Basics of Heat Conduction in Bulk Solids and Nanostructures

We first define the main quantities and outline the nanoscale size effects on heat conduction. Thermal conductivity is introduced through Fourier’s law, q = −k∇T, where is the heat flux and is the temperature gradient. In this expression, K is treated as a constant, which is valid for small T variations. In a wide temperature range, K is a function of T. In anisotropic materials, K varies with crystal orientation and is represented by a tensor [1-2]. In solid materials, heat is carried by acoustic phonons, i.e. ion core vibrations in a crystal lattice, and electrons so that K = K_p + K_e, where Kp and K_e are the phonon and electron contributions, respectively. In metals, K_e is dominant due to the large concentration of free carriers. Measurements of the electrical conductivity σ define K_e via the Wiedemann-Franz law, K_e/(σT) = π²k_B²/(3e²), where k_B is Boltzmann constant and e is the charge of an electron. Heat conduction in carbon materials is usually dominated by phonons even for graphite, which has metal-like properties. The latter is explained by the strong covalent sp² bonding resulting in efficient heat transfer by the acoustic phonons, i.e., lattice vibrations. However, K_e can become significant in doped materials.

The phonon thermal conductivity is expressed as, K_p = ∑_j∫C_j(ω)ν_j²(ω)τ_j(ω)dω where j is the phonon polarization branch, i.e. two transverse acoustic (TA) and one longitudinal acoustic (LA) branches, ν is the phonon group velocity, which, in many solids, can be approximated by the sound velocity, τ_j is the phonon relaxation time and C_j is the contribution to heat capacity from the given branch j. The phonon mean free path Λ is related to the relaxation time as Λ = τν. In the relaxation-time approximation, various scattering mechanisms, which limit MFP, are additive, i.e. τ⁻¹ = ∑τ_i⁻¹. In typical solids, the acoustic phonons, which carry the bulk of heat, are scattered by other phonons, lattice defects, impurities, conduction electrons, and interfaces [1-4]. A simpler equation for K, derived from the kinetic theory of gases, is K_p = (⅓)C_pνΛ. It is important to distinguish between diffusive and ballistic phonon transport regimes. The thermal transport is called diffusive if the size of the sample L is much larger than Λ, i.e. phonons undergo many scattering events. When L > Λ the thermal transport is termed ballistic. Fourier law assumes diffusive transport. Thermal conductivity is called intrinsic when it is limited by the crystal lattice anharmonicity. The crystal lattice is anharmonic when its potential energy has terms higher than the second order with respect to the ion displacements from equilibrium. The intrinsic K limit is reached when the crystal is perfect, without defects or impurities, and phonons can only be scattered by other phonons, which “see” each other due to anharmonicity. The anharmonic phonon interactions, which lead to finite K in 3D, can be described by the Umklapp processes [2]. Thermal conductivity is extrinsic when it is mostly limited by extrinsic effects such as phonon – rough boundary or phonon–defect scattering.

In nanostructures, K is reduced by scattering from boundaries, which can be evaluated as 1/τ_B = (ν/D)[(1 − p)/(1 + p)]. Here D is the nanostructure or grain size and p is the specularity parameter defined as a probability of specular scattering at the boundary. The momentum-conserving specular scattering (p = 1) does not add to thermal resistance. Only diffuse phonon scattering from rough interfaces (p → 0), which changes the momentum, limits the phonon MFP. One can find p from the surface roughness or use it as a fitting parameter for experimental data. When the phonon-boundary scattering is dominant, K scales with D as K_p ∼ C_pνΛ ∼ C_pν²τ_B ∼ C_pνD. In nanostructures with D<<Λ, phonon dispersion can undergo modifications due to spatial confinement resulting in changes in and more complicated size dependence [5]. The specific heat Cp is defined by the phonon density of states (DOS), which leads to different C_p(T) dependence in 3D, 2D, and 1D systems, and is reflected in K(T) dependence at low T. For example, in bulk at low T, K(T) ∼ T³ while it is K(T) ∼ T² in 2D systems.

Balandin Group Optothermal Technique for Thermal Studies of Graphene

Methods of measuring thermal conductivity K can be divided into two groups: steady-state and transient. In transient methods, the thermal gradient is recorded as a function of time, enabling fast measurements of the thermal diffusivity DT over large T ranges. The specific heat Cp and mass density ρm have to be determined independently to calculate K = D_TC_pρ_m. If K determines how well materials conduct heat, DT tells how fast materials conduct heat. Although many methods rely on electrical means to supply heating power and measure T, there are other techniques where the power is provided with light. In many steady-state methods, T is measured by thermocouples.

Professor Balandin conducted the first experimental study of heat conduction in graphene using an original optothermal technique he developed based on a conventional micro-Raman spectrometer (Figure 1 (a-d)) [6]. The heating power ΔP was provided with laser light focused on a suspended graphene layer connected to heat sinks at its ends (e.g. the Figure shows graphene of rectangular shape suspended across a trench in a Si wafer). Temperature rise ΔT in response to ΔP was determined with a micro-Raman spectrometer. The G peak in graphene’s Raman spectrum exhibits strong T dependence. Figure 1 (c) presents the temperature shift in bilayer graphene. The calibration of the spectral position of G peak with T was performed by changing the sample T while using low laser power to avoid local heating. The calibration curve ωG(T) allows one to convert a Raman spectrometer into an “optical thermometer”. During the thermal conductivity measurements, the suspended graphene layer is heated by increasing laser power. Local temperature rise in graphene is determined as ΔT=ΔωG/ξG, where ξG is the temperature coefficient of G peak. The amount of heat dissipated in graphene can be determined either via measuring the integrated Raman intensity of the G peak, as in the original experiments, or by a detector placed under the graphene layer, as in the follow-up experiments. Since optical absorption in graphene depends on the light wavelength and can be affected by strain, defects, contaminations, and near-field or multiple reflection effects for graphene flakes suspended over the trenches it is essential to measure absorption under the conditions of the experiment. A correlation between ΔT and ΔP for graphene samples with a given geometry gives the thermal conductivity, K, value via the solution of the heat diffusion equation. Large sizes of graphene layers ensure the diffusive transport regime. The suspended portion of graphene is essential for determining ΔP, forming a 2D heat front propagating toward the heat sinks, and reducing thermal coupling to the substrate. The method allows one to monitor the temperature of the Si and SiO2 layer near the trench with suspended graphene from the shift in the position of Si and SiO2 Raman peaks. This can be used to determine the thermal coupling of graphene to the SiO2 insulating layer. The optothermal technique for measuring the thermal conductivity of graphene, developed by Professor Balandin, is a direct steady-state method. It can be extended to other suspended films, e.g. graphene films, made of materials with pronounced temperature-dependent Raman signatures.

Figure 1: Illustration of the Balandin optothermal technique for measuring the thermal conductivity of graphene, two-dimensional materials, and thin films. (a) Schematic of the technique. (b) Scanning electron microscopy image of bilayer graphene suspended across the gap in Si/SiO2 wafer. (c) The shift in the G peak position of graphene with temperature is used for determining the local temperature of graphene heated with the laser light. The inset shows that the optical absorption of graphene is not constant but depends on the wavelength. (d) A macroscopic version of the optothermal technique for measuring the thermal conductivity of thin films.

A detailed description of the Balandin optothermal technique and the specifics of heat conduction in graphene and few-layer graphene can be found in original papers and reviews [6 -11]. The technique was extended to other two-dimensional materials and thin films [12-13]. To measure the thermal conductivity using this technique one needs to have a conventional Raman spectrometer, preferably with several laser excitation lines, a hot-cold cell for temperature-dependent calibration measurements, and a detector for the power measurements. The sample preparation normally includes suspending the 2D materials or thin film and attaching the sample sides to heat sinks. The data extraction requires an analytical or numerical solution of the heat diffusion equation with proper boundary conditions. Balandin group features an advanced Raman spectrometer with special holders and axillary equipment for measuring the thermal conductivity of van der Waals materials. The group has expertise in heat-diffusion and phonon transport modeling for accurate calculation of the resulting thermal conductivity.

References

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[2] Klemens, P. G. Solid State Physics, edited by F. Seitz and D. Turnbull (Academic, New York, 1958) Vol. 7, p. 1-98.
[3] Parrott, J. E. & Stuckes, A. D. Thermal Conductivity of Solids (Methuen, New York, 1975).
[4] Ziman, J.M. Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford University Press, New York, 2001).
[5] Balandin, A. and Wang, K.L. Significant decrease of the lattice thermal conductivity due to phonon confinement in a free-standing semiconductor quantum well. Phys. Rev. B, 58, 1544-1549 (1998).
[6] A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano Lett., 8, 902 (2008).
[7] S. Ghosh, W. Bao, D. L. Nika, S. Subrina, E. P. Pokatilov, C. N. Lau, and A. A. Balandin, “Dimensional crossover of thermal transport in few-layer graphene,” Nat. Mater., 9, 555 (2010).
[8] Balandin, A. A. Thermal Properties of Graphene and Nanostructured Carbon Materials, Nature Materials 2011, 10 (8), 569–581.
[9] Nika, D. L.; Balandin, A. A. Phonons and Thermal Transport in Graphene and Graphene-Based Materials. Reports on Progress in Physics 2017, 80 (3), 036502.
[10] Chen, S.; Wu, Q.; Mishra, C.; Kang, J.; Zhang, H.; Cho, K.; Cai, W.; Balandin, A. A.; Ruoff, R. S. Thermal Conductivity of Isotopically Modified Graphene. Nature Materials 2012, 11 (3), 203–207.
[11] A. A. Balandin, “Phononics of graphene and related materials,” ACS Nano, 14, 5170 (2020).
[12] H. Malekpour, P. Ramnani, S. Srinivasan, G. Balasubramanian, D. L. Nika, A. Mulchandani, R. K. Lake, and A. A. Balandin, “Thermal conductivity of graphene with defects induced by electron beam irradiation,” Nanoscale, 8, 14608 (2016).
[13] H. Malekpour and A. A. Balandin, “Raman-based technique for measuring the thermal conductivity of graphene and related materials,” J. Raman Spectroscopy, 49, 106 (2018).