Quasiballistic Thermal Transport from Nanoscale Heaters and the Role of the Spatial Frequency

Quasiballistic heat conduction from nanoscale heat sources of size comparable to phonon mean free paths has recently become of intense interest both scientifically and for its applications. Prior work has established that, in the quasiballistic regime, the apparent thermal properties of materials depend both on intrinsic mechanisms and the characteristics of the applied thermal gradient. However, many aspects of this regime remain poorly understood. Here, we experimentally study the thermal response of crystals to large thermal gradients generated by optical heating of nanoline arrays. Our experiments reveal the key role of the spatial frequencies and Fourier series amplitudes of the heating profile for thermal transport in the quasiballistic regime, in contrast to the conventional picture that focuses on the geometric dimensions of the individual heaters. Our work provides the insight needed to rationally mitigate local hot spots in modern applications by manipulating the spatial frequencies of the heater patterns.

Figure 1

Schematic of experimental geometry and example data from nanoline arrays. (a) Schematic of sample geometry. (b) Spatial heating profile on the substrate in real space (top) and versus spatial frequency (bottom). In the experiment, the probe beam that measures the thermal response has an identical geometry. (c) Representative AFM topography of 627-nm line arrays with a period of 1.2 $\mu \mathrm{m}$. The dashed green line indicates the cross section used for the height profile shown in the inset. (d) TEM cross-section profile of a 62-nm-width line. (e),(f) Representative TDTR amplitude and phase experimental data and best fit using a standard heat diffusion model [16] on line arrays at (e) 5.3-MHz modulation frequency, $w=1\mu \mathrm{m}$, and $L=2\mu \mathrm{m}$ at 294 K and (f) 3.1 MHz, $w=117$ nm, and $L=200$ nm at 150 K. In (e), the best-fit curve matches the data and yields a fitted thermal conductivity and interface conductance of 38 W ${\mathrm{m}}^{-1}$ ${\mathrm{K}}^{-1}$ and 196 MW ${\mathrm{m}}^{-2}$ ${\mathrm{K}}^{-1}$, respectively. In (f), the phase fitting is poor. In this situation, fitting to amplitude [16] or phase [17] will give different results.

Figure 2

Model to interpret TDTR data. (a) Amplitude and (b) phase of the thermal response of a thin film on a substrate versus temporal frequency for three spatial frequencies calculated using Fourier’s law (lines) and BTE (symbols). (c) Refit of experimental data for the case in Fig. 1 with the four-parameter fitting model. The best fit gives an excellent fit to the data with ${\kappa }_0=142$ W ${\mathrm{m}}^{-1}$ ${\mathrm{K}}^{-1}$, ${\kappa }_1=4$ W ${\mathrm{m}}^{-1}$ ${\mathrm{K}}^{-1}$, $G_0=137$ MW ${\mathrm{m}}^{-2}$ ${\mathrm{K}}^{-1}$, $G_1=155$ MW ${\mathrm{m}}^{-2}$ ${\mathrm{K}}^{-1}$, respectively. (d) Centerline temperature profile versus depth into the sample for a line array at a single temporal frequency of 10 MHz calculated using the four-parameter model (dashed lines) and the exact solution from the BTE (solid lines). The four-parameter model reasonably explains the spatial temperature profile into the sample.

Figure 3

Thermal conductivities of nanoline arrays. (a) Thermal conductivity ${\kappa }_0$ (open symbols) and ${\kappa }_1$ (filled symbols) versus period at 294 K (blue circles) and 150 K (red squares) for nanoline arrays with a 50% duty cycle. (b) ${\kappa }_1$ normalized to ${\kappa }_0$ versus period at different temperatures from measurements (symbols) and synthetic TDTR data (dotted lines). (c) ${\kappa }_1$ normalized to ${\kappa }_0$ from measurements (symbols) and synthetic TDTR data (dotted lines) versus period for nanoline arrays with $w\approx 115$ nm at various temperatures. (d) ${\kappa }_1$ normalized to ${\kappa }_0$ versus period for duty cycles of $50\mathrm{\%}$ and $67\mathrm{\%}$, respectively. The temperature is 150 K. In (c), the linewidth is fixed and the period changes, while in (d), the linewidth and period both change so as to keep the duty cycle fixed. In both (c) and (d), the synthetic TDTR data show similar trends. The synthetic data are the TDTR data sets computed from the BTE and then fit with the four-parameter model to obtain thermal conductivities and interface conductances.

Figure 4

Normalized thermal conductivity versus spatial frequency. ${\kappa }_1$ normalized to ${\kappa }_0$ at each temperature versus spatial frequency at different temperatures from measurements (open symbols) and synthetic TDTR data (dotted lines) for line arrays with a 50% duty cycle. Measurements for line arrays with other duty cycles are also shown with filled symbols. For all the measurements, the thermal conductivity decreases monotonically as the spatial frequency increases. At a given temperature, the duty cycle has only a little effect on the thermal conductivity for a given spatial frequency, indicating that spatial frequency is the key parameter. The synthetic TDTR data show similar trends as those observed experimentally.

Figure 5

Heat dissipation for different heater patterns. (a) Apparent thermal conductivity of the nanoline array with a 50% duty cycle and a Gaussian spot versus characteristic size from experiments and synthetic TDTR data normalized by ${\kappa }_0$. The thermal conductivity strongly decreases as the Gaussian diameter decreases, but only weak dependence is observed for the nanoline array. (b) Weighted thermal response versus dimensionless spatial frequency, at 5-MHz heating frequency for a Gaussian spot (top) and the nanoline array (bottom, red symbols). The grey lines show the continuous function from which the discrete points for the nanoline array are obtained. The zero spatial frequency response dominates the overall response for the line array, while it is zero for the Gaussian heating pattern due to the weighting factor of ${\xi }_r$. The absence of the dc response is the reason that quasiballistic effects are readily observed with a Gaussian spot. $L_c$ is the characteristic length used to normalize the spatial frequency, which is the period and root-mean square of the pump and probe $1/e^2$ diameters for the nanoline array and Gaussian spot, respectively. Note that the x axis is the dimensionless spatial frequency and that patterns with smaller $L_c$ have larger fundamental spatial frequency.