simulations.heat

Classes:

Heat(S, force_recalc, **kwargs)

Heat simulations including laser excitation and N-temperature model.

class udkm1Dsim.simulations.heat.Heat(S, force_recalc, **kwargs)

Bases: Simulation

Heat simulations including laser excitation and N-temperature model.

Parameters:

• S (Structure) – sample to do simulations with.

• force_recalc (boolean) – force recalculation of results.

Keyword Arguments:

• save_data (boolean) – true to save simulation results.

• cache_dir (str) – path to cached data.

• disp_messages (boolean) – true to display messages from within the simulations.

• progress_bar (boolean) – enable tqdm progress bar.

• heat_diffusion (boolean) – true when including heat diffusion in the calculations.

• intp_at_interface (int) – number of additional spacial points at the interface of each layer.

• backend (str) – pde solver backend - either default scipy or matlab.

Attributes:

• S (Structure) – sample structure to calculate simulations on.

• force_recalc (boolean) – force recalculation of results.

• save_data (boolean) – true to save simulation results.

• cache_dir (str) – path to cached data.

• disp_messages (boolean) – true to display messages from within the simulations.

• progress_bar (boolean) – enable tqdm progress bar.

• heat_diffusion (boolean) – true when including heat diffusion in the calculations.

• intp_at_interface (int) – number of additional spacial points at the interface of each layer.

• backend (str) – pde solver backend - either default scipy or matlab.

• excitation (dict{ndarray[float, Quantity]}) – excitation parameters fluence, delay_pump, pulse_width, wavelength, theta, polarization, multilayer_absorption, backside

• boundary_conditions (dict{str, float, Quantity}) – boundary conditions of the top and bottom boundary for the heat diffusion calculation. top_type or bottom_type must be one of boundary_types. For the last two types the corresponding value, top_value and bottom_value have to be set as \(K \times 1\) array, where \(K\) is the number of sub-systems.

• boundary_types (list[str]) – description of boundary types.

  • isolator

  • temperature

  • flux

• distances (ndarray[float, Quantity]) – spatial grid for heat diffusion [m]

• ode_options (dict) – options for scipy solve_ivp ode solver

• ode_options_matlab (dict) – dict with options for the MATLAB pdepe solver.

• matlab_engine (module) – MATLAB to Python API engine required for calculating heat diffusion.

Methods:

get_hash(delays, init_temp, **kwargs)	Returns a unique hash given by the delays and init_temp as well as the sample structure hash for relevant thermal parameters.
check_initial_temperature(init_temp[, distances])	Initial temperature for heat simulations can be either a scalar temperature which is assumed to be valid for all layers in the structure or a temperature profile is given with one temperature for each layer in the structure and for each subsystem.
check_excitation(delays)	The optical excitation is a dictionary with fluence \(F\) [J/m²], delays \(t\) [s] of the pump events, and pulse width \(\tau\) [s].
get_absorption_profile([distances, backside])	Returns the differential absorption profile \(\mbox{d}A/\mbox{d}z\).
get_Lambert_Beer_absorption_profile([...])	The transmission is given by:
get_multilayers_absorption_profile([...])	Calculates the intensity, differential absorption and temperature increase profiles in each layer of a multilayers structure for \(p\) -polarized light.
get_temperature_after_delta_excitation(...)	Calculate the final temperature and temperature change for each layer of the sample structure after an optical excitation with a fluence \(F\) [J/m^2] and an initial temperature \(T_1\) [K]:
get_temp_map(delays, init_temp)	Returns a temperature profile for the sample structure after optical excitation.
calc_temp_map(delays, input_init_temp)	Calculates the temperature profile for the sample structure after optical excitation.
calc_heat_diffusion(init_temp, distances, ...)	Calculates a temperature profile including heat diffusion for a given delay array and initial temperature profile.
odefunc(t, u, N, K, d_x_grid, x, ...)	Ordinary differential equation that is solved for 1D heat diffusion.
conv_with_function(y, x, handle)	Convolutes the array \(y(x)\) with a function given by the handle on the argument array \(x\).
disp_message(message)	Wrapper to display messages for that class.
save(full_filename, data, *args)	Save data to file.

get_hash(delays, init_temp, **kwargs)

Returns a unique hash given by the delays and init_temp as well as the sample structure hash for relevant thermal parameters.

Parameters:

• delays (ndarray[float]) – delay grid for the simulation.

• init_temp (ndarray[float]) – initial spatial temperature profile.

• **kwargs (float, optional) – optional parameters.

Returns:

hash (str) – unique hash.

check_initial_temperature(init_temp, distances=[])

Initial temperature for heat simulations can be either a scalar temperature which is assumed to be valid for all layers in the structure or a temperature profile is given with one temperature for each layer in the structure and for each subsystem. Alternatively a spatial grid can be provided.

Parameters:

• init_temp (float, Quantity, ndarray[float, Quantity]) – initial temperature scalar or array [K].

• distances (ndarray[float, Quantity], optional) – spatial grid of the initial temperatrue.

Returns:

init_temp (ndarray[float]) – checked initial temperature as array on the according spatial grid.

check_excitation(delays)

The optical excitation is a dictionary with fluence \(F\) [J/m²], delays \(t\) [s] of the pump events, and pulse width \(\tau\) [s]. \(N\) is the number of pump events.

Traverse excitation vector to update the delay_pump \(t_p\) vector for finite pulse durations \(w(i)\) as follows

\[[t_p(i)-\mbox{window}\cdot w(i)]:[t_p(i)+\mbox{window}\cdot w(i)]: [w(i)/\mbox{intp}]\]

and to combine excitations which have overlapping intervals.

Moreover the incidence angle \(\vartheta\) is taken into account for the user-defined incidence fluence in order to project the laser footprint onto the sample surface.

Parameters:

delays (ndarray[Quantity]) – delays range of simulation [s].

Returns:

(tuple) –

• res (list[ndarray[float]]) - resulting list of excitations with interpolated delay density around excitations.

• fluence (ndarray[float]) - projected excitation fluences.

• delay_pump (ndarray[float]) - delays of the excitations.

• pulse_width (ndarray[float]) - pulse widths of the excitations.

get_absorption_profile(distances=[], backside=False)

Returns the differential absorption profile \(\mbox{d}A/\mbox{d}z\).

Parameters:

• distances (ndarray[float], optional) – spatial grid for calculation.

• backside (boolean, optional) – backside or frontside excitation.

Returns:

dAdz (ndarray[float]) – differential absorption within each layer calculated either by Lambert-Beers law or by a multilayer absorption formalism.

get_Lambert_Beer_absorption_profile(distances=[], backside=False)

The transmission is given by:

\[\tau = \frac{I}{I_0} = \exp(-z/ \zeta)\]

and the absorption by:

\[A = 1 - \tau = 1 - \exp(-z/\zeta)\]

The absorption profile can be derived from the spatial derivative:

\[\frac{\mbox{d}A(z)}{\mbox{d}z} = \frac{1}{\zeta} \exp(-z/\zeta)\]

Parameters:

• distances (ndarray[float], optional) – spatial grid for calculation.

• backside (boolean, optional) – backside or frontside excitation.

Returns:

dAdz (ndarray[float]) – differential absorption within each layer calculated by Lambert-Beers law.

get_multilayers_absorption_profile(distances=[], backside=False)

Calculates the intensity, differential absorption and temperature increase profiles in each layer of a multilayers structure for \(p\) -polarized light.

Calculation based on the method in Ref [5] and code developed Matlab by L. Le Guyader, see Ref [6].

Copyright (2012-2014) Loïc Le Guyader <loic.le_guyader@helmholtz-berlin.de>

Parameters:

• distances (ndarray[float], optional) – spatial grid for calculation.

• backside (boolean, optional) – backside or frontside excitation.

Returns:

(tuple) –

• dAdz (ndarray[float]) - differential absorption within each layer.

• Ints (ndarray[float]) - intensity profiles within each layer.

• R_total (float) - total amount of reflection from the multilayer.

• T_total (float) - total transmission in the last layer of the multilayer.

References:

[5]

K. Ohta & H. Ishida, Matrix formalism for calculation of the light beam intensity in stratified multilayered films, and its use in the analysis of emission spectra, Appl. Opt. 29, 2466 (1990).

[6]

L. Le Guyader, A. Kleibert, F. Nolting, L. Joly, P.M. Derlet, R.V. Pisarev, A. Kirilyuk, Th. Rasing & A.V. Kimel, Dynamics of laser-induced spin reorientation in Co/SmFeO_3 heterostructure, Phys. Rev. B 87, 054437 (2013).

get_temperature_after_delta_excitation(fluence, init_temp, distances=[])

Calculate the final temperature and temperature change for each layer of the sample structure after an optical excitation with a fluence \(F\) [J/m^2] and an initial temperature \(T_1\) [K]:

\[\Delta E = \int_{T_1}^{T_2} m \, c(T)\, \mbox{d}T\]

where \(\Delta E\) is the absorbed energy in each layer and \(c(T)\) is the temperature-dependent heat capacity [J/kg K] and \(m\) is the mass [kg].

The absorbed energy per layer can be linearized from the absorption profile \(\mbox{d} \alpha / \mbox{d}z\) as

\[\Delta E = \frac{\mbox{d} \alpha}{\mbox{d}z} E_0 \Delta z\]

where \(E_0\) is the initial energy impinging on the first layer given by the fluence \(F = E / A\). \(\Delta z\) is equal to the thickness of each layer.

Finally, one has to minimize the following modulus to obtain the final temperature \(T_2\) of each layer:

\[\left| \int_{T_1}^{T_2} m c(T)\, \mbox{d}T - \frac{\mbox{d}\alpha} {\mbox{d}z} E_0 \Delta z \right| \stackrel{!}{=} 0\]

Parameters:

• fluence (float, Quantity) – incident fluence [J/m²].

• init_temp (float, Quantity, ndarray[float, Quantity]) – initial temperature scalar or array [K].

• distances (ndarray[float], optional) – spatial grid for calculation.

Returns:

(tuple) –

• final_temp (ndarray[float]) - final temperature after delta excitation.

• delta_T (ndarray[float]) - temperature change.

get_temp_map(delays, init_temp)

Returns a temperature profile for the sample structure after optical excitation. The result can be saved using an unique hash of the sample and the simulation parameters in order to reuse it.

Parameters:

• delays (ndarray[Quantity]) – delays range of simulation [s].

• init_temp (float, Quantity, ndarray[float, Quantity]) – initial temperature scalar or array [K].

Returns:

(tuple) –

• temp_map (ndarray[float]) - spatio-temporal temperature map.

• delta_temp_map (ndarray[float]) - spatio-temporal temperature change map.

calc_temp_map(delays, input_init_temp)

Calculates the temperature profile for the sample structure after optical excitation. Heat diffusion can be included if heat_diffusion = true.

Parameters:

• delays (ndarray[Quantity]) – delays range of simulation [s].

• input_init_temp (float, Quantity, ndarray[float, Quantity]) – initial temperature scalar or array [K].

Returns:

(tuple) –

• temp_map (ndarray[float]) - spatio-temporal temperature map.

• delta_temp_map (ndarray[float]) - spatio-temporal temperature change map.

• checked_excitations (list[ndarray[float]]) - resulting list of checked excitations.

calc_heat_diffusion(init_temp, distances, delays, delay_pump, pulse_width, fluence)

Calculates a temperature profile including heat diffusion for a given delay array and initial temperature profile. Here we have to solve the 1D heat diffusion equation for \(N\) subsystems:

\[\begin{split}c_1(T_1) \rho \frac{\partial T_1}{\partial t} & = & \frac{\partial}{\partial z} \left( k_1(T_1) \frac{\partial T_1}{\partial z} \right) + G_1(T_1,...,T_N) + S(z,t) \\ & \vdots & \\ c_N(T_N) \rho \frac{\partial T_N}{\partial t} & = & \frac{\partial}{\partial z} \left( k_N(T_N) \frac{\partial T_N}{\partial z} \right) + G_N(T_1,...,T_N)\end{split}\]

where \(T_i\) is the temperature [K], \(z\) the distance [m], \(t\) the delay [s], \(c_i(T)\) the temperature dependent heat capacity [J/kg K], \(\rho\) the density [kg/m³] and \(k_i(T)\) is the temperature-dependent thermal conductivity [W/m K] and \(S(z,t)\) is a source term [W/m³]. The energy flow between the subsystems is given by the sub_system_coupling parameter \(G_i(T_1,...,T_N)\) of the individual layers. The index \(i\) refers to the \(i\)-th subsystem.

The 1D heat diffusion equation can be either solved with SciPy or Matlab as backend.

Parameters:

• init_temp (float, Quantity, ndarray[float, Quantity]) – initial temperature scalar or array [K].

• distances (ndarray[float]) – spatial grid for calculation.

• delays (ndarray[Quantity]) – delays range of simulation [s].

• delay_pump (ndarray[float]) – delays of the excitations.

• pulse_width (ndarray[float]) – pulse widths of the excitations.

• fluence (ndarray[float]) – excitation fluences.

Returns:

temp_map (ndarray[float]) – spatio-temporal temperature map.

static odefunc(t, u, N, K, d_x_grid, x, thermal_conds, heat_capacities, sub_system_coupling, densities, indices, dAdz, fluence, delay_pump, pulse_length, bc_top_type, bc_top_value, bc_bottom_type, bc_bottom_value, pbar, state)

Ordinary differential equation that is solved for 1D heat diffusion.

Parameters:

• t (ndarray[float]) – internal time steps of the ode solver.

• u (ndarray[float]) – internal variable of the ode solver.

• N (int) – number of spatial grid points.

• K (int) – number of sub-systems.

• d_x_grid (ndarray[float]) – derivative of spatial grid.

• x (ndarray[float]) – start point of actual layers.

• thermal_conds (ndarray[@lambda]) – T-dependent thermal conductivity function handles.

• heat_capacities (ndarray[@lambda]) – T-dependent heat capacity function handles.

• sub_system_coupling (ndarray[@lambda]) – T-dependent sub-system coupling.

• densities (ndarray[float]) – density of layers.

• indices (ndarray[int]) – indices of actual layers in respect to interpolated spatial grid.

• dAdz (ndarray[float]) – differential absorption profile.

• fluence (ndarray[float]) – excitation fluences.

• delay_pump (ndarray[float]) – delay of excitations.

• pulse_length (ndarray[float]) – pulse widths of excitations.

• bc_top_type (int) – top boundary type.

• bc_top_value (ndarray[float]) – top boundary value.

• bc_bottom_type (int) – bottom boundary type.

• bc_bottom_value (ndarray[float]) – bottom boundary value.

• pbar (tqdm) – tqdm progressbar.

• state (list[float]) – state variables for progress bar.

Returns:

dudt (ndarray[float]) – temporal derivative of internal variable.

static conv_with_function(y, x, handle)

Convolutes the array \(y(x)\) with a function given by the handle on the argument array \(x\).

Parameters:

• y (ndarray[float]) – y data.

• x (ndarray[float]) – x data.

• handle (@lamdba) – convolution function.

Returns:

y_conv (ndarray[float]) – convoluted data.

disp_message(message)

Wrapper to display messages for that class.

Parameters:

message (str) – message to display.

save(full_filename, data, *args)

Save data to file. The variable name can be handed as variable argument.

Parameters:

• full_filename (str) – full file name to data file.

• data (ndarray) – actual data to save.

• *args (str, optional) – variable name within the data file.