Thermal Characteristics of an Aluminum Thin Film due to Temperature Disturbance at Film Edges

Abstract

Phonon transport in an aluminum thin film is simulated due to a temperature disturbance across the film. The Boltzmann equation is introduced to formulate the radiative transport in the electron and lattice sub-systems. The transient and frequency dependence of the phonon transport is considered, and dispersion relations are accommodated to account for the group velocities in the analysis. Electron-phonon coupling is employed to couple the energy transport across the electron and lattice sub-systems. An equivalent equilibrium temperature is presented to assess the characteristics of the phonon intensity in the film. Temperature predictions are validated with data presented in a previous study. It is found that the equivalent equilibrium temperature differs significantly from that obtained from the two-equation model. The film thickness influences the transport characteristics of the film, in which case the time to reach an almost quasi-steady temperature is shorter for the thin film ($L_{x}= 0.25 \,\upmu \hbox {m}$, where $L_{x}$ is the film thickness) than that corresponding to the thick film ($L_{x}= 2 \,\upmu \hbox {m}$). In the diffusion limit (when the Knudsen number $Kn=\varLambda /{L_x }\rightarrow 0$, where $\varLambda$ is the mean free path), it is demonstrated that the radiative transport equation reduces to the formulation of the two-equation model.

Appendix 1: Diffusive Limit Consideration of Radiative Transport Energy

In the diffusion limit when the Knudsen number $Kn\;=\;\varLambda /L_x \rightarrow \;0$ (where $\varLambda$ is the mean free path and $L_x$ is the film thickness), Eqs. 3 and 4 should reduce to the two-equation model. This is can be verified as follows. After multiplying Eqs. 3 and 4 throughout by $2\pi \mu \hbox {d}\mu$ and then integrating from $-$1 to 1, it yields

$$\begin{aligned}&\frac{2\pi }{v_{\mathrm{p}} }\int _{-1}^1 {\mu \frac{\partial I_{\mathrm{p}} }{\partial t}\hbox {d}\mu } +2\pi \int _{-1}^1 {\mu ^{2}\frac{\partial I_{\mathrm{p}} }{\partial x}\hbox {d}\mu }\nonumber \\&\quad =\frac{2\pi \left( {\int _{-1}^1 {\mu \hbox {d}\mu } } \right) \left( {\frac{1}{2}\int _{-1}^1 {I_{\mathrm{p}} \hbox {d}\mu } } \right) -2\pi \int _{-1}^1 {\mu I_{\mathrm{p}} \hbox {d}\mu } }{\varLambda _{\mathrm{p}} } \nonumber \\&\quad \quad -2\pi \frac{G}{4\pi }\left( {T_{\mathrm{p}} -T_{\mathrm{e}} } \right) \left( {\int _{-1}^1 {\mu \hbox {d}\mu } } \right) ,\end{aligned}$$

(34)

$$\begin{aligned}&\frac{2\pi }{v_{\mathrm{e}} }\int _{-1}^1 {\mu \frac{\partial I_{\mathrm{e}} }{\partial t}\hbox {d}\mu } +2\pi \int _{-1}^1 {\mu ^{2}\frac{\partial I_{\mathrm{e}} }{\partial x}\hbox {d}\mu }\nonumber \\&\quad =\frac{2\pi \left( {\int _{-1}^1 {\mu \hbox {d}\mu } } \right) \left( {\frac{1}{2}\int _{-1}^1 {I_{\mathrm{e}} \hbox {d}\mu } } \right) -2\pi \int _{-1}^1 {\mu I_{\mathrm{e}} \hbox {d}\mu } }{\varLambda _{\mathrm{e}} } \nonumber \\&\quad \quad -2\pi \frac{G}{4\pi }\left( {T_{\mathrm{p}} -T_{\mathrm{e}} } \right) \left( {\int _{-1}^1 {\mu \hbox {d}\mu } } \right) . \end{aligned}$$

(35)

However, the second term on the left-hand-side needs some elaboration. In the diffusive limit, the behavior of $I_{\mathrm{e,p}} \left( {x,\mu ,t} \right)$ is mainly linear in $\mu$. This means that we may neglect the terms of order 2 or higher in the Taylor series expansion of $I_{\mathrm{e,p}} \left( {x,\mu ,t} \right)$ in $\mu$:

$$\begin{aligned} I_{\mathrm{e,p}} \left( {x,\mu ,t} \right) \;&= \;I_{\mathrm{e,p}} \left( {x,0,t} \right) \;+\;\mu \left. {\frac{\partial I_{\mathrm{e,p}} }{\partial \mu }} \right| _{\mu =0} \;+\;O\left( \mu \right) \nonumber \\&\approx I_{\mathrm{e,p}}^\mathrm{o} \left( {x,t} \right) +\mu \left. {\frac{\partial I_{\mathrm{e,p}} }{\partial \mu }} \right| _{\mu =0}. \end{aligned}$$

(36)

This is because $I_{\mathrm{e,p}}^\mathrm{o} \left( {x,t} \right) \;=\;\frac{1}{2}\int _{-1}^1 {I_{\mathrm{e,p}} \left( {x,\mu ,t} \right) \hbox {d}\mu } \;\approx \;I_{\mathrm{e,p}} \left( {x,0,t} \right) +\;\frac{1}{2}$ $\left( {\int _{-1}^1 {\mu \hbox {d}\mu } } \right) \left. {\frac{\partial I_{\mathrm{e,p}} }{\partial \mu }} \right| _{\mu =0} =\;I_{\mathrm{e,p}} \left( {x,0,t} \right) .$

Then,

$$\begin{aligned} 2\pi \int _{-1}^1 {\mu ^{2}\frac{\partial I_{\mathrm{e,p}} }{\partial x}\hbox {d}\mu } \;&= \;2\pi \int _{-1}^1 {\mu ^{2}\frac{\partial }{\partial x}\left( {I_{\mathrm{e,p}}^\mathrm{o} \left( {x,t} \right) \;+\;\mu \left. {\frac{\partial I_{\mathrm{e,p}} }{\partial \mu }} \right| _{\mu =0} } \right) \hbox {d}\mu } \nonumber \\&= \;2\pi \int _{-1}^1 {\mu ^{2}\frac{\partial I_{\mathrm{e,p}}^\mathrm{o} \left( {x,t} \right) }{\partial x}\hbox {d}\mu } \;+\;2\pi \frac{\partial }{\partial x}\left( {\left. {\frac{\partial I_{\mathrm{e,p}} }{\partial \mu }} \right| _{\mu =0} } \right) \int _{-1}^1 {\mu ^{3}\hbox {d}\mu } \nonumber \\&= 2\pi \frac{2}{3}\frac{\partial I_{\mathrm{e,p}}^\mathrm{o} \left( {x,t} \right) }{\partial x}. \end{aligned}$$

(37)

Since $\frac{C_{\mathrm{e,p}} v_{\mathrm{e,p}} T_{\mathrm{e,p}} }{4\pi }\;=\;I_{\mathrm{e,p}}^\mathrm{o} \left( {x,t} \right) \;=\;\frac{1}{2}\int _{-1}^1 {I_{\mathrm{e,p}} \left( {x,\mu ,t} \right) {}\hbox {d}\mu }$, so that finally

$$\begin{aligned} 2\pi \int _{-1}^1 {\mu ^{2}\frac{\partial I_{\mathrm{e,p}} }{\partial x}\hbox {d}\mu } \;=\;2\pi \frac{2}{3}\frac{\hbox {d}I_{\mathrm{e,p}}^\mathrm{o} }{\hbox {d}T_{e,p} }\frac{\partial T_{\mathrm{e,p}} }{\partial x}. \end{aligned}$$

Equations 34 and 35 are now simplified to

$$\begin{aligned} \frac{1}{v_{\mathrm{p}} }\frac{\partial {{q}_{\mathrm{p}}^{\prime \prime }} }{\partial t}+\frac{4\pi }{3}\frac{\hbox {d}I_{\mathrm{p}}^\mathrm{o} }{\hbox {d}T_{\mathrm{p}} }\frac{\partial T_{\mathrm{p}} }{\partial x}=\frac{0-{{q}_{\mathrm{p}}^{\prime \prime }} }{\varLambda _{\mathrm{p}} }-0 \end{aligned}$$

(38)

and

$$\begin{aligned} \frac{1}{v_{\mathrm{e}} }\frac{\partial {{q}_{\mathrm{e}}^{\prime \prime }} }{\partial t}+\frac{4\pi }{3}\frac{\hbox {d}I_{\mathrm{e}}^\mathrm{o} }{\hbox {d}T_{\mathrm{e}} }\frac{\partial T_{\mathrm{e}} }{\partial x}=\frac{0-{{q}_{\mathrm{e}}^{\prime \prime }} }{\varLambda _{\mathrm{e}} }-0+0 \end{aligned}$$

(39)

or

$$\begin{aligned} \frac{\varLambda _{\mathrm{p}} }{v_{\mathrm{p}} }\frac{\partial {{q}_{\mathrm{p}}^{\prime \prime }} }{\partial t}+{{q}_{\mathrm{p}}^{\prime \prime }} =-\frac{C_{\mathrm{p}} v_{\mathrm{p}} \varLambda _{\mathrm{p}} }{3}\frac{\partial T_{\mathrm{p}} }{\partial x} \end{aligned}$$

(40)

and

$$\begin{aligned} \frac{\varLambda _{\mathrm{e}} }{v_{\mathrm{e}} }\frac{\partial {{q}_{\mathrm{e}}^{\prime \prime }} }{\partial t}+{{q}_{\mathrm{e}}^{\prime \prime }} =-\frac{C_{\mathrm{e}} v_{\mathrm{e}} \varLambda _{\mathrm{e}} }{3}\frac{\partial T_{\mathrm{e}} }{\partial x}. \end{aligned}$$

(41)

The kinetic theory formula for the thermal conductivity is given as

$$\begin{aligned} k_{\mathrm{p}} =\frac{C_{\mathrm{p}} v_{\mathrm{p}} \varLambda _{\mathrm{p}} }{3},\qquad k_{\mathrm{e}} =\frac{C_{\mathrm{e}} v_{\mathrm{e}} \varLambda _{\mathrm{e}} }{3}. \end{aligned}$$

Hence, Eqs. 39 and 40 can be transformed into

$$\begin{aligned} \frac{\varLambda _{\mathrm{p}} }{v_{\mathrm{p}} }\frac{\partial {{q}_{\mathrm{p}}^{\prime \prime }} }{\partial t}+{{q}_{\mathrm{p}}^{\prime \prime }} =-k_{\mathrm{p}} \frac{\partial T_{\mathrm{p}} }{\partial x} \end{aligned}$$

(42)

and

$$\begin{aligned} \frac{\varLambda _{\mathrm{e}} }{v_{\mathrm{e}} }\frac{\partial {{q}_{\mathrm{e}}^{\prime \prime }} }{\partial t}+{{q}_{\mathrm{e}}^{\prime \prime }} =-k_{\mathrm{e}} \frac{\partial T_{\mathrm{e}} }{\partial x}. \end{aligned}$$

(43)

To eliminate ${{q}_{\mathrm{p}}^{\prime \prime }}$ and ${{q}_{\mathrm{e}}^{\prime \prime }}$ between Eqs. 40 and 42 as well as Eqs. 41 and 43, one can proceed as follows. After differentiating Eqs. 41 and 43 once with respect to $x$, it gives

$$\begin{aligned} \frac{\varLambda _{\mathrm{p}} }{v_{\mathrm{p}} }\frac{\partial }{\partial t}\left( {\frac{\partial {{q}_{\mathrm{p}}^{\prime \prime }} }{\partial x}} \right) +\frac{\partial {{q}_{\mathrm{p}}^{\prime \prime }} }{\partial x}=-k_{\mathrm{p}} \frac{\partial ^{2}T_{\mathrm{p}} }{\partial x^{2}} \end{aligned}$$

(44)

and

$$\begin{aligned} \frac{\varLambda _{\mathrm{e}} }{v_{\mathrm{e}} }\frac{\partial }{\partial t}\left( {\frac{\partial {{q}_{\mathrm{e}}^{\prime \prime }} }{\partial x}} \right) +\frac{\partial {{q}_{\mathrm{e}}^{\prime \prime }} }{\partial x}=-k_{\mathrm{e}} \frac{\partial ^{2}T_{\mathrm{e}} }{\partial x^{2}}. \end{aligned}$$

(45)

Now, substitute for ${\partial {{q}_{\mathrm{p}}^{\prime \prime }} }/{\partial x}$ and ${\partial {{q}_{\mathrm{e}}^{\prime \prime }} }/{\partial x}$ from Eqs. 40 and 41 into Eqs. 44 and 45, respectively, and simplify to obtain

$$\begin{aligned} \frac{\varLambda _{\mathrm{p}} }{v_{\mathrm{p}} }\frac{\partial }{\partial t}\left[ {C_{\mathrm{p}} \frac{\partial T_{\mathrm{p}} }{\partial t}+G\left( {T_{\mathrm{p}} -T_{\mathrm{e}} } \right) } \right] +C_{\mathrm{p}} \frac{\partial T_{\mathrm{p}} }{\partial t}=k_{\mathrm{p}} \frac{\partial ^{2}T_{\mathrm{p}} }{\partial x^{2}}-G\left( {T_{\mathrm{p}} -T_{\mathrm{e}} } \right) \end{aligned}$$

(46)

and

$$\begin{aligned} \frac{\varLambda _{\mathrm{e}} }{v_{\mathrm{e}} }\frac{\partial }{\partial t}\left[ {C_{\mathrm{e}} \frac{\partial T_{\mathrm{e}} }{\partial t}+G\left( {T_{\mathrm{e}} -T_{\mathrm{p}} } \right) } \right] +C_{\mathrm{e}} \frac{\partial T_{\mathrm{e}} }{\partial t}=k_{\mathrm{e}} \frac{\partial ^{2}T_{\mathrm{e}} }{\partial x^{2}}-G\left( {T_{\mathrm{e}} -T_{\mathrm{p}} } \right) . \end{aligned}$$

(47)

Noting that $\tau _{\mathrm{p}} ={\varLambda _{\mathrm{p}} }/{v_{\mathrm{p}} }$ and $\tau _{\mathrm{e}} ={\varLambda _{\mathrm{e}} }/{v_{\mathrm{e}} }$ are the relaxation times of the phonons and electrons, respectively, the above equations may be written as

$$\begin{aligned} \tau _{\mathrm{p}} \frac{\partial }{\partial t}\left[ {C_{\mathrm{p}} \frac{\partial T_{\mathrm{p}} }{\partial t}+G\left( {T_{\mathrm{p}} -T_{\mathrm{e}} } \right) } \right] +C_{\mathrm{p}} \frac{\partial T_{\mathrm{p}} }{\partial t}=k_{\mathrm{p}} \frac{\partial ^{2}T_{\mathrm{p}} }{\partial x^{2}}-G\left( {T_{\mathrm{p}} -T_{\mathrm{e}} } \right) \end{aligned}$$

(48)

and

$$\begin{aligned} \tau _{\mathrm{e}} \frac{\partial }{\partial t}\left[ {C_{\mathrm{e}} \frac{\partial T_{\mathrm{e}} }{\partial t}+G\left( {T_{\mathrm{e}} -T_p } \right) } \right] +C_{\mathrm{e}} \frac{\partial T_{\mathrm{e}} }{\partial t}=k_{\mathrm{e}} \frac{\partial ^{2}T_{\mathrm{e}} }{\partial x^{2}}-G\left( {T_{\mathrm{e}} -T_{\mathrm{p}} } \right) \end{aligned}$$

(49)

or,

$$\begin{aligned} \tau _{\mathrm{p}} C_{\mathrm{p}} \frac{\partial ^{2}T_{\mathrm{p}} }{\partial t^{2}}+\left( {C_{\mathrm{p}} +\tau _{\mathrm{p}} G} \right) \frac{\partial T_{\mathrm{p}} }{\partial t}-\tau _{\mathrm{p}} G\frac{\partial T_{\mathrm{e}} }{\partial t}=k_{\mathrm{p}} \frac{\partial ^{2}T_{\mathrm{p}} }{\partial x^{2}}-G\left( {T_{\mathrm{p}} -T_{\mathrm{e}} } \right) \end{aligned}$$

(50)

and

$$\begin{aligned} \tau _{\mathrm{e}} C_{\mathrm{e}} \frac{\partial ^{2}T_{\mathrm{e}} }{\partial t^{2}}+\left( {C_{\mathrm{e}} +\tau _{\mathrm{e}} G} \right) \frac{\partial T_{\mathrm{e}} }{\partial t}-\tau _{\mathrm{e}} G\frac{\partial T_{\mathrm{p}} }{\partial t}=k_{\mathrm{e}} \frac{\partial ^{2}T_{\mathrm{e}} }{\partial x^{2}}-G\left( {T_{\mathrm{e}} -T_{\mathrm{p}} } \right) . \end{aligned}$$

(51)

It should be noted that when the overall time scale of the heat transfer process is much larger than the relaxation time, then Eqs. 48 and 49 or Eqs. 50 and 51 can be further simplified to

$$\begin{aligned} C_{\mathrm{p}} \frac{\partial T_{\mathrm{p}} }{\partial t}=k_{\mathrm{p}} \frac{\partial ^{2}T_{\mathrm{p}} }{\partial x^{2}}-G\left( {T_{\mathrm{p}} -T_{\mathrm{e}} } \right) \end{aligned}$$

(52)

and

$$\begin{aligned} C_{\mathrm{e}} \frac{\partial T_{\mathrm{e}} }{\partial t}=k_{\mathrm{e}} \frac{\partial ^{2}T_{\mathrm{e}} }{\partial x^{2}}-G\left( {T_{\mathrm{e}} -T_{\mathrm{p}} } \right) . \end{aligned}$$

(53)

Equations 52 and 53 constitute the most familiar equations of the modified two-equation model [21].