Ballistic-phonon heat conduction at the nanoscale as revealed by time-resolved x-ray diffraction and time-domain thermoreflectance

We use time-resolved measurements of the evolution of surface and buried layer temperatures to quantify the contribution of ballistic phonons to heat transport on nanometer length scales. A laser pulse heats a $100\mathrm{nm}$ thick Al film which cools by conduction into a GaAs substrate. The top $120–250\mathrm{nm}$ of the GaAs substrate is doped with In to create a buried layer with a distinct lattice constant. The cooling of the Al film is monitored by time-domain thermoreflectance and, in the second set of experiments, the heating and cooling of the GaAs:In buried layer are monitored by time-resolved x-ray diffraction. The combination of these data shows that thermal transport by ballistic phonons accounts for nearly 20% of the heat flow across the buried layer on nanosecond time scales.

Figure 1

The ratio of the in-phase and out-of-phase signals from the rf lock-in amplifier used in TDTR measurements versus delay time between the pump and probe laser pulses for the $246\mathrm{nm}$ ${\left(\mathrm{Ga}\mathrm{As}\right)}_{0.993}{\left(\mathrm{In}\mathrm{As}\right)}_{0.007}$, $161\mathrm{nm}$ thick ${\left(\mathrm{Ga}\mathrm{As}\right)}_{0.986}{\left(\mathrm{In}\mathrm{As}\right)}_{0.014}$ layer, and $126\mathrm{nm}$ ${\left(\mathrm{Ga}\mathrm{As}\right)}_{0.981}{\left(\mathrm{In}\mathrm{As}\right)}_{0.019}$ samples. The dashed lines are fits of the data using a one-channel heat flow model in which the thermal conductivity of the ${\left(\mathrm{Ga}\mathrm{As}\right)}_{1-x}{\left(\mathrm{In}\mathrm{As}\right)}_x$ layers and the thermal conductance of the $\mathrm{Al}\text{−}{\left(\mathrm{Ga}\mathrm{As}\right)}_{1-x}{\left(\mathrm{In}\mathrm{As}\right)}_x$ interfaces are free parameters. The solid lines are fits of the data using a two-channel heat flow model which accounts for both diffusive and ballistic heat flows. In the two-channel model, the effusivity of the second channel is an additional free parameter.

Figure 2

Two scans of the 004 x-ray diffraction peak from a thin layer with composition ${\left(\mathrm{Ga}\mathrm{As}\right)}_{0.986}{\left(\mathrm{In}\mathrm{As}\right)}_{0.014}$. The filled circles show the diffraction peak before the arrival of the laser pulse and the open circles show the diffraction peak $4.4\mathrm{ns}$ after the laser pulse has reached the sample.

Figure 3

The time evolution of the average temperature of (a) the $246\mathrm{nm}$ ${\left(\mathrm{Ga}\mathrm{As}\right)}_{0.993}{\left(\mathrm{In}\mathrm{As}\right)}_{0.007}$ layer, (b) the $161\mathrm{nm}$ thick ${\left(\mathrm{Ga}\mathrm{As}\right)}_{0.986}{\left(\mathrm{In}\mathrm{As}\right)}_{0.014}$ layer, and (c) the $126\mathrm{nm}$ ${\left(\mathrm{Ga}\mathrm{As}\right)}_{0.981}{\left(\mathrm{In}\mathrm{As}\right)}_{0.019}$ layer as determined from shifts of the diffraction peak, see Eq. (12). The dashed lines are the evolution of the weighted average temperature of the buried layer predicted by a solution to the diffusion equation using the values of $\Lambda$ and $G$ measured using a one-channel model to fit the TDTR data. The solid lines are fits of the data using a two-channel heat flow model in conjunction with TDTR data to determine $\Lambda$, $G$, and $\epsilon$.

Figure 4

Sensitivity [see Eqs. (14, 15)] of the TDTR (solid lines) and TRXRD data (dashed lines) to the ${\left(\mathrm{Ga}\mathrm{As}\right)}_{1-x}{\left(\mathrm{In}\mathrm{As}\right)}_x$ layer thermal conductivity $\Lambda$, the thermal conductance $G$ of the $\mathrm{Al}\text{−}{\left(\mathrm{Ga}\mathrm{As}\right)}_{1-x}{\left(\mathrm{In}\mathrm{As}\right)}_x$ interface, and the effusivity $\epsilon$ of the ballistic channel plotted as a function of the delay time between pump and probe. This calculation was based on typical values of the three free parameters, $G=100\mathrm{MW}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$, $\Lambda =15\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, and $\epsilon =1150\mathrm{J}{\mathrm{m}}^{-2}{\mathrm{K}}^{-2}{\mathrm{s}}^{-1∕2}$.

Figure 5

Thermal conductivity (filled circles) of the ${\left(\mathrm{Ga}\mathrm{As}\right)}_{1-x}{\left(\mathrm{In}\mathrm{As}\right)}_x$ layers and thermal conductance (open triangles) of the $\mathrm{Al}\text{−}{\left(\mathrm{Ga}\mathrm{As}\right)}_{1-x}{\left(\mathrm{In}\mathrm{As}\right)}_x$ interfaces for three samples as measured by TDTR and TRXRD versus percentage of InAs content. These values were extracted using a two-channel heat flow model. Also shown are predicted values (open circles) for the thermal conductivity of the ${\left(\mathrm{Ga}\mathrm{As}\right)}_{1-x}{\left(\mathrm{In}\mathrm{As}\right)}_x$ layers based on the lattice dynamics model described in Sec. 4C. For the sample with 1.4% InAs content, the points for the measured and predicted values of the thermal conductivity overlap.

Figure 6

The thermal effusivity of the ballistic heat flow channel (open triangles) used in the two-channel heat flow model to consistently fit both the TDTR and the TRXRD data plotted as a function of the percentage of the InAs content of the buried layer. Predicted values for the thermal effusivity of the ballistic channel based on the lattice dynamics model (LD model) described in Sec. 4C are plotted as open circles. The effusivity of the diffusive channel calculated from the full heat capacity of the alloy layer and the thermal conductivity of the alloy layer are plotted as open squares.