phonon scattering

For phonon-phonon scattering, effects by normal processes (processes which conserve the phonon wave vector - N processes) are ignored in favor of Umklapp processes (U processes). Since normal processes vary linearly with $\backslash omega$ and umklapp processes vary with $\backslash omega^2$, Umklapp scattering dominates at high frequency.{{Cite journal

| last = Mingo | first = N

| year = 2003

| title = Calculation of nanowire thermal conductivity using complete phonon dispersion relations

| journal = Physical Review B

| volume = 68

| issue = 11

| pages = 113308

| arxiv = cond-mat/0308587

|bibcode = 2003PhRvB..68k3308M |doi = 10.1103/PhysRevB.68.113308 | s2cid = 118984828

| url = https://zenodo.org/record/1233747}} $\backslash tau\_U$ is given by:

:$\backslash frac\{1\}\{\backslash tau\_U\}=2\backslash gamma^2\backslash frac\{k\_B\; T\}\{\backslash mu\; V\_0\}\backslash frac\{\backslash omega^2\}\{\backslash omega\_D\}$

where $\backslash gamma$ is the Gruneisen anharmonicity parameter, {{mvar|μ}} is the shear modulus, {{mvar|V₀}} is the volume per atom and $\backslash omega\_\{D\}$ is the Debye frequency.{{Cite journal

|last = Zou

|first = Jie

|author2 = Balandin, Alexander

|year = 2001

|title = Phonon heat conduction in a semiconductor nanowire

|journal = Journal of Applied Physics

|volume = 89

|issue = 5

|pages = 2932

|doi = 10.1063/1.1345515

|url = http://www.ndl.ee.ucr.edu/jap-zou-1.pdf

|bibcode = 2001JAP....89.2932Z

|url-status = dead

|archiveurl = https://web.archive.org/web/20100618011126/http://ndl.ee.ucr.edu/jap-zou-1.pdf

|archivedate = 2010-06-18

}}

Thermal transport in non-metal solids was usually considered to be governed by the three-phonon scattering process,{{cite book|title=Electrons and Phonons: The Theory of transport phenomena in solids |series=Oxford Classic Texts in the Physical Sciences |publisher=Oxford University Press|year=1960|last1=Ziman|first1=J.M.}} and the role of four-phonon and higher-order scattering processes was believed to be negligible. Recent studies have shown that the four-phonon scattering can be important for nearly all materials at high temperature {{cite journal|doi = 10.1103/PhysRevB.93.045202|title = Quantum mechanical prediction of four-phonon scattering rates and reduced thermal conductivity of solids|journal = Physical Review B|volume = 93|issue = 4|pages = 045202|year = 2016|last1 = Feng|first1 = Tianli|last2 = Ruan|first2 = Xiulin|bibcode = |arxiv = 1510.00706| s2cid=16015465 }} and for certain materials at room temperature.{{cite journal|doi = 10.1103/PhysRevB.96.161201|title = Four-phonon scattering significantly reduces intrinsic thermal conductivity of solids|journal = Physical Review B|volume = 96|issue = 16|pages = 161201|year = 2017|last1 = Feng|first1 = Tianli|last2 = Lindsay|first2 = Lucas|last3 = Ruan|first3 = Xiulin|bibcode =2017PhRvB..96p1201F |doi-access = free}} The predicted significance of four-phonon scattering in boron arsenide was confirmed by experiments.

Mass-difference impurity scattering is given by:

:$\backslash frac\{1\}\{\backslash tau\_M\}=\backslash frac\{V\_0\; \backslash Gamma\; \backslash omega^4\}\{4\backslash pi\; v\_g^3\}$

where $\backslash Gamma$ is a measure of the impurity scattering strength. Note that $\{v\_g\}$ is dependent of the dispersion curves.

Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation rate is given by:

:$\backslash frac\{1\}\{\backslash tau\_B\}=\backslash frac\{v\_g\}\{L\_0\}(1-p)$

where $L\_0$ is the characteristic length of the system and $p$ represents the fraction of specularly scattered phonons. The $p$ parameter is not easily calculated for an arbitrary surface. For a surface characterized by a root-mean-square roughness $\backslash eta$, a wavelength-dependent value for $p$ can be calculated using

:$p(\backslash lambda)\; =\; \backslash exp\backslash Bigg(-16\backslash frac\{\backslash pi^2\}\{\backslash lambda^2\}\backslash eta^2\backslash cos^2\backslash theta\; \backslash Bigg)$

where $\backslash theta$ is the angle of incidence.{{cite journal |last1=Jiang |first1=Puqing |last2=Lindsay |first2=Lucas |date=2018 |title=Interfacial phonon scattering and transmission loss in > 1 um thick silicon-on-insulator thin films |journal=Phys. Rev. B |volume=97 |issue=19 |pages=195308 |doi=10.1103/PhysRevB.97.195308|arxiv=1712.05756 |bibcode=2018PhRvB..97s5308J |s2cid=118956593 }} An extra factor of $\backslash pi$ is sometimes erroneously included in the exponent of the above equation.{{cite journal |last1=Maznev |first1=A. |date=2015 |title=Boundary scattering of phonons: Specularity of a randomly rough surface in the small-perturbation limit |journal=Phys. Rev. B |volume=91 |issue=13 |pages=134306 |doi=10.1103/PhysRevB.91.134306|arxiv=1411.1721 |bibcode=2015PhRvB..91m4306M |s2cid=54583870 }} At normal incidence, $\backslash theta=0$, perfectly specular scattering (i.e. $p(\backslash lambda)=1$) would require an arbitrarily large wavelength, or conversely an arbitrarily small roughness. Purely specular scattering does not introduce a boundary-associated increase in the thermal resistance. In the diffusive limit, however, at $p=0$ the relaxation rate becomes

:$\backslash frac\{1\}\{\backslash tau\_B\}=\backslash frac\{v\_g\}\{L\_0\}$

This equation is also known as Casimir limit.{{Cite journal

| last = Casimir | first = H.B.G

| year = 1938

| title = Note on the Conduction of Heat in Crystals

| journal = Physica

| volume = 5

| bibcode = 1938Phy.....5..495C

| issue = 6

| doi = 10.1016/S0031-8914(38)80162-2

| pages = 495–500

}}

These phenomenological equations can in many cases accurately model the thermal conductivity of isotropic nano-structures with characteristic sizes on the order of the phonon mean free path. More detailed calculations are in general required to fully capture the phonon-boundary interaction across all relevant vibrational modes in an arbitrary structure.

Phonon-electron scattering can also contribute when the material is heavily doped. The corresponding relaxation time is given as:

:$\backslash frac\{1\}\{\backslash tau\_\backslash text\{ph-e\}\}=\backslash frac\{n\_e\; \backslash epsilon^2\; \backslash omega\}\{\backslash rho\; v\_g^2\; k\_B\; T\}\backslash sqrt\{\backslash frac\{\backslash pi\; m^*\; v\_g^2\}\{2k\_B\; T\}\}\; \backslash exp\; \backslash left(-\backslash frac\{m^*v\_g^2\}\{2k\_B\; T\}\backslash right)$

The parameter $n\_\{e\}$ is conduction electrons concentration, ε is deformation potential, ρ is mass density and m* is effective electron mass. It is usually assumed that contribution to thermal conductivity by phonon-electron scattering is negligible {{Citation needed|date=May 2022|reason=Questionable, as for instance, in https://www.nature.com/articles/s41467-020-19938-9, it is stated that "(...)the photo-excited carriers can significantly reduce thermal conductivity by more than 30% on a nanosecond timescale. This is a direct experimental verification of the large impact of electron–phonon interactions on the lattice thermal conductivity at room temperature"}}.

• Lattice scattering
• Umklapp scattering
• Electron-longitudinal acoustic phonon interaction

{{Reflist}}

Category:Condensed matter physics

Category:Scattering