1 Introduction

Electric and thermal properties of metal–metal interfaces and multilayers are of great interest for many technological applications including modern electronics and direct thermal to electrical energy conversion [1–3]. Under the thermal excitation of an external heat source, the heating process of a metal occurs in two steps: (1) first, the external energy increases the temperature of the electron gas and (2) second, the hot electrons heat up the phonons through the electron–phonon interactions, until equilibrium is reached [3]. Thus, at the microscopic level, the electrons and phonons have different thermal energy levels, and therefore, they are not at the same temperature (in general), and the condition of local thermal equilibrium between these two gases does not exist. This condition together with the fact that the dominant energy carriers in metals are the electrons imply energy transfer among electrons and phonons. In fact, the coupling between electrons and phonons inside metals and interfaces plays a significant role in the energy transport and conversion in metal–metal-layered systems. Because of this effect, these kinds of systems have been proposed as promising materials for solid-state thermionic power generation and refrigeration with the potential to achieve an efficiency that might not be feasible with single-phase materials [4–7].

The description of heat conduction through materials is governed by their thermal conductivity (under steady-state conditions) and thermal diffusivity (under transient conditions). These well-known thermal properties are proportional to each other [8], in such a way that the thermal diffusivity can be determined from the thermal conductivity if the volumetric heat capacity is provided [8]. However, theoretical models on the thermal conductivity of solid–solid-layered systems are scarce [9–12]. Majumdar and Reddy [10] derived an analytical expression to quantify the total thermal conductance of a metal–nonmetal interface. Even though they have shown that the temperature dependence of the experimental thermal conductivity of TiN/MgO interfaces can be explained by taking into account the electron–phonon coupling, their results do not take into account the size effects (i.e., finite thickness) of the layers. Yoo and Anderson [11] previously derived a similar expression for the thermal impedance of interfaces between superconductors and dielectrics, whose validity is restricted to layers of thickness much larger than their intrinsic electron–phonon coupling length. Recently, Hopkins et al. [9] showed that the electron–phonon coupling in gold films have a dominant role in the heat transport through them, when the electron and phonon are not in equilibrium. More recently, based on the two-temperature model of heat conduction [10, 13], Ordonez-Miranda et al. [12] proposed an analytical expression for the thermal conductivity of metal–nonmetal multilayers, where the size effects and the electron–phonon interactions in the metallic layers are considered. These last results reduce to those previously reported [14], which are in good agreement with experimental data. Therefore, the modeling of the thermal conductivity of a metal–metal-layered system is also desirable.

In this study, the two-temperature model of heat conduction [15–17] is used to determine the effective thermal conductivity and the total interface thermal resistance of a metal–metal-bilayer system, where the electron–electron and phonon–phonon interactions across the interface are modeled by the Kapitza interfacial thermal resistance [14, 18, 19]. It is shown that the thermal conductivity of the system has a simple interpretation in terms of the total interfacial thermal resistance, which considers the size effects of both metallic layers and generalizes the expressions derived previously [10, 11]. Our results could be useful for interpreting experimental data and guiding the design of novel metal–metal-layered structures for energy applications.

2 Mathematical Formulation and Solutions

Let us consider the metal–metal-layered system shown in Fig. 1a, which is thermally excited in such a way that the heat propagates along the \(x\) direction. The problem to be solved is to find the effective thermal conductivity of the system. Inside each metallic layer, the heat transport is due to the streams of the electron and phonon gases, and their interactions, which indicate that the coupling between electrons and phonons should be considered. This coupling is described approximately by the two-temperature model (TTM) of heat conduction [17], which considers that the diffusive Fourier law of heat conduction is valid for both the electron and phonon gases, separately; and its predictions have shown acceptable agreement with experimental data for the nonequilibrium transport between the electron and phonon gases under short-pulsed laser excitations [20]. This model considers the nonequilibrium between the electrons and phonons by defining the electron temperature \(T_\mathrm{e}\) and the phonon temperature \(T_\mathrm{p} \), and describes their spatial evolution by the following coupled differential equations [10, 13]:

$$\begin{aligned} \frac{\text{ d}^{2}T_\mathrm{e}}{\text{ d}x^{2}}-\frac{G}{k_\mathrm{e}}(T_\mathrm{e} -T_\mathrm{p} )=0,\quad \frac{\text{ d}^{2}T_\mathrm{p}}{\text{ d}x^{2}}+\frac{G}{k_\mathrm{p}}(T_\mathrm{e} -T_\mathrm{p} )=0, \end{aligned}$$
(1)

where \(k_\mathrm{e}\) and \(k_\mathrm{p} \) are the electron and phonon thermal conductivities, respectively; and \(G\) is the electron–phonon coupling factor, which takes into account the electron–phonon interactions. Note that if the electrons gas is in equilibrium with the phonons gas (\(T_\mathrm{e} =T_\mathrm{p})\), both Eq.  1 reduce to the differential equation established by the Fourier law. This indicates that the difference between TTM and the Fourier law is due to the nonequilibrium between electrons and phonons inside the metallic layers. According to Eq. 1, the thermal equilibrium between electrons and phonons is reached when \(G\rightarrow \infty \)  (perfect coupling), which shows that in this limit, the predictions of the current approach should reduce to the results obtained under the Fourier law, as shown below. Typical values of the coupling factor for a wide variety of metals (such as copper, silver, gold, and others) at room temperature are of the order of \(G\approx \left({10^{16}\;\text{ to}\;\;10^{17}}\right)\;~\text{ W}\cdot \text{ m}^{-3}\cdot \text{ K}^{-1 }\) [10, 13, 21]. In general, it may depend on the temperature and geometry of the materials; however, in this study, to keep our approach tractable, we are going to consider that all the thermal properties are constants, as is usually the case in many problems of practical interest.

Fig. 1
figure 1

(a) Schematic diagram of the metal–metal bilayer system, which is made up of layers with electron thermal conductivity \(k_{\mathrm{e}i}\), phonon thermal conductivity \(k_{\mathrm{p}i} \), electron–phonon coupling factor \(G_i\), and thickness \(l_i \), for \(i=1\) and 2 and (b) normalized temperature profiles as a function of the normalized position. Calculations were performed for \(k_{\mathrm{e}1} =5k_{\mathrm{p}1}, k_\mathrm{e2} =4k_{\mathrm{p}2},3k_\mathrm{p2} =2k_{\mathrm{p}1} , l_1 =l_2 , d_1 =d_2\), and \(R_\mathrm{ee} =5R_\mathrm{pp} =5{d_1 }/{k_1 }\)

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The solution of Eq. 1 is easy to obtain and is given by [12]

$$\begin{aligned} T_\mathrm{e} (x)=A+Bx+\frac{k_\mathrm{a}}{k_\mathrm{e}}\left[ {C\sinh \left({x/d}\right)+D\cosh \left({x/d}\right)}\right],\end{aligned}$$
(2a)
$$\begin{aligned} T_\mathrm{p} (x)=A+Bx-\frac{k_\mathrm{a}}{k_\mathrm{p}}\left[ {C\sinh \left({x/d}\right)+D\cosh \left({x/d}\right)}\right], \end{aligned}$$
(2b)

where \(A\), \(B\), \(C\), and \(D\) are constants that depend on the boundary conditions of a particular problem, and \(d\) is the intrinsic electron–phonon coupling length of metals defined by

$$\begin{aligned} d^{2}=\frac{k_\mathrm{a}}{G},\quad \frac{1}{k_\mathrm{a}}=\frac{1}{k_\mathrm{e}}+\frac{1}{k_\mathrm{p}}, \end{aligned}$$
(3)

with \(k_\mathrm{a}\) half of the harmonic mean of the electron and phonon thermal conductivities. Given that the microscopic electron–phonon interactions require space in order to take place, in general, the TTM is suitable to study the heat conduction in layers with thicknesses larger than the mean free path of the energy carriers. This constraint was the assumption done by Qiu and Tien [13] to derive this model from the Boltzmann transport equation, by evaluating its scattering term using quantum mechanical and statistical considerations. It was found that the coupling length for a wide variety of metals at room temperature is of the order of hundreds of nanometers (\(10^{-7}\text{ m}\)) [10, 13]. The mean free path of electrons and phonons in metals are of the order of tens of nanometers and a few nanometers, respectively, at room temperature [22–24]. We can thus conclude that the TTM could be useful for studying the impact of an electron–phonon coupling effect on the thermal conductivity of metal–metal materials with a thickness on the order of nanometers and upward [12].

$$\begin{aligned} T_{\mathrm{e}i} (x)=A_i +B_i x+\frac{k_{\mathrm{a}i}}{k_{\mathrm{e}i}}\left[ {C_i\sinh \left({x/{d_i }}\right)+D_i\cosh \left({x/{d_i }} \right)}\right],\end{aligned}$$
(4a)
$$\begin{aligned} T_{\mathrm{p}i} (x)=A_i +B_i x-\frac{k_{\mathrm{a}i}}{k_{\mathrm{p}i}}\left[ {C_i\sinh \left({x/{d_i }}\right)+D_i\cosh \left({x/{d_i}} \right)}\right], \end{aligned}$$
(4b)

where \(i=1 \text{ and} 2\), for the first and second layer, respectively. After writing out the general solutions for the temperature profiles inside the layers, it is necessary to establish boundary conditions to find the specific solutions. In general, the energy transport across metal–metal interfaces is determined by the electron–electron, phonon–phonon, electron–phonon, and phonon–electron interfacial couplings. Even though these last two channels of interfacial heat transport are always present, experimental or theoretical methodologies to quantify their contributions are scarce [9, 10]. However, it was suggested that for a wide variety of metals, their contribution to the total heat flux through the metal–metal interface could be small compared to those of the electron–electron and phonon–phonon interfacial interactions [10]. Based on these facts and for keeping the problem analytically solvable, we are going to consider the electron–electron and phonon–phonon interfacial interactions only, and focus our study on the effect of electron–phonon coupling on the effective thermal conductivity of metal–metal bilayer systems. Under these conditions, the boundary conditions obtained from the usual requirement of the temperature discontinuity and heat flux continuity at the interfaces are given by

$$\begin{aligned}&\left. {k_\mathrm{e1}\frac{\text{ d}T_\mathrm{e1}}{\text{ d}x}} \right|_{x=l_1 } =\left. {k_\mathrm{e2}\frac{\text{ d}T_\mathrm{e2}}{\text{ d}x}}\right|_{x=l_1 } ,\left. \quad {k_{\mathrm{p}1} \frac{\text{ d}T_{\mathrm{p}1}}{\text{ d}x}}\right|_{x=l_1 } =\left. {k_{\mathrm{p}2}\frac{\text{ d}T_{\mathrm{p}2}}{\text{ d}x}} \right|_{x=l_1},\end{aligned}$$
(5a)
$$\begin{aligned}&\left. {\left({T_\mathrm{e1} -T_\mathrm{e2}}\right)}\right|_{x=l_1 } =\left. {-R_\mathrm{ee} k_{\mathrm{e}2}\frac{\text{ d}T_\mathrm{e2}}{\text{ d}x}}\right|_{x=l_1 } ,\left.\quad {\left({T_\mathrm{p1} -T_\mathrm{p2}}\right)}\right|_{x=l_1 } =\left. {-R_\mathrm{pp} k_\mathrm{p2}\frac{\text{ d}T_\mathrm{p2}}{\text{ d}x}}\right|_{x=l_1},\nonumber \\ \end{aligned}$$
(5b)

where \(R_\mathrm{ee} \) and \(R_\mathrm{pp} \) are the electron–electron and phonon–phonon interfacial thermal resistances, respectively, which establish a jump in the electron and phonon temperatures at the interface. Typical values for both \(R_\mathrm{ee}\) and \(R_\mathrm{pp}\) were reported on the order of \(\text{10}^{-9 } \, \text{ m}^{2}\cdot \text{ K}\cdot \text{ W}^{-1}\) for a wide variety of metals at room temperature [10, 14]. In addition, by fixing the temperature at the surfaces \(x=0\) and \(x=l=l_1 +l_2\) of the layered system, at the constant values \(T_0 \) and \(T_1\), respectively; it can be considered that the electrons and phonons are in equilibrium at those positions and therefore the external boundary conditions can be written as

$$\begin{aligned} \left. {T_\mathrm{e1}}\right|_{x=0} =\left. {T_\mathrm{p1}}\right|_{x=0} =T_0 ,\left. \quad {T_{\mathrm{e}2}}\right|_{x=l} =\left. {T_\mathrm{p2}}\right|_{x=l} =T_1, \end{aligned}$$
(6)

where without loss of generality, it will be assumed that \(T_0 >T_1\) for the rest of the paper, such that the heat flows in the positive \(x\) direction (see Fig. 1a). The thermal equilibrium of the electron and phonon gases at the surfaces \(x=0\) and \(x=l\) could be obtained by placing these surfaces in thermal contact with a thermal bath system, as usually the steady-state conditions are imposed. After inserting Eqs. 4a and 4b into the eight boundary conditions given by Eqs. 5a–6, all the constants involved in Eqs. 4a and 4b are determined. The explicit expressions for the electron and phonon temperature profiles can then be written as

$$\begin{aligned} T_{\mathrm{e}1} (x)&= T_0 -\frac{\Delta T}{\lambda }\frac{d_1 }{k_1 }\left({\frac{x}{d_1 }-\frac{\beta _{12}}{Rk_\mathrm{e1}}\frac{\sinh \left({x/d_1 }\right)}{\cosh \left({{l_1 }/{d_1 }}\right)}}\right),\end{aligned}$$
(7a)
$$\begin{aligned} T_{\mathrm{p}1} (x)&= T_0 -\frac{\Delta T}{\lambda }\frac{d_1 }{k_1 }\left({\frac{x}{d_1 }+\frac{\beta _{12}}{Rk_{\mathrm{p}1}}\frac{\sinh \left({x/{d_1 }}\right)}{\cosh \left({{l_1 }/{d_1 }}\right)}}\right),\end{aligned}$$
(7b)
$$\begin{aligned} T_\mathrm{e2} (x)&= T_1 +\frac{\Delta T}{\lambda }\frac{d_2 }{k_2 }\left({\frac{l-x}{d_2 }-\frac{\beta _{21}}{Rk_\mathrm{e2}}\frac{\sinh \left({{\left({l-x}\right)}/{d_2 }}\right)}{\cosh \left({{l_2 }/{d_2 }}\right)}}\right),\end{aligned}$$
(7c)
$$\begin{aligned} T_{\mathrm{p}2} (x)&= T_1 +\frac{\Delta T}{\lambda }\frac{d_2 }{k_2 }\left({\frac{l-x}{d_2 }+\frac{\beta _{21}}{Rk_{\mathrm{p}2}}\frac{\sinh \left({{\left({l-x}\right)}/{d_2 }}\right)}{\cosh \left({{l_2 }/{d_2 }}\right)}}\right), \end{aligned}$$
(7d)

where \(\Delta T=T_0 -T_1 , k_i =k_{\mathrm{e}i} +T_{\mathrm{p}i} \), and

$$\begin{aligned} \lambda&= \frac{l_1 }{k_1 }+\frac{l_2 }{k_2 }+R_{12} ,\end{aligned}$$
(8a)
$$\begin{aligned} RR_{12}&= R_\mathrm{ee} R_\mathrm{pp} +\chi _\mathrm{ep} R_\mathrm{ee} +\chi _\mathrm{pe} R_\mathrm{pp} +\alpha _{12}^2\frac{k_{\mathrm{e}2}}{k_\mathrm{e1}}\frac{k_{\mathrm{p}2}}{k_{\mathrm{p}1}}\delta _1\delta _2 ,\end{aligned}$$
(8b)
$$\begin{aligned} R&= R_\mathrm{ee} +R_\mathrm{pp} +\frac{d_1 }{k_\mathrm{a1}}\tanh \left({{l_1 }/{d_1 }}\right)+\frac{d_2 }{k_{\mathrm{a}2}}\tanh \left({{l_2 }/{d_2 }}\right),\end{aligned}$$
(8c)
$$\begin{aligned} \alpha _{ij}&= \frac{k_{\mathrm{e}i}}{k_{\mathrm{e}j}}-\frac{k_{\mathrm{p}i}}{k_{\mathrm{p}j}},\end{aligned}$$
(8d)
$$\begin{aligned} \beta _{ij}&= R_\mathrm{ee} k_{\mathrm{e}i} -R_\mathrm{pp} k_{\mathrm{p}i} +\alpha _{ij} d_j\tanh \left({{l_j }/{d_j }}\right),\end{aligned}$$
(8e)
$$\begin{aligned} \chi _{ij}&= \frac{k_{i1}}{k_{j1}}\delta _1 +\frac{k_{i2}}{k_{j2}}\delta _2 ,\end{aligned}$$
(8f)
$$\begin{aligned} \delta _i&= \frac{d_i }{k_i }\tanh \left({{l_i }/{d_i }}\right), \end{aligned}$$
(8g)

for \(i,j=1\) and 2. Equations 7a–8g indicate that for any position inside the first metallic layer, the electron temperature is larger (smaller) than the phonon temperature, when the parameter \(\beta _{12}\) is positive (negative). The opposite situation holds for the temperature profiles of the second layer, through its parameter \(\beta _{21}\). Note that for \(R_\mathrm{ee} k_{\mathrm{e}1}>R_\mathrm{pp} k_{\mathrm{p}1}\), the sign of \(\beta _{12} \) is determined by the ratios of the electron and phonon thermal conductivities of the metallic layers, as shown by Eq. 8d.

3 Results and Discussion

Figure 1b shows the normalized temperature of the electrons \(\theta _{\mathrm{e}i} (x)\!=\!{\left({T_{\mathrm{e}i} (x)\!-\!T_1 }\right)}/{\Delta T}\) and phonons \(\theta _{\mathrm{p}i} (x)={\left({T_{\mathrm{p}i} (x)-T_1 }\right)}/{\Delta T}\) as a function of the normalized position \(x/l_1 \), for typical values of the electron and phonon thermal conductivities [17]. The nonlinear decrease of the electron and phonon temperatures indicates that the electron–phonon interaction (coupling factor) plays an important role in the thermal transport process. The jumps (\(T_\mathrm{e1} -T_\mathrm{e2} >0\) and \(T_{\mathrm{p}1} -T_{\mathrm{p}2} >0)\) of the electron and phonon temperatures, at the interface (\(x=l_1\)), are due to the effects of the electron and phonon interfacial thermal resistances, such that \(T_{\mathrm{e}1} -T_{\mathrm{e}2} >T_{\mathrm{p}1} -T_\mathrm{p2} \) because of \(R_\mathrm{ee} =\;5R_\mathrm{pp} \), for Fig. 1b. The normalized temperatures \(\theta _i \) correspond to the linear extrapolation of the electron and phonon temperatures, which would be obtained if the electrons and phonons were in equilibrium (\(T_{\mathrm{e}i} =T_{\mathrm{p}i} )\).

Considering that under steady-state conditions (see Eq. 6), the heat flux \(Q_0\) through the layered system is uniform, according to the Fourier law, the effective thermal conductivity \(k\) of the metal–metal-layered system can be defined as

$$\begin{aligned} Q_0 =k\frac{\Delta T}{l}. \end{aligned}$$
(9)

After calculating the heat flux \(Q_0\) using the linear extrapolation (common term) \(T_1 \) (or \(T_2\)) of the electron and phonon temperature profiles of the first (or second) layer, and combining the obtained result with Eq. 9, it is found that the effective thermal conductivity \(k\) is given by \(l/k=\lambda \), which, according to Eq. 8a, yields

$$\begin{aligned} \frac{l}{k}=\frac{l_1 }{k_1 }+\frac{l_2 }{k_2 }+R_{12}. \end{aligned}$$
(10)

Equation 10 resembles the well-known formula of the sum of thermal resistances in series [25], and suggests that the term \(R_{12}\) defined in Eq. 8b plays the role of an effective or total interfacial thermal resistance [10]. In fact, this is the case, given that it is easy to verify that this interfacial parameter defines the jump of the linearly extrapolated temperatures, \(T_1\) and \(T_2 \), such that \(T_1 (l_1 )-T_2 (l_1 )=R_{12} Q_0 \). It is worthwhile to point out the following remarks on Eqs. 8 and 10: (1) the effective thermal conductivity \(k\) depends on the electron–phonon coupling factors \(G_i\) of the metallic layers through the total interfacial thermal resistance \(R_{12}\), which can be interpreted as an average of the electron and phonon interfacial thermal resistances and a volumetric electron–phonon thermal resistance (the last term in Eq. 8b). (2) The contribution of \(G_i\) appears through the coupling length \(d_i\) only, whose effects are determined by its relative values with respect to the physical thickness \(l_i\) of the metallic layer, \(i=1\) and 2. (3) The total interfacial thermal resistance is invariant under the interchange of electrons by phonons [\(R_{12} (\text{ e},\text{ p})=R_{12} (\text{ p,e})\)]. This feature of \(R_{12} \) and \(k\) is due to the symmetry of the problem. Furthermore, the fact that \(R_{12} =R_{21}\), indicates that the total interfacial thermal resistance is independent of the direction in which the heat flows through the interface, which agrees with the definition of the Kapitza thermal resistance [14].

For a metallic thin film deposited on a dielectric substrate (\(k_\mathrm{e2} <<k_\mathrm{p2}, G_2\rightarrow \infty ,\) and \(R_\mathrm{ee}\rightarrow \infty )\), Eq. 8b reduces to our previous result [12], as expected. In order to have further insight of the effects of the electron–phonon coupling factors \(G_i\) on the total interfacial thermal resistance, the following limiting cases are considered:

3.1 Electron–Phonon Equilibrium

This case corresponds to the Fourier law approach and is determined by the condition \(G_i\rightarrow \infty (d_i\rightarrow 0)\), for \(i=1\) and 2. In this limit, the total interfacial thermal resistance defined in Eq. 8b reduces to

$$\begin{aligned} R_{12} =\frac{R_\mathrm{ee} R_\mathrm{pp}}{R_\mathrm{ee} +R_\mathrm{pp}}, \end{aligned}$$
(11)

which is just half of the harmonic mean of the electron and phonon interfacial thermal resistances, which are on the order of \(10^{-9}\,\text{ m}^{2}\cdot \text{ K}\cdot \text{ W}^{-1}\) at room temperature [10, 14]. Equation 11 combines the electron and phonon interfacial mismatches in just one parameter, which indicates that the electron and phonon gases can be considered as just one gas, given that the electrons and phonons are in equilibrium.

3.2 Large Electron Thermal Conductivity

This case is determined by the condition \(k_{\mathrm{e}i} >>k_{\mathrm{p}i} \), for \(i=1\) and 2, which is usually the case for a wide variety of metals [10]. Under these conditions \(k_{\mathrm{a}i}\approx k_{\mathrm{p}i}, \chi _{\mathrm{p}e}\rightarrow 0\) and the total interfacial thermal resistance \(R_{12} \) becomes less sensitive to \(R_\mathrm{pp} \) as follows (see Eq. 8a):

$$\begin{aligned} RR_{12} =R_\mathrm{ee}\left({R_\mathrm{pp} +\chi _\mathrm{ep}}\right)+\alpha _{12}^2\frac{k_{\mathrm{e}2}}{k_{\mathrm{e}1}}\frac{k_{\mathrm{p}2}}{k_{\mathrm{p}1}}\delta _1\delta _2. \end{aligned}$$
(12)

Note that Eq. 12 becomes independent of the electron–phonon couplings, for \(l_i <<d_i\), which indicates that for these small thicknesses the electron–phonon interactions do not take place, as expected. On the other hand, taking into account that the phonon thermal conductivity of metals is in the range of (10 to 20)  \(\text{ W}\cdot \text{ m}^{-1}\cdot \text{ K}^{-1}\) [10], which is one order of magnitude lower than their electronic thermal conductivity [8]; Eq. 12 shows that for thicknesses of the same order of magnitude or larger than the coupling length (\(l_i\ge d_i\approx 10^{-7}\text{ m})\), the total interfacial thermal resistance and therefore the effective thermal conductivity depends strongly on the electron–phonon coupling through the parameters \(R, \chi _\mathrm{ep} \), and \(\delta _i\). Thus, it is clear that the influence of the electron–phonon coupling is not only determined by the ratio of electron and phonon thermal conductivities, but also by the relative thickness of the layers.

3.3 Minimum of the Total Interfacial Thermal Resistance

According to Eqs. 8a and 8d, the total interfacial thermal resistance can be minimized by adjusting the electron and phonon thermal conductivities of both layers, such that they satisfy the following constraint:

$$\begin{aligned} \frac{k_\mathrm{e1}}{k_\mathrm{e2}}=\frac{k_{\mathrm{p}1}}{k_{\mathrm{p}2}}, \end{aligned}$$
(13)

which does not necessarily imply that both layers are of the same materials. In this case, Eq. 8b reduces to

$$\begin{aligned} R_{12} =\frac{R_\mathrm{ee} R_\mathrm{pp} +\chi _\mathrm{ep} R_\mathrm{ee} +\chi _\mathrm{pe} R_\mathrm{pp}}{R}, \end{aligned}$$
(14)

which vanishes in the absence of the electron and phonon interfacial thermal resistances. Taking into account that for a wide variety of metals, the ratio between the electron and phonon thermal conductivities \({k_{\mathrm{e}i}}/{k_{\mathrm{p}i}\approx 10}\) [10], Eq. 14 establishes that the total interfacial thermal resistance \(R_{12} \) depends strongly on the relative thickness \({l_i }/{d_i }\) of the metallic layers. Given that \(d_i\approx 10^{-7}\text{ m}\) [10], for micro-sized layers this ratio \({l_i}/{d_i }\approx 10\). In this way, Eqs. 11–14 show explicitly that the total interfacial thermal resistance \(R_{12} \) is strongly determined by the electron and phonon interfacial thermal resistances.

The normalized effective thermal conductivity \(k/{k_1 }\) as a function of the normalized thickness of the first and second layers is shown in Figs. 2a, b, respectively; for typical values of the electron and phonon thermal conductivities [13, 17]. Figure  2a shows that \(k/{k_1 }\) increases with the thickness \({l_1 }/{d_1}\). This behavior was expected, given that the total thermal conductivity of the first layer is larger than that of the second layer (\(5k_1 =9k_2 )\). For very large thicknesses of the first layer (\(l_1\approx 10^{3}d_1)\), the thermal conductivity becomes independent of both thicknesses \(l_1\) and \(l_2\) of the metallic layers and reaches its maximum value \(k_{\max } =k_1 \), which is independent of the thermal conductivity of the second layer, as expected for \(l_1 >>l_2\). Note also that for a small thickness \(l_1 <<d_1\), the effective thermal conductivity becomes independent of the thickness of the first layer and increases with the thickness of the second layer. In contrast, for \(l_1 >>d_1 ; k/{k_1}\) decreases when the thickness of the second layer increases. This indicates that for a fixed value of the thickness of the first layer, the overall thermal conductivity of the system can be maximized or minimized by adjusting the thickness of the second layer. Figure 2b shows that the thermal conductivity of the two-layer system can also be kept equal to that of the second layer, by choosing it large enough, such that \(l_2>>l_1 \).

Fig. 2
figure 2

Normalized effective thermal conductivity as a function of the normalized thickness of the (a) first and (b) second metallic layers. Calculations were performed for \(k_\mathrm{e1} =5k_{\mathrm{p}1} , k_{\mathrm{e}2} =4k_{\mathrm{p}2} , 2k_{\mathrm{p}1} =3k_{\mathrm{p}2} , d_2 =2d_1\), and \(R_\mathrm{ee} =5R_\mathrm{pp} =5{d_1 }/{k_1 }\)

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4 Conclusions

An analytical expression for the thermal conductivity of a metal–metal bilayer system has been derived under the framework of the two-temperature model of heat conduction. By using the concept of the total interface thermal resistance, which takes into account the electron–phonon coupling as well as the relative size of the metallic layers, it was shown that the overall thermal conductivity of the system can be interpreted as the sum of resistances in series. The obtained result depends strongly on ratios between the thickness of the metallic layers and their corresponding intrinsic coupling length and between the electron and phonon thermal conductivities of the layers. It has been demonstrated that the effective thermal conductivity of the system can be smaller, equal to, or larger than the thermal conductivity of the second layer. The obtained results could be useful in guiding the design of novel metal–metal-layered systems for electronics and energy applications.