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Submitted to Second AEQUITAS Workshop on Fairness and Bias in AI, Co-located
with ECAI 2024

[orcid=0000-0001-5307-997X<EMAIL_ADDRESS>url=https://fahimsikder.com/, ] [1]

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[1]Corresponding author.

# FairX: A comprehensive benchmarking tool for model analysis using fairness,
utility, and explainability

Md Fahim Sikder  Department of Computer and Information Science (IDA),
Linköping University, Sweden  Resmi Ramachandranpillai  Institute for
Experiential AI, Northeastern University, USA  Daniel de Leng  Fredrik Heintz

(2024)

###### Abstract

We present FairX, an open-source Python-based benchmarking tool designed for
the comprehensive analysis of models under the umbrella of fairness, utility,
and eXplainability (XAI). FairX enables users to train benchmarking bias-
removal models and evaluate their fairness using a wide array of fairness
metrics, data utility metrics, and generate explanations for model
predictions, all within a unified framework. Existing benchmarking tools do
not have the way to evaluate synthetic data generated from fair generative
models, also they do not have the support for training fair generative models
either. In FairX, we add fair generative models in the collection of our fair-
model library (pre-processing, in-processing, post-processing) and evaluation
metrics for evaluating the quality of synthetic fair data. This version of
FairX supports both tabular and image datasets. It also allows users to
provide their own custom datasets. The open-source FairX benchmarking package
is publicly available at https://github.com/fahim-sikder/FairX.

###### keywords:

Fair evaluation Benchmarking tool Synthetic data Data utility Explainability

##  1 Introduction

With the rapid development of artificial intelligence-based systems to aid us
in our daily lives, it is important for these systems to give outcomes that is
acceptable for all users, including—but not limited to—from demographic
perspective. Troublingly, as the available data is filled with human or
machine bias, models trained with these dataset often gives unfair outcome
towards some demographic [1]. It is therefore critical to mitigate bias in the
dataset and model. Over the years, researchers have used different techniques
to achieve this. These techniques can be roughly grouped in three families: 1)
_Pre-processing_ , i.e. where the dataset is processed in such a manner that
it produces less biased outcomes, before passing it to a model for training;
2) _In-processing_ , i.e. where the model learns the original data
distribution and shifts the data distribution to a fair distribution by adding
some constraints during the training process; and 3) _Post-processing_ , i.e.
where the model’s outcome is changed in such a manner that it gives fair
outcomes relative to protected attributes. The performance of these models or
datasets can be measured by the evaluation metrics that reflect both the
fairness and data utility. To ease up the work for training models and
evaluating them, researchers has developed benchmarking tool that bring the
training and evaluation in one framework. Recently, research on fair
generative models has found a lot of spotlight and measuring the quality of
the synthetic data is as crucial as evaluating fairness and data utlity.

Existing fairness-related benchmarking tools focus on creating benchmarks and
measuring their fairness on different datasets. For example, FairLearn [2] by
Microsoft contains several fair models and evaluation metrics for checking
fairness and data utility. AI Fairness 360 (AIF360) [3] by IBM also contains
fairness evaluation metrics and basic data utility measuring metrics. But both
of these frameworks lack the ability to train fair generative models and
measure the data utility for synthetic data. For synthetic fair data, it is
important to validate the quality of the generated data alongside measuring
the fairness and other data utilities. Explainability is an essential property
of fair models because it aids in making the model’s decision-making process
more transparent. These modules should therefore be included in such
benchmarking tools.

Figure 1: A High-level overview of FairX. An input dataset (possibly custom)
is fed to the FairX data loading module followed by a debiasing module and an
extensive evaluation module providing multi-faceted evaluations.

In this work, we present _FairX_ , an open-source modular fairness
benchmarking tool, available to use at https://github.com/fahim-sikder/FairX.
A high-level system overview is given in Figure 1. FairX contains data
processing techniques and benchmarking fairness models (incorporating pre-
processing, in-processing, and post-processing), including generative fair
models. We evaluate these models in terms of fairness, data utility. We also
add evaluation methods for synthetic fair data (Advanced Utility) to check the
quality of the generated samples. FairX supports both tabular and image data
and can plot feature importance for down-streaming task using explainable
algorithms.

The remainder of this paper is organised as follows. In Section 2 we discuss
some background information that will help the reader understand the rest of
the paper. We then present FairX in Section 3. Section 4 shows some fairness
results obtained by FairX for a number of datasets and models. Details on how
to use FairX are presented in Section 5. Finally, the paper looks ahead
towards future improvements in Section 6.

##  2 Background

In this section, we provide the necessary details to follow the paper.

###  2.1 Bias mitigation methods

A variety of bias mitigation methods have been proposed in the literature
based on data, training, and predictions. These methods can be broadly
categorized into three main approaches: pre-processing, in-processing, and
post-processing techniques.

#### Pre-processing.

These techniques involve altering the training data to resolve any potential
causes of biases before it is fed to the model. There are various techniques
in the literature such as disparate impact remover [4], data cleaning and
augmentation, and fair representation learning [5]. This involves balancing
the representation of different groups or generating synthetic data to augment
underrepresented groups, assigning weights to uphold some minority groups, and
transforming the data representation in a format that obscures protected
features while maintaining feature attributions.

#### In-processing.

This involves mitigating biases during training. The techniques involve
fairness constraints, adversarial de-biasing [6], and fairness-aware learning.
In fairness constraints training, a multi-objective optimization combining a
prediction loss and a fairness penalty will be used such as adding
regularization terms to the objective function that penalizes unfairness or
incorporating fairness metrics as part of the optimization process. In
adversarial de-biasing [6], adversarial training is used to reduce bias. The
model is trained to perform well on the primary classification/prediction
tasks while simultaneously trying to prevent an adversary from predicting the
protected features, thus forcing the model to learn less biased
representations.

#### Post-processing.

These methods are applied to the predictions of a classifier. Techniques such
as threshold adjustment, calibration [7], and Reject Option Classifications
[8] fall under this category. In threshold adjustment the decision thresholds
of a trained model are adjusted to ensure that the outcomes meet the chosen
fairness metric. Calibration [7] ensures that the predicted probabilities
maintain the true likelihood of outcomes equally across different demographic
groups. Techniques like equalized odds post-processing is used where the
model’s outputs are adjusted to satisfy fairness constraints. Reject Option-
Based Classification (ROC) [8] allows the model to prevent from making a
decision when the confidence is low, for the chosen sensitive attributes. This
can reduce the likelihood of biased or unfair decisions in uncertain
instances.

###  2.2 Evaluation metrics

To measure the performance of models or dataset, various evaluation methods
are being used. For evaluating fair model or checking the dataset for
potential bias, different kinds of fairness metrics exists. For example,
demographic parity checks if the decision from a down-streaming task is equal
for each class in sensitive attributes. Fairness through unawareness [9]
checks how the accuracy of down-stream task effects if no-sensitive attributes
is used during the training and prediction phase. Adding fairness constraints
to the models or datasets may change the data distributions and thereby affect
the performance of the dataset or models [10]. To check the data utility
performance, we commonly use Accuracy score, F1-score, Precision and Recall.
To evaluate the quality of the synthetic data researchers use,
$\alpha$-precision [11], $\beta$-recall [11]. Also to check, is the generative
model is truly generating new contents or not, the metrics authenticity [11]
is being used.

###  2.3 Comparison of existing benchmarking tools

Over the years researchers have developed various fairness benchmarking tools
which commonly include a dataset loader, different bias removal techniques and
evaluation metrics. Fairlearn [2] by Microsoft is one such benchmarking tool.
It has support for different algorithms for bias removal and measuring the
fairness of a model. AIF360 [3] by IBM is another benchmarking tool. It
supports a wide range of evaluation metrics (both for fairness and data
utility) and bias-removal algorithms (in-processing, pre-processing and post-
processing). Another example is Jurity [12]. It contains recommender system
evaluations, and various fairness and data utility functions. More recently,
in the area of generative models, there has been an increased interest in
generating fair data in the image, tabular and medical domains [13, 14, 15, 1,
16, 17]. But the aforementioned benchmarking tools do not contain these
models. Also, when evaluating models, other benchmarking tools, only measure
the fairness and data utility of the models itself. But evaluation methods for
generated data is needed. We need to verify the quality of the synthetic data.
We also need to verify the authenticity of the synthetic data, to show the
generative models are actually generating new content rather than just copying
the data itself. FairX is bridging this gap. We add support for evaluating
synthetic data and add generative models in our benchmarking tool. Table 1
shows the comparison among the models with FairX.

Table 1: Comparison of existing benchmarking tools with FairX over different
key areas of interests: Fairness Evaluation; Synthetic Data Evaluation; Model
Explainability; and Generative Fair Model Training.

Benchmarking | Fairness Evaluation | Synthetic Data | Explainability | Generative Model
---|---|---|---|---
Tools |  | Evaluation |  | Training
Fairlearn [2] | ✓ | ✗ | ✗ | ✗
AIF360 [18] | ✓ | ✗ | ✓ | ✗
Jurity [12] | ✓ | ✗ | ✗ | ✗
FairX (ours) | ✓ | ✓ | ✓ | ✓

##  3 FairX

In this section we present FairX in detail. FairX is built on three primary
modules, 1) the _Data Loading Module_ , 2) the _Bias-mitigating Techniques
Module_ , and 3) the _Evaluation Module_. The main pipeline (shown in Figure
1) works as follows. Given a dataset, FairX will pre-processed it in a way
that is compatible with the benchmarking model. Next the model will train
itself using the dataset. After the training the evaluation module will give
the results based on fairness, data utility and explain the outcome using
explainability.

###  3.1 Data loading module

The BaseDataClass handles the internal processing of datasets and make it
compatible with the bias-mitigating models that are present on our framework
as well as making it easier to handle for other bias-mitigating models that
are not present in this tool. This class contains different methods for
handling different kinds of data extension (CSV, and others). We add three
widely used tabular datasets (Adult-Income, COMPASS and Credit Card) and two
image datasets (Colored MNIST and Celeba) in the benchmarking tool, and we
plan to add more. Upon calling the BaseDataClass for a specific dataset, it
downloads the dataset from a predefined URL. Then it processes it based on
numerical and categorical features. The BaseDataClass provides methods to
normalize the dataset and is equipped with functionality for various encodings
(e.g. One-hot encoding, QuantileTransformer). It also has a dataset-splitting
function to split the dataset for training and testing purpose. We also added
functionality to prepare the dataset for explainability algorithms. Sample
usage of datasets are described in section 5, Listing 1.

#### Custom Dataset Loader.

Besides adding widely used benchmarking datasets for fair data research, we
also provide the option to use custom dataset. By using the CustomDataClass,
users can load their own dataset (CSV, TXT, etc.) and train the models. Users
need to specify the sensitive attributes and target attributes while using the
CustomDataClass. Pre-processing and other functionalities are also available
in this class, like in the BaseDataClass. We present sample usage of
CustomDataClass in Listing 4 of section 5.

###  3.2 Bias-mitigating techniques module

One of FairX’s main aims is to benchmark different bias-mitigation techniques
on various datasets. Over the years, different techniques have been proposed,
and we add models from these techniques to the tool. For the benchmarking
process, we use the same hyper-parameters used in their respective works. We
create a common format for all the bias-mitigation techniques to make it easy
for the users. For example, each bias-mitigation technique has its own class,
which has model.fit() function. This fit() function takes the dataset and
processes it (if needed for the specific model). For the generative models
(in-processing techniques), this function also generates synthetic data and
saves it as a Pandas dataframe. Sample usage of models is described in section
5, Listing 2.

#### Pre-processing.

We add the support for Correlation remover (CorrRemover in FairX) in the
benchmarking. Correlation Remover removes the correlation between the
sensitive attributes with other data features by using a linear transformation
while keeping as much information as possible. It is also possible to control
on how much correlation we want remove by using the remove_intensity parameter
while the value $1.0$ will result maximum correlation removal while $0.0$ will
do the opposite. We can access the pre-processing algorithm by using
fairx.models.preprocessing.

#### In-processing.

Most recent in-processing bias removal techniques are based on generative
models. And the fairness benchmarking tools we mentioned in this work does not
contain these models. One of our contribution of FairX is that, we add several
fair generative models, such as, TabFairGAN [16], Decaf [17] and Fairdisco
[1]. We can access the in-processing algorithm by using
fairx.models.inprocessing module. After training, these models will generate
and save the samples automatically.

#### Post-processing.

For the post-processing bias removal technique, We add Threshold Optimizer.
This technique operates on a classifier and improve the output of its based on
a fairness constraint. In this case, we use demographic_parity as a fairness
constraint to improve the outcome of the classifier. For using the post-
processing algorithm, we can use fairx.models.postprocessing module.

Table 2: Breakdown of FairX-supported features.

Dataset |  |  | Adult-Income, Compass, Credit-card
---
Colored MNIST (Image)
Celeba (Image)
| Pre-processing |  | Correlation Remover
---
Models | In-processing |  | TabFairGAN [16]
---
FairDisco [1]
Decaf [17]
| Post-processing | Threshold Optimizer
| Fairness |  | Demographic Parity Ratio (DPR)
---
Equilized Odds Ratio (EOR)
Fairness through Unawareness (FTU)
Metrics | Data Utility |  | AUROC, F1-score, Precision
---
Recall, Accuracy
| Synthetic Data Evaluation |  |  $\alpha$-precision [11]
---
$\beta$-recall [11]
Authenticity [11]
Plotting |  |  | PCA [19] & t-SNE [20] plots
---
Feature Importance
Fairness vs Accuracy
Intersectional Bias
Explainability |  |  | Explain prediction of a model
---
Feature Importance

###  3.3 Evaluation module

In FairX, we aim to evaluate the performance of model or dataset using wide
range of evaluation metrics. We evaluate in terms of fairness, data utility.
Other existing fairness benchmarking tools, lacks the capability to measure
the data quality of the synthetic data. It is necessary to check the data
quality of the synthetic data as well as the fairness criteria. Here, we
present the evaluation module FairX has and we use XGBoost as a classifier,
also we keep the option to use scikit-learn’s LogisticRegression.

#### Fairness Evaluation.

We create the FairnessUtils class to accommodate fairness evaluation metrics.
In this class, currently we add the support for checking the Demographic
Parity Ratio, Equalized Odds Ratio, Fairness Through Unawareness (FTU)
metrics. We also have plan to add more metrics over the time. Fairness metrics
can be accessed using the fairx.metrics.FairnessUtils module.

#### Data Utility.

Beside checking the fairness criteria of the datasets or models, we also add
the functionality to check the data utility using FairX. We add the support
for checking the Accuracy, Precision, Recall, AUROC, and F1-score. And these
functions can be accessed by using the fairx.metrics.DataUtilsMetrics module.

#### Synthetic Data Evaluation.

In FairX, we add the functionality to evaluate the quality of the generated
data by the fair generative models. It is important to validate the quality of
the synthetic data along with the validation of fairness and data utility
criteria. Existing fairness measuring benchmark do not have the functionality
to evaluate the synthetic data quality. We evaluate the synthetic data quality
in terms of fidelity, diversity and check if the synthetic data has any trace
of original data in it [21]. We use $\alpha$-precision [11] to evaluate the
fidelity of the synthetic data, $\beta$-recall [11] to check the diversity and
Authenticity [11] is used to check if the generative models are just
memorising the training data or not. Synthetic data evaluation module can be
accessed from fairx.metrics.SyntheticEvaluation. We also add the t-SNE and PCA
plots to check the fidelity and diversity of the synthetic data too, more
about the plots are discussed in section 3.4.

#### Explainability.

We add the explainability functionality in FairX to explain the prediction of
a model. We train a classifier (XGBoost) on the benchmarking datasets, and
then we explain the prediction using the fairx.explainability.ExplainUtils
module. This module is based on the TreeExplainer of SHAP [22]. Beside this,
we give the functionality to show the feature importance while making a
decision. This functionality is especially useful when we want to see how much
importance is given to the sensitive attributes while making a decision.

###  3.4 Plotting

We add various plotting support in FairX. They can be accessed under the
fairx.utils.plotting module. We add support to show the performance trade-off
of model accuracy and their fairness performance. Also, we plot the feature
importance to show which features are responsible for prediction outcome. This
comes in handy analyzing original data and synthetic fair data to see how much
the fair model reduce the feature importance for the sensitive attributes.

To show the quality of the synthetic data generated by the fair generative
models, we add PCA and t-SNE plots. These plots shows how close the synthetic
data is from the original data.

Table 3: Evaluation on the Adult-Income dataset using different models
presented at the FairX. Bold indicates best result, and all the metrics score
are higher as better. Synthetic Data Evaluation is only applicable to the Fair
Generative Models (i.e. TabFairGAN and Decaf).

|  | Fairness Metrics | Data Utility | Synthetic Data Evaluation
---|---|---|---|---
| Protected | DPR | EOR | ACC | AUC | F1- |  $\alpha$- |  $\beta$- | Authenticity
| Attribute |  |  |  |  | Score | precision | recall |
Correlation- | Gender | 0.99 $\pm$ .01 | 0.88 $\pm$ .01 | 0.86 $\pm$ .01 | 0.79 $\pm$ .01 | 0.71 $\pm$ .01 | n/a | n/a | n/a
Remover | Race | 0.71 $\pm$ .01 | 0.45 $\pm$ .01 | 0.86 $\pm$ .01 | 0.80 $\pm$ .01 | 0.71 $\pm$ .01 | n/a | n/a | n/a
TabFairGAN | Gender | 0.69 $\pm$ .01 | 0.68 $\pm$ .01 | 0.84$\pm$ .01 | 0.76 $\pm$ .01 | 0.65 $\pm$ .01 | 0.91 $\pm$ .01 | 0.50 $\pm$ .01 | 0.537 $\pm$ .01
| Race | 0.026 $\pm$ .01 | 0.00 $\pm$ .00 | 0.84 $\pm$ .01 | 0.77 $\pm$ .01 | 0.67 $\pm$ .01 | 0.98 $\pm$ .01 | 0.489 $\pm$ .01 | 0.52 $\pm$ .01
Decaf | Gender | 0.52 $\pm$ .01 | 0.42 $\pm$ .01 | 0.75 $\pm$ .01 | 0.63 $\pm$ .01 | 0.44 $\pm$ .01 | 0.07 $\pm$ .01 | 0.08 $\pm$ .01 | 0.009 $\pm$ .01
| Race | 0.55 $\pm$ .01 | 0.46 $\pm$ .01 | 0.77 $\pm$ .01 | 0.69 $\pm$ .01 | 0.53 $\pm$ .01 | 0.06 $\pm$ .01 | 0.08 $\pm$ .01 | 0.009 $\pm$ .01
FairDisco | Gender | 0.98 $\pm$ .01 | 0.81 $\pm$ .01 | 0.78 $\pm$ .01 | 0.63 $\pm$ .01 | 0.86 $\pm$ .01 | n/a | n/a | n/a
| Race | 0.95 $\pm$ .01 | 0.92 $\pm$ .01 | 0.812 $\pm$ .01 | 0.71 $\pm$ .01 | 0.88 $\pm$ .01 | n/a | n/a | n/a
Threshold | Gender | 0.93 $\pm$ .01 | 0.29 $\pm$ .01 | 0.86 $\pm$ .01 | 0.75 $\pm$ .01 | 0.65 $\pm$ .01 | n/a | n/a | n/a
Optimizer | Race | 0.87 $\pm$ .01 | 0.40 $\pm$ .01 | 0.87 $\pm$ .01 | 0.80 $\pm$ .01 | 0.71 $\pm$ .01 | n/a | n/a | n/a
Original Data | Gender | 0.32 $\pm$ .01 | 0.22 $\pm$ .01 | 0.88 $\pm$ .01 | 0.80 $\pm$ .01 | 0.72 $\pm$ .01 | n/a | n/a | n/a
| Race | 0.19 $\pm$ .01 | 0.00 $\pm$ .00 | 0.88 $\pm$ .01 | 0.80 $\pm$ .01 | 0.71 $\pm$ .01 | n/a | n/a | n/a

Table 4: Evaluation on the Compass dataset using different models presented at
the FairX. Bold indicates best result, and all the metrics score are higher as
better. Synthetic Data Evaluation is only applicable to the Fair Generative
Models (i.e. TabFairGAN and Decaf).

|  | Fairness Metrics | Data Utility | Synthetic Data Evaluation
---|---|---|---|---
| Protected | DPR | EOR | ACC | AUC | F1- |  $\alpha$- |  $\beta$- | Authenticity
| Attribute |  |  |  |  | Score | precision | recall |
Correlation- | Gender | 0.43 $\pm$ .01 | 0.33 $\pm$ .01 | 0.64 $\pm$ .01 | 0.64 $\pm$ .01 | 0.59 $\pm$ .01 | n/a | n/a | n/a
Remover | Race | 0.58 $\pm$ .01 | 0.63 $\pm$ .01 | 0.65 $\pm$ .01 | 0.64 $\pm$ .01 | 0.60 $\pm$ .01 | n/a | n/a | n/a
TabFairGAN | Gender | 0.52 $\pm$ .01 | 0.42 $\pm$ .01 | 0.68 $\pm$ .01 | 0.68 $\pm$ .01 | 0.66 $\pm$ .01 | 0.84 $\pm$ .01 | 0.70 $\pm$ .01 | 0.37 $\pm$ .01
| Race | 0.50 $\pm$ .01 | 0.49 $\pm$ .01 | 0.69 $\pm$ .01 | 0.68 $\pm$ .01 | 0.64 $\pm$ .01 | 0.94 $\pm$ .01 | 0.75 $\pm$ .01 | 0.33 $\pm$ .01
Decaf | Gender | 0.87 $\pm$ .01 | 0.84 $\pm$ .01 | 0.45 $\pm$ .01 | 0.45 $\pm$ .01 | 0.42 $\pm$ .01 | 0.77 $\pm$ .01 | 0.45 $\pm$ .01 | 0.61 $\pm$ .01
| Race | 0.99 $\pm$ .01 | 0.96 $\pm$ .01 | 0.45 $\pm$ .01 | 0.45 $\pm$ .01 | 0.42 $\pm$ .01 | 0.77 $\pm$ .01 | 0.45 $\pm$ .01 | 0.61 $\pm$ .01
FairDisco | Gender | 0.99 $\pm$ .01 | 0.99 $\pm$ .01 | 0.612 $\pm$ .01 | 0.61 $\pm$ .01 | 0.58 $\pm$ .01 | n/a | n/a | n/a
| Race | 0.99 $\pm$ .01 | 0.99 $\pm$ .01 | 0.59 $\pm$ .01 | 0.59 $\pm$ .01 | 0.56 $\pm$ .01 | n/a | n/a | n/a
Threshold | Gender | 0.92 $\pm$ .01 | 0.98 $\pm$ .01 | 0.65 $\pm$ .01 | 0.65 $\pm$ .01 | 0.61 $\pm$ .01 | n/a | n/a | n/a
Optimizer | Race | 0.99 $\pm$ .01 | 0.76 $\pm$ .01 | 0.63 $\pm$ .01 | 0.63 $\pm$ .01 | 0.60 $\pm$ .01 | n/a | n/a | n/a
Original Data | Gender | 0.37 $\pm$ .01 | 0.28 $\pm$ .01 | 0.66 $\pm$ .01 | 0.65 $\pm$ .01 | 0.61 $\pm$ .01 | n/a | n/a | n/a
| Race | 0.54 $\pm$ .01 | 0.58 $\pm$ .01 | 0.66 $\pm$ .01 | 0.65 $\pm$ .01 | 0.61 $\pm$ .01 | n/a | n/a | n/a

##  4 Results and discussion

We now consider the fairness, data utility and synthetic data evaluation (only
for in-processing generative models) of the models presented in this
benchmarking tool. We also present the explainability analysis where we use
the generated data by in-processing generative models and show how the fair
generated data perform on down-streaming task and how the prediction is
affected by the sensitive attributes. We also show the feature importance by
using these explainability analysis.

Table 3 and 4 shows the performance of the bias removal algorithm for the
Adult-Income dataset and Compass dataset respectively. We run experiment using
different Protected attributes111For the sake of brevity, we could not include
additional results using other datasets—we refer the reader to the FairX
repository for these results. Some metrics like precision, recall, fairness
through unawareness (FTU), and plots like fairness-accuracy trade-offs were
similarly omitted.. Besides, fairness and data utility, we add synthetic data
evaluation for the output of TabFairGAN, and Decaf.

Figure 2: PCA and t-SNE plots of the original data and Synthetic data
generated by TabFairGAN. Here each dot represents a record, if the generative
model learns the original data distribution then the dots should overlap with
each other. Dataset: Adult-Income, Protective attribute: ‘sex’.

From the table, we see for the generative fair models, TabFairGAN is
performing well comparing with the Decaf in both datasets with both protected
attributes. The $\alpha-$precision, $\beta-$recall scores of TabFairGAN is
better than Decaf, this represents the synthetic data quality of TabFairGAN is
superior than Decaf. On the other hand, TabFairGAN perform poorly in the
fairness evaluation for the ‘race’ protected attribute of the Adult-Income
dataset. Pre-processing technique Correlation Remover and in-processing
technique FairDisco also perform well in terms of fairness and data utility.

On the visual evaluation of fair synthetic data, we use the synthetic data
generated by TabFairGAN. Figure 2 shows the PCA and t-SNE plots of the
synthetic data generated by TabFairGAN. We show how closely the synthetic data
distribution is matching with the original data. If the generative model can
capture the original data distribution, original and synthetic data should
overlap with each other on the PCA and t-SNE plot. Figure 2 shows that data
generated by TabFairGAN partially learned the data distribution of the
original data.

Figure 3: Feature Importance on Prediction task on the Original Data (left)
and Synthetic Data (right) by TabFairGAN, the Sensitive Feature here is ‘sex’,
The Feature Value of Sensitive Attribute in Synthetic Data is less than
Original Data.

In Figure 3, we show the feature importance for a down-streaming task to
predict the target attribute of the Adult-Income dataset where the ‘Sensitive
attribute’ is ‘sex’. We compared the feature importance of original data with
the synthetic data generated by TabFairGAN. We can see the feature importance
of the synthetic data is lower than the original data. This means the
synthetic data generated by the TabFairGAN is less biased towards entity.

Figure 4: Representation of ‘sex’ and ‘race’ features on the target class,
here we can see the dataset is heavily in favor of white people.

Finally, Figure 4 shows the intersectional bias on the Adult-Income dataset.
We plot the percentage of ‘salary-income’ for both ‘race’ and ‘sex’ protected
attributes. We see in the dataset, decisions are given in favor towards white
people.

##  5 Usage

In this section, we present different sample code example of our tool. We give
a brief description of each module and their corresponding class description
and function details.

#### Dataset usage.

To use the dataset already pre-loaded with the tool, we need to use the
BaseDataClass. This class takes three hyperparameters as input; dataset_name,
sensitive_attirbute and a boolean flag for attaching the target variable with
the main dataframe. BaseDataClass has two functions, preprocess_data() and
split_data() to preprocess the dataset using categorical, numerical
transformation and split the dataset for training and testing purpose
respectively.

⬇

1 from fairX.dataset import BaseDataClass

2

3 dataset_name = ’Adult-Income’

4 sensitive_attribute = ’race’

5 attach_target = True

6 data_module = BaseDataClass(dataset_name, sensitive_attribute,
attach_target)

Listing 1: Using BaseDataset Class.

#### Model usage.

We add three kinds of bias-removal techniques under the models folder of
FairX. The list of available models can be found in Table 2. Here is an
example usage of in-processing algorithm called TabFairGAN. After initializing
the Model, we train the it by calling the fit() function which takes the
dataset, batch size and number of epochs as parameters. After training, for
the fair generative models (TabFairGAN and Decaf), synthetic data will be
automatically saved in the working directory.

⬇

1 from fairX.models.inprocessing import TabFairGAN

2

3 data_module = BaseDataClass(dataset_name, sensitive_attribute,
attach_target)

4 under_prev = ’Female’

5 y_desire = ’>50K’

6 tabfairgan = TabFairGAN(under_prev, y_desire)

7 tabfairgan.fit(data_module, batch_size = 256, epochs = 1000)

Listing 2: Using Models.

#### Metrics usage.

Here, we give a sample code for measuring the fairness and data utilities with
a dataset that is already part of the FairX system. Both FairnessUtils and
DataUtilsMetrics class takes the dataset as input and then we call the
evaluate_fairness() and evaluate_utility() function to measure the fairness
data utilities respectively. The result is stored as a dictionary file.

⬇

1 from fairX.metrics import FairnessUtils

2 from fairX.metrics import DataUtilsMetrics

3 from fairX.dataset import BaseDataClass

4

5 data_module = BaseDataClass(dataset_name, sensitive_attribute,
attach_target)

6 cat_transformer, num_scaler, transformed_data =
data_module.preprocess_data()

7 splitted_data = data_module.split_data(transformed_data)

8 fairness_measurement = FairnessUtils(splitted_data)

9 utility_measurement = DataUtilsMetrics(splitted_data)

10 fairness_res = fairness_measurement.evaluate_fairness()

11 datautils_res = utility_measurement.evaluate_utility()

12 print(fairness_res)

13 print(datautils_res)

Listing 3: Using Fairness & Data utility Metrics.

The following code example is to use the CustomDataClass to load custom
dataset in FairX. We need to give the dataset path, list of sensitive
attributes and a boolean operator for attaching the target. This code also
shows the usage of synthetic data evaluation using the SyntheticEvaluation
class.

⬇

1 from fairX.metrics import SyntheticEvaluation

2 from fairX.dataset import BaseDataClass

3 from fairX.dataset import CustomDataClass

4

5 original_data = BaseDataClass(dataset_name, sensitive_attribute,
attach_target)

6 generated_data = CustomDataClass(generated_data_path, sensitive_attribute,
attach_target)

7 synthetic_evaluation_class = SyntheticEvaluation(original_data,
generated_data)

8 synthetic_data_measurement = synthetic_evaluation_class.evaluate_synthetic()

9 print(synthetic_data_measurement)

Listing 4: Using Synthetic Data Evaluation Metrics with Custom Data Loader.

##  6 Conclusion and future work

Massive amounts of data are being produced everyday. Unfortunately, much of
this data contains human or machine biases. Furthermore, the usage of
recommendation system has increased with advancements in artificial
intelligence. But if we use biased data to train a recommendation system,
there is a high chance that the recommendation system will yield unfair
decision towards some demographics. To mitigate this issue, researchers have
developed various measure to remove the bias from the dataset or models; or to
train the model in such a way that the model produces bias-free data. To help
in this process, benchmarking tools equipped with different bias-removal
techniques and evaluation metrics were developed over the years. But these
benchmarking tools commonly lack the option to evaluate generative models or
to train them. We therefore presented FairX, an open-source, modular, fairness
benchmarking tool. FairX comes with a data-loader, supports model training,
and has an evaluation module. FairX provides support for training fair
generative models and for evaluating the synthetic data created by them. FairX
also contains various fairness evaluation metrics, data utility evaluation
metrics and different plotting techniques to help users to evaluate models and
visualize outcomes. FairX comes with support for explainability analysis of a
prediction using the dataset (both original and synthetic) and shows feature
importance. We believe FairX will help the researchers and mitigate the gap of
not having fair generative models and way of evaluating synthetic data.

In the future, we intend to extend FairX to be able to handle other modalities
in addition to tabular and image data, for example text and video. Also, we
will add wider range of evaluation metrics for both synthetic data utility and
fairness metrics. For the models, we plan to add text based and more tabular
and image based fair generative models [14, 15, 23, 13]. In this version of
FairX, we do not have option to add custom models, but we plan to add this
features in future version, so users can use their own model and use all the
functionalities of FairX for their model. We also plan to add hyper-parameter
optimization feature for the models so, we can find the optimal parameters and
best result. Finally, we plan to add functionalities to evaluate the output of
large language models.

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# Standing spin waves in Permalloy-NiO bilayers as a probe of the interfacial
exchange coupling

Diego Caso  Departamento de Física de la Materia Condensada C03, Universidad
Autónoma de Madrid, Madrid 28049, Spain  Ana García-Prieto  Spanish CRG
BM25-SpLine at the European Synchrotron, 71 Av. des Martyrs, 38000 Grenoble,
France  Instituto de Ciencia de Materiales de Madrid, Consejo Superior de
Investigaciones Cientf́icas, Madrid 28049, Spain  Eugenia Sebastiani-Tofano
Spanish CRG BM25-SpLine at the European Synchrotron, 71 Av. des Martyrs, 38000
Grenoble, France  Instituto de Ciencia de Materiales de Madrid, Consejo
Superior de Investigaciones Cientf́icas, Madrid 28049, Spain  Akashdeep Kamra
Departamento de Física Teorica de la Materia Condensada, C-05, Universidad
Autónoma de Madrid, Madrid 28049, Spain  IFIMAC, Universidad Autónoma de
Madrid, Madrid 28049, Spain  Cayetano Hernández  Departamento de Física
Aplicada, M-12, Universidad Autónoma de Madrid, Madrid 28049, Spain  Pilar
Prieto  Departamento de Física Aplicada, M-12, Universidad Autónoma de Madrid,
Madrid 28049, Spain  Instituto Nicolás Cabrera (INC), Universidad Autónoma de
Madrid, Madrid 28049, Spain  Farkhad G. Aliev<EMAIL_ADDRESS>Departamento de Física de la Materia Condensada C03, Universidad Autónoma de
Madrid, Madrid 28049, Spain  IFIMAC, Universidad Autónoma de Madrid, Madrid
28049, Spain  Instituto Nicolás Cabrera (INC), Universidad Autónoma de Madrid,
Madrid 28049, Spain

###### Abstract

Ferromagnetic/Antiferromagnetic (FM/AFM) bilayers dynamics have been a recent
topic of interest due to the interaction occurring at the interface, where the
magnetic moments of the AFM can be imprinted into the FM, and the exchange
bias field can affect these dynamics. Here, we investigate Permalloy (Py) and
NiO (Py/NiO) hybrids and for comparison single Py films in the broad Py
thickness range varied from few nm to 200 nm by using static (Kerr effect) and
dynamic (spin waves) measurements along with micromagnetic simulations. We
observe hybrid modes between uniform (ferromagnetic resonance FMR, n=0) and
perpendicular standing spin waves (PSSWs, n=1, 2) and a clear enhancement of
the PSSWs modes frequencies upon interfacing Py with NiO both from experiments
and simulations. This enhancement becomes less pronounced as the thickness of
the film increases, demonstrating its interfacial origin rooted in the
exchange coupling between the FM and AFM layers. Besides, through
micromagnetic simulations, we investigate and correlate changes in spatial
profiles of the PSSWs with the interfacial exchange coupling. As the thickness
is increased, we see that the n=1 and n=2 modes begin to couple with the
fundamental FMR mode, resulting in asymmetric (with respect the Py layer
center) modes. Our results suggest that PSSWs detection in a ferromagnet
offers a means of probing the interfacial exchange coupling with the adjacent
AFM layer. Furthermore, the controlled spatial symmetry breaking by the AFM
layer enables engineering of PSSWs with different spatial profiles in the FM.

##  I Introduction

Over the last few decades, one focus in the investigation of ferromagnet-
antiferromagnet bilayers has been concentrated around the induction of the
uniaxial anisotropy asymmetry due to the influence of exchange bias Nogués and
Schuller (1999); Smardz (2002); Blachowicz and Ehrmann (2021) Recently also an
opposite limit has been explored when a thin ferromagnet is coupled to a much
thicker epitaxial antiferromagnet Bommanaboyena  _et al._ (2021) to operate as
a readout mechanism for antiferromagnetic spintronics.

An important factor present in both limits is the exchange coupling at the
AFM/FM interface, since it not only shifts the static magnetization hysteresis
curves Smardz (2002), but may also affect magnetization dynamics. For example,
exchange coupling to antiferromagnets was used to increase the zero field FMR
frequency Phuoc  _et al._ (2010). On the other hand, excitation of spin waves
(SWs) in a ferromagnet coupled to an antiferromagnet was proposed as a method
to propagate spin excitation into the antiferromagnet Jenkins  _et al._
(2020). Exchange field was reported to influence the lowest frequency uniform
FMR mode Stamps  _et al._ (1996) but could even have an stronger influence in
the short wavelength perpendicular standing spin waves (PSSWs) when
magnetization symmetry in the direction perpendicular to the interfaces is
broken.

Recently, PSSWs have been studied in other symmetry breaking systems such as
heavy metal/magnetic insulator (HM/MI) bilayers, where the frequency of the
standing wave modes experiences a shift Lee  _et al._ (2023). The effect was
attributed to an additional confinement of the waves upon a local increase of
the interfacial damping due to spin pumping Lee  _et al._ (2023). A number of
other recent experiments also used PSSWs to probe coupling in ferromagnetic
bilayers Kostylev (2009); Qin  _et al._ (2018); Zhang  _et al._ (2021);
Kennewell  _et al._ (2010) or individual surface anisotropies on epitaxial
ferromagnetic layers with nonequivalent interfaces Solano  _et al._ (2022).
Interestingly, the PSSW modes excited in two interacting ferromagnetic films
have been proposed to enable quantum entanglement between distant magnon
sources for quantum information applications Sheng  _et al._ (2023).

Application of PSSWs to explore and understand interfacial spin dynamics in
AFM/FM bilayers has been however much less explored. To our best knowledge,
such studies have only been done by varying AFM thickness (fixing thickness of
FM layer) and only by using a single microwave excitation frequency Magaraggia
_et al._ (2010). As regards broadband magnetization dynamics studies, they
have been carried out only for the strongly coupled AFM/FM bilayers, where the
FMR mode is influenced and split into two separated modes by the adjacent AFM
Al-Hamdo  _et al._ (2023). Interestingly, recent studies also discovered the
appearance of an additional (to the uniform FMR and PSSWs modes) low frequency
(below GHz) modes in the coupled AFM/FM bilayers, attributed to the excitation
of the interfacial layers with reduced magnetic moment Polishchuk  _et al._
(2021). The emergence of an additional mode at the interface has also been
discussed in a ferrimagnet/ferromagnet bilayer Lüthi  _et al._ (2023). These
recent advances demonstrate a high potential for investigating and engineering
spin dynamics in AFM/FM bilayers, which is the goal of the present article.

As to the possible interaction between two different magnon modes, it was
Grünberg et al. Grünberg  _et al._ (1982) who first predicted that Damon
Eshbach (DE) surface modes would not interact with PSSWs in the dipolar limit.
However, if large exchange was considered, the modes do interact repulsing
each other and causing the elimination of the crosspoints Grünberg  _et al._
(1982). On the other side, Klingler et al. reported anti-crossings of YIG
PSSWs and the Co uniform precession FMR mode Klingler  _et al._ (2018),
modeled by a mutual spin pumping and exchange interaction Gou  _et al._
(2023); Mondal  _et al._ (2023). Generally, the investigation of magnon-magnon
coupling in the same material and with a shared propagation direction such as
PSSW of different order remains under-explored. To our best knowledge, only
Wang et al. reported modes anti-crossings for interactions in quasi-uniform
in-plane propagating dipole-exchange spin waves modes with first order PSSWs
in a FM/AFM heterostructure Wang  _et al._ (2022).

Figure 1: (a) Sketch of the experimental setup, depicting the MW and DC field
distributions on the film placed on the coplanar waveguide (CPW). The high
frequency signal travels from Port 1 to Port 2 through the waveguide (S), in
between the two grounds (G). (b) Sketch of a typical Py/NiO film investigated
here. Figure 2: (a) Hysteresis loops for some of the used Py/NiO films on its
easy axis, the Py thickness is depicted in the legend. (b) Coercive field of
all of our films. (c) Exchange bias of the films extracted from the asymmetry
of the hysteresis loops.

Here, we investigate the influence of the exchange field on the static and
dynamic response in Py(dPy)/NiO(240 nm) bilayers in the broad frequency (up to
30 GHz) and Permalloy thickness (24-200 nm) ranges (details of the sample
growth and characterization are given in Appendix A). The thickness of the NiO
layer was high enough to avoid AFM thickness effect. We observe both the
emergence of the surface layer’s magnetization dynamic response and the
influence of the exchange coupling on the perpendicular standing spin waves in
Permalloy. Our observation of frequency shifts in the PSSW modes resulting
from the interfacial exchange offers a dynamic means to probe and characterize
the interface properties. Furthermore, via micromagnetic simulations, we
demonstrate the ability to engineer the PSSW modes’ spatial profiles via the
interfacial exchange.

##  II Methods

###  II.1 High frequency experimental setup

To probe the dynamic magnetic properties of the different investigated films,
we use a basic vector network analyzer (VNA) high frequency setup (see for
details Ref. Aliev  _et al._ (2011)). By placing the films directly on top of
a 50$\Omega$ coplanar waveguide (CPW) connected to the VNA, the related small
MW field will oscillate in the same plane as the sample (see Figure 1(a)). A
DC magnetic field up to 2.7 kOe is simultaneously applied perpendicularly to
the microwave field to generate torque in the magnetic moments of the film.
The measurements were made by sweeping the frequency for constant DC magnetic
fields. Microwave frequency was swept from 10 MHz up to 30 GHz and MW power
was set at 0 dBm. The scattering matrix element S21, which relates the
magnitude and phase of the input signal sent to the CPW and the received
signal, is an indicative of the microwave permeability in the system Müller
_et al._ (2021). The real part of S21 is then calculated and normalized at
maximum DC field to obtain the frequency-applied field maps.

###  II.2 Simulation details

Dynamic simulations were carried out using the Mumax3 Vansteenkiste  _et al._
(2014) software for micromagnetic simulations. The discretization cell size
was set at 2 nm $\times$ 2 nm $\times$ 1 nm, which is far below the Permalloy
(Py) exchange length (5.3 nm) Wang  _et al._ (2017). The decreased cell size
in the $z$ direction is deliberately imposed to obtain an enhanced precision
on this coordinate and accurately characterize the thickness modes. The size
of the simulated structure was 128 nm$\times$ 128 nm $\times$ dPy, and
periodic boundary conditions (PBCs) were imposed on the in-plane dimensions to
avoid edge effects. We used typical magnetic parameters of Py in our
simulations Lara  _et al._ (2013): $M_{s}=1.16\,T;\quad A_{ex}=14\times
10^{-12}\,J/m;\quad\alpha=0.001\quad,$ where Ms is the saturation
magnetization of the Py and Aex and $\alpha$ specify the exchange stiffness
constant and the damping constant respectively.

Lower $M_{s}$ is set near the top surface of our films (0.25 T on the 2 top
layers). This translates to a lower demagnetizing field and a higher effective
potential in these layers, thus enabling the pinning of the PSSWs modes Waring
_et al._ (2023). This effect is plausible due to the surface oxidation
typically experimented by ferromagnetic films or a magnetization symmetry
breaking in the normal direction due to a capping layer. For Py specifically,
the thickness of this surface oxidation layer has been reported to be close to
1.5 nm Fitzsimmons  _et al._ (2006). In the case of the Py films, the
variation of $M_{s}$ is the most conceivable way to induce changes in the
surface pinning parameter Waring  _et al._ (2023), therefore generating
magnetization symmetry breaking and allowing the excitation of PSSWs. However
in the FM/AFM system, the simulations include an interfacial surface
anisotropy ($K_{s}$) due to the influence of the exchange bias. In that case,
the magnetization symmetry breaking is accomplished on both surfaces of the
films.

In order to uncover the main SW modes, a homogeneous magnetic field pulse of
the following form: $\vec{H}_{pulse}=H_{0}sinc\left(\frac{2\pi
t}{t_{0}}\right)\hat{u}_{x}$ was employed over a 10 ns duration. Here, H0
represents the amplitude of the applied field pulse, 20 Oe, and t0 denotes the
half width of the pulse, 10-12 s. The used excitation is homogeneous over the
thickness of the material thickness. The averaged out-of-plane (OOP)
magnetization of the system was converted into the frequency domain after the
use of a Fourier Transform Park  _et al._ (2003). The process was repeated for
each of the presented magnetic fields.

##  III Experimental results and discussion

###  III.1 Static characterization of the Py and Py/NiO films

Figure 3: (a) PSSWs on the Py and Py/NiO for 73, 48 and 24 nm thick Py against
frequency and bias field. (b) and (c) represent the n = 0 mode of the 73 and
24 nm films respectively against the magnetic field. Lines in (a, b, c) are
the corresponding fits to Eq. 1.

Kerr magneto-optical experiments (MOKE, see Appendix A) reveal a clear
enhancement of the coercive field of all of our ferromagnetic films upon the
addition of NiO (see Figure 2(b)). This phenomenon has been documented in both
experimental and analytical studies of FM/AFM films Mauri  _et al._ (1987); Li
and Zhang (2000); Hu (2003). Moreover, as the experimental data suggests, this
effect is suppressed once the thickness of the FM layer becomes large enough,
which can be attributed to the reduced effect of the interface exchange
coupling strength relative to the overall magnetization of the film.

The shift in the hysteresis loops from a field-centered position indicates the
presence of an exchange bias field in the Py/NiO films. The interplay between
the exchange bias and the thickness of the film is depicted in Fig. 2(c),
yielding also in a reduction of the field as the Py layer becomes thicker,
once again, an effect related to the reduced influence on the overall film
magnetization. These findings collectively point to the existence of a
quantifiable exchange coupling within our FM/AFM films.

###  III.2 Effect of the NiO on the frequency of the observed standing waves

Performed VNA-FMR experiments (see Methods section II.1) in both sets of Py
and Py/NiO films revealed the presence of an uniform FMR mode, characteristic
of the FM layers. For films of 24 nm and above, we also found other set of
higher order spin wave resonances above the frequency of the FMR mode. These
spin wave modes are identified as exchange-dominated thickness-dependant modes
related to perpendicular standing spin waves (PSSWs) in our films Dabrowski
_et al._ (2021). Several factors contribute to the identification of the
detected modes as PSSWs. For instance, the dependence of the modes with the
thickness of the films is a clear indicator, as well as the alignment with the
performed simulations and the extensive previous literature and reports of
these modes on thick ferromagnets.

For all of our films, the presence of the NiO layer shifted the standing spin
wave modes to higher frequencies (see Figure 3(a)). Our experimental data
indicates that this frequency shift decreases with the enhancement of the
thickness of the Py layer, as depicted in Figure 8. This suggests a direct
relation between the magnitude of the exchange coupling effects on the overall
behavior of the magnetic moments in the film as the FM layer grows thicker and
the frequency shift of the PSSW modes experienced when the NiO layer is added.
We notice that the FMR mode, which is not thickness-dependent, also
experiences a slight frequency shift in the same direction for all of our pair
of samples when the NiO is added, as Figures 3(b, c) illustrate.

As briefly mentioned in the introduction, we believe that the frequency shift
experienced by our films is a direct consequence of the Py magnetic moments
pinning at the interface due to the exchange bias. This absence of free spins
results in an additional confinement of the PSSWs. Furthermore, an estimation
of the thickness reduction of the Py layer due to the pinned magnetic moments
is presented in Figure 8(b), by letting the thickness variable to be free in
Equation 1. A 2 nm consistent effective thickness reduction can be appreciated
for all of the films. This additional confinement is further supported by the
experiments done on extremely thin films (see Appendix B).

The resonances of the PSSW modes were initially predicted by Kittel Kittel
(1958), and Seavey et al. Seavey and Tannenwald (1958) were the first to
experimentally observe PSSW modes. The following dispersion relation Gui  _et
al._ (2007); Conca  _et al._ (2014); Navabi  _et al._ (2017); Qin  _et al._
(2018); Liu  _et al._ (2020) has been broadly used to characterize the
standing spin waves of a ferromagnetic material assuming perfectly rigid
boundary conditions as an extension of the Kittel formula Kittel (1948):

$\displaystyle\begin{split}f_{PSSW}=\frac{\gamma}{2\pi}\sqrt{\left[(H_{ext}+H_{k})+\frac{2A_{ex}}{M_{s}}\left(\frac{n\pi}{d}\right)^{2}\right]\ldots}\\\
\overline{\ldots\times\left[(H_{ext}+H_{k})+\frac{2A_{ex}}{M_{s}}\left(\frac{n\pi}{d}\right)^{2}+4\pi
M_{s}\right]}\end{split}$ (1)

Where n is the quantized order of the PSSW mode along the thickness direction
(n=0 for the FMR mode), $d$ the thickness of the film, Aex the exchange
stiffness constant, Ms is the saturation magnetization, which is obtained by
fitting the uniform mode, Hext the external magnetic field, $\gamma$ the
gyromagnetic ratio and Hk corresponds to the in-plane uniaxial anisotropy
field.

Figure 4: (a) Sketch of a cross-section in the case of the simulated Py film,
and (b) introducing a FM/AFM interface. Simulated and experimental results via
VNA-FMR of the field derivative of the transmission at 2 kOe in the case of
Py(48 nm), (c), and Py(48 nm)/NiO, (d) with an IP uniaxial anisotropy of Ku =
10-5 J/m3. For the simulated Py(48 nm)/NiO film, an exchange bias field of 1
kOe IP was manually introduced in the bottom layer of the system. Figure 5:
Magnetic parameters of the Py/NiO bilayers and control Py samples. (a)
Saturation magnetization of the films. (b) Gilbert damping. (c) Calculated IP
anisotropy field. (d) Exchange stiffness constant calculated following a fit
to Eq. 1. Inset in (d) corresponds to the gyromagnetic ratio, also obtained as
a free parameter from a fit to Eq. 1.

The experimentally detected PSSWs modes in our films were fitted to eq. 1.
Solid lines in Figure 3(a, b, c) prove a great accuracy of the fit. Figure
5(a, b, c, d) represent the magnetic parameters obtained by fitting the PSSWs.
While the saturation magnetization remains very consistent at all thicknesses,
the anisotropy field of the Py/NiO samples progresses similarly as the
exchange bias and coercive fields extracted from the static hysteresis loops.
This is expected since the exchange bias of the films have less influence on
the overall magnetization as the films grow thicker (Figure 2(c)). In the
control samples of Py, however, the anisotropy field remains practically
constant for low thicknesses, proving once again the quantifiable aspect of
the interface coupling between the Py and the NiO in our study.

The damping parameter was obtained by fitting the FMR linewidth ($\Delta H$)
to the following formula:

$\Delta H=\Delta H_{0}+\alpha\frac{4\pi}{\gamma}f$ (2)

Where $\alpha$ is the Gilbert damping, $f$ corresponds to the resonance
frequency, and $\Delta H_{0}$ is the intrinsic damping contribution, which
denotes the frequency independent linewidth coming from the film
inhomogeneities. The Py/NiO samples exhibit a higher magnetic damping (almost
an order of magnitude for smaller Py thickness) in comparison with the control
Py samples, as depicted in Figure 5(c). The damping parameter could be
influenced by the spin pumping to antiferromagnet Wang  _et al._ (2020) and
exchange bias at the interface, freezing the magnetic moments of the Py and
preventing larger precessions.

Figure 6: (a) Evolution of the primary SW mode frequencies at a constant
magnetic field of H = 2 kOe as the Py film thickness varies from 20 nm to 200
nm. The FFT power is normalized at each film thickness. The inset exhibits the
three detected modes until 110 nm, where the lines in the n=1 and n=2 curves
represent a fit of the behavior of the modes using the equation f(GHz) = A +
B/d${}_{Py}^{2}$(nm), A and B being free parameters. (b, c, d) Snapshots of
the n=0, n=1 and n=2 PSSWs modes respectively at 50 nm of thickness. (e)
Snapshot of the hybrid n=0 mode at 90 nm (f). Snapshot of the asymmetric main
mode at 100 nm. (g) Snapshot of the main mode at 150 nm.

##  IV Numerical Results of PSSW presence on Py films

In order to deeper understand the magnetization dynamics we numerically
investigate via micromagnetic simulations (see the Methods section II.2 for
details) the presence of PSSWs on 20-200 nm-thick Py films. The films are
thick enough to accurately resolve the PSSW modes (the number of cells in the
OOP direction is directly correlated to the thickness of our simulated
samples) and to observe their possible shifts in the frequency when the
interaction with the AFM is introduced.

With a surface magnetization gradient (see Methods section II.2) caused by a
possible oxidation and/or capping, we aim to manipulate the interface pinning
parameter, which strongly affects the spin configuration of the PSSW modes
across the film Waring  _et al._ (2023). We disregard surface anisotropies in
our simulations. The pinning parameter has the following form:

$\xi=\frac{2K_{s}}{qM_{s}^{2}}$ (3)

where $K_{s}$ is the surface anisotropy, $M_{s}$ the saturation magnetization,
and $q=\nabla^{2}\textbf{M}$ is related to the non-uniform part of the
exchange field. Most of the studies involving PSSWs use the case of surface
rigid pinning ($\xi\rightarrow\infty$) introduced by Kittel Kittel (1958), in
which equation 1 is a solution. The other limit case, no surface pinning
($\xi=0$), in which the magnetic moments are free in the surface is the other
instance under extensive study Ament and Rado (1955). However, experimentally,
in both of these limits it would be difficult to excite the n = 1 PSSW since
there is no net magnetization in the mode profiles Gurevich and Melkov (2020).

Waring et al. Waring  _et al._ (2023) recently demonstrated that dynamic
pinning, i.e., $\xi$ being a nonzero value, describes better the behavior of
the excited PSSWs without the need to see major changes in the exchange
stiffness constant $A_{ex}$ from the fundamental mode, n=0, to higher order
PSSWs (n=1, 2…). On this topic, some works have introduced magnetization
graded ferromagnetic films to generate perpendicular standing spin wave modes
in their simulations Gallardo  _et al._ (2019); Waring  _et al._ (2023). This
practice accomplishes an enhancement of the pinning on the surface in which
the $M_{s}$ gradient is introduced, see Eq. 3.

Our micromagnetic simulations performed on a Py layer with an enhanced pinning
created by the reduction of the saturation magnetization on the top layers
revealed field-dependent modes (see Figure 4(b)). These modes resemble the
frequency of the PSSWs observed experimentally, as the blue and black lines
suggest for the 48 nm Py sample in Figure 4. In fact, a full spectrum over the
thickness of the films from 20 to 200 nm (Figure 6) reveals n = 0, 1 and 2
(and weak n = 3, 4, 5, see Appendix C) PSSW modes, which appear at every
thickness, 20 nm, and 55 nm respectively for the frequency range that is
explored in our VNA-FMR experiments (up to 30 GHz). These results match with
the experimentally observed modes in our Py films, see Figure 3(a), which show
n = 2 modes only in the case of 73 nm thick Py and the emergence of the first
order PSSW only in the films of 24 nm and above. Weaker modes above 30 GHz are
not observed experimentally in our system due to the strong losses experienced
at high frequencies.

The detected n$>$0 modes in our simulations are proportional to
1/d${}^{2}_{Py}$, as seen in the inset of Figure 6. This behavior is
indicative of PSSWs described by Eq. 1 and agrees with the reports on PSSWs in
other FM films Grünberg  _et al._ (1982). On the other hand, the frequency of
bulk (FMR) mode remains almost constant when the thickness of the film is
increased, slightly decreasing in frequency once the frequency of n=1 mode
approaches the fundamental mode. Eventually, the two modes combine at
d${}_{Py}>$110 nm.

###  IV.1 Simulating the Py/NiO interface. Optimization of the uniaxial
anisotropy

The simulations of magnetization dynamics in FM/AFM bilayer were performed by
introducing an extra layer in the system as an uncompensated AFM (see Figure
4(b)). This produces uncompensated spins at the interface that lead to an
exchange bias between the FM and the AFM Clercq  _et al._ (2016, 2017). Such
approach is valid as spin dynamics observed experimentally is related to the
FM layer interacting with an uncompensated antiferromagnet (NiO).

Figure 7: Difference in frequency of the first PSSW mode (n=1) for the Py(48
nm)/NiO film obtained experimentally via the dynamic VNA-FMR experiments and
the simulated Py(48 nm) film with an antiferromagnetic interface in the
bottom. The darker region in the figure indicates the limit where the PSSWs
modes are not excited due to the high Ku affecting the ground state
configurations.

Shifts in the frequency of the modes, especially in the experimentally
observed first order PSSW (n=1), are also present in all of the performed
simulations featuring an AFM/FM interface. These alterations move towards
higher frequencies upon introducing the AFM, which also aligns with the
empirical results. Specifically, the uniaxial anisotropy introduced in the
interface has a direct influence on the frequency of the PSSWs. As depicted in
Figure 7, a greater uniaxial unisotropy induced by the AFM leads to higher
frequency modes. This trend continues until the system reaches a threshold
(K${}_{u}>$1.7$\times$105 J/m3) where the system relaxes into different ground
state configurations, rendering the detection of PSSWs no longer feasible. In
our simulations employing solely uniaxial anisotropy, for a 48 nm thick film,
a difference in the first order perpendicular standing wave of less than 0.5
GHz for uniaxial anisotropy constants of 1.5-1.7$\times$105 J/m3 was found
between the experimental and the simulated results. However, the frequency of
the modes can be further enhanced by introducing a local in-plane exchange
bias field in the bottom layer of the order of Heb= 1 kOe. This enhancement
brought the modes closer to the experimentally observed values by less than
0.4 GHz, as illustrated in Figure 7. The significant discrepancy between the
exchange bias values obtained in simulations of the FM/AFM heterostructure,
notably an order of magnitude greater than the exchange bias detected via MOKE
(see Figure 2(c)), is rationalized by recognizing that MOKE spectroscopy
specifically probes the top surface of the sample. However, the exchange bias
effect is interfacial, as accounted for in the simulations, making it
effectively larger than the measurements taken several nanometers above (see
Appendix D).

Figure 8: (a) Frequency shift of the first order PSSW against the thickness of
our films upon the addition of the NiO layer at an applied magnetic field of 2
kOe and simulated frequency shift at the same thicknesses and field for a Py
film with an antiferromagnetic interface. (b) Effective thickness reduction of
the FM film upon the addition of NiO against the thickness of the Py layer.

Additionally, the frequency shifts of the first order PSSWs are tracked at a
constant magnetic field of 2 kOe for each of the films (see Figure 8). As the
thickness of the films decreases, we notice a decline in the frequency shifts
due to the decreasing impact of the AFM/FM interface on the overall
magnetization of the film in our experiments, which is to be expected. Our
simulations also yield similar results, showing a consistent trend of
diminishing the shifts upon increasing the thickness of the Py film, thereby
confirming a valid interpretation of the Py/NiO system. To examine the
evolution to the main SW modes in the Py/AFM system, see Appendix E.

###  IV.2 Hybrid modes and effect of the interaction with the antiferromagnet

Our simulations also revealed that the higher order PSSW modes couple with the
uniform FMR mode as the sample thickness increases, depicted in Figure 6(a).
This phenomenon is observed in our simulations for the n=1 mode at 80-100 nm
and between 140-160 nm in the case of the n=2 PSSW mode. Consequently, the
mode profiles exhibit significant alterations with respect to the symmetrical
n=1 and n=2 modes depicted for 50 nm (see Figure 6(c, d)), where there is no
impact from the uniform mode.

Figure 9: (a) Asymmetry of the n=1 mode in relation with the thickness of the
Py film. Asymmetry equal to 0 means a perfectly symmetrical placement of the
spin wave node along the thickness of the film (perfect n=1). On the other
hand, an asymmetry close to one means that the coupling of the fundamental and
the n=1 mode have taken place, and that the mode profile is closer to that of
the FMR mode. (b, c, d, e) are the one-dimensional mode profiles of the n=1
mode at 50, 100, 110 and 125 nm of thickness respectively. Clear blue parts at
(b, c, d, e) correspond to the reduced Ms layers in the film. (f) Mode
contribution to the mode profiles obtained against the thickness of the film.
(g) Histogram of the position of the node in a MW excitation of the main SW
mode for a film thickness of 150 nm over 10 ns.

However, as the influence of the FMR mode becomes more pronounced, the
mentioned asymmetry is more apparent. The nodes become displaced their
symmetrical position, as is evident in Figure 6(f) for the n=1 in a 100 nm
film and Figure 6(g) for the n=2 in a 150 nm sample. The FMR mode, seemingly
reducing its frequency as the thickness increases, becomes similar to a n=1
mode profile for significant thicknesses, see Figure 6(e) for the mode profile
on the 100 nm film. This mode profile is clearly distinct from the one
observed when there is no coupling of the modes, shown on Figure 6(b).

We have investigated in more details the asymmetry (see Section II for
explanation of the analysis) exhibited by the n=1 mode profile as the
thickness of the film is increased. Figure 9(a) shows the asymmetry of the n=1
mode profile, i.e, a quantification of the shift from the symmetric position
of the node. As the thickness of the simulated Py is increased, the asymmetry
progressively increases, indicating a stronger influence of the FMR mode in
relation to the n=1 mode. This evolution becomes evident in the cross-sections
of the modes profiles depicted in Figures 9(b, c, d, e) for film thicknesses
of 50, 100, 110 and 125 nm respectively. Beyond 50 nm, the n=1 mode gradually
diverges from the symmetric position as the hybrid modes are generated,
eventually resembling more and more the characteristic n=0 mode profile.

The emergence of hybrid modes was evidenced not only through micromagnetic
simulations, but also in VNA-FMR experiments at sufficiently high magnetic
fields and for thicknesses above 120 nm (see Fig. 10(c)). Notably, the
external in plane magnetic field helps the formation of hybrid modes; in the
Py(155nm)/NiO sample the FMR uniform mode and the first order PSSW fully
combine at fields above 1.8 kOe. Below 1 kOe, a distinct separation between
them is observed. This is in line with the simulations performed, which
demonstrate coupling of the modes at 2 kOe.

Experimentally, although the frequency shift is present upon the addition of
the AFM, it is easier to hybridize the n=0 and n=1 modes in this case. This is
reflected in Figures 10(a,b), where there is a complete combination of the two
modes to form a hybrid mode in the case of Py(155 nm)/NiO. However, for just
Py, the hybrid modes are not established in the same range of thicknesses.

Figure 10: (a) Resonance frequency of the n=0 uniform mode and n=1 against the
thickness of control Py films for a bias field of 2kOe. (b) Same graph as
panel (a), but for Py/NiO films. Insets in both panels represent the frequency
FWHM of the modes. (c) Frequency-field map of the transmission in a
Py(155nm)/NiO sample. Inset displays cross-sections at bias fields of 500 Oe
and 2kOe.

##  V Conclusions

In conclusion, we investigated experimentally and by simulations the static
and dynamic magnetization in Py(dPy)/NiO(240 nm) and control Py(dPy) films
with thickness (dPy) varied between 24 and 200 nm. We detected both
perpendicular PSSWs spin waves and quasi uniform modes and found that in
sufficiently thick Py films both exhibit a clear frequency shift when Py is
exchange coupled to NiO. The frequency shift is attributed to the frozen
magnetic moments at the Py/AFM interface, induced by the exchange bias,
reducing the effective thickness of the ferromagnetic sample.

Additionally, we found that hybrid (n=0,1,2) PSSW modes emerge for
sufficiently thick samples, which manifest (as detected by simulations) in
asymmetric character modes with the n=0 mode influenced by the contribution of
the different order PSSWs (n=1, 2…). The exchange coupling between Py and NiO
apparently stimulates such hybrid modes to emerge at reduced (in comparison
with control Py films) thickness. Our findings are expected to provide a
deeper understanding of the relation between exchange coupling and
magnetization dynamics in ferromagnet-antiferromagnet hybrids. Overall, our
results will be important for the applications of AFM/FM heterostructures
which hold great promise for enabling ultrafast spintronics devices with low
power consumption, high efficiency, as well as novel functionalities, paving
the way for the next generation of information processing and storage
technologies.

##  VI acknowledgments

The work in Madrid was supported by Spanish Ministry of Science and Innovation
(PID2021-124585NB-C32, TED2021-130196B-C22) and Consejería de Educación e
Investigación de la Comunidad de Madrid (NANOMAGCOST-CM Ref. P2018/NMT-4321)
grants. F.G.A. acknowledges financial support from the Spanish Ministry of
Science and Innovation, through the María de Maeztu Programme for Units of
Excellence in R&D (CEX2018-000805-M) and ”Acción financiada por la Comunidad
de Madrid en el marco del convenio plurianual con la Universidad Autónoma de
Madrid en Linea 3: Excelencia para el Profesorado Universitario”. We
acknowledge the Spanish Ministerio de Ciencia, Innovación y Universidades and
the Consejo Superior de Investigaciones Científicas for provision of
synchrotron radiation at the BM25-SpLine beamline at The European Synchrotron.

## Appendix

##  Appendix A Sample preparation and structural characterization

NiO(AFM)/Fe20Ni80(FM) bilayers were deposited on Al2O3 (0001) substrates by
ion beam sputtering using Ar+ ions from a 3 cm Kaufmann-type ion source in a
vacuum chamber with a base pressure of 2$\times$10-5 Pa. The thin nickel oxide
films were obtained from a pure nickel (99.99$\%$) target in a controlled
atmosphere of oxygen and argon with P(O2) = 2.6$\times$10-2 Pa and P(Ar)=
1.4$\times$10-2 Pa. The substrate temperature during NiO deposition was
maintained at 500∘C and the sputter ion energy and current density were 650 eV
and 1.1 mA/cm2. The thickness of the NiO layer was high enough (240$\pm$20 nm)
to avoid AFM thickness effect. Fe20Ni80 thin films with thicknesses ranging
between 4 and 200 nm have been obtained from a 99.99$\%$ pure Fe20Ni80 target
in an Ar atmosphere with P(Ar)= 1.4$\times$10-2 Pa and 100∘C substrate
temperature. In this case, the sputter ion energy and current density were 550
eV and 0.8 mA/cm2 respectively. The substrates rotated at 2 rpm to increase
the homogeneity of the deposit. An Al capping layer 2 nm thick was deposited
on top of the FM layer to prevent oxidation for “ex situ” studies. The crystal
structure and texture of the different films was analyzed by XRD (X-ray
difraction) in a $\theta$/2$\theta$ configuration using a PANanalytical X’Pert
MPD system with Cu-K$\alpha$ radiation. XRR (X-ray reflectivity) and GIXRD
(grazing angle XRD) were also performed at ESRF Synchrotron (European
Synchrotron Radiation Facility) at the BM25-Spline beamline. In-plane and out-
of-plane measurements were performed at room temperature and ambient pressure
conditions using a six-circle diffractometer in vertical geometry to profit
from the high collimation of the synchrotron beam. A photon energy of 15keV
(0.0827 nm) was used. In-depth composition and thickness of the different
layers were determined by RBS at the CMAM 5 MV tandem accelerator using 4He+
at 1.8 MeV Redondo-Cubero  _et al._ (2021) A silicon barrier detector, at a
scattering angle of 170.5∘, measured the backscattering ions while a three-
axis goniometer was employed to control the crystal position. The elemental
distribution and thickness quantification of the different layers were
determined with the SIMNRA simulation software package.

Figure 11: Random RBS spectra of a NiO/Py bilayer obtained with 4He+ ions at
1.8 MeV and its simulation. The fitting signal for O, Ni, Al and Fe are
included. Figure 12: (a) complete XRD $\theta$/2$\theta$ scan at a photon
energy of 15 KeV of a NiO/Py bilayer (dPy = 12 nm) (b, c, d, e) XRD
$\theta$/2$\theta$ scans of the (0006) Al2O3 and (111) Py diffraction peaks
for dPy= 48, 24, 12 and 6 nm respectively.

The angular dependence of magnetization was investigated at room temperature
by high-resolution vectorial-Kerr magneto-optical measurements (MOKE) in
longitudinal configuration. MOKE hysteresis loops were recorded by changing
the in-plane angular orientation of the sample, keeping fixed the external
magnetic field direction. The angular orientation, ranging from 0 to 360∘ was
probed at intervals of 9∘ with a maximum applied magnetic field of 1.1 kOe.

In-depth elemental composition and thickness of the different Py layers as
well as the complete heterostructure NiO/Py/Al capping were determined by
Rutherford backscattering spectroscopy (RBS) in a 5 MV tandem accelerator
using 4He+ at 1.8 MeV. As an example, the experimental spectrum of a NiO/Py
bilayer have been shown on Figure 11 that also include the fitting by using
SIMNRA simulation program assuming a three layer model, i.e., NiO, Py and Al
layers. The estimated composition and thickness for this particular sample is
Ni0.49O0.51 (d = 223 nm)/Fe0.19O0.81 (d = 11.6 nm)/Al (d = 2 nm). Figure 12(a)
shows the $\theta$/2$\theta$ XRD scans of a NiO/Py bilayer grown on (0001)
Al2O3 substrates obtained with a photon energy of 15 KeV (0.0827 nm). The
diffraction pattern shows the NiO (111) diffraction peak at 19.49∘, the only
one from NiO that can be distinguished in the complete diffractogram, the
Al2O3 (0006) peak at 21.89∘ and the Py (111) peak at 23.17∘ that indicate a
single oriented growth along the 111 direction. Fig. 12(b) shows just the
Al2O3 (0006) peak and the Py (111) peak for bilayers with different Py
thickness. It can be observed in the kissing fringes around the Py (111) film
Bragg peak, indicating a high quality with smooth surfaces and abrupt
interfaces. With the help of the position of successive Kiessig fringes, it is
possible to determine the Py film thickness, with results in good agreement
with the obtained by RBS quantification. Figure 13 displays the low-angle XRR
result for different NiO/Py bilayers deposited on Al2O3 (0001) substrates. The
presence of intense Kiessig fringes, which result from interference of the
X-ray beams reflected on the film surface and on the interface to the NiO
layer, is also observed in the reflectivity data, indicating a sharp NiO/Py
interface and smooth Py surface. The oscillations observed allow also an
estimation of the thickness of the ferromagnetic Py layer in good agreement
with the ones obtained by RBS. In Fig. 13(c), it is also possible to observe
very small oscillations that allow to estimate the thickness of the NiO layer
for this particular sample with a value of 230 nm. Fig. 14 shows
representative reciprocal space map for a NiO/Py bilayer with dPy = 24 nm
obtained by off-specular GIXRD using synchrotron beam. An incommensurate
epitaxial growth is evidenced from the non-coincidence of the Py and NiO layer
in-plane diffraction peaks respect to those of the Al2O3 (0001) substrate. The
substrate in-plane diffraction peak is obtained at an integer in-plane H/K
value, as the orientation matrix has been defined using the substrate (ALO)
lattice parameters, while the NiO and Py in-plane diffraction peaks are
obtained at H=K=0.92 and 1.1 substrate reciprocal lattice units, respectively.
This XRD pattern can be explained by a non–coincidence epitaxial growth in
which the in-plane lattice of the single oriented NiO/Py (111) rotates 30∘
respect to the underlying Al2O3 (0001) plane in order to reduce the lattice
mismatch minimizing the epitaxial energy. Calculations reveal an in–plane
lattice parameter of 4.22(4) Å (cubic notation) for the NiO and 3.55(1) Å
(cubic notation) for the Py. These results indicate that the NiO grows with a
tensile stress while the Py grows fully relaxed. The line profile scans along
the in-plane substrate H/K value (CTR) and the corresponding to the NiO and Py
(ROD) are also shown. A c-lattice parameter of 4.20(2) and 3.53(2) A are
obtained for NiO and Py, respectively.

Figure 13: Low angle reflectivity for different NiO/Py bilayers grown on Al2O3
(0001) substrates. Figure 14: (a) H = K reciprocal space maps for NiO/Py
bilayer (dPy = 24 nm) grown Al2O3 (0001) substrate. (b, c, d) CTR and RODs
scans obtained for H = K = 0.92, H = K = 1 and H = K = 1.1. Figure 15: (a)
Py(4nm) and (b) Py(4nm)/NiO VNA-FMR results of the field derivative of the
transmission implemented as frequency-field maps. Note that the colormap scale
range in (b) has been deliberately decreased to appreciate the low FMR signal.

##  Appendix B Low thickness VNA-FMR measurements

We conducted VNA-FMR experiments to analyze the dynamic response of extremely
thin Py and Py/NiO films. In Figure 15(a), a 4 nm thick Py measurement is
presented, where the uniform mode is clearly observed. However, when the same
nominal thickness Py film is coupled with NiO, the signal coming from the FMR
mode signal greatly diminished (Figure 15(b)).

This result provides further validation of the effective film thickness
reduction when coupling an AFM and an F, a consequence of the exchange bias
freezing the magnetic moments of the Py at the interface. This, in turn, leads
to an enhancement of the frequency of the PSSW modes. Our calculations
indicate that the frozen moments in our films reduce the effective thickness
by just over 2 nm upon the NiO addition, a value that remains largely
consistent across all of our films (see Figure 8(b)). Hence, these dynamic
measurements align with the expected outcomes for films as thin as these,
where a significant portion of the thickness is reduced, and the signal of the
ferromagnetic modes is nearly completely suppressed.

##  Appendix C Resolved standing spin waves modes above n=2

The resolved main SW modes on our micromagnetic simulations outcome clearly
showed strong PSSWs dynamics corresponding to n=0, 1 and 2 (see Figure 6 in
the main text). However, an in-depth examination of the dynamics of the system
indicated weak modes above the second order PSSW as the thickness of the film
increases, as shown in Figure 16. These modes, distinguishable up until n=5,
also exhibit a similar pattern as the lower order PSSWs (f(GHz)= A +
B/d${}_{Py}^{2}$(nm)).

Figure 16: Evolution of the PSSW modes above n=2 at a constant in-plane
magnetic field of H = 2 kOe.

##  Appendix D Simulations on the exchange bias along the thickness of the
film

As mentioned in section IV.1, the difference between the values obtained of
the exchange bias via MOKE (see Figure 2(c)), on the tenths or hundreds of Oe,
and the applied exchange bias at the interface, 1 kOe, are distant in value.
As Figure 17 shows via simulations, this order of magnitude is recovered due
to the different location of the field detected/applied: MOKE spectroscopy on
top of the Py film, whereas the FM/AFM coupling that provides the exchange
bias, on the interface.

These simulations are done on a Py(48nm) sample and the exchange bias field is
applied on the first 2 nm of the sample, which is coherent with the reduced
effective thickness calculations displayed on Figure 8(b).

Figure 17: Thickness evolution of the effective field in the exchange bias
direction on a 48 nm Py simulated sample. The exchange bias is applied at the
interface (d/dPy=0).

##  Appendix E Evolution of the SW modes in a Py/AFM bilayer with the
thickness of the film

Figure 18: (a) Evolution of the primary SW mode frequencies at a constant
magnetic field of H = 2 kOe as the Py layer thickness varies from 20 nm to 200
nm in a Py/AFM bilayer. The FFT power is normalized at each film thickness.

Adding an AFM/FM interface with Ku = 10-5 J/m3 and an exchange bias field of
Heb = 1 kOe, as suggested by the parameter optimization described in Section
IV.1 of the main text, leads to noticeable shifts in the frequencies of the
PSSWs modes for low thicknesses (refer to Figure 18) in comparison with the Py
system lacking the AFM/FM interface depicted in Figure 6(a). These frequency
shifts are towards higher frequencies, consistent with our experimental
findings using the VNA-FMR system collected in Figure 3(a).

The resolved antisymmetric modes, particularly n=3, exhibit significantly
greater amplitude compared to the Py system. This amplification is attributed
to the induced out-of-plane asymmetry caused by the uniaxial anisotropy and
the exchange bias of the system, which are localized on the bottom layer where
the AFM is placed. Consequently, the applied sinc-shaped magnetic pulse
excites the antisymetric modes more effectively. Another interesting
characteristic of the resolved modes is that at larger thicknesses of the Py
layer (150-165 nm), there is an abrupt shift towards lower frequencies,
suggesting a diminished impact of the AFM. In these cases, the influence of
the uniaxial anisotropy and the exchange bias is reduced. The precise values
of these interface parameters determine the sharp, step-like transition
observed in the modes.

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