Thermal conductance of interfaces with amorphous ${\mathbf{SiO}}_2$ measured by time-resolved magneto-optic Kerr-effect thermometry

We use time-resolved magneto-optic Kerr effect and ultrathin Co/Pt transducer films to perform thermal-transport experiments with higher sensitivity and greater time resolution than typically available in studies of interfacial thermal transport by time-domain thermoreflectance. We measure the interface conductance between Pt and amorphous ${\mathrm{SiO}}_2$ using Pt/Co/Pt ferromagnetic transducer films with thicknesses between 4.2 and 8.2 nm and find an average value of $G_{\mathrm{Pt}}\approx 0.3\mathrm{GW}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$. This result demonstrates that interfaces between metals and amorphous dielectrics can have a conductance corresponding to Kapitza lengths of the order of 4 nm, and are thus of relevance when engineering nanoscale devices. For thin ${\mathrm{SiO}}_2$ layers, our method also provides sensitivity to the interface conductance between ${\mathrm{SiO}}_2$ and Si and we find $G_{\mathrm{Si}}=0.6\mathrm{GW}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ as the lower limit.

Figure 1

(a) Schematic of Co/Pt transducer films on ${\mathrm{SiO}}_2/\mathrm{Si}$ samples. (b) Hysteresis of Co/Pt thin films with total thicknesses of 4.2 nm (red), 5.5 nm (green), and 8.2 nm (blue) on 440-nm ${\mathrm{SiO}}_2$ measured by polar magneto-optic Kerr effect at a wavelength of $\lambda =635$ nm. The magnetic field was applied along the easy axis perpendicular to the sample plane.

Figure 2

Experimental setup. Pump and probe beam paths are split by polarizing beam splitters (PBS) and spectrally using short-pass (blue) and long-pass (orange) filters. Time-domain thermoreflectance (TDTR) is measured with a Si photodiode. Part of the beams reflected from the sample is redirected by a beam splitter (BS) towards a balanced photodetector to measure the time-dependent changes in Kerr rotation (TR-MOKE).

Figure 3

Contour intervals marking the best agreement between TR-MOKE data and thermal model simultaneously adjusting conductance $G_{\mathrm{Pt}}$ between transducer and sample and heat capacity $C_{\mathrm{ad}}$ of adsorbates. For the 4.2-nm Co/Pt film on 26-nm ${\mathrm{SiO}}_2/\mathrm{Si}$ (orange) the contour is much wider compared to the Co/Pt films on 440-nm-thick oxide (others). The reason for this is the increased noise in the data arising from the smaller out-of-phase signal due to the higher thermal conductivity of the substrate.

Figure 4

(a) Sensitivity coefficients with respect to interface conductance $G_{\mathrm{Pt}}$ (red), transducer thickness $h_{\mathrm{Pt}}$ (blue), and thermal conductivity ${\mathrm{\Lambda }}_{{\mathrm{SiO}}_2}$ (orange) versus delay time $t$ for a 5-nm-thick Pt transducer on thermally thick oxide. Simulation parameters are listed in Table 1, $G_{\mathrm{Pt}}=0.25\mathrm{GW}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$. (b) Maximum sensitivity coefficients $S_{\max }$ at delay time of maximum sensitivity $t_{\max }$ calculated for various thicknesses $h_{\mathrm{Pt}}$.

Figure 5

Data measured by time-domain thermoreflectance (TDTR, open squares) with 4.2-nm Co/Pt (blue) and with 43-nm Pt (black) transducer, and by time-resolved magneto-optic Kerr effect (TR-MOKE, blue circles) with 4.2-nm Co/Pt transducer on ${\mathrm{Al}}_2{\mathrm{O}}_3$. (a) Normalized in-phase signals. (b) Ratio signals, blue squares are multiplied by $-1$ as ratio is negative. Red lines are the best fits obtained using the interface conductance $G$ between Pt and sapphire as only free parameter. Model parameters are listed in Table 1. $C_{\mathrm{ad}}=2.8\mathrm{mJ}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ was considered for both samples.

Figure 6

(a) TR-MOKE ratio data versus delay time $t$ measured using Co/Pt transducers with thicknesses of 4.2 nm (red), 5.5 nm (green), and 8.2 nm (blue) on 440-nm (thermally thick) ${\mathrm{SiO}}_2$. Black lines are fits of thermal models to the data obtained treating $G_{\mathrm{Pt}}$ as only free parameter and considering an adsorbed layer with a heat capacity of $C_{\mathrm{ad}}=2.8\mathrm{mJ}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$. Black numbers are conductances in $\mathrm{MW}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$. (b) Results for conductances from this work [orange point for 4.2-nm Co/Pt on 26-nm ${\mathrm{SiO}}_2$, other color coding as in panel (a)] and data by Liu et al. [16]. All error bars are calculated assuming same relative uncertainty in experimental parameters.

Figure 7

TR-MOKE ratios normalized by models for $C_{\mathrm{ad}}=0\mathrm{mJ}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$, no $\mathrm{Pt}/{\mathrm{SiO}}_2$ interface, and no adjusted parameters to highlight deviations between data and model curves. Data points and black curves are same as in Fig. 6. Dashed lines are models representing experimental uncertainties. Red lines are best fits obtained not considering the additional heat capacity of adsorbates. $C_{\mathrm{ad}}$ values are in $\mathrm{mJ}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$, numbers next to model lines are conductance values $G_{\mathrm{Pt}}$ in $\mathrm{MW}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$. Green model line in (a) includes an interface with conductance $G_{\mathrm{ad}}=400\mathrm{MW}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ between Co/Pt transducer and adsorbed layer.

Figure 8

TR-MOKE ratio data measured with 4.2-nm-thin Co/Pt transducers on 26-nm ${\mathrm{SiO}}_2$. Thermal models include an adsorbed layer with a heat capacity of $C_{\mathrm{ad}}=2.8\mathrm{mJ}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ added on top of the Co/Pt. Conductance $G_{\mathrm{Pt}}=320\pm 110\mathrm{MW}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ of the $\mathrm{Pt}/{\mathrm{SiO}}_2$ interface was fitted between 30 and 500 ps. Black numbers are conductances $G_{\mathrm{Si}}$ between ${\mathrm{SiO}}_2$ and Si in $\mathrm{MW}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ fitted between 800 and 3600 ps. All curves are normalized by models for $C_{\mathrm{ad}}=0\mathrm{mJ}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$ and no $\mathrm{Pt}/{\mathrm{SiO}}_2$ and ${\mathrm{SiO}}_2/\mathrm{Si}$ interfaces.

Figure 9

Absorptions $A$ (dotted) and reflectances $R$ (dashed lines) calculated with a transfer-matrix model [36] for (a) Co/Pt transducers with varying thickness $h_{\mathrm{Co}/\mathrm{Pt}}$ on 440-nm ${\mathrm{SiO}}_2$. (b) Models and measured reflectance values for $h_{\mathrm{Co}/\mathrm{Pt}}=8.2\mathrm{nm}$ (blue), 5.5 nm (green), and 4.2 nm (red) for varying ${\mathrm{SiO}}_2$ thicknesses. (c) Kerr rotation angles ${\theta }_{\mathrm{K}}$ versus $h_{\mathrm{Co}/\mathrm{Pt}}$ and (d) versus $h_{\mathrm{SiO}2}$ including experimental values. If not stated differently, $Q=0.0063-0.062i$ was used as magneto-optic constant.