It is well known that normal processes (i.e., scattering processes that conserve the total crystal momentum) cannot by themselves produce a finite thermal conductivity. Therefore, one cannot simply add the inverse relaxation time of normal processes (N processes) to those of momentum-destroying processes. Examples of the latter include Umklapp processes, impurity scattering, and boundary scattering. Throughout this article, all such momentum-nonconserving processes will be collectively referred to as R processes.
For a phonon mode $\lambda \equiv (\mathbf{q}, j)$, we have
The equilibrium Bose–Einstein distribution is
\[n_{\lambda}^{(0)}(T) = \frac{1}{\exp(\hbar\omega_\lambda/k_B T) - 1}.\]In the presence of a temperature gradient $\nabla T$, the occupation becomes
\[n_\lambda = n_{\lambda}^{(0)} + n_{1,\lambda}, \qquad |n_{1,\lambda}|\ll n_{\lambda}^{(0)}.\]Scattering is classified into two categories — the core of the Callaway model:
Key point: N processes do not generate thermal resistance.
They merely establish a “drifting” equilibrium state of the phonon gas, while the actual dissipation of heat flux is governed by R processes.
In steady state, driven only by a temperature gradient, the linearized BTE can be written as
\[\mathbf{v}_\lambda \cdot \nabla T \,\frac{\partial n_{\lambda}^{(0)}}{\partial T} = \left(\frac{\partial n_\lambda}{\partial t}\right)_{\text{coll}}. \tag{1}\]Callaway splits the collision term into two parts:
\[\left(\frac{\partial n_\lambda}{\partial t}\right)_{\text{coll}} = -\frac{n_\lambda - n_{\lambda}^{(d)}}{\tau_{N,\lambda}} -\frac{n_\lambda - n_{\lambda}^{(0)}}{\tau_{R,\lambda}}. \tag{2}\]The “displaced Planck distribution” is
\[n_{\lambda}^{(d)} = \left[ \exp\left( \frac{\hbar\omega_\lambda - \boldsymbol{\Lambda}\cdot\mathbf{k}_\lambda}{k_B T} \right) -1 \right]^{-1}, \tag{3}\]where $\boldsymbol{\Lambda}$ is a small vector describing the collective drift of the phonon system, aligned with $\nabla T$.
Introduce the dimensionless variable
\[x_\lambda = \frac{\hbar\omega_\lambda}{k_B T}.\]Expanding Eq. (3) to first order in $\boldsymbol{\Lambda}\cdot\mathbf{k}_\lambda$, we obtain
\[n_{\lambda}^{(d)} \simeq n_{\lambda}^{(0)} + \frac{\partial n_{\lambda}^{(0)}}{\partial(\hbar\omega_\lambda)}\, (\boldsymbol{\Lambda}\cdot\mathbf{k}_\lambda). \tag{4}\]Convert the derivative with respect to energy into a temperature derivative:
\[n_{\lambda}^{(d)} \simeq n_{\lambda}^{(0)} - \frac{\partial n_{\lambda}^{(0)}}{\partial T}\, \frac{\boldsymbol{\Lambda}\cdot\mathbf{k}_\lambda}{k_B T}. \tag{5}\]Insert
\[n_\lambda = n_{\lambda}^{(0)} + n_{1,\lambda}\]and Eq. (5) into Eqs. (1) and (2), keeping only first-order terms.
Left-hand side:
Right-hand side:
\[-\frac{n_\lambda - n_{\lambda}^{(d)}}{\tau_{N,\lambda}} -\frac{n_\lambda - n_{\lambda}^{(0)}}{\tau_{R,\lambda}} = -\left(\frac{1}{\tau_{N,\lambda}}+\frac{1}{\tau_{R,\lambda}}\right)n_{1,\lambda} + \frac{n_{\lambda}^{(d)}-n_{\lambda}^{(0)}}{\tau_{N,\lambda}}.\]Thus,
\[\mathbf{v}_\lambda \cdot \nabla T \,\frac{\partial n_{\lambda}^{(0)}}{\partial T} = -\left(\frac{1}{\tau_{N,\lambda}}+\frac{1}{\tau_{R,\lambda}}\right)n_{1,\lambda} + \frac{n_{\lambda}^{(d)}-n_{\lambda}^{(0)}}{\tau_{N,\lambda}}. \tag{6}\]Define the combined relaxation time (“N+R combined”)
\[\frac{1}{\tau_{c,\lambda}} = \frac{1}{\tau_{N,\lambda}}+\frac{1}{\tau_{R,\lambda}}. \tag{7}\]Solve for $n_{1,\lambda}$:
\[n_{1,\lambda} = \tau_{c,\lambda}\,\mathbf{v}_\lambda\cdot\nabla T\, \frac{\partial n_{\lambda}^{(0)}}{\partial T} + \frac{\tau_{c,\lambda}}{\tau_{N,\lambda}}\left(n_{\lambda}^{(d)}-n_{\lambda}^{(0)}\right). \tag{8}\]Insert Eq. (5):
\[n_{1,\lambda} = \tau_{c,\lambda}\,\frac{\partial n_{\lambda}^{(0)}}{\partial T} \left[ \mathbf{v}_\lambda\cdot\nabla T - \frac{\boldsymbol{\Lambda}\cdot\mathbf{k}_\lambda}{k_B T} \frac{1}{\tau_{N,\lambda}} \right]. \tag{9}\]Interpretation:
- First term: RTA-like, except the relaxation time becomes $\tau_c$.
- Second term: the correction involving “$\boldsymbol{\Lambda}$”, originating entirely from momentum conservation of N processes.
Physical condition: normal processes do not change the total crystal momentum, hence
\[\sum_\lambda \hbar \mathbf{k}_\lambda \left(\frac{\partial n_\lambda}{\partial t}\right)_{N} = -\sum_\lambda \hbar \mathbf{k}_\lambda \frac{n_\lambda - n_{\lambda}^{(d)}}{\tau_{N,\lambda}} =0. \tag{10}\]When the temperature gradient is one-dimensional, $\boldsymbol{\Lambda}$ is parallel to $\nabla T$, so we write
\[\boldsymbol{\Lambda} = -\beta\,\nabla T, \tag{11}\]where $\beta$ is an “effective time–like parameter”.
Substituting Eqs. (9) and (11) into Eq. (10), and using
\[C_\lambda = \hbar\omega_\lambda\,\frac{\partial n_{\lambda}^{(0)}}{\partial T}\](the modal heat capacity), one can obtain $\beta$ in the discrete-sum form
\[\beta = \frac{ \displaystyle\sum_\lambda C_\lambda v_{\lambda x}^2\,\tau_{c,\lambda}^2/\tau_{N,\lambda} }{ \displaystyle\sum_\lambda C_\lambda v_{\lambda x}^2 \left(1-\tau_{c,\lambda}/\tau_{N,\lambda}\right) }. \tag{12}\]This is the “locking mechanism” of the Callaway model, linking the drift strength $\beta$ to the modal relaxation times $\tau_N$ and $\tau_R$.
The heat flux along the $x$ direction is
\[J_x = \frac{1}{V}\sum_\lambda \hbar\omega_\lambda v_{\lambda x}\,n_{1,\lambda}.\]Together with Fourier’s law,
\[J_x = -\kappa\,\frac{\partial T}{\partial x},\]Eqs. (9) and (11) lead to the decomposition
\[\kappa = \kappa_1 + \kappa_2.\]This term resembles the conventional RTA expression, except that $\tau$ is replaced with $\tau_c$:
\[\kappa_1 = \frac{1}{V} \sum_\lambda C_\lambda v_{\lambda x}^2 \tau_{c,\lambda}. \tag{13}\]In the absence of N processes ($\tau_{N,\lambda}\to\infty$),
$\tau_{c,\lambda} \to \tau_{R,\lambda}$, and Eq. (13) reduces to the familiar RTA formula.
The contribution induced by normal processes is
\[\kappa_2 = \frac{1}{V} \frac{ \left[ \displaystyle\sum_\lambda C_\lambda v_{\lambda x}^2\,\tau_{c,\lambda}^2/\tau_{N,\lambda} \right]^2 }{ \displaystyle\sum_\lambda C_\lambda v_{\lambda x}^2 \left(1-\tau_{c,\lambda}/\tau_{N,\lambda}\right) }. \tag{14}\]Intuitive picture:
- Numerator: roughly the squared sum of all drift-type contributions enabled by N processes.
- Denominator: the total strength of R processes that pull the distribution back from this drift.
- When R is strong and N weak, the correction collapses, $\kappa_2\to 0$.
- When N dominates and R is weak, $\kappa_2$ increases significantly, enhancing the total conductivity — reminiscent of phonon hydrodynamic behavior.
Starting point (BTE):
A temperature gradient drives phonon occupations out of equilibrium; the collision term pulls the system toward some “equilibrium-like” state.
References:
[1] Phys. Rev. 113, 1046–1051 (1959).
[2] Phys. Rev. B 88, 144302 (2013).
[3] Phys. Rev. B 90, 035203 (2014).