capbtabboxtable[][]

# MiDAS: Multi-integrated Domain Adaptive Supervision for Fake News Detection

Abhijit Suprem
School of Computer Science
Georgia Institute of Technology
Atlanta, GA 30313
<EMAIL_ADDRESS>
&Calton Pu
School of Computer Science
Georgia Institute of Technology
Atlanta, GA 30313
<EMAIL_ADDRESS>

###### Abstract

Covid-19 related misinformation and fake news, coined an ‘infodemic’, has
dramatically increased over the past few years. This misinformation exhibits
concept drift, where the distribution of fake news changes over time, reducing
effectiveness of previously trained models for fake news detection. Given a
set of fake news models trained on multiple domains, we propose an adaptive
decision module to select the best-fit model for a new sample. We propose
MiDAS, a multi-domain adaptative approach for fake news detection that ranks
relevancy of existing models to new samples. MiDAS contains 2 components: a
doman-invariant encoder, and an adaptive model selector. MiDAS integrates
multiple pre-trained and fine-tuned models with their training data to create
a domain-invariant representation. Then, MiDAS uses local Lipschitz smoothness
of the invariant embedding space to estimate each model’s relevance to a new
sample. Higher ranked models provide predictions, and lower ranked models
abstain. We evaluate MiDAS on generalization to drifted data with 9 fake news
datasets, each obtained from different domains and modalities. MiDAS achieves
new state-of-the-art performance on multi-domain adaptation for out-of-
distribution fake news classification.

##  1 Introduction

The misinformation and fake news associated with the COVID-19 pandemic, called
an ‘infodemic’ by WHO (Enders et al., 2020), have grown dramatically, and
evolved with the pandemic. Fake news has eroded institutional trust (Ognyanova
et al., 2020) and have increasingly negative impacts outside social
communities (Quinn et al., 2021). The challenge is to filter active fake news
campaigns while they are raging, just like today’s online email spam filters,
instead of offline, retrospective detection long after the campaigns have
ended. We divide this challenge to detect fake news online into two parts: (1)
the variety of data (both real and fake), and (2) the timeliness of data
collection and processing (both real and fake). In this paper, we focus on the
first (variety) part of challenge, with the timeliness (which depends on
solutions to handle variety) in future work (Pu et al., 2020).

The infodemic, and fake news more generally, evolves with a growing variety of
ephemeral topics and content, a phenomenon called real concept drift (Gama et
al., 2014). However, the excellent results on single-domain classification
(Chen et al., 2021), have generalization difficulties when applied to cross-
domain experiments (Wahle et al., 2022; Suprem and Pu, 2022). A benchmark
study over 15 language models shows reduced cross-domain fake news detection
accuracy (Wahle et al., 2022). A generalization study in (Suprem and Pu, 2022)
finds significant performance deterioration when models are used on unseen,
non-overlapping datasets. Intuitively, it is entirely reasonable that state-
of-the-art models trained on one dataset or time period will have reduced
accuracy on future time periods. Real concept drift is introduced into fake
news as content changes (Gama et al., 2014), camouflage (Shrestha and
Spezzano, 2021), linguistic drift (Eisenstein et al., 2014), and adversarial
adaptation by fake news producers when faced with debunking efforts such as
CDC on the pandemic (Weinzierl et al., 2021).

To catch up with concept drift, the classification models need to be expanded
to cover a wide variety of data sets (Li et al., 2021; Suprem and Pu, 2022;
Kaliyar et al., 2021), or augmented with new knowledge on true novelty such as
the appearance of the Omicron variant (Pu et al., 2020). In this paper, we
assume the availability of domain-specific authorative sources such as CDC and
WHO that provide trusted up-to-date information on the pandemic.

A key challenge of such multi-domain classifiers is a decision module to
select the best-fit model amongst a set of existing models to classify new
samples. This degree of knowledge is defined by the overlap between an
unlabeled sample and existing models’ training datasets (Suprem and Pu, 2022).
Intuitively, a best-fit model better captures a sample point’s neighborhood in
its own training data Urner and Ben-David (2013); Chen et al. (2022).

MiDAS..MiDAS.MiDAS.. We propose MiDAS, a multi-domain adaptative approach for
early fake news detection, with potential for online filtering. MiDAS
integrates multiple pre-trained and fine-tuned models along with their
training data to create a domain-invariant representation. On this
representation, MiDAS uses a notion of local Lipschitz smoothness to estimate
the overlap, and therefore relevancy, between a new sample and model training
datasets. This overlap estimate is used to rank models on relevancy to the new
sample. Then, MiDAS selects the highest ranked model to perform
classification. We evaluate MiDAS on 9 fake news datasets obtained from
different domains and modalities. We show new state-of-the-art performance on
multi-domain adaptation for early fake news classification.

Contributions..Contributions.Contributions.. Our contributions are as follows:

  * 1\.

MiDAS: a framework for adaptive model selection by using sample-to-data
overlap to measure model relevancy

  * 2\.

Experimental results of MiDAS on 9 fake news datasets with state-of-the-art
results using unsupervised domain adaptation.

##  2 Related Work

###  2.1 Multi-Domain Adaptation

Domain adaptation maps a target domain into a source domain. This allows a
classifier learned from the source domain to predict the target domain samples
(Farahani et al., 2021). Some approaches focus on a domain invariant
representation between source and target (Huang et al., 2021). Then, a new
classifier can be trained on this invariant representation for both source and
target samples. Domain invariance is scalable to multiple source domains by
fusing their latent representations with an adversarial encoder-discriminator
framework (Li et al., 2021). For multi-source domain adaptation (MDA),
classifiers for each source have different weights: static weights using
distance (Li et al., 2021) or per-sample weights on l2 norm (Suprem et al.,
2020).

###  2.2 Label Confidence

Alongside domain adaptation, weak supervision (WS) is also common for
propagating labels from source domains to a target domain (Ratner et al.,
2017). Both approaches estimate labels closest to the true label of the target
domain sample. This works with the assumption that the source domains or
labeling functions, respectively, are correlated to the true labels due to
expertise and domain knowledge. In each case, whether MDA or WS, domains or
labeling functions need to be weighted to ensure reliance on the best-fit
source. Snorkel, from (Ratner et al., 2017), uses expert labeling functions
and weighs them on conditional independence. Similarly, approaches in (Chen et
al., 2020; Fu et al., 2020) use coverage of expert foundation models and weigh
on distance to embedded sample. EEWS from (Rühling Cachay et al., 2021)
directly combines source data and labeling function in estimator
parametrization to generate dynamic weights for each sample. MDA approaches
weigh sources with weak supervision (Li et al., 2021), distance (Suprem and
Pu, 2022), or as team-of-experts (Pu et al., 2020).

##  3 Problem Setup and Strategy

Let there be $k$ source data domains, with labels
$\\{X_{i},Y_{i}\\}_{i=1}^{k}\in\\{D\\}_{(}i=1)^{k}$. Each of these source has
an associated source model SM, with a total of $k$ SMs:
$\\{f_{i}\\}_{(}i=1)^{k}$, where we have access to the training data $X_{i}$
and weights $w_{i}$. Each SM yields hidden embeddings through a feature
extractor backbone, or foundation model (Bommasani et al., 2021). Embeddings
are projected to class probabilities with any type of classification
layer/module.

MiDAS adaptively weights the $k$ SM predictions for some unlabeled,
potentially drifted target data domain $X^{\prime}$. We accomplish this by
using local embedding smoothness of the SMs as a proxy for model relevance to
a sample $x^{\prime}\in X^{\prime}$. SMs are typically smooth in the embedding
space (Urner and Ben-David, 2013); further, smoothness is correlated with
local accuracy (Chen et al., 2022).

With MiDAS, we rank each $f_{k}$ on a smoothness measure around the embedding
for $x^{\prime}$, i.e. $f_{k}(x^{\prime})$. Then, MiDAS can directly use the
top-ranked $f_{k}$, or the smoothest $f_{k}$ under a smoothness threshold, as
the best-fit relevant models for $x^{\prime}$, with the remaining models
abstaining. Because we are directly measuring smoothness on the embedding
space, MiDAS can use already fine-tuned, state-of-the-art classifiers for each
task, allowing off-the-shelf, plug-and-play usage. These classifiers are
foundation models (Bommasani et al., 2021) that have been fine-tuned with
architectural changes, learned weights, and hyperparameter tuning for their
specific dataset.

There are two key challenges in MiDAS:

  * 1\.

How do we compare smoothness of SMs that have been trained on different
domains?

  * 2\.

How can be measure the smoothness itself for unlabeled samples in the
embedding space of SMs?

We address (1) with an encoder $E$ that generates a domain invariant
representation on $\\{X_{i}\\}_{i=1}^{k}$. This unifies the data domains,
allowing comparisons for different SMs to start from the same source domain.
For (2), we extend the idea of local Lipschitz smoothness from (Chen et al.,
2022) and (Urner and Ben-David, 2013) to randomized Lipschitz smoothness. In
randomized Lipschitz smoothness, we randomly perturb $E(x^{\prime})$, the
domain invariant representation of $x^{\prime}$. Then, we compute the local
lipschitz constant $L$ on these perturbations $E(x^{\prime})+\epsilon$, with
respect to $E(x^{\prime})$ to measure smoothness. This allows us to calculate
an $L_{k}$ for each $f_{k}$ and use the local Lipschitz constant, a measure
for the local smoothness, as a measure of relevancy.

##  4 MiDAS

Figure 1: MiDAS architecture: The encoder generates a domain invariant
representation. We add perturbation to this representation. Then, fine-tuned
models $\\{M\\}_{(}i=1)^{k}$ process the sample and perturbations. After
computing the local Lipschitsz constant for each model, we can rank their L
scores and select the best-fit model’s label from the label matrix.

We now describe MiDAS components and implementation details. The MiDAS
architecture is shown in Figure 1. First, we cover the encoder-decoder
framework to generate the domain invariant representations of source and
target datasets. Then we present the randomized Lipschitz smoothness measure
to generate SM relevancy rankings.

###  4.1 Domain Invariant Encoder

To compare different $f_{k}$ relevancy, we require a common source domain (Li
et al., 2021). We achieve this with a single-encoder multiple-decoders design,
where we have a single encoder to generate a domain invariant representation
from all source domains. Then we use $k$ decoders to reconstruct the invariant
representation for each $f_{k}$. To train $E$, we use an adversarial
discriminator $D^{\prime}$ to enforce invariance with a min-max game, where
the discriminator tries to identify the source domain of the invariant
representation, and the encoder tries to fool it:

$\min_{E}\max_{D^{\prime}}-\sum_{i=1}^{k}\mathds{E}_{(x,y)\sim
X_{k}}[l(D^{\prime}(E(x)),k)]$ (1)

We use a gradient reversal layer $R$ to convert the min-max to single-step
minimization. $R$ is the identity matrix during the forward pass, and the
discriminator gradient during the backward pass, scaled by $\lambda=-1$. Then,
the adversarial optimization becomes:

$Loss=\min_{E,R(D^{\prime})}-\sum_{i=1}^{k}\mathds{E}_{(x,y)\sim
X_{k}}[l(D^{\prime}(R(E(x))),k)]$ (2)

To train $D_{k}$, we use the masked language modeling loss from BERT and
AlBERT pretraining. Parameters are initialized with AlBERT’s albert-base-
uncased weights (Lan et al., 2019).

In summary, we train a single encoder and $k$ decoders. The encoder projects
our training data in the form of SentencePiece tokens (Kudo and Richardson,
2018) into a domain-invariant representation, trained with a domain
discriminator. Each decoder then reconstructs the original input training
tokens from the invariant representation. We use decoders because MiDAS is
designed to work with our existing BERT and AlBERT classifiers, which expect
SentencePiece token input. Decoders are trained with masked language modeling,
where we randomly mask up to 15% of words or tokens in the input. Then, during
prediction, an unlabeled, potentially drifted sample $x^{\prime}$ from an
unseen distribution $X^{\prime}$ is converted to a domain invariant
representation $E(x^{\prime})$. Each decoder $D_{k}$ reconstructs from this
invariant representation the input to its corresponding SM $f_{k}$.

###  4.2 Randomized Lipschitz Smoothness

With a common embedding space, we can now compare relevancy of each model to
an unlabeled, potentially drift sample $x^{\prime}$. To present our approach,
we need to introduce Lipschitz continuity.

Definition (Lipshitz Continuity).Definition (Lipshitz Continuity)Definition
(Lipshitz Continuity). A function $f:\mathds{R}^{n}\rightarrow\mathds{R}^{m}$
is Lipschitz continuous if, for some metric space $(X,\theta)$, there exists a
constant $L$ such that

$\theta(f(x_{1}),f(x_{2}))\leq L\cdot\theta(x_{1},x_{2})$

We can extend this to define Lipschitz-smooth with respect to SM predictions
using Lipschitzness from (Chen et al., 2022).

Definition (Lipshitzness).Definition (Lipshitzness)Definition (Lipshitzness).
An SM is Lipschitz smooth if, for some class label $C$,

$|\Pr(f_{k}(x_{1})=C)-Pr(f_{k}(x_{2})=C)|\leq L_{k}\theta(x_{1},x_{2})$ (3)

That is, with $L_{k}$ smoothness, the diference in $f_{k}$’s predicted labels
on $x_{1}$ and $x_{2}$ is bounded by $L_{k}$ for all $x\in X$. However, the
local value of $L_{k}$ can vary across the embedding space. Consequently,
$f_{k}$ is smoother wherever $L_{k}$ is smaller 111As $L_{k}\rightarrow 0$,
the embedding function approaches mode collapse, where every input point is
projected to the same embedding point. As such, we want small $L_{k}$ for
samples in the same class, and large $L_{k}$ for samples from different
classes.

With these definitions, we can present our approach for finding the best-fit
relevant $f_{k}$ for $x^{\prime}$, defined as the $f_{k}$ with the smoothest
embedding space around $x^{\prime}$.

Theorem 1.Theorem 1Theorem 1. Let the best-fit $f_{k}$ for a sample
$x^{\prime}$ be the SM that is smoothest around $x^{\prime}$. We can find the
best-fit $f_{k}$ for a particular sample $x^{\prime}$, given a distance
threshold $\epsilon$, by solving:

$\operatorname*{arg\,min}_{k}\max_{\theta(x^{\prime},x_{r})\leq\epsilon|_{r=1}^{N}}\frac{\theta(\Pr(f_{k}(x^{\prime})),\Pr(f_{k}(x_{r})))}{\theta(x^{\prime},x_{r})}$
(4)

The $\max$ term estimates the $L_{k}$ value for each $f_{k}$ by sampling $N$
random points in an $\epsilon$-Ball around $x^{\prime}$. Then, we find the
$f_{k}$ that has the smallest $L_{k}$.

A key insight is that adversarial attacks exploit non-smoothness of a model’s
embedding space to fool classifiers, by generating a noise $\epsilon$ such
that $f_{k}(x+\epsilon)\neq f_{k}(x)$. This non-smoothness occurs when $f_{k}$
does not capture enough training data in the region around $x$ properly; in
GANs, this causes ‘holes’ in the latent space (Suprem et al., 2020) during
image synthesis. Conversely, adversarial defenses either enforce smoothness
around embedding space or on potentially perturbed inputs themselves (Das et
al., 2018). Similarly, GANs can enforce 1-Lipschitzness to improve coverage of
sample generation (Qin et al., 2020) .

So, given several $f_{k}$ with different local $L_{k}$ around the embedding
$f_{k}(x+\epsilon)$, a lower $L_{k}$ indicates smoother embedding space
because that SM has captured more training data in the region surrounding $x$
relative to other SMs, similar to the overlap metric calculated in (Suprem and
Pu, 2022) However, even with a low $L_{k}$, the classification labels
$y_{k}=f^{\prime}_{k}(f_{k}(x))$, obtained from the classification module
$f^{\prime}_{k}$ of the $k$th SM, can change on perturbations around $x$. We
can use probabilistic Lipschitzness to bound the probability of $y_{k}$
changing in the neighborhood $x+\epsilon$ as a function of the perturbation
$\epsilon$.

Definition (Probabilistic Lipschitzness).Definition (Probabilistic
Lipschitzness)Definition (Probabilistic Lipschitzness). Let
$\phi:\mathds{R}\rightarrow[0,1]$. Given $x^{\prime},x_{r}\sim P_{X}$, we say
that $f_{k}$ is $\phi$-Lipschitz if, for all $\epsilon>0$, there is an
increasing function $\phi(\epsilon)$ such that:

$\Pr_{x^{\prime},x_{r}\sim
P_{X}}[\theta(f_{k}(x^{\prime}),f_{k}(x_{r}))-\frac{1}{\epsilon}\theta(x^{\prime},x_{r})>0]\leq\phi(\epsilon)$
(5)

That is, a function that is Lipschitz by Definition 1 (L-Lipschitz) satisfies
Definition 3 ($\phi$-Lipschitz) with $\phi(\epsilon)=1$ if $\epsilon\geq 1/L$
and $\phi(\epsilon)=0$ if $\epsilon<1/L$.

From this, it follows that if $f_{k}$ satisfies the $\phi$-Lipschitz
condition, then the number of samples within an $\epsilon$-Ball of
$x^{\prime}$ that have a different label from $x^{\prime}$ is bounded by
$\phi(\epsilon)$, per (Urner and Ben-David, 2013). As we move further from
$x^{\prime}$, the probability of a label change increases. Let $a^{\prime}$ be
the accuracy for $f_{k}$ at $x^{\prime}$. If we know the accuracy drop
$\alpha$ at the edge of the $\epsilon$-Ball where labels change values, we can
bound the accuracy of predictions between $x^{\prime}$ and some perturbed
point $x_{r}$ in an $\epsilon$-Ball around $x^{\prime}$ to:

$a_{r}\geq a^{\prime}-\alpha\cdot\phi(\epsilon)$

If we calculate $\alpha$ for an SM using training examples of different labels
within the margins allowed by probabilistic Lipschitzness, we find that
accuracy bound depends only on choice of $\epsilon$. Consequently we can
approximate $f_{k}$’s accuracy on $x^{\prime}$ if $x_{r}$ is within a small
$\epsilon$-Ball around $x^{\prime}$ to be at least:

$\theta(\Pr(f_{k}(x^{\prime})),\Pr(f_{k}(x_{r})))\geq\Pr(f_{k}(x^{\prime})=y_{k})-\alpha\cdot\phi(\epsilon)$
(6)

###  4.3 Putting it Together: MiDAS

Recall that the encoder projects all samples to a common invariant domain
$E:\\{X\\}_{i=1}^{k}\rightarrow G$. Each SM-specific decoder $D_{k}$ then
converts from the invariant domain to the $k$-th source domain
$D_{k}:G\rightarrow X_{k}$. With this encoder-decoders framework, we can use
the invariant domain $G$ as the common source domain for all SMs.

Per Definition 3, accuracy is best estimated in an $\epsilon<1\L$ ball. Once
$E$ is trained, we can use the class cluster centers (obtained from K-Means
clustering on the embeddings) from the training data to compute a local
cluster-specific $L$ for each cluster in each $f_{k}$. This partitioning
allows us to compute local smoothness characteristics of the embedding space,
simialr to the concurrent work in (Chen et al., 2022). We use the cluster
centers because we want the strongest measure of smoothness for each $f_{k}$,
and this occurs near the cluster center; for example (Suprem and Pu, 2022) and
(Chen et al., 2022) both use the centroid to set up accuracy thresholds. Then,
we can estimate a local $L_{k}$ for each $f_{k}$ at the cluster centers with
the $m$ nearest point to the center using Equation 3. We take the maximum
$L_{k}$ among all $f_{k}$ to obtain the upper bound on the local smoothness
among the SMs. On this maximum, we can calculate
$\epsilon=1/\max({L_{k}}_{(}i=1)^{k})$. So, the only hyperparameter here is
$m$, the number of neighbors to compute the local $L_{k}$ for each label
cluster in each $f_{k}$. We explore impact of $m$ in Section 5.2.

Now, for an unlabeled sample $x^{\prime}$, we first generate the domain
invariant representation $x^{\prime}_{G}=E(x^{\prime})$. Then we perturb
$x^{\prime}_{G}$ to generate $r$ points $\\{x^{\prime}_{G}i\\}_{i=1}^{r}$ in
an $\epsilon$-Ball around $x^{\prime}_{G}$. Using $x^{\prime}_{G}$ and
$\\{x^{\prime}_{G}i\\}_{i=1}^{r}$, we can compute the local Lipschitz constant
$L^{\prime}_{k}$ for each $f_{k}$ using Equation 3. We have 2 possibilities:

  * 1\.

$L^{\prime}_{k}\geq 1/\epsilon$: $f_{k}$ that satisfy this condition abstain
from providing predictions, since the accuracy drop is unbounded.

  * 2\.

$L^{\prime}_{k}<1/\epsilon$: $f_{k}$ that satisfy this condition can provide
labels, because their accuracy is bounded per Equation 6.

##  5 Evaluation

###  5.1 Experimental Setup and Datasets

We implemented and evaluated MiDAS on PyTorch 1.11 on a server running NVIDIA
T100 GPUs. We have released our implementation code.

Table 1: We use 9 COVID-19 fake news datasets to evaluate MiDAS. Here we also present a motivating experiment with respect to concept drift and generalizability: for each dataset, we train an ‘oracle’ on the training data. The oracle’s performance on the test data is compared to an ensemble of the other 8 datasets. The latter tests the concept drift case, where models need to generalize to data distributions they have not yet encountered. There is significant accuracy drop due to concept drift, domain shift, and label overlap Dataset | Training | Testing |  | Oracle Acc.
---
(Fine-Tuning)
| Generalization Acc.
---
(Held-Out)
Decrease
kaggle_short (Patel, 2021) | 31k | 9K | 0.97 | 0.57 | 42%
kaggle_long (Patel, 2021) | 31k | 9K | 0.98 | 0.53 | 46%
coaid (Cui and Lee, 2020) | 5K | 1K | 0.97 | 0.56 | 42%
cov19_text (Agarwal, 2020) | 2.5K | 0.5K | 0.98 | 0.58 | 41%
cov19_title (Agarwal, 2020) | 2.5K | 0.5K | 0.95 | 0.62 | 35%
rumor (Cheng et al., 2021) | 4.5K | 1K | 0.83 | 0.54 | 35%
cq (Mutlu et al., 2020) | 12.5K | 2K | 0.71 | 0.51 | 28%
miscov19 (Memon and Carley, 2020) | 4K | 0.6K | 0.68 | 0.50 | 26%
covid_fake (Das et al., 2021) | 4K | 2K | 0.96 | 0.52 | 45%

MiDAS Datasets.MiDAS DatasetsMiDAS Datasets. We use 9 fake news datasets,
shown in Table 1. Where available, we used provided train-test splits;
otherwise, we performed class-balanced 70:30 splits. We performed a
preliminary motivating evaluation, shown in Table 1. Here, we have compared an
oracle case to the concept drift case. In the oracle case, we train 9 models,
one on each dataset, and then evaluate this model on its corresponding
dataset’s test set. This is the case where the prediction data matches the
training distribution. In the concept drift case, we trained a model on 8
datasets, then evaluated on the held-out dataset. In this case, the prediction
dataset, even though on the same topic of Covid-19 fake news detection, does
not match the training distribution. We see significant accuracy drops,
between 20% to 50%. This matches the observations in generalizability in
(Suprem and Pu, 2022) and (Wahle et al., 2022).

MiDAS Evaluation.MiDAS EvaluationMiDAS Evaluation. We evaluate MiDAS with
held-one-out testing at the dataset level similar to the generalization
studies approach in (Suprem and Pu, 2022). Our results are presented in
Section 5.2 To test MiDAS, we first train the encoder to learn a domain
invariant representation with all but one dataset. Then we evaluate MiDAS’
performance on classifying the unseen, i.e. drifted, dataset. We repeat this
for each dataset in Table 1. MiDAS’ performance is compared to Snorkel (Ratner
et al., 2017), EEWS (Rühling Cachay et al., 2021), an ensemble, and an
‘oracle’ fine-tuned AlBert model trained on the held-out dataset.

Ablation Study.Ablation StudyAblation Study. To further evaluate MiDAS’
efficacy, we also conducted an ablation study in Section 5.3 by varying the
number of training datasets, types of loss functions, masked language model
trainig, and loss weights.

Adjustment of $m$.Adjustment of $m$Adjustment of $m$. For each experiment in
Section 5.2, we sampled points in an $\epsilon$-ball around $x$, where
$\epsilon$ calculated using steps Section 4.3. We explore effects of adjusting
the radius of this sampling ball by changing $m$, the the number of nearest
neighbors, in Section 5.4. Specifically, we show that as the
sampling/perturbation radius increases beyond $\epsilon$, MiDAS’ accuracy
decreases. Conversely, as sampling radius is reduced, MiDAS increases accuracy
while sacrificing coverage.

(a) Overlapping clusters before MiDAS

(b) Domain invariance after MiDAS

(c) L-Values for different $m$

(d) $\epsilon$-Ball radius for different $m$

Figure 2: MiDAS characteristics. Figure 2(a) shows tSNE projection of all 9
datasets’ pretrained BERT encoder’s embeddings. Figure 2(b) shows dataset
embeddings when generated with MiDAS’ encoder. The label overlaps between true
news and fake news have been separated; in this case, enforcing domain
invariance forces true and fake labels of all datasets to cluster. In Figure
2(c) and Figure 2(d), we show impact of L-Value and resulting $\epsilon$
calculation for different values of $m$. We select $m=50$ for remaining
experiments.

###  5.2 MiDAS Evaluation

We now present evaluation results for MiDAS. In each experiment, we designate
a single dataset as the target dataset without labels, and the remaining
datasets act as source domains. In these cases, the held-one-out dataset acts
as the drifted dataset, similar to the generalization experiments in (Suprem
and Pu, 2022). We follow the steps in Section 4.3 to test MiDAS, and compare
classification accuracy to 5 approaches: (i) an ensemble of the training
models with equal weights, (ii) a Snorkel labeler (Ratner et al., 2017) that
treats each model as a labeling function, (iii) an EEWS labeler (Rühling
Cachay et al., 2021) that treats each model as a labeling function, (iv) a
KMP-model (Suprem and Pu, 2022) that uses KMeans clustering with proxies to
compute overlap, and (v) an ‘oracle’ AlBERT model fine-tuned on the held-out
dataset.

Figure 2 shows several characteristics of the MiDAS encoder. Figure 2(b) shows
domain invariance in the labels. Each point is a sample from the 9 datasets;
before applying MiDAS, there is significant label overlap because each dataset
exists in separate domains. After applying MiDAS, datasets are projected to a
domain invariance embedding. This forces sampes with the same label,
irrespective of source domain, to cluster together and reduce the label
overlap observed in (Suprem and Pu, 2022).

After training MiDAS’ encoder, we need to compute the $\epsilon$ radius using
$m$ nearest neighbors to the label cluster centers. We examine the impact of
different $m$ values for computing $\epsilon$ in Figure 2(c) and Figure 2(d).
As we increase the number of neighbors used in estimating local $L$, the
estimate for $L$ increases, indicating reduced smoothness the further we
deviate from the cluster center. In turn, this reduces the acceptable
$\epsilon$-ball radius to bound probability of label change, per (Urner and
Ben-David, 2013), since $\epsilon=1/L$. A large $m$ would significantly reduce
the size of the sampling $\epsilon$-ball, and perturbations would be
negligible. A small $m$ would yield a poor estimate for local $L$ and a large
sampling ball. We further explore impact of changing $m$ directly on accuracy
in Section 5.4. Here, we fix $m=50$ for remaining experiments, since we
observed the $\epsilon$-Ball radius generally stabilized around this value.

Table 2: MiDAS Evaluation: MiDAS outperforms on generalizing to unseen, drifted data points. For each held-out dataset, given 8 fine-tuned models trained on the remaining 8 datasets, MiDAS is able to select the best-fit model for each sample. Using this best-fit model, MiDAS outperforms an equal-weighted ensemble by over 30%. Dataset | Oracle Labels | No access to Labels
---|---|---
FT-AlBert | Ensemble | Snorkel | EEWS | KMP | MiDAS
kaggle_short | 0.97 | 0.57 | 0.61 | 0.67 | 0.69 | 0.86
kaggle_long | 0.98 | 0.53 | 0.63 | 0.68 | 0.70 | 0.71
coaid | 0.97 | 0.56 | 0.64 | 0.74 | 0.81 | 0.84
cov19_text | 0.98 | 0.58 | 0.59 | 0.68 | 0.75 | 0.79
cov19_title | 0.95 | 0.62 | 0.69 | 0.75 | 0.61 | 0.81
rumor | 0.83 | 0.54 | 0.59 | 0.62 | 0.45 | 0.67
cq | 0.71 | 0.51 | 0.54 | 0.52 | 0.56 | 0.57
miscov19 | 0.68 | 0.50 | 0.52 | 0.56 | 0.45 |  0.54
covid_fake | 0.96 | 0.52 | 0.56 | 0.61 | 0.50 | 0.75

In Table 2, we show results of MiDAS compared to the 5 approaches discussed
above. In all but one of our experiments, MiDAS outperforms other labeling
schemes in classifying the unseen, drifted samples. On average, MiDAS sees a
30% increase in accuracy compared to an ensemble. Further, by using the
training data itself to adaptively guide SM selection, MiDAS improves by 21%
on Snorkel, 10% on EEWS and KMP.

###  5.3 Ablation Study

We evaluated the impact of several design and training choices for MiDAS in an
ablation study in Figure 4 We use a version of MiDAS trained with half of the
sources with the most data points for each experiment (MiDAS-Half). This
yields near-random accuracy, since this is a modified ensemble on different
source datasets. Adding the remaining sources improves MiDAS’ coverage and
improves accuracy by 15%. We add a center loss term (He et al., 2018) to the
encoder output to encourage clustering on the labels between multiple sources;
increasing accuracy by 5%. Next, we added language masking to the input during
the encoder-decoder training to further fine-tune the encoder for the fake-
news tasks, yielding a 4% improvement. Finally, we increased the weights for
the discriminator loss compared to encoder loss to emphasize domain
invariance, yielding a 5% improvement for MiDAS’ accuracy on fake news
detection for unseen, drifted data. We compare convergence for different
experiments in Figure 4: the encoder converges faster in each case. Further,
adding the center and weighted losses contribute to discriminator fooling and
stabilizing the discriminator loss.

floatrowsep=none

Dataset | MiDAS-Half | +Sources | +Center Loss | \+ Masking | +Weighted Loss
---|---|---|---|---|---
kaggle_short | 0.56 | 0.68 | 0.75 | 0.79 | 0.86
kaggle_long | 0.55 | 0.62 | 0.65 | 0.68 | 0.71
coaid | 0.53 | 0.71 | 0.76 | 0.79 | 0.84
cov19_text | 0.54 | 0.68 | 0.72 | 0.75 | 0.79
cov19_title | 0.58 | 0.71 | 0.76 | 0.78 | 0.81
rumor | 0.57 | 0.61 | 0.63 | 0.65 | 0.67
cq | 0.52 | 0.54 | 0.55 | 0.55 | 0.57
miscov19 | 0.51 | 0.53 | 0.53 | 0.54 | 0.54
covid_fake | 0.51 | 0.59 | 0.64 | 0.69 | 0.75

Figure 3: MiDAS Ablation Study. We examined impact of different design choices
here. Of note is that using masked language modeling is significantly useful
in improving end-to-end accuracy. Further, adding a center loss term ensures
the domain-invariance representations have separable clusters, as we see in
Figure 2(b). Finally, with weighted loss, we give the discriminator 2x the
importance of the encoder masking loss to focus on domain invariance.

[4cm]

Figure 4: Convergence of MiDAS with each component in ablation study.

###  5.4 Adjustment of $\epsilon$

Table 3: Impact of $m$ values: As we increase the number of nearest neighbors, we get a higher estimate for L, per Figure 2(c), which reduces the $\epsilon$-Ball sampling radius and relaxes threshold for model predictions. This leads to lower accuracy with higher coverage. Decreasing $m$, in turn, increases accuracy at the cost of lower coverage. m-value | kaggle_short | coaid | rumor | cq
---|---|---|---|---
F1-Score | Coverage | F1-Score | Coverage | F1-Score | Coverage | F1 | Coverage
m=1 | 0.97 | 0.03 | 0.97 | 0.01 | 0.86 | 0.05 | 0.81 | 0.03
m=10 | 0.91 | 0.35 | 0.92 | 0.29 | 0.82 | 0.48 | 0.74 | 0.34
m=20 | 0.87 | 0.65 | 0.85 | 0.73 | 0.73 | 0.63 | 0.61 | 0.59
m=50 | 0.86 | 0.86 | 0.84 | 0.89 | 0.67 | 0.82 | 0.57 | 0.78
m=75 | 0.73 | 0.91 | 0.78 | 0.94 | 0.62 | 0.89 | 0.53 | 0.85
m=100 | 0.69 | 0.95 | 0.73 | 0.98 | 0.59 | 0.97 | 0.51 | 0.91
m=150 | 0.57 | 0.99 | 0.56 | 1.00 | 0.54 | 1.00 | 0.43 | 0.95

Finally, we investigate $\epsilon$ with respect to $m$, which was fixed at
$m$=50. For these experiments, we investigated increasing and decreasing $m$
to, respectively, increase and decrease $\epsilon$. Increasing the neighbors
increases the computed $L$, since we are using points further from the smooth
cluster center. In turn, this reduces the sampling $\epsilon$-ball, so the
perturbations we apply will be smaller, and in some cases, negligible.
Furthermore, threshold value of $L$ needed to accept an SM’s prediction is
higher (since it is $1/\epsilon$), so MiDAS tolerates lower smoothness for
each model, and accepts predictions from more models, resulting in higher
coverage and lower overall accuracy. On the other hand, using fewer neighbors
means larger sampling ball and smaller threshold for acceptance. It is more
likely for perturbed samples to be further away, yielding a higher value of L
unless a corresponding model is especially smooth around that point. This
would, as a result, reduce coverage, but increase accuracy.

We show this in Table 3 for several values of $m$ across 4 of our datasets.
Using only the nearest neighbor yields minimal coverage. As we increase the
$m$, coverage increases significantly, and accuracy approaches ensemble
accuracy. Conversely, as we reduce $m$, we also reduce $L$ and consequently,
the smoothness threshold to accept a prediction. This increases accuracy,
since MiDAS rejects predictions that do not satisfy the threshold. However,
coverage decreases as well: we show in Table 3 fewer unseen samples from the
target domain can be labeled as we decrease $m$. We also see that $m$ can have
outsized impact on accuracy as well: ‘coaid’ f1 scores drop from 0.73 to 0.56,
even though coverage increases only slightly, from 0.98 to 1.0 when we
increase $m$ from 100 to 150. This occurs because at $m=150$, MiDAS’ relaxed
thresholds allow poorer models to provide predictions as well, reducing
accuracy in the final averaged result.

###  5.5 Limitations and Future Directions

We tested MiDAS in the scenario where fine-tuned models already exist. This
constrains the MiDAS encoder, which must also train a decoder so match the
inputs of the fine-tuned models that expect tokenized input. A more flexible
approach would deploy models and MiDAS together, with each fine-tuned model
directly trained with the MiDAS encoder. This would improve both training
time, convergence, as well as accuracy, since each model would directly use
the MiDAS generated encodings, instead of domain-specific reconstructions.

Furthermore, we selected $m$ using empirical observations. However, there can
be technically grounded approaches as well, such as using the high-density
bands from (Suprem et al., 2020; Jiang et al., 2018). In these cases, the
$\alpha$-high density region of each cluster can be used to estimate a good
$m$. We leave further exploration of $m$ as well as integration of fine-tuned
models into the encoder training framework to future work.

##  6 Conclusion

We have presented MiDAS, a system for adaptively selecting best-fit model for
a set of samples from drifting distributions. MiDAS uses a domain-invariance
embedding to estimate local smoothness for fine-tuned models around drifting
samples. By using local smoothness as a proxy for accuracy and training data
relevancy, MiDAS improves on generalization accuracy across 9 fake news
datasets. With MiDAS, we can detect COVID-19 related fake news with over 10%
accuracy improvement over weak labeling approaches. We hope MiDAS will lead
further exploration into the tradeoff between generalizability and fine-
tuning, as well as research into mitigating generalization difficulties of
pre-trained models.

## References

  * Agarwal (2020) Isha Agarwal.  Covid 19 Fake News Dataset.  6 2020.  doi: 10.6084/m9.figshare.12489293.v1.  URL https://figshare.com/articles/dataset/COVID19FN_Dataset_csv/12489293.
  * Bommasani et al. (2021) Rishi Bommasani, Drew A Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S Bernstein, Jeannette Bohg, Antoine Bosselut, Emma Brunskill, et al.  On the opportunities and risks of foundation models.  _arXiv preprint arXiv:2108.07258_ , 2021.
  * Chen et al. (2021) Ben Chen, Bin Chen, Dehong Gao, Qijin Chen, Chengfu Huo, Xiaonan Meng, Weijun Ren, and Yang Zhou.  Transformer-based language model fine-tuning methods for covid-19 fake news detection.  In _International Workshop on Combating Hostile Posts in Regional Languages during Emergency Situation_ , pages 83–92. Springer, 2021.
  * Chen et al. (2020) Mayee F Chen, Daniel Y Fu, Frederic Sala, Sen Wu, Ravi Teja Mullapudi, Fait Poms, Kayvon Fatahalian, and Christopher Ré.  Train and you’ll miss it: Interactive model iteration with weak supervision and pre-trained embeddings.  _arXiv preprint arXiv:2006.15168_ , 2020.
  * Chen et al. (2022) Mayee F Chen, Daniel Y Fu, Dyah Adila, Michael Zhang, Frederic Sala, Kayvon Fatahalian, and Christopher Ré.  Shoring up the foundations: Fusing model embeddings and weak supervision.  _arXiv preprint arXiv:2203.13270_ , 2022.
  * Cheng et al. (2021) Mingxi Cheng, Songli Wang, Xiaofeng Yan, Tianqi Yang, Wenshuo Wang, Zehao Huang, Xiongye Xiao, Shahin Nazarian, and Paul Bogdan.  A covid-19 rumor dataset.  _Frontiers in Psychology_ , 12:1566, 2021.
  * Cui and Lee (2020) Limeng Cui and Dongwon Lee.  Coaid: Covid-19 healthcare misinformation dataset.  _arXiv preprint arXiv:2006.00885_ , 2020.
  * Das et al. (2018) Nilaksh Das, Madhuri Shanbhogue, Shang-Tse Chen, Fred Hohman, Siwei Li, Li Chen, Michael E Kounavis, and Duen Horng Chau.  Shield: Fast, practical defense and vaccination for deep learning using jpeg compression.  In _Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining_, pages 196–204, 2018.
  * Das et al. (2021) Sourya Dipta Das, Ayan Basak, and Saikat Dutta.  A heuristic-driven uncertainty based ensemble framework for fake news detection in tweets and news articles.  _Neurocomputing_ , 2021.
  * Eisenstein et al. (2014) Jacob Eisenstein, Brendan O’Connor, Noah A Smith, and Eric P Xing.  Diffusion of lexical change in social media.  _PloS one_ , 9(11):e113114, 2014.  ISSN 1932-6203.
  * Enders et al. (2020) Adam M Enders, Joseph E Uscinski, Casey Klofstad, and Justin Stoler.  The different forms of covid-19 misinformation and their consequences.  _The Harvard Kennedy School Misinformation Review_ , 2020.
  * Farahani et al. (2021) Abolfazl Farahani, Sahar Voghoei, Khaled Rasheed, and Hamid R Arabnia.  A brief review of domain adaptation.  _Advances in Data Science and Information Engineering_ , pages 877–894, 2021.
  * Fu et al. (2020) Daniel Fu, Mayee Chen, Frederic Sala, Sarah Hooper, Kayvon Fatahalian, and Christopher Ré.  Fast and three-rious: Speeding up weak supervision with triplet methods.  In _International Conference on Machine Learning_ , pages 3280–3291. PMLR, 2020.
  * Gama et al. (2014) Joao Gama, Indre Zliobaite, Albert Bifet, Mykola Pechenizkiy, and Abdelhamid Bouchachia.  A survey on concept drift adaptation.  _ACM computing surveys (CSUR)_ , 46(4):44, 2014\.  ISSN 0360-0300.
  * He et al. (2018) Xinwei He, Yang Zhou, Zhichao Zhou, Song Bai, and Xiang Bai.  Triplet-center loss for multi-view 3d object retrieval.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 1945–1954, 2018.
  * Huang et al. (2021) Yinqiu Huang, Min Gao, Jia Wang, and Kai Shu.  Dafd: Domain adaptation framework for fake news detection.  In _International Conference on Neural Information Processing_ , pages 305–316. Springer, 2021.
  * Jiang et al. (2018) Heinrich Jiang, Been Kim, Melody Guan, and Maya Gupta.  To trust or not to trust a classifier.  _Advances in neural information processing systems_ , 31, 2018.
  * Kaliyar et al. (2021) Rohit Kumar Kaliyar, Anurag Goswami, and Pratik Narang.  Mcnnet: generalizing fake news detection with a multichannel convolutional neural network using a novel covid-19 dataset.  In _8th ACM IKDD CODS and 26th COMAD_ , pages 437–437. 2021.
  * Kudo and Richardson (2018) Taku Kudo and John Richardson.  Sentencepiece: A simple and language independent subword tokenizer and detokenizer for neural text processing.  _arXiv preprint arXiv:1808.06226_ , 2018.
  * Lan et al. (2019) Zhenzhong Lan, Mingda Chen, Sebastian Goodman, Kevin Gimpel, Piyush Sharma, and Radu Soricut.  Albert: A lite bert for self-supervised learning of language representations.  _arXiv preprint arXiv:1909.11942_ , 2019.
  * Li et al. (2021) Yichuan Li, Kyumin Lee, Nima Kordzadeh, Brenton Faber, Cameron Fiddes, Elaine Chen, and Kai Shu.  Multi-source domain adaptation with weak supervision for early fake news detection.  In _2021 IEEE International Conference on Big Data (Big Data)_ , pages 668–676. IEEE, 2021.
  * Memon and Carley (2020) Shahan Ali Memon and Kathleen M Carley.  Characterizing covid-19 misinformation communities using a novel twitter dataset.  _arXiv preprint arXiv:2008.00791_ , 2020.
  * Mutlu et al. (2020) Ece C Mutlu, Toktam Oghaz, Jasser Jasser, Ege Tutunculer, Amirarsalan Rajabi, Aida Tayebi, Ozlem Ozmen, and Ivan Garibay.  A stance data set on polarized conversations on twitter about the efficacy of hydroxychloroquine as a treatment for covid-19.  _Data in Brief_ , page 106401, 2020.
  * Ognyanova et al. (2020) Katherine Ognyanova, David Lazer, Ronald E Robertson, and Christo Wilson.  Misinformation in action: Fake news exposure is linked to lower trust in media, higher trust in government when your side is in power.  _Harvard Kennedy School Misinformation Review_ , 2020.
  * Patel (2021) Sameer Patel.  Covid-19 Fake News Dataset (Kaggle).  2021\.  URL https://www.kaggle.com/code/therealsampat/fake-news-detection/.
  * Pu et al. (2020) Calton Pu, Abhijit Suprem, Rodrigo Alves Lima, Aibek Musaev, De Wang, Danesh Irani, Steve Webb, and Joao Eduardo Ferreira.  Beyond artificial reality: Finding and monitoring live events from social sensors.  _ACM Transactions on Internet Technology (TOIT)_ , 20(1):1–21, 2020.
  * Qin et al. (2020) Yipeng Qin, Niloy Mitra, and Peter Wonka.  How does lipschitz regularization influence gan training?  In _European Conference on Computer Vision_ , pages 310–326. Springer, 2020.
  * Quinn et al. (2021) Emma K Quinn, Sajjad S Fazel, and Cheryl E Peters.  The instagram infodemic: cobranding of conspiracy theories, coronavirus disease 2019 and authority-questioning beliefs.  _Cyberpsychology, Behavior, and Social Networking_ , 24(8):573–577, 2021.
  * Ratner et al. (2017) Alexander Ratner, Stephen H Bach, Henry Ehrenberg, Jason Fries, Sen Wu, and Christopher Ré.  Snorkel: Rapid training data creation with weak supervision.  In _Proceedings of the VLDB Endowment. International Conference on Very Large Data Bases_ , volume 11, page 269. NIH Public Access, 2017.
  * Rühling Cachay et al. (2021) Salva Rühling Cachay, Benedikt Boecking, and Artur Dubrawski.  End-to-end weak supervision.  _Advances in Neural Information Processing Systems_ , 34, 2021.
  * Shrestha and Spezzano (2021) Anu Shrestha and Francesca Spezzano.  Textual characteristics of news title and body to detect fake news: a reproducibility study.  In _European Conference on Information Retrieval_ , pages 120–133. Springer, 2021.
  * Suprem and Pu (2022) Abhijit Suprem and Calton Pu.  Exploring generalizability of fine-tuned models for fake news detection, 2022.  URL https://arxiv.org/abs/2205.07154.
  * Suprem et al. (2020) Abhijit Suprem, Joy Arulraj, Calton Pu, and Joao Ferreira.  Odin: Automated drift detection and recovery in video analytics.  _Proc. VLDB Endow._ , 13(12), 2020.  ISSN 2150-8097.  URL https://doi.org/10.14778/3407790.3407837.
  * Urner and Ben-David (2013) Ruth Urner and Shai Ben-David.  Probabilistic lipschitzness a niceness assumption for deterministic labels.  In _Learning Faster from Easy Data-Workshop@ NIPS_ , volume 2, page 1, 2013.
  * Wahle et al. (2022) Jan Philip Wahle, Nischal Ashok, Terry Ruas, Norman Meuschke, Tirthankar Ghosal, and Bela Gipp.  Testing the generalization of neural language models for covid-19 misinformation detection.  In _International Conference on Information_ , pages 381–392. Springer, 2022.
  * Weinzierl et al. (2021) Maxwell Weinzierl, Suellen Hopfer, and Sanda M Harabagiu.  Misinformation adoption or rejection in the era of covid-19.  In _Proceedings of the International AAAI Conference on Web and Social Media (ICWSM), AAAI Press_ , 2021.


# Probing spin textures in atomically thin CrSBr through tunneling
magnetoresistance

Ziqi Liu1 Chengfeng Zhu1 Yuchen Gao1 Zuxin Chen2<EMAIL_ADDRESS>Pingfan Gu1<EMAIL_ADDRESS>Yu Ye1,3,4<EMAIL_ADDRESS>1State Key
Laboratory for Artificial Microstructure $\rm{\&}$ Mesoscopic Physics and
Frontiers Science Center for Nano-optoelectronics, School of Physics, Peking
University, Beijing, 100871, China
2School of Semiconductor Science and Technology, South China Normal
University, Foshan 528225, China
3Yangtze Delta Institute of Optoelectronics, Peking University, Nantong 226010
Jiangsu, China
4Liaoning Academy of Materials, Shenyang, 110167, China

###### Abstract

The exploration of spin configurations and magnetoresistance in van der Waals
magnetic semiconductors, particularly in the realm of thin-layer structures,
holds paramount significance for the development of two-dimensional spintronic
nanodevices. In this Letter, we conducted comprehensive magnetotransport
measurements on a few-layer CrSBr using a vertical tunneling device
configuration. Notably, our investigation revealed that tunneling
magnetoresistance possesses a distinctive capability to discern spin
configurations that would otherwise remain indistinguishable through
alternative techniques such as photoluminescence. We observed the existence of
energy-degenerate states exhibiting identical net magnetization and comparable
spin configurations, which could be differentiated based on their
rectification properties, reminiscent of a diode-like behavior at positive and
negative bias voltages. Specifically, in devices comprising 5-layer CrSBr, we
observed an intriguing positive magnetoresistive state when subjected to an
in-plane magnetic field along the $b$-axis. To gain a deeper understanding of
the underlying mechanisms, we developed a one-dimensional linear chain model
that successfully computed the magnetic state, thereby elucidating the
underlying spin configurations responsible for the observed transport
phenomena. These findings not only provide novel perspectives into the
intricate spin textures of two-dimensional CrSBr but also underscore the
sensitivity of tunneling as a probing technique for investigating the magnetic
order in van der Waals materials.

##  I Introduction

Magnetic semiconductors play a crucial role in the field of condensed matter
physics and materials science owing to their unique combination of
semiconducting and magnetic properties [1, 2, 3, 4]. This amalgamation
presents promising opportunities for advances in spintronics, magnetic
storage, and quantum computing [5, 6, 7, 8]. The ability to manipulate both
electrical current flow and magnetic moment orientation within a single
material has great potential to improve the efficiency and versatility of
electronic devices, with potential applications in energy-efficient
electronics [9, 10, 11, 12]. However, the controlled and stable realization of
the desired combination of semiconducting and magnetic properties in materials
remains a formidable challenge.

The recently discovered van der Waals (vdW) material CrSBr has garnered
significant attention as a highly promising candidate due to its distinctive
direct bandgap semiconductor characteristics and layered A-type
antiferromagnetic (AFM) order [13]. Notably, CrSBr exhibits a pronounced spin-
charge interaction, where its band structure exhibits a strong dependence on
its magnetic order through interlayer interactions [14]. Consequently,
obtaining a comprehensive understanding of the evolution of spin textures in
atomically thin CrSBr under external magnetic fields is of utmost importance
for understanding the spin-charge interaction in a $d$-band hosting magnetic
semiconductor, as well as for designing vdW AFM-based spintronic devices.
Experimental investigations have demonstrated that below the Néel temperature
of 132 K, CrSBr adopts an A-type AFM state [13]. Further studies have unveiled
an intermediate ferromagnetic (iFM) state above the Néel temperature,
characterized by intralayer ferromagnetic (FM) order and random interlayer
magnetic order along the $c$-axis, with the critical temperature for
intralayer magnetization ($T_{\rm{C}}^{\rm{intra}}$) decreasing from $\sim$
160 K in bulk to $\sim$146 K in monolayers [15, 16]. Moreover, zero-field muon
spin rotation (ZF-$\mu$SR) studies have indicated a deceleration of magnetic
fluctuations below $\sim$100 K, with a spin freezing process occurring around
40 K [17]. Electrical transport and specific heat measurements have identified
two additional magnetic phase transitions at higher temperatures (156 K and
185 K) [18], underscoring the intricate magnetic nature of CrSBr. Previous
studies have reported layer-by-layer spin flips and overall spin-flop
processes in few-layer CrSBr under an external magnetic field along the
$b$-axis [19, 14]. However, the detailed mechanisms of each magnetic phase
transition still remain unclear based on the present experimental data.

Within the domain of vdW magnetic materials, particularly AFM semiconductors
or insulators, the absence of a discernible magnetic moment necessitates the
utilization of tunneling magnetoresistance as an effective and sensitive
method for characterizing their magnetic properties [20, 21]. Variations in
the band structure and material thickness resulting from magnetic phase
transitions can influence the tunneling probability [2, 12, 19]. As tunneling
probability is exponentially related to the barrier height and thickness of
the material, slight changes can induce significant modifications in tunneling
probability [22, 23]. Furthermore, the tunneling process has the ability to
distinguish between degenerate magnetic states, since the tunneling barrier is
influenced by both the bias voltage and the magnetic structure [24]. Given
that the tunneling current is obtained by integrating the tunneling
probability within the energy range that spans the Fermi levels of two metal
contacts, thermal excitation has minimal impact on it [25].

This study delves into the investigation of spin configurations and
magnetoresistance in few-layer CrSBr using vertical tunneling devices. When
subject to an in-plane external magnetic field along the $b$-axis, CrSBr
predominantly adopts colinear magnetic states, exhibiting a distinctive spin-
flip evolution process during field sweeping. Notably, our measurements of
tunneling magnetoresistance reveal spin configurations that are otherwise
indistinguishable using alternative techniques. We also observe the presence
of energy-degenerate states with identical magnetization but differing
rectification behaviors under positive and negative biases. Surprisingly, in
devices comprising 5-layer CrSBr, a positive magnetoresistive state appeared
when an in-plane field was applied along the $b$-axis. To gain a deeper
understanding of these phenomena, we develop a one-dimensional (1D) chain
model that computes the magnetic states and elucidates the underlying spin
textures that govern the transport properties. Overall, our findings provide
novel insight into the intricate spin structure of CrSBr and underscore the
sensitivity of tunneling as a probing technique for investigating magnetic
order in van der Waals materials.

Figure 1: Device structure and basic characterizations. (a), Top view of the
atomic structure of CrSBr. (b), Side view of the CrSBr atomic structure with
arrows indicating the spin orientation of each Cr atom, revealing an A-type
AFM ground state. (c, d), Schematic and optical image of the device. (e),
Tunneling resistance of a 4-layer CrSBr device from 300 K to 2 K at a bias
voltage of 10 mV. The dashed lines in the figure mark the two inflection
points at 40 K and 128 K. The inset shows the $I$-$V$ curve of the device at 2
K. (f), Tunneling resistance of the device with an in-plane magnetic field.
The angles of the magnetic field corresponding to the $a$\- and $b$-axes are
labeled in the figure. The bias voltage is 80 mV, and the magnitude of the
magnetic field is 1 kG.

##  II Results and Discussion

Single crystals of CrSBr are synthesized using the chemical vapor transport
(CVT) technique and subsequently exfoliated into few-layer flakes using the
conventional scotch tape method (see Methods). CrSBr possesses an orthorhombic
lattice with cell dimensions of $a$=3.50 ${\mathrm{\SIUnitSymbolAngstrom}}$,
$b$=4.76 ${\mathrm{\SIUnitSymbolAngstrom}}$, and $c$=7.96
${\mathrm{\SIUnitSymbolAngstrom}}$ [16], where each CrSBr layer is stacked
along the $c$-axis (Fig. 1a, b). Notably, CrSBr exhibits intralayer
ferromagnetism along the $b$-axis, while interlayer A-type AFM coupling occurs
along the $c$-axis (Fig. 1b). To investigate the magnetic transitions of
CrSBr, we fabricated vertical tunneling devices based on few-layer CrSBr
flakes (Fig. 1c). Cross-structured few-layer graphite stripes are employed to
establish contact with the CrSBr tunneling layer, and the entire device is
encapsulated by hexagonal boron nitride (h-BN, see Methods). The intentionally
narrow width of the graphite layer aims to minimize the area of the tunneling
junction, thereby reducing the impact of multi-domain effects in the junction
area [23, 19]. The number of CrSBr layers is determined through atomic force
microscope measurements (Supplementary Fig. S1).

Initially, the resistance of the devices was measured over a temperature range
of 300 K to 2 K. Due to the semiconducting nature of CrSBr, the devices
exhibited a resistance in the range of a few thousand ohms at room temperature
(Fig. 1e). However, contrary to expectations based on typical semiconductor
behavior, the device resistance did not show an exponential increase as the
temperature decreased. Instead, a minor kink was observed at the Néel
temperature ($\sim$128 K), indicating the emergence of AFM magnetic order [15,
26, 16]. Notably, the device resistance displayed a significant increase as
the temperature decreased below 40 K, suggesting the dominance of the
tunneling process (inset of Fig. 1e). This behavior may be associated with the
spin-freezing process below 40 K [17]. Due to the strong in-plane anisotropy
of CrSBr, the tunneling resistance exhibited precise 180° periodic symmetry
when a small external magnetic field was applied and rotated in the $ab$-plane
(Fig. 1f), providing a means to determine the orientation of the $a$\- and
$b$-axes (as confirmed by polarized Raman spectra in Supplementary Fig. S1d).

Following the determination of the $a$\- and $b$-axes, the tunneling
resistance was measured by sweeping a magnetic field along the $a$\- or
$b$-axis at a fixed bias voltage to investigate the magnetic behavior of the
few-layer CrSBr. The magnetoresistance (MR) was calculated using the formula:

${\rm{MR}}=\frac{R(B)-R(B=0)}{R(B=0)}\times 100\%$ (1)

When a magnetic field was applied along the $a$-axis, the 4- and 5-layer CrSBr
devices exhibited consistent behavior. Specifically, the tunneling resistance
decreases with increasing magnetic field and eventually saturates, indicating
a negative MR (Fig. 2a, c). This observation aligns with the magnetic ground
state of CrSBr that magnetized along the $b$-axis, which features interlayer
A-type antiferromagnetic. In the presence of an applied magnetic field, the
magnetization of the layers gradually tilts towards the $a$-axis and
eventually becomes fully magnetized along the $a$-axis [14]. In the FM state,
the spin-up or spin-down electrons encounter lower tunneling barriers due to
the uniform direction of magnetization in all CrSBr layers, resulting in
significantly lower tunneling resistance than in the AFM state.

Figure 2:  Layer-dependent tunneling MR with magnetic field sweeping along the
$a$\- and $b$-axes. (a), Tunneling resistance of a 4-layer CrSBr device with
external magnetic field sweeping back and forth along the $a$-axis at 2 K and
70 mV bias voltage. (b), Tunneling resistance of the 4-layer CrSBr with the
external magnetic field sweeping along the $b$-axis at 60 mV bias voltage. Two
intermediate states with different resistance values are observed. (c),
Tunneling resistance of a 5-layer CrSBr with the external magnetic field
sweeping along the $a$-axis at 100 mV basis voltage. (d), Tunneling resistance
of the 5-layer CrSBr with the external magnetic field sweeping along the
$b$-axis at 100 mV bias voltage. A positive magnetoresistive state with an MR
of 50% is observed.

Nevertheless, when a magnetic field is applied along the $b$-axis, the 4- and
5-layer devices display distinct and unexpected transport properties, with
each device undergoing a multistep spin-flip process. These transitions are
accompanied by sharp changes in tunneling resistance (Figs. 2b and d). In the
case of the 4-layer CrSBr device, the material undergoes a transition from an
A-type AFM state (high-resistance state) to an FM state (low-resistance
state), resulting in a negative MR. Between the AFM and FM states, the
tunneling resistance assumes an intermediate value, displaying significant
hysteresis. Notably, at the same bias voltage and magnetic field, the
tunneling resistance of the device exhibits two distinct resistance values
during the ascending-field sweep and descending-field sweep, while the
photoluminescence spectra display consistent exciton emission energy [14].
Similarly, in this intermediate state, the tunneling resistance displays two
different values, low (high) and high (low), under negative and positive
magnetic fields, respectively, during the ascending-field (descending-field)
sweep (Fig. 2b).

Figure 3: 1D linear chain model and tunneling model. (a), Calculated results
of the 1D linear chain model for the 4-layer CrSBr. The red and blue lines
represent the magnetic evolution process for ascending and descending field
sweeps starting from the AFM state, respectively. The background dashed lines
indicate the calculated energy variations of the steady states with the
external magnetic field during the evolution process. (b), Calculated results
of the 1D linear chain model for 5-layer CrSBr. All lines have the same
meaning as in (a). (c), Schematic diagram illustrating the evolution of the
spin configuration of the 4-layer CrSBr with the magnetic field. The green and
orange arrows indicate the spin direction of each layer of CrSBr. The black
triangles indicate the ascending and descending field processes. (d),
Schematic diagram illustrating the evolution of the spin configuration of the
5-layer CrSBr with the magnetic field. (e), Schematic diagrams of the
tunneling barriers for the spin-up (orange) and spin-down (green) electrons in
the 4-layer CrSBr devices in state 1 and state 2. The directions of electron
tunneling are indicated by the black arrows.

In the case of the 5-layer device, a total of five distinct tunneling
magnetoresistive states are observed during the evolution under a magnetic
field along the $b$-axis. Taking the ascending-field sweep as an illustrative
example, the device transitions from the low-resistance state of the FM phase
(FM1) to a spin-flip intermediate resistive state (state1) and then to the
high-resistance state of the AFM phase (AFM1) during the field’s ascent. As
the positive magnetic field increases, the device enters another magnetic
state (state 2) with even higher resistance than the AFM state, exhibiting a
positive MR of up to 50%. Subsequently, it undergoes another spin-flip
intermediate resistance state (state 3) to reach the FM low-resistance state
(FM2) polarized along the positive magnetic field. It should be noted that the
descending field process is a time-reversal process of the ascending field
process (Fig. 2d).

To analyze the observed phenomena, a 1D linear chain model is employed to
compute the magnetic state of CrSBr under an external magnetic field. The
total energy of the CrSBr magnetic system is divided into three components:
the AFM coupling between adjacent layers, the magnetic anisotropy energy, and
the magnetization energy induced by the external field.

$\begin{split}E=&\sum_{i=1}^{n-1}{J_{i,i+1}\cos{(\varphi_{i+1}-\varphi_{i})}}+\sum_{i=1}^{n}{K_{i}\sin^{2}{(\varphi_{i})}}\\\
&-\mu H\sum_{i=1}^{n}{\cos{(\varphi_{i})}}\end{split}$ (2)

where $\varphi_{i}$ represents the angle between the spin orientation of the
$i$-th layer and the $b$-axis. We set $J_{ij}=J=1$, $K_{i}=K=1$ and $\mu=1$ in
our model, consistent with previous studies [27]. Due to the significant
magnetic anisotropy energy ($K$) of CrSBr, the magnetic state evolution
encounters formidable barriers. Recent studies have suggested that the
evolution of the magnetic state in CrSBr may involve domain nucleation and the
propagation of domain walls[28], enabling real systems to ”leap” over these
barriers. To address this, Monte Carlo and conjugate gradient descent methods
are utilized to determine all stable magnetic states and compare their
energies, thereby identifying the actual path of magnetic evolution. The
actual evolutionary path involves a transition from a high-energy state to a
low-energy state, with the Zeeman energy favoring the alignment of the
magnetic moment parallel to the external magnetic field (Figs. 3a and b).

Here, we take the ascending-field sweep of the 4-layer CrSBr as an example
(Figs. 3a and c), when subjected to a substantially large negative external
magnetic field, the spin orientation of each layer aligns with the external
field, exhibiting an FM (FM1, $\downarrow\downarrow\downarrow\downarrow$)
state along the $b$-axis. As the external field increases, one of the two
middle layers undergoes a spin flip as an initial step to minimize the total
energy by lowering the AFM coupling energy between the adjacent layers. Due to
the device’s asymmetric character, one of these two layers is more likely to
undergo the initial flip, reaching state 1 (e.g.,
$\downarrow\uparrow\downarrow\downarrow$). With a further increase of the
external field, one surface layers also flips its spin orientation, causing
CrSBr to reach the A-type AFM ground state (AFM1,
$\downarrow\uparrow\downarrow\uparrow$). As the magnetic field sweeps towards
positive values, the surface spin-flip process flips the surface layer that is
antiparallel to the magnetic field, transitioning CrSBr into state 2
($\uparrow\uparrow\downarrow\uparrow$). Subsequently, the spin orientation of
the intermediate layer undergoes a flip, resulting in CrSBr transforming into
an FM state aligned with the external field (FM2,
$\uparrow\uparrow\uparrow\uparrow$). The descending-field sweep represents a
time-reversal operation of the ascending-field sweep, during which the 4-layer
CrSBr goes through state 1′ and state 2′ (Fig. 3c).

It is worth emphasizing that state 1 and state 2 exhibit significantly
different tunneling resistances, which cannot be distinguished by
photoluminescence [14]. A toy model of the two-spin channel demonstrates that
the tunneling barrier on each layer of CrSBr is modulated in the presence of a
bias voltage (Supplementary Note S1), leading to a discrepancy in tunneling
resistance between the two states (Fig. 3e). State 1′ (state 2′) represents
the time-reversed counterpart of state 1 (state 2), and according to the two-
spin channel model, it is expected to have the same tunneling resistance as
state 1 (state 2), which was also confirmed in the experimental measurements
(Fig. 2b). Notably, state 1 (state 2) and state 2′ (state 1′) are
energetically degenerate states with identical net magnetization and similar
spin configurations, but they can be differentiated based on their tunneling
resistance. Ideally, reversing the current direction in these states would
result in a change in their tunneling resistance, leading to rectification of
the $I$-$V$ curve.

Similarly, in the case of a 5-layer CrSBr device during an ascending-field
sweep, the initial FM state transitions to the AFM ground state through two
sequential spin-flip processes. For the odd-layer samples, the uncompensated
layer’s net magnetization in the AFM state breaks the energy degeneracy under
a non-zero external magnetic field. As the positive external magnetic field
increases, the spin orientations of both surface layers align in the opposite
direction to the external field. With further increase of the magnetic field,
the spins of the two surface layers flip one after the other, followed by the
spin of the intermediate layer, leading to the sample transitioning into the
FM state, magnetized according to the direction of the magnetic field. The
descending-field sweep represents the time-reversal process of the ascending-
field sweep (Fig. 3d). However, while the spin configurations during the field
sweep have been determined, the underlying cause of positive MR in the
$\downarrow\downarrow\uparrow\downarrow\uparrow$
($\uparrow\uparrow\downarrow\uparrow\downarrow$) magnetic state remains
unknown. The two-spin channel model provides unconventional insights,
suggesting a more complex origin. First, there is a one-to-one correspondence
between the spin configuration and the band structure in this material, with
certain spin configurations potentially having larger energy band gaps,
leading to increased tunneling barriers [29]. Secondly, the charge transfer
resulting from the difference in work functions between CrSBr and graphite may
influence the spin-filtering effect of CrSBr in close proximity to the
graphite layer, resulting in positive MR. Additionally, the proximity effect
[30] caused by the strong spin-orbit coupling of CrSBr in graphite may also
impact the MR of the tunneling junction. The specific reasons for this
phenomenon require further investigation, and a more detailed discussion can
be found in the Supplementary Note. S2. Due to the asymmetry of the spin
configurations, reversing the current direction in the
$\downarrow\uparrow\downarrow\downarrow\downarrow$ and
$\uparrow\uparrow\downarrow\uparrow\downarrow$ states leads to two additional
tunneling resistances (Supplementary Fig. S2).

Figure 4: Rectification properties of asymmetric spin configurations and
evolution of magnetic states with temperature and magnetic field direction.
(a), $I$-$V$ curve mapping of a 4-layer CrSBr device under the external
magnetic field along the $b$-axis descending-field (upper panel) and
ascending-field (lower panel) sweeps. (b), $I$-$V$ curve mapping of a 5-layer
CrSBr device under the external magnetic field along the $b$-axis descending-
field (upper panel) and ascending-field (lower panel) sweeps. MR values near
zero bias voltage are noisy due to the measurement resolution limit. (c),
Tempearture-dependent tunneling resistance of a 5-layer CrSBr device along the
$b$-axis magnetic field. The bias voltages are 100 mV at 2 K, 5 K, and 7 K, 80
mV at 10 K, and 50 mV at 15 K. (d), Tunneling resistance of a 5-layer CrSBr
device as a function of the magnetic field direction during rotation of the
external magnetic field from the $b$-axis to the $a$-axis. The angle labeled
in the figure represents the angle between the magnetic field direction and
the CrSBr $b$-axis. The bias voltage is 100 mV.

To further confirm the evolution of the magnetic state, we performed
measurements of the $I$-$V$ curves while sweeping the magnetic field (Figs. 4a
and b). In both the 4-layer and 5-layer CrSBr devices, all asymmetric spin
configurations exhibit a distinct diode-like rectification feature, indicating
different tunneling resistances under positive and negative bias voltages.
Importantly, this rectification behavior is determined by the spin
configuration dictated by the magnetization history and is not influenced by
the voltage sweeping history (Supplementary Fig. S3). The rectification ratio
of the device depends on the specific magnetic configuration (Supplementary
Note 1); for instance, state 1 and state 2 in the 4-layer sample display
opposing rectification ratios. Additionally, certain asymmetric resistances at
positive and negative bias voltages observed in the 4- and 5-layer CrSBr
devices, which remain constant irrespective of the magnetization history, may
arise from device asymmetry or discrepancies between the two graphite
electrodes.

To provide further information of the above-mentioned magnetic states, we
conducted tunneling MR measurements on a 5-layer CrSBr device at different
temperatures (Fig. 4c). As the temperature increases, a significant decrease
in the tunneling resistance ratio between the AFM and FM states is observed.
Simultaneously, the magnitude of the positive MR decreases and eventually
disappears at $\sim$10 K, leaving only a minor negative MR component akin to
the 4-layer CrSBr device. This result suggests that the plateaus of positive
MR may be related to new emergent phases at low temperatures and susceptible
to thermal fluctuations. Next, we investigated the MR behavior as a function
of the in-plane magnetic field direction (Fig. 4d). Notably, the flip field of
the magnetic state shows slight variation following the variable temperature
measurements, which could be attributed to structural changes in magnetic
domains or defects. As the external magnetic field direction rotates from the
$b$-axis to the $a$-axis, the spin orientation of CrSBr undergoes a tilting
transition during the field sweep (Supplementary Fig. S4). Consequently, the
magnitude of the positive MR diminishes as the external magnetic field
deviates from the $b$-axis, while the spin tilt of the state becomes more
pronounced. With further increase in the rotation angle, the positive MR state
gradually transitions into a negative MR state and eventually disappears.
Since this positive MR phenomenon has not been previously observed in reported
CrSBr tunneling devices [2, 12, 21, 22, 24, 31], it signifies that the origin
of positive MR is complex and requires further investigation.

In conclusion, our study investigated the magnetic properties and spin
textures of few-layer CrSBr using tunneling magnetoresistance measurements. We
identified distinct spin configurations that evolve with an in-plane magnetic
field applied along different crystal axes. Notably, we observed energy-
degenerate states that exhibited rectification behaviors under positive and
negative biases. Intriguingly, a positive magnetoresistive state emerged in
5-layer CrSBr devices with a magnetic field along the $b$-axis. To gain
insights into these phenomena, we developed a 1D chain model to compute the
magnetic states, which helped to elucidate the underlying spin textures
responsible for the observed transport characteristics. Our findings provide
novel insights into the complex spin structure of CrSBr, revealing subtle spin
textures that are not easily discernible by other techniques. Furthermore, our
work underscores the potential of tunneling magnetoresistance as a sensitive
probe for investigating magnetic order in 2D materials. Further investigations
focusing on spin dynamics and proximity effects in CrSBr heterostructures may
unveil additional opportunities for spintronic applications.

##  III Methods

Crystal synthesis. CrSBr single crystals were synthesized using the CVT
method. Disulfur dibromide and chromium metal were mixed in a molar ratio of
7:13, and the mixture was sealed in a silica tube under vacuum conditions.
Subsequently, the evacuated silica tube was inserted into a two-zone tubular
furnace. The growth of CrSBr crystals took place over a period of 7 days,
utilizing a temperature gradient ranging from 950 ${\degreecelsius}$ to 880
${\degreecelsius}$, with a controlled heating and cooling rate of 1
${\degreecelsius}$/min.

Device fabrication. Graphite, h-BN, and CrSBr flakes were exfoliated using the
scotch tape method in ambient conditions and transferred onto a Si substrate
covered with a 285 nm thick SiO2 layer. The heterostructures were then stacked
using a pick-up and release method based on a PPC/PDMS polymer stamp on a
glass slide. Subsequently, the device underwent an aceton wash to remove PPC
residual from the sample. A layer of poly methyl methacrylate (PMMA) was spin-
coated and patterned using electron-beam lithography (EBL). The PMMA was
developed using isopropyl alcohol, and the device was etched in a reaction-ion
etching (RIE) system under O2 and CHF3 gas flow. Finally, Cr/Au electrodes,
with a thickness of 5/35 nm were deposited on the device followed by a lift-
off process to define metal contacts.

Electrical measurements. Transport measurements were performed using a Heliox3
He insert system equipped with a 14 T superconducting magnet, allowing
temperature control down to 1.8 K. To analyze the $I$-$V$ characteristics of
the tunneling barrier and investigate the MR, a Keithley 2636B source meter
was used to apply the bias voltage, while the tunneling current was measured
using a standard two-probe module. To ensure the detection of intrinsic
signals and minimize the impact of Joule heating, the tunneling current was
limited to several hundreds of nA, resulting in a total power of several
hundreds of nW in a junction with an approximate area of 1 µm2.

## References

  * Dietl [2010] T. Dietl, Nature Materials 9, 965 (2010), ISSN 1476-4660, URL https://doi.org/10.1038/nmat2898.
  * Wang et al. [2018] Z. Wang, I. Gutiérrez-Lezama, N. Ubrig, M. Kroner, M. Gibertini, T. Taniguchi, K. Watanabe, A. Imamoğlu, E. Giannini, and A. F. Morpurgo, Nature Communications 9, 2516 (2018), ISSN 2041-1723, URL https://doi.org/10.1038/s41467-018-04953-8.
  * Huang et al. [2017] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, et al., Nature 546, 270 (2017), ISSN 1476-4687, URL https://doi.org/10.1038/nature22391.
  * Gong et al. [2017] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, et al., Nature 546, 265 (2017), ISSN 1476-4687, URL https://doi.org/10.1038/nature22060.
  * Huang et al. [2018] B. Huang, G. Clark, D. R. Klein, D. MacNeill, E. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A. McGuire, D. H. Cobden, D. Xiao, et al., Nature Nanotechnology 13, 544 (2018), ISSN 1748-3387, 1748-3395, URL https://www.nature.com/articles/s41565-018-0121-3.
  * Jiang et al. [2018] S. Jiang, L. Li, Z. Wang, K. F. Mak, and J. Shan, Nature Nanotechnology 13, 549 (2018), ISSN 1748-3395, URL https://doi.org/10.1038/s41565-018-0135-x.
  * Verzhbitskiy et al. [2020] I. A. Verzhbitskiy, H. Kurebayashi, H. Cheng, J. Zhou, S. Khan, Y. P. Feng, and G. Eda, Nature Electronics 3, 460 (2020), ISSN 2520-1131, URL https://www.nature.com/articles/s41928-020-0427-7.
  * Wang et al. [2022] Q. H. Wang, A. Bedoya-Pinto, M. Blei, A. H. Dismukes, A. Hamo, S. Jenkins, M. Koperski, Y. Liu, Q.-C. Sun, E. J. Telford, et al., ACS Nano p. acsnano.1c09150 (2022), ISSN 1936-0851, 1936-086X, URL https://pubs.acs.org/doi/10.1021/acsnano.1c09150.
  * Krempaský et al. [2023] J. Krempaský, G. Springholz, S. W. D’Souza, O. Caha, M. Gmitra, A. Ney, C. A. F. Vaz, C. Piamonteze, M. Fanciulli, D. Kriegner, et al., Nature Communications 14, 6127 (2023), ISSN 2041-1723, URL https://www.nature.com/articles/s41467-023-41718-4.
  * Zhang et al. [2021] K. Zhang, Y. Lee, M. J. Coak, J. Kim, S. Son, I. Hwang, D. Ko, Y. Oh, I. Jeon, D. Kim, et al., Advanced Functional Materials 31, 2105992 (2021), ISSN 1616-301X, 1616-3028, URL https://onlinelibrary.wiley.com/doi/10.1002/adfm.202105992.
  * Gu et al. [2022] P. Gu, Y. Sun, C. Wang, Y. Peng, Y. Zhu, X. Cheng, K. Yuan, C. Lyu, X. Liu, Q. Tan, et al., Nano Letters 22, 1233 (2022), ISSN 1530-6984, publisher: American Chemical Society, URL https://doi.org/10.1021/acs.nanolett.1c04373.
  * Gu et al. [2023] P. Gu, C. Wang, D. Su, Z. Dong, Q. Wang, Z. Han, K. Watanabe, T. Taniguchi, W. Ji, Y. Sun, et al., Nature Communications 14, 3221 (2023), ISSN 2041-1723, URL https://www.nature.com/articles/s41467-023-39004-4.
  * Göser et al. [1990] O. Göser, W. Paul, and H. Kahle, Journal of Magnetism and Magnetic Materials 92, 129 (1990), ISSN 03048853, URL https://linkinghub.elsevier.com/retrieve/pii/030488539090689N.
  * Wilson et al. [2021] N. P. Wilson, K. Lee, J. Cenker, K. Xie, A. H. Dismukes, E. J. Telford, J. Fonseca, S. Sivakumar, C. Dean, T. Cao, et al., Nature Materials (2021), ISSN 1476-1122, 1476-4660, URL http://www.nature.com/articles/s41563-021-01070-8.
  * Lee et al. [2021] K. Lee, A. H. Dismukes, E. J. Telford, R. A. Wiscons, J. Wang, X. Xu, C. Nuckolls, C. R. Dean, X. Roy, and X. Zhu, Nano Letters 21, 3511 (2021), ISSN 1530-6984, 1530-6992, URL https://pubs.acs.org/doi/10.1021/acs.nanolett.1c00219.
  * Telford et al. [2020] E. J. Telford, A. H. Dismukes, K. Lee, M. Cheng, A. Wieteska, A. K. Bartholomew, Y. Chen, X. Xu, A. N. Pasupathy, X. Zhu, et al., Advanced Materials 32, 2003240 (2020), ISSN 0935-9648, 1521-4095, URL https://onlinelibrary.wiley.com/doi/10.1002/adma.202003240.
  * López-Paz et al. [2022] S. A. López-Paz, Z. Guguchia, V. Y. Pomjakushin, C. Witteveen, A. Cervellino, H. Luetkens, N. Casati, A. F. Morpurgo, and F. O. Von Rohr, Nature Communications 13, 4745 (2022), ISSN 2041-1723, URL https://www.nature.com/articles/s41467-022-32290-4.
  * Liu et al. [2022] W. Liu, X. Guo, J. Schwartz, H. Xie, N. U. Dhale, S. H. Sung, A. L. N. Kondusamy, X. Wang, H. Zhao, D. Berman, et al., ACS Nano p. acsnano.2c02896 (2022), ISSN 1936-0851, 1936-086X, URL https://pubs.acs.org/doi/10.1021/acsnano.2c02896.
  * Ye et al. [2022] C. Ye, C. Wang, Q. Wu, S. Liu, J. Zhou, G. Wang, A. Söll, Z. Sofer, M. Yue, X. Liu, et al., ACS Nano 16, 11876 (2022), ISSN 1936-0851, 1936-086X, URL https://pubs.acs.org/doi/10.1021/acsnano.2c01151.
  * Dong et al. [2022] J. Dong, X. Li, G. Gurung, M. Zhu, P. Zhang, F. Zheng, E. Y. Tsymbal, and J. Zhang, Physical Review Letters 128, 197201 (2022), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/PhysRevLett.128.197201.
  * Klein et al. [2018] D. R. Klein, D. MacNeill, J. L. Lado, D. Soriano, E. Navarro-Moratalla, K. Watanabe, T. Taniguchi, S. Manni, P. Canfield, J. Fernández-Rossier, et al., Science 360, 1218 (2018), ISSN 0036-8075, 1095-9203, URL https://www.science.org/doi/10.1126/science.aar3617.
  * Yang et al. [2024] B. Yang, T. Patel, M. Cheng, K. Pichugin, L. Tian, N. Sherlekar, S. Yan, Y. Fu, S. Tian, H. Lei, et al., Nature Communications 15, 4982 (2024), ISSN 2041-1723, URL https://www.nature.com/articles/s41467-024-49261-6.
  * Boix-Constant et al. [2023] C. Boix-Constant, S. Jenkins, R. Rama-Eiroa, E. J. G. Santos, S. Mañas-Valero, and E. Coronado, Nature Materials (2023), ISSN 1476-1122, 1476-4660, URL https://www.nature.com/articles/s41563-023-01735-6.
  * Fu et al. [2024] Z. Fu, P. I. Samarawickrama, J. Ackerman, Y. Zhu, Z. Mao, K. Watanabe, T. Taniguchi, W. Wang, Y. Dahnovsky, M. Wu, et al., Nature Communications 15, 3630 (2024), ISSN 2041-1723, URL https://www.nature.com/articles/s41467-024-47820-5.
  * Heath et al. [2020] J. J. Heath, M. Costa, M. Buongiorno-Nardelli, and M. A. Kuroda, Physical Review B 101, 195439 (2020), ISSN 2469-9950, 2469-9969, URL https://link.aps.org/doi/10.1103/PhysRevB.101.195439.
  * Telford et al. [2022] E. J. Telford, A. H. Dismukes, R. L. Dudley, R. A. Wiscons, K. Lee, D. G. Chica, M. E. Ziebel, M.-G. Han, J. Yu, S. Shabani, et al., Nature Materials 21, 754 (2022), ISSN 1476-1122, 1476-4660, URL https://www.nature.com/articles/s41563-022-01245-x.
  * Xie et al. [2023] K. Xie, X.-W. Zhang, D. Xiao, and T. Cao, ACS Nano 17, 22684 (2023), ISSN 1936-0851, 1936-086X, URL https://pubs.acs.org/doi/10.1021/acsnano.3c07125.
  * Yu et al. [2024] J. Yu, D. Liu, Z. Ding, Y. Yuan, J. Zhou, F. Pei, H. Pan, T. Ma, F. Jin, L. Wang, et al., Advanced Functional Materials 34, 2307259 (2024), ISSN 1616-301X, 1616-3028, URL https://onlinelibrary.wiley.com/doi/10.1002/adfm.202307259.
  * Long et al. [2020] G. Long, H. Henck, M. Gibertini, D. Dumcenco, Z. Wang, T. Taniguchi, K. Watanabe, E. Giannini, and A. F. Morpurgo, Nano letters 20, 2452 (2020).
  * Ghiasi et al. [2021] T. S. Ghiasi, A. A. Kaverzin, A. H. Dismukes, D. K. de Wal, X. Roy, and B. J. van Wees, Nature Nanotechnology 16, 788 (2021), ISSN 1748-3395, URL https://www.nature.com/articles/s41565-021-00887-3.
  * Song et al. [2018] T. Song, X. Cai, M. W.-Y. Tu, X. Zhang, B. Huang, N. P. Wilson, K. L. Seyler, L. Zhu, T. Taniguchi, K. Watanabe, et al., Science 360, 1214 (2018), ISSN 0036-8075, 1095-9203, URL https://www.science.org/doi/10.1126/science.aar4851.