Information and citation refere to NTMpy: An open source package for solving coupled parabolic differential equations in the framework of the three-temperature model paper.
Further information on the solver package itself can be found here: NTMpy.
Documentation and example sessions can be found in the Wiki.
This is a code providing a solution to the heat diffusion in a 1D structure in a 3-temperature model approximation.
We consider:
- Material comprising piecewise homogeneous layers
- Heating of electron system with an energy source with Gaussian or custom shape (i.e. an ultrashort laser pulse)
- Three temperature model: electron, lattice and spin systems are considered and coupled
- Transfer Matrix Method to account for local absorbed energy in the multi-layer material
The equation under consideration is:
where C = specific heat, k = conductivity, G is the coupling constant between the two systems (Electron and Lattice)
and 
are the respective temperatures of the electron lattice and spin system with respect to space x and time t. The superscripts L and E indicate whether a parameter belongs to the electron (E) lattice (L) or spin (S) system and the subindex i denotes to which layer the parameter belongs.
A sketch of the model:
Our approach is to use a combination of finite elements (B-Splines) to approximate the derivation in space and Explicit Euler to approximate the evolution in time. To stay within the stability region of the Explicit Euler algorithm, a well-suited time step is automatically computed.
Using the material parameters, reported in Ultrafast Spin Dynamics in Ferromagnetic Nickel we consider a 3- 3-temperature system in a Nickel layer. We are able to simulate the reported temperature map for all systems, in accordance with the paper.
| Temperature evolution of 3-coupled systems | Gaussian laser pulse S(x,t) hitting sample |
|---|---|
 |
 |
Here we consider an energy source S(x,t), that is Gaussian in time and decays exponentially, representing a laser pulse hitting the material from the left. Note, that the implemented Transfer Matrix Method makes it easily able to simulate realistic laser sources, taking multiple reflection, different incident angles and polarization into account.
The animation clearly shows the following effects:
- The electron system heats up immediately, as expected looking at the equation above.
- The heat in the electronic system diffuses along the x-axis.
- The heat transfers to the subsystems on longer time scales.
A short example of code to simulate a nonlinear 2-temperature model with 2 layers
from NTMpy import NTMpy as ntm
# Initialize source
s = ntm.source() # default option, Gaussian pulse
s.fluence = 1 # Energy per meter square
s.FWHM = 2e-12 # Full Width Half Maximum (duration of the pulse)
s.t0 = 2e-12 # time the maximum intensity hits
s.lambda_vac = 400 # Wavelength of source (in nanometers)
# initialize simulation: ntm.simulation(number of temperatures, source)
sim = ntm.simulation(2,s) # Default initial temperature: 300 Kelvin
# add material layers:
# > non constant quantities are inserted as lambda functions
# > if more temperature are used, conductivity and heat capacity are vectors
# addlayer ( Length, refractive index, conductivity, Heat Capacity, density, coupling)
sim.addLayer( 30e-9, 1+3j, [ 8, 0], [lambda T: .112*T, 450], 6500, 6e17) # first layer
sim.addLayer( 80e-9, 1+3j, [24, 12], [lambda T: .025*T, 730], 5100, 6e17) # second layer
# set final simulation time (in seconds)
sim.final_time = 50e-12
# Run simulation
[x,t,phi] = sim.run()
# Link the plotting library to the simulation
vs = ntm.visual(sim)
# Show animation (like the one in the github page)
vs.animation(1) # input is the animation speedThe ouput phi is a 3D array with the following structure:
- the first index selects the temperature.
For example,
phi[0]is the electron temperature,phi[1]is the lattice temperature. - the second index is the time instant.
phi[0][0]contains the initial temperature of the electron system,phi[0][100]contains the electron temperature after 100 time steps. - the third index is the space position
phi[0][100][10]is the temperature of the electron system after 100 time steps in the 10th point of the grid.
Complete documentation and example sessions can be found in the Wiki
This is a project from the Ultrafast Dynamics in Condensed Matter group at Stockholm University and at the Ca' Foscari University of Venice. The main contributors are:
@article{alber2020ntmpy, title={NTMpy: An open source package for solving coupled parabolic differential equations in the framework of the three-temperature model}, author = {Lukas Alber and Valentino Scalera and Vivek Unikandanunni and Daniel Schick and Stefano Bonetti}, journal = {Computer Physics Communications}, year={2021}, volume = {265}, pages = {107990}, issn = {0010-4655}, doi = {https://doi.org/10.1016/j.cpc.2021.107990}, url = {https://www.sciencedirect.com/science/article/pii/S0010465521001028} }
Since finding the thermophysical parameters needed to solve the equation under consideration (see above) is currently subject to research, we want to encourage our users to contribute with their knowledge of parameters. That is, please send us the data of the parameters in use, together with references from literature. We are working on providing an open-source data base. Also, see NTMpy page.
Fork from the Developer- branch and pull request to merge back into the original Developer- branch.
Working updates and improvements will then be merged into the Master branch, which will always contain the latest working version.
Note that by downloading the package via pip install NTMpy a routine, which automatically checks if all the required packages exist on the local machine is implemented. If one of the dependent pip packages, listed here, is missing an automatic download is initiated.