[1]:

import numpy as np
import matplotlib.pyplot as plt

from abtem.ionization import SubshellTransitions, TransitionPotential, EELSDetector
from abtem import SMatrix, Potential, GridScan

from ase import units


# Core energy-loss with SrTiO3 (early experimental version)¶

We calculate an “energy-filtered” STEM image of SrTiO$$_3$$ targeting oxygen, specifically we target its K-edge (i.e. the 1s subshell; quantum numbers $$(n, \ell) = (1, 0)$$)). In a hypothetical experiment, this would be roughly equivalent to setting your energy filter to 456 eV.

The following code calculates projected overlap integrals following to Dwyer et al. (see https://doi.org/10.1016/j.ultramic.2005.03.005 or https://doi.org/10.1103/PhysRevB.57.3273), dynamical scattering following Brown et al. (https://doi.org/10.1103/PhysRevResearch.1.033186) and uses the density functional theory code GPAW for calculating wave functions. Please see our citation guide if you use this code in a publication.

[3]:

Z = 8 # atomic number
n = 1 # principal quantum number
l = 0 # azimuthal quantum number
xc = 'PBE' # exchange-correlation functional

O_transitions = SubshellTransitions(Z = Z, n = n, l = l, xc = 'PBE')

print('bound electron configuration:', O_transitions.bound_configuration)
print('ionic electron configuration:', O_transitions.excited_configuration)

bound electron configuration: 1s2 2s2 2p4
ionic electron configuration: 1s1 2s2 2p4


Applying the selection rules

$\Delta \ell = \pm 1 \quad\mathrm{and}\quad \Delta m = 0, \pm 1$

we obtain the following dipole transitions.

[4]:

for bound_state, continuum_state in O_transitions.get_transition_quantum_numbers():
print(f'(l, ml) = {bound_state} -> {continuum_state}')

(l, ml) = (0, 0) -> (1, -1)
(l, ml) = (0, 0) -> (1, 0)
(l, ml) = (0, 0) -> (1, 1)


For a fast electron with an energy of 100 keV and a specified grid, we obtain the following transition potentials; the code will run an all-electron density function theory calculation using GPAW.

[5]:

atomic_transition_potentials = O_transitions.get_transition_potentials(extent = 5,
gpts = 256,
energy = 100e3)

[6]:

fig, axes = plt.subplots(1,3, figsize = (10,5))

for ax, atomic_transition_potential in zip(axes, atomic_transition_potentials):
atomic_transition_potential.show(ax = ax, title = str(atomic_transition_potential))


We have created transition potentials for single atoms, now we need to put them together in a multislice simulation.

We import our Atoms as usual and make a 2$:nbsphinx-math:times$2 supercell.

[7]:

atoms = read('data/srtio3_100.cif') * (2,2,1)
atoms.center(axis = 2)

from abtem import show_atoms
fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize = (10,2))

show_atoms(atoms, ax = ax1, title = 'Beam view')
show_atoms(atoms, ax = ax2, plane = 'yz', title = 'Side view')
show_atoms(atoms[atoms.numbers == 8], ax = ax3, plane = 'xy', title = 'Beam view (Oxygen)');


Next, we create a TransitionPotential. The overlap integrals depend on the incoming energy, hence we have to provide the acceleration voltage. We also provide a sampling in order to show the potential below.

[8]:

transition_potential = TransitionPotential(O_transitions,
atoms = atoms,
sampling = .05,
energy = 100e3,
slice_thickness = 2)


We can show the projected intensity of the transition potential.

[9]:

transition_potential.show()


Finally, we can do a full “energy-filtered” STEM simulation targeting the oxygen K edge.

We create a scattering matrix SMatrix as usual (note: interpolation is not yet implemented!), and an EELSDetector (interpolation is implemented), as well as a standard electrotrostatic potential.

We also create a new TransitionPotential, which will automatically adopt the appropriate atoms and energy from the other simulation objects.

[10]:

S = SMatrix(energy = 100e3, semiangle_cutoff = 25) # interpolation not implemented!

detector = EELSDetector(collection_angle = 100, interpolation = 4)

potential = Potential(atoms, sampling = .05, slice_thickness = .5,
projection = 'infinite', parametrization = 'kirkland')

transition_potential = TransitionPotential(O_transitions)

scan = GridScan((0,0), potential.extent, sampling = .9*S.ctf.nyquist_sampling)

measurement = S.coreloss_scan(scan, detector, potential, transition_potential)


We show the final (tiled, interpolated) energy-filtered image below.

[11]:

measurement.tile((2,2)).interpolate(.02).show(figsize = (6,6));


We further target the K-edge of oxygen as above and the L$$_{23}$$-edge of titanium and strontium. We use the PBE functional to calculate the transitions.

[12]:

O_transitions = SubshellTransitions(Z = 8, n = 1, l = 0, xc = 'PBE')
Ti_transitions = SubshellTransitions(Z = 22, n = 2, l = 1, xc = 'PBE')
Sr_transitions = SubshellTransitions(Z = 38, n = 2, l = 1, xc = 'PBE')

transitions = [O_transitions, Ti_transitions, Sr_transitions]

transition_potential = TransitionPotential(transitions)

[13]:

print('Oxygen:')
for bound_state, continuum_state in O_transitions.get_transition_quantum_numbers():
print(f'(l, ml) = {bound_state} -> {continuum_state}')
print('Titanium:')
for bound_state, continuum_state in Ti_transitions.get_transition_quantum_numbers():
print(f'(l, ml) = {bound_state} -> {continuum_state}')
print('Strontium:')
for bound_state, continuum_state in Sr_transitions.get_transition_quantum_numbers():
print(f'(l, ml) = {bound_state} -> {continuum_state}')

Oxygen:
(l, ml) = (0, 0) -> (1, -1)
(l, ml) = (0, 0) -> (1, 0)
(l, ml) = (0, 0) -> (1, 1)
Titanium:
(l, ml) = (1, -1) -> (0, 0)
(l, ml) = (1, -1) -> (2, -2)
(l, ml) = (1, -1) -> (2, -1)
(l, ml) = (1, -1) -> (2, 0)
(l, ml) = (1, 0) -> (0, 0)
(l, ml) = (1, 0) -> (2, -1)
(l, ml) = (1, 0) -> (2, 0)
(l, ml) = (1, 0) -> (2, 1)
(l, ml) = (1, 1) -> (0, 0)
(l, ml) = (1, 1) -> (2, 0)
(l, ml) = (1, 1) -> (2, 1)
(l, ml) = (1, 1) -> (2, 2)
Strontium:
(l, ml) = (1, -1) -> (0, 0)
(l, ml) = (1, -1) -> (2, -2)
(l, ml) = (1, -1) -> (2, -1)
(l, ml) = (1, -1) -> (2, 0)
(l, ml) = (1, 0) -> (0, 0)
(l, ml) = (1, 0) -> (2, -1)
(l, ml) = (1, 0) -> (2, 0)
(l, ml) = (1, 0) -> (2, 1)
(l, ml) = (1, 1) -> (0, 0)
(l, ml) = (1, 1) -> (2, 0)
(l, ml) = (1, 1) -> (2, 1)
(l, ml) = (1, 1) -> (2, 2)

[14]:

measurements = S.coreloss_scan(scan, detector, potential, transition_potential)


By abTEM convention, the first dimensions always represent the scan or navigation dimensions. Hence, in our case the third dimension represents the subshell of an electron (or, experimentally, a specific energy loss).

[15]:

fig, (ax1, ax2, ax3)= plt.subplots(1, 3, figsize = (10,2.7))

measurements[0].tile((2, 2)).interpolate(.1).show(ax = ax1)
measurements[1].tile((2, 2)).interpolate(.1).show(ax = ax2)
measurements[2].tile((2, 2)).interpolate(.1).show(ax = ax3);

[16]:

fig, (ax1, ax2) = plt.subplots(1, 2, figsize = (10,4))

measurements[0].interpolate_line((0,0), (0, potential.extent[1]),
sampling = .1).show(ax = ax1, label = 'O')
measurements[1].interpolate_line((0,0), (0, potential.extent[1]),
sampling = .1).show(ax = ax1, label = 'Ti')
measurements[2].interpolate_line((0,0), (0, potential.extent[1]),
sampling = .1).show(ax = ax1, label = 'Sr')
ax1.legend()

measurements[0].interpolate_line((atoms[3].x, 0), (atoms[3].x,potential.extent[1]),
sampling = .1).show(ax = ax2, label = 'O')
measurements[1].interpolate_line((atoms[3].x, 0),(atoms[3].x, potential.extent[1]),
sampling = .1).show(ax = ax2, label = 'Ti')
measurements[2].interpolate_line((atoms[3].x, 0),(atoms[3].x,potential.extent[1]),
sampling = .1).show(ax = ax2, label = 'Sr')
ax2.legend();

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