Thermal conductivity

Calculation of thermal conductivity via Fourier law or MFP

FreePATHS can calculate the thermal conductivity of the structure using two different approaches, via Fourier law or via mean free path. Although both quantities are called thermal conductivity, they are different quantities obtained in very different ways and capture different nanoscale effects. This page will explain both of these approaches.

Fourier law approach

Traditionally, the thermal conductivity is a coefficient in the Fourier law:

$$J = \kappa \nabla T,$$

In this approach, we use the main algorithm of FreePATHS to obtain heat flux (J) and temperature gradient (dT/dL) in the steady state regime. An example of such simulation is described here. To use this approach, one must understand the time parameters of the simulation. Specifically, that over several timeframes, the simulation is trying to achieve steady state thermal flow and then measured the profiles used for the thermal conductivity calculation.

Schematic of the simulation time.

We can see the convergence of the temperature profile to the linear gradient as the time progresses and the simulation reaches steady state in the last few time frames:

Temperature profiles at different time segments.

Likewise, heat flux profiles converge to the flat line in the last few time frames:

Heat flux profiles converge to the flat line in the steady state.

Then, the temperature gradient is obtained by a linear regression on the temperature profile and the heat flux value through the mean over the heat flux profile. The thermal conductivity is then obtained from Fourier law. As the simulation progresses and reaches the steady state, the thermal conductivity converges to a stable value:

Thermal conductivity converges as the system reaches steady state.

In this regime, two values of the thermal conductivity are calculated, the effective and material thermal conductivity. Although they are the same in this example, it is important to understand the difference, which is explained below.

Mean free path approach

Alternatively, you can run FreePATHS in the mean free path sampling mode, which is designed to calculate the thermal conductivity by integrating phonon dispersion and sampling the phonon relaxation time (or phonon mean free paths) for each frequency. To run the program in this mode, reduce the number of phonons to about 30 and add the -s flag in the command:

freepaths -s your_input_file.py

In this mode, the thermal conductivity at a given temperature (T) is calculated as:

$$\kappa = \frac{1}{6\pi^2} \sum_{j} \int \frac{\hslash^2\omega_j^2(q)}{k_bT^2} \frac{\exp\left[\hslash\omega_j(q)/k_bT\right]}{(\exp\left[\hslash\omega_j(q)/k_bT\right]-1)^2} v_j^2(q) \tau_j(q,T) q^2 dq$$

where k is the Boltzmann constant, ω(q) and v(q) are the frequency and group velocity on the branch j of the phonon dispersion at the wavevector q. The phonon relaxation time τ (or the phonon mean free path Λ = v(q)·τ) is measured by running phonons through the structure and recording the average of the distances between diffuse scattering events.

For example, if we take the same input file as in the nanowire example, reduce the number of phonons to 30 and run it with -s flag:

freepaths -s nanowire.py

We obtain the thermal conductivity of about 51 W/m·K, consistent with the Fourier law approach above.

Effective vs Material thermal conductivity

One must distinguish two different quantities, material and effective thermal conductivity. For simple structures without holes (like a nanowire) these quantities are the same. However, if the structure has non-uniform volume or pores or pillars, these quantities are not the same and the effective thermal conductivity is lower. For example, for a structure with some holes, FreePATHS will typical output a plot showing both quantities, as follows:

Example of material and effective thermal conductivities of porous structure.

The effective thermal conductivity is essentially the conductivity of a "black box" structure with a given thickness, width, and length, regardless of how much material has been removed inside the box. This quantity is useful in engineering, but note that it can be physically misleading as one can achieve the effective conductivity even below the amorphous limit simply removing lots of material, for example by making dense holes.

The material thermal conductivity takes into account the volume reduction due to material removal and essentially represents the thermal conductivity of the material between the holes. This quantity typically reflects the reduction in thermal conductivity caused by phonon scattering on the holes.

Likewise, you'll see that there are profiles for effective and material heat flux, which follows the same logic that material heat flux is the flux that was corrected by the absent volume of the holes, which usually creates jumps of the heat flux around the holes.