Akhiezer mechanism limits coherent heat conduction in phononic crystals

Heat in phononic crystals (PnCs) is carried by phonons, which can behave coherently (wave-like) or incoherently (particle-like) depending on the modes, temperature, and length scales. By comparing the measured thermal conductivity of PnCs with theories, recent works suggest that thermal conductivity of PnCs can be explained by considering only surface and boundary scatterings, which not only backscatter phonons but also break their coherence. The logic here is that since average phonon wavelength at room temperature is only a few nanometers, the roughness at the surfaces and boundaries make the scattering diffusive (break the phase coherence of phonons), and thus only very long wavelength (low frequency) phonons with negligible contribution to total thermal conductivity remain coherent. Here, we theoretically show that in a thin film PnCs, the low frequency coherent phonons could significantly contribute to thermal conductivity when assuming the three-phonon scattering model for intrinsic scattering because of their extremely large density of states that result from the low dimensional nature. Yet, further analysis shows the contribution of the low frequency coherent phonons is still negligible within temperature range from 130 to 300 K due to the Akhiezer mechanism, which properly answers the question why the thermal conductivity of PnCs can be explained by considering only scattering of incoherent phonons at these temperatures.

Figure 1

(a) Schematic of a two-dimensional silicon phononic crystal (PnC). $t=150\mathrm{nm},w=100\mathrm{nm}$, and $d=80\mathrm{nm}$ denote the height, width, and hole diameter of the PnC, respectively. (b) Phonon dispersion relation of PnC along G-X. (c) Frequency-dependent phonon group velocity of PnCs and thin film.

Figure 2

(a) Frequency and temperature dependent phonon relaxation time for bulk silicon crystals. The dashed lines are calculated results for longitudinal modes (LA) at different temperatures. Experimental data are measured results for LA modes and are taken from Refs. [15, 27, 28, 29] (b) Frequency-dependent relaxation time of thin film and PnCs at 300 K, and a comparison with bulk phonon relaxation time for LA modes and modes in full Brillion zone (FB). The boundary of Akhiezer and three-phonon-scattering regimes at 300 K is around 200 GHz. Note that “three-phonon scattering+Akhiezer” in Fig. 2 means the relaxation time calculated using Eq. (7).

Figure 3

A comparison of thermal conductivity of thin films and PnCs with two different relaxation time $\tau$ models at 130 and 300 K for a 2-nm roughness. (a) $\tau$ model: only three-phonon scattering; (b) $\tau$ model: three-phonon scattering and Akhiezer damping. The coherent regime is determined as 0–0.2 THz for 300 and 130 K when there is a 2-nm roughness.

Figure 4

Frequency-dependent density of states $D\left(\omega \right)$ of thin films and PnCs, and a comparison with density of states of bulk silicon crystals.

Figure 5

Contributions of coherent and incoherent phonons to total thermal conductivity of thin films and PnCs as a function of switching frequency at 300 and 130 K. The maximum coherent regime is determined as 0–3 THz at 300 K, and 0–4 THz at 130 K.

Figure 6

A comparison of thermal conductivity of a thin film and PnC with two different relaxation time $\tau$ models at 130 and 300 K for the most ideal case (no roughness). (a) $\tau$ model: only three-phonon scattering; (b) $\tau$ model: three-phonon scattering and Akhiezer damping. The coherent regime is, respectively, determined as 0–3 THz and 0–4 THz for 300 and 130 K when there is no roughness.

Figure 7

Switching frequency as a function of roughness size $R$ (surface roughness, hole wall roughness, lattice site displacement, hole disorder, etc.) for selected specularity parameters $P=0.3$.