# Improved singing voice separation with chromagram-based pitch-aware remixing

###### Abstract

Singing voice separation aims to separate music into vocals and accompaniment
components. One of the major constraints for the task is the limited amount of
training data with separated vocals. Data augmentation techniques such as
random source mixing have been shown to make better use of existing data and
mildly improve model performance. We propose a novel data augmentation
technique, chromagram-based pitch-aware remixing, where music segments with
high pitch alignment are mixed. By performing controlled experiments in both
supervised and semi-supervised settings, we demonstrate that training models
with pitch-aware remixing significantly improves the test signal-to-distortion
ratio (SDR).

Index Terms—  Singing voice separation, augmentation, pitch-aware, chromagram,
self-training

##  1 Introduction

Singing voice separation is the task of separating vocals from music. It is
often a crucial first step for many applications including music editing,
singer identification, lyrics alignment, and transcription, singing voice
synthesis training, and tone analysis. Recent work has primarily focused on
using various deep neural network architectures in a supervised manner [1, 2,
3, 4, 5, 6]; training on music libraries with paired vocal and accompaniment
as ground truth.

Despite significant progress, the bottleneck of further improving the
performance of supervised models is primarily the lack of music libraries with
isolated sources as ground truth labels. Multi-track datasets that are
publicly available for singing voice separation, like MIR-1K [7], ccMixter
[8], and MUSDB [9], are limited to just hours of audio. This limited amount of
training data constrains the use of larger networks due to overfitting issues.
Artificially increasing the size and variety of the datasets through data
augmentation presents an opportunity to enhance large models’ ability to
generalize. Previous augmentation methods include remixing audio recordings,
swapping left and right channels, shifting pitches, and scaling and stretching
audio recordings [10, 2, 11]. However, these methods, individually or
combined, have been shown empirically to enhance the model performance only by
a marginal amount [10]. The work [12] proposes a data augmentation method,
mix-audio augmentation, that randomly mix audio segments from the same source.
Although it was shown to be effective, the improvement over random mixing is
still limited.

This paper introduces a novel data augmentation method, _chromagram-based
pitch-aware remixing_ , where sources in song segments with similar pitch
content are mixed producing mixtures that are more “realistic” than random
mixing, and more diverse than mix-audio augmentation outputs. [13] had
experimented with a similar idea for violin/piano, but with a thresholding of
match scores to select segments. We instead use the softmax of the match
scores as probabilities of selection, and use a temperature parameter, $T$, to
adjust the diversity of the mixing from “realistic” to “random”. To show the
effectiveness of our technique, we compare pitch-aware mixing in a supervised
setting with the random mixing and mix-audio augmentations. Benchmarking on
the MUSDB-18 test set, we show that pitch-aware remixing is significantly more
effective. We also find that the best results come from setting the
temperature component at a non-zero value; meaning that it is beneficial to
remix songs that match well in pitch, but not in a way that limits the
diversity of remixes.

Besides data augmentation, noisy self-training [14] is another direction to
compensate for the limited amount of labeled data by utilizing a large amount
of publicly available unseparated, or separated but noisy data by self-
labeling them using a teacher model. In this setting, the quality of the
initial teacher model tends to be important for starting the bootstrapping
process. We show the effectiveness of chromagram-based pitch-aware remixing
also in a student-teacher setting. We add the chromagram-based remixing to the
workflow of [15], where we use the above-discussed supervised-trained model as
teacher to significantly improve on the student baseline. We obtain further
gains by incorporating chromagram-based remixing into the student-training.

##  2 Proposed Method

In this section, we describe our chromagram-based pitch-aware augmentation
strategy, and review the teacher-student framework in [15].

###  2.1 Chromagram-based Pitch-aware Remixing

####  2.1.1 Chromagram

Chromagram or chroma-based features are a widely used and powerful technique
for music alignment and synchronization. Chromagram is closely related to the
twelve different pitch classes. The main idea is to aggregate each pitch class
across octaves for a given local time window to obtain a 1-D vector expressing
how the representation’s pitch content within the time window is spread over
the twelve chroma bands. Shifting the time window across the music results in
a 2-D Time-Chroma representation. We leverage chromagram correlation between
song segments as a metric to quantify song similarities because of its high
robustness to variations in timbre and closely connected to the musical
harmony.

####  2.1.2 Incorporating pitch-aware re-mixing into network training

We perform chromagram matching of music segments on the fly. For each input
mixture, $x_{0}$, we compute its vocal (or accompaniment) 2-D chromagram,
$C(t,\mathrm{pitch})$ using a python package, librosa, and then take the
average along time, $t$, to obtain a twelve-dimensional chroma vector,
$c_{0}(\mathrm{pitch})$. We then load $n$ random $t$-second song segments from
the same dataset, and compute their vocal or accompaniment chroma vectors,
$c_{j}$, where $j=1...n$. By performing normalized cross-correlation between
$c_{0}$ and $c_{j}$, we obtain $n$ scores, $s_{0j}$, where $j=1...n$. Song
segments having similar pitch content to $x_{0}$ would have a higher score.
Taking softmax with the temperature of $T$ on $(s_{0j})_{j=1}^{n}$, we obtain
a probability distribution $(p_{j})_{j=1}^{n}$, from which we draw an index
$j^{\prime}$, and conduct source mixing to obtain a new mixture,
$\tilde{x_{0}}=x_{0,voc}+x_{j^{\prime},acc}$. New mixtures are more likely to
be obtained by mixing song segments with identical pitch content if we lower
the softmax temperature. With higher temperature, the song segments to be
mixed are more likely to be randomly chosen. Figure 1 show examples of
chromagram and chroma vectors for three randomly chosen 10-second
accompaniment segments from MUSDB. $s_{ac}=0.98>s_{ab}=0.88$. Therefore,
segments, (a) and (c), are more likely to remix due to the similar chromagram
features.

Fig. 1: Accompaniment chromagram $C$ and chroma vectors $c$ of three randomly
chosen 10-second segments from MUSDB. Normalized chroma-vector cross-
correlation scores, $s_{ab}=0.88$, $s_{ac}=0.98$, indicates segments (a) and
(c) share more similarities than (a) and (b).

Note that there are two options to compute chroma vectors of the song segments
by using either vocal or accompaniment. Using vocal would mean that we rely on
the vocal-to-vocal (voc2voc) match to quantify segment similarity, otherwise
we rely on accompaniment-to-accompaniment (acc2acc) match. We experiment with
both options.

###  2.2 Noisy self-training framework

The framework consists of the following steps:

  1. 1.

Train a teacher separator network $M_{0}$ on a small labeled dataset $D_{l}$.

  2. 2.

Assign pseudo-labels for the large unlabeled dataset $D_{u}$ with $M_{0}$ to
obtain the self-labeled dataset $D_{0}$.

  3. 3.

Filter data samples with a pretrained voice activity detector (VAD) from
$D_{0}$ to obtain $D_{f0}$.

  4. 4.

Train a student network $M_{1}$ with $D_{l}\cup D_{f0}$.

###  2.3 Separator Network

We adopt the same PoCoNet [16] architectures for both teacher and student
models as [15]. The inputs are the concatenation of real and imaginary parts
of the mixture’s STFT spectrogram. The output is the complex ratio masks for
each source. The wave-form signal is obtained by applying inverse STFT
transform on the estimated spectrograms.
The separator is a fully-convolutional 2D U-Net architecture with DenseNet and
attention blocks. Each DenseNet block contains three convolutional layers,
each followed by batch normalization and Rectified Linear Unit (ReLU).
Convolutional operations are causal in the time direction but not in the
frequency direction. We choose a kernel size of 3 × 3 and a stride size of 1,
and the number of channels increases from 32, 64, 128 to 256. We control the
size of the network by varying the number of levels in U-Net and the maximum
number of channels. In the attention module, the number of channels is set to
5 and the encoding dimension for key and query is 20. Frequency-positional
embeddings are applied to each time-frequency bin of the input spectrogram.

####  2.3.1 Loss function

For each output source, the loss function is the weighted sum of waveform and
spectral loss:

$\mathcal{L}_{s}(y,\hat{y})=\lambda_{\mathrm{audio}}\mathcal{L}_{\mathrm{audio}}(y,\hat{y})+\lambda_{\mathrm{spec}}\mathcal{L}_{\mathrm{spec}}(Y,\hat{Y})$
(1)

where $s$ is the output source $\mathrm{voc}$ or $\mathrm{acc}$, $y$ and
$\hat{y}$ are time domain groundtruth and the network output. $Y$ and
$\hat{Y}$ are the corresponding STFT magnitude spectrograms. We choose both
$L_{audio}(\cdot)$ and $L_{spectral}(\cdot)$ to be $l1$ loss. The total loss
is the weighted sum of the two sources:

$\mathcal{L}(y,\hat{y})=\lambda_{\mathrm{voc}}\mathcal{L}_{\mathrm{voc}}(y,\hat{y})+\lambda_{\mathrm{acc}}\mathcal{L}_{\mathrm{acc}}(Y,\hat{Y})$
(2)

##  3 Experimental setup

###  3.1 Training Dataset

Following [15], we use MIR-1K [7], ccMixter [8], and the training partition of
MUSDB [9] as the labeled dataset for supervised training with roughly 11 hours
of recordings. To train the student model with a large unlabeled dataset with
more than 300 hours of recordings from a karaoke app. We train and test at
16kHz. For preprocessing, we compute the STFT spectrograms with a DFT size of
1024 and a hop size of 256.

###  3.2 Noisy self-training with pitch-aware mixing

For both the supervised (teacher) training and student training, we minimize
Equation 2 with an Adam optimizer with an initial learning rate of $10^{-4}$,
and we decrease the learning rate by half for every 100k iterations until it’s
no greater than $10^{-6}$. We set $\lambda_{audio}=\lambda_{spec}=1$ in
Equation 1 and $\lambda_{voc}=\lambda_{acc}=1$ in Equation 2. We set the input
window size to 10 seconds and the batch size to 1 following the optimal
configuration in [15].

####  3.2.1 Supervised (teacher) training

The prior work [15] conducts supervised (teacher) training experiments using
random mixing for data augmentation, and shows that random mixing with
probability of 1 leads to the best teacher model. Here, we experiment with two
additional data augmentation strategies: mix-audio augmentation [17] and the
proposed pitch-aware mixing. For a fair comparison, the model architectures
and hyper-parameters are the same as the best random mixing models in [15]. We
implemented the chromagram-based pitch-aware mixing based on the description
in Section 2.1.2. For each training sample, we load 8 other random 10-second
segments in the same dataset to compute matching scores. We experiment with
both voc2voc and acc2acc chromagram matching. We also experiment with the
softmax temperatures $T\in\\{0,\,0.33,\,1,\,3\\}$. For comparison, we also
experiment with the mix-audio strategy: for each 10-second training samples,
we randomly select another 10-second segment from the same song, and mix the
sources.

####  3.2.2 Student training

Student model is trained on both labeled data and DAMP dataset self-labeled by
the chroma teacher model. We integrate pitch-aware mixing on student training,
and experiment with $T\in\\{0,\,0.33,\,1,\,3\\}$.

##  4 Evaluation results and discussion

###  4.1 Evaluation Framework

Following the SiSEC separation campaign [18], and for the purpose of
continuity with other works (c.f. Table 3), we use Signal-to-Distortion Ratio
(SDR) to evaluate the separation performance, computed using the python
package museval, which partitions each of the 50 songs from the test partition
of MUSDB test set into non-overlapping ten-second segments, and takes the
median of segment-wise SDR for each song and reports the median from all 50
songs. Using a standalone validation set to choose $T$ would strengthen our
arguments; however, given the limited size of the training set, the knowledge
gains from using a validation split would be offset by the reduction in the
quality of the training set.

###  4.2 Supervised (teacher) Model Performance

The model architecture and hyperparameters we use here are consistent with the
best teacher model in [15]. Instead of using random mixing, we experiment with
the pitch-aware mixing using different softmax temperatures and experiment
with voc2voc versus acc2acc matching. Table 1 shows the test SDR for the
experiments. We can see that both mix-audio augmentation and pitch-aware
mixing outperform the random mixing baseline, indicating the effectiveness of
both our approach and the mix-audio augmentation. It is also clear that pitch-
aware mixing outperforms mix-audio in that all chroma teachers outperform the
mix-audio augmentation except the high-temperature chroma teacher ($T=3$). We
observe that for the chroma teachers (acc2acc), SDR increases by 0.62 dB as
$T$ decreasing from 3 to 0.33. High temperatures make chromagram-mixing closer
to random mixing. So, higher $T$ leading to poorer performance is in line with
the random mixing results; which can be interpreted as diverse but less
“realistic” training data limiting model performance. However, we see that
decreasing $T$ from 0.33 to 0 doesn’t improve the model further, which can be
explained with $T=0$, always mixing the best matching songs, causing a less
diverse dataset. With $T=0.33$ seems to reach a balanced state obtaining a
both diverse and “realistic” dataset that leads to the best teacher model
improving the average SDR by 1.05 dB compared to the baseline teacher model
with random mixing. Our best supervised model even surpasses the student
baseline by 0.48 dB.

Table 1: Test performance metrics (SDR in dB) for teacher models. $T$ refers to the softmax temperature, and ‘voc’/‘acc’ corresponds to voc2voc/acc2acc matching strategy in section 2.1.2. The best performance is highlighted in bold. Experiments |  SDR (V) |  SDR (A) |  Mean
---|---|---|---
Teacher [15] (Random mixing; i.e. $T=\infty$) |  6.91 |  13.66 |  10.29
Student [15] (Random mixing) |  7.8 |  13.92 |  10.86
Teacher + mix-audio aug |  7.48 |  14.06 |  10.77
Chroma Teacher (voc; T=1) |  7.76 |  14.02 |  10.89
Chroma Teacher (acc; T=3) |  7.57 |  13.88 |  10.72
Chroma Teacher (acc; T=1) |  7.75 |  14.08 |  10.92
Chroma Teacher (acc; T=0.33) |  7.92 |  14.77 |  11.34
Chroma Teacher (acc; T=0) |  7.79 |  14.31 |  11.05

###  4.3 Student Model Performance

We labeled the mixtures from the unlabeled DAMP dataset using the supervised
chroma teacher models (acc2acc) with temperatures of 1 and 0.33, respectively.
Correspondingly, we train two student models with pitch-aware mixing using
temperatures of 1 and 0.33. Test results are shown in Table 2. We can see that
our chroma student outperforms the student baseline with random mixing by 0.69
dB. The lower temperature student model performs the best, implying similar
conclusions on teacher training that low temperature training results in both
diverse and more realistic dataset leading to higher performance boost.

Table 2: Test performance metrics (SDR in dB) for student models. Experiments |  SDR (V) |  SDR (A) |  Mean
---|---|---|---
Best student model in [15] |  7.8 |  13.92 |  10.86
Chroma Student (T=1) |  8.55 |  14.67 |  11.61
Chroma Student (T=0.33) |  8.39 |  15.0 |  11.70

###  4.4 Comparisons with Other Models

In Table 3, we compare our best model with other models. We can see that our
chroma student model achieves the second highest mean SDR score, indicating
the effectiveness of out augmentation approach. Although, the recent
ResUnetDecouple+ performs the best, the data augmentation approach is mix-
audio augmentation, which is same as in the previous work it builds upon
CatNet[12] with lower SDR than we obtain here. The improvement of
ResUnetDecouple+ over CatNet is mainly due to the innovative model
architecture and decoupling of the magnitude and phase. Therefore, we believe
that applying our data augmentation to the ResUnetDecouple+ architecture has
the potential for further improvements.

Table 3: Comparison of the proposed method and other baseline models. [17] is a follow-up work on [12] with the same mix-audio augmentation but with architecture improvement. Its contribution is orthogonal to our data method. Therefore, considering our improvement over [15], it is worth experimenting with combining chromagram-based remixing and the ResUNetDecouple+ architecture. Name |  SDR (V) |  SDR (A) |  Mean
---|---|---|---
Demucs [6] |  7.05 |  N/A |  N/A
MMDenseLSTM[19] |  4.94 |  16.4 |  10.67
MT U-Net[20] |  5.28 |  13.04 |  9.16
Wang _et. al._ [15] |  7.8 |  13.92 |  10.86
CatNet[12] |  7.54 |  15.18 |  11.36
ResUNetDecouple+[17] |  8.98 |  16.63 |  12.81
Ours(chroma student) |  8.39 |  15.0 |  11.70

###  4.5 Discussion

The experimental results show that our pitch-aware mixing demonstrates
noticeable improvement over the random mixing baseline and mix-audio
augmentation. A potential problem with random mixing is that it produces a
considerable amount of mixtures by songs with dramatically different pitch
content. These mixtures could sound “unrealistic” and are more likely to incur
domain shift to the underlying distribution of real songs. Besides, these
samples could be easier than real songs to separate considering that vocal and
accompaniment contain mismatch spectral patterns. These “unrealistic” training
data could make it difficult for the separator to perform well on unseen
realistic test data. While the mix-audio strategy is better than random
mixing, the training samples generated are not as diverse as our method, since
mix-audio only considers segments from the same song to form mixtures. In
contrast, with a larger sample space as well as the temperature component, the
proposed method is able to better adjust the trade-off between being diverse
and realistic.

##  5 Conclusion

We introduced a novel data augmentation strategy for singing voice separation.
Our approach is chromagram-based and pitch-aware; aiming to mix song segments
with similar pitch content to form mixtures that are more likely to resemble
real songs while maintaining diversity. Experimental results on the noisy-self
training framework show that pitch-aware mixing improves model training
compared to random mixing and mix-audio augmentation. Future work would be
investigating the effectiveness of our approach on other architectures and
other music-related tasks.

## References

  * [1] Aditya Arie Nugraha, Antoine Liutkus, and Emmanuel Vincent,  “Multichannel music separation with deep neural networks and deep neural network based instrument extraction from music,”  in EUSIPCO, 2016.
  * [2] Stefan Uhlich, Marcello Porcu, Franck Giron, Michael Enenkl, Thomas Kemp, Naoya Takahashi, and Yuki Mitsufuji,  “Improving music source separation based on deep neural networks through data augmentation and network blending,”  in IEEE ICASSP, 2017.
  * [3] Po-Sen Huang, Minje Kim, Mark Hasegawa-Johnson, and Paris Smaragdis,  “Joint optimization of masks and deep recurrent neural networks for monaural source separation,”  in IEEE ICASSP, 2015.
  * [4] Fabian-Robert Stöter, Stefan Uhlich, Antoine Liutkus, and Yuki Mitsufuji,  “Open-unmix - a reference implementation for music source separation,”  in Journal of Open Source Software, 2019, vol. 4.
  * [5] Naoya Takahashi and Yuki Mitsufuji,  “Open-unmix - a reference implementation for music source separation,”  in IEEE WASPAA, 2017.
  * [6] Alexandre Défossez, Nicolas Usunier, Léon Bottou, and Francis Bach,  “Demucs: Deep extractor for music sources with extra unlabeled data remixed,”  in ArXiv:1909.01174, 2019.
  * [7] C. Hsu and J. R. Jang,  “On the improvement of singing voice separation for monaural recordings using the mir-1k dataset,”  in IEEE/ACM TASLP, 2010, vol. 18, pp. 310–319.
  * [8] A. Liutkus, D. Fitzgerald, Z. Rafii, B. Pardo, and L. Daudet,  “Kernel additive models for source separation,”  in IEEE Transactions on Signal Processing, 2014, vol. 62, pp. 4298–4310.
  * [9] Zafar Rafii, Antoine Liutkus, Fabian-Robert Stöter, Stylianos Ioannis Mimilakis, and Rachel Bittnert,  “The musdb18 corpus for music separation,”  2017\.
  * [10] Laure Prétet, Romain Hennequin, Jimena Royo-Letelier, and Andrea Vaglio,  “Singing voice separation: A study on training data,,”  in IEEE ICASSP, 2019, pp. 506–510.
  * [11] Alice Cohen-Hadria, Axel Röbel, and Geoffroy Peeters,  “Improving singing voice separation using deep u-net and waveu-net with data augmentation,,”  in EUSIPCO, 2019, pp. 1–5.
  * [12] Xuchen Song, Qiuqiang Kong, Xingjian Du, and Yuxuan Wang,  “Catnet: music source separation system with mix-audio augmentation,”  in ArXiv:2102.09966, 2021.
  * [13] Ching-Yu Chiu, Wen-Yi Hsiao, Yin-Cheng Yeh, yi-hsuan Yang, and Alvin Su,  “Mixing-specific data augmentation techniques for improved blind violin/piano source separation,”  in ArXiv:2008.02480, 2020.
  * [14] Qizhe Xie, Eduard H. Hovy, Minh-Thang Luong, and Quoc V. Les,  “Self-training with noisy student improves imagenet classification,,”  in ArXiv, vol. abs/1911.04252, 2019.
  * [15] Zhepei Wang, Ritwik Giri, Umut Isik, Jean-Marc Valin, and Arvindh Krishnaswamy,  “Semi-supervised singing voice separation with noisy self-training,”  in IEEE ICASSP, 2021.
  * [16] Umut Isik, Ritwik Giri, Neerad Phansalkar, Jean-Marc Valin, Karim Helwani, and Arvindh Krishnaswamy,  “Poconet: Better speech enhancement with frequency-positional embeddings, semi-supervised conversational data, and biased loss,”  in INTERSPEECH, 2020.
  * [17] Qiuqiang Kong, Yin Cao, Haohe Liu, Keunwoo Choi, and Yuxuan Wang,  “Decoupling magnitude and phase estimation with deep resunet for music source separation,”  in ISMIR, 2021.
  * [18] Fabian-Robert Stöter, Antoine Liutkus, and Nobutaka Ito,  “The 2018 signal separation evaluation campaign,,”  2018\.
  * [19] N. Takahashi, N. Goswami, and Y. Mitsufuji,  “Mmdenselstm: An efficient combination of convolutional and recurrent neural networks for audio source separation,”  in IWAENC, 2018, pp. 106–110.
  * [20] Venkatesh S. Kadandale, Juan F. Montesinos, Gloria Haro, and Emilia Gómez,  “Multi-channel u-net for music source separation,”  2020\.


# Role of substrate clamping on anisotropy and domain structure in the canted
antiferromagnet $\alpha$-Fe2O3

Angela Wittmann1,2 Olena Gomonay2 Kai Litzius1,3 Allison Kaczmarek1 Alexander
E. Kossak1 Daniel Wolf4 Axel Lubk4 Tyler N. Johnson5 Elizaveta A. Tremsina1,6
Alexandra Churikova1 Felix Büttner7 Sebastian Wintz3,7 Mohamad-Assaad Mawass7
Markus Weigand7 Florian Kronast7 Larry Scipioni8 Adam Shepard8 Ty Newhouse-
Illige8 James A Greer8 Gisela Schütz3 Norman O. Birge1,9 Geoffrey S. D. Beach1
1Department of Materials Science and Engineering, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA  2Institute of Physics,
Johannes Gutenberg-University Mainz, 55128 Mainz, Germany  3Max Planck
Institute for Intelligent Systems, 70569 Stuttgart, Germany  4Leibniz IFW
Dresden, 01069 Dresden, Germany  5Department of Chemical Engineering and
Materials Science, Michigan State University, East Lansing, Michigan 48824,
USA  6Department of Electrical Engineering and Computer Science, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA  7Helmholtz-
Zentrum Berlin für Materialien und Energie GmbH, 14109 Berlin, Germany  8PVD
Products, Wilmington, Massachusetts 01887, USA  9Department of Physics and
Astronomy, Michigan State University, East Lansing, Michigan 48824, USA

###### Abstract

Antiferromagnets have recently been propelled to the forefront of spintronics
by their high potential for revolutionizing memory technologies. For this,
understanding the formation and driving mechanisms of the domain structure is
paramount. In this work, we investigate the domain structure in a thin-film
canted antiferromagnet $\alpha$-Fe2O3. We find that the internal destressing
fields driving the formation of domains do not follow the crystal symmetry of
$\alpha$-Fe2O3, but fluctuate due to substrate clamping. This leads to an
overall isotropic distribution of the Néel order with locally varying
effective anisotropy in antiferromagnetic thin films. Furthermore, we show
that the weak ferromagnetic nature of $\alpha$-Fe2O3 leads to a qualitatively
different dependence on magnetic field compared to collinear antiferromagnets
such as NiO. The insights gained from our work serve as a foundation for
further studies of electrical and optical manipulation of the domain structure
of antiferromagnetic thin films.

##  I Introduction

Antiferromagnets (AFMs) exhibit highly favorable properties such as robustness
to external magnetic field, no stray field, and ultra-fast switching in the
terahertz regime making them promising candidates for novel memory
technologies [1, 2, 3, 4, 5]. Recent interest in this field, and the canted
antiferromagnet $\alpha$-Fe2O3 in particular, was sparked by the experimental
demonstration of electrical control and readout of the Néel order in metallic
AFMs [6, 7, 8], insulating AFM/heavy-metal bilayers [9, 10, 11, 12, 13, 14,
15], and the observation of topological spin textures [16, 17, 18] paving the
path towards using AFMs as active materials for spintronic devices.

Analogously to demagnetizing fields in ferromagnets, destressing fields have
been identified as the dominant driving force behind domain formation in AFMs
[19, 20]. So far, it was assumed that the domains are oriented along the
magnetocrystalline easy-axes. However, the underlying model is strictly true
only for free layers. Our experimental findings on $\alpha$-Fe2O3 thin films
imply that substrate clamping results in a far more complex and disordered
domain structure. We show that this leads to long-range interactions that
impose a locally-varying effective anisotropy. The local fluctuations of the
easy-axis orientations qualitatively impact the domain structure and
magnetization process. These previously unidentified effects should generally
be present and in competition with the magnetocrystalline anisotropy in
antiferromagnetic oxide thin films based on their similarities in magnetic,
chemical and structural properties [21, 22]. Therefore, the insights and the
refined model developed in this work will allow for a more informed
understanding of magnetic switching and complex spin textures in easy-plane
antiferromagnets.

Figure 1: (a) Long-range (006)/(0012) symmetric scan of $\alpha$-Fe2O3 (10 nm)
on c-axis (001) oriented Al2O3. (b) (1 0 10)- reciprocal space map of
$\alpha$-Fe2O3 (10 nm) on c-axis (001) oriented Al2O3. (c) Phi-scan of
$\alpha$-Fe2O3 (20 nm) on c-axis (001) oriented Al2O3 with inset of hexagonal
crystal structure. (d) Raman spectrum of $\alpha$-Fe2O3 (10 nm) thin film on a
Al2O3 substrate. (e) Cross-sectional HRTEM image of $\alpha$-Fe2O3 (20 nm) on
c-axis (001) oriented Al2O3. The inset shows the Fourier transform of the
image.

##  II Thin film characterization

Our epitaxial $\alpha$-Fe2O3 films (10 nm thick unless mentioned otherwise)
were grown at 640 ∘C on $c$-axis (001) oriented Al2O3 with off-axis magnetron
sputtering. In order to validate the quality of the thin films, we have
performed structure and composition characterization.

Figure 1a-c show the structure characterization of the $\alpha$-Fe2O3 thin
films using x-ray diffraction (XRD) confirming the high epitaxial quality of
the thin films (see supplemental material for details [23]). The (006)/(0012)
symmetric 2$\theta$ scan shown in Fig. 1a was used to determine the c lattice
parameters of the magnetic thin film and the substrate (c${}_{\alpha-
Fe_{2}O_{3}}$ = 13.79 Å and c${}_{Al_{2}O_{3}}$ = 12.99 Å). The film thickness
was confirmed to be 11.88 nm (nominally 10 nm) using the Scherrer method which
relates peak broadening to the crystallite size. The peaks from gold appear
due to the gold contacts on the sample. The strain state and degree of
relaxation was determined from a reflection with in-plane and out-of-plane
components. In the (1 0 10)- reciprocal space map (Fig. 1b), shifting of the
$\alpha$-Fe2O3 film peak around q${}_{x}\approx-0.234$ (red) towards the
origin is indicative of film relaxation with respect to the substrate at
q${}_{x}\approx-0.247$ (green). At the substrate-film interface, the film
experiences strain due to clamping on the substrate, as shown by the shadow
peak located along q${}_{x}\approx-0.247$. Through the thickness of the
$\alpha$-Fe2O3 film, the strain relaxes by 93.7 % to a nearly fully relaxed
film. The phi-scan (Fig. 1c) confirms the six-fold symmetry in the
$\alpha$-Fe2O3 and Al2O3 lattices.

To characterize the film-substrate interface in more detail, we acquired
cross-sectional high-resolution transmission electron microscopy (HRTEM)
images on a 20 nm $\alpha$-Fe2O3 film (see supplemental material for details
[23]). Fig. 1e shows an HRTEM image exhibiting clear atomic lattice plane
contrast. The Fourier transform shown as inset in Fig. 1e confirms the high
crystallinity of the $\alpha$-Fe2O3 film. To quantify the strain distribution
along the thickness of the $\alpha$-Fe2O3 film, we conducted a geometric phase
analysis [24]. We find a sharp relaxation after 1 nm of the $\alpha$-Fe2O3
lattice clamped at the Al2O3 interface. This is facilitated by the presence of
misfit dislocations and other crystal defects indicated with circles in the
HRTEM image in Fig. 1e. Moreover, we observe an almost constant in- and out-
of-plane dilation in the $\alpha$-Fe2O3 lattice with respect to the smaller
Al2O3 substrate lattice. The relative dilation by 6 % agrees well with the
ratio of the lattice plane distances for in-plane and out-of-plane directions.
The sharp relaxation and dilation of the $\alpha$-Fe2O3 film support the
findings from the XRD measurements described above.

Figure 1d shows the Raman measurement data (black dots) taken using an
excitation laser wavelength of 532 nm with fits using Voigt functions (solid
line). The center of the peaks are labelled by vertical dashed lines
corresponding to the identified Raman modes based on previous reports in
literature [25, 26, 27, 28, 29] (see supplemental material for detailed peak
information [23]). In particular, we note the presence of a wide peak at 660
cm-1 corresponding to the Raman-forbidden longitudinal optical (LO) mode in
$\alpha$-Fe2O3 due to defect-induced scattering [27, 30, 31]. We find that the
experimental data is accurately fitted with Raman peaks corresponding to
$\alpha$-Fe2O3 and Al2O3, with a small shift of up to 6 cm-1 for
$\alpha$-Fe2O3 compared to the bulk values. This shift agrees well with
previous reports that compressive strain as observed in our in $\alpha$-Fe2O3
films in the XRD measurements leads to higher peak center values [25, 32].

We note that the structure and composition characterization based on XRD,
HRTEM, and Raman measurements presented here confirm the high epitaxial
quality and phase-purity of the $\alpha$-Fe2O3 thin films grown on Al2O3
substrates investigated in this work. Importantly, we do not observe any
evidence of impurity phases of Fe3O4 (see reference marks in Fig. 1a,d).

##  III Experiment

At room temperature, there is a strong easy-plane anisotropy in $\alpha$-Fe2O3
forcing the magnetic moments to lie in the basal plane. Additionally, there is
a weak six-fold magnetocrystalline anisotropy within this easy-plane with an
anisotropy field of $H_{a}\approx 1$ $\mu$T [33, 34, 35, 36, 37]. The easy-
axes are labeled E1, E2, and E3 in the inset of Fig. 3b.

The two magnetic sublattices $M_{A}$ and $M_{B}$ of $\alpha$-Fe2O3 are
slightly canted due to the Dzyaloshinskii-Moriya interaction (DMI) [38, 39]
with a canting angle $\delta=0.13\pm 0.01^{\circ}$[40], giving rise to a small
FM moment $m$ in the basal plane of the hexagonal cell (see inset of Fig. 2b).
Moreover, $\alpha$-Fe2O3 exhibits pronounced magnetoelastic coupling with a
magnetostriction constant $\lambda\approx 10^{-5}$ originating from
spontaneous strain [41, 42].

The AFM state is commonly read electrically by measuring the angle-dependent
spin Hall magnetoresistance (SMR) in $\alpha$-Fe2O3/Pt bilayers where the
perpendicular alignment of the Néel vector with respect to the external
magnetic field results in a negative sign of the SMR signal [43, 44, 45, 46].
The electrical measurements presented in this work were performed by applying
an AC probing current $I_{p}=500\mu$A and measuring the transverse resistance
$R_{xy}$ of the Pt (5 nm thick and 20 $\mu$m wide) crossbar grown on
$\alpha$-Fe2O3 by magnetron sputtering.

Figure 2: (a) Angle-dependent SMR signal for magnetic field varying from 50 mT
(light orange) to 1 T (dark red). Inset shows a schematic of the SMR
measurement geometry. (b) SMR signal as function of field at
$\theta=45^{\circ}$. The gray line shows the initial field sweep from a
demagnetized state $0_{d}$ to $+H_{sat}$, the dark and light red lines show
the hysteresis between $+H_{sat}\rightarrow-H_{sat}$ and
$-H_{sat}\rightarrow+H_{sat}$ field sweeps, respectively. The inset
illustrates the canting of the sublattice magnetizations MA and MB.

As shown in Fig. 2a, the SMR shows the characteristic $\sin(2\theta)$
dependence, where $\theta$ denotes the angle between the current direction
(along easy-axis E2) and the external in-plane magnetic field. A magnetic
field smaller than 500 mT is not sufficient to fully reorient the Néel order
perpendicular to the field, resulting in a smaller amplitude of the angle-
dependent SMR.

Fig. 2b shows a measurement of the field dependence of the SMR at a fixed
angle $\theta=45^{\circ}$. Starting from a demagnetized state at zero field
$0_{d}$, $R_{xy}$ initially increases linearly with magnetic field and
saturates at $H_{sat}=\pm 600$ mT. Above $H_{sat}$, the Néel order is oriented
fully perpendicular to the magnetic field. When the field is reduced back to
zero, the SMR signal decreases to nearly the initial value at $0_{d}$,
signaling that the sample has demagnetized into a multi-domain state with
nearly zero net Néel vector. This implies the existence of strong internal
fields driving the formation of domains. We have confirmed the strong
dependence of the AFM order on the magnetic field by measuring x-ray magnetic
linear dichroism (XMLD) contrast images of the domain structure of
$\alpha$-Fe2O3 by total electron yield using a scanning x-ray microscope
(Maxymus) (see Fig. 3a).

Figure 3: (a) Positive field loop at $45^{\circ}$. The SMR signal shows small
hysteresis below $H_{irr}\approx 200$ mT and is fully reversible above
$H_{irr}$. Insets show scanned XMLD images of the $\alpha$-Fe2O3 domain
structure at 0 mT and 250 mT. Dark/ bright contrast corresponds to horizontal/
vertical orientation of the Néel vector. (b) Transverse remanent resistance
(dark red) after $H$ has been relaxed from a saturated state at 0.6 T (light
red) to zero field as a function of angle of the applied magnetic field. The
inset shows the orientation of the magneto-crystalline easy-axes within the
hexagonal unit cell.

In contrast to collinear AFMs such as NiO, we observe clear hysteresis in the
SMR signal in $\alpha$-Fe2O3/Pt. We define the location of the minima of
$R_{xy}$ to be the coercive field $H_{c}\approx 50$ mT. The hysteresis is a
signature of the weak ferromagnetic nature of $\alpha$-Fe2O3. In collinear
AFMs with an applied field greater than the spin-flop field, the Néel vector
can adopt either of the two directions orthogonal to the field. In canted
AFMs, on the contrary, the small FM moment prefers to align with the external
magnetic field, and hence the $180^{\circ}$ degeneracy of the orientation of
the Néel order is lifted due to the coupling of the AFM and FM moments by DMI.
As a consequence, the interpretation of angle-dependent SMR at low fields,
where hysteresis plays a significant role, is not straightforward in
$\alpha$-Fe2O3.

Therefore, we focus on a field regime with a single polarity, where all Néel
vectors are oriented within a semicircle defined by the direction of the
preceding saturating magnetic field. Fig. 3a shows the SMR signal for an in-
plane field sweep from $+H_{sat}$ to zero and back to $+H_{sat}$ oriented at
$\theta=45^{\circ}$. $R_{xy}$ initially increases linearly with magnetic
field. This linear dependence is in stark contrast to the experimentally
observed and theoretically modeled quadratic dependence on the magnetic field
in the low-field regime previously reported in collinear AFMs such as NiO
[44]. We note that there is a small hysteresis below the irreversibility field
$H_{irr}\approx 200$ mT. At low fields $0\leq H<H_{irr}$, the reorientation of
the Néel order is dominated by domain wall movement. Pinning of the AFM domain
walls induces irreversibility which manifests in the slight hysteresis of the
SMR signal. We can extract an effective pinning field of the AFM domain walls
of $H_{p}\approx 20$ mT. For $H>H_{irr}$, $R_{xy}$ is independent of the
preceding magnetic field. In this regime, the reorientation of the Néel order
is dominated by coherent rotation of the Néel vectors within the domains and
is, hence, fully reversible.

To elucidate further details of the equilibrium AFM domain structure, we have
investigated the angular dependence of the remanent state. We applied an
external in-plane magnetic field $H=1.2$ T $>H_{sat}$ (light red curve in Fig.
3b) and measured the remanent SMR signal $R^{rem}_{xy}$ (dark red curve) as a
function of the angle $\theta$ of the preceding field. If magneto-crystalline
anisotropy were the driving mechanism for relaxation in this system, one would
expect the Néel vector to relax towards the closest anisotropy axis depending
on the direction of the preceding magnetic field. This would result in maxima
and minima in R${}^{rem}_{xy}$ along E1 and E3 respectively. However,
$R^{rem}_{xy}$ does not show any signature of the crystal symmetry. Fitting
$R^{rem}_{xy}$ to $\sin 2\theta$ (gray) shows good agreement, implying a
simple scaling $R^{rem}_{xy}(\theta)=wR^{sat}_{xy}(\theta)$ with $w\approx
0.2$ between the two curves. This suggests that the remanent signal arises
from a small fraction of domains that are oriented perpendicular to the
preceding saturating magnetic field direction.

Figure 4: (a,b) Schematic illustrating the deflection of the anisotropy axis
away from the magnetocrystalline easy-axis in the central domain depending on
the local surroundings. (c) Schematic of distribution of orientation of Néel
vectors ranging from strongly aligned with a magnetocrystalline anisotropy
(dark gray) to isotropic (red). (d) Slope of the linear SMR signal at small
$H$ as a function of angle of the field calculated using the modified
destressing model based on a six-fold anisotropy (black line) and isotropic
distribution of Néel vector orientations (red line). The isotropic model shows
good agreement with the experimental data for a field sweep from a
demagnetized state and saturated state (dots and triangles respectively).

##  IV Theoretical discussion

In order to identify the different contributions to the formation of the
domains, we have investigated the underlying driving mechanisms in more
detail. The formation of a multi-domain state when the magnetic field is
reduced has been investigated recently in several AFMs [47, 43, 48, 44, 49,
13, 50]. So far, it has been presumed that the domains are oriented along the
magneto-crystalline anisotropy axes. The hysteresis and formation of well-
defined small domains in zero external magnetic field are evidence for the
existence of anisotropy in our system. The lack of signatures of the six-fold
symmetry of the magnetocrystalline anisotropy despite the high-quality
crystallinity of the epitaxially grown films, however, implies that different
contributions are dominant for the effective anisotropy. In the following
section, we show that the effective anisotropy is composed of a competition of
the short-range magnetocrystalline anisotropy and a long-range contribution of
magnetoelastic nature.

First, we note that the spontaneous magnetoelastic strains in the AFM layer
are incompatible with the non-strained nonmagnetic substrate and create a
long-range so-called destressing field which is similar to the stray field in
ferromagnets [19, 20] (see supplemental material for details [23]). The local
orientation of the easy magnetic axes is parameterized by the angle
$\varphi(\mathbf{r})$ that is calculated from the easy-axis E2 within the film
plane. By minimizing the energy of the antiferromagnetic layer, we get the
following equation for $\varphi(\mathbf{r})$:

$\displaystyle H_{\mathrm{an}}\sin
6\varphi+H_{\mathrm{destr}}\left[\psi_{1}\sin 2\varphi-\psi_{2}\cos
2\varphi\right]=0,$
$\displaystyle\psi_{1}(\mathbf{r})\equiv\frac{1}{4\pi}\int\frac{\cos
2\varphi(\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\delta^{\prime}(z)d\mathbf{r}^{\prime},$
(1) $\displaystyle\psi_{2}(\mathbf{r})\equiv\frac{1}{4\pi}\int\frac{\sin
2\varphi(\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\delta^{\prime}(z)d\mathbf{r}^{\prime},$

where $H_{\mathrm{an}}$ denotes the in-plane magnetic anisotropy and
$H_{\mathrm{destr}}$ is the destressing field which is proportional to the
magnetoelastic constant $\lambda$, and $\delta^{\prime}(z)$ is the derivative
of the Dirac delta function. Here, we neglect the contributions from the AFM
domain walls within the plane.

In the second step, we analyse the possible distribution of the effective
anisotropy by solving Eq. (IV). The first term in Eq. (IV) stems from the six-
fold crystallographic anisotropy and turns to zero for $\varphi_{0}=0,\pi$
(direction along E2), $\pi/3,4\pi/3$ (E1), or $2\pi/3,5\pi/3$ (E3). The second
term originates from the magnetoelastic coupling and depends on the
distribution of the magnetic texture in the whole sample. To estimate this
term, we note that the destressing fields $\psi_{1}(\mathbf{r})$,
$\psi_{2}(\mathbf{r})$ satisfy the Poisson equations

$\displaystyle-\Delta\psi_{1}$ $\displaystyle=$
$\displaystyle\delta^{\prime}(z)\cos 2\varphi(\mathbf{r}),$ (2)
$\displaystyle-\Delta\psi_{2}$ $\displaystyle=$
$\displaystyle\delta^{\prime}(z)\sin 2\varphi(\mathbf{r}),$

in which the functions $\cos 2\varphi(\mathbf{r})$, $\sin
2\varphi(\mathbf{r})$ play the role of magnetoelastic charges. Based on the
analogy with electrostatics, we conclude that the crystalline anisotropy
defined by the first term in Eq. (IV) can only be restored by either a
vanishingly small destressing field $H_{\mathrm{destr}}$ or an equiprobable
distribution of domains in which case the average charges in Eq. (2) vanish.
However, in AFM thin-films with finite-sized domains, such averaging can only
take place by considering a sufficiently large area including enough different
domains. On a smaller scale, the local anisotropy is very sensitive to the
direct surroundings. A local imbalance between the domains E1, E2, and E3
creates a local nonzero density of charges leading to a rotation of the local
anisotropy axis away from the magnetocrystalline anisotropy axis, as
illustrated schematically in Fig. 4a,b. In this example, the Néel vector in
the central domain in Fig. 4a is oriented along E2 as long as the areas
$S_{1}$ and $S_{3}$ of the surrounding domains along E1 and E3 respectively
are equal. In this configuration, the magnetoelastic charge $\sin 2\varphi$ is
fully compensated.111 The noncompensated charge $\cos 2\varphi$ in this case
creates a field $\psi_{1}$ that affects the value of the anisotropy but not
the direction. However, an imbalance $S_{1}\neq S_{3}$ between the areas along
E1 and E3 creates a nonzero charge $\sin
2\varphi\propto|S_{1}-S_{3}|/(S_{1}+S_{3})$ and a related field $\psi_{2}$. As
a result, the equilibrium orientation of the Néel vector in the central domain
rotates away from the crystalline axis E2 by
$\varphi\propto(H_{\mathrm{destr}}/H_{\mathrm{an}})|S_{1}-S_{3}|/(S_{1}+S_{3})$
(Fig. 4b). Hence, spatially fluctuating magnetoelastic charges result in local
anisotropy axes which do not have to coincide with the crystalline anisotropy
axes [35]. Depending on the relative strength of the crystalline anisotropy
$H_{\mathrm{an}}$ and the fluctuating destressing fields, the domains in an
AFM thin-film can range from distributed sharply around the crystalline
anisotropy axes up to an isotropic distribution as illustrated in Fig. 4c.

Based on our model, we can introduce a characteristic ”Debye radius” $D$ at
which the effect of fluctuations of the magnetoelastic charges is fully
screened. To estimate $D$, we assume that the orientations of different
domains are statistically independent. We can model the probability to find a
domain with a given orientation $\varphi$ using the Boltzmann distribution
with effective temperature $T_{\mathrm{eff}}$. In analogy to the Debye model,
we estimate the Debye radius
$D=t_{\mathrm{AFM}}\sqrt{T_{\mathrm{eff}}/(H_{\mathrm{destr}}M_{s})}$, where
$t_{\mathrm{AFM}}$ is the thickness of the AFM layer, and $M_{s}/2$ is the
sublattice magnetization.

We associate the statistical independence of the domain distribution with the
process of the magnetic ordering when cooling through the Néel temperature.
However, on a short length scale (below the Debye radius $D$) temperature-
induced redistribution of the domains can be blocked e.g. by domain wall
pinning. In this case correlations between the different domain types result
in local fluctuations around the equilibrium distribution and rotation of the
Néel vector away from the crystallographic directions, as explained above.

In order to verify this model, we investigate the response of the domain
structure to an external in-plane magnetic field. For this, we have refined
the existing theoretical framework [52, 44, 53], which was developed for a
compensated three-domain collinear AFM. Here, we consider the competition
between the destressing energy and Zeeman energy. Starting from an
equiprobable domain distribution at zero field, the existing theory predicts
that the dominant domain fraction increases as $H^{2}/(H_{dest}H_{ex})$ (see
Eq. S27 in SI) [45]. The DMI in $\alpha$-Fe2O3 and the corresponding nonzero
FM moment $m$ give rise to an additional Zeeman term proportional to
$mH/(MH_{dest})$. Our estimations show that the linear Zeeman term dominates
over the quadratic term for magnetic fields $H<8$ T. This agrees with our
experimental findings that the SMR signal depends linearly on magnetic field
in the low field regime (see Figs. 2b and 3a).

To calculate the angular dependence of the SMR signal, we add the Zeeman
contribution to Eq. (IV) (see SI). We consider two limiting cases of the
equilibrium domain distribution schematically shown in Fig. 4d: i) isotropic
distribution of all $\varphi$ values due to destressing effects (red line);
ii) sharp peaks corresponding to six-fold crystalline anisotropy (dark grey
line). Fig. 4d shows the results of the theoretical models and the
experimental data of the slope $dR_{xy}/dH$ at low field. The experimentally
extracted slope d$R_{xy}/$d$H$ of the field-dependent SMR signal in the linear
regime (triangles and dots) shows the qualitative $\sin 2\theta$ angle-
dependence irrespective of the preceding magnetic field history. This
dependence is in agreement with the theoretical prediction

$R_{xy}\propto(mH/H_{\mathrm{destr}})\sin 2\theta$ (3)

for an isotropic distribution with the maxima located at $\theta=45^{\circ}$
and $225^{\circ}$. In contrast, the six-fold anisotropy leads to peaks at
$\theta=30^{\circ}$ and 150∘. This further confirms that the orientation of
the domains in AFM thin films does not follow the crystal symmetry but lies
along isotropically distributed local anisotropy axes due to substrate
clamping.

Figure 5: (a) Vector map of the $\alpha$-Fe2O3 domain structure constructed
from angle-dependent XMLD XPEEM images. The orientation of the Néel vector
represented by different colors is shown in the histogram in (b). The height
of the bars in the circular histogram mark the frequency of occurrence of the
given orientation of the Néel vectors on a linear scale. Pixels for which the
uncertainty is larger than 15∘ are marked in gray.

##  V Experimental verification

In order to investigate the orientation of the Néel vectors in the multi-
domain state, we have performed angle-dependent XMLD x-ray photoemission
electron microscopy (XPEEM) on $\alpha$-Fe2O3 (20 nm with 2 nm Carbon
capping). Varying the angle between the linear polarization of the incident
x-rays and the sample allows reconstruction of the orientation of the Néel
orientation (see supplemental material for details of analysis [23]).

Fig. 5a shows a vector map of the antiferromagnetic domain structure. The
orientation of the Néel vectors encoded by different colors is shown in the
histogram in Fig. 5b. While the histogram does not show any signatures of the
hexagonal crystal structure of $\alpha$-Fe2O3, the distribution of domain
orientation is not fully isotropic either in the limited field of view of the
figure. This agrees well with the model of locally varying effective
anisotropies we introduce above.

##  VI Conclusions

This work highlights the impact of strain effects on the equilibrium domain
structure in thin-film AFMs. Combining imaging of the AFM domain structure
with electrical measurements, we find that the formation of domains is
predominantly driven by destressing fields which do not follow the hexagonal
crystal symmetry of $\alpha$-Fe2O3 but rather result in an overall isotropic
distribution of the Néel vector orientations with fluctuating easy-axes
depending on the local surrounding domains. Our work demonstrates that
understanding the formation of the domain structure and effective anisotropy
in thin-films is paramount for realizing AFM-based memory technologies.

###### Acknowledgements.

The authors thank M. Birch, S. Meyer, D. Bono, and B. Neltner for their
valuable contributions to this work. We thank A. Lee for the use of his Raman
spectrometer and P. Askeland for technical assistance. The x-ray imaging
measurements were carried out at the UE-49-PGM-SPEEM (XPEEM) and UE46 MAXYMUS
(STXM) beamlines at Helmholtz-Zentrum Berlin. We thank HZB for the allocation
of synchrotron radiation beamtime. This work is supported in part by SMART,
one of seven centers of nCORE, a Semiconductor Research Corporation program,
sponsored by the National Institute of Standards and Technology (NIST), and by
the DARPA TEE program. A. W. acknowledges financial support from the Swiss
National Science Foundation. O.G. acknowledges support of the ERC Synergy
Grant SC2 (No. 610115), the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - TRR 173 – 268565370 (project B12) and TRR 288 –
422213477 (project A09). A.L. and D.W. acknowledge funding from the Würzburg-
Dresden Cluster of Excellence on Complexity and Topology in Quantum
Matter—ct.qmat (EXC 2147, project no. 390858490). F.B. acknowledges financial
support from the Helmholtz Young Investigator Group Program. N. B. thanks
G.S.D. Beach and his group for their hospitality during his sabbatical visit
to MIT.

## References

  * Olejník  _et al._ [2018] K. Olejník, T. Seifert, Z. Kašpar, V. Novák, P. Wadley, R. P. Campion, M. Baumgartner, P. Gambardella, P. Němec, J. Wunderlich, J. Sinova, P. Kužel, M. Müller, T. Kampfrath, and T. Jungwirth, Science Advances 4, eaar3566 (2018).
  * Jungwirth  _et al._ [2016] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nature Nanotechnology 11, 231 (2016).
  * Baltz  _et al._ [2018] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Reviews of Modern Physics 90, 015005 (2018).
  * Jungwirth  _et al._ [2018] T. Jungwirth, J. Sinova, A. Manchon, X. Marti, J. Wunderlich, and C. Felser, Nature Physics 14, 200 (2018).
  * Železný  _et al._ [2018] J. Železný, P. Wadley, K. Olejník, A. Hoffmann, and H. Ohno, Nature Physics 14, 220 (2018).
  * Wadley  _et al._ [2016] P. Wadley, B. Howells, J. Elezny, C. Andrews, V. Hills, R. P. Campion, V. Novak, K. Olejnik, F. Maccherozzi, S. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich, F. Freimuth, Y. Mokrousov, J. Kune, J. S. Chauhan, M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds, B. L. Gallagher, and T. Jungwirth, Science 351, 587 (2016).
  * Bodnar  _et al._ [2018] S. Y. Bodnar, L. Šmejkal, I. Turek, T. Jungwirth, O. Gomonay, J. Sinova, A. A. Sapozhnik, H.-J. Elmers, M. Kläui, and M. Jourdan, Nature Communications 9, 348 (2018).
  * Godinho  _et al._ [2018] J. Godinho, H. Reichlová, D. Kriegner, V. Novák, K. Olejník, Z. Kašpar, Z. Šobáň, P. Wadley, R. P. Campion, R. M. Otxoa, P. E. Roy, J. Železný, T. Jungwirth, and J. Wunderlich, Nature Communications 9, 4686 (2018).
  * Chen  _et al._ [2018] X. Z. Chen, R. Zarzuela, J. Zhang, C. Song, X. F. Zhou, G. Y. Shi, F. Li, H. A. Zhou, W. J. Jiang, F. Pan, and Y. Tserkovnyak, Physical Review Letters 120, 207204 (2018).
  * Gray  _et al._ [2019] I. Gray, T. Moriyama, N. Sivadas, G. M. Stiehl, J. T. Heron, R. Need, B. J. Kirby, D. H. Low, K. C. Nowack, D. G. Schlom, D. C. Ralph, T. Ono, and G. D. Fuchs, Physical Review X 9, 041016 (2019).
  * Baldrati  _et al._ [2019] L. Baldrati, O. Gomonay, A. Ross, M. Filianina, R. Lebrun, R. Ramos, C. Leveille, F. Fuhrmann, T. R. Forrest, F. Maccherozzi, S. Valencia, F. Kronast, E. Saitoh, J. Sinova, and M. Kläui, Physical Review Letters 123, 177201 (2019).
  * Cheng  _et al._ [2020] Y. Cheng, S. Yu, M. Zhu, J. Hwang, and F. Yang, Physical Review Letters 124, 027202 (2020).
  * Zhang  _et al._ [2019] P. Zhang, J. Finley, T. Safi, and L. Liu, Physical Review Letters 123, 247206 (2019).
  * Chiang  _et al._ [2019] C. C. Chiang, S. Y. Huang, D. Qu, P. H. Wu, and C. L. Chien, Physical Review Letters 123, 227203 (2019).
  * Churikova  _et al._ [2020] A. Churikova, D. Bono, B. Neltner, A. Wittmann, L. Scipioni, A. Shepard, T. Newhouse-Illige, J. Greer, and G. S. D. Beach, Applied Physics Letters 116, 022410 (2020).
  * Chmiel  _et al._ [2018] F. P. Chmiel, N. Waterfield Price, R. D. Johnson, A. D. Lamirand, J. Schad, G. van der Laan, D. T. Harris, J. Irwin, M. S. Rzchowski, C.-b. Eom, and P. G. Radaelli, Nature Materials 17, 581 (2018).
  * Jani  _et al._ [2021] H. Jani, J.-C. Lin, J. Chen, J. Harrison, F. Maccherozzi, J. Schad, S. Prakash, C.-B. Eom, A. Ariando, T. Venkatesan, and P. G. Radaelli, Nature 590, 74 (2021).
  * Ross  _et al._ [2020] A. Ross, R. Lebrun, C. Ulloa, D. A. Grave, A. Kay, L. Baldrati, F. Kronast, S. Valencia, A. Rothschild, and M. Kläui, Physical Review B 102, 094415 (2020).
  * Gomonay and Loktev [2007] H. V. Gomonay and V. M. Loktev, Physical Review B 75, 174439 (2007).
  * Gomonay  _et al._ [2014] O. Gomonay, S. Kondovych, and V. Loktev, Journal of Magnetism and Magnetic Materials 354, 125 (2014).
  * Rao [1989] C. N. R. Rao, Annual Review of Physical Chemistry 40, 291 (1989).
  * Duò  _et al._ [2010] L. Duò, M. Finazzi, and F. Ciccacci,  _Magnetic Properties of Antiferromagnetic Oxide Materials: Surfaces, Interfaces, and Thin Films_, edited by L. Duò, M. Finazzi, and F. Ciccacci (Wiley, 2010).
  * [23] See supplemental Material at xxx for more detailed information .
  * Hÿtch  _et al._ [2011] M. Hÿtch, F. Houdellier, F. Hüe, and E. Snoeck, Ultramicroscopy 111, 1328 (2011).
  * Lübbe  _et al._ [2010] M. Lübbe, A. M. Gigler, R. W. Stark, and W. Moritz, Surface Science 604, 679 (2010).
  * Cvejic  _et al._ [2009] Z. Cvejic, B. Antic, A. Kremenovic, S. Rakic, G. Goya, H. Rechenberg, C. Jovalekic, and V. Spasojevic, Journal of Alloys and Compounds 472, 571 (2009).
  * Jubb and Allen [2010] A. M. Jubb and H. C. Allen, ACS Applied Materials and Interfaces 2, 2804 (2010).
  * Massey  _et al._ [1990] M. J. Massey, U. Baier, R. Merlin, and W. H. Weber, Physical Review B 41, 7822 (1990).
  * Porto and Krishnan [1967] S. P. S. Porto and R. S. Krishnan, The Journal of Chemical Physics 47, 1009 (1967).
  * Chernyshova  _et al._ [2007] I. V. Chernyshova, M. F. Hochella Jr, and A. S. Madden, Physical Chemistry Chemical Physics 9, 1736 (2007).
  * Xu  _et al._ [2009] Y. Xu, D. Zhao, X. Zhang, W. Jin, P. Kashkarov, and H. Zhang, Physica E: Low-dimensional Systems and Nanostructures 41, 806 (2009).
  * Angel  _et al._ [2019] R. J. Angel, M. Murri, B. Mihailova, and M. Alvaro, Zeitschrift fur Kristallographie \- Crystalline Materials 234, 129 (2019).
  * Tasaki and Iida [1963] A. Tasaki and S. Iida, Journal of the Physical Society of Japan 18, 1148 (1963).
  * Flanders and Remeika [1965] P. J. Flanders and J. P. Remeika, The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics 11, 1271 (1965).
  * Besser  _et al._ [1967] P. J. Besser, A. H. Morrish, and C. W. Searle, Physical Review 153, 632 (1967).
  * Chen  _et al._ [2012] P. Chen, N. Lee, S. McGill, S. W. Cheong, and J. L. Musfeldt, Physical Review B - Condensed Matter and Materials Physics 85, 1 (2012).
  * Cheng  _et al._ [2019] Y. Cheng, S. Yu, A. S. Ahmed, M. Zhu, Y. Rao, M. Ghazisaeidi, J. Hwang, and F. Yang, Physical Review B 100, 220408 (2019).
  * Dzyaloshinsky [1958] I. Dzyaloshinsky, Journal of Physics and Chemistry of Solids 4, 241 (1958).
  * Moriya [1960] T. Moriya, Physical Review 120, 91 (1960).
  * Morrish [1995] A. H. Morrish,  _Canted Antiferromagnetism: Hematite_ (WORLD SCIENTIFIC, 1995).
  * Urquhart and Goldman [1956] H. M. A. Urquhart and J. E. Goldman, Physical Review 101, 1443 (1956).
  * Voskanyan  _et al._ [1968] R. Voskanyan, R. Levitin, and V. Shchurov, Soviet Journal of Experimental and Theoretical Physics 27, 790 (1968).
  * Hoogeboom  _et al._ [2017] G. R. Hoogeboom, A. Aqeel, T. Kuschel, T. T. M. Palstra, and B. J. van Wees, Applied Physics Letters 111, 052409 (2017).
  * Fischer  _et al._ [2018] J. Fischer, O. Gomonay, R. Schlitz, K. Ganzhorn, N. Vlietstra, M. Althammer, H. Huebl, M. Opel, R. Gross, S. T. B. Goennenwein, and S. Geprägs, Physical Review B 97, 014417 (2018).
  * Lebrun  _et al._ [2019] R. Lebrun, A. Ross, O. Gomonay, S. A. Bender, L. Baldrati, F. Kronast, A. Qaiumzadeh, J. Sinova, A. Brataas, R. A. Duine, and M. Kläui, Communications Physics 2, 50 (2019).
  * Geprägs  _et al._ [2020a] S. Geprägs, M. Opel, J. Fischer, O. Gomonay, P. Schwenke, M. Althammer, H. Huebl, and R. Gross, Journal of Applied Physics 127, 243902 (2020a).
  * Han  _et al._ [2014] J. H. Han, C. Song, F. Li, Y. Y. Wang, G. Y. Wang, Q. H. Yang, and F. Pan, Physical Review B 90, 144431 (2014).
  * Baldrati  _et al._ [2018] L. Baldrati, A. Ross, T. Niizeki, C. Schneider, R. Ramos, J. Cramer, O. Gomonay, M. Filianina, T. Savchenko, D. Heinze, A. Kleibert, E. Saitoh, J. Sinova, and M. Kläui, Physical Review B 98, 024422 (2018).
  * Hajiri  _et al._ [2019] T. Hajiri, L. Baldrati, R. Lebrun, M. Filianina, A. Ross, N. Tanahashi, M. Kuroda, W. L. Gan, T. O. Menteş, F. Genuzio, A. Locatelli, H. Asano, and M. Kläui, Journal of Physics: Condensed Matter 31, 445804 (2019).
  * Fischer  _et al._ [2020] J. Fischer, M. Althammer, N. Vlietstra, H. Huebl, S. T. Goennenwein, R. Gross, S. Geprägs, and M. Opel, Physical Review Applied 13, 014019 (2020).
  * Note [1] The noncompensated charge $\mathop{cos}\nolimits 2\varphi$ in this case creates a field $\psi_{1}$ that affects the value of the anisotropy but not the direction.
  * Gomonay and Loktev [2002] H. Gomonay and V. M. Loktev, Journal of Physics: Condensed Matter 14, 3959 (2002).
  * Geprägs  _et al._ [2020b] S. Geprägs, M. Opel, J. Fischer, O. Gomonay, P. Schwenke, M. Althammer, H. Huebl, and R. Gross, Journal of Applied Physics 127, 243902 (2020b).