The thermal conductivity is decreased by scattering of the phonons, and phonons in some modes will tend to scatter more.

If we are thinking of a pristine material, there can be several reasons for the scattering, either these modes are more strongly anharmonic (leading to scattering against other phonons) or there can be strong features in the electronic density of states in some region of the Brillouin zone leading to phonon-electron scattering (the Kohn anomaly in the phonon spectrum of graphene near the K-point is a well-known example there). Phonon anharmonicity is also typically connected with some region in the Brillouin zone. If you have a polar insulator you should be able to get scattering of optical phonons against photons, if you shine light of an appropriate wavelength at the material, but I guess this would not affect the thermal conductivity very strongly since such a process would transfer little momentum. In a magnetic material you can have scattering against magnons. And so on.

But in a real material the scattering has very large components of scattering from different defects and surfaces (like grain boundaries and other microstructure) rather than by other excitations of the pristine solid.

The thermal conductivity is decreased by scattering of the phonons, and phonons in some modes will tend to scatter more.

If we are thinking of a pristine material, there can be several reasons for the scattering, either these modes are more strongly anharmonic (leading to scattering against other phonons) or there can be strong features in the electronic density of states in some region of the Brillouin zone leading to phonon-electron scattering (the Kohn anomaly in the phonon spectrum of graphene near the K-point is a well-known example there). Phonon anharmonicity is also typically connected with some region in the Brillouin zone. If you have a polar insulator you should be able to get scattering of optical phonons against photons, if you shine light of an appropriate wavelength at the material, but I guess this would not affect the thermal conductivity very strongly since such a process would transfer little momentum. In a magnetic material you can have scattering against magnons. And so on.

But in a real material the scattering has very large components of scattering from different defects and surfaces (like grain boundaries and other microstructure) rather than by other excitations of the pristine solid.

thank you so much for your help,

what does the Fourier transformation of heat current autocorrelation function gives us exactly?

To complement Torbjörn's answer, so-called Umklapp-scattering strongly influences (suppresses) heat conduction. Rather large phonon momenta are required, and for their contribution to be substantial, elevated temperature is a requirement.

This makes sapphire an excellent heat conductor at low temperature and a bad thermal conductor at elevated temperature.

At this point I have to confess that I know very little about the uses of the heat current autocorrelation function, but speaking of autocorrelation functions (of time) generally, the fourier transform will give you a power spectrum. In your case, peaks in the power spectrum will tell you at which frequencies you have large contributions to the thermal conductivity, for example at peaks in the phonon density of states. That's what little I know about this.

wow you are amazing Torbjörn,

you know, i was read about it totally vice versa. It said the Fourier transform of heat current autocorrelation function gives the power spectrum of heat current, which can tell the distribution of the vibrational frequencies (phonon frequencies) that contribute to the reduction of thermal conductivity !!!

I am completely confused. please helppppppppp me

Fourier transform of heat current autocorrelation function gives you the so called VDOS(Vibrational Density of States), and I think VDOS mainly means that there are how many phonon modes for each frequency. And the total phonon modes for the system is 3N, where N mean the total number of atoms. There are specific numbers of phonon modes for each frequency. Generally, low-frequency phonon modes contribute more to heat transfer than high-frequency modes. To check whether the thermal conductivity of A material is lower than B, we can compare the VDOS between them.

Thank u so much dear Xiaoliang

So you believe that non of phonons could be caused in reduction of thermal conductivity?

Your question looks a little strange. I think VDOS just tell us the frequency distribution for the phonon modes, actually more or less, each phonon modes contributes to thermal conductivity. You ask that "Why do some normal modes (phonon frequencies) lead to a reduction in thermal conductivity? ", I think the question is not so good. If you want to say "reduction", there should be a comparison, e.g., material A compared to B, etc. If you want to use VDOS to explain the difference for the thermal conductivity of A and B, then you can compare their value of VDOS.

Thank you dear Xiaoliang. your opinions are a good news for me.

Maybe a few more words for clarification (well, hopefully).

(1) The pure DOS of vibrational states gives no clue about heat conductivity, since it does not reveal information on the nature of these states.

(2) Phonons are the (quantized) normal modes of vibration of a crystal. Strictly speaking, a phonon does not conduct heat, since it is everywhere in the crystal at the same time (since it is a normal mode, fully delocalized).

(3) As with light propagation, we can construct wave packets of vibrations, these then represent localized vibrational energy.

(4) wave packets around some average wave vector shall then transport vibrational energy with the approx. group velocity of the phonon dispersion relation at that average wave vector. Nonlinear and/or anisotropic dispersion relation results in "smearing" of the wave packet.

(5) For wave packets at small group velocity, the heat conduction may be poor. However, the phonon dispersion relation of acoustical phonons is linear in a good portion of the Brillouin zone (around the zone center), the group velocity equalling the phase velocity and the velocity of sound.

(6) Another limitation to heat conductivity is the non-harmonicity of the interatomic interaction. It leads to finite lifetimes of phonons which can be described as phonon-phonon scattering. Of course, this is the stronger, the higher the temperature (or the less good the harmonic approximation is even at small amplitudes in the first place).

(7) The last point gets yet another amplification, when phonon-phonon scattering involves states in higher Brillouin zones. Not only "sideward scattering" may then occur, but also backscattering ("Umklapp processes"). I had mentioned this point before.

Edit: (8) Phonon scattering may additionally occur due to interaction with other degrees of freedom. See Torbjörns answer(s).

Thank you Kai

So do you agree with Xiaoliang who said "actually more or less, each phonon modes contributes to thermal conductivity"? and do you believe that my question (Why do some normal modes lead to a reduction in thermal conductivity?) is wrong?

Sayyed,

you should try to work it out for yourself from here. My conclusions are no guarantee! I am improvising my answer, which roughly would be as follows:

I do not think your question is bad, because evidently

(a) as you raise the temperature, more phonons are excited, and a growing portion of the Brillouin zone is "significantly populated" with acoustical phonons.

(b) you can turn a good heat conductor (sapphire at low temperature) into a bad one (sapphire at high temperature).

So here you could have a point with your formulation, that some phonons appear detrimental to heat conductivity.

But how does it occur? Because of what I wrote before I tend to disagree with Xiaoliang's notion that all phonons contribute similarly to the heat conductance (more or less). They contribute according to the combination of group velocity and lifetime, which you could view as an energy propagation velocity and some mean free path. [This leads to a very typical formulation of heat conductivity, btw., when discussing heat conductivity via phonons and electrons in metals at the undergrad level.]

The phonon excitation, however, depends only on temperature via Boltzmann factors (Bose statistics).

At sufficiently low temperature, you will only excite phonons in the linear portions of the dispersion relations. Here, group velocity = sound velocity ~constant (well, maybe anisotropic and depend on polarization - think of some suitable average, corresponding to polycrystalline material for simplicity). In this temperature regime there is little variation of heat conductance with temperature, since the relevant phonon dispersions always look the same.

As you heat more and more, you cannot help but excite phonons on the nonlinear portions of the phonon brances, eventually also reaching the extrema of the dispersion curves with zero group velocity. This portion of the heat energy is thus much less efficiently transported away and your heat conductance is reduced. Add to this all the other mechanisms discussed before: most of them become only important above some temperature (resp. wave number). At low excitation (temperature), the harmonic crystal is usually good enough an approximation and the dispersion relations linear enough to yield a good thermal conductivity in a crystal.

Kai

Thank you for your time. I'm working on it and my incomplete results indicate that it seems my first question is not a good one. However i'll let u know when it's over.