1. Background : Introduction to thermal conductivity, of materials
2. NEMD simulations : Calculation of using molecular dynamics simulations
3. System size scaling : Effect of simulation cell size on calculated
4. Temperature scaling : Effect of system temperature on calculated
5. Thermal conductivity plugins : Calculating of MoS₂
6. Quantum corrections : Beyond classical thermal conductivity simulations
   • Velocity autocorrelation in equilibrium simulations
   • Phonon density of states
   • Specific heat of materials
   • Quantum-corrected thermal conductivity
7. Summary : Things to remember when you do your own thermal conductivity simulations
8. Current research applications : Engineering thermal conductivity of materials. Brief look at thermal conductivity of fractal and alloyed systems
9. References

• Thermal conductivity (denoted by the symbol ) is a fundamental property of materials that determines their ability to conduct (i.e. transmit) heat (See Refs 1,2). Materials with a higher values conduct heat well and low- materials are more insulating (Ref 9.).

Material	(W/m-K)
Diamond	1000
Silver	406
Copper	401
Water	0.591
Wood	0.12
Wool	0.0464
Air	0.025
Silica Aerogel	0.003

• High- materials are commonly used in heat-sink and thermal-dissipation applications and materials with low thermal conductivity are used primarily for insulation. Low insulating materials are also used for thermoelectric energy harvesting applications (See Section 8).

• Both high and low materials are extremely useful for engineers.

As described previously, thermal conductivity is calculated by measuring the temperature gradient along the material. We establish the thermal gradient by adding and removing a predefined quantity of heat, E₀, at and respectively. Since we can control the amount, E₀ and frequency of heat input, , we effectively control the heat flux in the system. Once the steady-state temperature profile is established in the simulation cell, we combine the temperature profile with the known heatflux to calculate the thermal conductivity.

Specifically,

Heat Flux,

The factor comes from the fact that heat conduction happens along both the +x and -x directions away from the heat source at .

Also, from Fourier's law of thermal conduction, we have , where A is the cross sectional area of heat transfer. (Ref 6)

Putting these equations together, we have

Note here that the thermal conductivity goes as the inverse of the temperature gradient. More conducting material will have 'flatter' temperature profiles.

Image 1: Schematic of NEMD simulations for measuring thermal conductivity of 2D materials

Length Scaling is a direct consequence because of MD. The thermal transport in NEMD happens over two square regions. Phonons having mean free path largest than the size of the square do not take part in thermal transport in our system setup. As the system increases in size, more and more phonons are being involved in thermal transport. Studies have shown the mean free path for Transition metal Dichalcogenides is of the order of 1000 nm \cite{gandi2016thermal}. It is therefore essential to scale systems to at least one third its value.

These simulations are performed inside the Casimir Limit (See Ref.13). This results in thermal conductivity values having a strong size dependence. Following the scaling scheme of Schon which uses Matthiessen's Rule . To predict the thermal conductivity of the infinite system we make use of a linear function such as

Here, is the inverse scattering length relating to the heat source-sink distance and is the material's inverse scattering length because of phonon phonon scattering.

It is important to note that length scaling happens in two dimensions. It is important to maintain a 2:1 ratio between length and width while increasing system size. The major difference is that one dimensional scaling ignores the contribution of phonons with wavelengths larger than the smallest dimension of the solid, whereas two dimensional scaling is inclusive.

The system temperature reflects the population of phonons in the material. At higher temperatures, there is a greater chance of phonon scattering (by other phonons) leading to reduced thermal conductivity.

Now that we have the temperature gradient data for our length-scaling runs, we will see how to calculate the thermal conductivity values using our Python-based thermal conductivity plugin, calthermal_conductivity.py

The next input file, Temperature.txt, is a direct output of our MD simulations. This file contains the temperature of the atoms, calculated as averaged over bins along the thermal transport, i.e. x, direction. The second column is the coordinate in the system and the fourth column is the corresponding average temperature.

The main discrepancy arises because classical molecular dynamics simulations (like ours) do not correctly describe the specific heat of materials (denoted by C_V) at low temperatures. There is a method to correct for this deficiency by calculating the quantum-mechanically accurate specific heat of materials separately using equilibrium molecular dynamics simulations. This involves computing the velocity autocorrelation functions, the vibrational density of states and the specific heat, in that order, from the MD simulations, as shown below.

Image 5: Workflow for calculating specific heat from equilibrium MD simulations

The velocity autocorrelation function Z(t) for atom type is defined as

The density of states is given by the equation

Here Z is the velocity autocorrelation for each element. G() is the density of states. If you observe the following equation you can see that the vibrational density of state is simply proportional to the Fourier transform of the velocity autocorrelation.

With the density of states available, we can calculate specific heat using the following equation. This simply multiplies the vibrational energy of each state with the temperature derivative of the probability of occupation of each vibrational state.

where

1. Create a relaxed system at any given temperature : We use a LAMMPS Script to create a dump file.
2. Compute the VACF, VDOS and C_v from the LAMMPS dump file : Use the file caldos.py which does the calculation for you

The directory also has a file caldos.py. This is a python script that calls a C program dos.c that performs the VDOS calculations for us and plots the output. We won't actually be modifying that file in today's session. dos.c requires an input file input.txt that contains values for the various parameters required for the calculation. Let us take a look at that file.

It is essential to understand the variables in this file. We will talk about a few important ones.

1. dT refers to the time-step of the equilibrium MD simulations.
2. TFREQ refers to frequency with which snapshots are written out by LAMMPS. I
3. massMo and massS are the atomic masses of Molybdenum (95.94) and Sulfur (32.065) respectively. We can change them according to the atoms in the system.
4. Corrlength is the length of the trajectory over which VACF is calculated
5. Ninitial is the number of initial conditions for calculating VACF and Ngap is the time-delay between consecutive initial conditions. These two sampling parameters are described in detail below.

One of the most significant improvements enabled by the thermal conductivity plugin tools is the usage of multiple initial conditions for calculation of velocity autocorrelation functions. This relies on the fact that the crystal at equilibrium 'loses memory' of its velocities after a few ps. Therefore, a single long MD trajectory can be broken into multiple overlapping independent sub-trajectories, each Corrlength steps long and separated from each successive sub-trajectory by Ngap steps. This significantly improves the sample-size for the calculation of atomic velocity correlations, often by over 1 order of magnitude.

• N_Frame is the total number of frames, which is equal to the number of steps that we run the simulation for.

• We must ensure that

• If this condition isn't satisfied we will get an error, because the program does not get enough data

The caldos.py script also calculates the temperature-dependent specific heat, C_v, of the material using the density of states. This value can be used to correct our thermal conductivity value that was over-estimated at low temperatures using the equation below.

• NEMD is a simple method to compute lattice thermal conductivity of nanomaterials
• The thermal conductivity plugin, calthermal_conductivity.py helps with size-scaling and temperature-scaling calculations to obtain the material-intrinsic thermal conductivity at different temperatures
• The thermal conductivity plugin, caldos.py helps include quantum-mechanical corrections to computed $\kappa$ to obtain experimentally-realistic values at low temperatures

• Thermal conductivity simulations must be performed for a sufficiently long time (usually ~10s of ns) to ensure that a good steady state has been reached
• We must ensure good sampling (i.e. number of independent initial conditions) in the VACF and VDOS calculation for smooth curves.

• Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop driven by recursion (See Ref. 12).

Image 7: The Sierpinski carpet is a model fractal system in two-dimensions

• Here is a picture depicting the structure and heat flux in such a fractal system composed of MoSe₂ and WSe₂ materials.

Image 8: Atomic structure and heat-flux through a fractal two-dimensional material

• Local heat flux . It is clear from the above figure that majority of the heat flux moves through the MoSe₂ lattice and the interface acts as a source of phonon scattering.

1. Cahill, D.G., et al., Nanoscale thermal transport. Journal of Applied Physics, 2003. 93(2): p. 793-818.
2. Cahill, D.G., et al., Nanoscale thermal transport. II. 2003-2012. Applied Physics Reviews, 2014. 1(1).
3. Payam, N. and J.S. David, Thermal conductivity of single-layer WSe 2 by a Stillinger–Weber potential. Nanotechnology, 2017. 28(7): p. 075708.
4. Sahoo, S., et al., Temperature-Dependent Raman Studies and Thermal Conductivity of Few-Layer MoS2. Journal of Physical Chemistry C, 2013. 117(17): p. 9042-9047.
5. Yan, R.S., et al., Thermal Conductivity of Monolayer Molybdenum Disulfide Obtained from Temperature-Dependent Raman Spectroscopy. ACS Nano, 2014. 8(1): p. 986-993.
6. Wu, X.F., et al., How to characterize thermal transport capability of 2D materials fairly? - Sheet thermal conductance and the choice of thickness. Chemical Physics Letters, 2017. 669: p. 233-237.
7. Klemens, P.G., Thermal Conductivity and Lattice Vibrational Modes. Solid State Physics, 1958. 7: p. 1-98.
8. Thermal Conductivity Wiki page: https://en.wikipedia.org/wiki/Thermal_conductivity
9. Fourier's Law: https://en.wikipedia.org/wiki/Thermal_conduction
10. Thermal Insulation in a Space Shuttle: https://www.nasa.gov/sites/default/files/atoms/files/shuttle_tiles_5_8v2.pdf
11. Space Shuttle thermal protection system: https://en.wikipedia.org/wiki/Space_Shuttle_thermal_protection_system
12. Sierpinski carpet: https://en.wikipedia.org/wiki/Sierpinski_carpet
13. HBG Casimir. Note on the conduction of heat in crystals Physica, 5(6):495-500, 1938