Next: 3. Thermoelectric Properties of Graphene-Based Nanostructures
Up: 2.4 Phonon Transport
Previous: 2.4.1 Landauer Formula
Contents
At equilibrium, the distribution of phonons in branch
and wavevector
is given by the Bose-Einstein distribution function
:
![$\displaystyle n\left( \omega_{\alpha}(\vec{q}) \right)=\frac{1}{\mathrm{e}^{\hbar \omega_{\alpha}(\vec{q})/k_{\mathrm{B}}T}-1}$](img308.png) |
(2.59) |
Under non-equilibrium conditions, the distribution of phonons deviates from its equilibrium distribution, and transport of phonons is computed using the Boltzmann transport formalism. The non-equilibrium distribution function
, in general, is a function of time
and position
. The BTE can be written as:
![$\displaystyle \frac{\partial \overline{n}}{\partial t}+\vec{v} \cdot \nabla_{\v...
...rline{n}=\left( \frac{\partial \overline{n}}{\partial t}\right)_{\mathrm{scat}}$](img312.png) |
(2.60) |
and for the steady state:
![$\displaystyle \vec{v} \cdot \nabla_{\vec{r}}\overline{n}=\left( \frac{\partial \overline{n}}{\partial t}\right)_{\mathrm{scat}}$](img313.png) |
(2.61) |
Under a temperature gradient, the BTE can be written as [60]:
![$\displaystyle \vec{v} \cdot \nabla_{\vec{r}}T \frac{\partial \overline{n}}{\partial T}=\left( \frac{\partial \overline{n}}{\partial t}\right)_{\mathrm{scat}}$](img314.png) |
(2.62) |
In the relaxation time approximation, the change of the distribution function due to the scattering events can be given by:
![$\displaystyle \left( \frac{\partial \overline{n}}{\partial t}\right)_{\mathrm{scat}}=\frac{n-\overline{n}}{\tau_{\alpha}(\vec{q})}$](img315.png) |
(2.63) |
and therefore
![$\displaystyle \vec{v} \cdot \nabla_{\vec{r}}T \frac{\partial \overline{n}}{\partial T}=\frac{n-\overline{n}}{\tau_{\alpha}(\vec{q})}$](img316.png) |
(2.64) |
where
is the relaxation time of phonons of frequency
. In this work we use a linearized
form of Eq. 2.64, which assumes that the temperature gradient causes only a
small deviation from Bose-Einstein distribution function [61,62], so that:
![$\displaystyle \frac{\partial \overline{n}}{\partial T}\approx \frac{\partial n}{\partial T}=\frac{\hbar \omega_{\alpha}(\vec{q})}{k_{\mathrm{B}}T^2}n(n+1)$](img319.png) |
(2.65) |
and
![$\displaystyle \overline{n}=\frac{1}{\mathrm{e}^{\left( \hbar \omega_{\alpha}(\v...
...l (\hbar \omega)}\right)=n+\frac{\Psi_{\alpha}(\vec{q})n(n+1)}{k_{\mathrm{B}}T}$](img320.png) |
(2.66) |
where
shows the deviation from the equilibrium distribution. Then, one may eliminate the temperature gradient using
and write:
![$\displaystyle \overline{n}=n-\frac{n(n+1)}{k_{\mathrm{B}}T}\psi_{\alpha}(\vec{q})\nabla_{\vec{r}}T$](img323.png) |
(2.67) |
Since the equilibrium distribution does not carry any heat flux, the heat flux equals
to [62]:
![$\displaystyle I_{\mathrm{q}}=\sum_{\alpha,\vec{q}}\hbar \omega (\overline{n}-n)...
...\vec{q}) \frac{n(n+1)}{k_{\mathrm{B}}T}\psi_{\alpha}(\vec{q})\nabla_{\vec{r}} T$](img324.png) |
(2.68) |
On the other hand, it holds the differential form of Fourier's law:
![$\displaystyle I_{\mathrm{q}}=-\kappa_l \nabla T$](img325.png) |
(2.69) |
Therefore, one can obtain the lattice thermal conductivity as:
![$\displaystyle \kappa_l=\sum_{\alpha,\vec{q}}\hbar \omega v_{\alpha}(\vec{q}) \frac{n(n+1)}{k_{\mathrm{B}}T}\psi_{\alpha}(\vec{q})$](img326.png) |
(2.70) |
Under the single-mode relaxation time (SMRT) approximation [62],
follows from the linearized BTE (Eqs. 2.64-2.66) as:
![$\displaystyle \psi_{\alpha}(\vec{q})=\frac{\hbar \omega_{\alpha}(\vec{q})}{T}v_{\alpha}(\vec{q}) \tau_{\alpha}(\vec{q})$](img328.png) |
(2.71) |
Here,
is the scattering time in SMRT approximation. Therefore, Eq. 2.70 becomes
![$\displaystyle \kappa_l=\sum_{\alpha,\vec{q}}\hbar \omega v_{\alpha}(\vec{q})^2 \tau_{\alpha}(\vec{q}) \frac{\partial n}{\partial T}$](img329.png) |
(2.72) |
Next: 3. Thermoelectric Properties of Graphene-Based Nanostructures
Up: 2.4 Phonon Transport
Previous: 2.4.1 Landauer Formula
Contents
H. Karamitaheri: Thermal and Thermoelectric Properties of Nanostructures