Figure 4. We again show the diffusivities for the diamond nanothread with 20% defect density and present the 50% defect density results in the Supporting Information. The corresponding results for the modal diffusivities of 3D amorphous carbon are also shown in the Supporting Information. The low-D systems exhibit features similar to what is found for 3D disordered systems, with the glassy nanothreads and the amorphous graphene both exhibiting a propagon/diffuson boundary and a mobility edge. These are marked on the figure. Although the modal diffusivity plots for both systems indicate that Anderson localized modes are present, the boundary occurs at frequencies >40 THz, near the vibrational Debye frequency cutoff ∼45 THz. The observed mobility edge is consistent with the results from the PRs in Figure 2, which also show a drop around 40 THz. We observe a mobility edge in 3D amorphous carbon as well (Supporting Information), although it occurs in an extended range around 30−50 THz. This behavior is also similar to 3D amorphous silicon, for which the mobility edge has been reported to occur around 17 THz,13,14 also close to the corresponding Debye cutoff frequency (∼19 THz). Therefore, although we can rigorously observe a mobility edge in the low-D systems, it is physically of little consequence, since these highest frequency 4768


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Figure 5. Thermal conductivity of (a) 1D diamond nanothreads and (b) 2D amorphous graphene as a function of temperature, obtained from equilibrium molecular dynamics. Note that amorphous graphene melts above T = 600 K. The thermal conductivity in both cases scale inversely with temperature (κ ∝ T−m, m > 0), akin to crystalline cases, due to the dominant contributions of propagons. (c,d) Generalized model prediction of the temperature dependence of the thermal conductivity for 1D nanothreads and 2D amorphous graphene, respectively, using best fit propagon/diffuson boundary ωξ (solid line) and for ±10% change to ωξ (dashed lines). The insets show the individual contributions of propagons, further broken down by polarization, and diffusons. Diffusons contribute negligibly to the thermal conductivity for both low-D systems. Inset: Adapted with permission from ref 21. Copyright 2016 American Physical Society.

formalism to assess the thermal conductivity κ of the disordered systems, as well as its temperature dependence. The second is an adaptation of the approaches of Debye/Peierls18−20 and Allen/Feldman13,14 into an analytical framework specific to low-dimensional materials, to which we compare the atomistic molecular dynamics results. This analytical framework is described in detail in ref 21. Our molecular dynamics simulations are carried out using LAMMPS.36 For consistency with the modal analysis, we again used the AIREBO potential23 for the 1D systems and the optimized Tersoff potential22 for the 2D systems. In all cases we use an 0.1 fs time step. Both systems are initially thermalized to the desired temperature with a Nose-Hoover thermostat (typically for 500 ps), and the computational supercell dimensions are relaxed so as to yield zero stress. Once the target temperature is attained, equilibrium molecular dynamics simulations are initiated and run for >50 ns in all cases. For each value of thermal conductivity reported, we have performed five independent microcanonical runs using statistically independent initial ensemble realizations, which is necessary for proper sampling due to the weak ergodicity of the systems. The thermal conductivity is obtained using the Green−Kubo approach37 from the time integral of the autocorrelation of the heat flux. Details and convergence tests for the molecular

dynamics simulation parameters and the Green−Kubo integration are provided in the Supporting Information. The error bars shown in our results indicate 95% confidence intervals, and the small magnitude suggests reasonable convergence in all cases. Figure 5a,b shows the EMD results for κ versus T for diamond nanothreads and graphene. For both low-D cases, we compare the disordered systems to the crystalline counterparts. The results are surprising, but consistent with several aspects of the localization analysis. In both cases κ is suppressed by the disorder, as expected. In the glassy nanothreads with both 20% and 50% defect density, κ is reduced by a factor of ∼5 across the full temperature range sampled. For amorphous graphene, the suppression is less evident, with κ retaining more than 75% of its crystalline value throughout the full temperature range sampled. Thus, the suppression of κ in graphene appears to be particularly mild when disorder is introduced. To probe these observations further, we designed and implemented a generalized model to capture the nature of κ versus T in low-dimensional, disordered materials. The model is described in detail elsewhere,21 but it weaves the ideas of Allen and Feldman into a framework reminiscent of the Debye/ Peierls approach18−20 to the calculation of κ. According to this framework, heat is carried by quanta of propagating modes, and 4769


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1

∑ ∫ vg2(ω)τ(ω)Cph(ω)g(ω) dω d m

κ=

statistics from quantum Bose−Einstein statistics. The Debye temperature of graphene is ∼2100 K,39 so the simulation temperature of T = 200 K is low in comparison. By contrast the Debye temperature of a carbon nanotube is typically lower, e.g., reported to be 473 K40 in one case, and hence the discrepancy is not as large.] This best match is obtained for propagon/ diffuson boundaries of ωξ ≈ 0.45 THz for the glassy nanothreads and ωξ ≈ 0.8 THz for amorphous graphene. Notably, these estimates of the propagon/diffuson boundaries are reasonably consistent with the boundaries suggested by the modal diffusivities in Figure 4. The match is very good for the nanothreads (both 0.45 THz), but less so for graphene (0.6 THz vs 0.8 THz). The discrepancy in the latter case can be attributed to the fact that several modes with frequencies >0.6 THz still retain a well-defined flexural polarization. Thus, the best effective measure of the boundary occurs at a larger frequency of 0.8 THz according to our generalized model. The dashed blue lines in Figure 4c,d show κ versus T obtained for a ± 10% change in ωξ, to show the sensitivity of our results to the crossover boundary. As the effective propagon/diffuson boundary for modes in amorphous graphene does not occur until ∼0.8 THz, a large portion of the low-frequency flexural carriers in amorphous graphene retain their propagon likenature, which is consistent with the markedly small reduction of κ in Figure 5a,b and the localization analysis. The propagon/ diffuson boundary for diamond nanothreads occurs at lower frequency of 0.45 THz, and the reduction in κ is greater than for amorphous graphene. The generalized model also allows us to isolate the contributions to κ coming from propagons (further broken down by their polarization) and diffusons. These individual contributions are shown in the insets of Figure 5c,d. For amorphous graphene, the model clearly captures the dominant contribution of the flexural ZA modes (the red dashed line is nearly superposed over the solid blue line) suggested by the polarization spheres. For the glassy nanothreads, it captures the dominant contributions of longitudinal (LA) and twist (TW) modes, also consistent with the polarization spheres. An interesting observation from Figure 5c,d is the negligibly small contribution of the diffusons (green lines) for both glassy nanothreads and amorphous graphene. For these materials, propagons turn out to be the dominant carriers of heat at all temperatures. This contrasts, for example, with the behavior observed in 3D amorphous silicon, for which diffusons become the dominant contributors to κ for temperatures above ∼200 K. It also contrasts with our predictions for 3D amorphous carbon which also indicates that diffusons become activated at higher temperatures T > 100 K (Supporting Information, Figure S5). One consequence of the negligible contribution of diffusons would be a different scaling of κ versus T in low-D systems compared to 3D. For example, in amorphous silicon at low temperatures before diffusons are activated (T < 200 K), the suppression of κ, relative to the crystalline system, can be very large (a factor of ∼103−104). As the temperature increases, the diffuson contribution helps to recover κ and reduce the degree of suppression 101−102 (depending on the temperature and particular sample). Similarly, our predictions of 3D amorphous carbon also show that the suppression is ∼104 at the lowest temperatures before diffuson activation but then decreases to ∼10 at room temperature. In both cases, the suppression decreases as κ recovers with diffuson activation. By contrast, if diffusons are negligible contributors in low-D, the degree of suppression would be steady throughout the full temperature

∑ m

1 d

∫ vg Λ(ω)Cph(ω)g(ω) dω

(7)

(8)

where d denotes the dimension, vg the group velocity, τ(ω) the scattering time, Λ(ω) = vgτ(ω) the mean free path, Cph(ω) the heat capacity, and g(ω) the phonon density of states of the perfect crystal. Our model generalizes the Debye/Peierls approach to calculate κ and its temperature dependence in two principal ways. First, it explicitly accounts for the presence of modes with parabolic as well as linear dispersion, since the parabolic modes that appear in low-dimensions (such as the flexural modes of graphene) have different group velocities and density of states. This can have an important consequences on the scaling behavior of κ versus T. Second, since the analysis of modal PRs, polarization spheres, and diffusivities (Figures 2−4) suggests the presence of both propagons (low-frequency vibrational members that retain typical phonon-like features) as well as diffusons (higher frequency members that are spatially delocalized but lack a welldefined polarization), we introduce a crossover frequency ωξ and associated wave vector kξ that represents the boundary between propagons and diffusons. Vibrational modes with frequency ω < ωξ are classified as propagons, and their contribution to κ is obtained from eq 7. Since these modes are most similar to phonons (delocalized and characterized by a well-defined polarization), they are subject to typical phonon scattering mechanisms arising from boundaries, defects, and phonon−phonon interactions (both Normal and Umklapp). For these modes we use scattering descriptions commonly invoked in single-mode relaxation time modeling to describe τ(ω).21 By contrast, the modes with frequency ω > ωξ are designated as diffusons (delocalized, but without a well-defined polarization). The physical picture of their transport mechanism is that of a random walk of energy between localized and decoupled oscillators of different sizes and frequencies.38 To obtain their contribution to κ, we followed the approach of Cahill and Pohl38 in which we choose an effective mean free path step Λ ≈ λ/2 = πv/ω in eq 8 to capture the “random walk” nature of the diffusive vibrational transport (λ refers to the wavelength of the original mode of the crystalline system). The crossover frequency ωξ is typically the only adjustable parameter in our generalized model, and it gives insight into the propagon/diffuson boundary for a given disordered system. We previously validated our approach by comparing its predictions to available experimental results of κ versus T for 3D vitreous silica and find excellent agreement for the choice ωξ = 0.25 THz.21 The model is able to capture the temperature dependence of κ versus T across the full temperature regime, including the plateau region which represents a transition from propagon-dominated to diffuson-dominated transport. We then applied the model to the low-D systems of interest here, to better capture the physics underlying the trends. A comparison of the predictions of the model to the equilibrium molecular dynamics results are presented in Figure 5c,d, which shows a very good match. [Note that the discrepancy observed at low temperature T = 200 K in graphene is typical for classical molecular dynamics results due to the deviation of Boltzmann 4770


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regime. We further predict that in low-D amorphous systems, in contrast to 3D, κ will simply drop with increasing T, as the dominant effect will be the increasing effects of phonon− phonon scattering rather than the activation of diffusons. One of the important predictions of our work, therefore, is that unlike 3D amorphous materials their 1D and 2D analogues will not exhibit a plateau in the κ versus T curve. It is interesting that the degree of suppression in the low-D systems is similar to or milder than (especially for graphene) the 3D value, even without the help of diffusons. We attribute this to the large overall contribution to κ from the long wavelength modes in the low-D systems (i.e., ballistic carriers in 1D nanothreads and flexural ZA modes in graphene), which is captured by our generalized model. We can obtain some practical insights into the reason for the negligible contributions of diffusons by considering the scaling of the integrand21 of eq 8. The integrand decays with frequency in 1D, is constant in 2D, and grows in 3D for diffusons with mean free path set by the random walk step Λ ≈ λ/2 = πv/ω. Diffusons transport heat in a diffusive random walk manner, and we speculate that inherent differences in the nature of random walks in one, two, and three dimensions may influence their effectiveness as heat carriers. For example, according to Polya’s recurrence theorem, a well-known theorem from combinatorial mathematics, a simple random walk on a ddimensional lattice is recurrent for d = 1,2 but transient for d ≥ 3. Recurrent means that the walker returns to the starting point with probability one, and transient means that it does not. This can be intuitively understood by considering that, in a 1D network, after a random walker takes the first step, it has a probability of 1/2 of returning to its starting point in second step. In 2D the probability is reduced to 1/4, and in 3D it is reduced to 1/6. Since the number of paths going away from the starting point increases as dimension increases, as the dimension increases, the walkers are more likely to diffuse away from the starting point. An intriguing possibility is that the space-filling nature of random walks in 3D make diffusons more effective heat carriers, in comparison to the recurrent nature of random walks in 1D and 2D. In conclusion, we have analyzed the thermal transport properties of amorphous graphene and glassy diamond nanothreads. We find several unique aspects of the vibrational modes in these low-dimensional, disordered systems. In analogy with 3D we observe the presence of modes similar in nature to propagons, diffusons, and locons. In the glassy diamond nanothreads, despite the presence of disorder the lowest frequency modes exhibit well-defined longitudinal and twist-like polarization. For amorphous graphene, the lowest frequency modes maintain a well-defined flexural out of plane polarization. The presence of these modes despite the disorder, combined with their large relative contribution to the total thermal conductivity in the crystalline counterparts, results in a suppression that is similar to or milder than typical 3D materials (especially at low T), even without the contributions of the diffusons. We suggest that the negligible diffuson contribution may reflect a fundamental difference in the nature of random walks in low dimensions in comparison to three dimensions. These results present an intriguing set of predictions for the experimental measurement of thermal conductivity in disordered, low-dimensional materials.

Letter

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b00557. Assessment of different defect densities in diamond nanothreads, equilibrium molecular dynamics simulation parameters and convergence, discussion of the calculation of Green−Kubo thermal conductivity, and predictions for 3D amorphous carbon (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 We gratefully acknowledge fruitful discussions with David Cahill and Sanjiv Sinha. We acknowledge financial support from National Science Foundation under Grant No. CBET9122625 and Grant No. DMR-1555278. We also acknowledge support from various computational resources: this research is part of the Blue Waters sustained petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana−Champaign and the National Center for Supercomputing Applications. Additional resources were provided by (i) the Extreme Science and Engineering Discovery Environment (XSEDE) allocation DMR-140007, which is supported by National Science Foundation Grant No. ACI-1053575 and (ii) the Illinois Campus Computing Cluster.

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