Effect of temperature oscillation on thermal characteristics of an aluminum thin film

Abstract

Energy transport in aluminum thin film is examined due to temperature disturbance at the film edge. Thermal separation of electron and lattice systems is considered in the analysis, and temperature variation in each sub-system is formulated. The transient analysis of frequency-dependent and frequency-independent phonon radiative transport incorporating electron–phonon coupling is carried out in the thin film. The dispersion relations of aluminum are used in the frequency-dependent analysis. Temperature at one edge of the film is oscillated at various frequencies, and temporal response of phonon intensity distribution in the film is predicted numerically using the discrete ordinate method. To assess the phonon transport characteristics, equivalent equilibrium temperature is introduced. It is found that equivalent equilibrium temperature in the electron and lattice sub-systems oscillates due to temperature oscillation at the film edge. The amplitude of temperature oscillation reduces as the distance along the film thickness increases toward the low-temperature edge of the film. Equivalent equilibrium temperature attains lower values for the frequency-dependent solution of the phonon transport equation than that corresponding to frequency-independent solution.

Appendix: Diffusive limit consideration

In the diffusion limit, when the Knudsen number Kn = Λ/L → 0 (where Λ is the mean free path and L is the film thickness), Eqs. (3) and (4) should reduce to the two-equation model. This is can be demonstrated as follows. Multiplying Eqs. (3) and (4) throughout by 2πμdμ and then integrating from −1 to 1, it yields,

$$ \begin{aligned} \frac{2\pi }{{v_{\text{p}} }}\int_{ - 1}^{1} {\mu \frac{{\partial I_{\text{p}} }}{\partial t}\,{\text{d}}\mu } + 2\pi \int_{ - 1}^{1} {\mu^{2} \frac{{\partial I_{\text{p}} }}{\partial x}{\text{d}}\mu } & = \frac{{2\pi \left( {\int_{ - 1}^{1} {\mu \,{\text{d}}\mu } } \right)\left( {\tfrac{1}{2}\int_{ - 1}^{1} {I_{\text{p}} {\text{d}}\mu } } \right) - 2\pi \int_{ - 1}^{1} {\mu I_{\text{p}} \,{\text{d}}\mu } }}{{\Lambda _{\text{p}} }} \\ & \quad - 2\pi \frac{G}{4\pi }\left( {T_{\text{p}} - T_{\text{e}} } \right)\left( {\int_{ - 1}^{1} {\mu \,{\text{d}}\mu } } \right) \\ \end{aligned} $$

(35)

and

$$ \begin{aligned} \frac{2\pi }{{v_{\text{e}} }}\int_{ - 1}^{1} {\mu \frac{{\partial I_{\text{e}} }}{\partial t}{\text{d}}\mu } + 2\pi \int_{ - 1}^{1} {\mu^{2} \frac{{\partial I_{\text{e}} }}{\partial x}\,{\text{d}}\mu } & = \frac{{2\pi \left( {\int_{ - 1}^{1} {\mu \,{\text{d}}\mu } } \right)\left( {\tfrac{1}{2}\int_{ - 1}^{1} {I_{\text{e}} \,{\text{d}}\mu } } \right) - 2\pi \int_{ - 1}^{1} {\mu I_{\text{e}} \,{\text{d}}\mu } }}{{\Lambda _{\text{e}} }} \\ & \quad - 2\pi \frac{G}{4\pi }\left( {T_{\text{p}} - T_{\text{e}} } \right)\left( {\int_{ - 1}^{1} {\mu \,{\text{d}}\mu } } \right) \\ \end{aligned} $$

(36)

However, the second term on the left-hand side needs some elaboration. In the diffusive limit, the behavior of I _e,p(x, μ, t) is mainly linear in μ. This means that we may neglect the terms of order 2 or higher in the Taylor series expansion of I _e,p(x, μ, t) in μ:

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

(37)

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

Then,

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

(38)

Since, $\frac{{C_{\text{e,p}} v_{\text{e,p}} T_{\text{e,p}} }}{4\pi } = I_{\text{e,p}}^{o} \left( {x,t} \right) = \tfrac{1}{2}\int_{ - 1}^{1} {I_{\text{e,p}} \left( {x,\mu ,t} \right){\kern 1pt} {\text{d}}\mu }$, so that finally:

$$ 2\pi \int_{ - 1}^{1} {\mu^{2} \frac{{\partial I_{\text{e,p}} }}{\partial x}{\text{d}}\mu } = 2\pi \frac{2}{3}\frac{{{\text{d}}I_{\text{e,p}}^{o} }}{{{\text{d}}T_{\text{e,p}} }}\frac{{\partial T_{\text{e,p}} }}{\partial x} $$

Equations (35) and (36) are now simplified to:

$$ \frac{1}{{v_{\text{p}} }}\frac{{\partial q_{\text{p}}^{{\prime \prime }} }}{\partial t} + \frac{4\pi }{3}\frac{{{\text{d}}I_{\text{p}}^{o} }}{{{\text{d}}T_{\text{p}} }}\frac{{\partial T_{\text{p}} }}{\partial x} = \frac{{0 - q_{\text{p}}^{{\prime \prime }} }}{{\Lambda _{\text{p}} }} - 0 $$

(39)

and

$$ \frac{1}{{v_{\text{e}} }}\frac{{\partial q_{\text{e}}^{{\prime \prime }} }}{\partial t} + \frac{4\pi }{3}\frac{{{\text{d}}I_{\text{e}}^{o} }}{{{\text{d}}T_{\text{e}} }}\frac{{\partial T_{\text{e}} }}{\partial x} = \frac{{0 - q_{\text{e}}^{{\prime \prime }} }}{{\Lambda _{\text{e}} }} - 0 + 0 $$

(40)

or,

$$ \frac{{\Lambda _{\text{p}} }}{{v_{\text{p}} }}\frac{{\partial q_{\text{p}}^{{\prime \prime }} }}{\partial t} + q_{\text{p}}^{{\prime \prime }} = - \frac{{C_{\text{p}} v_{\text{p}}\Lambda _{\text{p}} }}{3}\frac{{\partial T_{\text{p}} }}{\partial x} $$

(41)

and

$$ \frac{{\Lambda _{\text{e}} }}{{v_{\text{e}} }}\frac{{\partial q_{\text{e}}^{{\prime \prime }} }}{\partial t} + q_{\text{e}}^{{\prime \prime }} = - \frac{{C_{\text{e}} v_{\text{e}}\Lambda _{\text{e}} }}{3}\frac{{\partial T_{\text{e}} }}{\partial x} $$

(42)

The kinetic theory formula for the thermal conductivity is given as,

$$ k_{\text{p}} = \frac{{C_{\text{p}} v_{\text{p}}\Lambda _{\text{p}} }}{3},\quad k_{\text{e}} = \frac{{C_{\text{e}} v_{\text{e}}\Lambda _{\text{e}} }}{3} $$

Hence, Eqs. (40) and (41) can be transformed into,

$$ \frac{{\Lambda _{\text{p}} }}{{v_{\text{p}} }}\frac{{\partial q_{\text{p}}^{{\prime \prime }} }}{\partial t} + q_{\text{p}}^{{\prime \prime }} = - k_{\text{p}} \frac{{\partial T_{\text{p}} }}{\partial x} $$

(43)

and

$$ \frac{{\Lambda _{\text{e}} }}{{v_{\text{e}} }}\frac{{\partial q_{\text{e}}^{{\prime \prime }} }}{\partial t} + q_{\text{p}}^{{\prime \prime }} = - k_{e} \frac{{\partial T_{\text{e}} }}{\partial x} $$

(44)

To eliminate $q_{\text{p}}^{\prime \prime }$ and $q_{\text{e}}^{\prime \prime }$ between Eqs. (41) and (43) as well as Eqs. (44), one can proceed as follows. After differentiating Eqs. (42) and (44) once with respect to x, it gives:

$$ \frac{{\Lambda _{\text{p}} }}{{v_{\text{p}} }}\frac{\partial }{\partial t}\left( {\frac{{\partial q_{\text{p}}^{{\prime \prime }} }}{\partial x}} \right) + \frac{{\partial q_{\text{p}}^{{\prime \prime }} }}{\partial x} = - k_{\text{p}} \frac{{\partial^{2} T_{\text{p}} }}{{\partial x^{2} }} $$

(45)

and

$$ \frac{{\Lambda _{\text{e}} }}{{v_{\text{e}} }}\frac{\partial }{\partial t}\left( {\frac{{\partial q_{\text{e}}^{{\prime \prime }} }}{\partial x}} \right) + \frac{{\partial q_{\text{e}}^{{\prime \prime }} }}{\partial x} = - k_{\text{e}} \frac{{\partial^{2} T_{\text{e}} }}{{\partial x^{2} }} $$

(46)

Now, substitute for $\partial q_{\text{p}}^{\prime \prime } /\partial x$ and $\partial q_{\text{e}}^{\prime \prime } /\partial x$ from Eqs. (41) and (42) into Eqs. (45) and (46), respectively, and simplify to obtain:

$$ \frac{{\Lambda _{\text{p}} }}{{v_{\text{p}} }}\frac{\partial }{\partial t}\left[ {C_{\text{p}} \frac{{\partial T_{\text{p}} }}{\partial t} + G\left( {T_{\text{p}} - T_{\text{e}} } \right)} \right] + C_{\text{p}} \frac{{\partial T_{\text{p}} }}{\partial t} = k_{\text{p}} \frac{{\partial^{2} T_{\text{p}} }}{{\partial x^{2} }} - G\left( {T_{\text{p}} - T_{\text{e}} } \right) $$

(47)

and

$$ \frac{{\Lambda _{\text{e}} }}{{v_{\text{e}} }}\frac{\partial }{\partial t}\left[ {C_{\text{e}} \frac{{\partial T_{\text{e}} }}{\partial t} + G\left( {T_{\text{e}} - T_{\text{p}} } \right)} \right] + C_{\text{e}} \frac{{\partial T_{\text{e}} }}{\partial t} = k_{\text{e}} \frac{{\partial^{2} T_{\text{e}} }}{{\partial x^{2} }} - G\left( {T_{\text{e}} - T_{\text{p}} } \right) $$

(48)

Noting that τ _p = Λ_p/v _p and τ _e = Λ_e/v _e are the relaxation times of the phonons and electrons, respectively, the above equations may be written as,

$$ \tau_{\text{p}} \frac{\partial }{\partial t}\left[ {C_{\text{p}} \frac{{\partial T_{\text{p}} }}{\partial t} + G\left( {T_{\text{p}} - T_{\text{e}} } \right)} \right] + C_{\text{p}} \frac{{\partial T_{\text{p}} }}{\partial t} = k_{\text{p}} \frac{{\partial^{2} T_{\text{p}} }}{{\partial x^{2} }} - G\left( {T_{\text{p}} - T_{\text{e}} } \right) $$

(49)

and

$$ \tau_{\text{e}} \frac{\partial }{\partial t}\left[ {C_{\text{e}} \frac{{\partial T_{\text{e}} }}{\partial t} + G\left( {T_{\text{e}} - T_{\text{p}} } \right)} \right] + C_{\text{e}} \frac{{\partial T_{\text{e}} }}{\partial t} = k_{\text{e}} \frac{{\partial^{2} T_{\text{e}} }}{{\partial x^{2} }} - G\left( {T_{\text{e}} - T_{\text{p}} } \right) $$

(50)

or,

$$ \tau_{\text{p}} C_{\text{p}} \frac{{\partial^{2} T_{\text{p}} }}{{\partial t^{2} }} + \left( {C_{\text{p}} + \tau_{\text{p}} G} \right)\frac{{\partial T_{\text{p}} }}{\partial t} - \tau_{\text{p}} G\frac{{\partial T_{\text{e}} }}{\partial t} = k_{\text{p}} \frac{{\partial^{2} T_{\text{p}} }}{{\partial x^{2} }} - G\left( {T_{\text{p}} - T_{\text{e}} } \right) $$

(51)

and

$$ \tau_{\text{e}} C_{\text{e}} \frac{{\partial^{2} T_{\text{e}} }}{{\partial t^{2} }} + \left( {C_{\text{e}} + \tau_{\text{e}} G} \right)\frac{{\partial T_{\text{e}} }}{\partial t} - \tau_{\text{e}} G\frac{{\partial T_{\text{p}} }}{\partial t} = k_{\text{e}} \frac{{\partial^{2} T_{\text{e}} }}{{\partial x^{2} }} - G\left( {T_{\text{e}} - T_{\text{p}} } \right) $$

(52)

It should be noted that when the overall time scale of the heat transfer process is much larger than the relaxation time, Eqs. (49) and (50) or Eqs. (51) and (52) can be further simplified to,

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

(53)

and

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

(54)

Equations (53) and (54) constitute the familiar of two-equation model [21].