Imaging and structure analysis of ferroelectric domains, domain walls, and vortices by scanning electron diffraction#
Ursula Ludacka1, Jiali He1, Shuyu Qin3,4, Manuel Zahn1,5, Emil Frang Christiansen2, Kasper A. Hunnestad1, Zewu Yan6,7, Edith Bourret7, István Kézsmárki5, Antonius T. J. van Helvoort2, Joshua Agar3,4, Dennis Meier1
Email: dennis.meier@ntnu.no, jca92@drexel.edu
1 Department of Materials Science and Engineering, NTNU Norwegian University of Science and Technology, Trondheim, Norway
2 Department of Physics, NTNU Norwegian University of Science and Technology, Trondheim, Norway
3 Department of Materials Science and Engineering, Lehigh University, Bethlehem, USA
4 Department of Mechanical Engineering and Mechanics, Drexel University, Philadelphia, USA
5 Experimental Physics V, University of Augsburg, Augsburg, Germany
6 Department of Physics, ETH Zurich, Zürich, Switzerland.
7 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, USA.
Abstract#
Direct electron detectors in scanning transmission electron microscopy give unprecedented possibilities for structure analysis at the nanoscale. In electronic and quantum materials, this new capability gives access to, for example, emergent chiral structures and symmetry-breaking distortions that underpin functional properties. Quantifying nanoscale structural features with statistical significance, however, is complicated by the subtleties of dynamic diffraction and coexisting contrast mechanisms, which often results in low signal-to-noise and the superposition of multiple signals that are challenging to deconvolute. Here we apply scanning electron diffraction to explore local polar distortions in the uniaxial ferroelectric Er(Mn,Ti)O3. Using a custom-designed convolutional autoencoder with bespoke regularization, we demonstrate that subtle variations in the scattering signatures of ferroelectric domains, domain walls, and vortex textures can readily be disentangled with statistical significance and separated from extrinsic contributions due to, e.g., variations in specimen thickness or bending. The work demonstrates a pathway to quantitatively measure symmetry-breaking distortions across large areas, mapping structural changes at interfaces and topological structures with nanoscale spatial resolution.
Imports and settings#
# !pip install m3_learning
import matplotlib.pyplot as plt
import numpy as np
import hyperspy.api as hs
import torch
from m3_learning.nn.random import random_seed
from m3_learning.nn.STEM_AE.Viz import Viz
from m3_learning.nn.STEM_AE.STEM_AE import ConvAutoencoder
from m3_learning.viz.style import set_style
from m3_learning.viz.printing import printer
from m3_learning.viz.Movies import make_movie
from m3_learning.nn.STEM_AE.Dataset import STEM_Dataset
from m3_learning.util.file_IO import download_and_unzip
# Specify the filename and the path to save the file
save_path = './../../../Data/2023_STEM_AE'
fig_path = save_path.replace("Data", "Figures") + '/'
# builds the printer object
printing = printer(basepath=fig_path)
# Set the style of the plots
set_style("printing")
# Set the random seed for reproducibility
random_seed(seed=42)
%matplotlib inline
printing set for seaborn
Pytorch seed was set to 42
Numpy seed was set to 42
tensorflow seed was set to 42
# list of files to download from zenodo
files = [
"11.1_log_epoch_0017_L1loss_0.00181_l1coef_0.0050_lr_0.00009_contrasPara_0.000050_contrasLoss_0.00020_maxiPara_0.00070_maxiLoss_0.0019_trainloss_0.2289.pkl",
"11.23_log_epoch_0376_coef_1e-05_trainloss_0.1362.pkl",
"256p_3w_step1k4_a5_cl12_50ms_nmf_4-001.hspy",
"4.5_log_epoch_0285_L1loss_0.0001__lr_0.00003_trainloss_0.2279.pkl",
"SED_256x256x12_30x30_alpha3_NBD_spot0p5nm_CL12cm_vortex.hspy",
]
# downloads all the files
for file in files:
# Download the data file from Zenodo
url = f"https://zenodo.org/record/7837986/files/{file}?download=1"
# download the file
download_and_unzip(file, url, save_path)
Using files already downloaded
Using files already downloaded
Using files already downloaded
Using files already downloaded
Using files already downloaded
Domains#
Loads the Dataset#
# loads the dataset
# this object will also compute the log of the dataset
# this is available as self.log_data
domain = STEM_Dataset(
data_path=f"{save_path}/256p_3w_step1k4_a5_cl12_50ms_nmf_4-001.hspy"
)
WARNING:hyperspy.io:This file contains a signal provided by the pyxem Python package that is not currently installed. The signal will be loaded into a generic HyperSpy signal. Consider installing pyxem to load this dataset into its original signal class.
WARNING:hyperspy.io:`signal_type='electron_diffraction'` not understood. See `hs.print_known_signal_types()` for a list of installed signal types or https://github.com/hyperspy/hyperspy-extensions-list for the list of all hyperspy extensions providing signals.
/home/ferroelectric/anaconda3/envs/STEM/lib/python3.9/site-packages/hyperspy/misc/utils.py:471: VisibleDeprecationWarning: Use of the `binned` attribute in metadata is going to be deprecated in v2.0. Set the `axis.is_binned` attribute instead.
warnings.warn(
/home/ferroelectric/anaconda3/envs/STEM/lib/python3.9/site-packages/hyperspy/io.py:572: VisibleDeprecationWarning: Loading old file version. The binned attribute has been moved from metadata.Signal to axis.is_binned. Setting this attribute for all signal axes instead.
warnings.warn('Loading old file version. The binned attribute '
Instantiates the Visualizer#
# instantiates the visualization class
viz = Viz(printer=printing, labelfigs_=True)
viz.scalebar_ = {"width": 250, "scale length": 100, "units": "nm", "channel": -1}
Visualizes the Raw Data#
bright_field_ = (117, 135, 119, 135)
dark_field_ = (102, 117, 114, 130)
filename = "STEM_raw_and_virtual_domain"
viz.STEM_raw_and_virtual(
domain,
bright_field_=bright_field_,
dark_field_=dark_field_,
scalebar_=True,
filename=filename,
shape_=domain.data.shape,
)
./../../../Figures/2023_STEM_AE/STEM_raw_and_virtual_domain.png
./../../../Figures/2023_STEM_AE/STEM_raw_and_virtual_domain.svg
Builds the Neural Network Model#
encoder_step_size = [256, 256]
pooling_list = [4, 4, 2]
decoder_step_size = [8, 8]
upsampling_list = [2, 4, 4]
embedding_size = 32
conv_size = 128
# set device to load model
device = "cpu"
if torch.cuda.is_available():
device = "cuda:1"
model = ConvAutoencoder(
encoder_step_size=encoder_step_size,
pooling_list=pooling_list,
decoder_step_size=decoder_step_size,
upsampling_list=upsampling_list,
embedding_size=embedding_size,
conv_size=conv_size,
device=device,
learning_rate=3e-5,
)
Trains or Loads Pretrained Weights#
# set retrain parameter
train = False
if train:
model.Train(
domain.log_data,
coef_1=1e-5,
epochs=225,
with_scheduler=False,
folder_path="Domain_Weights",
batch_size=256,
)
else:
model.load_weights(
f"{save_path}/11.23_log_epoch_0376_coef_1e-05_trainloss_0.1362.pkl"
)
Computes the Embeddings#
embedding = model.get_embedding(domain.log_data)
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Visualizes the Embeddings#
viz.channels = None
viz.scalebar_ = {"width": 250, "scale length": 100, "units": "nm"}
viz.model = model
viz.embeddings(
mod=4,
scalebar_=viz.scalebar_,
name="domain wall",
shape_=domain.data.shape,
figsize=(5, 8),
)
./../../../Figures/2023_STEM_AE/domain wall_embedding_maps.png
./../../../Figures/2023_STEM_AE/domain wall_embedding_maps.svg
Generates Data from the Latent Space#
model.embedding = embedding
viz.model = model
viz.channels = [20, 29]
ranges = [[0, 0.018], [0, 0.022]]
folder_name = "Generator_of_Figure_4_a_d"
viz.generator_images(
ranges=ranges,
folder_name=folder_name,
figsize=(5, 1.25),
cbar_number_format="%d",
shape_=domain.data.shape,
divider_=False,
)
Show code cell output
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Builds a Movie#
# Builds a Movie
folder = f"{fig_path}/generator_images_Generator_of_Figure_4_a_d"
output_folder = "./movies/"
# function to make the movie
make_movie("Domain Generator", folder, output_folder, "png", 3)
Vortex Domains#
Loads the Dataset#
# loads the dataset
# this object will also compute the log of the dataset
# this is available as self.log_data
vortex = STEM_Dataset(
data_path=f"{save_path}/SED_256x256x12_30x30_alpha3_NBD_spot0p5nm_CL12cm_vortex.hspy"
)
WARNING:hyperspy.io:This file contains a signal provided by the pyxem Python package that is not currently installed. The signal will be loaded into a generic HyperSpy signal. Consider installing pyxem to load this dataset into its original signal class.
WARNING:hyperspy.io:`signal_type='electron_diffraction'` not understood. See `hs.print_known_signal_types()` for a list of installed signal types or https://github.com/hyperspy/hyperspy-extensions-list for the list of all hyperspy extensions providing signals.
Visualizes the Raw Data#
bright_field_ = (150, 162, 152, 165)
dark_field_ = (106, 123, 110, 125)
filename = "STEM_raw_and_virtual_vortex_domains"
viz.scalebar_ = {"width": 291, "scale length": 100, "units": "nm"}
viz.STEM_raw_and_virtual(
vortex,
bright_field_=bright_field_,
dark_field_=dark_field_,
scalebar_=True,
filename=filename,
shape_=vortex.data.shape,
)
./../../../Figures/2023_STEM_AE/STEM_raw_and_virtual_vortex_domains.png
./../../../Figures/2023_STEM_AE/STEM_raw_and_virtual_vortex_domains.svg
Builds the Neural Network Model#
encoder_step_size = [256, 256]
pooling_list = [4, 4, 2]
decoder_step_size = [8, 8]
upsampling_list = [2, 4, 4]
embedding_size = 32
conv_size = 128
# set device to load model
device = "cpu"
if torch.cuda.is_available():
device = "cuda:1"
model = ConvAutoencoder(
encoder_step_size=encoder_step_size,
pooling_list=pooling_list,
decoder_step_size=decoder_step_size,
upsampling_list=upsampling_list,
embedding_size=embedding_size,
conv_size=conv_size,
device=device,
learning_rate=3e-5,
)
Trains or Loads Pretrained Weights#
# set retrain parameter
train = False
if train:
model.Train(
vortex.log_data,
coef_1=1e-5,
epochs=225,
with_scheduler=False,
folder_path="vortex_domains_Weights",
batch_size=256,
)
model.Train(
vortex.log_data,
max_learning_rate=5e-5,
coef_1=5e-4,
epochs=286,
epoch_=224,
with_scheduler=True,
folder_path="vortex_domains_Weights",
batch_size=256,
)
else:
model.load_weights(
f"{save_path}/4.5_log_epoch_0285_L1loss_0.0001__lr_0.00003_trainloss_0.2279.pkl"
)
Computes the Embeddings#
embedding = model.get_embedding(vortex.log_data)
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Visualizes the Embeddings#
viz.model = model
viz.channels = None
viz.scalebar_ = {"width": 291, "scale length": 100, "units": "nm"}
viz.embeddings(
mod=4,
scalebar_=viz.scalebar_,
figsize=(5, 8),
name="vortex_domains",
shape_=vortex.data.shape,
)
./../../../Figures/2023_STEM_AE/vortex_domains_embedding_maps.png
./../../../Figures/2023_STEM_AE/vortex_domains_embedding_maps.svg
Generates Data from the Latent Space#
model.embedding = embedding
viz.model = model
viz.channels = [13, 31]
ranges = [[0, 0.0225], [0, 0.03]]
folder_name = "Generator_of_Figure_4_vortex_domains"
viz.generator_images(
ranges=ranges,
folder_name=folder_name,
figsize=(5, 1.25),
cbar_number_format="%d",
shape_=vortex.data.shape,
divider_=False,
)
Show code cell output
0%| | 0/200 [00:00<?, ?it/s]
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./../../../Figures/2023_STEM_AE/generator_images_Generator_of_Figure_4_vortex_domains/0008_maps.svg
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./../../../Figures/2023_STEM_AE/generator_images_Generator_of_Figure_4_vortex_domains/0009_maps.svg
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./../../../Figures/2023_STEM_AE/generator_images_Generator_of_Figure_4_vortex_domains/0010_maps.png
./../../../Figures/2023_STEM_AE/generator_images_Generator_of_Figure_4_vortex_domains/0010_maps.svg
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./../../../Figures/2023_STEM_AE/generator_images_Generator_of_Figure_4_vortex_domains/0011_maps.png
./../../../Figures/2023_STEM_AE/generator_images_Generator_of_Figure_4_vortex_domains/0011_maps.svg
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./../../../Figures/2023_STEM_AE/generator_images_Generator_of_Figure_4_vortex_domains/0012_maps.png
./../../../Figures/2023_STEM_AE/generator_images_Generator_of_Figure_4_vortex_domains/0012_maps.svg
6%|▋ | 13/200 [00:13<03:06, 1.00it/s]
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./../../../Figures/2023_STEM_AE/generator_images_Generator_of_Figure_4_vortex_domains/0013_maps.svg
7%|▋ | 14/200 [00:14<03:05, 1.00it/s]
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./../../../Figures/2023_STEM_AE/generator_images_Generator_of_Figure_4_vortex_domains/0014_maps.svg
8%|▊ | 15/200 [00:15<03:04, 1.00it/s]
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10%|▉ | 19/200 [00:20<04:47, 1.59s/it]
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10%|█ | 20/200 [00:21<04:14, 1.41s/it]
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Builds the Movie#
# Builds a Movie
folder = f"{fig_path}/generator_images_Generator_of_Figure_4_vortex_domains/"
output_folder = "./movies/"
# function to make the movie
make_movie("Vortex Domains Generator", folder, output_folder, "png", 3)
Vortex Domain Walls#
Builds the Neural Network Model#
encoder_step_size = [256, 256]
pooling_list = [4, 4, 2]
decoder_step_size = [8, 8]
upsampling_list = [2, 4, 4]
embedding_size = 32
conv_size = 128
# set device to load model
device = "cpu"
if torch.cuda.is_available():
device = "cuda:1"
model = ConvAutoencoder(
encoder_step_size=encoder_step_size,
pooling_list=pooling_list,
decoder_step_size=decoder_step_size,
upsampling_list=upsampling_list,
embedding_size=embedding_size,
conv_size=conv_size,
device=device,
learning_rate=3e-5,
)
Trains or Loads Pretrained Weights#
In this model we have additional custom regularization to the embedding to help with disentangling the vortices from the walls.
We have an l1 activity regularization:
\[ \sum\_{i=1}^{n} |a_i| \]
We have a contrastive regularization:
\[ L = \frac{\lambda*1}{2N} \sum\limits*{i=1}^{N} \left[\left(\sum\limits_{j=1, j\neq i}^{N} \frac{latent_i \cdot latent_j}{\left\lVert latent_i \right\rVert \left\lVert latent_j \right\rVert}\right] - 1\right)\]
Divergence regularization:
\[L = \frac{1}{B} \sum\limits_{i=1}^{B} \left(\frac{\lambda_2}{2} \sum\limits_{j=1}^{n} \sum\limits_{k=j+1}^{n} \left| latent_{i,j} - latent_{i,k} \right|\right)\]
# set retrain parameter
train = False
if train:
model.Train(
vortex.log_data,
coef_0=5e-3,
coef_1=5e-5,
coef_2=2e-4,
epochs=18,
with_scheduler=False,
folder_path="Vortex_walls_Weights",
batch_size=256,
)
else:
model.load_weights(
f"{save_path}/11.1_log_epoch_0017_L1loss_0.00181_l1coef_0.0050_lr_0.00009_contrasPara_0.000050_contrasLoss_0.00020_maxiPara_0.00070_maxiLoss_0.0019_trainloss_0.2289.pkl"
)
Computes the Embeddings#
embedding = model.get_embedding(vortex.log_data)
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Visualizes the Embeddings#
viz.channels = None
viz.model = model
viz.scalebar_ = {"width": 291, "scale length": 100, "units": "nm"}
viz.embeddings(
mod=4,
scalebar_=viz.scalebar_,
figsize=(5, 8),
name="Vortex_walls",
shape_=vortex.data.shape,
)
./../../../Figures/2023_STEM_AE/Vortex_walls_embedding_maps.png
./../../../Figures/2023_STEM_AE/Vortex_walls_embedding_maps.svg
Generates Data from the Latent Space#
model.embedding = embedding
viz.model = model
viz.channels = [0, 2]
ranges = [[0, 0.025], [0, 0.04]]
folder_name = "Generator_of_Figure_4_e_f_vortex_walls"
viz.generator_images(
ranges=ranges,
folder_name=folder_name,
figsize=(5, 1.25),
cbar_number_format="%d",
shape_=vortex.data.shape,
divider_=False,
)
Show code cell output
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Builds a Movie#
folder = f"{fig_path}/generator_images_Generator_of_Figure_4_e_f_vortex_walls/"
output_folder = "./movies/"
# function to make the movie
make_movie("Vortex Wall Generator", folder, output_folder, "png", 3)