Nonequilibrium Energy Transfer in Nanostructures

Abstract

This chapter begins with a description of the phenomenological theories in which the energy transport processes are represented by a single differential equation or a set of differential equations that can be solved with appropriate initial and boundary conditions. These equations are often called non-Fourier heat equations, which can be considered as extensions of the conventional heat diffusion equation based on Fourier’s law. The limitations of the phenomenological theories are discussed. While the BTE, Monte Carlo method, and MD simulations have been presented in previous chapters, this chapter stresses the application in solid nanostructures, including thermal boundary resistance (TBR) and multilayer structures. The equation of phonon radiative transfer (EPRT) is introduced and used to delineate the diffusive and ballistic heat conduction regimes in thin films. A heat conduction regime with respect to length and time scale is presented, followed by a summary of the contemporary methods for measuring thermal transport properties of solids, thin films, and nanostructures.

Problems

• 7.1. What are the characteristic lengths for heat conduction along a thin film? Why is local equilibrium a good assumption in this case, even though the film thickness is less than the mean free path of heat carriers? Why does the thermal conductivity depend on the thickness of the film?

• 7.2. Why do we say that Fourier’s law is a fundamental physical law, like Newton’s laws in mechanics, but Cattaneo’s equation is not? Comment on the paradox of infinite speed of heat diffusion by considering the feasibility of exciting the surface temperature or depositing a heat flux to the surface instantaneously.

• 7.3. Consider a 1D semi-infinite medium, initially at uniform temperature $T_{i}$, where the surface temperature is suddenly changed to a constant temperature, $T(0,t) = T_{\text{s}}$. The analytical solution of the heat diffusion equation gives $\theta (x,t) = \frac{{T(x,t) - T_{\text{i}} }}{{T_{\text{s}} - T_{\text{i}} }}$ $= {\text{erfc}}\left( {\frac{x}{{2\sqrt {\alpha t} }}} \right)$. For silicon at various temperatures, use the properties given in Example 5.6 to estimate how long it will take for a given location to gain a temperature rise that is 10⁻¹², or one part per trillion of the maximum temperature difference. Estimate the average thermal diffusion speed in terms of x and $T_{\text{i}}$. Hint: ${\text{erfc}}(5.042) = 1.00 \times 10^{ - 12}$.

• 7.4. Repeat Problem 7.3, using copper instead of silicon as the material, based on the properties given in Example 5.5. Discuss why the average thermal diffusion speed is different under different boundary conditions, i.e., constant heat flux and constant temperature. From an engineering point of view, do you think heat diffusion is a fast or slow process? Why?

• 7.5.

  • (a) Derive Eq. (7.4), the hyperbolic heat equation from Cattaneo’s equation

  • (b) Derive Eq. (7.14), the lagging heat equation, based on the dual-phase-lag model.

• 7.6. Take GaAs as an example. How would you compare the speed of sound with the average thermal diffusion speed, at different temperatures and length scales? This problem requires some literature search on the properties.

• 7.7. Assume the hyperbolic heat equation would work for transient heat transfer in glass (Pyrex), at room temperature. Given $\kappa = 1.4{\text{ W/m}} \, {\text{K}}$, $\rho = 2500{\text{ kg/m}}^{ 3}$, $c_\text{{p}} = 835{\text{ J/kg}} \, {\text{K}}$, and $v_{\text{a}} = 5640{\text{ m/s}}$.

  • (a) At what speed would the temperature wave propagate?

  • (b) For an excimer laser with a pulse width $t_{\text{p}} = 10{\text{ ns}}$, 0.1 ns after the pulse starts, could the hyperbolic equation be approximated by the parabolic equation?

  • (c) Suppose we have an instrument available to probe the timescale below $\tau_{\text{q}}$, will the hyperbolic heat equation be able to describe the observation?

• 7.8. Derive Eq. (7.13b) from Eq. (7.13a). Discuss the conditions for these equations to be reduced to Fourier’s law or Cattaneo’s equation.

• 7.9. Show that Eq. (7.17) satisfies Eq. (7.16). Discuss the conditions for Eq. (7.17) to represent Fourier’s law or Cattaneo’s equation.

• 7.10. Derive Eqs. (7.18a), (7.18b), and (7.18c).

• 7.11. Derive Eqs. (7.27a) and (7.27b). Calculate $\tau$, $\tau_{\text{q}}$, and $\tau_{\text{T}}$ of copper, for $T_{\text{e}}$ = 300, 1000, and 5000 K, assuming the lattice temperature $T_{\text{s}} = 300{\text{ K}}$.

• 7.12. Calculate the electron–phonon coupling constant G for aluminum, copper, gold, and silver, near room temperature. Discuss the dependence of κ and G upon the electron and lattice temperatures $T_{\text{e}}$ and $T_{\text{s}}$.

• 7.13. At $T_{\text{e}}$ = 1000, 3000, and 6000 K, estimate the energy transfer by thermionic emission from the copper surface, assuming that the electrons obey the equilibrium distribution function at $T_{\text{e}}$.

• 7.14. Based on Example 7.3, evaluate the heat flux in a thin silicon film. How thin must it be in order for it to be considered as in the radiative thin limit? Calculate the medium temperature T. Plot the left-hand side and the right-hand side of Eq. (7.43). Furthermore, assuming Eq. (7.43) to be true for each frequency, find a frequency-dependent temperature $T(\omega )$ of the medium. At what frequency does $T(\omega ) = T$? Is there any physical significance of $T(\omega )$?

• 7.15. Derive Eq. (7.53), using Eqs. (7.38), (7.49a), (7.49b), and (7.50).

• 7.16. In principle, one should be able to study nonequilibrium electrical and thermal conduction in the direction perpendicular to the plane and use the BTE to determine the effective conductivities. This could be a team project, for a few students, to formulate the necessary equations. As an individual assignment, describe how to set up the boundary conditions, as well as the steps you plan to follow, without actually deriving the equations.

• 7.17. For a diamond type IIa film, $v_{l} = 17,500{\text{ m/s}}$, $v_{t} = 12,800{\text{ m/s}}$, and $\kappa = 3,300{\text{ W/m}} \, {\text{K}}$, near 300 K. Assume that the boundaries can be modeled as blackbodies for phonons. For boundary temperatures $T_{1} = 350{\text{ K}}$ and $T_{2} = 250{\text{ K}}$, calculate and plot the heat flux $q_{x}^{{\prime \prime }}$ and the effective thermal conductivity $\kappa_{\text{eff}}$ across the film of thickness L, varying from 0.05 to 50 μm.

• 7.18. Calculate the TBR between high-temperature superconductor YBa₂Cu₃O_7-δ and MgO substrate, at an average temperature between 10 and 90 K, using both the AMM and the DMM without considering the electronic effect. The following parameters are given for YBa₂Cu₃O_7-δ: $v_{l} = 4780{\text{ m/s}}$, $v_{t} = 3010{\text{ m/s}}$, $\rho = 6338{\text{ kg/m}}^{ - 3}$, and $\Theta_{\text{D}} = 450{\text{ K}}$; and for MgO: $v_{l} = 9710{\text{ m/s}}$, $v_{t} = 6050{\text{ m/s}}$, $\rho = 3576{\text{ kg/m}}^{ - 3}$, and $\Theta_{\text{D}} = 950{\text{ K}}$.

• 7.19. Evaluate the effective thermal conductivity near room temperature of a GaAs/AlAs superlattice, with a total thickness of 800 nm, using the DMM to compute the transmission coefficient. Assume the end surfaces are blackbodies to phonons; consider that (a) each layer is 4 nm thick and (b) each layer is 40 nm thick. The following parameters are given, considering phonon dispersion on thermal conductivity, for GaAs: $C = 880{\text{ kJ/m}}^{ 3} \, {\text{K}}$, $v_{\text{g}} = 1024{\text{ m/s}}$, and $\Lambda = 145{\text{ nm}}$; and for AlAs: $C = 880{\text{ kJ/m}}^{ 3} \, {\text{K}}$, $v_{\text{g}} = 1246{\text{ m/s}}$, and $\Lambda = 236\,{\text{nm}}$. How is the result compared with a single layer of either GaAs or AlAs?

• 7.20. Evaluate the effective thermal conductivity near room temperature of a Si/Ge superlattice, with a total thickness of 1000 nm, using the DMM to compute the transmission coefficient. Assume the end surfaces are blackbodies to phonons; consider that (a) each layer is 5 nm thick and (b) each layer is 50 nm thick. The following parameters are given, considering phonon dispersion on thermal conductivity, for Si: $C = 930{\text{ kJ/m}}^{ 3} \, {\text{K}}$, $v_{\text{g}} = 1804\,{\text{m/s}}$, and $\Lambda = 260\,{\text{nm}}$; and for Ge: $C = 870{\text{ kJ/m}}^{ 3} \, {\text{K}}$, $v_{\text{g}} = 1042\,{\text{m/s}}$, and $\Lambda = 199\,{\text{nm}}$. How is the result compared with a single layer of either Si or Ge?

• 7.21. Make a comparison of the different methods for measuring the thermal conductivity of a thin film.

• 7.22. Suppose one wishes to measure the thermal conductivity of a graphene sheet of 10 μm × 10 μm, what method(s) would you recommend?

• 7.23. Suppose one wishes to measure the thermal conductivity of a superlattice Si/Ge nanowire of length 50 μm and diameter 3 nm, what method would you suggest?

• 7.24. What is the mechanism of transient thermal grating? What properties can be measured by the TTG method?