Phonon Mean Research Articles

Reply to Comment on “Boundary-limited and Glassy-like Phonon Thermal Conduction in EtMe3Sb[Pd(dmit)2]2” [J. Phys. Soc. Jpn. 89, 086001 (2020)

In molecular-based quantum-spin-liquid candidate EtMe$_3$Sb[Pd(dmit)$_2$]$_2$ with two-dimensional $S$=1/2 triangular lattice, a finite residual linear term in the thermal conductivity, $\kappa_0/T\equiv\kappa/T (T \rightarrow 0)$, has been observed and attributed to the presence of itinerant gapless excitations. Here we show that the data of $\kappa$ measured in several single crystals are divided into two groups with and without the residual linear term. In the first group with finite $\kappa_0/T$, the phonon thermal conductivity $\kappa_{ph}$ is comparable to that of other organic compounds. In these crystals, the phonon mean free path $\ell_{ph}$ saturates at low temperatures, being limited by sample size. On the other hand, in the second group with zero $\kappa_0/T$, $\kappa_{ph}$ is one order of magnitude smaller than that in the first group, comparable to that of amorphous solids. In contrast to the first group, $\ell_{ph}$ shows a glassy-like non-saturating behavior at low temperatures. These results suggest that the crystals with long $\ell_{ph}$ are required to discuss the magnetic excitations by thermal conductivity measurements.

Nanostructured planar-type uni-leg Si thermoelectric generators

We demonstrate the power enhancement of planar-type uni-leg poly-Si thermoelectric generators by nano-patterning. The thermoelectric generators were fabricated on an SOI wafer by a conventional lithography-based process. The size of the phononic nano-patterning (300 nm in period) was designed based on the thermal phonon mean free path spectrum. The thermal conductivity of the Si membrane was reduced by 60% compared with an unpatterned one, which resulted in the doubled thermoelectric figure-of-merit. The nano-patterning increased the temperature gradient in the Si membrane and resulted in a four times higher voltage and ten times larger power density.

Large lattice thermal conductivity, interplay between phonon-phonon, phonon-electron, and phonon-isotope scatterings, and electrical transport in molybdenum from first principles

We describe an ab initio phonon Boltzmann transport equation (BTE) approach accounting for phonon-electron scattering in addition to the well-established phonon-phonon and isotope scatterings. The phonon BTE is linearized and can be exactly solved beyond the relaxation time approximation (RTA). We use this approach to study the lattice thermal conductivity (${\ensuremath{\kappa}}_{\text{ph}}$) of molybdenum (Mo). ${\ensuremath{\kappa}}_{\text{ph}}$ of Mo is found to possess several anomalous features: (1) like in another group VI element tungsten (W), ${\ensuremath{\kappa}}_{\text{ph}}$, with a large value of 37 W ${\mathrm{m}}^{\ensuremath{-}1}$ ${\mathrm{K}}^{\ensuremath{-}1}$ at room temperature, follows weak temperature dependence due to interplay between phonon-phonon (ph-ph), phonon-electron (ph-el), and phonon-isotope (isotope) scatterings; and (2) compared with W, though Mo is much lighter in mass, Mo has a smaller ${\ensuremath{\kappa}}_{\text{ph}}$. This is attributed to weaker interatomic bonding, larger isotope mixture, and larger density of states at Fermi level in Mo. In isotopically pure samples, ${\ensuremath{\kappa}}_{\text{ph}}$ increases from 37 to 48 W ${\mathrm{m}}^{\ensuremath{-}1}$ ${\mathrm{K}}^{\ensuremath{-}1}$ at room temperature. Considering the similarity of the phonon dispersion, our work suggests that chromium should also have a large ${\ensuremath{\kappa}}_{\text{ph}}$, which, rather than the complexity of the electronic band structure argued in the literature, accounts for the significant deviation of measured Lorenz number $L$ from the expected Sommerfeld value. The electrical conductivity ($\ensuremath{\sigma}$) and electronic thermal conductivity (${\ensuremath{\kappa}}_{\text{e}}$) of Mo are also calculated by using an ab initio electron BTE approach. $\ensuremath{\sigma}$ and the total thermal conductivity ($\ensuremath{\kappa}$) agree with the experimental data reasonably. These results demonstrate that the ab initio calculations can quantify the lattice and electronic contributions to $\ensuremath{\kappa}$. We also look into the cumulative $\ensuremath{\sigma}$ and ${\ensuremath{\kappa}}_{\text{ph}}$ with respect to electron and phonon mean free paths (MFPs), respectively, in order to reveal the size effect in Mo. The MFPs of electrons contributing to conductivity range from 5 to 22 nm, whereas the MFPs of phonons primarily distribute between 5 and 73 nm with more than 80% contribution to ${\ensuremath{\kappa}}_{\text{ph}}$. This suggests that a reduced Lorenz number can be observed in Mo nanostructures when the relevant size goes below $\ensuremath{\sim}70$ nm.

Heat conduction in oxide glasses: Balancing diffusons and propagons by network rigidity

Understanding the variation of thermal conduction in disordered oxides is important for applications related to energy saving and electronics but currently lacks fundamental insight into the phonon propagation mechanisms. In this Letter, we report a strong correlation between thermal conductivity and the speed of sound within two families of modified oxide glass formers, in agreement with phonon-gas-theory predictions for crystalline materials. Based on calculations of diffuson-mediated heat transport, we then show that thermal conductivity in borate glasses is dominated by diffusive vibrational modes, while both diffusive and propagative modes contribute to heat conduction in silicate glasses. This fundamental difference in the heat propagation mechanism originates from differences in the phonon mean free path of low-frequency modes caused by the inherent variation of atomic rigidity between silicate and borate glasses.

Thermal conduction mechanism based on microstructural transformations of molten slag: The role of calcium oxide

The built up of the slag layer is a character of the entrained-flow gasifier, and its thickness distribution and heat transfer characteristics affect the stable operation of gasification, which is closely related to the thermal conductivity of molten slag. A non-stationary hot wire experimental method and a reverse non-equilibrium molecular dynamics simulation method were combined to investigate the thermal conductivity of the SiO2-Al2O3CaO-Fe2O3−MgO molten slag with a mass ratio of CaO ranging from 5 to 25 wt.%. The slag microstructure information of the distribution of oxygen and tetrahedron Qn was calculated by molecular dynamics simulation. A good agreement within the deviation of 18% between the measured and calculated thermal conductivity over the entire operating temperature range is observed, and one can find that thermal conductivity decreases significantly as the CaO content increased. The principal pathway for CaO modifying the network structure is the connection between two tetrahedral structures by generation of the Si-O-Ca and Al-O-Ca through the fracture of Si-O-Si and Si-O-Al. The rise of CaO promotes the formation of non-bridging oxygen (NBO) and the decomposition of bridging oxygen. The existence of NBOs leads to an enhancement of anharmonicity and shortens the interatomic distance in the tetrahedral structure, which aggravates phonon scattering to reduce phonon mean free path, thereby decreasing the thermal conductivity. Since the phonon propagation is restricted to short-range tetrahedron separated by structural disorder, the Q4 structure is the most sensitive to the addition of CaO. An equivalent molar concentration of the Q4 tetrahedron (XQ4) is defined and the approximate positive linear relationships between the thermal conductivities and XQ4 are obtained.

Thermal Conductivity of Graphene Oxide: A Molecular Dynamics Study

The thermal properties of graphene oxide containing hydroxyl and epoxy functional groups were studied using non-equilibrium molecular dynamics to understand the thermal transport phenomena involved and the structure factors limiting heat conduction. Estimates were given in terms of phonon mean free paths for the reduction in thermal conductivity by interior defects due to scattering. The mechanism of phonon transport in the graphene oxide was discussed. The results indicated that the degree of oxidation can significantly affect the thermal performance of graphene oxide. A low degree of oxidation is necessary to enhance the phonon transport properties of graphene oxide and reduce the probability of phonon-defect scattering. Phonon transport in graphene oxide with a high degree of oxidation is governed by the mean free path of phonons associated with scattering from interior defects. Oxygen-containing functional groups can adversely affect performance and reduce the efficiency of phonon transport in graphene oxide due to phonon mean free paths limited mainly by interior defects. The calculated intrinsic thermal conductivity of graphene oxide at room temperature is about 72 W/(m K) with an oxidation degree of 0.35 and about 670 W/(m K) with an oxidation degree of 0.05. The phonon mean free path decreases with increasing the degree of oxidation due to enhanced phonon-defect scattering, making the thermal conductivity very sensitive to the concentration of oxygen-containing functional groups.

Effect of biaxial strain on thermal transport in WS2 monolayer from first principles calculations

Abstract Recently, tungsten disulfide (WS2) monolayer has emerged as a two-dimensional material due to its outstanding physical properties. In this work, the thermal transports in strained WS2 monolayer are studied systematically by combining the first principles calculations and the Boltzmann transport equation. It can be found that the lattice thermal conductivity (k) decreases from 262.78 to 190.66 W m−1 K−1 for biaxial tensile strain (from 0% to 8%) and from 262.78 to 217.40 W m−1 K−1 for biaxial compressive strain (from 0% to −4%), respectively. The softened transverse/longitudinal acoustic phonon mode and the stiffened flexural acoustic phonon mode in strained WS2 monolayer change the phonon group velocities. We attribute the reduction of k in tensilely strained WS2 monolayer to the declining phonon group velocity, heat capacity and phonon lifetime of acoustic phonon modes. The comprehensive competition among the increasing phonon group velocity, heat capacity and the decreasing phonon lifetime suppresses the lattice thermal conductivity of compressively strained WS2 monolayer. Furthermore, the increasing phase space proves that the enhancing anharmonic phonon scattering, which results in the suppression of the k in tensilely strained WS2 monolayer. Additionally, the critical phonon mean free path (MFP) under different biaxial strain levels is calculated to demonstrate the contribution of the phonons with different MFPs to the total k of WS2 monolayer. This work provides a phonon behavior analysis of the thermal conduction in WS2 monolayer and paves the path for potential applications of WS2-based devices.

Bulk-like Intrinsic Phonon Thermal Conductivity of Micrometer-Thick AlN Films.

Aluminum nitride (AlN) has garnered much attention due to its intrinsically high thermal conductivity. However, engineering thin films of AlN with these high thermal conductivities can be challenging due to vacancies and defects that can form during the synthesis. In this work, we report on the cross-plane thermal conductivity of ultra-high-purity single-crystal AlN films with different thicknesses (∼3-22 μm) via time-domain thermoreflectance (TDTR) and steady-state thermoreflectance (SSTR) from 80 to 500 K. At room temperature, we report a thermal conductivity of ∼320 ± 42 W m-1 K-1, surpassing the values of prior measurements on AlN thin films and one of the highest cross-plane thermal conductivities of any material for films with equivalent thicknesses, surpassed only by diamond. By conducting first-principles calculations, we show that the thermal conductivity measurements on our thin films in the 250-500 K temperature range agree well with the predicted values for the bulk thermal conductivity of pure single-crystal AlN. Thus, our results demonstrate the viability of high-quality AlN films as promising candidates for the high-thermal-conductivity layers in high-power microelectronic devices. Our results also provide insight into the intrinsic thermal conductivity of thin films and the nature of phonon-boundary scattering in single-crystal epitaxially grown AlN thin films. The measured thermal conductivities in high-quality AlN thin films are found to be constant and similar to bulk AlN, regardless of the thermal penetration depth, film thickness, or laser spot size, even when these characteristic length scales are less than the mean free paths of a considerable portion of thermal phonons. Collectively, our data suggest that the intrinsic thermal conductivity of thin films with thicknesses less than the thermal phonon mean free paths is the same as bulk so long as the thermal conductivity of the film is sampled independent of the film/substrate interface.

Evaluation of Structural and Thermal Properties of Ce1-yGdyO2-x Solid Solution

In the present study, the systematic evaluation of thermal properties was performed for CeO2–Gd2O3 (Gd2O3 = 0–15 wt%) samples. Gd2O3 doping into CeO2 reduced the grain size of CeO2–Gd2O3 solid solution and caused the expansion of the lattice parameter. The thermal conductivity of CeO2–Gd2O3 solid solutions decreased with increasing Gd2O3 content and decreased with temperature up to nearly 900 K, though the decreasing rates became smaller at a higher temperature. The values of the thermal conductivity were mostly attributed to phonon mean free path which decreased due to the Umklapp processes at high temperatures and reduced by phonon scattering due to Gd dopants and oxygen vacancies at low temperatures. The phonon scattering caused by grain boundaries, and chemical impurities can be neglected, and the isotope effect is considered to be small.

Ultra-low thermal conductivity and super-slow hot-carrier thermalization induced by a huge phononic gap in multifunctional nanoscale boron pnictides

Abstract Recent synthesis of thin films of boron monophosphide (BP) motivates us to report herewith the lattice thermal conductivity calculated in nanoscale boron pnictides. The lattice thermal conductivity in BX (X = P, As, Sb) monolayers are calculated to be 51.3, 20.7 and 5.4 Wm−1K−1 respectively at 300 K. The ultra-low lattice thermal conductivity in BSb, which is comparable to that in SnSe, is attributed to low elastic and bond stiffness, low Debye temperature, small group velocity and large mode Gruneisen parameter near zone center. Phonon group velocities in BSb are comparable to that in SnSe and Bi2Te3. Up to 50% reduction in lattice thermal conductivity is noticed at 300 K upon shortening the phonon mean free path, which is realizable, e.g., in 1D nanoribbons, or via defects, vacancies, nanoengineering, etc. A comparative study based on different models to compute the lattice thermal conductivity will provide useful insights into the experimentally measured ones. The widest ever phononic gap (436 & 438 cm−1) is found in BAs and BSb monolayers, where the transverse and longitudinal optical phonons meet the criterion for the prevention of Klemens decay and hence, they can be gainfully exploited in hot-carrier solar cells. BX (X = P, As, Sb) monolayers show a direct bandgap and piezoelectricity. Nanoscale boron pnictides are promising materials in futuristic thermoelectrics, bolometers, third-generation solar cells and nanopiezotronics.

Thermal and oxygen transport properties of complex pyrochlore RE2InTaO7 for thermal barrier coating applications

Pyrochlore-type rare earth zirconate (RE2Zr2O7) has been considered as a new generation thermal barrier coating (TBC) material. We substitute Zr4+ with a pair of In3+ and Ta5+ in RE2Zr2O7 (RE = Pr, Nd, Sm, Eu, Gd), forming a complex pyrochlore material RE2InTaO7. Combined with the structural disordering, the thermal conductivities of RE2InTaO7 were progressively reduced to the amorphous limit. Phonon scattering due to the structural complexity and disordering was confirmed by the reduced phonon mean free path and further demonstrated by the broadened Raman modes. Meanwhile, the oxygen ion conductivities were reduced by two orders of magnitude compared with that of normal pyrochlore structure. This paper demonstrates the approach of structural complexing could simultaneously tune thermal and oxygen transport properties toward high-efficiency and durable TBC applications.

Kapitza resistance at water-graphene interfaces.

Heat transfer across fluid-solid interfaces in nanoconfinement has received significant attention due to its relevance in nanoscale systems. In this study, we investigate the Kapitza resistance at the water-graphene interface with the help of classical molecular dynamics simulation techniques in conjunction with our recently proposed equilibrium molecular dynamics (EMD) method [S. Alosious et al., J. Chem. Phys. 151, 194502 (2019)]. The size effect of the Kapitza resistance on different factors such as the number of graphene layers, the cross-sectional area, and the width of the water block was studied. The Kapitza resistance decreases slightly with an increase in the number of layers, while the influence of the cross-sectional area and the width of the water block is negligible. The variation in the Kapitza resistance as a function of the number of graphene layers is attributed to the large phonon mean free path along the graphene cross-plane. An optimum water-graphene system, which is independent of size effects, was selected, and the same was used to determine the Kapitza resistance using the predicted EMD method. The values obtained from both the EMD and the non-equilibrium molecular dynamics (NEMD) methods were compared for different potentials and water models, and the results are shown to be in good agreement. Our method allows us to compute the Kapitza resistance using EMD simulations, which obviates the need to create a large temperature gradient required for the NEMD method.

First-principles calculations of phonon transport in two-dimensional penta-X2C family

Two-dimensional (2D) materials exhibit enhanced thermoelectric (TE) performance compared to bulk materials, which relies heavily on lattice thermal conductivity. Penta-X2C (X = P, As, and Sb) is a newly predicted 2D material family with promising potential applications in photocatalytic water splitting and photovoltaic and optoelectronic devices. To achieve a combination of photovoltaic and TE technologies and further boost the energy utilization rate, in this paper, we systematically investigate the thermal transport of the penta-X2C family. Density functional theory combined with semiclassical Boltzmann transport approach was used to evaluate the thermal transport. Interestingly, the calculated lattice thermal conductivities (kl) of penta-Sb2C are two orders of magnitude smaller than that of penta-P2C, despite that they share similar atomic structure. The calculated kl of penta-P2C, penta-As2C, and penta-Sb2C are 75.27 W m−1 K−1, 19.11 W m−1 K−1, and 0.72 W m−1 K−1, respectively. Penta-Sb2C also exhibits low average acoustic group velocity, large Grüneisen parameters, strong optical–acoustic phonon coupling, and short phonon mean free path. These results qualify penta-Sb2C as a promising candidate for building outstanding TE devices.

Phononic transport and simulations of annealing processes in nanometric complex structures

Modeling thermal transport at the nanoscale is a difficult task, especially when external time-varying heating sources and the complexity of the studied systems make computational schemes that rely on accurate particlelike simulations nonaffordable. Alternative strategies based on corrections of the Fourier law could satisfy the trade-off between accuracy and computational efficiency, since they can be implemented in partial differential equation solvers. This continuum approach could also allow for the coupling between thermal transport and other evolving fields related to the generalized temperature field. Here we demonstrate that corrections due to the finite phonon mean free paths can be suitably included in annealing process simulations of three-dimensional nanosystems. Quantitative predictions can be obtained and readily compared with the experimental characterization of the processed samples.

Role of geometry and surface roughness in reducing phonon mean free path and lattice thermal conductivity of modulated nanowires

Phonon transport simulations were performed to investigate the effects of geometry and surface roughness on the thermal conductivity of modulated nanowires. The finite volume method and ray tracing method were utilized to determine the phonon mean free path. The simulation results showed that the mean free path of the nanowire can be reduced by fabricating intricate structures under a constant ratio of surface area to volume. Then the specularity of the parts of the surfaces of the modulated nanowires was tuned. The simulations revealed that the impact of the specularity on the mean free path depends on the location of the surface whose specularity was tuned; in particular, the mean free path was sensitive to the specularity of the surfaces of neck parts but insensitive to that of other parts. Finally, the thermal conductivity was evaluated by using the simulated mean free path and other phonon transport properties obtained from first-principle calculations. It was found that optimization of the geometry and surface roughness can reduce sufficiently large thermal conductivities; this effect is beneficial in some semiconductor devices.

Thermal Characterization and Modelling of AlGaN-GaN Multilayer Structures for HEMT Applications

To optimize the thermal design of AlGaN-GaN high-electron-mobility transistors (HEMTs), which incorporate high power densities, an accurate prediction of the underlying thermal transport mechanisms is crucial. Here, a HEMT-structure (Al0.17Ga0.83N, GaN, Al0.32Ga0.68N and AlN on a Si substrate) was investigated using a time-domain thermoreflectance (TDTR) setup. The different scattering contributions were investigated in the framework of phonon transport models (Callaway, Holland and Born-von-Karman). The thermal conductivities of all layers were found to decrease with a temperature between 300 K and 773 K, due to Umklapp scattering. The measurement showed that the AlN and GaN thermal conductivities were a magnitude higher than the thermal conductivity of Al0.32Ga0.68N and Al0.17Ga0.83N due to defect scattering. The layer thicknesses of the HEMT structure are in the length scale of the phonon mean free path, causing a reduction of their intrinsic thermal conductivity. The size-effect of the cross-plane thermal conductivity was investigated, which showed that the phonon transport model is a critical factor. At 300 K, we obtained a thermal conductivity of (130 ± 38) Wm−1K−1 for the (167 ± 7) nm thick AlN, (220 ± 38) Wm−1K−1 for the (1065 ± 7) nm thick GaN, (11.2 ± 0.7) Wm−1K−1 for the (423 ± 5) nm thick Al0.32Ga0.68N, and (9.7 ± 0.6) Wm−1K−1 for the (65 ± 5) nm thick Al0.17Ga0.83N. Respectively, these conductivity values were found to be 24%, 90%, 28% and 16% of the bulk values, using the Born-von-Karman model together with the Hua–Minnich suppression function approach. The thermal interface conductance as extracted from the TDTR measurements was compared to results given by the diffuse mismatch model and the phonon radiation limit, suggesting contributions from inelastic phonon-scattering processes at the interface. The knowledge of the individual thermal transport mechanisms is essential for understanding the thermal characteristics of the HEMT, and it is useful for improving the thermal management of HEMTs and their reliability.

Thermoelectric properties of holey silicon at elevated temperatures

Compatibility to the semiconductor industry makes silicon thin-film devices attractive for thermoelectric applications. Silicon has a competitive thermoelectric power factor but a large thermal conductivity, which results in an overall small thermoelectric figure of merit, ZT. By patterning arrays of nano-sized holes with spacing less than the average phonon mean free path into the Si thin films, their thermal conductivity can be greatly suppressed, whereas their electronic properties are less affected. We fabricated and measured the electronic and thermal transport properties of such holey Si devices from 300 K to 650 K. Heat diffusion imaging, a hybrid approach that combines thermoreflectance imaging and the heat spreader method was used for the in-plane thermal conductivity measurement and gives a value of 6.00 ± 1.83 W/mK at room temperature. The power factor times temperature is about 0.52 ± 0.04 W/mK at 300 K and 1.10 ± 0.09 W/mK at 650 K. Therefore, ZT of the holey Si device is approximately 0.09 at room temperature and is at least 0.29 at 650 K. Further improvement is possible by optimizing the feature sizes and using surface doping.

Strongly anisotropic thermal conductivity in planar hexagonal borophene oxide sheet

The flat hexagonal borophene oxide (B2O) has the highest Li storage capacity among existing two-dimensional materials. Thermal conductivity is an important parameter for the safety of Li-ion batteries. We investigate the lattice thermal conductivity of B2O by solving phonon Boltzmann transport equation combined with the first-principles calculations. We found that the relaxation time approximation remarkably underestimate the thermal conductivity (κ) of monolayer B2O, revealing phonon hydrodynamics characteristic. The κ of B2O from the exact solution of Boltzmann transport equation is 53 Wm−1K−1 and 130 Wm−1K−1 along armchair-direction and zigzag-direction at 300 K, respectively. B2O exhibits strong thermal transport anisotropy due to anisotropic phonon group velocity, obviously larger than that of other borophene allotropes. At room temperature, the phonon mean free path of B2O is about 231 nm and 49 nm along armchair-direction and zigzag-direction, respectively. The highly anisotropic thermal conductivity of B2O offers new possibilities for its applications in thermal management.

Enargite Cu3PS4: A Cu–S‐Based Thermoelectric Material with a Wurtzite‐Derivative Structure

AbstractCompound semiconductors derived from ZnS (zincblende and wurtzite) with tetrahedral framework structures have functions for various applications. Examples of such materials include Cu–S‐based materials with zincblende‐derivative structures, which have attracted attention as thermoelectric (TE) materials over the past decade. This study illuminates superior TE performance in polycrystalline samples of enargite Cu3P1−xGexS4 with a wurtzite‐derivative structure. The substitution of Ge for P dopes holes into the top of the valence band composed of Cu‐3d and S‐3p, whereby its multiband characteristic leads to a high TE power factor. Furthermore, a reduction in the grain size to 50–300 nm can effectively decrease phonon mean free paths, leading to low thermal conductivity. These features result in a dimensionless TE figure of merit ZT of 0.5 at 673 K for the x = 0.2 sample. Environmentally benign and low‐cost characteristics of the constituent elements of Cu3PS4, as well as its high‐performance thermoelectricity, make it a promising candidate for large‐scale TE applications. Furthermore, this finding extends the development field of Cu–S‐based TE materials to those with wurtzite‐derivative structures.

Thermal Conductivity of Graphene on Nanostructure Size from Nonequilibrium Molecular Dynamics Simulations

Thermal transport of graphene occupies a unique place in thermal management of electronic devices, especially for nanosize devices with high-density integration and high dissipated power. The structure of graphene on nanometer scale changes its thermal conductance. Here, the thermal characters of graphene have been researched by nonequilibrium molecular dynamics simulation (NEMDS) at room temperature. Special attention is focused on the edge type (zigzag or armchair) and nanostructure size dependence of conductivity for heat. The consequences suggest that the thermal conductivity of zigzag edge has been higher than that of armchair, which is because of the higher phonon group velocities. Furthermore, thermal conductivity shows a rising tendency, when the model is calculated from length of 21.84 nm to 43.78 nm. The result indicates that the thermal property performs a strong dependence on nanostructure size which is less than phonon mean free path (775 nm). Our research highlights the significance of structure attribute relationships together with providing useful guideline in calculations for nanosize devices thermal management.