Length-dependent Thermal Conductivity Research Articles

Length dependent thermal conductivity of silicon and copper nanowire: a molecular dynamics study

The miniaturization and higher power density of modern electronics pose a significant challenge in thermal management. A key focus in addressing this challenge revolves around the advancement of thermal interfaces within microchip packaging, aiming to enhance thermal energy dissipation and optimization of performance. Copper nanowires are extensively employed in the chip industry as interconnects for signal transmission and thermal management purposes. Investigating the impact of reduced cross-section on the thermal transport properties of nanowires is crucial. In this study, the thermal conductivity of copper and silicon nanowires is studied with variations in the length of the nanowires. The simulation is conducted with the Equilibrium Molecular Dynamics (EMD) process. The cross-section of the nanowire is kept fixed (10 × 10 nm) and with the increase in length, its thermal conductivity is studied. At room temperature for a 50 nm length, the lattice thermal conductivity value is 1.68 and 0.037 W m − 1 K − 1 for silicon and copper nanowires, respectively. We further studied the phonon scattering, mean free path, and group velocity of silicon and copper lattices. Our study may help to design more thermally efficient microchips and innovate new cooling methods of microelectronics.

Dependence of thermal conductivity of graphene on the coefficient of linear expansion and temperature

Graphene attracts particular interest recently due to its extraordinary properties. Xu et al. have studied length-dependent thermal conductivity in suspended single-layer graphene. They observed that the thermal conductivity of graphene increases and remains logarithmic divergence with sample length. Yoon et al. observed that the coefficient of linear expansion (α) for graphene is always negative between at least up to 2500 K. In this paper, we study the dependence of thermal conductivity of graphene on the coefficient of linear expansion (CLE). We find that the thermal conductivity of graphene bears an exponential relationship with its CLE and is directly proportional to the sample length. Using the Slack equation, we have also found that the thermal conductivity of graphene is inversely proportional to the temperature and verified the relationship between the thermal conductivity of graphene and CLE. Due to this property graphene can be utilized to remove fish plates in railroads which can increase the efficiency of railway tracks.

Ballistic heat conduction in semiconductor nanowires

Ballistic thermal transport is a remarkable nanoscale phenomenon with possible applications in microelectronics. In the past decade, research on ballistic thermal transport focused on the measurements of length-dependent thermal conductivity in semiconductor nanowires. In this Perspective article, we review the experimental demonstrations of this phenomenon in nanowires of various materials and sizes and at different temperatures. Our literature analysis reveals a controversy between works reporting two distinct pictures of ballistic conduction: perfectly ballistic conduction even at room temperature and weak quasi-ballistic conduction mainly below room temperature. Simulations seem to support the weaker version of the effect. Thus, future experiments are expected to resolve this controversy.

Thermal transport characteristics of supported carbon nanotube: Molecular dynamics simulation and theoretical analysis

With the rapid development of carbon nanotube (CNT)-based devices, thermal transport characteristics of CNT supported on substrate is to be urgently explored. Herein, we systematically study the thermal transport properties of supported single-wall CNT including length, temperature and substrate coupling strength dependence of thermal conductivity using molecular dynamics (MD) simulation. Interestingly, a theoretical model is proposed to describe the length dependent thermal conductivity of supported SWCNT, which matches well with our MD results. Moreover, the phenomenon of non-monotonic change of thermal conductivity with the variation of coupling coefficient is first discovered in supported CNT. The underlying mechanism is revealed by spectral energy density (SED), phonon transport and stress distribution analysis. Our study provides valuable insights towards the thermal dissipation performance of CNT-based electronic and photonic devices.

Scaling behavior of thermal conductivity in single-crystalline α-Fe2O3 nanowires**Project supported by the Key-Area Research and Development Program of Guangdong Province, China (Grant No. 2020B010190004), the National Natural Science Foundation of China (Grant Nos. 11674245, 11775158, 11890703, and 11935010), and the Open Fund of Zhejiang Provincial Key Laboratory of Quantum Technology and Device, China (Grant No. 20190301), and the Shanghai Committee

Unveiling the thermal transport properties of various one-dimensional (1D) or quasi-1D materials like nanowires, nanotubes, and nanorods is of great importance both theoretically and experimentally. The dimension or size dependence of thermal conductivity is crucial in understanding the phonon–phonon interaction in the low-dimensional systems. In this paper, we experimentally investigate the size-dependent thermal conductivity of individual single crystalline α-Fe2O3 nanowires collaborating the suspended thermal bridge method and the focused electron-beam self-heating technique, with the sample diameter (d) ranging from 180 nm to 661 nm and length (L) changing from 4.84 μm to 20.73 μm. An empirical relationship for diameter-/length-dependent thermal conductivity is obtained, which shows an approximately linear dependence on the aspect ratio (L/(1 + Cd)) at T = 300 K, where C is a fitting parameter. This is related to the boundary scattering and diameter effect of α-Fe2O3 nanowires although rigorous calculations are needed to confirm the result.

Thermal conductivity and scattering models for graphene: From intrinsic scattering of pristine graphene to strong extrinsic scattering of functionalized graphene

The present study used various scattering models in a Callaway framework to elucidate the relationship between the scattering mechanisms and thermal conductivity of graphene. As well as intrinsic scattering, the scattering caused by external conditions was constructed using a combination of boundary, imperfection, and interface scattering. The effect of each type of scattering on the length-dependent thermal conductivity of graphene, which is a subject of fundamental importance, was evaluated. The change in specularity shifted the logarithmic length-dependent regime from the micrometer scale for a diffuse boundary to the submicrometer scale for high specularity. Imperfection scattering had little effect on the intrinsically determined length dependence of thermal conductivity. Substrate scattering, that is, interface scattering by van der Waals interaction, caused large decreases of both the thermal conductivity and length scale. A rough model for functionalized graphene was constructed by considering both impurity and interface scattering. The spring constant in interface scattering with functionalizing bond was nearly 40 times larger than that for substrate scattering, in combination with an imperfection source, explained the length-independent low thermal conductivity of functionalized graphene. This model was extended to nonsubstitutionally functionalized graphene, for which imperfection scattering strength was proportional to the mass of the adatom.

Enhancement of thermoelectric figure-of-merit of graphene upon BN-doping and sample length reduction

Using first-principles density functional perturbation theory based calculations of length-dependent lattice thermal conductivity (κL) and using our previously calculated results [R. D’Souza and S. Mukherjee, Phys. Rev. B 95, 085435 (2017)] of electrical transport, we report results of thermoelectric figure-of-merit (ZT) of monolayer and bilayer graphene. We find a nearly tenfold increase in ZT for the graphene sample doped with boron nitride and reduced sample length. We also compare κL calculated using the iterative real space method with conventional analytical Callaway-Klemens method and obtain the flexural (ZA) phonon modes to be dominant in thermal transport unlike in the latter method. Our calculations are in good agreement with available experimental data.

Equivalence of the equilibrium and the nonequilibrium molecular dynamics methods for thermal conductivity calculations: From bulk to nanowire silicon

Molecular dynamics simulations play an important role in studying heat transport in complex materials. The lattice thermal conductivity can be computed either using the Green-Kubo formula in equilibrium MD (EMD) simulations or using Fourier's law in nonequilibrium MD (NEMD) simulations. These two methods have not been systematically compared for materials with different dimensions and inconsistencies between them have been occasionally reported in the literature. Here we give an in-depth comparison of them in terms of heat transport in three allotropes of Si: three dimensional bulk silicon, two-dimensional silicene, and quasi-one-dimensional silicon nanowire. By multiplying the correlation time in the Green-Kubo formula with an appropriate effective group velocity, we can express the running thermal conductivity in the EMD method as a function of an effective length and directly compare it with the length-dependent thermal conductivity in the NEMD method. We find that the two methods quantitatively agree with each other for all the systems studied, firmly establishing their equivalence in computing thermal conductivity.

Length-dependent lattice thermal conductivity of single-layer and multilayer hexagonal boron nitride: A first-principles study using the Callaway-Klemens and real-space supercell methods

The phonon dispersion, density of states, Gr\"{u}neisen parameters, and the lattice thermal conductivity of single- and multi-layered boron nitride were calculated using first-principles methods. For the bulk {\it h}-BN we also report the two-phonon density of states. We also present simple analytical solutions to the acoustic vibrational mode-dependent lattice thermal conductivity. Moreover, computations based on the elaborate Callaway-Klemens and the real space super cell methods are presented to calculate the sample length and temperature dependent lattice thermal conductivity of single- and multi-layered hexagonal boron nitride which shows good agreement with experimental data.

Reactive force field simulation on thermal conductivities of carbon nanotubes and graphene

Carbon nanotubes (CNTs) and graphene are thought to be building bricks for next-generation thermal management devices due to their ultra-high thermal conductivities. However, in practical applications they do not exhibit their full potentials due to issues from interfaces, dopants, and functional groups. Thus, deep understanding of these effects on the thermal conductivities is needed. Molecular dynamics (MD) simulation can provide such insights into these material systems. However, prediction capability is limited by a narrow choice of chemical elements in traditional equilibrium MD potentials. Herein, non-equilibrium MD (NEMD) simulations were implemented to study the thermal conductivities of CNTs and graphene by using ReaxFF potentials (Reax-03, Reax-12, and Reax-15), all of which enable a wider choice of the element category. In the simulations, CNT and graphene have maximum lengths of 400nm. The length-dependent thermal conductivities were studied in those systems. The ReaxFF potentials predicted higher thermal conductivities and longer phonon mean free paths than the traditional AIREBO potential. The geometry and location of G-peaks shown in the phonon density of states calculated by the Reax-03 and Reax-12 potentials were similar to the results obtained from the AIREBO potential. In contrast, a wider G-peak was obtained from the Reax-15 potential. The phonon dispersion curves based on each potential showed that Reax-03 and Reax-12 can predict accurate out-of-plane modes when compared with experiments. The Grüneisen parameters of the ZA mode based on the Reax-03 and Reax-12 potentials are smaller than those based on the AIREBO potential, indicating a lower anharmonicity of lattice vibrations. Radical distribution functions from the Reax-03 and Reax-12 potentials showed a higher lattice stiffness and longer phonon mean free paths (MFPs) than those obtained from the AIREBO potential. These results illustrate the capability of ReaxFF in predicting thermal conductivities of low dimensional carbon allotropes and their derivatives for advanced heat management.

Exploiting length-dependent effects for the design of single-material systems with enhanced thermal transport properties

Topology optimization approaches for thermal transport problems generally rely on the presence of high contrast in thermal conductivity between constituent materials. Even though this might result in topological configurations that are optimal according to some desired thermal metric, the resulting systems might be impractical from a mechanical standpoint: high contrast in thermal conductivity generally translates into disparate ranges of thermal expansion, which in turn may lead to mechanical failure. In this paper, we focus on the combination of multi-scale modeling and topology optimization techniques for the design of polycrystalline single-material systems with enhanced thermal transport properties. Spatial variation of the length-dependent thermal conductivity is achieved by grading the characteristic grain size of the microstructure. We adopt a multi-scale approach to compute the attainable range of thermal conductivity due to grain-size effect, given maximum and minimum grain sizes and an operating temperature range. Then, we utilize an adaptive topology optimization technique to obtain an optimal grain size distribution, while accounting for manufacturing constraints. Through a series of specific examples, we demonstrate the extent to which thermal transport properties can be optimized without resorting to the adoption of multi-material systems.

Measuring Phonon Mean Free Path Distributions by Probing Quasiballistic Phonon Transport in Grating Nanostructures.

Heat conduction in semiconductors and dielectrics depends upon their phonon mean free paths that describe the average travelling distance between two consecutive phonon scattering events. Nondiffusive phonon transport is being exploited to extract phonon mean free path distributions. Here, we describe an implementation of a nanoscale thermal conductivity spectroscopy technique that allows for the study of mean free path distributions in optically absorbing materials with relatively simple fabrication and a straightforward analysis scheme. We pattern 1D metallic grating of various line widths but fixed gap size on sample surfaces. The metal lines serve as both heaters and thermometers in time-domain thermoreflectance measurements and simultaneously act as wire-grid polarizers that protect the underlying substrate from direct optical excitation and heating. We demonstrate the viability of this technique by studying length-dependent thermal conductivities of silicon at various temperatures. The thermal conductivities measured with different metal line widths are analyzed using suppression functions calculated from the Boltzmann transport equation to extract the phonon mean free path distributions with no calibration required. This table-top ultrafast thermal transport spectroscopy technique enables the study of mean free path spectra in a wide range of technologically important materials.

Thermal Transport in Fullerene Derivatives Using Molecular Dynamics Simulations.

In order to study the effects of alkyl chain on the thermal properties of fullerene derivatives, we perform molecular dynamics (MD) simulations to predict the thermal conductivity of fullerene (C60) and its derivative phenyl-C61-butyric acid methyl ester (PCBM). The results of non-equilibrium MD simulations show a length-dependent thermal conductivity for C60 but not for PCBM. The thermal conductivity of C60, obtained from the linear extrapolation of inverse conductivity vs. inverse length curve, is 0.2 W m−1 K−1 at room temperature, while the thermal conductivity of PCBM saturates at ~0.075 W m−1 K−1 around 20 nm. The different length-dependence behavior of thermal conductivity indicates that the long-wavelength and low-frequency phonons have large contribution to the thermal conduction in C60. The decrease in thermal conductivity of fullerene derivatives can be attributed to the reduction in group velocities, the decrease of the frequency range of acoustic phonons, and the strong scattering of low-frequency phonons with the alkyl chains due to the significant mismatch of vibrational density of states in low frequency regime between buckyball and alkyl chains in PCBM.

(Invited) Phonon Transport in Holey Silicon Nanostructures

When the size of semiconductors is smaller than the phonon mean free path, phonons can carry heat with no internal scattering. Ballistic phonon transport has received great attention for understanding heat conduction in nanomaterials and managing heat dissipation in nanoelectronics. While past experiments on thin films and nanowires focused on the size effect in lateral dimensions such as film thickness and nanowire diameter, few reports demonstrated the size dependence in the same direction as heat flow. Recent experiments are showing the length dependent thermal conductivity and the possible presence of ballistic phonons in SiGe nanowires and silicon membranes in the length scale of 1 µm or greater. However, the thermal conductivity demonstration in the length scale of 10-100 nm range, or below the phonon mean free path in silicon, is not available due to challenges associated with sample preparation and low measurement sensitivity. Here we demonstrate the length dependence through cross-plane thermal conductivity measurements of holey silicon in the length scale of 35-200 nm, which has direct relevance to nanoscale transistors and potential phononic devices. The thermal conductivity scales linearly with the length (i.e. thickness of holey silicon layer) up to 200 nm at room temperature, even though the lateral dimension is as narrow as 20 nm. We attribute this to the strong presence of ballistic phonon transport and the large contribution of long-wavelength phonons that can carry the heat from top to bottom surfaces through the narrow channels. We assess the impact of long-wavelength phonons and predict a transition from ballistic to diffusive regime using scaling models based on the frequency dependent mean free path. The frequency dependent boundary scattering may be responsible for the unique size effect and the non-classical low-temperature dependence because low-frequency phonon is less susceptible to surface disorder. We also demonstrate the in-plane thermoelectric measurement results using a fully integrated experimental platform that allows simultaneous characterization of electrical conductivity, Seebeck coefficient, and thermal conductivity. The integrated platform improves the thermoelectric measurement accuracy and allows systematic investigation of electron and phonon transport through varying neck size, porosity, and doping concentration. The holey silicon nanostructure devices are fabricated with a pitch size of 60 nm and neck sizes in the range of 16 -34 nm, which opens up the possibility of studying incoherent and coherent phonon transport in periodic structures. The detailed analysis on the thermoelectric figure of merit indicates that the doping concentration, the pitch, and the neck size are critical parameters for fabricating all-silicon based thermoelectric devices. The holey silicon nanostructures offer unique opportunities to study phonon transport at the nanoscale and beyond the limit of classical approaches, which can provide new pathways of manipulating phonons for thermoelectric energy harvesting and potential phononic information-processing applications. Figure 1

Length, width and roughness dependent thermal conductivity of graphene nanoribbons

This letter reports on the investigation of temperature dependent thermal conductivity along a long free-standing graphene nanoribbon (GNR) by a one-dimensional direction-dependent phonon-boundary scattering in the frame of linearized phonon Boltzmann transport method. It is observed that GNR has highly length and width dependent thermal conductivity due to the difference between in-plane and out-of-plane phonon modes in the phonon dispersion curve. We show that phonon mediated thermal conduction increases with size, while it decreases with increasing edge roughness. These results are relevant for tailoring and engineering of the thermal conductivity of graphene nanoribbons in thermoelectronic and photonic devices.

Length dependent thermal conductivity measurements yield phonon mean free path spectra in nanostructures.

Thermal conductivity measurements over variable lengths on nanostructures such as nanowires provide important information about the mean free paths (MFPs) of the phonons responsible for heat conduction. However, nearly all of these measurements have been interpreted using an average MFP even though phonons in many crystals possess a broad MFP spectrum. Here, we present a reconstruction method to obtain MFP spectra of nanostructures from variable-length thermal conductivity measurements. Using this method, we investigate recently reported length-dependent thermal conductivity measurements on SiGe alloy nanowires and suspended graphene ribbons. We find that the recent measurements on graphene imply that 70% of the heat in graphene is carried by phonons with MFPs longer than 1 micron.

Length-dependent thermal transport and ballistic thermal conduction

Probing length-dependent thermal conductivity of a given material has been considered as an important experimental method to determine the length of ballistic thermal conduction, or equivalently, the averaged phonon mean free path (l). However, many previous thermal transport measurements have focused on varying the lateral dimensions of samples, rendering the experimental interpretation indirect. Moreover, deducing l is model-dependent in many optical measurement techniques. In addition, finite contact thermal resistances and variations of sample qualities are very likely to obscure the effect in practice, leading to an overestimation of l. We point out that directly investigating one-dimensional length-dependent (normalized) thermal resistance is a better experimental method to determine l. In this regard, we find that no clear experimental data strongly support ballistic thermal conduction of Si or Ge at room temperature. On the other hand, data of both homogeneously-alloyed SiGe nanowires and heterogeneously-interfaced Si-Ge core-shell nanowires provide undisputed evidence for ballistic thermal conduction over several micrometers at room temperature.

Multiscale modeling of heat conduction in graphene laminates

We developed a combined atomistic-continuum hierarchical multiscale approach to explore the effective thermal conductivity of graphene laminates. To this aim, we first performed molecular dynamics simulations in order to study the heat conduction at atomistic level. Using the non-equilibrium molecular dynamics method, we evaluated the length dependent thermal conductivity of graphene as well as the thermal contact conductance between two individual graphene sheets. In the next step, based on the results provided by the molecular dynamics simulations, we constructed finite element models of graphene laminates to probe the effective thermal conductivity at macroscopic level. A similar methodology was also developed to study the thermal conductivity of laminates made from hexagonal boron-nitride (h-BN) films. In agreement with recent experimental observations, our multiscale modeling confirms that the flake size is the main factor that affects the thermal conductivity of graphene and h-BN laminates. Provided information by the proposed multiscale approach could be used to guide experimental studies to fabricate laminates with tunable thermal conduction properties.

Coherent and incoherent phonon thermal transport in isotopically modified graphene superlattices

Understanding the nature of phonon transport is of both fundamental and technological importance. In this paper, we demonstrate unambiguous coherent and incoherent phonon transport in 12C/13C graphene superlattices using large-scale molecular dynamics simulations. Coherent phonon modes preserve their phases as they propagate through multiple interfaces. For these phonons, the superlattice can be treated as a homogeneous material with its own unit cell and phonon dispersion. We observe length-dependent thermal conductivity of the 12C/13C graphene superlattices, which indicates the existence of coherent phonons that transport ballistically over large distances. By changing the period length of the superlattices and thus the interface density, we observe a minimum in thermal conductivity, which implies the crossover from incoherent to coherent phonon transport. The thermal conductivity of the superlattices can be further decreased as we disrupt the coherence of phonons by manipulating and randomizing the superlattice structure. Our results show that graphene – a two-dimensional material with intrinsically weak anharmonic phonon scattering – is an ideal platform for studying the nature of phonons. The ability of manipulating thermal conductivity using superlattice-based two-dimensional materials can also potentially open up opportunities for thermoelectric applications given existing reports on their high thermoelectric power factors.

Lattice thermal conductivity of crystalline and amorphous silicon with and without isotopic effects from the ballistic to diffusive thermal transport regime

Thermal conductivity of a material is an important physical parameter in electronic and thermal devices, and as the device size shrinks down, its length-dependence becomes unable to be neglected. Even in micrometer scale devices, materials having a long mean free path of phonons, such as crystalline silicon (Si), exhibit a strong length dependence of the thermal conductivities that spans from the ballistic to diffusive thermal transport regime. In this work, through non-equilibrium molecular-dynamics (NEMD) simulations up to 17 μm in length, the lattice thermal conductivities are explicitly calculated for crystalline Si and up to 2 μm for amorphous Si. The Boltzmann transport equation (BTE) is solved within a frequency-dependent relaxation time approximation, and the calculated lattice thermal conductivities in the BTE are found to be in good agreement with the values obtained in the NEMD. The isotopic effects on the length-dependent lattice thermal conductivities are also investigated both in the crystalline and amorphous Si.