Microscopic description of thermal-phonon coherence: From coherent transport to diffuse interface scattering in superlattices

We demonstrate the existence of a coherent transport of thermal energy in superlattices by introducing a microscopic definition of the phonon coherence length. A criterion is provided to distinguish the coherent transport regime from diffuse interface scattering and discuss how these can be specifically controlled by several physical parameters. Our approach provides a convenient framework for the interpretation of previous thermal conductivity measurements and calculations; it also paves the way for the design of a new class of thermal interface materials.

Figure 1

Illustration of a wave packet extending on a range of atomic planes on a superlattice composed of two materials $A$ and $B$. $l_c$ corresponds to its spatial phonon coherence length. The reference plane is depicted with a dashed line.

Figure 2

(a) The LDOS for $d_{\mathrm{SL}}$ from 1 to 4 nm; (b) spatial phonon coherence length normalized by the system length for $d_{\mathrm{SL}}=1$, 2, and 4 nm; (c) the LDOS for $d_{\mathrm{SL}}$ for 8 and 16 nm; (d) spatial phonon coherence length normalized by the system length for $d_{\mathrm{SL}}=8$ and 16 nm.

Figure 3

${log}_{10}[l_c\left(\omega \right)/d_{\mathrm{SL}}]$ is calculated to determine the phonon transport regime. When ${log}_{10}[l_c\left(\omega \right)/d_{\mathrm{SL}}]>$0 (respectively $<$0), the phonon transport is coherent (respectively incoherent), the wave-packet spatial extension is larger (respectively smaller) than $d_{\mathrm{SL}}$. The horizontal dotted line indicates the threshold between the coherent and the incoherent regime.

Figure 4

Effect of the system size on argon superlattices in the coherent regime with $d_{\mathrm{SL}}=1$ nm.

Figure 5

Effect of the system size on argon superlattices in the incoherent regime with $d_{\mathrm{SL}}=16$ nm.

Figure 6

(a) ${log}_{10}[l_c\left(\omega \right)/d_{\mathrm{SL}}]$ for the longitudinal component of the velocity field for Si/hSi superlattices with period thickness ranging 1–8 nm. (b) ${log}_{10}[l_c\left(\omega \right)/d_{\mathrm{SL}}]$ for the transverse components of the velocity field for Si/hSi superlattices with period thickness ranging 1–8 nm. Coherent transport is observed below $d_{\mathrm{SL}}=4$ nm.

Figure 7

Effect of the system size on Si/hSi superlattices with $d_{\mathrm{SL}}=1$ nm.

Figure 8

Effect of the system size on Si/hSi superlattices with $d_{\mathrm{SL}}=8$ nm.

Figure 9

(a) Spatial phonon coherence length $l_c\left(\omega \right)$ normalized by the system length $L$ for silicon superlattices with $d_{\mathrm{SL}}=1$ nm, for $T=300$, 600, 800, and 1000 K. (b) Phonon DOS of the silicon superlattice with $d_{\mathrm{SL}}=1$ nm at 300 K.

Figure 10

(a) Spatial phonon coherence length $l_c\left(\omega \right)$ normalized by the system length $L$ for Si/hSi superlattices with $d_{\mathrm{SL}}=2$ nm for perfect interfaces $(N=0)$, then 10$\%$ $(N=1$; $m=10\%)$ and 50$\%$ $(N=1$; $m=50\%)$ interfacial mixing on 1 atomic layer away from each interface. (b) Spatial phonon coherence length $l_c\left(\omega \right)$ normalized by the system length $L$ for Si/hSi superlattices with $d_{\mathrm{SL}}=2$ nm for perfect interfaces $(N=0)$, then 10$\%$ $(m=10\%)$ interfacial mixing on $N$ atomic layers away from each interface, where $N\in \{1,4\}$.

Figure 11

The cross-plane thermal conductivity $\kappa$ for perfect Si/hSi superlattices at 500 K from Green-Kubo calculations.

Figure 12

Schematic representation of all phonon characteristic lengths that are involved in the phonon transport in superlattices. $l_c$ is the phonon coherence length, $\lambda$ is the wavelength associated to the wave packet, $d_{\mathrm{SL}}$ is the period thickness of the superlattice, and $L$ is its length. For simplification, the two bulk materials 1 and 2 are assumed to have a similar mean free path, noted ${\Lambda }_{\mathrm{bulk}1,2}$. Finally, ${\Lambda }_{\mathrm{SL}}$ represents the mean free path of the wave packet in the superlattice. For each of these six cases, two trends for the thermal conductivity $\kappa$ are depicted: one as a function of the period thickness $d_{\mathrm{SL}}$ with a constant length $L$ and one with respect to $L$ with a constant $d_{\mathrm{SL}}$.