- Source: Phonon scattering
Phonons can scatter through several mechanisms as they travel through the material. These scattering mechanisms are: Umklapp phonon-phonon scattering, phonon-impurity scattering, phonon-electron scattering, and phonon-boundary scattering. Each scattering mechanism can be characterised by a relaxation rate 1/
τ
{\displaystyle \tau }
which is the inverse of the corresponding relaxation time.
All scattering processes can be taken into account using Matthiessen's rule. Then the combined relaxation time
τ
C
{\displaystyle \tau _{C}}
can be written as:
1
τ
C
=
1
τ
U
+
1
τ
M
+
1
τ
B
+
1
τ
ph-e
{\displaystyle {\frac {1}{\tau _{C}}}={\frac {1}{\tau _{U}}}+{\frac {1}{\tau _{M}}}+{\frac {1}{\tau _{B}}}+{\frac {1}{\tau _{\text{ph-e}}}}}
The parameters
τ
U
{\displaystyle \tau _{U}}
,
τ
M
{\displaystyle \tau _{M}}
,
τ
B
{\displaystyle \tau _{B}}
,
τ
ph-e
{\displaystyle \tau _{\text{ph-e}}}
are due to Umklapp scattering, mass-difference impurity scattering, boundary scattering and phonon-electron scattering, respectively.
Phonon-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
ω
{\displaystyle \omega }
and umklapp processes vary with
ω
2
{\displaystyle \omega ^{2}}
, Umklapp scattering dominates at high frequency.
τ
U
{\displaystyle \tau _{U}}
is given by:
1
τ
U
=
2
γ
2
k
B
T
μ
V
0
ω
2
ω
D
{\displaystyle {\frac {1}{\tau _{U}}}=2\gamma ^{2}{\frac {k_{B}T}{\mu V_{0}}}{\frac {\omega ^{2}}{\omega _{D}}}}
where
γ
{\displaystyle \gamma }
is the Gruneisen anharmonicity parameter, μ is the shear modulus, V0 is the volume per atom and
ω
D
{\displaystyle \omega _{D}}
is the Debye frequency.
Three-phonon and four-phonon process
Thermal transport in non-metal solids was usually considered to be governed by the three-phonon scattering process, 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 and for certain materials at room temperature. The predicted significance of four-phonon scattering in boron arsenide was confirmed by experiments.
Mass-difference impurity scattering
Mass-difference impurity scattering is given by:
1
τ
M
=
V
0
Γ
ω
4
4
π
v
g
3
{\displaystyle {\frac {1}{\tau _{M}}}={\frac {V_{0}\Gamma \omega ^{4}}{4\pi v_{g}^{3}}}}
where
Γ
{\displaystyle \Gamma }
is a measure of the impurity scattering strength. Note that
v
g
{\displaystyle {v_{g}}}
is dependent of the dispersion curves.
Boundary scattering
Boundary scattering is particularly important for low-dimensional nanostructures and its relaxation rate is given by:
1
τ
B
=
v
g
L
0
(
1
−
p
)
{\displaystyle {\frac {1}{\tau _{B}}}={\frac {v_{g}}{L_{0}}}(1-p)}
where
L
0
{\displaystyle L_{0}}
is the characteristic length of the system and
p
{\displaystyle p}
represents the fraction of specularly scattered phonons. The
p
{\displaystyle p}
parameter is not easily calculated for an arbitrary surface. For a surface characterized by a root-mean-square roughness
η
{\displaystyle \eta }
, a wavelength-dependent value for
p
{\displaystyle p}
can be calculated using
p
(
λ
)
=
exp
(
−
16
π
2
λ
2
η
2
cos
2
θ
)
{\displaystyle p(\lambda )=\exp {\Bigg (}-16{\frac {\pi ^{2}}{\lambda ^{2}}}\eta ^{2}\cos ^{2}\theta {\Bigg )}}
where
θ
{\displaystyle \theta }
is the angle of incidence. An extra factor of
π
{\displaystyle \pi }
is sometimes erroneously included in the exponent of the above equation. At normal incidence,
θ
=
0
{\displaystyle \theta =0}
, perfectly specular scattering (i.e.
p
(
λ
)
=
1
{\displaystyle p(\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
{\displaystyle p=0}
the relaxation rate becomes
1
τ
B
=
v
g
L
0
{\displaystyle {\frac {1}{\tau _{B}}}={\frac {v_{g}}{L_{0}}}}
This equation is also known as Casimir limit.
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
Phonon-electron scattering can also contribute when the material is heavily doped. The corresponding relaxation time is given as:
1
τ
ph-e
=
n
e
ϵ
2
ω
ρ
v
g
2
k
B
T
π
m
∗
v
g
2
2
k
B
T
exp
(
−
m
∗
v
g
2
2
k
B
T
)
{\displaystyle {\frac {1}{\tau _{\text{ph-e}}}}={\frac {n_{e}\epsilon ^{2}\omega }{\rho v_{g}^{2}k_{B}T}}{\sqrt {\frac {\pi m^{*}v_{g}^{2}}{2k_{B}T}}}\exp \left(-{\frac {m^{*}v_{g}^{2}}{2k_{B}T}}\right)}
The parameter
n
e
{\displaystyle 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 .
See also
Lattice scattering
Umklapp scattering
Electron-longitudinal acoustic phonon interaction
References
Kata Kunci Pencarian:
- Magnesium oksida
- Phonon scattering
- Umklapp scattering
- Electron mobility
- Phonon
- Surface phonon
- Brillouin spectroscopy
- Rayleigh scattering
- Thermal conductivity and resistivity
- Thermoelectric materials
- Raman scattering
