Structure#
In order to carry out any simulations using the udkm1Dsim package, an according one-dimensional Structure needs to be created in advance.
This Structure object can consists of one or many sub-structures and/or Layers of type UnitCell or AmorphousLayer.
UnitCells and AmorphousLayers consist of the fundamental building blocks, namely Atoms and AtomMixed.
In this example the basic concepts of creating the above mentioned objects are introduced. Furthermore one should easily see how to set and access all the physical properties of these physical objects.
Setup#
Do all necessary imports and settings.
import udkm1Dsim as ud
u = ud.u # import the pint unit registry from udkm1Dsim
import scipy.constants as constants
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
u.setup_matplotlib() # use matplotlib with pint units
Atoms#
The atoms module contains two classes: Atom and AtomMixed.
Atom#
The Atom object represents a real physical atom as it can be found in the periodic table.
Accordingly, it is initialized with the required symbol of the element.
Then all necessary data are loaded from parameter files linked to this element.
An optional ID can be given as keyword parameter if atoms of the same element but with different properties are used.
Another keyword argument is the ionicity of the atom, which has to be present in the parameter files.
The magnetization of every atom can be set by the three keyword parameters
mag_amplitudemag_phimag_gamma
If no individual paths to the atomic- and/or magnetic scattering factors are given, by atomic_form_factor_path and magnetic_form_factor_path, respectively, the defaults parameters are used from the
Chantler tables:
C.T. Chantler, K. Olsen, R.A. Dragoset, J. Chang, A.R. Kishore, S.A. Kotochigova, & D.S. Zucker,
Detailed Tabulation of Atomic Form Factors, Photoelectric Absorption and Scattering Cross Section, and Mass Attenuation Coefficients for Z = 1-92 from E = 1-10 eV to E = 0.4-1.0 MeV
NIST Standard Reference Database 66.
as well as Project Dyna, respectively
O = ud.Atom('O')
Om1 = ud.Atom('O', id='Om1', ionicity=-1)
Om2 = ud.Atom('O', id='Om2', ionicity=-2)
Fe = ud.Atom('Fe', mag_amplitude=1, mag_phi=0*u.deg, mag_gamma=90*u.deg)
Cr = ud.Atom('Cr')
Ti = ud.Atom('Ti')
Sr = ud.Atom('Sr')
Ru = ud.Atom('Ru')
Pb = ud.Atom('Pb')
Zr = ud.Atom('Zr')
One can easily print all properties of a single atom:
print(Sr)
Atom with the following properties
================= =================================
id Sr
symbol Sr
name Strontium
atomic number Z 38
mass number A 87.62
mass 1.455×10⁻²⁵ kg
ionicity 0
Cromer Mann coeff [38. 0. 17.5663 9.8184]
.. [ 5.422 2.6694 1.5564 14.0988]
.. [ 0.1664 132.376 2.5064]
magn. amplitude 0
magn. phi 0.0 deg
magn. gamma 0.0 deg
================= =================================
Or just a single property:
print(Ti.mass)
7.948502350094219×10⁻²⁶ kg
Mixed Atom#
The AtomMixed class allows for solid solutions that can easily achieved by the following lines of code.
The input for the initialization of the AtomMixed object are the symbol, id, and name, whereas only the first is required.
ZT = ud.AtomMixed('ZT', id='ZT', name='Zircon-Titan 0.2 0.8')
ZT.add_atom(Zr, 0.2)
ZT.add_atom(Ti, 0.8)
print(ZT)
AtomMixed with the following properties
=============== ====================
id ZT
symbol ZT
name Zircon-Titan 0.2 0.8
atomic number Z 25.6
mass number A 56.538399999999996
mass 9.388×10⁻²⁶ kg
ionicity 0.0
magn. amplitude 0
magn. phi 0.0 deg
magn. gamma 0.0 deg
=============== ====================
2 Constituents:
--------- ------
Zirconium 20.0 %
Titanium 80.0 %
--------- ------
Layers#
The atoms created above can be used to build Layers.
There are two types of layers available: AmorphousLayer and crystalline UnitCell.
Both share many common physical properties which are relevant for the later simulations.
Please refer to a complete list or properties and methods in the API documentation.
Amorphous Layers#
The AmorphousLayer must be initialized with an id, name, thickness, and density.
All other properties are optional and must be set to carry out the according simulations.
amorph_Fe = ud.AmorphousLayer('amorph_Fe', 'amorph_Fe',
20*u.nm, 7.874*u.g/u.cm**3)
# print the layer properties
print(amorph_Fe)
Amorphous layer with the following properties
======================== ===============
parameter value
======================== ===============
id amorph_Fe
name amorph_Fe
thickness 20.0 nm
area 0.01 nm²
volume 0.2 nm³
mass 1.5748×10⁻²⁴ kg
mass per unit area 1.5748×10⁻²⁴ kg
density 7.874×10³ kg/m³
roughness 0.0 nm
Debye Waller Factor 0 m²
sound velocity 0.0 m/s
spring constant [0.0] kg/s²
phonon damping 0.0 kg/s
opt. pen. depth 0.0 nm
opt. refractive index 0
opt. ref. index/strain 0
thermal conduct. 0 W/(m K)
linear thermal expansion 0
heat capacity 0 J/(kg K)
subsystem coupling 0 W/m³
no atom set
======================== ===============
The physical properties can be also given during initialization using a dict:
params = {
'opt_pen_depth': 10*u.nm,
'sound_vel': 5*(u.nm/u.ps),
}
amorph_Cr = ud.AmorphousLayer('amorph_Cr', 'amorph_Cr',
40*u.nm, 7.14*u.g/u.cm**3, atom=Cr, **params)
# print the layer properties
print(amorph_Cr)
Amorphous layer with the following properties
======================== ===========================
parameter value
======================== ===========================
id amorph_Cr
name amorph_Cr
thickness 40.0 nm
area 0.01 nm²
volume 0.4 nm³
mass 2.856×10⁻²⁴ kg
mass per unit area 2.856×10⁻²⁴ kg
density 7.14×10³ kg/m³
roughness 0.0 nm
Debye Waller Factor 0 m²
sound velocity 5×10³ m/s
spring constant [0.04462500000000001] kg/s²
phonon damping 0.0 kg/s
opt. pen. depth 9.999999999999998 nm
opt. refractive index 0
opt. ref. index/strain 0
thermal conduct. 0 W/(m K)
linear thermal expansion 0
heat capacity 0 J/(kg K)
subsystem coupling 0 W/m³
atom Chromium
magnetization
amplitude 0
phi [°] 0.0 deg
gamma [°] 0.0 deg
======================== ===========================
Unit Cells#
The UnitCell requires an id, name, and c_axis upon initialization.
Multiple atoms can be added to relative positions along the c-Axis in the 1D UnitCell.
Note that all temperature-dependent properties can be given either as scalar (constant) value or as string that represents a temperature-dependent lambda-function:
# c-axis lattice constants of the two layers
c_STO_sub = 3.905*u.angstrom
c_SRO = 3.94897*u.angstrom
# sound velocities [nm/ps] of the two layers
sv_SRO = 6.312*u.nm/u.ps
sv_STO = 7.800*u.nm/u.ps
# SRO layer
prop_SRO = {}
prop_SRO['a_axis'] = c_STO_sub # a-Axis
prop_SRO['b_axis'] = c_STO_sub # b-Axis
prop_SRO['deb_Wa l_Fac'] = 0 # Debye-Waller factor
prop_SRO['sound_ve l'] = sv_SRO # sound velocity
prop_SRO['opt_ref_index'] = 2.44+4.32j
prop_SRO['therm_ond'] = 5.72*u.W/(u.m*u.K) # heat conductivity
prop_SRO['lin_therm_ exp'] = 1.03e-5 # linear thermal expansion
prop_SRO['heat_capacity'] = '455.2 + 0.112*T - 2.1935e6/T**2' # [J/kg K]
SRO = ud.UnitCell('SRO', 'Strontium Ruthenate', c_SRO, **prop_SRO)
SRO.add_atom(O, 0)
SRO.add_atom(Sr, 0)
SRO.add_atom(O, 0.5)
SRO.add_atom(O, 0.5)
SRO.add_atom(Ru, 0.5)
print(SRO)
Unit Cell with the following properties
======================== =========================================
parameter value
======================== =========================================
id SRO
name Strontium Ruthenate
a-axis 0.3905 nm
b-axis 0.3905 nm
c-axis 0.3949 nm
area 0.1525 nm²
volume 0.06022 nm³
mass 3.93×10⁻²⁵ kg
mass per unit area 2.577×10⁻²⁶ kg
area 0.1525 nm²
volume 0.06022 nm³
mass 3.930300027032341×10⁻²⁵ kg
mass per unit area 2.5774107046400294×10⁻²⁶ kg
density 6.527×10³ kg/m³
roughness 0.0 nm
Debye Waller Factor 0 m²
sound velocity 0.0 m/s
spring constant [0.0] kg/s²
phonon damping 0.0 kg/s
opt. pen. depth 0.0 nm
opt. refractive index (2.44+4.32j)
opt. ref. index/strain 0
thermal conduct. 0 W/(m K)
linear thermal expansion 0
heat capacity 455.2 + 0.112*T - 2.1935e6/T**2 J/(kg K)
subsystem coupling 0 W/m³
======================== =========================================
5 Constituents:
========= ========== =================== ======= =========== ========= ===========
atom position position function magn. amplitude phi [°] gamma [°]
========= ========== =================== ======= =========== ========= ===========
Oxygen 0 0 0 0 0
Strontium 0 0 0 0 0
Oxygen 0.5 0.5 0 0 0
Oxygen 0.5 0.5 0 0 0
Ruthenium 0.5 0.5 0 0 0
========= ========== =================== ======= =========== ========= ===========
Non-Linear Strain Dependence#
In general the position of each atom in a unit cell depends linearly from an external strain. In some cases this linear behavior has to be altered. This can be easily achieved by providing a string representation of a strain-dependent lambda-function for the atom position when the atom is added to the unit cell.
# STO substrate
prop_STO_sub = {}
prop_STO_sub['a_axis'] = c_STO_sub # a-Axis
prop_STO_sub['b_axis'] = c_STO_sub # b-Axis
prop_STO_sub['deb_Wal_Fac'] = 0 # Debye-Waller factor
prop_STO_sub['sound_vel'] = sv_STO # sound velocity
prop_STO_sub['opt_ref_index'] = 2.1+0j
prop_STO_sub['therm_cond'] = 12*u.W/(u.m *u.K) # heat conductivity
prop_STO_sub['lin_therm_exp'] = 1e-5 # linear thermal expansion
prop_STO_sub['heat_capacity'] = '733.73 + 0.0248*T - 6.531e6/T**2' # [J/kg K]
STO_sub = ud.UnitCell('STOsub', 'Strontium Titanate Substrate',
c_STO_sub, **prop_STO_sub)
STO_sub.add_atom(O, '0.1*(s**2+1)')
STO_sub.add_atom(Sr, 0)
STO_sub.add_atom(O, 0.5)
STO_sub.add_atom(O, 0.5)
STO_sub.add_atom(Ti, 0.5)
print(STO_sub)
Unit Cell with the following properties
======================== =========================================
parameter value
======================== =========================================
id STOsub
name Strontium Titanate Substrate
a-axis 0.3905 nm
b-axis 0.3905 nm
c-axis 0.3905 nm
area 0.1525 nm²
volume 0.05955 nm³
mass 3.047×10⁻²⁵ kg
mass per unit area 1.998×10⁻²⁶ kg
area 0.1525 nm²
volume 0.05955 nm³
mass 3.046843427429143×10⁻²⁵ kg
mass per unit area 1.9980578610298977×10⁻²⁶ kg
density 5.117×10³ kg/m³
roughness 0.0 nm
Debye Waller Factor 0 m²
sound velocity 7.8×10³ m/s
spring constant [7.971777885147345] kg/s²
phonon damping 0.0 kg/s
opt. pen. depth 0.0 nm
opt. refractive index (2.1+0j)
opt. ref. index/strain 0
thermal conduct. 12.0 W/(m K)
linear thermal expansion 1e-05
heat capacity 733.73 + 0.0248*T - 6.531e6/T**2 J/(kg K)
subsystem coupling 0 W/m³
======================== =========================================
5 Constituents:
========= ========== =================== ======= =========== ========= ===========
atom position position function magn. amplitude phi [°] gamma [°]
========= ========== =================== ======= =========== ========= ===========
Strontium 0 0 0 0 0
Oxygen 0.1 0.1*(s**2+1) 0 0 0
Oxygen 0.5 0.5 0 0 0
Oxygen 0.5 0.5 0 0 0
Titanium 0.5 0.5 0 0 0
========= ========== =================== ======= =========== ========= ===========
A simple visualization is also available:
SRO.visualize()
Structure#
The AmorphousLayer and UnitCell can now be added to an actual structure:
S = ud.Structure('Single Layer')
S.add_sub_structure(SRO, 100) # add 100 layers of SRO to sample
S.add_sub_structure(STO_sub, 1000) # add 1000 layers of STO substrate
print(S)
S.visualize()
Structure properties:
Name : Single Layer
Thickness : 429.99 nanometer
Roughness : 0.00 nanometer
----
100 times Strontium Ruthenate: 39.49 nanometer
1000 times Strontium Titanate Substrate: 390.50 nanometer
----
no substrate
There are various methods available to determine specific properties of a Structure, please also refer to the API documentation for more details:
[d_start, d_end, d_mid] = S.get_distances_of_layers()
K = S.get_number_of_sub_structures()
L = S.get_number_of_unique_layers()
M = S.get_number_of_layers()
P = S.get_all_positions_per_unique_layer()
I = S.get_distances_of_interfaces()
c_axis = S.get_layer_property_vector('c_axis')
The Structure class also allows to nest multiple substructures in order to build more complex samples easily:
S2 = ud.Structure('Super Lattice')
# define a single double layer
DL = ud.Structure('Double Layer')
DL.add_sub_structure(SRO, 15) # add 15 layers of SRO
DL.add_sub_structure(STO_sub, 20) # add 20 layers of STO substrate
# add the double layer to the super lattice
S2.add_sub_structure(DL, 10) # add 10 double layers to super lattice
S2.add_sub_structure(STO_sub, 500) # add 500 layers of STO substrate
print(S2)
S2.visualize()
Structure properties:
Name : Super Lattice
Thickness : 332.58 nanometer
Roughness : 0.00 nanometer
----
sub-structure 10 times:
Structure properties:
Name : Double Layer
Thickness : 13.73 nanometer
Roughness : 0.00 nanometer
----
15 times Strontium Ruthenate: 5.92 nanometer
20 times Strontium Titanate Substrate: 7.81 nanometer
----
no substrate
500 times Strontium Titanate Substrate: 195.25 nanometer
----
no substrate
There a few more interfaces than before (includes also the top and bottom interface):
I2 = S2.get_distances_of_interfaces();
print(len(I2))
21
Static Substrate#
Mainly for X-ray scattering simulations it might be helpful to add a static substrate to the structure, which is not included in the dynamic simulations of heat, phonons, and magnetization.
Hence the simulation time can be kept short while the scattering result include thick substrate contributions.
substrate = ud.Structure('Static Substrate')
substrate.add_sub_structure(STO_sub, 1000000)
S2.add_substrate(substrate)
print(S2)
S2.visualize()
Structure properties:
Name : Super Lattice
Thickness : 332.58 nanometer
Roughness : 0.00 nanometer
----
sub-structure 10 times:
Structure properties:
Name : Double Layer
Thickness : 13.73 nanometer
Roughness : 0.00 nanometer
----
15 times Strontium Ruthenate: 5.92 nanometer
20 times Strontium Titanate Substrate: 7.81 nanometer
----
no substrate
500 times Strontium Titanate Substrate: 195.25 nanometer
----
Substrate:
----
1000000 times Strontium Titanate Substrate: 390500.00 nanometer