# Quantitative Imaging Principles Improves Medical Image Learning

Lambert T. Leong
Department of Molecular Bioscience and Bioengineering
University of Hawaii
<EMAIL_ADDRESS>
Michael C. Wong
Department of Epidemiology
University of Hawaii Cancer Center
<EMAIL_ADDRESS>
Yannik Glaser
Department of Computer Science
University of Hawaii
<EMAIL_ADDRESS>
Thomas Wolfgruber
Department of Epidemiology
University of Hawaii
<EMAIL_ADDRESS>
Steven B. Heymsfield
Pennington Biomedical Research Center
Louisiana State University
<EMAIL_ADDRESS>
Peter Sadowski
Department of Computer Science
University of Hawaii
<EMAIL_ADDRESS>
John A. Shepherd
Department of Epidemiology
University of Hawaii Cancer Center
<EMAIL_ADDRESS>
https://github.com/LambertLeong/DXA-VAE

###### Abstract

Fundamental differences between natural and medical images have recently
favored the use of self-supervised learning (SSL) over ImageNet transfer
learning for medical image applications. Differences between image types are
primarily due to the imaging modality and medical images utilize a wide range
of physics based techniques while natural images are captured using only
visible light. While many have demonstrated that SSL on medical images has
resulted in better downstream task performance, our work suggests that more
performance can be gained. The scientific principles which are used to acquire
medical images are not often considered when constructing learning problems.
For this reason, we propose incorporating quantitative imaging principles
during generative SSL to improve image quality and quantitative biological
accuracy. We show that this training schema results in better starting states
for downstream supervised training on limited data. Our model also generates
images that validate on clinical quantitative analysis software.

##  1 Introduction

Deep learning is a promising methodology for medical imaging applications.
However, acquiring large labeled medical training datasets needed to train
robust models is difficult and major barriers result from necessary laws that
protect patient privacy. Transfer learning has been the most common method
used to address data scarcity in which models would first leverage ImageNet
[1], for instance, and subsequently be fine-tuned with the limited medical
domain data. While some have reported benefits from transfer learning [2] its
efficiency and efficacy for medical image learning problems is questionable
[3]. Medical images are inherently different from natural images contained in
transfer learning datasets like ImageNet. For example, subtle textures, pixel
intensity, scale, and local structures are significant in medical images [4,
5] while large-scale shapes are of relevance in natural images where a global
subject is often present. Stark differences between natural and medical images
highlight the need for improved domain specific methods.

Curation efforts of larger public medical imaging datasets [6, 7, 8, 9, 10]
have made unsupervised and self-supervised learning (SSL) a more promising
alternative method than natural image transfer learning alone. SSL enables the
leveraging of unlabeled image data from the medical domain and often results
in better starting states for downstream task-specific tuning with limited
data [11]. Despite performance improvement as a result of SSL, most medical
imaging SSL implementations mirror implementations used for natural images
thus not considering the differences in medical images. In other words,
medical image learning problems are often framed similar to learning problems
involving natural images and use similar tools, network architectures, or loss
functions. Consequently, medical image model learning may be hindered by the
way in which some medical images, as a data type, are misperceived and the
method in which learning tasks are constructed.

Images from modalities which capture internal body components such as X-ray,
computed tomography (CT), or magnetic resonance imaging (MRI) are often viewed
and used by model developers in a qualitative way similar to the way natural
images are used. However, many medical image types should be used as
quantitative maps instead of just qualitative anatomical images [12]. Internal
imaging modalities rely on specific biophysical interactions to generate
images in which each pixel corresponds to a specific biological, physical, and
physiological property [13]. This is especially relevant to the case of image
reconstruction learning task which includes generative self-supervised
learning and cycle generative adversarial networks (GANs) [14]. Often, image
quality metrics such as peak signal to noise ratios (PSNR) or structural
similarity indexes (SSIM) [15] are reported for reconstructions of medical
images yet the accuracy as it relates quantitative imaging measurements or
biology is underexplored. In fact, the lack of quantitative accuracy has
limited the adoption of deep learning generated images into clinical practice
since it fails to validate quantitatively on commercial clinical systems and
software.

In this work, we explore an SSL training method to incorporate quantitative
medical imaging principles with the main objective of improving biological
accuracy for image reconstruction and subsequent learning tasks. This work
focuses on the dual energy X-ray absorptiometry (DXA) image type to illustrate
the significance of our contributions which are as follows:

  1. 1.

Using quantitative imaging knowledge, we constructed a loss function to
constrain the SSL of a variational auto-encoder (VAE) which resulted in
improved image reconstruction quality as well as improved quantitative
biological accuracy.

  2. 2.

The SSL of the VAE with the custom loss function results in a trained encoder
subnetwork which outperformed an ImageNet transfer learning model on a medical
imaging regression task.

  3. 3.

The trained VAE generator subnetwork resulting from SSL demonstrated superior
performance on a cross modality image translation task compared to a model not
utilizing quantitative medical imaging knowledge during pretraining.

##  2 Related Work

We provide an overview of prior work in medical imaging pretraining methods as
well as works on quantitative DXA image deep learning.

#### Pretraining

strategies are employed in medical image deep learning to overcome a lack of
available training images, labels, or both. Reviews have highlighted the major
differences between medical and non-medical imaging datasets and conclude that
the benefits of natural image (i.e. ImageNet) transfer learning are not clear
[11, 16, 17]. Recent focus has turned to self-supervised methods which include
methods such as contrastive learning [18, 19], auto-encoding [20, 21], and
adversarial [22, 23, 24] learning as alternatives strategies. The
aforementioned methods allow models to leverage more domain specific data,
i.e. medical images, and these SSL methods have led to improved image
analysis, segmentation, and classification [4, 25, 26, 27, 28]. Generative
image tasks such as denoising, image reconstruction, and artifact removal [29,
30] have also benefited from SSL methods such as multimodal imaging models and
unpaired data learning by means of cycleGANs [31, 32, 33, 34, 35]. This work
builds upon the notion that medical images are unique to images used in common
transfer learning datasets and that SSL provides better results in performance
for downstream medical imaging tasks [19, 36]. This work differs from previous
works because our generative SSL method utilizes the knowledge of medical
imaging physics [13] and model generated images are evaluated on a
quantitative biological imaging basis and not just for image quality [15, 37,
24].

#### DXA

is a quantitative imaging modality and has long been considered the criterion
method for measuring body composition [38] in addition to its primary and
popular use of measuring bone density and quality for osteoporosis [39]. Deep
learning DXA modeling primarily focuses on predicting bone density metrics
from images [40, 41, 42] and work on body composition evaluation or DXA image
reconstruction or generation is limited [43]. Clinical software is used to
derive body composition from DXA images and while work has been done to
reconstruct high quality DXA images with deep learning [44], none have
performed quantitative body composition analysis on such images. Generating
quantitatively accurate images is a more difficult problem and we turn to
informed deep learning methods to address this problem. Domain informed
learning such as physics, biology, or etc. [45, 46], has shown to
significantly improve model learning and performance. These works demonstrate
the benefit of further constraining the learning of models to the concepts,
principles, and law of the domain for which the task it is being trained to
accomplish [47, 48].

##  3 Approach

We first detail the DXA-only dataset used for SSL and the paired dataset used
to compare subnetworks trained with and without a custom DXA loss function to
networks utilizing ImageNet transfer learning. We then describe the SSL model
architecture, followed by our custom DXA loss function. Last, we describe how
quantitative biological body composition is obtained from our actual and
predicted DXA scans.

###  3.1 Datasets

Data used for modeling and experiments were curated from multiple National
Institutes of Health (NIH) studies. All participants gave informed consent,
and the study protocols were approved by the Institutional Review Board (IRB)
for each respective study. All DXA images used for modeling, testing, and
subsequent experiments were acquired on Hologic DXA systems (Hologic Inc., MA,
USA) and used in the raw image format. The Fit3D Proscanner (Fit3D Inc., CA,
USA) was used to acquire 3D body scans which were then standardized to the
same pose and number of vertices with the Meshcapade API (Meshcapade GmbH,
Tübingen, Germany).

The SSL VAE was trained and evaluated using 16,002 whole-body DXA scans
acquired as a part of the Health ABC Study [49, 50] and Bone Mineral Density
in Childhood Study [51, 52]. Data was split into a train, validation, and
holdout test set using an 80, 10, 10% split which was stratified by patient
age. Splits were also performed by patient key to avoid data leakage.
Therefore duplicate scans of the same patient, if any, remained together in
the same split.

Experiments conducted to validate our SSL VAE network utilized two additional
datasets that had no overlap with those used to train the VAE. Paired images
of DXA and 3D body surface scans were acquired as a part of the Shape Up!
Adults (R01DK109008) and Shape Up! Kids (R01DK111698) studies. These scans
were used to evaluate the trained VAE encoder and generator in two separate
experiments. The 3D body surface scan yielded many 3D anthropometric
measurements such as waist circumference, body volume, and torso length, to
name a few. The combined dataset consisted of 1107 unique DXA/3D scan pairs
which were split by patient key into a train, validation, and holdout test set
using an 80, 10, 10% split.

###  3.2 Self-supervised Variational Auto-Encoder (VAE)

The decision to use a VAE was due to the modularity of the architecture and
VAEs are commonly used for SSL generative reconstruction [20]. For instance,
the VAE consists of two main subnetwork components which include the encoder
and the decoder or generator. We used the trained subcomponents to investigate
possible improved starting points for training with our small paired dataset.
The Densenet121 [53] architecture was used as the encoder with a single
convolutional layer prepended to handle the six channel DXA input. The
generator consisted of consecutive units of bilinear upsampling and 2D
convolutional layers.

###  3.3 DXA Loss Function

Raw DXA images from this particular manufacturer are acquired by cycling
between two X-rays energies, low and high, and three filters, air, tissue, and
bone, which results in a six channel image with a pixel depth of 14-bits. The
search space of possible pixel values is large and therefore, we exploit known
relationships between the raw DXA channels to constrain the pixel level
predictions of the VAE for improved learned reconstructions.

Different attenuation values are obtained when X-ray photons at both low and
high energies pass through the same material; a body in this case [54, 55,
56]. The observed attenuation at low energy can be expressed as a ratio (R) to
the observed attenuation at high energy [57]. The voltages used to generate
the X-rays on a DXA system are predetermined and fixed and thus, extensive
studies have been performed to map chemical elements to corresponding R values
[58, 59, 60]. R values for molecular components such as fatty acids, protein,
and triglycerides found in soft tissue as well as calcium hydroxyapatite
composing bone are also well characterized [61]. As a result, there are
theoretically-derived R values that are specific to biological composition and
this knowledge can be used to constrain possible pixel values.

Every pixel on all DXA image channels represent X-ray attenuations at that
given point in 2D space [62]. Therefore, the R value can be calculated at each
pixel location by dividing the low energy image channels by the high energy
image channels to obtain R images or quantitative maps of varying fractions of
biological molecules. Our DXA loss function evaluates both the reconstruction
of the raw six channel DXA images (low and high energy images plus 4
additional calibration images) and the R images generated from each
reconstruction. Using a perceptual loss [61], feature maps are obtained for
actual and predicted raw DXA and R images using a pretrained VGG-16 with
frozen weights. The mean absolute error (MAE) is computed for both the raw DXA
image and the R image feature maps and the DXA loss is the sum of that MAE for
both image types.

###  3.4 Quantitative DXA Analysis for Body Composition Evaluation

In clinical practice, commercial software is used to analyze DXA images to
obtain measures of body composition for the entire body as well as for
standardized and adhoc-defined subregions. Compositional analysis includes the
quantification of fat, lean, and bone mass as well as bone mineral density
(BMD) and bone mineral content (BMC). In this work, Hologic’s Apex 5.4
software was used to assess biological accuracy, with respect to body
composition, of deep learning generated DXA images. This software is meant to
be used in a clinical setting and not designed for high throughput analysis.
Additionally, the automated analysis features still requires trained personnel
to assess that regions of interest segmented the anatomical landmarks
properly. Failure to properly identify regions of interest will result in
inaccurate body composition. This manual component limited the number of
images that could be analyzed quantitatively. Qualitative metrics were
reported on the entire testset for all experiments however, results tables for
quantitative body composition comparisons only report on 117 images.

##  4 Experiments

In this section we first compare image reconstructions from VAE models trained
with and without the DXA loss function. We then conduct two experiments to
show that utilizing trained VAE subnetworks results in better model
performance when fine-tuned for specific tasks on smaller datasets, See Figure
1. Hyperparameters for all following models were tuned using the SHERPA [63]
python module and all modeling was performed on NVIDIA DGX-1 tesla v100 GPUs.

Figure 1: Training setup for SSL of the VAE model on unpaired DXA scans
(left), pretraining encoder experimental setup to predict 3D anthropometry
from DXA scans (middle), and generator pretraining experimental setup to
predict DXA scans from 3D scans (right).

###  4.1 SSL of VAE with DXA Loss

The primary motivation for incorporating the DXA loss function was to improve
biological accuracy of pixel values by enforcing specific relationships
between image channels. Using the same dataset and data split, two identical
VAEs were trained starting from randomly initialized weights. The noDXA-VAE
was trained using only a perceptual loss while the DXA-VAE utilized the DXA
loss detailed in Section3.3. We compared the quality of the image
reconstructions by computing the PSNR, SSIM, and normalized mean absolute
error (NMAE) for both the actual and reconstructed images. Image quality
metrics were computed for images resulting from both the noDXA-VAE and the
DXA-VAE and the average values obtained by the test set are shown in Table 1.

Table 1: Image quality comparisons of DXA image predicted by self-supervised
models when trained without (noDXA-VAE) and with (DXA-VAE) the DXA loss
functions.

Model | PSNR ↑ | SSIM ↑ | NMAE ↓
---|---|---|---
noDXA-VAE | 49.52 | 0.997 | 0.038
DXA-VAE | 51.15 | 0.998 | 0.032

The DXA-VAE model resulted in better PSNR, SSIM, and NMAE of 51.15, 0.998, and
0.032, respectively when compared to the noDXA-VAE metrics of 49.52, 0.997,
and 0.038, respectively, shown in Table 1. These results suggest that the DXA
loss improves the generation of DXA image however, the quality metrics
indicate only a slight improvement.

Figure 2: Predicted DXA scan imaging channels and corresponding error maps
(represented as percentages) from VAE models strained without and with the DXA
loss function

Figure 2 shows representative results test images generated with and without
training with the DXA loss. Each column represents one of the six raw DXA
channels which. The first row contains the real DXA scan, the second and
fourth rows contain a predicted DXA from each SSL VAE training scenario (no
DXA loss and DXA loss), and the third and fifth rows contain the error maps
computed from the percent difference between the actual and predicted images.
Error maps indicate more difference in the prediction from the noDXA-VAE when
compared to the network which utilized the DXA loss.

To fully understand the benefit of training with the DXA loss, we evaluate the
biological composition of the predicted image using clinical software. We
report quantifications of lean and fat tissue as well as bone for the entire
body as well as subregions which include just the arms and just the legs
alone. Compositional measurements are reported in Table 2.

Table 2: Quantitative body composition analysis comparing images predicted
from the noDXA-VAE and DXA-VAE.

Measurements |  | R2
---
noDXA-VAE
| R2
---
DXA-VAE
| RMSE
---
noDXA-VAE
| RMSE
---
DXA-VAE
Arm BMC | 0.61 | 0.88 | 0.04 | 0.02
Arm BMD | 0.49 | 0.84 | 0.09 | 0.07
Arm Fat | 0.22 | 0.63 | 0.66 | 0.64
Arm Lean | 0.68 | 0.89 | 0.73 | 0.57
Leg BMC | 0.67 | 0.87 | 0.06 | 0.04
Leg BMD | 0.47 | 0.76 | 0.08 | 0.03
Leg Fat | 0.31 | 0.70 | 1.43 | 1.51
Leg Lean | 0.68 | 0.88 | 1.68 | 1.39
Total BMC | 0.63 | 0.89 | 0.34 | 0.21
Total BMD | 0.34 | 0.75 | 0.10 | 0.05
Total Fat | 0.42 | 0.80 | 6.84 | 7.88
Total Lean | 0.73 | 0.91 | 8.95 | 8.46
Total Mass | 0.79 | 0.99 | 9.82 | 1.43

Despite only a modest increase with respect to image quality, training with
the DXA loss results in images with more accurate biological composition. DXA-
VAE images resulted in better quantitative accuracy as captured by higher R2
values for all composition measurements. Using the DXA loss resulted in an
overall better model and reconstructed images were of higher quality and more
accurate with respect to body composition. The DXA-VAE which was trained in a
SSL fashion was then used to investigate the effectiveness of pretraining
using domain specific data, i.e. medical images.

###  4.2 SSL VAE Encoder Performance Evaluation

While DXA scans are 2D image types, the pixel values are related to X-ray
attenuation through varying thickness of soft tissue and bone. As such, the
objective of this modeling was to predict 3D anthropometry from DXA scans.
This dataset consisted of DXA scans and corresponding anthropometrics
measurements which include circumferences, volumes, and surface areas of
various parts of the body. We trained three models, identical in architecture,
that differed only in their pretraining methodology. One model utilized a
common transfer learning strategy by loading the Densenet121 portion of the
network with ImageNet weights while the other two models were created from the
encoder portion of our trained DXA-VAE, one trained without the DXA loss
function and the other trained with. Hyperparameters, architectures, and data
splits were kept consistent across all models to ensure fair comparisons.
Prediction results from the holdout test set are shown in Table 3.

Table 3: Comparisons of ImageNet transfer learning, SSL of VAE without the DXA
loss function, and SSL of VAE with the DXA loss function for predicting 3D
anthropometry from DXA scans.

Measurment |  | Anatomical
---
Location
| R2
---
ImageNet
| R2
---
noDXA-VAE
Encoder
| R2
---
DXA-VAE
Encoder
| RMSE
---
ImageNet
| RMSE
---
noDXA-VAE
Encoder
| RMSE
---
DXA-VAE
Encoder
Circumference (cm) | Bicep Left | 0.84 | 0.85 | 0.88 | 2.00 | 1.88 | 1.63
Circumference (cm) | Bicep Right | 0.86 | 0.88 | 0.94 | 1.80 | 1.63 | 1.22
Circumference (cm) | Calf Left | 0.72 | 0.77 | 0.85 | 1.75 | 1.51 | 1.26
Circumference (cm) | Calf Right | 0.76 | 0.80 | 0.88 | 1.65 | 1.44 | 1.17
Circumference (cm) | Hip | 0.88 | 0.89 | 0.93 | 3.80 | 3.69 | 3.17
Circumference (cm) | Thigh Left | 0.78 | 0.80 | 0.82 | 2.18 | 2.12 | 2.01
Circumference (cm) | Thigh Right | 0.79 | 0.81 | 0.85 | 2.26 | 2.12 | 1.99
Circumference (cm) | Waist | 0.92 | 0.92 | 0.94 | 3.53 | 3.51 | 3.40
Surface Area (cm2) | Whole Body | 0.96 | 0.96 | 0.96 | 0.40 | 0.39 | 0.39
Volume (L) | Arm Left | 0.82 | 0.83 | 0.86 | 0.53 | 0.50 | 0.39
Volume (L) | Arm Right | 0.86 | 0.86 | 0.91 | 0.40 | 0.39 | 0.29
Volume (L) | Leg Left | 0.67 | 0.71 | 0.76 | 0.78 | 0.74 | 0.70
Volume (L) | Leg Right | 0.70 | 0.75 | 0.78 | 0.78 | 0.71 | 0.67
Volume (L) | Torso Volume | 0.94 | 0.94 | 0.96 | 0.35 | 0.34 | 0.27
Volume (L) | Total Volume | 0.97 | 0.97 | 0.97 | 0.31 | 0.31 | 0.30

The results in Table 3 indicate that transfer learning with ImageNet may be a
reasonable approach for some learning problems involving medical imaging.
However, the results also indicate that SSL on domain specific data will yield
superior performance when compared to transfer learning from natural images.
SSL VAE encoder models trained with the DXA loss resulted in the best R2 for
all anthropometry measurements with the exception of total volume and total
surface area. All models achieved similar performance R2 of 0.97 and 0.96,
respectively. The best anthropometry predictions (bolded in Table 3) resulted
from the model which included the VAE encoder pretrained with the DXA loss
function.

###  4.3 SSL VAE Generator Performance Evaluation

The motivation for this experiment was to explore the notion that one’s
exterior 3D body shape is directly related to the underlying bone and soft
tissue distribution or composition. Here we again used a paired dataset
consisting of 3D body scans and DXA scans from the same individual taken at
the same time point. The objective was to derive a mapping to generate DXA
scans for a given 3D scan. Just as in the previous section, we are comparing
performance improvements brought about by SSL which incorporated quantitative
imaging physics. In this experiment, however, we used the trained VAE decoder
or generator instead of the encoder subnetwork. The architecture of the 3D
scan or mesh network was modeled after the Pointnet++ [64] model and it mapped
to the latent space of the DXA generator. For fair comparison, a second model
was trained using the same hyperparameters, data split, and architecture with
the only difference being that the generator resulted from VAE SSL without the
DXA loss function. We evaluate both 3D to DXA or Pseudo-DXA models on the
basis of image quality (Table 4) and biological composition (Table 5).

Table 4: Image quality comparisons of DXA image predicted from 3D body surface
by networks generators trained without (noDXA-VAE) and with (DXA-VAE) the DXA
loss functions.

Model | PSNR ↑ | SSIM ↑ | NMAE ↓
---|---|---|---
noDXA-VAE Generator | 35.14 | 0.897 | 0.201
DXA-VAE Generator | 38.49 | 0.938 | 0.130

(a) Example of a participant 3D body surface scan.

(b) Predicted DXA scan imaging channels and corresponding error maps
(represented as percentages) resulting from the 3D body scan (Figure 3(a)) and
differing modeling methods.

Figure 3: Representative participant 3D scan and predicted DXA image with
error map produced by Pseudo-DXA models in which generators were pretrained
with and without the DXA loss.

The Pseudo-DXA which incorporated the trained VAE generator yielded better
image quality metrics when compared to the Pseudo-DXA models with randomly
initialized weights. Figure 3 contains the input 3D body scan and Figure 3(a)
contains the actual corresponding DXA image and predicted images from both
models accompanied by their error maps. Figure 3(b) shows better DXA image
predictions are produced from the model using the SSL VAE generator. The
randomly initialized weights model fails to produce clear DXA images and large
prediction errors can be seen in the low energy tissue channel. Positioning
differences account for some of the significant errors (red and orange
regions) visualized on the error maps in Figure 3. Some image quality metrics
are influenced by positioning difference however, body composition assessment
is invariant to differences in position.

Table 5: Body composition analysis of DXA image predicted from 3D body surface
by networks generators trained without (noDXA-VAE) and with (DXA-VAE) the DXA
loss functions

Measurments |  | R2
---
noDXA-VAE
Generator
| R2
---
DXA-VAE
Generator
| RMSE
---
noDXA-VAE
Generator
| RMSE
---
DXA-VAE
Generator
Arm BMC | 0.31 | 0.71 | 0.05 | 0.03
Arm BMD | 0.17 | 0.67 | 0.14 | 0.08
Arm Fat | 0.28 | 0.52 | 0.74 | 0.66
Arm Lean | 0.40 | 0.74 | 1.18 | 0.65
Leg BMC | 0.29 | 0.73 | 0.12 | 0.07
Leg BMD | 0.06 | 0.52 | 0.22 | 0.15
Leg Fat | 0.38 | 0.54 | 1.65 | 1.55
Leg Lean | 0.31 | 0.79 | 2.68 | 1.51
Total BMC | 0.20 | 0.66 | 0.60 | 0.37
Total BMD | 0.02 | 0.41 | 0.23 | 0.14
Total Fat | 0.34 | 0.68 | 10.05 | 7.44
Total Lean | 0.39 | 0.80 | 1.18 | 8.58
Total Mass | 0.53 | 0.90 | 12.99 | 5.28

We deduce from Table 5 that the DXA-VAE resulted in a better generator which
was used to construct a Pseudo-DXA model that can produce more biologically
accurate images. Pseudo-DXA containing the trained generator out performed the
model containing the noDXA-VAE generator on all body composition values. It
should be noted that image quality and body composition were lower when
comparing outputs from Pseudo-DXA models to outputs from the VAE models. This
is likely caused by a relatively weak mapping between the 3D and 2D DXA space
and overfitting due to the smaller paired dataset.

##  5 Conclusion

To our knowledge, this is the first instance in which a deep learning model
can generate medical images on which subsequent quantitative imaging analysis
can be performed accurately. In this work we frame medical images as
quantitative maps rather than just anatomical representations. In doing so, we
were able to introduce a simple DXA loss function which constrained SSL of a
VAE to learn the specific pixel relationships between image channels. These
specific relationships correspond to biological quantities of soft tissue and
bone which allow for the derivation of body composition. Subnetworks of the
trained DXA-VAE demonstrated superior performance when trained for downstream
specific tasks which includes predicting 3D anthropology from DXA scans and
producing real analyzable DXA scans from 3D body scans. The broader impact of
this work is to demonstrate that medical image deep learning should consider
the principles used to generate or produce images. This will lead towards more
accurate synthesized or reconstructed images that can then be better
integrated into clinical practice.

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# Room Temperature Ferrimagnetism, Magnetodielectric and Exchange Bias Effect
in CoFeRhO4

P. Mohanty  Indian Institute of Science Education and Research Bhopal, Bhopal,
462066, India  N. Sharma  School of Physics and Materials Science, Thapar
Institute of Engineering and Technology, Patiala 147004, India  D. Singh
Indian Institute of Science Education and Research Bhopal, Bhopal, 462066,
India  Y. Breard  Laboratoire CRISMAT, UMR 6508 CNRS ENSICAEN, 6 bd du
Marechal Juin, 14050 Caen Cedex 4, France  D. Pelloquin  Laboratoire CRISMAT,
UMR 6508 CNRS ENSICAEN, 6 bd du Marechal Juin, 14050 Caen Cedex 4, France  S.
Marik<EMAIL_ADDRESS>School of Physics and Materials Science, Thapar
Institute of Engineering and Technology, Patiala 147004, India  R. P. Singh
<EMAIL_ADDRESS>Indian Institute of Science Education and Research
Bhopal, Bhopal, 462066, India

###### Abstract

Geometrically frustrated structures combined with competing exchange
interactions that have different magnitudes are known ingredients for
achieving exotic properties. Herein, we studied detailed structural, magnetic,
thermal (specific heat), magneto-dielectric, and magnetic exchange bias
properties of a mixed 3d - 4d spinel oxide with composition CoFeRhO4. Detailed
magnetization, heat capacity, and neutron powder diffraction studies (NPD)
highlight long-range ferrimagnetic ordering with an onset at 355 K. The
magnetic structure is established using a ferrimagnetic model (collinear-type)
that has a propagation vector k = 0, 0, 0. The magneto-dielectric effect
appears below the magnetic ordering temperature, and the exchange bias (EB)
effect is observed in field cooled (FC) conditions below 355 K. The magneto-
dielectric coupling in CoFeRhO4 originates due to the frustration in the
structure, collinear ferrimagnetic ordering, and uncompensated magnetic
moments. The unidirectional anisotropy resulting from the uncompensated
magnetic moments causes the room-temperature exchange bias effect. Remarkably,
the appearance of technologically important properties (ferromagnetism,
magnetodielectric effect, and EB) at room temperature in CoFeRhO4 indicates
its potential use in sensors or spintronics.

##  I Introduction

Spinel oxides are an intriguing class of materials, not only for their
potentiality for a wide range of applications but also because of a variety of
new and exciting physics (such as frustrated magnetism, multiferroic
properties, orbital glass system, spintronics applications, and spin-orbital
liquids) that continues to arise from the strong interactions among spin,
orbital, and structural degrees of freedom 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ;
10 ; 11 ; 12 ; 13 ; 14 . They have a unique structure with a general formula
of AB2O4, where A and B are metal ions (Fig.1). This structure comprises an
array of metal cations in octahedral and tetrahedral coordination, surrounded
by oxygens, creating two sets of magnetic sublattices 15 . B cations generally
form a pyrochlore-like lattice by residing in the octahedral sites and
originate frustrated magnetic interactions 16 ; 17 ; 18 ; 19 . On the other
hand, the metal A ions occupy the tetrahedral sites (eightfold) and construct
a diamond lattice 20 ; 21 ; 22 ; 23 ; 24 . This bipartite lattice can be
interpreted as two face-centered interpenetrated cubic (fcc) sublattices.
These sublattices are shifted diagonally by one-quarter. Variation of magnetic
and nonmagnetic cations on the tetrahedral A sites and the octahedral B sites
can originate complex magnetic interactions by affecting the magnitudes of
superexchange interactions (JAA, JBB, and JAB). Therefore, exotic properties
(and states) such as a spiral spin liquid phase 25 , unique, glassy magnetic
behavior 21 ; 26 , and spin-orbital liquids 27 ; 28  emerge in spinel
materials. The pyrochlore lattice is a fertile playground for theoretical and
experimental research to explore new physics. At the same time, the bipartite
diamond lattice (in spinel systems) is a fruitful platform for realizing
exotic quantum behavior. In particular, the bipartite nature of the diamond
lattice can be useful in designing the recently proposed 3D topological
paramagnetism for the frustrated S = 1 diamond lattice 21 ; 22 .

In addition to geometrical frustration and competing exchange interactions
having different magnitudes, spin-orbit coupling (SOC) is a known ingredient
that favors more exotic spin order and dynamics. In this connection, 4d and 5d
elements containing oxides attracted remarkable research interest in recent
times. Being spatially more extended (4d/5d orbitals), the on-site Coulomb
repulsion energy (U) is smaller in 4d and 5d elements than their 3d analogues.
However, in the spinel oxide family, only one 5d-containing material has been
reported so far. The iridium-containing spinel oxide with composition
Cu[Ir1.5Cu0.5]O4 highlights a highly frustrated magnetic state 29 . Among
4d-rhodium-based diamond lattice spinel oxides, in diamond-lattice Heisenberg
antiferromagnet CoRh2O4, a combined experimental and theoretical work showed
that the S = 3/2 spins are unfrustrated and display static and dynamic
properties 20 . Tetragonally distorted CuRh2O4 shows an incommensurate
magnetic order for the S = 1/2 spins and the presence of sizable quantum
effects 20 . A spin-orbit entangled paramagnetic state is suggested in NiRh2O4
21 ; 30 ; 31 . Studies on magnetically diluted Cu1-xZnxRh2O4 highlight spin
transition triggered by an enhancement of preceding spin fluctuations 32 . At
the same time, it shows the suppression of orbital order on the octahedral
sites through the percolative manner. In general, geometrically frustrated
structures with 4d/5d elements (SOC) are ideal for realizing exotic phenomena.
Therefore, exploring new frustrated structures (geometrical frustration and
competing magnetic interactions) coupled with SOC is essential.

In this article, we present the detailed structural (using X-ray powder
diffraction, neutron powder diffraction, and electron microscopy), magnetic
(magnetization), thermal (specific heat), and magneto-dielectric studies on a
mixed 3d - 4d spinel oxide with composition CoFeRhO4. In this material, non-
magnetic Rh3+ cations occupy octahedral B sites; however, they can be a source
of SOC in the structure. An insulating and ferrimagnetic ground state is
observed near room temperature (TC = 355 K). Detailed magnetic and magneto-
dielectric measurements highlight the room-temperature exchange bias effect
and the magnetodielectric effect in this material.

##  II Experimental Details

Synthesis. We have used the standard solid-state reaction route to synthesize
the polycrystalline CoFeRhO4 materials. Stoichiometric amounts of Co3O4
(99.9%) Fe2O3 (99.999%) and Rh (99.9%) metal powders were used for the
synthesis. The stoichiometric amounts of the starting materials were mixed
with a mortar pestle. The materials were heated several times and the final
sintering was performed at 1473 K for 36 h.

X-ray Powder Diffraction and Neutron Diffraction. X-ray diffraction (XRD)
using the powder form of the material was collected at ambient temperature
(RT) using a PANalytical diffractometer (Cu-Kα, $\lambda$ = 1.54056 Å).
Neutron diffraction (NPD) data for the powdered CoFeRhO4 sample were collected
at various temperatures (12 K, 100 K and 250 K) in JEEP II, Kjeller Reactor,
Norway ($\lambda$ = 1.5538 Å). We have performed the Rietveld refinements of
the diffraction patterns (XRD and NPD) using the fullProf suite software.

Electron Microscopy. We have performed electron microscopy measurements (high-
angle annular dark-field scanning transmission electron microscopy (HAADF-
STEM) and electron diffraction) to explore the crystal structure of CoFeRhO4.
HAADF-STEM and electron diffraction were carried out using a JEOL ARM-200F
cold FEG probe image aberration corrected 200 kV microscope. The instrument is
also equipped with a solid-angle CENTURIO EDX detector.

Magnetism and Specific Heat. Direct current magnetization (temperature and
magnetic field dependent) measurements were done in a Quantum Design
superconducting quantum interference device (MPMS 3). Magnetic measurements
were conducted in field-cooled (FC) and zero-field-cooled (ZFC) modes.
Specific heat was measured (2 K - 400 K) in a physical property measurement
system (PPMS, Quantum Design) without a magnetic field.

Dielectric Measurements. The temperature and magnetic field-dependent
dielectric measurements were performed employing an LCR meter (Agilent 4284A)
and a sample insert for a PPMS (Quantum Design).

Figure 1: crystal structure for the CoFeRhO4. Green spheres represent the
Fe/Rh cations (Octahedral B sites in the AB2O4 structure), Blue spheres
represent the Co cations (Tetrahedral A sites), and red spheres represent
oxygen.

##  III Results and Discussion

Preliminary Rietveld refinement uses the room-temperature XRD pattern of
CoFeRhO4. It shows that the sample crystallizes in a cubic structure with a
space group $Fd\overline{3}m$, which is isostructural with CoFe2O4 33 ; 34 .
However, a small amount of Rh is detected (main peak at 2$\theta$ $\simeq$ 41
degrees, Fig. S1 in the supporting information (SI)) SI  in the RT XRD pattern
for CoFeRhO4. Further, to explore the detailed nuclear and magnetic structure,
we have collected the NPD patterns at 12 K, 100 K, and 250 K. Therefore, we
will use the results of the NPD Rietveld refinements to describe the crystal
structure of CoFeRhO4. Similar to the XRD pattern, a small amount of Rh is
also detected in the NPD patterns. Fig.2 illustrates the plot of the NPD
Rietveld refinement for the pattern collected at 12 K. The plots of the NPD
Rietveld refinements for the patterns collected at different temperatures are
provided in the supporting information (Figure S2 in SI) SI . The structural
parameters obtained from the 12 K - NPD refinement for CoFeRhO4 are summarized
in Table 1. The crystal structure for CoFeRhO4 is highlighted in Fig.1. Rh
atoms and most of the Fe cations occupy the octahedrally coordinated B site
(16d (0.5, 0.5, 0.5)) of the structure. Co-cations occupy the eightfold
tetrahedral A sites (8a (0.125, 0.125, 0.125)). Oxygen atoms occupy the 32e
(x, x, x) Wycoff positions. The refinement of the oxygen occupancy indicates
that there is no deviation from the full occupancy. Rh shows a small deviation
from full occupancy (Occupancy = 0.98 (2)). Fig.3 shows the STEM- HAADF image
along the [011] direction for CoFeRhO4. The observed STEM images agree with
the expected crystal structure of the spinel material. The lattice parameter
obtained from STEM images (8.4 Å) matched the values found from the NPD (Table
1) and RT-XRD refinements.

Figure 2: Rietveld refinement plot of the Neutron powder diffraction (NPD)
pattern collected at 12 K for CoFeRhO4. The lower ticks highlight the magnetic
peak positions (k = 0, 0, 0). Insets in the figure show the enhancement of the
[111] and [220] NPD peaks with a lowering of the temperature.

The structural (Rietveld) refinement of the 250 K NPD highlights the existence
of magnetic peaks at that temperature. These magnetic ordering-related peaks
show an increase in intensity with decreasing temperature (inset in Fig.2).
The position of all magnetic peaks for CoFeRhO4 coincides with the allowed
nuclear reflections. Therefore, we describe the magnetic structure using the
propagation vector k = (0,0,0). For the long-range magnetic order, the best
fit to the NPD magnetic reflections is achieved using a collinear
ferrimagnetic ordering model (Fig. 4). A similar magnetic ordering scheme was
proposed for the CoFe2O4 material 33 . The absence of a (200) magnetic peak
discards the non-collinear arrangement of the spins. In general, any long-
range spin canting would result in the appearance of the (200) magnetic Bragg
peak. We do not observe the (200) peak at 21.27 $\degree$ ((400) peak is
observed at $2\theta$ = 43.43 $\degree$).

Figure 3: HAADF image and crystal structure along the [011] zone axis for
CoFeRhO4, collected at room temperature. Fe, Rh, and Co cations are
highlighted in the figure. The corresponding electron diffraction pattern is
shown in the inset. Figure 4:  The observed magnetic structure (k = 0, 0, 0)
for CoFeRhO4. A collinear ferrimagnetic ordering scheme is observed in the NPD
refinements. Table 1: Structural parameters for Ba2ScRuO6 at RT extracted from
Rietveld refinement of powder XRD diffraction data

Space group $Fd\overline{3}m$, a = b = c = 8.4207 (1) Å
RP = 6.98, RWP = 9.01, $\chi$2 = 1.03,RBragg = 2.79
RMag = 5.29, M (Td) = 3.4 (1) $\mu$B, M (Oct.) = 4.1 (1) $\mu$B

Atom | Wyck. Pos. | x | y | z | Occu. |  Biso
---|---|---|---|---|---|---
Co(Td) | 8a | 0.125 | 0.125 | 0.125 | 0.88 (2) | 0.1 (1)
Fe(Td) | 8a | 0.125 | 0.125 | 0.125 | 0.12(2) | 0.1(1)
Co(Oct.) | 16d | 0.5 | 0.5 | 0.5 | 0.065(5) | 0.11(3)
Fe(Oct.) | 16d | 0.5 | 0.5 | 0.5 | 0.44(1) | 0.11(3)
Rh(Oct.) | 16d | 0.5 | 0.5 | 0.5 | 0.49(1) | 0.11(3)
O | 32e | 0.249(1) | 0.249(1) | 0.249(1) | 1 | 0.10(3)

Magnetic moments in different sublattices (M (Td) - tetrahedral A sites, M
(Oct) - octahedral B sites) obtained from refinements are listed in Table 1.
The resultant magnetic moment per formula unit obtained from the NPD
refinements is 0.6 (1) $\mu$B at T = 12 K. However, a discrepancy is observed
between the magnetic moments’ experimental and theoretical values in the
tetrahedral and octahedral sites. Theoretically calculated magnetic moments in
different sublattices are MA = $0.88\times 3.5+0.12\times 5$ = 3.66 $\mu$B and
MB = $0.87\times 5+0.13\times 3.5)$ = 4.81 $\mu$B, calculated with MFe3+ = 5
$\mu$B and MCo2+ = 3.5 $\mu$B. Reduced magnetic moments could result from
local disorder (and/or local canting) of spins. The complex cationic
distribution and the existence of nonmagnetic Rh in the structure can
originate competing magnetic interactions and can create nonuniform spin
canting. A similar reduced magnetic moment due to the local spin canting is
also observed for the CoFe2O4 and Ti-doped CoFe2O4 materials 33 ; 35 .
However, a collinear ferrimagnetic long-range magnetic ordering scheme is
suggested for both of these materials.

Figure 5:  (a) Temperature-dependent magnetization (M - T) and (b) magnetic
field variation of the magnetization (M - H) for CoFeRhO4. Ferrimagnetic
behavior is observed for CoFeRhO4.

The FC and ZFC magnetic moment (M-T) temperature variation measured at 0.1 T
for CoFeRhO4 is presented in 5. The M-T shows a sharp upturn at 355 K,
highlighting the onset of the ferrimagnetic transition at that temperature.
Upon further lowering the temperature bifurcation between FC and ZFC
magnetization (MFC and MZFC) is visible at 250 K. Below 250 K, MFC shows an
increasing trend with decreasing temperature. Several competing magnetic
interactions exist in CoFeRhO4. The complex crystal structure (diamond and
pyrochlore lattice) and the distribution of magnetic and nonmagnetic cations
in two different sublattices can create competing magnetic interactions and
frustration. For example, A-site magnetic cations are surrounded by 12 B-site
magnetic and non-magnetic first neighbours, which can be Fe3+ or Co2+ or Rh3+;
any B-cation is surrounded by 6 A and 6 B-sited cations. Therefore, the
bifurcation and the rise in MFC at a lower temperature could be due to the
frustration in the spinel structure. However, the ferrimagnetic transition in
CoFeRhO4 is much lower than in CoFe2O4. In CoFe2O4, the collinear
ferrimagnetic transition is observed at 800 K with a strong intersublattice
AFM superexchange interaction (JAB = - 12.39 kB) 33 . Introducing non-magnetic
Rh into the structure could dilute the magnetic interactions by decreasing the
strength of inter- and intra-sublattice superexchange interactions and,
therefore, can reduce the magnetic ordering temperature. Figure 5(b) shows the
magnetization as a function of the magnetic field (M - H) at various
temperatures measured in the ZFC mode. Above the magnetic transition
temperature (at 400 K), a linear paramagnetic type M-H behavior is observed.
However, at 350 K, the M-H loop illustrates non-linearity at low magnetic
fields, indicating the appearance of the magnetically ordered state. As we
further lower the temperature, saturated ferrimagnetic-type M - H loops are
observed. The saturation magnetization and coercive field (HC) enhance with
decreasing the temperature.

Figure 6 highlights the temperature variation of total specific heat for
CoFeRhO4. Magnetic ordering usually involves an entropy change, resulting in
specific heat anomalies. The CP vs T plot shows an anomaly at 355 K,
indicating a true phase transition. This also confirms the magnetic ordering
temperature for CoFeRhO4. The low-temperature part of the specific heat (2-20
K) can be well represented by eqn. 1:

$C_{P}=\gamma T+\beta T^{3},$ (1)

where $\gamma$ = Sommerfeld coefficient and $\beta$ = lattice contributions to
the specific heat. The obtained $\gamma$ = 0.015 Jmol-1K-2. However, the total
specific heat at higher temperatures contains both the phonon and magnetic
parts ($C_{P}=C_{ph}+C_{m}$). A combined Einstein-Debye model can generally
estimate the total phonon contribution in the specific heat. Therefore, to
estimate the magnetic contributions ($C_{m}$) in the specific heat, we fit the
data with the Einstein-Debye model 36 :

$C_{ph}=9Rx_{D}^{-3}\int_{0}^{x_{D}}\frac{x^{4}e^{x}}{({e^{x}-1})^{2}}dx+R\sum_{i=1}^{2}a_{i}\frac{x_{Ei}^{2}e_{Ei}^{x}}{({e_{Ei}^{x}-1})^{2}}$
(2)

Figure 6:  Temperature variation of the specific heat (C) for CoFeRhO4. An
anomaly due to the magnetic ordering is observed at 355 K. The green line
highlights the fitted curve using Eqn.2. Low-temperature C is also fitted
using eqn.1 and shown in the inset of the figure.

Where xD = $\theta$D/T and xEi = $\theta$Ei/T and, Debye temperature =
$\theta$D and $\theta$${}_{E}i$ = Einstein temperature. R = 8.31 J/mol K, and
ai = degree of freedom for each Einstein mode. The fitted curve is shown in
figure 6. The best fit is obtained with $\theta$D = 632 K, $\theta$E1 = 177 K
and $\theta$E2 = 740 K. Then the contribution related to the magnetic ordering
(Cm) in the specific heat is calculated by subtracting C${}_{p}h$ from the
measured total specific heat data. Figure 7 highlights the Cm \- T plot,
showing a sharp maximum at T = 355 K. The entropy change due to the magnetic
ordering can be calculated by using the following formula:

$S_{m}(T)=\int_{0}^{T}\frac{C_{m}(T)}{T}dT$ (3)

The $S_{m}(T)$ vs T plot is shown in figure 7. Magnetic entropy increases with
increasing temperature and shows a saturation value of $\simeq$ 4.4 J mole-1
K-1 (above 355 K), and this is less than the theoretically estimated magnetic
entropy for CoFeRhO4 ($S_{m}(T)$ = R $\ln{(2S+1)}$ = 26.41 J mole-1 K-1, Co2+
adopts the eg4t${}_{2}g$3 electronic configuration with S = 3/2, Fe3+ adopts
the eg2t${}_{2}g$3 electronic configuration with S = 5/2). In general, the
frustration of the magnetic cations near and above the magnetic transition
temperature can reduce the entropy contribution to the magnetic ordering. In
CoFeRhO4, geometrical frustration in the structure (diamond and pyrochlore
lattice) and the distribution of magnetic and non-magnetic cations in two
different sublattices could cause frustration of magnetic cations. The reduced
magnetic moments in the neutron powder diffraction experiments point to spins’
local disorder (and/or local canting). Therefore, the reduced entropy related
to magnetic ordering can be attributed to the frustration of the magnetic
cations 37 .

Figure 7:  Temperature variation of the magnetic component of the specific
heat (C) (left axis) and calculated magnetic entropy (right axis) for CoFeRhO4

Oxides geometrical frustration and ferrimagnetic transition often result in
exciting magnetoelectric/magnetodielectric coupling. Geometric frustration
(local spin canting) in a magnetic system is proven to be a key ingredient for
magneto(di)electric coupling. For example, the triangular Ising lattice
Ca3Co2O6 shows magneto dielectric coupling below the ferrimagnetic ordering
temperature (24 K) 38 . Haldane spin-chain system Dy2BaNiO5 shows a
magneto-(di)electric effect below the long-range ordering temperature (58 K)
39 . In spinels, the magnetodielectric effect is observed from the beginning
of the ferrimagnetic ordering in CoCr2O4 (TC = 96 K) 40 ; 41  and NiCr2O4 42 .
MnCr2O4 shows magneto dielectric effect below the ferrimagnetic ordering
temperature (43 K) 43 . However, the magneto-(di)electric effect in
ferrimagnetic materials near room temperature is rare. The near-room-
temperature magnetodielectric effect is important due to its possible
applications in spintronics devices, magnetically accessible ferroelectric
random-access memories, and communication technology. Figure 8 shows the
temperature variation of the real part of the dielectric constant
($\epsilon_{r}$) for CoFeRhO4. The dielectric loss (tan$\delta$) is
highlighted in the inset of Figure 8. The extremely low values of dielectric
loss (tan$\delta$) highlight the insulating behaviour of CoFeRhO4. It also
excludes the possibility of extrinsic Maxwell-Wagner-like behaviour appearing
because of the leakage currents. However, the frequency dispersion of
$\epsilon_{r}$ and tan$\delta$ started above 320 K, indicating the growing
Maxwell-Wagner-type relaxation or other sources of conductivity around 320 K.
Nonetheless, the dielectric constant is intrinsic below 320 K. In addition to
a small hump near the magnetic ordering temperature (335 K), two anomalies at
220 K and 50 K are observed in the $\epsilon_{r}$ \- T plots. In particular,
the magnetisation measurements also observed similar anomalies at 220 K and 50
K.

Figure 8:  Temperature variation of the real part of the dielectric constant
($\epsilon_{r}$) at several fixed frequencies for CoFeRhO4. The inset in the
figure shows the dependence of the dielectric loss as a temperature function.

As observed from our nuclear and magnetic structure refinements, heat
capacity, and magnetization studies, CoFeRhO4 hosts an exciting combination of
complex crystal structure (diamond and pyrochlore lattice) and competing
magnetic interactions. The reduced magnetic moments observed in the NPD
refinements highlight the possibility of local spin canting. Therefore, the
geometrical frustration, competing magnetic interactions, and the local spin
canting can originate magnetic frustration in the structure. The magnetic
frustration could be why these anomalies are observed below the long-range
magnetic ordering. To understand the effect of the magnetic field on the
dielectric constant, we have performed magnetic field dependence measurements
of the dielectric constant. Figure 9 shows the temperature variation of
$\epsilon_{r}$ measured in the presence of different magnetic fields.
Interestingly, the dielectric constant increases in the presence of a magnetic
field below the magnetic ordering temperature, and a positive
magnetodielectrictance effect is observed. The starting of the
magneto(di)electric effect below the long-range ferrimagnetic ordering
temperature indicates that the observed magneto(di)electric effect is related
to the magnetic ordering of CoFeRhO4. The change in the dielectric data with
the magnetic field is most pronounced near 220 K. The magneto-dielectric
constant
($\Delta\epsilon_{r}=[\epsilon_{r}(H)-\epsilon_{r}(H=0)]/\epsilon_{r}(H=0)$)
and its dependence on the magnetic field are shown in the inset of figure 9.
As discussed, the magnetic frustration in the structure (geometric
frustration, competing magnetic interactions, and local spin canting) could be
the origin of the increase in $\epsilon_{r}$ with the magnetic field at that
temperature.

Figure 9:  Temperature dependent dielectric constant (real part,
$\epsilon_{r}$) at different magnetic fields (H = 0 T, 5 T, and 14 T) for
CoFeRhO4. The inset highlights the magnetic field variation of the dielectric
constant
($\Delta\epsilon_{r}=[\epsilon_{r}(H)-\epsilon_{r}(H=0)]/\epsilon_{r}(H=0)$)
at different temperatures.

In recent times, complex magnetic materials (for instance, ferrimagnetic
systems Mn3-xPtxGa 44 , Ba2Fe1.12Os0.88O6 45  and SrFe0.15Co0.85O2.62) 46 , in
particular, those having a magnetic transition temperature near room
temperature have attracted significant interest in exploring the exchange bias
effect (EB). The EB effect shifts the isothermal magnetization loop, becoming
asymmetric and shifting along the field axis. This effect has important
technological applications, such as in the development of magnetic sensors,
magnetic recording read heads 47 , random access memories 48 , and other
spintronic devices 49 ; 50 . Figure 10 highlights the magnetic field
dependence of magnetization at 100 K, measured in FC mode (cooling field,
$H_{FC}$ = 3 T). It shows a clear shift towards the left field (negative
magnetic field). The EB field (HEB) can be calculated from the shift of the
hysteresis loop using the following equation

$H_{EB}=-(H_{C(L)}+H_{C(R)})/2$ (4)

$(H_{C(L)}$ and $H_{C(R)}$ are the intercepts with the positive (right) and
negative (left) field axis. The calculated exchange bias field HEB = 65 Oe at
100 K. The exchange bias field is almost constant with varying cooling fields
($H_{FC}$, ranging from 1 T to 7 T). To investigate the temperature evolution
of the exchange bias effect and to explore whether the exchange bias property
in CoFeRhO4 is correlated with magnetic ordering, we have measured the
temperature variation of the EB field. The material was cooled to the
measuring temperatures in a magnetic field of 3 T. Then the M-H loops were
measured between $\pm$0.5 T. Fig. 11 highlights the temperature evolution of
the EB field for CoFeRhO4. Interestingly, the EB effect emerges just below the
long-range ferrimagnetic ordering (350 K); however, with decreasing
temperature, HEB shows almost constant behavior down to 10 K. In ferrimagnetic
systems, the presence of uncompensated magnetic moments in a compensated AFM
host is proven to be very effective in designing the exchange bias effect (for
instance, Mn3-xPtxGa 44 , Ba2Fe1.12Os0.88O6) 45 .

Figure 10:  $M$-$H$ loop of CoFeRhO4 at 100 K measured in field cooled (FC)
mode ($H_{FC}$ = 3 T). The inset highlights the enlarged central part of the
FC $M$-$H$ loop. A shift along the left field axis is highlighted here.

In this spinel structure, there is an uncompensated magnetic moment due to the
distribution of two different magnetic cations (Co2+ and Fe3+) in two
different sublattices. In CoFeRhO4, as observed from the neutron diffraction
refinements (table 1), below the magnetic ordering temperature, the magnetic
moment of B sublattice (M (Oct)) dominates over the M (Td). Under FC
conditions, the magnetic field orients the net magnetic moment of the
sublattices along the field direction. Now, as we gradually reverse the
direction of the magnetic field, the magnetic moments of the B sublattice (M
(Oct)) do not easily reverse its direction as they are antiferromagnetically
coupled with the A-sublattice (M (Td)). This pinning effect of irreversible
uncompensated spins (Octahedral B sublattice) develops the unidirectional
anisotropy, and therefore, exchange bias is observed in CoFeRhO4.

##  IV Conclusion

To summarize, we synthesized and examined the detailed structural, magnetic,
thermal (specific heat), magneto-dielectric, and magnetic exchange bias
properties of CoFeRhO4, a mixed 3d-4d spinel oxide. We used various
diffraction techniques, such as RT-XRD, NPD, RT-electron diffraction, and
STEM, to explore the material’s structural details. The material crystallizes
in a cubic structure with a space group of $Fd\overline{3}m$. Our measurements
show a long-range ferrimagnetic ordering with an onset at 355 K, which is
explained by a collinear ferrimagnetic ordering model with k = [0, 0, 0].
However, we observe reduced magnetic moments in the refinement, possibly due
to the spins, local disorder, local canting, or frustration. Magnetic
frustration can also reduce magnetic entropy, as reflected in the specific
heat measurement. The magnetic entropy increases with temperature and reaches
a saturation value of 4.4 J mole-1K-1 (above 355 K), but it is less than the
theoretically estimated value. Furthermore, our dielectric and FC
magnetization measurements show the appearance of two technologically
significant phenomena near room temperature: the magnetodielectric effect and
the exchange bias effect. These effects appear below the magnetic ordering
temperature (355 K) and can originate from magnetic frustration, collinear
ferrimagnetic ordering, and uncompensated magnetic moments. The uncompensated
magnetic moments create unidirectional anisotropy, resulting in an exchange
bias effect. Importantly, the presence of room temperature ferrimagnetism, EB,
and the magneto(di)electric effect demonstrates the potential of CoFeRhO4 in
developing materials for sensors or spintronics applications operating at room
temperature.

Figure 11:  Temperature variation of the exchange bias field ($H_{EB}$) for
CoFeRhO4. The exchange bias effect started to appear below the long-range
ferrimagnetic ordering (350 K).

##  V Acknowledgments

R. P. S. acknowledges the Science and Engineering Research Board (SERB),
Government of India, for the CRG/2019/001028 Core Research Grant. S. M.
acknowledges the SERB, Government of India, for the SRG/2021/001993 Start-up
Research Grant.

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