Two-channel model for nonequilibrium thermal transport in pump-probe experiments

We present an analytic solution for heat flow in a multilayer two-channel system for the interpretation of time-domain thermoreflectance (TDTR) experiments where nonequilibrium effects are important. The two-channel solution is used to analyze new room temperature TDTR measurements of Al/Cu and Al/Si${}_{0.99}$Ge${}_{0.01}$ systems. Cu and Si${}_{0.99}$Ge${}_{0.01}$ are examples of materials well suited for analysis with a two-channel model because thermal excitations responsible for the vast majority of the heat capacity in these solids contribute little to their thermal conductivity. Nonequilibrium effects are found to be unimportant for the interpretation of the Al/Cu TDTR data but dramatic for the Al/Si${}_{0.99}$Ge${}_{0.01}$ TDTR data. The two-channel model predicts a significant reduction in the effective thermal conductivity of Si${}_{0.99}$Ge${}_{0.01}$ in a region within 150 nm of the Al/Si${}_{0.99}$Ge${}_{0.01}$ interface. The extra thermal resistance in this region is a result of the disparate heat flux boundary conditions for low- and high-frequency phonons in combination with weak coupling between low- and high-frequency phonons. When the experimental data are analyzed with a single-channel model, both the conductance and thermal conductivity appear to depend on pump-modulation frequency, consistent with the two-channel model's predictions. Finally, we compare the results of our diffusive two-channel model to a nonlocal description for steady-state heat flow near a boundary and show they yield nearly identical results.

Figure 1

Calculated frequency response of 1D semi-infinite two-channel Cu and Si${}_{0.99}$Ge${}_{0.01}$ layers. For Cu, $T_1$ and $T_2$ are the temperature of the electrons and phonons, while in Si${}_{0.99}$Ge${}_{0.01}$, $T_1$ and $T_2$ describe the low- and high-frequency phonons. Thermal properties of both systems are described in the text. (a) Amplitude of temperature oscillations for each channel as a function of depth at a heating frequency of 20 MHz. The temperature rise of each channel is normalized with the prediction for surface temperature from a one channel model with bulk properties, $T_B$(0). (b) Surface temperature of channel 2 as a function of heating frequency. Solid lines are the in-phase temperature and dashed lines are the out-of-phase temperature. A one-channel model predicts no difference in the in-phase and out-of-phase temperature response.

Figure 2

(a) Calculated dependence of TDTR data on coupling strength, $g$, at fixed modulation frequency of 20 MHz and 1/$e^2$ laser spot size of 10 μm for an 80 nm Al/Si${}_{0.99}$Ge${}_{0.01}$ two-channel system. The units of $g$ are W m${}^{-3}$ K${}^{-1}$. The apparent interface conductance and thermal conductivity are defined as the values that result from analyzing the two-channel data with a single-channel model. The apparent interface conductance is primarily determined by the time decay of the in-phase signal, while the apparent thermal conductivity is primarily determined by the magnitude of the out-of-phase signal. (b) The apparent interface conductance $G_A$ versus coupling strength. (c) The apparent thermal conductivity ${\Lambda }_A$ versus coupling strength.

Figure 3

Apparent thermal conductivity and interface conductance from TDTR measurements of Al/Cu and Al/Si${}_{0.99}$Ge${}_{0.01}$ systems as a function of pump-modulation frequency. Lines indicate the results of fitting TDTR signals generated by the two-channel model with a single-channel model. The units of $g$ are W m${}^{-3}$ K${}^{-1}$.

Figure 4

The ratio of heat flow for ballistic phonons, $q_B$, to the value predicted by Fourier's law, $q_F=C{v}_z{\ell }_z$, for steady-state conditions and a boundary that transmits and reflects phonons at $z$ $=$ 0. The curves were calculated with Eq. (17) for a linear temperature gradient at $z$ > 0. The temperature is assumed to be continuous at $z$ $=$ 0 and constant at $z$ < 0. Solid lines represent the radiation limit ($R$ $=$ 0), while dashed lines represent the adiabatic limit ($R$ $=$ 1).