Thomas Pellegrini
email: thomas.pellegrini@irit.fr

Deep-learning-based central African primate species classification with MixUp and SpecAugment

Keywords: automated species classification, primate vocalizations, audio data augmentation

In this paper, we report experiments in which we aim to automatically classify primate vocalizations according to four primate species of interest, plus a background category with forest sound events. We compare several standard deep neural networks architectures: standard deep convolutional neural networks (CNNs), MobileNets and ResNets. To tackle the small size of the training dataset, less than seven thousand audio files, the data augmentation techniques SpecAugment and MixUp proved to be very useful. Against the very unbalanced classes of the dataset, we used a balanced data sampler that showed to be efficient. An exponential moving average of the model weights allowed to get slight further gains. The best model was a standard 10-layer CNN, comprised of about five million parameters. It achieved a 93.6% Unweighted Average Recall (UAR) on the development set, and generalized well on the test set with a 92.5% UAR, outperforming an official baseline of 86.6%. We quantify the performance gains brought by the augmentations and training tricks, and report fusion and classification experiments based on embeddings that did not bring better results.

Introduction

These last years, the domain of bioacoustics has greatly benefited from the impressive performance breakthroughs of deep learning and deep neural networks (DNNs), alike many other audio-related applications such as speech recognition and audio event detection [START_REF] Stowell | State of the art in computational bioacoustics and machine learning: How far have we come?[END_REF]. Automatic detection of animal vocalizations, besides being a scientific challenge in itself, can be helpful for monitoring biodiversity. Researchers involved in bioacoustics and/or the recent field named ecoacoustics [START_REF] Sueur | Ecoacoustics: the ecological investigation and interpretation of environmental sound[END_REF] gather ever-growing quantities of in-situ recordings that need to be manually analyzed. Automatic tools that label the recordings accurately are very much in demand to ease the time-consuming task of listening to hours of them [START_REF] Guyot | Sinusoidal modelling for ecoacoustics[END_REF].

A variety of tasks have been addressed with automatic tools: "simply" detecting the absence/presence of animal vocalizations in recordings (bird singing [START_REF] Stowell | Automatic acoustic detection of birds through deep learning: the first bird audio detection challenge[END_REF], for instance), identifying species of a given animal (the yearly LifeCLEF Bird Identification Task [START_REF] Goëau | LifeCLEF Bird Identification Task 2016: The arrival of Deep learning[END_REF], for instance), recognizing individuals in multiple species [START_REF] Stowell | Automatic acoustic identification of individuals in multiple species: improving identification across recording conditions[END_REF]. All the previous examples dealt with singing birds but many other animals have been considered, such as bats [START_REF] Prat | Everyday bat vocalizations contain information about emitter, addressee, context, and behavior[END_REF], and primates [START_REF] Heinicke | Assessing the performance of a semiautomated acoustic monitoring system for primates[END_REF][START_REF] Versteegh | Classification and automatic transcription of primate calls[END_REF][START_REF] Zwerts | Introducing a Central African Primate Vocalisation Dataset for Automated Species Classification[END_REF].

A number of machine learning techniques have been used in the literature, such as Support Vector Machines either with raw acoustic features or feature representations issued from deep neural networks [START_REF] Versteegh | Classification and automatic transcription of primate calls[END_REF][START_REF] Schuller | The INTERSPEECH 2021 Computational Paralinguistics Challenge: COVID-19 Cough, COVID-19 Speech, Escalation & Primates[END_REF], kernel methods, such as Extreme Learning Machine (ELM) [START_REF] Zwerts | Introducing a Central African Primate Vocalisation Dataset for Automated Species Classification[END_REF], and, of course, a variety of deep neural networks: convolutional and recurrent convolutional neural networks [START_REF] Grill | Two convolutional neural networks for bird detection in audio signals[END_REF], densely connected networks [START_REF] Pellegrini | Densely connected cnns for bird audio detection[END_REF],

Residual Neural Networks (ResNets) [START_REF] Kahl | Birdnet: A deep learning solution for avian diversity monitoring[END_REF], etc.

In this work, we are interested in comparing a number of DNN architectures in the task of distinguishing between four primate species based on their vocalizations: chimpanzees, mandrills, red-capped mangabeys and a mixed group of guenon species. We use the audio dataset provided by the INTER-SPEECH 2021 Computational Paralinguistics Challenge [START_REF] Versteegh | Classification and automatic transcription of primate calls[END_REF][START_REF] Schuller | The INTERSPEECH 2021 Computational Paralinguistics Challenge: COVID-19 Cough, COVID-19 Speech, Escalation & Primates[END_REF][START_REF] Zwerts | Introducing a Central African Primate Vocalisation Dataset for Automated Species Classification[END_REF]. Besides the four primate species of interest, a fifth class is considered in this classification task: natural background noise. In real applications, this "reject" class would be very interesting to have in a classifier if we aim at spotting out primate calls within lengthy audio field recordings.

Besides comparing different models, we also investigate the use of audio data augmentation, as it is key for training networks. We will report significant performance gains using SpecAugment [START_REF] Park | Specaugment: A simple augmentation method for automatic speech recognition[END_REF] and MixUp [START_REF] Zhang | mixup: Beyond Empirical Risk Minimization[END_REF].

The best system reported by the challenge organizers is based on extracting feature representations from a recurrent autoencoder in an unsupervised fashion and then use these embeddings, named "auDeep", as input to an SVM classifier. We therefore tried to use the auDeep embeddings within one of our best models. Although this improved the auDeep-SVM performance, no gain was obtained with our networks.

Models

In this section, the four different model architectures are described. All the models make use of blocks of convolution layers followed by global pooling before a classification head comprised of two fully-connected layers. The difference between the architectures lies in the convolutional blocks' definition.

In the present work, all the models take log-mel spectrograms of 64 coefficients as input. More details on data preprocessing will be given in Section 3.

Standard CNNs: CNN6 and CNN10

By standard CNNs we mean networks comprised of blocks of a convolution layer, followed by batch-normalization (BN), the rectifier linear unit (ReLU) activation function, and 2 × 2 average pooling. Dropout of rate 20% is applied after each convolution block. The blocks are followed by, first, an average pooling over the frequency dimension, and second, summation of mean and max-pooling on the time axis. The pooled representations (vectors) are given to a first fully-connected layer (FC) of 512 units, with ReLU, and second to the classification FC layer of dimension the number of classes of interest (five classes).

CNN6 and CNN10 follow this architecture with six and ten layers, respectively. CNN6 consists of four convolution layers with a 5 × 5 kernel size, and 2 × 2 padding. CNN10, shown in Table 1, uses kernels of size 3 × 3 and there are two consecutive convolution layers in each block. CNN6 and CNN10 totalize 4.57 M and 4.95 M parameters, respectively.

MobileNetV1

MobileNets [START_REF] Howard | Mobilenets: Efficient convolutional neural networks for mobile vision applications[END_REF] make use of depthwise separable convolutions to reduce computation costs and compensate with 1 × 1 kernel sized convolutions, also called point-wise convolutions, in order to exchange information between different channels.

In this work, we used MobileNetV1, comprised of 13 Mo-bileNetV1 convolutional blocks. Each block consists of a depthwise separable convolution, followed by average pooling, BN, ReLU, and then a pointwise convolution with BN and ReLU. The same pooling and FC layers as in the CNNs and ResNet22 are used after the convolutional blocks. In total, this network comprises 4.26 M parameters.

ResNet22

A ResNet is comprised of convolutional blocks, each of two convolution layers with a 3 × 3 kernel size, and a shortcut connection between input and output that allows to train efficient deep networks [START_REF] He | Deep residual learning for image recognition[END_REF]. We used ResNet22, a model comprised of 22 layers: two convolutional layers and a downsampling layer applied on the spectrograms and then eight convolutional blocks with residual connections. Again, the same classification head as the other models is used. The resulting model is much larger than the other architectures, with 62.60 M parameters. Despite this large size and the relatively small number of training samples, the test performance and generalization capability of this model is remarkable, as we shall see hereafter. No training strategy specific to this model was used.

WideResNet28-2

Wide Residual Networks [START_REF] Zagoruyko | Wide residual networks[END_REF] consist of an initial convolutional layer followed by three groups of residual blocks. After these blocks, follow average pooling and a linear layer that acts as a classifier. The residual blocks, each composed of two "Ba-sicBlocks" with a convolutional layer with 3 × 3 kernels, Batch Normalization and ReLU, are repeated three times. We used the official implementation available in PyTorch [START_REF] Paszke | Pytorch: An imperative style, high-performance deep learning library[END_REF] of the Wide ResNet 28-2 architecture, but slightly modified to be more similar to the other models in this study: after the convolution blocks, we use global max-and average-pooling, sum the result and apply two FC layers. It results in a small sized model of about 1.5 million parameters.

Data processing and setup

Feature extraction

The primate vocalization dataset [START_REF] Zwerts | Introducing a Central African Primate Vocalisation Dataset for Automated Species Classification[END_REF] consists of three subsets ("train", "devel" and "test") of comparable size: 6915, 6918 and 6923 audio files of variable duration. We padded all the files to reach three seconds duration, as it corresponds to the largest recording duration of all the files. 64 log-mel spectrograms were extracted with a 1024-sample-long Hanning window and 320 samples hop size. We ran experiments with the original 16 kHz sampled signal and a downsampled 8 kHz version to speed up model comparison. With the 16 kHz version, spectrograms are of size 150 × 64 in time and frequency.

Data balancing

The train and development subsets are highly imbalanced regarding the five classes of interest: 50.0%, 32.1%, 2.3%, 12.6%, 3.0% for Background, Chimpanzee, Guenon, Mandrill, and Red-capped mangabeys, respectively.

Training data are input to our models in mini-batches of size 32 samples during training, which is a relatively small size. Given the important imbalanced representation of the classes, it may happen that all the data in a mini-batch may belong to the same sound class Background. This will cause the models to overfit to this class, with more training clips, and underfit to the other sound classes with fewer training clips. To avoid this issue, Kong and colleagues proposed to use a balanced data sampler used during training [START_REF] Kong | Panns: Large-scale pretrained audio neural networks for audio pattern recognition[END_REF]. With this sampler, audio clips are equally sampled from all the five sound classes when constituting a mini-batch. We will compare the results obtained with this strategy to using a uniform standard sampler.

Data augmentation

Data augmentation is key to increase the generalization capabilities of DNNs. Nevertheless, finding the right augmentation strategies is a difficult task. In [START_REF] Salamon | Deep convolutional neural networks and data augmentation for environmental sound classification[END_REF], a number of audio-specific augmentations were tested in a systematic way for environmental sound classification: time stretching, pitch shifting, dynamic range compression and background noise addition. The authors show that the impact of using these techniques depends on the sound classes of interest: performance may decrease if the technique is not well suited for a given class.

In this work, several augmentations were tested: time stretching with a rate uniformly sampled in the [0.9, 1.1] interval, pitch shifting by 0 to 3 quarter-tones up or down, MixUp, and SpecAugment. In preliminary experiments, time stretching and pitch shifting did not bring performance improvements. On the contrary, MixUp and SpecAugment revealed very efficient.

MixUp [START_REF] Zhang | mixup: Beyond Empirical Risk Minimization[END_REF] is a successful data augmentation/regularization technique, in which we mix pairs of data samples (images, audio clips, etc.). If x1 and x2 are two different input samples (spectrograms in our case), and y1, y2 their respective one-hot encoded labels, then the mixed sample and target are obtained by a simple convex combination:

x mix = λx1 + (1 -λ)x2 y mix = λy1 + (1 -λ)y2
where λ is a scalar sampled from a symmetric Beta distribution at each mini-batch generation:

λ ∼ Beta(α, α)
where α is a real-valued hyper-parameter to tune (always smaller than 1.0 in our case).

SpecAugment [START_REF] Park | Specaugment: A simple augmentation method for automatic speech recognition[END_REF] is an occlusion augmentation technique, applied onto the log-mel spectrograms. In our case, we randomly choose to mask zero, one or two stripes both in the time and frequency axes. The width of the strides is also chosen randomly with a maximum of 16 and 8 bins in time and frequency, respectively. SpecAugment is applied at mini-batch level, meaning that the same random strides are masked in all the samples of a given mini-batch. SpecAugment has been originally proposed in automatic speech recognition but it has been rapidly used with success also for other audio-related tasks, such as audio tagging.

Experimental setup

The models were trained to optimize binary cross-entropy, when MixUp was used, and categorical cross-entropy otherwise. In all our experiments, we used the Adam [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF] optimizer to train the models with a learning rate of 1e-3. The size of the minibatches was set to 32. We trained our model for 60k iterations. Except WideResNet, all the model architectures are the ones proposed in [START_REF] Kong | Panns: Large-scale pretrained audio neural networks for audio pattern recognition[END_REF] for multilabel audio event detection and we largely used their open-sourced implementation1 . Our PyTorch [START_REF] Paszke | Pytorch: An imperative style, high-performance deep learning library[END_REF] code is available 2 .

We used Exponential Moving Average (EMA) to obtain our final models: during the training of any model, we create a separate model which is the EMA variant of the model being trained. We used a 0.995 decay and EMA updates were initiated after 15k training iterations.

Results

In this section, we report results in terms of mean average precision (mAP) and unweighted average recall (UAR). UAR is the metric used in the Primate sub-challenge, occurring in the framework of the yearly Paralinguistic Compare challenges.

Comparison between models

Table 2 shows the results obtained with the different models on the development subset. As a global comment, all the DNNs outperformed the 84.6% UAR reported in the official challenge paper [START_REF] Schuller | The INTERSPEECH 2021 Computational Paralinguistics Challenge: COVID-19 Cough, COVID-19 Speech, Escalation & Primates[END_REF]. The best model is CNN10-16k, the 10-layer standard CNN, that achieved a 93.6% UAR on the devel subset and 92.5% on the official test set. CNN10-16k is closely followed by ResNet22-16k. CNN10-16k's size is an order of magnitude smaller than ResNet22, thus, besides being slightly more accurate, it is also much more preferable in terms of computation. CNN6, the 6-layer standard CNN is ranked third closely followed by MobilenetV1. The least performing model was WideResNet28-2. This ranking between architectures should be considered with caution because we used the same learning hyper-parameters for all models. It may happen that the 1e-3 learning rate is not the best one for all the models, for instance. Nevertheless, the fact that a standard CNN is the best model in these experiments, is in line with the results obtained by Kong and colleagues on AudioSet [START_REF] Kong | Panns: Large-scale pretrained audio neural networks for audio pattern recognition[END_REF]. We tested CNNs with more layers (CNN14) on our task, but CNN10 proved to be better, probably because CNN14 is too deep to be correctly trained with our small dataset.

In Table 2, we also report three UAR values obtained on the official challenge test set. CNN10-16k reached a 92.5% UAR on the test set, showing that it generalized well on these new data. It outperformed the ELM model [START_REF] Zwerts | Introducing a Central African Primate Vocalisation Dataset for Automated Species Classification[END_REF] at 70.7%, and the 86.6% auDeep-SVM baseline provided by the challenge organizers [START_REF] Schuller | The INTERSPEECH 2021 Computational Paralinguistics Challenge: COVID-19 Cough, COVID-19 Speech, Escalation & Primates[END_REF].

Figure 1 shows the confusion matrix on the devel subset, obtained with our best model. The best recognized classes are guenon (G) and mandrills (M) with about 97% accuracy. Background (B) has the lowest accuracy, about 89%, and most confusions are made with samples of the chimpanzee class (C), and to a less extent with the other primate species. Finally, Redcapped mangabeys (R) are confused with Mandrills in 3.3% of the cases.

Ablation study

Table 3 shows the impact of several components: balanced sampler, data augmentation and EMA. This ablation study was performed with ResNet22-8k and CNN10-16k.

The first two lines compare the use of a random sampler (no-bal) and a balanced sampler (bal). Using a random sampler is worse than a balanced one in UAR, as expected. In the case of ResNet22, there is a two points difference in mAP and 4 points in UAR. This is because the background class is better recognized with no-bal, and it is by far the most represented class in the dataset (50% of the samples).

The next three lines show the impact of data augmentation: specAugment alone (specA), MixUp alone and specAugment and MixUp together. We can see that both techniques boost mAP and UAR significantly: about 3 points absolute in mAP and 5.5 points in UAR for ResNet22, or instance. Finally using both methods together gave the best results for both models.

The last line in the table shows the results of the counterpart of the EMA model of the preceding line. EMA shows slightly better results than their non-EMA counterparts. This tendency was observed with all the architectures.

Complementary experiments

In this section, we briefly describe complementary experiments that did not bring improvements, at least in UAR.

CNN10 embeddings and SVM. The challenge organizers reported their best results using Support Vector Machines with a linear kernel applied onto embeddings obtained with a deep recurrent neural network named auDeep: 84.6% and 86.6% of the development and test subsets respectively. In a similar fashion, we extracted CNN10 embeddings outputted from the penultimate fully-connected layer in CNN10. Figure 2 shows a twodimensional representation of the embeddings obtained with t-SNE [START_REF] Van Der Maaten | Visualizing data using t-sne[END_REF] on the development subset. We can observe that the five classes are mostly well separated in the plane, Background (in red) and Chimpanzee (in green) to a less extent though. Linear SVMs reached a 88.8% UAR on the development set.

Fusion between CNN10 and ResNet22. Since CNN10 and ResNet22 achieved very similar mAP and UAR values, we experimented an intermediate fusion: we defined a "ResNet22-CNN10" model that combines CNN10-8k and ResNet22-8k as ResNet22 branches were initialized with the weights of the best models reported above. We tried to freeze or not these branches, and we also tried with and without augmentation (specAugment and/or MixUp). Freezing the branches and using both augmentations led to the best results: 91.6% UAR and 96.2% mAP. This UAR value is below that of the CNN10-8k alone. Nevertheless, the 96.2% mAP is slightly larger than the CNN10-8k 95.4% value.

Conclusions

We reported experiments aiming to classify primate vocalizations into four primate species of interest, plus a background category with forest sound events. We compared standard DNN architectures: standard deep convolutional neural networks (CNNs), MobileNets, ResNets and WideResNets.

To tackle the relative small size of the training dataset, less than seven thousand audio files, the data augmentation techniques SpecAugment and MixUp proved to be very useful. Against the very unbalanced distribution of the classes, we used a balanced data sampler, which was efficient with a ResNet but not with our standard CNN. An exponential moving average of the model weights allowed to get slight further gains. The best model was a standard 10-layer CNN, comprised of about five million parameters, that achieved a 92.5% UAR on the test subset. We quantified the performance gains brought by the augmentations and training tricks, and report fusion and classification experiments based on embeddings that did not bring better results in UAR at least.

Most confusions occur between Background and all the primate species, thus, a hierarchical system would be worth trying. One can also imagine a single model but trained on both objectives.

Figure 1 :

 1 Figure 1: Confusion matrix on the devel subset, obtained with our best model CNN10-16k. B: Background, C: Chimpanzee, G: Guenon, M: Mandrille, R: Red-capped Mangabeys.

Figure 2 :

 2 Figure 2: 2-d representations obtained with t-SNE on the embeddings obtained with CNN10-16k on the devel samples.

Table 1 :

 1 CNN10 architecture[START_REF] Kong | Panns: Large-scale pretrained audio neural networks for audio pattern recognition[END_REF]. The number before @ indicates the number of output channels.

	log-Mel spectrograms
	75 frames × 64 Mel bins
	64@3 × 3 BN, ReLU	×2
	Pooling 2 × 2
	128@3 × 3 BN, ReLU	×2
	Pooling 2 × 2
	256@3 × 3 BN, ReLU	×2
	Pooling 2 × 2
	512@3 × 3 BN, ReLU	×2
	Global pooling
	FC, 512, ReLU
	FC, 5, Sigmoid or Softmax

Table 2 :

 2 Comparison between models. mAP: average precision, UAR: unweigthed average recall. 8k, 16k: audio sampling rate.

	devel	test

Table 3 :

 3 Ablation study mAP(%)/UAR(%) results using CNN10-16k on the development subset. bal: balanced data sampler, no-aug: no augmentation at all, specA: SpecAugment.

	setting	ResNet22-8k CNN10-16k
	no-bal, no-aug	87.9/78.7	94.7/90.9
	bal, no-aug	89.8/82.6	94.7/92.1
	bal, specA	92.6/88.1	94.1/92.1
	bal, MixUp	92.4/88.8	96.0/92.5
	bal, specA+MixUp	93.8/91.2	97.0/93.6
	w/o EMA	93.1/90.0	96.7/93.0
	two parallel branches to extract embeddings from their penul-
	timate fully-connected layer. Their embeddings, of size 512
	(CNN10) and 2048 (ResNet22) are then simply concatenated
	and given as input an FC classification layer. Both CNN10 and

https://github.com/qiuqiangkong/audioset tagging cnn

https://github.com/topel/ComParE2021 PRS

Acknowledgements

This work was partially supported by the French ANR agency within the LUDAU project (ANR-18-CE23-0005-01) and the French "Investing for the Future -PIA3" AI Interdisciplinary Institute ANITI (Grant agreement ANR-19-PI3A-0004). We used HPC resources from CALMIP (Grant 2020-p20022).


Françoise Damay
email: francoise.damay@cea.fr

Jonas Sottmann

François Fauth

Emmanuelle Suard

Antoine Maignan

Christine Martin

High temperature spin-driven multiferroicity in ludwigite chromocuprate Cu2CrBO5

We report spin-driven multiferroicity above 100 K in the ludwigite Cu(II) oxyborate Cu2CrBO5.

Spontaneous polarization, which reaches 35 µC.m -2 at 5 K, appears below 120 K, concomitantly with an incommensurate antiferromagnetic order and complex magnetodielectric effects. In magnetically induced ferroelectrics, multiferroicity usually appears at low temperature, because of the competing magnetic exchanges needed to stabilize a magnetic spiral : the remarkably high transition temperature observed in Cu2CrBO5 originates from the presence of strong Cu-O-Cu magnetic super-exchange interactions, which are not weakened by Cu/Cr cationic disorder. Our result provides an important contribution to the search for high temperature spindriven multiferroics amongst low-dimensional cuprates.

Introduction

To understand and optimize the coupling between magnetic and dielectric properties in multiferroics is one of the enduring challenge of condensed matter physics [1], [2]. Amongst the most promising multiferroic compounds are spin-driven ferroelectrics, in which ferroelectricity emerges concomitantly with a non-collinear spiral magnetic order breaking inversion symmetry [3]. Different microscopic mechanisms are at play depending on whether this complex magnetic ordering is cycloidal [4] or helicoidal [5], but in both cases, a strong coupling between the ferroic parameters is expected, hence their high potential for applications [6]. The design of spiral magnetic order is not straightforward, however, as it often relies on the existence of competing (frustrated) magnetic interactions, which also means that such orders ordinarily occur at low temperature, typically below 40 K [7]. To overcome this issue, compounds with large magnetic super-exchange interactions have been studied, as, for instance, hexaferrites, in which magnetic super-exchange between spin blocks can be tuned to produce long-wavelength magnetic structures close to or above room temperature (RT), and large magnetoelectric effects [8], [9], [10]. Another example is cupric oxide CuO, with one of the largest Curie temperature (TC) amongst spin-driven multiferroics so far, ~ 230 K [11], owing to the presence of a strong magnetic super-exchange J ~ 700 K. Yet, CuO is ferroelectric in a narrow temperature range of about 20 K only, which corresponds to the range of existence of its spiral magnetic order : this serious limitation calls for better candidates.

As rich as that of silicates or phosphates, borate chemistry has always provided interesting structural frameworks [12], with, sometimes, unique functionalities, especially in non-linear optics [13]. A notorious oxyborate crystal structure for transition metals (TM) is for instance the ludwigite, of formula M2M'BO5 (M and M' being divalent and trivalent TM, respectively).

Its intricate framework of edge-sharing TM octahedra, prone to frustration, has been studied in the past for a wide variety of uncommon features, including mixed valence and dimerization [14], [15], [16], [17], magnetic sub-lattices decoupling [18], [19], ferroelectricity [20] and for its potential use in lithium-ion battery [21]. Literature on heterometallic copper ludwigites Cu2M'BO5 (M' = Al, Fe, Ga, Mn) is in contrast somewhat scarcer. Indeed, Cu/M' ions nonrandom distribution over at least two out of the four crystallographically distinct TM sites of the ludwigite structure is always found to be an issue [22], [23], [24], [25], which, combined with the complex geometry of the magnetic exchange paths [26], is known to affect physical properties [27], [28] and hinder reproducibility [29].

To the best of our knowledge, ludwigite Cu2CrBO5 has only been investigated up to now for its redox activity [21]. In this article, we show that Cu2CrBO5 is a remarkable example amongst copper ludwigites of a perfect cationic order between Cu(II) and Cr(III) species, and the first instance in this family of a spin-driven multiferroic above 100 K with complex magnetodielectric properties. Perpendicular chains of strongly interacting magnetic spins, weakly coupled, are proposed to explain the complex non-centrosymmetric magnetic order at the origin of the spin driven multiferroic properties.

Experimental Methods

Cu2CrBO5 was prepared starting from a mixture of 2 CuO/0.5 Cr2O3/0.5 B2O3 ( 11 B enriched), heated in air at 900°C for 24 hrs. An impurity of CrBO3 (~0.5 wt %) was identified. Synchrotron X-ray powder diffraction (SXRPD) experiments were performed between 10 K and 260 K at BL04-MSPD at ALBA synchrotron (Barcelona, Spain) (λ = 0.44296 Å) [30].

Cu2CrBO5 powders were 20 m sieved and filled in 0.5 mm  borosilicate capillaries. Diffractograms were collected using the Position Sensitive Detector Mythen. Neutron powder diffraction (NPD) experiments were performed at room temperature on the D2B high-resolution diffractometer (λ = 1.5946 Å), at ILL (Grenoble, France), and between 1.5 K and 300 K on the G4.1 diffractometer (λ = 2.426 Å), at LLB-Orphée (Gif-sur-Yvette, France). Rietveld refinements were performed using the Fullprof Suite [31].

Magnetic susceptibility (χ) was derived from magnetization data recorded in a field of 1 T, on warming from 5 to 400 K after a zero-field cooling (MPMS, Quantum Design). Isothermal magnetic field (H) dependent magnetization (M) loops were recorded using the AC measurement system of a 9 T Physical Properties Measurement System (PPMS, Quantum Design). A homemade sample probe inserted in the PPMS was used to measure the dielectric permittivity (ε') and the electric polarization (P). For that purpose, square thin platelets (2 x 2 x 0.5 mm) were cut from the reacted bars. The electrical contacts were deposited on the opposite largest faces using silver paint. The electret was soldered to the sample probe using Cu wires. ε' was measured as a function of T or H using a LCR meter (Agilent 4284A) whereas P was obtained by integrating the pyroelectric current collected via an electrometer (Keithley 6514A). For the P(T) curves, the data were collected upon warming at 4 K.min -1 . S1, S2 andS3. Its framework can be described by TM oxygen polyhedra sharing edges to form zigzag walls, themselves connected to each other through apexes. These walls delimit triangular tunnels occupied by boron atoms (Figure 1a), with typical B-O bond lengths ~ 1.35-1.37(1) Å [23]. A few specifics of the Cu2CrBO5 structure, however, are noteworthy. A major structural difference lies in the shift of the Cu1 position to a 2c (0 ½ 0) site in Cu2CrBO5, which flattens the usually strongly corrugated Cu chains found in ludwigites. Moreover, no Cu/Cr mixing is observed on the TM sites, in contrast with other heterometallic copper ludwigites. In Cu2CrBO5, the oxygen coordination around all three Cu sites is (4 + 2), in agreement with a Jahn-Teller ion like Cu(II), with the axial Cu-O bonds being ~ 20 to ~ 50 % longer than the equatorial ones (Table S2). CrO6 octahedra are very regular in contrast, with the Cr-O bonds being equal within a few % (Table S2). Based on those considerations, the crystal structure of Cu2CrBO5 can be described as chains, running along b, of CuO4 squares sharing edges or corners, and connected through their edges or corners to chains (parallel to a) of edge-sharing CrO6 octahedra (Figure 1b). The shortest distance between TM corresponds to CuO4 sharing edges (Cu1-Cu3 ~ 2.80 Å, Table S3), forming trimerlike motifs (thick red segments on Figure 1a). The Cu2CrBO5 crystal structure is stable with decreasing temperature; it remains monoclinic P21/c down to 10 K, the lattice parameters decreasing with a less than 0.1 % variation between 275 K and 10 K (Figure 1c). Slight increases of the c parameter and of the monoclinic distortion are observed, however, below 60 K (inset of Figure 1c).

Results

Cu2CrBO5 crystallizes at

Figure 2a shows the temperature evolution of the zero-field cooled (zfc) magnetic susceptibility of Cu2CrBO5 in 1 T. A susceptibility maximum is clearly seen at TN = 117 K, suggesting a transition to an antiferromagnetic state. Above TN, the susceptibility does not follow a Curie-Weiss law in the temperature range investigated (T < 400 K, inset of Figure 2a), which could be due to low-dimensional effects or short range magnetic ordering, and prevented us from extracting a meaningful CW. At 119 K, concomitantly with the magnetic transition, a like peak is observed on the zero-field dielectric constant ' (Figure 2b, left axis), and is accompanied by a step on the dielectric loss, whose low values indicate very small losses in the vicinity of the transition temperature (Figure 2b, right axis). This transition is frequencyinvariant over the f-range studied (f = 5 to 100 kHz, not shown), thus excluding spurious experimental effects. Figure 2c shows the evolution with temperature of the electric polarization P obtained from measurements of the pyroelectric current. A finite P is measured in zero magnetic field below 122 K, and down to 5 K, where it reaches ~ 35 µC.m -2 for a poling field of 181.8 kV.m -1 . P can be switched by inverting the sign of the poling electric field, unambiguously confirming the ferroelectric nature of Cu2CrBO5 in its magnetically ordered state. The '(T)H curves (Figure 2d) reveal a clear magnetodielectric effect : as H increases, the dielectric peak shifts to lower temperature (-4 K between 0 and 9 T), with, up to 5 T, an increase of the peak amplitude. Above 5 T, the ' peak starts to broaden markedly with H, its amplitude decreasing concomitantly. Further measurements versus magnetic field H were therefore performed to characterize this effect. Three different behaviors are identified, as illustrated by the magnetodielectric curves MDE = (('H-'0)/ '0)(H) of Figure 2e ; just above TN, there is no magnetodielectric effect ; just below TN, a negative MDE is observed at 116 K, while a positive MDE is measured at 112 K. On the latter, a maximum of 0.6 % is reached at +/-5.9 T. This characteristic magnetic field of 5.9 T at 112 K corresponds to a change of slope on the magnetization M(H) curves, more clearly identified on the dM/dH(H) curves (Figure S2 A proper understanding of the microscopic mechanism behind the multiferroic properties of Cu2CrBO5 requires knowledge of its magnetic ordering. Cold neutron diffraction experiments were performed down to 1.5 K (Figure 3a). Below TN, new Bragg peaks of magnetic origin can be seen on the neutron diffractograms, whose incommensurate Q positions vary with temperature down to 30 K (inset of Figure 3a) and then remain invariant down to 1.5 K, but still incommensurate. Indexing of these peaks actually proved to be challenging, owing to the limited number of magnetic peaks. Several solutions with similar reliability factor could be obtained for the magnetic propagation vector k. One of them, with only one incommensurate modulation along a, corresponds to k = (0.09 0 0.25) (illustrated on Figure S3), while all the other solutions involve at least two incommensurate modulations along b and c (or a and c). It is not possible to characterize further the magnetic ordering at that stage, because of the low symmetry of the crystal structure, the existence of four distinct magnetic sites, and the uncertainty on the number of modulations for k. The fact that the magnetic state is ferroelectric nevertheless suggests a long wavelength helicoidal or cycloidal magnetic structure, rather than a spin amplitude modulation, arguably involving both Cr and Cu spins. This incommensurate magnetic propagation vector confirms that the ludwigite framework of Cu2CrBO5 is host to frustration. Owing to its different crystal structure, the magnetic topology in Cu2CrBO5 is distinct from that of other ludwigites, and can be seen as two perpendicular sublattices made of spin chains. One chain is made of Cu spins and runs along b; because it is made of an alternation of CuO4 units sharing corners or edges, magnetic interactions in these chains are either of the super-exchange type (through the angle 1 on Figure 1a) or directexchange like (through the angle 2 on Figure 1a). This chain thus corresponds to an alternation of magnetic exchanges -J1-J1-J2-J2-between Cu atoms (Figure 3b). The second magnetic sublattice is made of Cr chains. As CrO6 octahedra only share edges along a, Cr-Cr interactions are through a unique direct-exchange Ja (Figure 3b, in which only nearest-neighbor (NN) interactions only are considered for simplicity). A minimum of six inter-sublattice interactions have to be taken into account in addition: they involve triangular configurations in the bc plane and along a (Figure 3b and Table S3), so that competing interactions are conceivable in the three directions. This will be dwelt upon further in the discussion, after considerations on J1 and J2. S3). White (colored) arrows are for direct (super)-exchange configurations; intra-chains magnetic paths are in green or red, inter-sublattices exchange paths in dark/light grey. Cu/Cr atoms in blue/green for clarity. The thin arrow line on the left panel indicates the direction of the polyhedral block shown on the right.

The simultaneous occurrence of ferroelectricity and magnetic order designates copper ludwigite Cu2CrBO5 as a new member of the spin-driven multiferroic family. Compared with most known multiferroic compounds, Cu2CrBO5 exhibits in addition a high transition temperature of 120 K, amongst the highest known, but still behind tuned hexaferrites (295 K [32]), CuO (230 K [11]) or YBaCuFeO5 (200 K [33], [34]). Along CuO, and non-oxide CuBr2 [35] or Cu2OCl2 [36], Cu2CrBO5 confirms that low-dimensional divalent copper compounds have a potential to be high-temperature multiferroics. It is therefore of interest to rationalize our findings, using the extensive knowledge built on Cu(II) oxides of low dimensionality, since the discovery of high-Tc superconductivity in cuprates.

For more than 50 years, the Goodenough-Kanamori-Anderson (GKA) rules [37], [38] have been used to qualitatively relate magnetic exchange J with orbital overlap. In divalent copper oxide compounds, the NN Cu-Cu spin interaction J is thus expected to change from antiferromagnetic (AFM) to ferromagnetic (FM), when the angle  of the Cu-O-Cu bond goes from 180° to 90°. Such a correlation has indeed been shown in a series of low-dimensional Cu(II) compounds [39], [40], [41], including CuO [11], [42], and is recalled in Figure 4. In Cu2CrBO5, Cu chains are made of edge-sharing trimer units (2 close to 90°), connected through corners by a Cu-O-Cu angle 1 close to 120° (Figure 1a and Table S3). Using Figure 4, we can expect ferromagnetic Cu trimers (FM J2), antiferromagnetically coupled through J1 (Figure 3b). J1 could be as high as 20 meV in this case, which is the right order of magnitude to account for TN = 120 K (~ 10 meV).

For comparison with other copper ludwigites, two instances can be mentioned. In a very recent study on Cu2MBO5 (M = Ga, Al) [START_REF] Kulbakov | Destruction of long-range magnetic order in an external magnetic field and the associated spin dynamics in Cu2[END_REF], authors report for Cu2GaBO5 a complex non-collinear long-range magnetic order, with a propagation vector k = (0.45, 0, -0.7), below TN = 4.1 K.

Following the GKA rules for Cu-O-Cu bonds, they suggest that, since half of the TM sites (corresponding in this case to the Cu1 and Cr sites of Cu2CrBO5) are statistically occupied by 1/3 of the Cu 2+ , this disordered sublattice of Cu orphan spins provides only a very weak FM coupling between corner-sharing AFM units. Cu2MnBO5 [24], in which a k = 0 magnetic order below TN = 92 K was found, is built on FM trimer units roughly coupled AFM. The overall scenario of a FM J2 and AFM J1 seems to hold in this case, but this resemblance has to be taken with caution, as Cu/Mn disorder, along with the Jahn-Teller nature of Mn 3+ , will obviously alter magnetic exchanges. presented in [39], [41], [42], [44], obtained either from susceptibility measurements, neutron data or theoretical calculations). The dotted line is a guide to the eyes. The value of the predominant exchange J in Cu2CrBO5 can be estimated to be ~ 20 meV, from the 1 value obtained from SXRPD refinements (see Figure 1a and Table S3).

The role of Cr chains becomes crucial because of the complex inter-sublattice pathways exemplified earlier. It is not possible to establish a hierarchy of exchange at this stage, but the modulated magnetic propagation vector tells us that one scenario is for J2 and J'1 to be both AF, or J'2 and J1 to be of opposite signs. Based on the fact that J2 and J1, and J'2 and J'1 have very similar geometries, it is most likely that frustration arises from the phase shift between Cu chains. It is a similar scenario, based on two magnetic sublattices with strong intra-sublattice interactions and weakly frustrated inter-sublattice interactions, which is invoked to explain spin frustration in CuO [45]. Arguably, magnetic anisotropy could also be an important ingredient in the stabilization of the spiral ground state. To compare with Cu2MnBO5, one could argue that in Cu2CrBO5 the Heisenberg character of the Cr spins favors incommensurate orderings, an effect that disappears when axial magnetic anisotropy directs the moment, such as in a Jahn-Teller ion like Mn 3+ .

Conclusion

Our results emphasize that the ludwigite network of copper oxyborates displays the right balance of low dimension, magnetic frustration, and strong magnetic exchange, for high temperature multiferroicity. Cu2CrBO5 is the first member of this family to be evidenced, but other heterometallic Cu(II) compounds remain to be investigated, especially if the disorder issue can be controlled. Density functional calculations should prove a really useful tool to further explore this system, in particular to better apprehend the importance of the intersublattice coupling, and understand how it can be used to further optimize its multiferroic properties.

AUTHOR CONTRIBUTIONS

Sample synthesis and structural characterizations by C.M. and J.S., SXRPD experiment by J.S. Å 3 . RBragg = 1.8 %;  2 = 17.6). Atom labelling corresponds to that of Figure 1a. Thermal displacement parameters Biso were constrained to be equal for all oxygen atoms. Anisotropic strain (strain_model = 2 [31]) was introduced in the refinement.

Atom

  RT with the monoclinic space group P21/c (#14, a = 3.05487(2) Å, b = 12.18070(8) Å, c = 9.41432(7) Å and β = 94.5602(4)°), like Cu-based ludwigites Cu2FeBO5 and Cu2GaBO5 [23]. Rietveld refinements are shown in Figure S1, and corresponding structural parameters are given in Tables

Figure 1 .

 1 Figure 1. Projections of the monoclinic P21/c crystal structure of Cu2CrBO5 along a (A) or along b (B). Cu atoms are in blue, Cr in green, B in pink and O in red. Distances between transition metals at RT are indicated, as well as the Cu-O-Cu angles 1 and 2. C illustrates the temperature evolution of the cell parameters a, b, and c, and of the monoclinic angle  (inset) (from SXRPD Rietveld refinements).

  ), and is indicative of a metamagnetic transition in the vicinity of TN, more generally suggesting for Cu2CrBO5 a rich multiferroic phase diagram to explore.

Figure 2 .

 2 Figure 2. Temperature (T) dependence of the zfc magnetic susceptibility  in 1 T (A). The small bump observed around 70 K is of unknown origin. The inset shows 1/(T) in 0.01T, in the range 100 -400 K. Temperature evolution of the dielectric constant ' (B-right axis), of the corresponding loss (tan  (B-right axis) at 10 kHz in 0 T, and of the spontaneous polarization P after +/-181.8 kV.m -1 electric field poling (C). Magnetic field evolution of the dielectric constant '(T) measured at 10 kHz between 0 and 9 T (D). E shows the magnetodielectric coefficient MDE defined as (('H-'0)/'0) versus H at 112 K, 116 K and 120 K.

Figure 3 .

 3 Figure 3. Cu2CrBO5 neutron diffraction data at 1.5 K (A), with the inset illustrating the corresponding temperature evolution. Asterisks indicate the incommensurate Bragg magnetic peaks appearing below TN = 120 K. B schematizes the different magnetic exchange paths (as defined in TableS3). White (colored) arrows are for direct (super)-exchange configurations;

Figure 4 .

 4 Figure 4. Evolution of the main magnetic exchange interaction J as a function of the Cu-O-Cu bond angle  in some cuprates, including Cu(II) oxyborate Cu2CrBO5 (drawn from data

Figure S2 .

 S2 Figure S2. Magnetic field evolution (A) and corresponding first magnetic field derivative (B) of the magnetization M of Cu2CrBO5 at different temperatures (5 K, 110 K, 115 K and 120 K).

Figure S3 .

 S3 Figure S3. LeBail profile of the Cu2CrBO5 NPD data (λ = 2.426 Å) at 1.5 K. The chosen propagation vector in that case is k = (0.096(3), 0, 0.25) (experimental data: open red circles, calculated profile: continuous line, allowed Bragg (magnetic Bragg) reflections: vertical green (purple) ticks. The difference between the experimental and calculated profiles is displayed at the bottom of each graph as a blue line).

Table S2 .

 S2 M-O bond lengths in the TMO6 polyhedra of Cu2CrBO5 at 260 K (from SXRPD refinement results). In italic are shown bond lengths > 2.35 Å (not taken into account in the average calculation).

		Site	x	y	z	Biso (Å 2 )
	Cu1	2c	0	0.5	0	0.28(2)
	Cu2	2a	0	0.5	0.5	0.20(2)
	Cu3	4e	0.9979(3)	0.2709(5)	0.0244(7)	0.20(2)
	Cr	4e	0.4959(4)	0.3750(1)	0.2738(1)	0.15(2)
	B	4e	0.526(3)	0.3744(9)	0.7459(8)	0.4(1)
	O1	4e	0.015(1)	0.1453(3)	0.9095(4)	0.03(3)
	O2	4e	0.975(1)	0.3966(3)	0.1438(3)	0.03(3)
	O3	4e	0.432(2)	0.2232(3)	0.1853(4)	0.03(3)
	O4	4e	0.716(1)	0.3818(4)	0.8819(3)	0.03(3)
	O5	4e	0.458(2)	0.0312(3)	0.1650(4)	0.03(3)
			Cu1 (2c)	Cu2 (2a)	Cu3 (4e)	Cr (4e)
	TM-O		O2:1.855(3) x2	O1:1.966(4) x2	O1:1.877(3)	O2:1.943(4)
	distance (Å)	O4:1.977(4) x2	O5:2.042(4) x2	O2:1.904(3)	O1:1.972(4)
			O4:2.911(4) x2	O5:2.388(5) x2	O3:2.018(4)	O5:1.990(4)
					O4:2.045(4)	O2:1.998(4)
					O3:2.455(5)	O3:2.029(4)
					O4:2.983(4)	O1:2.036(4)
	Average (Å) 1.916(4)	2.004(4)	1.961(4)	1.995(4)

We acknowledge S. Dufourd (CRISMAT) for sample preparations and F. Veillon (CRISMAT) for the dielectric and polarization measurements. We thank N. Barrier (CRISMAT) for assistance with the crystal structure determination. This work was partially supported by the ANR grant BORA-BORA (ANR-16-CE08-0007-01).

SUPPLEMENTARY MATERIAL

See supplementary material for Rietveld refinement of the Cu2CrBO5 SXRPD data at 260 K and NPD data at RT ; Magnetic field evolution of the magnetization and first field derivative of the magnetization of Cu2CrBO5 at different temperatures ; LeBail profile of the Cu2CrBO5 NPD data at 1.5 K, and results of the SXRPD Rietveld refinement of Cu2CrBO5 at 260 K (including M-O bond lengths and selected atomic distances and angles).

DATA AVAILABILITY STATEMENT

The data that supports the findings of this study are available within the article and its supplementary material.

Supplementary Material

High temperature spin-driven multiferroicity in ludwigite chromocuprate Cu2CrBO5

Françoise Damay, Jonas Sottmann, François Fauth, Emmanuelle Suard, Antoine Maignan, Christine Martin


N Matsubara

S Petit

C Martin

F Fauth

E Suard

S Rols

F Damay

BiMnTeO 6 : a multi-axis Ising antiferromagnet

The crystal and magnetic properties of honeycomb BiMnTeO 6 have been studied between 1.5 K and 300 K using synchrotron X-rays and neutron scattering experiments. Commensurate magnetic ordering is observed below T N = 10 K, and corresponds to a non collinear spin arrangement, with spins tilted away from the anisotropy easy-axis set by the Jahn-Teller distortion of the MnO 6 octahedra. Inelastic scattering experiments show two main nearly localised magnetic excitations, which can be well described by an exchange Hamiltonian involving weak Mn-Mn magnetic interactions and a crystal field Hamiltonian characterizing the strong easy-axis anisotropy associated with the d z2 orbital ordering of Mn 3+ . The crystal field levels can be accurately calculated, taking into account a transverse molecular field imposed by the magnetic ordering of the neighbouring atoms below T N . This makes of BiMnTeO 6 an interesting example of a multi-axis Ising system in a self-imposed transverse magnetic field.

INTRODUCTION

In the transverse Ising magnet, a magnetic field applied transverse to the spin easy-axis induces a zero Kelvin phase transition from the magnetically ordered state to a disordered, yet correlated, state [1], [2], [3], [START_REF] Chakrabarti | Quantum Ising Phases and Transitions in Transverse Ising Models[END_REF]. The transverse field Ising magnet (TFIM) is now the archetypal example of a system displaying a quantum phase transition [START_REF] Sachdev | Quantum Phase Transitions[END_REF]. Its widespread use relies on the fact that the application of a transverse field is a simple -and controllable -way to introduce quantum dynamics in an otherwise classical (Ising) system [START_REF] Lin | [END_REF], [7]. It is a customary tool in many relevant topics like quantum magnetism [8], [9], or novel topological excitations [10], but also for understanding materials with unconventional properties, such as heavy fermion systems [11] and high-temperature superconductors [12]. It has been shown to successfully address anisotropic quantum phase transitions in spin chains, like for instance LiHoF 4 [13], [14], BaCo 2 V 2 O 8 [15] or the quantum critical point of CoNb 2 O 6 [9], [16], while being also at the heart of more subtle phenomena governing the stability of the quantum spin ice phase in some pyrochlore materials [17], [18], [19], [20], [21], or of the intertwined polar and multipolar order on the triangular lattice magnet TmMgGaO 4 [22].

In this article, we show that the low energy properties of the layered honeycomb tellurate BiMnTeO 6 fall into TFIM physics [23]. X-ray and neutron diffraction demonstrate that this material hosts a zig-zag-like pattern of orbital orientations, resulting in Mn 3+ site-dependent magnetic easy-axes. Below T N = 10 K, those anisotropies compete with exchange interactions, to stabilize eventually a non-collinear magnetic order, out of which emerges a specific spectrum of nearly localised magnetic excitations, as observed by inelastic neutron scattering. This spectrum can be correctly modelled by considering spins with large easy-axis magnetic anisotropy, yet coupled by a molecular field arising from the ordering of neighbouring Mn 3+ spins. This molecular field is characterized by a sizable component perpendicular (transverse) to the local easy-axis directions. Random Phase Approximation (RPA) and mean-field self-consistent calculations successfully explain both inelastic and elastic neutron scattering experiments, including the canting of the Mn 3+ spin away from the local easy-axis. AMTeO 6 layered tellurates could then prove a new way to design tunable TFIMs, playing with the nature of A and M, or with an applied external constraint like magnetic field or pressure. The obtained dark brown powder of BiMnTeO 6 was then characterized by laboratory X-ray powder diffraction (XRPD) on a D8 ADVANCE Vario1 (Bruker) diffractometer (Cu-K 1 radiation) ; the monoclinic cell parameters (SG : P2 1 /c, a = 5.173 (1) Å, b = 9.058 (1) Å, c = 9.914 (1) Å,  = 90.17 (1) °) used to index the diffraction pattern were found to be in excellent agreement with [23].

EXPERIMENTAL

Synthesis

In-situ high-temperature XRPD measurements were also carried out using an Anton Paar HTK1200N chamber, from room temperature (RT) to 760°C in O 2 flow, on a flat alumina sample holder.

Magnetic susceptibility measurements

Magnetic measurements versus temperature were performed with a Quantum Design 5 T SQUID magnetometer, in zero-field-cooled warming (zfcw) and field-cooled warming (fcw) modes, between 5 and 300 K, in 100 Oe.

Synchrotron X-ray and neutron diffraction

Synchrotron X-ray powder diffraction (SXRPD) experiments were performed on the BL04-MSPD beamline at the ALBA synchrotron, at RT and 6 K with the wavelength  = 0.442746 and 0.442942 Å, respectively, using a rotating glass (Ø 0.3mm) capillary tube and the position sensitive detector MYTHEN. Neutron powder diffraction (NPD) was performed on the D2B (ILL, Grenoble, λ = 1.595 Å) and G4.1 diffractometers (CEA-Saclay, France, λ = 2.426 Å), at RT and from 1.5 to 300 K, respectively. Symmetry analysis and Rietveld refinements were performed with tools from the FullProf Suite [24]. Symmetry analysis was carried out using the Bilbao Crystallographic Server [25], [26] and its AMPLIMODES routine [27], [28].

Inelastic neutron scattering

Time-of-flight (TOF) inelastic neutron scattering experiments were performed using the thermal spectrometer IN4 at the Institut Laue-Langevin (ILL, Grenoble), with two incident wavelength settings ( i = 2.2 Å and 3.4 Å), at 5 K and 20 K. At 3.4 Å, the instrumental resolution is ~ 0.5 meV. Inelastic neutron scattering experiments were also performed on a cold neutron triple-axis spectrometer at LLB-Orphée (4F2, k f = 1.55 Å -1 ) to study the temperature dependence of the inelastic scattering between 3 K and 50 K. Higher order contaminations were removed with a nitrogen-cooled Be filter placed in the scattered beam.

The calculations presented in the study have been carried out with the code developed in [29], [30].

RESULTS AND DISCUSSION

A. Crystal structure

The crystal structure of BiMnTeO 6 at RT has been refined in the monoclinic P2 1 /c space group (n°14), starting from the unit cell and atomic coordinates reported by Kim et al. [23].

This structural model provides excellent fitting to both the SXRPD and NPD data (Figure S.1a,b, S.2 and Tables S.I, S.II, S.III), and the corresponding structure is illustrated in Figure 1. BiMnTeO 6 can be described accordingly as a distorted honeycomb structure, closely related to trigonal BiFeTeO 6 or BiCrTeO 6 [23], [31]. This 2D crystal structure is made of edge-sharing MnO 6 and TeO 6 octahedra forming honeycomb layers in the ab plane, with Bi 3+ cations sandwiched between two such layers (Figure 1a). Importantly, the Mn and Te cations are fully ordered (within the refinement standard deviation), in contrast to Cr and Fe analogues [23] : the resulting topology of the Mn lattice is stacked distorted triangular planes (Figure 1b). As already stated in [23], the lower symmetry structure of BiMnTeO 6 (with respect to its trigonal P-31c parents) is caused by the Jahn-Teller (JT) distortion of Mn 3+ O 6 octahedra, which are strongly elongated, with 4 Mn-O distances ranging between ~1.87 Å and ~1.97 Å and two Mn-O distances larger than 2.1 Å (Table S.III). On the other hand, TeO 6 octahedra show very little distortion, confirming its "rigid compact unit" character also reported in Mn 2 TeO 6 [32]. The monoclinic P2 1 /c distortion decomposes into three distortion modes of different symmetries, corresponding to the irreducible representations (irreps)  1 + ,

 3 + and M 2 -
, with isotropy subgroups P-31c, C2/c and P2 1 /c, respectively. Their respective amplitudes show unambiguously that the larger amplitude is that of the M 2 -mode (Table S.IV), thus identified as the primary order parameter. It is associated with a sheer motion of the Mn rows, triggering a rotation of the TeO 6 octahedra and an elongation of the MnO 6 ones. This is reminiscent of the structural transition observed in Mn 2 TeO 6 [32], explained in terms of rotation of rigid TeO 6 units and distortion of more flexible JT active Mn 3+ octahedra. In BiMnTeO 6 , the resulting orbital ordering pattern is a zig-zag (or herringbone) one, consisting of alternating rows with two different orbital orientations, as illustrated on Figure 1b.

The P2 1 /c structural model also fits the synchrotron data at 6 K (Tables S corroborating long-range magnetic ordering. All the magnetic Bragg peaks can be indexed with a commensurate propagation vector k = (½ 0 0). To constrain the number of solutions for the magnetic models, a symmetry analysis was carried out, for the Wyckoff site 4e of Mn.

There are four irreducible representations of dimension 1, each containing 3 basis vectors:

 m = 3 1  3 2  3 3  3 4 .
The best agreement with the experimental data is obtained for the  1 representation, whose basis vectors are given in Table S.V. For the single independent magnetic atom of the asymmetric unit cell, the three magnetic components m x , m y , and m z can be refined independently. At 1.5 K, the Rietveld refinement (Figure 2c) yields m x = 1.7(1), m y = -1.6(1) and m z = 2.8(1), which gives an ordered magnetic moment value of of 3.7(2) µ B , close to the expected value of 4 µ B for Mn 3+ (S = 2). The resulting non-collinear magnetic structure is illustrated in Figure 2d. Its main features are antiferromagnetic chains of parallel spins running along a, the spin direction in two adjacent chains approximately following the zig-zag orbital pattern described earlier. This magnetic structure corresponds to the Shubnikov group P a 2 1 /c (#BNS 14.80), whose magnetic point group 2/m1' leads to zero polarization and magnetoelectric tensors.

C. Magnetic excitations

From Figure 2d, one can expect magnetic exchanges to be dependent on the relative orientations of orbitals on adjacent sites: indeed, one observes collinear antiferromagnetic ordering of the spins along a, which would correspond to antiferromagnetic Mn-O-Te-O-Mn super-super exchange interactions [START_REF] Goodenough | Magnetism and the Chemical Bond[END_REF]. Along the two other directions of the triangle, there is an alternation of up-up-down-down type configurations (although not strictly parallel), matching the alternation of short (5.093 Å) and long (5.348 Å) Mn-Mn paths. Rietveld refinement results also indicate that Mn 3+ moments are not exactly aligned along the longest axis of their octahedron, but are canted by about 20°.

To investigate these intriguing points, inelastic scattering experiments were performed.

Figure 3 shows the inelastic powder spectra of BiMnTeO 6 mapped over momentum and energy transfer at 5 K and 20 K. Remarkably, the dominant feature of the spectrum at 5 K is a strong almost flat band around 2 meV (Figure 3a and3c), whose intensity decreases with increasing Q as expected for magnetic scattering. This mode seems to slightly soften and broaden for 0.6 < Q < 0.9 Å -1 , indicating a very weak dispersion. There is no dispersive magnetic signal originating from the magnetic Bragg positions around Q = 0.6 and 0.9 Å -1 , however, as clearly seen on Figure 3c. Another weaker flat band is also observed around 3

meV. The evolution with temperature of constant Q = 0.8 Å -1 cuts is illustrated on Figure 3d and shows clearly the existence, as high as 50 K (that is, far above T N ) of a magnetic signal (peaking around 1.5 meV), which evolves continuously across the magnetic transition to split into two levels at 2 and 3 meV. This behaviour strongly suggests that the magnetic spectrum is not composed of classical dispersing spin waves, but, rather, of magnetic excitons, corresponding primarily to crystal field-like levels, which are modified by an internal molecular magnetic field as spins order below T N and thus acquire a weak dispersion. They are visible in neutron scattering data provided they are characterized by S = 1. This scenario can be modelled using the following Hamiltonian :

ℋ = ℋ 𝑒𝑥𝑐ℎ + ℋ 𝐶𝐸𝐹 (1)
Here

ℋ 𝑒𝑥𝑐ℎ = ∑ 𝐽 𝑖𝑗 J 𝑖 J 𝑗 𝑖,𝑗
describes the exchange interactions between spins J i and J j at sites i and j, with J ij the exchange coupling, and

ℋ 𝐶𝐸𝐹 = ∑ [∑ 𝐵 𝑙𝑚 𝑂 𝑖,𝑙𝑚 𝑙,𝑚 ] 𝑖
describes the crystal electric field (CEF)-like Hamiltonian and characterizes the anisotropy of the 3d electron cloud around Mn 3+ . From a general point of view, it is written in terms of the O nm Stevens operators [START_REF] Hutchings | [END_REF], [START_REF] Mackintosh | Magnetic properties of rare-earth metals[END_REF], [START_REF] Wybourne | Spectroscopic Properties of Rare Earth[END_REF], while B lm are the crystal field coefficients.

In the specific case of BiMnTeO 6 , all J ij exchanges are considered first-neighbour. They correspond to three interactions within the triangular plane, J a , J f , J af and to an inter-plane coupling J C , as drawn on Figure 1. Mn 3+ ions are located on the 4e site of the P2 1 /c structure, whose local symmetry is C 1 . To avoid excessive over parameterization, only a single B 20 term was taken into account in the CEF Hamiltonian. Such a choice can be argued to be rather crude, considering the low symmetry of the Mn 3+ site, it is solidly warranted, however, by other examples of Mn 3+ complexes with strong JT distortion, in which the axial parameter is at least ten times the rhombic one [START_REF] Ridier | [END_REF], [38]. This B 20 term thus corresponds to an easyaxis anisotropy (B 20 < 0), as anticipated for Mn 3+ in a JT distorted environment. Note that the main CEF z axis is set to locally correspond to the longest axis of the MnO 6 octahedron. It should be stressed again that this anisotropy axis changes from site to site according to the orbital pattern.

To determine the parameters of the model, the following procedure was applied. A first estimation of B 20 was assessed from the position of the broad energy level observed at 20 K (that is, when H exch = 0, above T N ), at ~ 1.5 meV. This corresponds to B 20 = -0.17 meV, in agreement with the expected axial magnetic anisotropy (see also Figure S.4). We then solved Eq (1) at the mean field level for each site within the magnetic unit cell: For each trial, the ground state ordered magnetic structure was determined by such self-consistent calculations. The excitonic spectrum was computed subsequently in the RPA approximation, along with the corresponding inelastic neutron cross section.

ℋ 𝑀𝐹 = ∑
Calculations show that if (|J f |+J af ) = 0, increasing (J a +J C ) leads to a linear increase of the energy position of the four Mn 3+ crystal field levels, with only one mode being neutron active (Figure S.4). For instance, as can be seen on Figure 4a, for (|J f |+J af ) = 0 and (J a +J C ) = 0.14 meV, four energy levels are calculated, at 2.09 meV (i), 2.24 meV (ii), 3.16 meV (iii) and 3.21 meV (iv), with only (i) having a non-zero neutron cross section. It is important to underline here that, in the (|J f |+J af ) = 0 case, the molecular field at a given site is purely longitudinal (i.e., along the local z axis) and proportional to (J a +J C ).

A much more complex picture emerges when (|J f |+J af ) ≠ 0. One example is given in Figure 4a, which illustrates the evolution of the calculated energy levels and intensities with increasing 4c).

(|J f |+J af ),
Using Figure 4a, a good modelling of the experimental data is obtained for (|J f |+J af ) = 0.08 meV (with B 20 = -0.17 meV and (J a +J C ) = 0.14 meV). The corresponding calculated powder averaged excitation spectrum, convoluted with the Mn 3+ magnetic form factor, is shown on Figure 4d, and gives excellent agreement with the experimental results of Figure 3a. A comparison between experimental and calculated Q = 0.8 Å -1 profiles is also given on Figure 3c, to illustrate the very good match between the observed and computed positions, and relative intensities, of the two modes. The slight dispersion of the 2 meV exciton level is also reproduced by the calculation, although it is somewhat blurred on Figure 4d, because of the powder average and the resolution function that is used to match experimental data. In the ordered spin configuration calculated as the ground state for this set of parameters, spins are tilted by 37° from their local anisotropy axis, which lies slightly above the range expected from diffraction results (Figure 4b).

DISCUSSION

The microscopic magnetic properties of electrons in transition metal and rare-earth magnetic compounds are determined mainly by exchange and crystal field effects and their relative magnitudes. In conventional transition metal compounds, magnetic exchanges being dominant, the elementary excitations are spin waves, which are dispersive by nature. Crystal field effects appear only as small correction terms, contributing to the magnetic anisotropy.

In compounds which contain JT active cations like Mn 3+ , magnetic anisotropy is imposed by the ordering of the occupied/empty electronic states (or orbital ordering) controlled by JT distortions. In extensively studied LnMnO 3 (Ln = La, to Gd) [39] or TlMnO 3 [40] perovskites, orbital ordering is not collinear, but the experimentally observed nearly collinear magnetic structures (T N ≥ 40 K) show that, in these compounds, Heisenberg exchange interactions are substantially larger than the energy of single ion magnetic anisotropy. The more immediate impact of orbital ordering is actually anisotropic next-nearest Heisenberg exchange interactions, which lead to in-plane ferromagnetic, but out-of-plane antiferromagnetic exchanges (so called A-structure) [41]. In contrast, in the layered triangular manganese oxide like CuMnO 2 [42] or NaMnO 2 [43], there is no competition between easy-axis anisotropy and magnetic exchanges, owing to the d z2 ferro orbital order.

Amongst tellurates, the prominent role of orbital ordering has been emphasized recently in Mn 2 TeO 6 , through the unusually complex set of structural and magnetic transitions observed [32], [44]. The relationship between spins and orbital ordering still remains to be understood however, as both crystal and magnetic structures could be incommensurate, like in CaMn 7 O 12 [45], pinpointing a delicate balance between Heisenberg exchanges and magnetic anisotropy in this compound.

On the other hand, in systems with weak magnetic exchanges and consequently low ordering temperatures (typically below 10 K), crystal fields effects are now expected to predominate :

excitations correspond to local transitions between energy levels, and characterize the ground state electronic structure. Typical examples of the latter case are 4f compounds, and abundant literature can be found on the subject [START_REF] Wybourne | Spectroscopic Properties of Rare Earth[END_REF]. With regards to transition metals, most studies are performed on metal complexes with very weak magnetic interactions, such as molecular magnets or clusters [38], to better apprehend magnetic anisotropies or intra-clusters exchanges, and high-spin to low-spin transitions. LaCoO 3 in its S = 0 state is a rare example of a transition metal oxide in which neutron spectroscopic techniques have investigated the CEF energy levels [46].

BiMnTeO 6 is therefore a rather unusual example of a transition metal oxide, in which non collinear magnetic ordering is set by the strong easy axis anisotropy imposed by the orbital ordering. Magnetic exchange arises as a perturbation, which is quantified through the temperature evolution of the crystal field excitations at T N . The choice of a single crystal field coefficient B 20 for the modelling, substantiated by the literature on Mn 3+ in JT environment, is justified a posteriori by the quality of the agreement between the number and energy positions of the experimental and calculated levels. In addition, the relative intensities of the 2 and 3 meV modes appearing below T N , as well as the fact that there is no magnetic excitation above 4 meV on the 20 K and 5 K spectra (up to 15 meV), are very strong constraints to the modelling, which sustain further this approximation, so that the introduction of non-diagonal terms in the BiMnTeO 6 CEF Hamiltonian appears unneeded at that stage.

An important outcome of this study is that it is the transverse components of the local magnetic field which are responsible for the splitting of the crystal field levels. BiMnTeO 6 can be therefore described as a full member of the TFIM family, whose originality resides in the zig-zag orbital pattern imposing non-collinear easy-axis anisotropies, with the transverse component of the magnetic field arising below T N only.

BiMnTeO 6 was recently reported as a new member of ordered AMTeO 6 layered honeycomb tellurates [23], [47]. Our results on BiMnTeO 6 underline the potential of this family as a tunable TFIM system, in which long-range exchange interactions and easy-axis anisotropy character could be adjusted, depending on the size and nature of the A and M cations. Other means, such as applied external magnetic field or pressure, could also be used to modify the ratio between the local transverse magnetic field and the easy-axis magnetic anisotropy. All these features could be used to further investigate the physics of TFIMs and the TFIM magnetic phase diagrams, to determine for instance the threshold value of the transverse field at which a quantum phase transition is observed.

BiMnTeO 6

 6 was prepared by conventional solid-state reaction, starting from a mixture of Bi 2 O 3 , Mn 2 O 3 and H 6 TeO 6 . Stoichiometric amounts of precursors were mixed in an agate mortar, before being calcined in alumina crucibles at 700°C for 24 h in O 2 flow. The powder was then thoroughly ground, pressed into bars (~ 2 x 2 x 12 mm) with a uniaxial press and sintered at 750°C for 12 h in O 2 flow.

  .I, S.II, S.III), showing the absence of a structural phase transition going across the magnetic ordering transition at T N . Close inspection of the data of Tables S.II shows remarkably little structural change (less than 0.13%) between 300 K and 6 K, indicating that the zig-zag orbital pattern remains unchanged in that temperature range. Above RT, the monoclinic distortion of BiMnTeO 6 decreases discernibly with increasing temperature (Figure S.1c). At 750°C in O 2 , just before BiMnTeO 6 starts to decompose, the deviation from the trigonal symmetry is slight but still visible at high Q. B. Magnetic structure From the Curie-Weiss fitting of the paramagnetic range (see Figure S.3 and [23]), weak but clearly predominantly antiferromagnetic interactions are expected to govern the magnetic ordering of BiMnTeO 6 . The evolution of the neutron diffraction patterns confirms this expectation, with the appearance of new Bragg peaks below T N = 10 K (Figure 2a and 2b),

CONCLUSIONA

  new experimental realisation of the transverse field Ising model has been evidenced inBiMnTeO 6 , by means of elastic and inelastic neutron scattering, combined with X-ray diffraction experiments. Magnetic ordering sets in at T N = 10 K, with no evidence of magnetoelastic effects. The magnetic ground state is made of non-collinear magnetic spins, which follow the zig-zag pattern of d z2 orbitals existing at RT, but deviating by ~20° from their local anisotropy axis. Above T N , the existence of a broad mode peaking at 1.5 meV is attributed to a crystal field exciton. When magnetic orders sets in, this broad mode splits into two separate barely dispersive modes at ~ 2 and ~3 meV, of very distinct intensities. By means of self-consistent calculations based on a conventional Hamiltonian, using three independent parameters only, B 20 = -0.17 meV, (J a +J C ) = 0.14 meV and (|J f |+J af ) = 0.08 meV, all these experimental observations can be well reproduced. Results show in addition that the temperature dependence of the crystal field excitations originates from the increasing transverse component of the molecular field, which arises as Mn 3+ spins order.

FIGURE CAPTIONS Figure 1 (

 CAPTIONS1 FIGURE CAPTIONS

Figure 2 (

 2 Figure 2 (color online) : (a) Temperature evolution of the neutron diffraction data (G4.1)

Figure 3 (

 3 Figure 3 (color online) : TOF inelastic scattering data (IN4) of BiMnTeO 6 at 5 K (a) and 20 K

Figure 4 (

 4 Figure 4 (color online) : (a) Calculated variation with (J af +|J f |) of the neutron cross section of Mn 3+ crystal fields transitions ((J a +J C ) = 0.14 meV, B 20 =-0.17 meV). (b) Corresponding

Figure 1 Figure 2

 12 Figure 1

Figure 4

 4 Figure 4

  From analytical results, it is quickly inferred that the relevant parameters are (J a +J C ) and (|J f |+J af ). A series of calculations was then performed, varying (J a +J C ) and (|J f |+J af ).

	𝐵 𝑙𝑚 𝑂 𝑖,𝑙𝑚	+ J 𝑖 . H 𝑙𝑜𝑐 = ∑ 𝐵 𝑙𝑚 𝑂 𝑖,𝑙𝑚	+ J 𝑖 . ∑ 𝐽 𝑖𝑗 <J 𝑗 >
	𝑙,𝑚	𝑙,𝑚

𝑗

H loc defines the molecular field experienced by a spin at site i.

  for a given (J a +J C ) = 0.14 meV. With increasing (|J f |+J af ), a progressive increase of the calculated intensities of the (ii) and (iii) levels is observed, so that they become visible for

(|J f |+J af ) > 0.06 meV. Further increasing (|J f |+J af ) leads eventually to a clear splitting of the calculated (i) and (ii) levels above (|J f |+J af ) ~ 0.09 meV, as the energy position of the (i) level decreases, while that of the (ii) level increases. When (|J f |+J af ) is ≠ 0, the molecular field is not purely longitudinal any more. Increasing (|J f |+J af ) is equivalent to increasing the transverse components of the molecular field : this effect is shown on Figure

4b

, which gives the evolution with (|J f |+J af ) of the angle between H loc and the local z direction (as schematized on Figure

ACKNOWLEDGEMENTS

This work was partially supported by the IUCr2017-STOE grant, awarded to Nami Matsubara, winner of the Dragons' Den contest at the IUCr2017 Congress, Hyderabad, India, 2017.