What you describe as "phonon diffusion" is indeed just another type of scattering, usually called anharmonic or phonon-phonon scattering (sometimes also called 3-phonon scattering as it typically involves 3 phonons, either by one phonon splitting into two, or two combining into one). The anharmonic process is an intrinsic property of a material and can only be controlled by changing the material structure/composition or temperature. As temperature increases, the number of available phonons increases too, thereby making the anharmonic phonon-phonon interaction stronger. Consequently, phonon-phonon interactions are more prominent at higher temperatures.

Other types of scattering, either through nanostructuring or alloying, can be quite strong and can be engineered to control thermal conductivity. Since such scattering is elastic (one phonon in, one out, conserving energy), it typically has a weak temperature dependence and usually the resulting thermal conductivity follows the heat capacity in terms of its temperature dependence, which means that the thermal conductivity increases as temperature increases, and eventually plateaus until phonon-phonon scattering takes over.

Alloy (sometimes called mass-difference) scattering can be enhanced by introducing atoms with a different mass. The scattering rate will depend on the magnitude of the difference in mass as well as the alloy composition. Alloy scattering can reduce thermal conductivity one to two orders of magnitude in some cases (SiGe alloys are a prime example where room temperature thermal conductivity is reduced from 146 W/m/K in Si to about 7 W/m/K in SiGe alloys, depending on the amount of Ge added). The difference can be seen in the temperature dependence: while the thermal conductivity of Si peaks around 30K and then goes as ~1/T above the peak, the thermal conductivity of SiGe alloys does not show a prominent peak, having mostly a positive temperature slope up to room temperature.

Various forms of nanostructuring can be also very effective in reducing thermal conductivity. For example, Si/SiGe superlattices (with alternating layers of few nm thick Si and SiGe alloy) have been found to have thermal conductivity below bulk SiGe alloys, as low as 2 W/m/K which is almost two orders of magnitude below Si and very close to the amorphous limit of SiO2 (which has thermal conductivity of ~1 W/m/K). The reduction was explained through phonon scattering at the roughness between consecutive layers in the superlattice (see attached paper).

In summary I would say that alloying (mass difference scattering) and nanostructuring (boundary roughness scattering) are most effective in reducing thermal conductivity, unless you are looking at very high temperatures above the Debye temperature of a material where phonon-phonon scattering could be prominent. In experiments, it is challenging to distinguish between different sources of scattering, but the dependence of thermal conductivity on temperature can help--a positive slope (increasing with temperature) usually implies some type of disorder such as boundary roughness scattering. A constant thermal conductivity is often linked to the presence of impurities, isotopes, alloying, and dislocations. Finally, a negative temperature slope (thermal conductivity decreasing with temperature) implies that phonon-phonon interactions are dominant. All of these mechanisms are usually present simultaneously but in varying amounts; temperature acts as a sort of "selector" between them. The location (and sharpness) of the peak in the thermal conductivity vs temperature curve (relative to the Debye temperature) is usually taken as an indicator of the relative contributions of these mechanisms--if the peak is at low T, then phonon-phonon scattering is prominent, while if the peak is at high T or absent, then disorder in the form of roughness or structural variations (impurities, isotopes, alloying, and dislocations) is the dominant force.

Article Thermal conductivity of Si 1−x Ge x /Si 1− y Ge y superlatti...

What you describe as "phonon diffusion" is indeed just another type of scattering, usually called anharmonic or phonon-phonon scattering (sometimes also called 3-phonon scattering as it typically involves 3 phonons, either by one phonon splitting into two, or two combining into one). The anharmonic process is an intrinsic property of a material and can only be controlled by changing the material structure/composition or temperature. As temperature increases, the number of available phonons increases too, thereby making the anharmonic phonon-phonon interaction stronger. Consequently, phonon-phonon interactions are more prominent at higher temperatures.

Other types of scattering, either through nanostructuring or alloying, can be quite strong and can be engineered to control thermal conductivity. Since such scattering is elastic (one phonon in, one out, conserving energy), it typically has a weak temperature dependence and usually the resulting thermal conductivity follows the heat capacity in terms of its temperature dependence, which means that the thermal conductivity increases as temperature increases, and eventually plateaus until phonon-phonon scattering takes over.

Alloy (sometimes called mass-difference) scattering can be enhanced by introducing atoms with a different mass. The scattering rate will depend on the magnitude of the difference in mass as well as the alloy composition. Alloy scattering can reduce thermal conductivity one to two orders of magnitude in some cases (SiGe alloys are a prime example where room temperature thermal conductivity is reduced from 146 W/m/K in Si to about 7 W/m/K in SiGe alloys, depending on the amount of Ge added). The difference can be seen in the temperature dependence: while the thermal conductivity of Si peaks around 30K and then goes as ~1/T above the peak, the thermal conductivity of SiGe alloys does not show a prominent peak, having mostly a positive temperature slope up to room temperature.

Various forms of nanostructuring can be also very effective in reducing thermal conductivity. For example, Si/SiGe superlattices (with alternating layers of few nm thick Si and SiGe alloy) have been found to have thermal conductivity below bulk SiGe alloys, as low as 2 W/m/K which is almost two orders of magnitude below Si and very close to the amorphous limit of SiO2 (which has thermal conductivity of ~1 W/m/K). The reduction was explained through phonon scattering at the roughness between consecutive layers in the superlattice (see attached paper).

In summary I would say that alloying (mass difference scattering) and nanostructuring (boundary roughness scattering) are most effective in reducing thermal conductivity, unless you are looking at very high temperatures above the Debye temperature of a material where phonon-phonon scattering could be prominent. In experiments, it is challenging to distinguish between different sources of scattering, but the dependence of thermal conductivity on temperature can help--a positive slope (increasing with temperature) usually implies some type of disorder such as boundary roughness scattering. A constant thermal conductivity is often linked to the presence of impurities, isotopes, alloying, and dislocations. Finally, a negative temperature slope (thermal conductivity decreasing with temperature) implies that phonon-phonon interactions are dominant. All of these mechanisms are usually present simultaneously but in varying amounts; temperature acts as a sort of "selector" between them. The location (and sharpness) of the peak in the thermal conductivity vs temperature curve (relative to the Debye temperature) is usually taken as an indicator of the relative contributions of these mechanisms--if the peak is at low T, then phonon-phonon scattering is prominent, while if the peak is at high T or absent, then disorder in the form of roughness or structural variations (impurities, isotopes, alloying, and dislocations) is the dominant force.

Article Thermal conductivity of Si 1−x Ge x /Si 1− y Ge y superlatti...

I think the most effective way of decrease the thermal conductivity is to combine nanostructuring and alloying because in this case all phonons are strongly scattered. Nanostructuring (embedded nanoparticles, boundaries (nanowires, nanograins), superlattices) scatters mainly low frequency phonons, while alloying scatters high frequency phonons. Typical examples are embedded nanoparticles in an alloy matrix (see for example ErAs nanoparticles in InxGa1-xAs Phys. Rev. Lett.), or nanocrystalline alloys (see BiSbTe, Science).

Thank you both for the detailed answers. As always in research every answer stimulates new questions.

The anharmonic process of phonon-phonon scattering, do you think is this mainly caused by anharmonicity in the ion-ion-interatomic potential curve, or can the (anharmonic) interaction between outer electrons and ions or even the interaction of outer electrons of both partner atom can be the reason too?

Is there a kind of digital database for thermal conductivity vs temperature curve available?

The second part of my initial question was concerning experimental confirmation. In our days coherent phonon sources are available. I wonder whether such experiments have been done, like monitoring the diffusion or dispersion of a phonon wave-packet with time and space?

may i know how to calculate thermal conductivity of the material while phonon scattering?

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