# Effect Of Personalized Calibration On Gaze Estimation Using Deep-Learning

Nairit Bandyopadhyay Sébastien Riou Didier Schwab

Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG, 38000 Grenoble, France

<EMAIL_ADDRESS>

{sebastien.riou<EMAIL_ADDRESS>
###### Abstract

With the increase in computation power and the development of new state-of-
the-art deep learning algorithms, appearance-based gaze estimation is becoming
more and more popular. It is believed to work well with curated laboratory
data sets, however it faces several challenges when deployed in real world
scenario. One such challenge is to estimate the gaze of a person about which
the Deep Learning model trained for gaze estimation has no knowledge about. To
analyse the performance in such scenarios we have tried to simulate a
calibration mechanism. In this work we use the MPIIGaze data set. We trained a
multi modal convolutional neural network and analysed its performance with and
without calibration and this evaluation provides clear insights on how
calibration improved the performance of the Deep Learning model in estimating
gaze in the wild.

_Keywords_ Gaze tracking $\cdot$ Deep Learning $\cdot$ Personal Calibration

##  1 Introduction

Increase in interactions between human and computing devices is leading to the
popularity of research in gaze-tracking. Specific domains like Human Computer
Interaction and Computer Vision are researching on the various applications of
appearance based gaze estimation or tracking. (Morimoto and Mimica, 2005).
Machine learning methods can be used to train gaze-estimators, provided we
have lots of reliable data, and head pose-independent training data (Sugano et
al., 2014; Schneider et al., 2014). These methods can bring appearance-based
methods into a certain state where these methods do not require any user or
device-specific training. The quantity of user data is sufficient to provide
information to the gaze-estimators. Gaze tracking using monocular cameras in
mobile phones, laptops and interactive displays are cost effective because of
their widespread availability. While appearance-based gaze estimation performs
well with machine learning-based methods, new techniques are still developed.
These new techniques are evaluated on curated data sets collected under
controlled laboratory conditions (Zhang et al., 2015). As a result there is
lack of availability of different eye shapes as well as we have to make
assumptions that the head pose is accurate. This often results in problems in
object recognition (Torralba and Efros, 2011) and object detection (Li et al.,
2014).
In our study we have used MPIIGaze data set. The data set and annotations are
publicly available online (Zhang et al., 2015).
Here in this article we are going to see how introduction of small amounts of
test data in the training set can significantly increase the accuracy of the
model. We are calling this approach as a personal calibration that is done in
gaze estimation software - when a user starts using it for the first time.
We have divided our work in two parts. First, we train a CNN model for
appearance-based gaze estimation. The data set we are using is one order of
magnitude larger than existing data sets and has more variation with respect
to illumination and appearance (Zhang et al., 2015). Second, we perform
experiments on the model with and without calibration to see the effect in
performance of the model.

##  2 Dataset and its features

We used MPIIGaze dataset that contains a total of 213,659 images from 15
participants. The number of images of each participant varied from 34,745 to
1,498. The data set contains larger variability in illumination and larger
appearance variations (Zhang et al., 2015). All these data were collected from
the laptop because it can be used for daily recordings and are an important
platform of eye-tracking devices (Majaranta and Bulling, 2014).

##  3 Method

Figure 1 provides a high level view of gaze estimation task using multi modal
convolutional neural networks (CNN). A monocular camera is used to capture the
image of the face. Then the face detection and facial landmarks detection
methods are used to locate landmarks in the image (Zhang et al., 2015). A 3D
facial shape model is used to estimate 3D poses of the detected faces and
apply the space normalisation technique to crop and warp the head pose and eye
images to the normalised training space (Sugano et al., 2014; Zhang et al.,
2015). Then we train a convolutional neural network to learn the mapping from
the head poses and eye images to 2D gaze-points in the camera coordinate
system.

Figure 1: Overview of our method for appearance-based gaze estimation using
multi modal convolutional neural networks (Zhang et al., 2015).

###  3.1 Preprocessing : Face Alignment and 3D Head Pose Estimation

Before using the images to train the model, the images present in the MPIIGaze
data set needs to be processed. The user’s face is detected in the image using
Li et al.’s SURF cascade method (Li and Zhang, 2013; Zhang et al., 2015).
Afterwards, Baltrušaitis et al.’s constrained local mode framework to detect
facial landmarks (Baltrušaitis et al., 2014; Zhang et al., 2015). The 3D
positions of six facial landmarks (eye and mouth corners, cf. Figure 1)
constitutes the facial model. The head coordinate system is defined according
to the triangle connecting three midpoints of the eyes and mouth. EPnP
algorithm is used to fit the model by estimating the initial solution and
further refining the pose via non-linear optimisation (Lepetit et al., 2009;
Zhang et al., 2015). 3D head rotation r is defined as the rotation from the
head coordinate system to the camera coordinate system, and the eye position t
is defined as the midpoint of eye corners for each eye (Zhang et al., 2015).

Figure 2: ResNet Model Summary

###  3.2 Normalizing the Preprocessed Data

The normalisation is done by two camera matrix transformation operations:
scaling and rotation of camera (Zhang et al., 2015). The x axes of the camera
coordinate space and that of head coordinate space is made parallel by
pointing the camera to one of the facial landmarks - the midpoint of eye
corners. After that, the eye images were cropped at a fixed resolution W × H
with a fixed focal length f in the normalised camera space, and histogram-
equalised (to improve the contrast of the image) to form the input eye image.
This results in a set of fixed-resolution eye images e and 2D head angle
vectors h (Zhang et al., 2015). In order to reduce the effect of different
lighting conditions, so that no extra noise gets inside the CNN model, eye
images e are histogram-equalised after the normalisation process (Zhang et
al., 2015). The setting for camera distance d, focal length f and the
resolution W ×H remains same (Sugano et al., 2014).

###  3.3 Using Multimodal CNN To Learn The Mapping

The CNN learns the mapping from the input features (2D head angle h and eye
image e) to 2D points g in the screen normalised space (Sugano et al., 2014;
Zhang et al., 2015).
Our model uses a pre-activated ResNet like architecture that consists of one
convolutional layer containing 3 stages, followed by batch normalisation and a
final connected layer. There is a linear regression layer on top of the final
connected layer to predict gaze positions g. The head pose information is
introduced in our CNN model by concatenating h with the output of the fully
connected layer. Our input for the network are the grey-scale eye images e
with a fixed size of 60 ×36 pixels (Zhang et al., 2015). The number of
parameters for the convolutional layer is 144. For more details on the
architecture please refer to Figure 2 The output of the network is a 2D gaze
position g that consists of x and y coordinates (normalised) of the 2D screen.
Our loss function is the sum of the individual losses that measure the
euclidean distance between the predicted g and the actual g. Thus with the
neural network we are solving an optimisation problem of minimising this loss
function
The normalized data set is available online for public use. Hence we did not
have to perform the techniques in Section 3.1 and Section 3.2. We mainly
worked on Section 3.3.

Figure 3: Higher level representation of personal calibration method for Deep
Learning technique (Cheng et al., 2021)

##  4 Experiments

Here we will be discussing about the person-independent gaze estimation task
with and without calibration to validate the effectiveness of the proposed CNN
based gaze estimation approach with personalized calibration. There are
various ways of personal calibration, like fine-tuning the model in target
domain (Krafka et al., 2016). Some researchers have added some target person
specific features for gaze tracking which are learned during fine tuning
(Linden et al., 2019). We will be retraining the model with the target feature
and observe if there are any changes in the performance of the model, with and
without personalized calibration. Please refer to this Figure 3 to understand
the pipeline of personalized calibration.

###  4.1 Equipment Used

All the training were done using GeForce GTX 1650 Max-Q. The average time to
train 1 epoch having 12 steps is around 20 seconds at around 90% GPU
utilisation.

###  4.2 Methodology

We conduct the experiments using the MPIIGaze dataset only. We choose 1,500
left eye samples and 1,500 right eye samples from each person so that we can
take into account the sample number bias in the data set. Since one
participant has only 1,448 images, we randomly over sampled the data to get
3,000. So for each person, we now have 3000 images (Zhang et al., 2015). We
split these 3000 images into 10 partitions, each containing 10% i.e 300 images
for calibration. We used leave-one-out evaluation strategy. This means, if we
want to evaluate the performance of a model on the data of person X, then we
train a model using data of the remaining persons and some small percentage of
data of person X. Here we have used 10% of the test data for calibration. Then
we test the resulting model A with the remaining 90% data of the person X.
Again, we also have to train a model B that is without calibration and test
with the same 90% test data that we used to test model A. This will now give
us the difference in estimation error with and without calibration. For each
person, we perform this experiment 10 times (because we have 10 partitions) so
as to get the mean estimation error and see which partition provides best
calibration for each person.
We trained every model for 40 epochs because we found that around 40 epochs,
the training loss and validation loss converges and also the loss function is
minimum at that time.

Figure 4: Leave-one-person-out evaluation on MPIIGaze. Mean estimation errors
are plotted for each person for the proposed method of personalised
calibration. Error bars indicate standard deviations when the experiments are
conducted with and without calibration.

##  5 Results

As described in Section 5.1, the experiments were conducted. We had to conduct
150 experiments and in each experiment we had to create two models - with and
without calibration. For each experiment we obtain two error values. It is
important to note that the error value here is unit less because it is a
relative error. Due to differences in screen size for each participant, the
ground truth pixel coordinates values were normalized by dividing the x and y
pixel coordinate by width and height of the screen in pixels respectively so
that the pixel coordinate values remain between 0 and 1. Hence the error value
also remains in between 0 and 1. For training and testing of model, such
normalization was necessary, or else the model was not getting trained due to
some out of range errors. However for real time setup where the model infers
the gaze position on the screen, in that case the x,y coordinate values can be
multiplied with dimensions of the screen so as to get the exact pixel
position. The results obtained showed an interesting observation that
calibration significantly reduces the mean error compared to without
calibration. This is because the model now has some prior information about
the person (See Figure 4).
Let’s look at the results of p00. The mean error is 0.25 before calibration.
It means this 0.25 is the difference in the ground truth value and the
inferred value of the pixel coordinate. As mentioned before, it has no unit
because the pixel coordinates were normalised. After calibration, it is now
0.20. There has been a significant improvement in case of p00 where the error
decreased by almost 20%.

##  6 Limitations

One of the major limitation is the process of head-pose estimation in real
time. Since we used the normalized dataset of MPIIGaze, we did not have to
perform the head-pose estimation. However if someone decides to create a new
dataset of their own, then they have to perform this head-pose estimation.
Sometimes the detection of facial landmarks and obtaining eye-images are also
challenging because of lighting conditions and presence of accessories like
when a person wears glasses. We did not have to do this as we used the clean
normalized MPIIGaze dataset.

(a) Facial landmark detection

(b) Predicted x,y coordinates - for left eye and right eye separately along
with the head position

Figure 5: Real Time 2D Gaze Points Estimation

##  7 Future Work

By using only 300 extra images (10% of the test data) we have seen significant
decrease in estimation errors. So our next work will be to create a real-time
software with this calibration method. Before the software starts gaze-
tracking, it will capture images of the face of the person using webcam. Most
modern day webcams can shoot 30 times in one second. So we can capture around
450 images in 15 seconds by which we can calibrate a pre-trained model. After
that we can start using that software to get a predicted 2-D position on the
screen.
We have already started creating such software which is still in development
stage. As of now, it does not have the calibration feature and also we have
not tested it rigorously. It can do the tasks of facial landmark detection and
predicting the x,y coordinates of the estimated eye-gaze on a 2D screen.
Please refer to Figure 5 which shows our software working in real time.
Apart from this, we can also try using other models like UNet etc. to see if
it improves the performance of learning by the model. Thus, there are multiple
scope of future works based on this work.

##  8 Conclusion

Appearance-based gaze estimation methods have so far been evaluated
exclusively under controlled laboratory conditions. In this work, we present
an extensive study on appearance-based gaze estimation along with a
calibration technique. We used a data set that contains images which has wide
variations. Our CNN-based estimation model significantly shows improvement in
performance when calibration is performed. This work provides a critical
insight on addressing the performance challenges of Deep Learning models in
daily-life gaze interaction.

## References

  * Morimoto and Mimica (2005) Morimoto, C.; Mimica, M. Eye gaze tracking techniques for interactive applications. _Computer Vision and Image Understanding_ 2005, _98_ , 4–24.
  * Sugano et al. (2014) Sugano, Y.; Matsushita, Y.; Sato, Y. Learning-by-Synthesis for Appearance-Based 3D Gaze Estimation. 2014; pp 1821–1828.
  * Schneider et al. (2014) Schneider, T.; Schauerte, B.; Stiefelhagen, R. Manifold Alignment for Person Independent Appearance-Based Gaze Estimation. 2014 22nd International Conference on Pattern Recognition. 2014; pp 1167–1172.
  * Zhang et al. (2015) Zhang, X.; Sugano, Y.; Fritz, M.; Bulling, A. Appearance-based gaze estimation in the wild. 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). 2015; pp 4511–4520.
  * Torralba and Efros (2011) Torralba, A.; Efros, A. A. Unbiased look at dataset bias. CVPR 2011. 2011; pp 1521–1528.
  * Li et al. (2014) Li, Y.; Hou, X.; Koch, C.; Rehg, J. M.; Yuille, A. L. The secrets of salient object segmentation. Proceedings of the IEEE conference on computer vision and pattern recognition. 2014; pp 280–287.
  * Majaranta and Bulling (2014) Majaranta, P.; Bulling, A. _Advances in physiological computing_ ; Springer, 2014; pp 39–65.
  * Li and Zhang (2013) Li, J.; Zhang, Y. Learning surf cascade for fast and accurate object detection. Proceedings of the IEEE conference on computer vision and pattern recognition. 2013; pp 3468–3475.
  * Baltrušaitis et al. (2014) Baltrušaitis, T.; Robinson, P.; Morency, L.-P. Continuous conditional neural fields for structured regression. European conference on computer vision. 2014; pp 593–608.
  * Lepetit et al. (2009) Lepetit, V.; Moreno-Noguer, F.; Fua, P. Epnp: An accurate o (n) solution to the pnp problem. _International journal of computer vision_ 2009, _81_ , 155.
  * Cheng et al. (2021) Cheng, Y.; Wang, H.; Bao, Y.; Lu, F. Appearance-based Gaze Estimation With Deep Learning: A Review and Benchmark. _arXiv preprint arXiv:2104.12668_ 2021,
  * Krafka et al. (2016) Krafka, K.; Khosla, A.; Kellnhofer, P.; Kannan, H.; Bhandarkar, S.; Matusik, W.; Torralba, A. Eye Tracking for Everyone. 2016,
  * Linden et al. (2019) Linden, E.; Sjostrand, J.; Proutiere, A. Learning to personalize in appearance-based gaze tracking. Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops. 2019; pp 0–0.


# Tuning magnetocaloric effect by optimizing thickness induced 3D strain state

Samir Kumar Giri1! Wasim Akram2! Manisha Bansal2 Tuhin Maity2
<EMAIL_ADDRESS>

! Equal Contribution first authors 1Kharagpur College, Kharagpur, Paschim
Medinipur, West Bengal 721305, India
2School of Physics, Indian Institute of Science Education and Research
Thiruvananthapuram, Thiruvananthapuram, Kerala 695551, India

###### Abstract

The effect of 3-dimensional strain state on the magnetocaloric properties of
epitaxial La0.8Ca0.2MnO3 (LCMO) thin films grown on two types of substrates,
SrTiO3 (001) (STO) and LaAlO3 (001) (LAO) has been studied as a function of
film thickness within the range of 25 – 300 nm. The STO substrate imposes an
in-plane tensile biaxial strain while LAO substrate imposes an in-plane
compressive biaxial strain. The in-plane biaxial strain on LCMO by STO
substrate gets relaxed more rapidly than that by LAO substrate but both
LCMO/STO and LCMO/LAO show a maximum entropy change ($\Delta{S}_{M}$) of
$\sim$ 12.1 JKg-1K-1 and $\sim$ 3.2 JKg-1K-1, respectively at a critical
thickness of 75 nm (at 6 T applied magnetic field). LCMO/LAO is found to
exhibit a wider transition temperature region with full width at half maxima
(FWHM) $\sim$ 40 K of the $\frac{dM}{dT}$ vs $T$ curve compared to LCMO/STO
with FWHM $\sim$ 33 K of that curve. This broadening of the transition region
indicates that the Table like magnetocaloric effect (MCE) is attainable by
changing the strain type. The maximum Relative Cooling Power, $\sim$ 361 JKg-1
of LCMO/STO and $\sim$ 339 JKg-1 of LCMO/LAO is also observed at the thickness
$\sim$ 75 nm. The Curie temperature varies with the thickness exploring the
variation of ferromagnetic interaction strength due to strain relaxation. The
film thickness and substrate induced lattice strain are proved to be the
significant parameters for controlling MCE. The highest MCE response at a
particular thickness shows the possibility of tuning MCE in other devices by
optimizing thickness.

††preprint: APS/123-QED

##  I Introduction

Depletion of energy resources is pushing the scientific community to
materialize the energy-efficient and environmental friendly technologies. To
sustain the nature and the running demands alongside, low power refrigeration
or cooling at both macro and micro scale has been proved to be one of the
future energy saving innovation[1, 2, 3, 4]. Existing cooling techniques
include absorption and adsorption refrigerators, thermoelectric cooling,
thermoacoustic refrigerators, ejector refrigeration systems, magnetocaloric
(MC) refrigeration, etc[5, 6]. Among these, magnetocaloric effect (MCE) is the
most promising candidate due to its efficiency and environmental friendly
approach. Use of thermal effects induced by the application of magnetic field
to produce efficient cooling, popularly known as MCE has gained recognition
among the researchers[7, 8, 9]. Room temperature cheap and efficient MC
devices lead to potent energy safeguarding household and industrial
applications outperforming conventional cooling or refrigeration
techniques[10]. However, only few materials can be used for MC applications
due to the complexity involved in studying the coupled magnetic and structural
parameters of the system[11, 12, 13]. Materials should have high MC constant,
wide range of operating temperature, low hysteresis loss, low specific heat,
high thermal conductivity, etc., to produce large MCE response which is
challenging as of significant technological importance. Recent material
researches show possible improvement in the MCE efficiency concomitant with
challenges like fabrication complexity, less resistance to corrosion, poor
conductivity, etc[2, 14, 15]. Different alloys (e.g., Iron based alloy
Gd5Ge1.9Si2Fe0.1)[16] and composites (e.g., La(Fe, Si)13Hy/In)[2] showed
potential to overcome such deficiencies, henceforth improving the performance
of MCE based magnetic refrigeration or cooling.

Moreover, most research activities on MCE are limited to bulk materials since
the studies on thin films are more complex due to the intricacies involved in
their fabrication and study. But the industries especially for micro and
nanoscale devices are in a huge demand of thin film based MCE because of their
high efficiency, less space consumption, flexibility, larger specific surface
area for higher exchange of heat, etc[17, 18]. Interestingly, epitaxial thin
films show several fascinating phenomena including colossal magnetoresistance
(CMR) due to the additional strain constraints[19, 20] which are of particular
importance to determine their physical properties[21, 22, 23, 24]. The giant
MCE has been observed in the CMR based epitaxial manganite systems like
La0.67Sr0.33MnO3 (LSMO)/BaTiO3 (BTO) ($\Delta{S_{M}}/\mu_{0}\Delta{H}$ = 1.95
JKg-1K-1T-1)[25], La0.7Ca0.3MnO3/BTO ($\Delta{S_{M}}/\mu_{0}\Delta{H}$ = -0.7
JKg-1K-1T-1 for intrinsic and -9.0 JKg-1K-1T-1 for extrinsic)[26], where the
substrate induced strain influences the temperature dependent magnetic
properties of the CMR materials resulting in adiabatic thermal process[27, 28,
29]. This substrate induced strain may be compressive (tensile) depending upon
larger (smaller) lattice parameter of the epitaxial film than that of the
substrate[30, 22, 31] which results in decrease (increase) of the Mn-O bond
length as well as altering the Mn-O-Mn bond angle[22], enhancing MCE.
Generally, under a tensile strain, ferromagnetism due to double exchange (D-E)
interaction gets weakened because of the reduction in $e_{g}$ electron
transfer integral, $t_{\alpha}$ = $t_{\alpha}^{\star}$sin($\phi$/2) where
$t_{\alpha}^{\star}$ is the bare transfer integral and $\phi$ is the Mn–O–Mn
bond angle[32] and decreases the Curie temperature ($T_{C}$)[32]. The
enhancement of $T_{C}$ under compressive strain[33] can also be explained by
the same mechanism. So, it’s expected that the strain relaxation by increasing
film thickness ($t$) should bring about the $T_{C}$ to higher values always
for the tensile strain and to lower values for the compressive strain. On the
contrary, several reports[34, 35, 36] were able to convey that the tensile
strain also drives up $T_{C}$ while the compressive strain reduces it[37] and
relaxation of these strains would do the opposite as expected.

In this work, we investigate the evolution of strain states on MCE of the
epitaxial La0.8Ca0.2MnO3 (LCMO) thin films grown on STO and LAO substrates
which give rise to tensile and compressive strain, respectively and show the
improvement of performance by large tuneable magnetic entropy change as well
as the variation of transition temperatures. It is observed that the
transition temperatures do not always decrease or increase under strain but
varies with $t$ witnessing a maximum value at 75 nm for both the tensile and
compressive strain. For cooling applications, distributed magnetic ordering
temperature is highly desiderated as it brings on to “Table like MCE”[38, 39]
which is the basis of ideal Ericsson-cycle based magnetic refrigeration[38].
Here we report the broadening of the transition region for LCMO/LAO than that
of the LCMO/STO suggesting the possibility of acquiring “Table like MCE”, a
goal towards the ideal Ericsson-cycle. To compare the viability of these LCMOs
grown on STO and LAO as an efficient refrigerant, the relative cooling power
(RCP) is calculated and plotted against $t$ and magnetic field ($H$). The bulk
La0.67Ca0.33MnO3 shows a magnetic entropy change, $\Delta{S_{M}}$ = 0.9
JKg-1K-1T-1[40] which underscores the need for efficient cooling application.
Here, we show $\Delta{S_{M}}$ = 2 JKg-1K-1T-1 in epitaxial La0.8Ca0.2MnO3 by
optimizing $t$ and substrate induced lattice strain.

##  II Experimental method

The epitaxial LCMO thin films were grown by pulsed laser deposition (PLD)
technique from a stoichiometric target at 750oC with an oxygen pressure of 300
mTorr and an estimated laser energy density $\sim$ 3.5 Jcm-2 with a repetition
rate of 5 Hz. Two different types of single crystal substrates i.e., LAO (001)
(aP $\sim$ 3.79Å) and STO (001) (aP $\sim$ 3.905Å) were used to attain two
different types of lattice mismatch for the growth of LCMO thin films. The
thicknesses of the films were varied from 25 nm to 300 nm in order to
correlate the strain relaxation process with MCE. The composition of the film
with target was confirmed from wavelength dispersive x-ray spectroscopy within
experimental error. Structural characterization was carried out using four-
circle x-ray diffractometer. Figs. 1(a) and 1(b) show XRD scans of 25 nm LCMO
films grown on STO and LAO substrates, respectively. The out-of-plane and in-
plane lattice parameters were determined by normal $\theta-2\theta$ and
grazing incidence diffraction (GID) scans at room temperature, respectively.
The variation of in-plane lattice parameters w.r.t. $t$ for the two different
substrates is shown in Figs. 1(c) and 1(d). The bulk value of LCMO lattice
parameter (3.881 Å) is obtained from the target pellet. Temperature-dependent
magnetizations of all LCMO thin films were measured, in order to explore the
effect of 3D strain states on magnetic properties. Magnetization measurements
were carried out using SQUID magnetometer in the temperature range 5-300 K and
field range of ±6 T. The temperature dependent zero field cooled (ZFC)
Magnetization ($M$) vs Temperature ($T$) of all the films are taken at $H$ = 5
mT. For slow magnetic field scan, 0.01 T per minute ramp rate was used in
SQUID to consider all the magnetization process as quasi equilibrium and
isothermal.

Figure 1: (a) and (b) Grazing incidence XRD scans of 25 nm LCMO films grown on
STO and LAO substrates, respectively. (c) and (d) show the change of lattice
parameters as a function of $t$ of LCMO/STO and LCMO/LAO, respectively

##  III Results and Discussion

It’s evident from the XRD scans that the epitaxially grown LCMO thin films are
subjected to two distinct strain states on LAO and STO substrates. In Figs.
1(c) and 1(d), the in-plane lattice parameters (aP) of the LCMO thin films
obtained from grazing incidence x-ray diffraction (GID) at room temperature
are found to be deviated from that of the substrates to the bulk value
(3.881Å) with the increase in $t$. Rao _et al._[41] have previously discussed
the variation of 3D strain state w.r.t. $t$ in detail. LAO substrate (aP
$\sim$ 3.79Å) imposes an in-plane compressive strain with lattice mismatch
-2.34 % while STO substrate (aP $\sim$ 3.905Å) imposes a corresponding tensile
strain with a lattice mismatch +0.62 % on the films.

The temperature dependence of the in-plane magnetization of LCMO/STO and
LCMO/LAO are shown in Figs. 2(a) and 2(b), respectively. The transition
temperatures are obtained from the extrema of $\frac{dM}{dT}$ vs $T$ curves
(insets of Figs. 2(a) and 2(b)). LCMO/STO shows a cusp at $T_{P}$ in lower
temperature regime as well as at Curie temperature $T_{C}$ in higher
temperature regime. The full width at half maxima (FWHM) of $\frac{dM}{dT}$ vs
$T$ curve for LCMO/LAO ($\sim$ 40 K) is larger than that of the LCMO/STO
($\sim$ 33 K) (Fig. 2(c)) elucidating the broadening of the transition region
for the former compared to the latter. Along with the absence of any thermal
hysteresis in $MT$ curve for LCMO/LAO, this suggests the probability of second
order magnetic phase transition (SOMPT) while the thermal hysteresis in $MT$
curve for LCMO/STO (Fig. 2(d)) between field cooling and field heating
elucidates the first order magnetic phase transition (FOMPT) at transition
temperature[42].

Figure 2: (a) and (b) $M$ vs $T$ curves of 25 nm LCMO thin film grown on STO
and LAO substrates, respectively under 5 mT magnetic field. Inset figures show
the $\frac{dM}{dT}$ vs $T$ curves and corresponding $T_{C}$. (c) Broadening of
transition region for LCMO/STO compared to LCMO/LAO. (d) Hysteresis between
warming up and cooling down cycle of magnetization for LCMO/STO under 5 mT
field.

For comparison, $T_{C}$ and $T_{P}$ of the epitaxial LCMO films are plotted as
a function of $t$ in Fig. 3. LCMO/LAO shows higher $T_{C}$ compared to
LCMO/STO of the same $t$ due to persistence of stronger ferromagnetism below
transition temperature $T_{C}$ under compressive strain[43]. However, the
transition temperatures $T_{C}$ and $T_{P}$ vary non-monotonously with the
strain relaxation on increasing $t$ below 100 nm thickness. Both of them at
first go on increasing for up to 75 nm thickness and then decrease. This can
be explained as dominating behavior of ferromagnetic (FM) phase up to 75 nm
thickness which has been discussed later. At 300 nm thickness both $T_{C}$ and
$T_{P}$ are higher which is usual for thin film[44, 45]. The increase in
$T_{C}$ for tensile strain relaxation up to $t$ $\sim$ 75 nm originates from
the strengthening of D-E interaction due to Mn-O bond length reduction[32].
However, the compressive strain induced ferromagnetism in LCMO system observed
here is in contrast as $T_{C}$ was supposed to be decreased on strain
relaxation due to weakening of D-E interaction. So, this anomaly needs to be
understood.

Figure 3: $T_{C}$ and $T_{P}$ vs $t$ plot for LCMO/LAO and LCMO/STO.

To understand this, two effects should be taken into consideration as the
result of the in-plane Mn-O bond length elongation (contraction) – one is the
decrease (increase) of $e_{g}$ electron transfer integral as mentioned before
and the another is electron localization in $d_{x^{2}-y^{2}}$ -orbital[22] due
to in-plane $d_{x^{2}-y^{2}}$ -orbital stabilization[32] which can enhance the
ferromagnetism and increase $T_{C}$. In particular, Mn3+ ions in LaMnO3 are
already known to be arranged in an A type planar antiferromagnetic structure
consisting of oppositely aligned FM {001} planes[46]. Hence, in-plane Mn3+
ions interact ferromagnetically with each other. When LaMnO3 is doped with
divalent Ca2+ ions, they replace some of the trivalent La3+ ions and form
mixed valence state in Mn ions (Mn3+ and Mn4+)[47]. The Mn3+ and Mn4+ ions
interact with each other ferromagnetically through D-E interaction mediated by
O-$2p$ orbital electrons[48]. The spacing between Mn4+ and Mn3+ gets increased
under tensile strain giving rise to reduction[32] in $e_{g}$ orbital electron
transfer integral and weakening ferromagnetism due to D-E interaction and the
reverse case happens for the compressive strain. But, at the same time, due to
elongation of the in-plane Mn-O bond length, the overlapping between the lobes
of the in-plane Mn-$3d$ orbitals and O-$2p$ orbitals gets decreased[49] giving
rise to lower coulomb repulsion and favoring the in-plane orbital electron
localization[22] which in turn induces ferromagnetism. So, due to the
competition between these two effects, the practical behavior depends on the
dominant effect or possibly a mixed effect. For tensile strain, the reduction
in D-E interaction dominates over orbital electron localization whereas the
latter one dominates over the first one for compressive strain and due to
compressive strain relaxation, the in-plane orbital electrons are getting
localized to give rise to stronger ferromagnetism. Above 75 nm thickness,
$T_{C}$ for each sample starts to fall from the maximum value indicating that
the dominating behavior is not always the same throughout the strain
relaxation process. Similarly $T_{P}$ achieves the highest value at 75 nm of
thickness as well.

The viability of these epitaxial LCMOs as the core refrigeration system can be
investigated by a comparative study of the MCE produced by these. The MCE is
parameterized by the isothermal magnetic entropy change
($(\Delta{S}_{M})_{iso}$) and/or adiabatic temperature change
($\Delta{T}_{ad}$) of a magnetic material under the variation of $H$[18]. Most
of the research reports publish $(\Delta{S}_{M})_{iso}$ data instead of
$\Delta{T}_{ad}$ for the former being easily measurable with standard
magnetometers, but $\Delta{T}_{ad}$ measurement has some technical difficulty
at the micro and nano scale due to uncertainty of adiabatic procedure and
occurrence of the rapid thermal diffusion from the studied micro/nanostructure
towards a heat/cold sink[6, 18]. The isothermal magnetic entropy change
$\Delta{S}_{M}(H)$ of a magnetic material on application of magnetic field $H$
at temperature $T$ can be written from the Maxwell’s thermodynamic relation
$(\frac{\partial S}{\partial H})_{T}$ = $(\frac{\partial M}{\partial T})_{H}$
as[26]

$\begin{split}\Delta{S}_{M}(T,H)&={S}_{M}(T,H)-{S}_{M}(T,0)\\\
&=\int_{0}^{H}(\frac{\partial M}{\partial T})_{H}\,dH\end{split}$ (1)

This expression ensures the maximum magnetic entropy change around the
transition temperature due to switching in magnetic ordering and rapid change
of magnetization. For these LCMO films, the values of $\Delta{S_{M}}$ are
calculated from the isothermal magnetization curves (not shown here) obtained
at different temperatures with appropriate interval of temperature
$\Delta{T}$. Hence Eq. (1) can be approximated by

$\mid{\Delta{S_{M}}}\mid=\sum{\frac{(M_{n}-M_{n+1})}{(T_{n+1}-T_{n})}\,{\Delta{H_{n}}}}$
(2)

where, $M_{n}$ and $M_{n+1}$ are the magnetization values measured in a field
$H$ at temperatures $T_{n}$ and $T_{n+1}$, respectively[50, 51]. The values of
temperature dependent $\Delta{S_{M}}$ are calculated by using Eq. (2) as shown
in Fig. 4 for all LCMO/STO and LCMO/LAO.

Figure 4: $\Delta{S_{M}}$ vs $T$ curve for (a) LCMO/STO and (b) LCMO/LAO on
varying $t$ from 25 nm to 300 nm. (c) and (d) are $\Delta{S_{M}}$ vs $T$ curve
for LCMO/STO and LCMO/LAO, respectively on varying $H$ from 6 T to 0.5 T

Here, the maximum value of $\Delta{S_{M}}$ (Fig. 5(a)) is found to be $\sim$
12.1 JKg-1K-1 for LCMO/STO of 75 nm thickness under a field change of 6 T.
Interestingly, the value of $\Delta{S_{M}}$ changes due to variation of strain
type on different substrates and strain amount of different $t$ (Fig. 5(a)).
The values of $-\Delta{S_{M}}$ derived from the Maxwell equation show a
negative peak (i.e., normal MCE) around $T_{P}$ (for LCMO/STO) and positive
peak (i.e., inverse MCE) around $T_{C}$ (for LCMO/STO and LCMO/LAO both),
respectively. Both the peaks increase with increasing $H$ (Figs. 4(c) and
4(d)). For LCMO/STO, the value of $-\Delta{S_{M}}$ (+ 12.1 JKg-1K-1) at
$T_{C}$ is almost three times larger in magnitude than at $T_{P}$
($-\Delta{S_{M}}$= - 4.2 JKg-1K-1) for $\mu_{0}H$ = 6 T and negative
$\mid$$\Delta{S_{M}}$$\mid$ remains in narrow temperature regime around
$T_{C}$ due to FOMPT while for LCMO/LAO, it gets broadened over a wide
temperature range due to the suppression of FOMPT with the peak value of 3.2
JKg-1K-1. To make sure that the temperature for maximum entropy change
($T_{P2}$) and the transition temperature ($T_{C}$) follow the similar trend,
$T_{P2}$ is plotted against $t$ (Fig. 5(b)).

Figure 5: (a) Plot of $\Delta{S_{max}}$ against $t$ for LCMO/STO and LCMO/LAO
(b) Temperature at $\Delta{S_{max}}$($T_{P2}$) vs $t$ plot

At 75 nm thickness both the $\Delta{S_{max}}$ and $T_{P2}$ show maxima and to
investigate this anomalous behavior, we plotted the JT distortion
($\epsilon_{JT}$) and in-plane biaxial strain ($\epsilon_{XX}$ =
$\epsilon_{YY}$) against $t$ (Figs. 6(a) and 6(b)). Both are decaying
exponentially following the equation

$\epsilon={\epsilon^{\star}}e^{-\frac{t}{t_{0}}}+\epsilon_{0}$ (3)

with the fitting parameters ${\epsilon^{\star}}$, $\epsilon_{0}$ and $t_{0}$
[for $\epsilon_{JT}$ curve (dark cyan colored) of LCMO/LAO (Fig. 6(a)) and
LCMO/STO (Fig. 6(b)), ${\epsilon^{\star}}$ = 4.42 % and -1.92 %,
$\epsilon_{0}$ = -0.3 % and -0.42 %, and $t_{0}$ = 30.9 nm and 25.5 nm,
respectively; for $\epsilon_{XX}$ curve (red colored) of LCMO/LAO (Fig. 6(a))
and LCMO/STO (Fig. 6(b), ${\epsilon^{\star}}$ = -2.25 % and 3.71 %,
$\epsilon_{0}$ = 0 % and 0 %, and $t_{0}$ = 30 nm and 15.5 nm, respectively.]
where $t_{0}$ is the thickness at which the strain drops to $1/e$ times of its
maximum value and indicates how fast the strain gets relaxed with increasing
$t$. The product ($\epsilon_{XX}\times\epsilon_{JT}$) which is the combined
effect on $\Delta{S_{max}}$ by JT distortion and in - plane strain also
follows the similar exponential decay curve

$(\epsilon_{XX}\times\epsilon_{JT})=(\epsilon_{XX}\times\epsilon_{JT})^{\star}e^{-\frac{t}{t_{0}}}$
(4)

with the fitting paramaters $(\epsilon_{XX}\times\epsilon_{JT})^{\star}$ =
-0.052 % and -0.075 %, and $t_{0}$ = 20 nm and 11.5 nm for LCMO/LAO (Fig.
6(c)) and LCMO/STO (Fig. 6(d)), respectively. All the strain components
mentioned here and their product for LCMO/STO decay more rapidly than those of
the LCMO/LAO, respectively.

For the given range, the $t$ dependence of $\Delta{S_{max}}$ can be well
fitted with Lorentz function,

$\Delta{S_{max}}=(\Delta{S_{max}})_{0}+\frac{a}{b^{2}+(t-t_{1})^{2}}$ (5)

with highest value at $t=t_{1}$ yielding the fitting parameters
$(\Delta{S_{max}})_{0}$ = 1.49 JKg-1K-1, $a$ = 805.15 JKg-1K-1nm2, $b$ = 20.3
nm, $t_{1}$ = 66.3 nm for LCMO/LAO (Fig. 6(c)) and $(\Delta{S_{max}})_{0}$ =
4.14 JKg-1K-1, $a$ = 3124.23 JKg-1K-1nm2, $b$ = 18.4 nm and $t_{1}$ = 66.8 nm
for LCMO/STO (Fig. 6(d)). From the Lorentz function fitting we observed that
the value of $t_{1}$ ($\sim$ 66 nm ± 5nm) is almost same for both LCMO/LAO and
LCMO/STO. Certainly more data points could give precise value of $t_{1}$ by
reducing the deviation of the parameters.

Figure 6: In-plane biaxial strain ($\epsilon_{XX}$) relaxation and decaying of Jahn-Teller distortion ($\epsilon_{JT}$) of (a) LCMO/LAO and (b) LCMO/STO. The vertical error bars are standard deviation. (c) and (d) $\epsilon_{XX}$ $\times$ $\epsilon_{JT}$ and $\Delta{S_{max}}$ vs $t$ plot of LCMO/LAO and LCMO/STO, respectively. Table 1: Comparison of $\Delta{S_{max}}$ (in JKg-1K-1) and RCPs (in JKg-1) of different MC materials Compound | $\Delta{S_{max}}$ | RCP-1 | RCP-2 | Ref
---|---|---|---|---
Pr5Ni1.9Si3 | 8.15111Under the field change of 5 T | 165 | - | [52]
PrNi | 6.151 | 56 | - | [52]
Er3Ni2 | 19.51 | 507 | - | [53]
TmGa | 34.21 | 485 | 364 | [54]
GdPd2Si | 6.01 | 329 | - | [55]
ErFeSi | 23.11 | 460 | 365 | [56]
LCMO/STO | 12.1222Under the field change of 6 T | 361 | 257 | This Work
LCMO/LAO | 3.22 | 339 | 258 | This Work
Figure 7: RCP for LCMO/STO and LCMO/LAO calculated around $T_{C}$ as a
function of (a) $t$ (b) $H$. Inset figures show the RCPs of LCMO/STO
calculated around $T_{P}$ as a function of $t$ and $H$, respectively.

Generally, the first order transition materials (FOTM) undergo larger magnetic
entropy change ($\Delta{S_{M}}$) with the smaller working temperature span (WT
span) than the second order transition materials (SOTM) and exhibit
temperature dependent magnetic hysteresis which diminishes the energy
efficiency of the magnetocaloric materials[57]. So, confusion may arise to
figure out the suitability in cooling application between FOTM with larger
$\Delta{S_{M}}$ but smaller WT span and SOTM with smaller $\Delta{S_{M}}$ but
larger WT span. To overcome this, instead of focusing only on WT span or only
on $\Delta{S_{M}}$, the net heat extracted in a cooling cycle known as RCP is
calculated[57]. Two methods can be used to calculate RCP[16], 1st method -
Wood and Potter method[58] by which RCP is defined by

$RCP-1=\Delta{S_{M}}\Delta{T}$ (6)

where $\Delta{S_{M}}$ is the magnetic entropy change at the hot and cold ends
of the cycle (defined equal) and $\Delta{T}$ = $T_{hot}–T_{cold}$ and the 2nd
method - the area under the $\Delta{S_{M}}$ vs $T$ curves taking the
temperatures of the cold and hot sinks as lower and upper limit of
integration, respectively[1].In both methods, we take $\delta{T}_{FWHM}$ as
the working temperature range following the protocol as $\Delta{S_{M}}$ almost
vanishes beyond this temperature range[59, 60]. So,

$RCP-2=\int_{T_{cold}}^{T_{hot}}\Delta{S_{M}}(T,H_{max})\,dT[50]$ (7)

where $T_{hot}–T_{cold}$= $\delta{T}_{FWHM}$ and $H_{max}$ is the maximum
value of the applied field. It shows a large RCP-1 value, i.e., 361 Jkg-1 for
LCMO/STO at $H$ = 6 T. The estimated values of RCP-2 and RCP-1 near $T_{C}$ as
a function of $t$ and $H$ are plotted in Figs. 7(a) and 7(b), respectively.
Insets of Fig. 7 show the RCPs around $T_{P}$ for LCMO/STO.

It can be observed that for both the temperature regimes, the value of RCP
linearly increases with increasing $H$ and has the largest value at $t$ $\sim$
75 nm. In practical cooling applications, the material in the same
refrigeration cycle with higher RCP is preferred as it would confirm the
transport of a greater amount of heat in an ideal refrigeration cycle. To make
the applicability of our results, i.e., to make the LCMO thin film as a
magnetic refrigerant, $\mid$$\Delta{S_{max}}$$\mid$ and the values of RCPs,
i.e., RCP-1 and RCP-2 are determined in this present study and they are
compared in Table-I with several other magnetic refrigerants reported earlier
in literatures. It is clearly evident from the Table-I that the obtained peak
values of $\mid$$\Delta{S_{max}}$$\mid$ and the RCPs at a critical $t$ of 75
nm in our work are also comparable with other listed magnetic refrigerants.

##  IV Conclusion

We reported an anomalous strain effect in LCMO/STO and LCMO/LAO showing that
MCE as well as transition temperature for both the tensile and compressive
strain at first increases with $t$ due to in-plane strain relaxation showing
maximum values at $t$ $\sim$ 75 nm and then, it decreases. To explain the
anomaly of increasing MCE for both the compressive and tensile strain, two
well-known effects have been considered simultaneously- D-E interaction
between Mn3+ and Mn4+ ions and $3d_{x^{2}-y^{2}}$ orbital stabilization. We
have also shown the in-plane strain ($\epsilon_{XX}$) relaxation and decaying
of JT distortion ($\epsilon_{JT}$). $\epsilon_{XX}$ , $\epsilon_{JT}$ and the
product ($\epsilon_{XX}\times\epsilon_{JT}$) for LCMO/STO decay more rapidly
than those of LCMO/LAO, respectively. These decay are exponential unlike $t$
dependence of $\Delta{S_{max}}$ which is the Lorentz function with the maximum
value at $t$ $\sim$ 66±5nm for both LCMO/LAO and LCMO/STO. Here, $t$ can be
optimized for larger MCE using this Lorentz function. Moreover, the variation
in transition temperatures with respect to $t$ and substrates makes the LCMO
thin films flexible for using at different operating temperatures.

## Acknowledgement

WA acknowledges DST-INSPIRE. MB acknowledges institute funding from IISER
Thiruvananthapuram.

## References

  * [1] KA Gschneidner Jr, VK Pecharsky, AO Pecharsky, and CB Zimm.  In Materials science forum, volume 315, pages 69–76. Trans Tech Publ, 1999.
  * [2] YiXu Wang, Hu Zhang, EnKe Liu, XiChun Zhong, Kun Tao, MeiLing Wu, ChengFen Xing, YaNing Xiao, Jian Liu, and Yi Long.  Advanced Electronic Materials, 4(5):1700636, 2018.
  * [3] Lingwei Li and Mi Yan.  Journal of Alloys and Compounds, 823:153810, 2020.
  * [4] Tino Gottschall, Konstantin P Skokov, Maximilian Fries, Andreas Taubel, Iliya Radulov, Franziska Scheibel, Dimitri Benke, Stefan Riegg, and Oliver Gutfleisch.  Advanced Energy Materials, 9(34):1901322, 2019.
  * [5] Andrej Kitanovski.  Advanced Energy Materials, 10(10):1903741, 2020.
  * [6] X Moya, Sohini Kar-Narayan, and Neil David Mathur.  Nature materials, 13(5):439–450, 2014.
  * [7] Xavier Moya, Emmanuel Defay, Volker Heine, and Neil D Mathur.  Nature physics, 11(3):202–205, 2015.
  * [8] X Moya and ND Mathur.  Science, 370(6518):797–803, 2020.
  * [9] V. Franco, J.S. Blázquez, J.J. Ipus, J.Y. Law, L.M. Moreno-Ramírez, and A. Conde.  Progress in Materials Science, 93:112–232, 2018.
  * [10] O Tegus, E Brück, KHJ Buschow, and FR De Boer.  Nature, 415(6868):150–152, 2002.
  * [11] Vitalij K Pecharsky and Karl A Gschneidner Jr.  Physical review letters, 78(23):4494, 1997.
  * [12] Vitalij K Pecharsky and Karl A Gschneidner Jr.  Applied Physics Letters, 70(24):3299–3301, 1997.
  * [13] Hirofumi Wada and Y Tanabe.  Applied physics letters, 79(20):3302–3304, 2001.
  * [14] Julia Lyubina, Ullrich Hannemann, Lesley F Cohen, and Mary P Ryan.  Advanced Energy Materials, 2(11):1323–1327, 2012.
  * [15] Julia Lyubina, Rudolf Schäfer, Norbert Martin, Ludwig Schultz, and Oliver Gutfleisch.  Advanced Materials, 22(33):3735–3739, 2010.
  * [16] Virgil Provenzano, Alexander J Shapiro, and Robert D Shull.  Nature, 429(6994):853–857, 2004.
  * [17] Vladimir V Khovaylo, Valeria V Rodionova, Sergey N Shevyrtalov, and Val Novosad.  physica status solidi (b), 251(10):2104–2113, 2014.
  * [18] João H Belo, Ana L Pires, João P Araújo, and André M Pereira.  Journal of Materials Research, 34(1):134–157, 2019.
  * [19] H. Röder, Jun Zang, and A. R. Bishop.  Phys. Rev. Lett., 76:1356–1359, Feb 1996.
  * [20] S Jin, M McCormack, TH Tiefel, and R Ramesh.  Journal of Applied Physics, 76(10):6929–6933, 1994.
  * [21] Eun-Mi Choi, Tuhin Maity, Ahmed Kursumovic, Ping Lu, Zenxhing Bi, Shukai Yu, Yoonsang Park, Bonan Zhu, Rui Wu, Venkatraman Gopalan, et al.  Nature communications, 11(1):1–9, 2020.
  * [22] M Zarifi, P Kameli, MH Ehsani, H Ahmadvand, and H Salamati.  Journal of Magnetism and Magnetic Materials, 420:33–38, 2016.
  * [23] A Baena, L Brey, and MJ Calderón.  Physical Review B, 83(6):064424, 2011.
  * [24] Anamitra Mukherjee, William S Cole, Patrick Woodward, Mohit Randeria, and Nandini Trivedi.  Physical review letters, 110(15):157201, 2013.
  * [25] SK Giri, JL MacManus-Driscoll, W Li, R Wu, TK Nath, and TS Maity.  Journal of Physics D: Applied Physics, 52(16):165302, 2019.
  * [26] X Moya, LE Hueso, F Maccherozzi, AI Tovstolytkin, DI Podyalovskii, C Ducati, LC Phillips, M Ghidini, O Hovorka, A Berger, et al.  Nature materials, 12(1):52–58, 2013.
  * [27] Manh-Huong Phan and Seong-Cho Yu.  Journal of Magnetism and Magnetic Materials, 308(2):325–340, 2007\.
  * [28] JS Amaral, MS Reis, VS Amaral, TM Mendonca, JP Araujo, MA Sa, PB Tavares, and JM Vieira.  Journal of magnetism and magnetic materials, 290:686–689, 2005\.
  * [29] ZM Wang, G Ni, QY Xu, H Sang, and YW Du.  Journal of Applied Physics, 90(11):5689–5691, 2001.
  * [30] A Antonakos, E Liarokapis, GH Aydogdu, and H-U Habermeier.  Materials Science and Engineering: B, 144(1-3):83–88, 2007.
  * [31] GH Aydogdu, Y Kuru, and H-U Habermeier.  Journal of crystal growth, 310(21):4521–4524, 2008.
  * [32] Jun Zhang, Hidekazu Tanaka, Teruo Kanki, and Tomoji Kawai.  Surface and Interface Analysis: An International Journal devoted to the development and application of techniques for the analysis of surfaces, interfaces and thin films, 32(1):62–65, 2001.
  * [33] FS Razavi, G Gross, H-U Habermeier, O Lebedev, S Amelinckx, G Van Tendeloo, and A Vigliante.  Applied Physics Letters, 76(2):155–157, 2000.
  * [34] GQ Gong, A Gupta, Gang Xiao, Ph Lecoeur, and TR McGuire.  Physical Review B, 54(6):R3742, 1996.
  * [35] W Prellier, AM Haghiri-Gosnet, B Mercey, Ph Lecoeur, M Hervieu, Ch Simon, and B Raveau.  Applied Physics Letters, 77(7):1023–1025, 2000.
  * [36] Teruo Kanki, Hidekazu Tanaka, and Tomoji Kawai.  Physical Review B, 64(22):224418, 2001.
  * [37] Jun Zhang, Hidekazu Tanaka, Teruo Kanki, Jae-Hyoung Choi, and Tomoji Kawai.  Physical Review B, 64(18):184404, 2001.
  * [38] Anurag Chaturvedi, Stevce Stefanoski, Manh-Huong Phan, George S Nolas, and Hariharan Srikanth.  Applied Physics Letters, 99(16):162513, 2011.
  * [39] Niraj K Singh, S Agarwal, KG Suresh, R Nirmala, AK Nigam, and SK Malik.  Physical Review B, 72(1):014452, 2005.
  * [40] XX Zhang, J Tejada, Y Xin, GF Sun, KW Wong, and X Bohigas.  Applied Physics Letters, 69(23):3596–3598, 1996.
  * [41] RA Rao, D Lavric, TK Nath, CB Eom, L Wu, and F Tsui.  Journal of applied physics, 85(8):4794–4796, 1999.
  * [42] Tran Dang Thanh, Nguyen Thi Dung, Nguyen Van Dang, Le Viet Bau, Hong-Guang Piao, The Long Phan, Pham Duc Huyen Yen, Kieu Xuan Hau, Dong-Hyun Kim, and Seong-Cho Yu.  AIP Advances, 8(5):056419, 2018.
  * [43] Yonglai Fu.  Applied Physics Letters, 77(1):118–120, 2000.
  * [44] Renjun Zhang and Roy F Willis.  Physical review letters, 86(12):2665, 2001.
  * [45] Emad Almahmoud, Igor Kornev, and L Bellaiche.  Physical review B, 81(6):064105, 2010.
  * [46] X Renshaw Wang, Chiang J Li, WM Lü, TR Paudel, DP Leusink, M Hoek, N Poccia, A Vailionis, T Venkatesan, JMD Coey, et al.  Science, 349(6249):716–719, 2015.
  * [47] JMD Coey, M Viret, and S Von Molnar.  Advances in physics, 48(2):167–293, 1999.
  * [48] John B Goodenough.  Physical Review, 100(2):564, 1955.
  * [49] PA Cox.  In Physics and Chemistry of Low-Dimensional Inorganic Conductors, pages 255–270. Springer, 1996.
  * [50] JS Amaral and VS Amaral.  Journal of magnetism and magnetic materials, 322(9-12):1552–1557, 2010.
  * [51] Luis Andrés Burrola Gándara, Lizeth Vázquez Zubiate, Diana M Carrillo Flores, José T Elizalde Galindo, Carlos Ornelas, and Manuel Ramos.  Nanomaterials, 10(3):435, 2020.
  * [52] Alexandra O Pecharsky, Yu Mozharivskyj, KW Dennis, Karl A Gschneidner Jr, R William McCallum, Gordon J Miller, and Vitalij K Pecharsky.  Physical Review B, 68(13):134452, 2003.
  * [53] QY Dong, J Chen, J Shen, JR Sun, and BG Shen.  Applied Physics Letters, 99(13):132504, 2011.
  * [54] Zhao-Jun Mo, Jun Shen, Li-Qin Yan, Cheng-Chun Tang, Jing Lin, Jian-Feng Wu, Ji-Rong Sun, Li-Chen Wang, Xin-Qi Zheng, and Bao-Gen Shen.  Applied Physics Letters, 103(5):052409, 2013.
  * [55] R Rawat and I Das.  Journal of Physics: Condensed Matter, 13(3):L57, 2001.
  * [56] H Zhang, BG Shen, ZY Xu, J Shen, FX Hu, JR Sun, and Y Long.  Applied Physics Letters, 102(9):092401, 2013.
  * [57] Varun Chaudhary and Raju V Ramanujan.  IEEE Magnetics Letters, 6:1–4, 2015.
  * [58] ME Wood and WH Potter.  Cryogenics, 25(12):667–683, 1985.
  * [59] VI Zverev, AM Tishin, and MD Kuz’Min.  Journal of Applied Physics, 107(4):043907, 2010.
  * [60] Rafael Caballero-Flores, Victorino Franco, Alejandro Conde, KE Knipling, and MA Willard.  Applied Physics Letters, 98(10):102505, 2011.


# Single-Collision Statistics Reveal a Global Mechanism Driven by Sample
History for Contact Electrification in Granular Media

Galien Grosjean<EMAIL_ADDRESS>Scott Waitukaitis  Institute
of Science and Technology Austria (ISTA), Lab Building West, Am Campus 1, 3400
Klosterneuburg, Austria

###### Abstract

Models for same-material contact electrification in granular media often rely
on a local charge-driving parameter whose spatial variations lead to a
stochastic origin for charge exchange. Measuring the charge transfer from
individual granular spheres after contacts with substrates of the same
material, we find instead a ‘global’ charging behavior, coherent over the
sample’s whole surface. Cleaning and baking samples fully resets charging
magnitude and direction, which indicates the underlying global parameter is
not intrinsic to the material, but acquired from its history. Charging
behavior is randomly and irreversibly affected by changes in relative
humidity, hinting at a mechanism where adsorbates, in particular water, are
fundamental to the charge-transfer process.

††preprint: SingleContact/0.0

Contact electrification (CE), the transfer of electrical charge when objects
touch, plays a crucial role in granular media [1]. In nature, ice crystals in
thunder clouds or ash particles in volcanic plumes collide and charge to help
create spectacular displays of lightning [2, 3]. In industrial settings, e.g.
fluidized beds [4] or pharmaceutical plants [5], CE adversely affects adhesion
and flow, but can also be harnessed for filtration [6]. In grain silos, sparks
from charged grains can ignite deadly explosions [7]. Charged dust is
important for space exploration, as landers and rovers must be engineered to
withstand its accumulation [8]. Further away still, charged grains are
suspected to play an essential role in rocky planet formation, speeding up the
process sufficiently to allow Earth-like planets to exist [9, 10, 11].

Regarding what causes CE, in particular for insulators where the effect is
strongest, there is no consensus on the mechanism or the species transferred
[1]. With different materials, it is widely assumed that charge transfer is
driven by a material parameter [12, 13, 14]. This model is ‘global’ in that
charge-transfer does not vary with the location of the contact. In granular
media, charging occurs between grains of the same material, seemingly
precluding a global mechanism. Hypotheses to overcome this historically resort
to a ‘local’ picture for charge exchange, i.e. where the charge-driving
parameter varies over the surface [15, 16, 17, 18]. This parameter would
average out over large scales to render grains identical globally, but
nonetheless change sufficiently over the scale of contacts to permit transfer.
Prominent recently are ‘patch models’, where surfaces are thought to consist
of nanoscale donor/acceptor regions and charging arises stochastically from
exchange between these [19, 20, 21, 22]. Relevant to any mechanism is the
omnipresent influence of adsorbed surface water. For global models, water is
generally seen as providing a conductive path that amplifies some other
underlying charge-driving mechanisms [23]. For local models, ‘islands’ of
adsorbed water have been implicated as the actors that define patches [24, 22,
25, 21].

While a local model might seem necessary to explain same-material CE in
granular media, we are not aware of any experiments that directly demonstrate
its occurrence. In principle, all that is needed are samples that are as
identical as possible, a careful preparation protocol to keep them so, and a
statistically significant number of charge-exchange measurements at random
surface locations. Local models predict this should lead to charge-exchange
distributions with zero average, while for any global mechanism it would be
non-zero. Yet this is not an easy task. It is straightforward to probe bulk
granular CE with Faraday cups, but this yields no information on individual
grains [26]. Some experiments address single grains, but are not precise
enough to measure charge exchange [27, 28, 9, 29]. A handful measure charge
exchange, but with different materials [30, 31, 32, 33] or with centimeter-
scale objects [34, 21, 35, 24] to enhance magnitude. We are not aware of any
experiments with sufficient resolution and flexibility to gather comprehensive
statistics of same-material CE at the scale of a single grain.

Figure 1:  Setup and protocol. (a) The setup consists of a Langevin transducer
above the target substrate and electrode. The spherical particle levitates in
the node of the acoustic standing wave. (b) Charge-exchanging contacts are
initiated by briefly interrupting the acoustic field, causing the sphere to
bounce exactly once on the surface before we ‘catch’ it. (c) To measure
charge, we frequency sweep a spatially uniform applied electric field through
the sphere’s resonance and track its position with a high-speed camera.
Fitting the acceleration to Eq. (1) yields the charge. (d) Trajectory of a
charged sphere in response to a harmonic E-field with the discharger off
($t<0$), and then on ($t>0$); fitting for $t>0$ to an exponential gives a time
constant of $\sim$8 s. (e) The three tasks shown in (b)-(d), contact (i),
sweep (ii) and discharge (iii), can be combined in different ways depending on
the measurement mode. For further details on the setup and videos
demonstrating contacts, charge measurement, and discharge, see the
Supplemental Material [37].

In this Letter, we dissect the global vs. local nature of same-material
granular CE by further pioneering charge measurement via acoustic levitation
[38, 25, 36, 39], which enables exquisite charge resolution and automated
contacts without physical handling. Observing the charge evolution over
sequential contacts and the charge-exchange distributions of initial contacts,
we demonstrate that the symmetry-breaking parameter is global. We find that
this parameter is not inherent to individual samples, but acquired during
their history—merely recleaning and rebaking samples can flip the charging
direction. Considering the ubiquitous influence of adsorbed water, we vary
relative humidity (RH), expecting to uniformly affect charging. Instead, we
find random shifts to the exchange: the same change in RH can cause charging
to either increase or decrease, and irreversibly so. Our results suggest that
same-material granular CE is determined by and extremely sensitive to
environmental history, pointing to adsorbates—and in particular water—as the
charge-driving agents.

Our samples are research-grade fused silica (SiO_2) spheres and substrates,
carefully selected to be as pure and identical as possible. Both are made from
a single traceable source material, Heraeus Spectrosil® 2000, which limits
bulk impurities to parts per billion. The spheres (Sandoz Fils SA., grade 25)
have diameters $D=500\pm$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The
substrates (UQG Optics Ltd WFS-252) are disks with
$25\text{\,}\mathrm{m}\mathrm{m}$ diameter and
$6\text{\,}\mathrm{m}\mathrm{m}$ thickness. AFM topography measurements on
spheres/substrates reveal roughness on the order of
$4\text{\,}\mathrm{n}\mathrm{m}$ and $1\text{\,}\mathrm{n}\mathrm{m}$,
respectively. Spheres and substrates are subjected to a rigorous cleaning
protocol before experiments: first sonicating for 30 minutes each in acetone
($>\,$99.5 %), methanol ($>\,$99.9 %), and Milli-Q® water, and then baking
overnight at 300∘C. A particular sphere/substrate pair always undergoes this
protocol jointly, i.e. together in the same beakers with the same solvent at
each step. Immediately after baking, samples enter a temperature
($\pm$$2^{\circ}$C) and RH ($\pm$1%) regulated environment, also fed by
Milli-Q® water.

The experimental apparatus is illustrated in Fig. 1(a), and builds upon the
acoustic levitation technique introduced in Refs. [25, 36]. We levitate a
sphere using an ultrasonic standing wave created by a resonant Langevin
transducer suspended above our target substrate. To initiate a contact, we
briefly interrupt the acoustic field, with the duration ($\sim$
$25\text{\,}\mathrm{ms}$) tuned so that the sphere falls and bounces exactly
once before it is recaught in the trap; see Fig. 1(b) and Supplemental
Material, Video 1 [37]. To measure charge, a spatially uniform electric field
is AC-swept to pass through the natural frequency of the sphere in the
acoustic trap ($f_{\mathrm{t}rap}\approx$50\text{\,}\mathrm{Hz}$$). We record
the sphere’s motion with a high-speed camera (Phantom VEO 640L) and use
particle tracking to obtain its vertical position as a function of time,
$y(t)$. Newton’s second law projected on the vertical direction can be written

$\ddot{y}=-g-a\sin 2ky-2\beta_{0}\dot{y}-2\beta_{1}|\dot{y}|\dot{y}+QE(t)/m.$
(1)

The electric field, $E(t)$, the acoustic wavenumber, $k$, and the sphere mass,
$m$, are known, and the first and second derivatives of $y$ can be numerically
calculated. The unknowns are the acoustic amplitude, $a$, the linear and
quadratic damping coefficients, $\beta_{0}$ and $\beta_{1}$, arising from air
drag, and charge, $Q$, which we obtain by fitting. Typical acceleration data
and a fit are shown in Fig. 1(c) (see also Supplemental Material, Video 1
[37]).

Several advances beyond previous works [11, 36] are required for our purposes.
First, we must be able to change the location of contact on both the sphere
and the substrate. With the sphere, symmetry prevents any preferred
orientation, causing it to rotate while levitating such that contacts occur at
a random locations. This rotation is visible in Supplemental Material, Video
2, and in the Supplemental Material we estimate the frequency to be of the
order of 100 Hz [37]. For the substrate, we incorporate a piezo-driven XYZ-
stage to laterally displace it between contacts [Fig. 1(a)]. We move it in a
square spiral with steps of $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$, just
larger than the estimated contact diameter
($d\approx$19.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$$). Second, to carry out
experiments with the same initial (zero charge) conditions, we introduce a
discharge mechanism. We place a photoionizer (Hamamatsu L12645) in the chamber
directed away from the sphere/substrate, which enhances the conductivity of
the surrounding air to cause rapid discharge. Figure 1(d) shows how the steady
trajectory of a sphere shaken harmonically at constant amplitude quickly
decays after the device is turned on. Fitting to an exponential yields a time
constant of $\sim$$8\text{\,}\mathrm{s}$ (see Supplemental Material, Video 1
[37]).

Figure 2:  Charge evolution. (a) The charge, $Q$, of a sphere is measured
after $n$ contacts with the substrate and with no discharge between, i.e. in
‘charge evolution mode,’ for a total of 1000 bounces. The steady charging
rate, $Q^{\prime}$, indicates a global difference between this
sphere/substrate pair. (b) Distribution of $Q^{\prime}$ measured for 25
sphere/subtrate pairs, which is centered on zero; this indicates that there is
no systematic difference between spheres and substrates.

The capacity to perform (i) contact, (ii) charge measurement, and (iii)
discharge gives us access to otherwise unattainable modes of experimentation.
The most direct is ‘charge evolution mode’, e.g. in Fig. 2(a), where we
perform 1000 cycles of contact then charge measurement
(i$\,\rightarrow\,$ii$\,\rightarrow\,$repeat). As is clear, the sphere’s
charge in this instance marches steadily upward at a constant rate per
collision, $dQ/dn\equiv Q^{\prime}$. In the standard patch model, net charge
is exchanged in a single collision due to fluctuations in the number of charge
donor/acceptor pairs facing each other at the contact location, but over many
locations the average is predicted to approach zero [19, 22]. Hence, the data
in Fig. 2(a) already indicate a global mechanism driving exchange in this
sphere/substrate pair. Similar trends were seen before, but this implication
was missed [25]. If all spheres charged with the same sign against all
substrates, one could argue that they differ in an intrinsic way, but Fig.
2(b) shows this is not the case. Calculating the distribution of the rates
$Q^{\prime}$ for an ensemble of sphere/substrate pairs shows that they are
centered around zero—each sphere is globally different from each substrate,
but the average difference is zero.

Figure 3:  Charge distributions. (a) In ‘charge distribuiton mode,’ the sphere
charge after the first contact, $Q_{1}$, is repeatedly measured for a given
sphere/substrate pair by discharging the system between collisions. The median
of the distribution can be either negative (pair #1), positive (pair #2) or
close to zero (pair #3). With pair #3, the distribution is composed of two
sets of measurements, taken 56 hours apart (colored dark gray and light
green), showing no significant drift of the distribution with time. Gaussian
fits are shown only as a guide to the eye. (b) Between contacts, the substrate
is moved along a square spiral. Plotting $Q_{1}$ vs. the contact location on
the substrate, no clear trend can be identified with either space or time.

Beyond charge evolution, we can also measure the distribution of charge
exchange, $Q_{1}$, for the initial (fully discharged) contact of a
sphere/substrate pair. In this ‘charge distribution mode’, we cycle over
discharging, performing a contact, and then measuring charge
(iii$\,\rightarrow\,$i$\,\rightarrow\,$ii$\,\rightarrow\,$repeat). As Fig. 2
shows, the typical magnitude of charge exchange is $\sim$$\,10^{5}\,e$, and as
we explain regarding ‘charge uncertainty mode’ in the Supplemental Material
[37] our measurement uncertainty is $\lesssim$$\,10^{3}\,e$—two orders of
magnitude lower. With this level of resolution, charge-exchange distributions,
even with our small samples and identical materials, are easily resolved.
Typical results for three sphere/substrate pairs are shown in Fig. 3(a). As is
immediately observed, the distributions are not constrained to be centered
around zero, as a local model would require. The distribution for sample pair
#1 is predominantly negative, pair #2 positive, and pair #3 close to zero.
Though we do not delve into their shapes [33], distributions are often
approximately Gaussian, with widths of around $10^{5}\,e$. We confirm that
distributions are stable over time by repeating the same measurement several
days apart. For instance, the distribution of pair #3 is comprised of two sets
of measurements, taken 56 hours apart and shown in different colors. Neither
the median value nor the standard deviation display any discernible change.

What we learned from charge evolution [Fig. 2(a)] is thus reconfirmed by the
charge distributions: the charging between a particular sphere/substrate pair
is driven by a global, not local, parameter. To make this point even stronger,
Fig. 3(b) shows $Q_{1}$ as a function of the contact location on the
substrate, following the square spiral from the center outward. Here the
global charging behavior becomes visually apparent—a substrate that charges
positive/negative does so over large regions of its surface. Positive/negative
regions are not spatially correlated, and no drift over time occurs. The fact
that charge distributions can be strictly positive or negative also suggests
that the first contact does not play any special role in breaking symmetry.
Electric fields could certainly influence charge exchange [25, 3], however
charging direction is preserved after discharge, suggesting that polarization
from the particle’s own field is not what determines the sign during
subsequent collisions [40].

Figure 4:  Resetting charging via recleaning and rebaking. If we reclean and
rebake a sphere/substrate pair, their charging behavior is ‘reset.’ Every
time, the median and width of the distribution is randomly altered; even the
sign can be flipped. This indicates that the global charge-driving parameter
is acquired, not intrinsic.

We now perform experiments to uncover the nature of the charge-driving
parameter, starting with the question: is it intrinsic to a given
sphere/substrate pair, or acquired during their history? To answer this, we
measure a $Q_{1}$ distribution for a particular pair, and then reclean and
rebake them together and measure the distribution again. As can be seen in
Fig. 4, repreparing samples changes the median value and width of the original
distribution. The difference is such that it would be impossible to tell
whether the same pair has been used or a different one—the conditions are
entirely reset. Most notably, as in Fig. 4(b), the sign of charging can be
flipped. We conclude that the global parameter driving charging is an acquired
one, with the most likely candidate being surface adsorbates. Considering that
baking at a few hundred degrees removes most (though not all [41]) organic
adsorbates, the implicated species are likely acquired afterward when samples
enter the experimental chamber, where controlled RH is maintained. We thus
consider the possibility, as have many others recently, that adsorbed surface
water is driving the charging [11, 42, 43, 44, 24, 21].

To find out how, we measure $Q_{1}$ distributions for sample pairs
before/after they jointly experience changes of RH. Several results in the
literature, using both different and same materials, have indicated that
ensemble averages for charge-exchange magnitude reach a maximum at $\sim$ 30 %
RH [5, 21, 45]. One might expect, then, that varying RH would uniformly affect
CE for our sphere/substrate pairs, increasing the magnitude when moving closer
to the optimum and decreasing it when moving away. This is not what is
observed. Increasing RH from 15 to 30 % causes random shifts to charging
magnitude—about half of the time it is increased [Fig. 5(a)], and half of the
time it is decreased [Fig. 5(b)]. The shift can be large—often comparable in
magnitude to the distribution widths. As shown in Fig. 5(b), lowering the RH
back to 15 % does not undo the shift—the changes are not reversible. Shifts
are strongest between 15 % to 30 % RH, and increasing RH beyond 30 % has
relatively little effect. We observe that exposure to RH $\gg\,$30 % largely
diminishes any shifts thereafter.

Figure 5:  Charge shifts via humidity changes. Changing the RH of the
experimental chamber (from 15 % to 30 %) while a sphere/substrate pair are
present causes their charging distributions to shift. This shift can either
(a) increase or (b) decrease the charging magnitude. Such shifts are not
reversible, as shown in (b) where the RH was decreased back to 15 %. These
effects are similar to those observed after recleaning and rebaking, though no
clear sign flips were observed.

If adsorbed water were merely a conduit for some other underlying global
mechanism, then the addition/removal via RH should affect charging uniformly.
Instead, we observe a significant, random and irreversible alteration of the
charging behavior. The shifts observed due to RH are typically smaller than
those caused by recleaning and rebaking, however they bear qualitative
similarities. While no statistically significant sign flips were observed due
to RH, charging sometimes went from clearly positive/ negative to essentially
zero [Fig. 5(b)], or vice versa. If adsorbed water drives the charging
behavior, then we indeed expect the changes from RH to be weaker than those
from recleaning and especially rebaking, after which samples would be
presumably devoid of most water [41] and hence completely ‘reset’, consistent
with what we see.

On the other hand, our data also show that water is not merely an actor in a
local patch model. Some new type of ‘global patch model’ could explain our
data, provided that water-patch coverage can (1) be different from one sample
to the next at a fixed RH (to explain Figs. 2 & 3) and (2) evolve differently
for each sample with preparation and/or RH (to explain Figs. 4 & 5). However,
if this is the correct interpretation, it requires a new aspect of surface
water to be considered—adsorption hysteresis. While we found no discussion of
this phenomenon in the CE literature, it is well documented in other contexts
[46, 47, 48, 49, 50]. It seems to be particularly important when multiple
adsorbates compete on a surface. For example, when water coadsorbs on an SiO_2
surface with different alcohols, the water surface coverage need not have a
single value for a given RH, and evolves differently with RH depending on
history [49]. The magnitude of the effect can be such that, under identical
environmental conditions, two same-material surfaces with different histories
can have surface water coverages that differ by up to a monolayer. Significant
adsorption hysteresis is also known to occur in porous materials, including
porous glasses, gels and polymers [51, 52, 53]. Considering that the scale of
charge exchange in CE typically only requires about one in every $10^{4}$
surface atoms/molecules ($\sim$10-4 monolayers) to participate, and that water
in atmospheric conditions coadsorbs with a complex mixture of many other
molecules, effects from adsorption hysteresis cannot by any means be excluded.
Nonetheless, to clearly establish that water drives charge exchange would
require to correlate charging behavior with direct measurements of adsorbed
water. Such measurements would have to be precise enough to resolve minute
differences, potentially down to the submonolayer scale.

Having recently published theoretical work on same-material CE based on a
local, patch-driven framework [22], we embarked upon these experiments with
the expectation that signatures of a local model could be observed. However,
despite extreme care with regard to sample purity and preparation, we only
find evidence of a global mechanism—the tendency to charge positive/negative
does not average to zero from one contact location to the next, but is stable
over large length scales. Our data tells us the charge-driving parameter is
acquired during sample history; it can be reset by cleaning and baking, and
randomly and irreversibly shifted via RH. These observations are difficult to
reconcile with mechanisms based on intrinsic parameters, including: work
functions, dielectric constants, specific heats, Seebeck coefficients, surface
roughness, flexoelectric constants, piezoelectric constants, mechanochemistry,
etc. Polarization [3, 40] does not seem to cause the initial symmetry
breaking, though we cannot rule out this effect without additional studies of
contacts under applied electric fields. The most consistent mechanism we can
propose is that the global charge-driving parameter is related to adsorbates
acquired during a sample’s history, in particular water. Though many other
authors have proposed that water plays an important role, our data suggest a
new twist—namely that minute deviations in conditions during water adsorption
lead to global differences in water coverage, which drive charging. Such a
twist is not outside reason considering the unpredictability and
irreproducibility of CE generally [1, 54]. Even more so when one considers the
well-documented existence of adsorption hysteresis causing coverage
differences up to a full monolayer [46, 47, 48, 49, 50]. Further
investigations that attempt to correlate surface water coverage and CE
directly would be extremely valuable in testing this hypothesis.

We would like to thank Troy Shinbrot, Victor Lee and Daniele Foresti for
helpful discussions. This project has received funding from the European
Research Council Grant Agreement No. 949120 and from the the Marie Skłodowska-
Curie Grant Agreement No. 754411 under the European Union’s Horizon 2020
research and innovation program.

## References

  * Lacks and Shinbrot [2019] D. J. Lacks and T. Shinbrot, Nat. Rev. Chem. 3, 465 (2019).
  * Gilbert  _et al._ [1991] J. S. Gilbert, S. J. Lane, R. S. J. Sparks, and T. Koyaguchi, Nature (London) 349, 598 (1991).
  * Pähtz  _et al._ [2010] T. Pähtz, H. J. Herrmann, and T. Shinbrot, Nat. Phys. 6, 364 (2010).
  * Grosshans and Papalexandris [2016] H. Grosshans and M. V. Papalexandris, Powder Technol. 301, 1008 (2016).
  * Rescaglio  _et al._ [2019] A. Rescaglio, F. De Smet, L. Aerts, and G. Lumay, Part. Sci. Technol. 37, 1024 (2019).
  * Liu  _et al._ [2020] H. Liu, C. Cao, J. Huang, Z. Chen, G. Chen, and Y. Lai, Nanoscale 12, 437 (2020).
  * Abbasi and Abbasi [2007] T. Abbasi and S. A. Abbasi, J. Hazard. Mater. 140, 7 (2007).
  * Calle  _et al._ [2011] C. Calle, C. Buhler, M. Johansen, M. Hogue, and S. Snyder, Acta Astronaut. 69, 1082 (2011).
  * Steinpilz  _et al._ [2020] T. Steinpilz, K. Joeris, F. Jungmann, D. Wolf, L. Brendel, J. Teiser, T. Shinbrot, and G. Wurm, Nat. Phys. 16, 225 (2020).
  * Singh and Mazza [2018] C. Singh and M. G. Mazza, Phys. Rev. E 97, 022904 (2018).
  * Lee  _et al._ [2015] V. Lee, S. R. Waitukaitis, M. Z. Miskin, and H. M. Jaeger, Nat. Phys. 11, 733 (2015).
  * Matsusyama and Yamamoto [2006] T. Matsusyama and H. Yamamoto, Chem. Eng. Sci. 61, 2230 (2006).
  * Ireland [2010] P. M. Ireland, Powder Technol. 198, 189 (2010).
  * Grosshans and Papalexandris [2017] H. Grosshans and M. V. Papalexandris, Powder Technol. 305, 518 (2017).
  * Lowell and Truscott [1986] J. Lowell and W. Truscott, J. Phys. D 19, 1281 (1986).
  * Lacks  _et al._ [2008] D. J. Lacks, N. Duff, and S. K. Kumar, Phys. Rev. Lett. 100, 188305 (2008).
  * Kok and Lacks [2009] J. F. Kok and D. J. Lacks, Phys. Rev. E 79, 051304 (2009).
  * Forward  _et al._ [2009] K. M. Forward, D. J. Lacks, and R. M. Sankaran, Phys. Rev. Lett. 102, 028001 (2009).
  * Apodaca  _et al._ [2010] M. M. Apodaca, P. J. Wesson, K. J. M. Bishop, M. A. Ratner, and B. A. Grzybowski, Angew. Chem., Int. Ed. Engl. 49, 946 (2010).
  * Yu  _et al._ [2017] H. Yu, L. Mu, and L. Xie, J. Electrostat. 90, 113 (2017).
  * Xie  _et al._ [2016] L. Xie, N. Bao, Y. Jiang, and J. Zhou, AIP Adv. 6, 035117 (2016).
  * Grosjean  _et al._ [2020] G. Grosjean, S. Wald, J. C. Sobarzo, and S. Waitukaitis, Phys. Rev. Mater. 4, 082602(R) (2020).
  * Zheng  _et al._ [2014] X. Zheng, R. Zhang, and H. Huang, Sci. Rep. 4, 4399 (2014).
  * Zhang  _et al._ [2015] Y. Zhang, T. Pähtz, Y. Liu, X. Wang, R. Zhang, Y. Shen, R. Ji, and B. Cai, Phys. Rev. X 5, 011002 (2015).
  * Lee  _et al._ [2018] V. Lee, N. M. James, S. R. Waitukaitis, and H. M. Jaeger, Phys. Rev. Mater. 2, 035602 (2018).
  * LaMarche  _et al._ [2009] K. R. LaMarche, X. Liu, S. K. Shah, T. Shinbrot, and B. J. Glasser, Powder Technol. 195, 158 (2009).
  * Waitukaitis and Jaeger [2013] S. R. Waitukaitis and H. M. Jaeger, Rev. Sci. Instrum. 84, 025104 (2013).
  * Waitukaitis  _et al._ [2014] S. R. Waitukaitis, V. Lee, J. M. Pierson, S. L. Forman, and H. M. Jaeger, Phys. Rev. Lett. 112, 218001 (2014).
  * Carter and Hartzell [2020] D. Carter and C. Hartzell, J. Electrostat. 107, 103475 (2020).
  * Matsuyama and Yamamoto [1995] T. Matsuyama and H. Yamamoto, J. Phys. D 28, 2418 (1995).
  * Matsuyama  _et al._ [2003] T. Matsuyama, M. Ogu, H. Yamamoto, J. C. Marijnissen, and B. Scarlett, Powder Technol. 135, 14 (2003).
  * Watanabe  _et al._ [2006] H. Watanabe, A. Samimi, Y. L. Ding, M. Ghadiri, T. Matsuyama, and K. G. Pitt, Part. Part. Syst. Charact. 23, 133 (2006).
  * Haeberle  _et al._ [2018] J. Haeberle, A. Schella, M. Sperl, M. Schröter, and P. Born, Soft Matter 14, 4987 (2018).
  * Xie  _et al._ [2013] L. Xie, G. Li, N. Bao, and J. Zhou, J. Appl. Phys. 113, 184908 (2013).
  * Hu  _et al._ [2012] W. Hu, L. Xie, and X. Zheng, Appl. Phys. Lett. 101, 114107 (2012).
  * Kline  _et al._ [2020] A. G. Kline, M. X. Lim, and H. M. Jaeger, Rev. Sci. Instrum. 91, 023908 (2020).
  * [37] See Supplemental Material at https://arxiv.org/src/2211.02488v4/anc for the supplemental videos and further experimental details. .
  * Andrade  _et al._ [2018] M. A. B. Andrade, N. Pérez, and J. C. Adamowski, Braz. J. Phys. 48, 190 (2018).
  * Harper  _et al._ [2022] J. M. Harper, D. Harvey, T. Huang, J. McGrath, D. Meer, and J. C. Burton, PNAS Nexus 1, pgac220 (2022).
  * Yoshimatsu  _et al._ [2017] R. Yoshimatsu, N. a. M. Araújo, G. Wurm, H. J. Herrmann, and T. Shinbrot, Sci. Rep. 7, 39996 (2017).
  * Zhuravlev [2000] L. Zhuravlev, Colloids Surf. A 173, 1 (2000).
  * Burgo  _et al._ [2016] T. A. Burgo, F. Galembeck, and G. H. Pollack, J. Electrostat. 80, 30 (2016).
  * Gil and Lacks [2019] P. S. Gil and D. J. Lacks, Phys. Chem. Chem. Phys. 21, 13821 (2019).
  * Jungmann  _et al._ [2022] F. Jungmann, F. C. Onyeagusi, J. Teiser, and G. Wurm, J. Electrostat. 117, 103705 (2022).
  * Schella  _et al._ [2017] A. Schella, S. Herminghaus, and M. Schröter, Soft Matter 13, 394 (2017).
  * Sullivan  _et al._ [2007] P. D. Sullivan, B. R. Stone, Z. Hashisho, and M. J. Rood, Adsorption 13, 173 (2007).
  * Taqvi  _et al._ [1999] S. M. Taqvi, W. S. Appel, and M. D. LeVan, Ind. Eng. Chem. Res. 38, 240 (1999).
  * Rudisill  _et al._ [1992] E. N. Rudisill, J. J. Hacskaylo, and M. D. LeVan, Ind. Eng. Chem. Res. 31, 1122 (1992).
  * Barnette and Kim [2012] A. L. Barnette and S. H. Kim, Langmuir 28, 15529 (2012).
  * Qi and LeVan [2005] N. Qi and M. D. LeVan, Ind. Eng. Chem. Res. 44, 3733 (2005).
  * Kierlik  _et al._ [2001] E. Kierlik, P. A. Monson, M. L. Rosinberg, L. Sarkisov, and G. Tarjus, Phys. Rev. Lett. 87, 055701 (2001).
  * Kierlik  _et al._ [2002] E. Kierlik, P. A. Monson, M. L. Rosinberg, and G. Tarjus, J. Phys. Condens. Matter 14, 9295 (2002).
  * Chen  _et al._ [2018] M. Chen, B. Coasne, R. Guyer, D. Derome, and J. Carmeliet, Nat. Commun. 9, 3507 (2018).
  * Lacks [2012] D. J. Lacks, Angew. Chem., Int. Ed. Engl. 51, 6822 (2012).


# Complex scalar field in $\kappa$-Minkowski spacetime ††thanks: Presented at
POTOR 8

Andrea Bevilacqua<EMAIL_ADDRESS>National Centre for Nuclear
Research, Pasteura 7, 02-093 Warszawa, Poland

###### Abstract

It is often expected that one cannot treat spacetime as a continuous manifold
as the Planck scale is approached, because of to possible effects due to a
quantum theory of gravity. There have been several proposals to model such a
deviation from the classical behaviour, one of which is noncommutativity of
spacetime coordinates. In this context, the non-commutativity scale is seen as
an observer-independent length scale. Of course, such a scale impose a
modification of ordinary relativistic symmetries, which now need to be
deformed to accommodate this fundamental scale. The $\kappa$-Poincaré algebra
is an example of this deformation. In what follows I will briefly describe a
construction of a $\kappa$-deformed complex scalar field theory, while at the
same time shedding light on the behaviour of discrete and continuous
symmetries in this formalism. This in turn will open the way to the study of
the application of this formalism to actual physical processes. I will then
conclude with some comments and prospects for the future.

##  I From non-commutative spacetime to the action

We will present a brief preview of the works in Refs. [1], [2]. For a more
detailed introduction to the framework and the formalism, see for example Ref.
[3]. The starting point is $k$-Minkowski spacetime, whose coordinates satisfy
the $\mathfrak{an}(3)$ Lie algebra defined as
$[\hat{x}^{0},\hat{\mathbf{x}}^{i}]=i\hat{\mathbf{x}}^{i}/\kappa$. Notice that
one recovers canonical Minkowski spacetime in the formal limit
$\kappa\rightarrow\infty$. To have a more physical understanding of the above
commutator, notice that $1/\kappa$ has dimensions of length, and it is
sometimes identified with the Plank length. In order to build fields in this
spacetime, we need to define plane waves first. To do so, one can proceed as
follows. First pick an explicit representation of the $\mathfrak{an}(3)$ Lie
algebra in terms of matrices (it turns out that the lowest dimensional one has
dimension 5). Then, one can define a plane wave as an element $\hat{e}_{k}$ of
the relative $AN(3)$ Lie group, i.e. for example111A different choice on the
ordering results in a change of coordinates in momentum space.
$\hat{e}_{k}=\exp(i\mathbf{k}_{i}\hat{\mathbf{x}}^{i})\exp(ik_{0}\hat{x}^{0})$.
Notice that, since the elements of the matrices $\hat{x}^{\mu}$ are
dimensionful, so are the parameters $k_{\mu}$. Written explicitly, one has

$\hat{e}_{k}=\left(\begin{array}[]{ccc}\frac{\bar{p}_{4}}{\kappa}&\;\frac{\mathbf{k}}{\kappa}&\;\frac{p_{0}}{\kappa}\\\
&&\\\ \frac{\mathbf{p}}{\kappa}&\mathbf{1}&\frac{\mathbf{p}}{\kappa}\\\ &&\\\
\frac{\bar{p}_{0}}{\kappa}&-\frac{\mathbf{k}}{\kappa}&\frac{p_{4}}{\kappa}\end{array}\right)$
(1)

where222The explicit expression of $\bar{p}_{4}$ and $\bar{p}_{0}$ in terms of
$k_{\mu}$ is not relevant for the present discussion.

$\displaystyle
p_{0}=\kappa\sinh\frac{k_{0}}{\kappa}+\frac{\mathbf{k}^{2}}{2\kappa}e^{k_{0}/\kappa}\quad\mathbf{p}_{i}=k_{i}e^{k_{0}/\kappa}\quad
p_{4}=\kappa\cosh\frac{k_{0}}{\kappa}-\frac{\mathbf{k}^{2}}{2\kappa}e^{k_{0}/\kappa}.$
(2)

Both $k_{\mu}$ and $p_{A}(k)$ ($A=0,1,2,3,4$) are two different coordinates of
momentum space. Notice that the $p_{A}$ are not independent, since
$-p_{0}+\mathbf{p}^{2}+p_{4}^{2}=\kappa^{2}$ (notice also that
$p_{+}=p_{0}+p_{4}>0$). It is therefore clear that momentum space is curved.
This is reflected in a deformed sum of momenta, which is defined through the
group property $\hat{e}_{k}\hat{e}_{l}=:\hat{e}_{k\oplus l}$. At the same
time, inverse momenta are defined in terms of group inverses
$\hat{e}_{k}^{-1}=:\hat{e}_{S(k)}$, and $S(k)$ is called the antipode of $k$.
In order to simplify the treatment of integrals and derivatives, we can use
a333There are several equivalent ways in which a suitable Weyl map can be
chosen, see Ref. [1] for more details. Weyl map $\mathcal{W}$ to send group
elements $\hat{e}_{k}$ into canonical plane waves
$e_{p}:=\exp(ip_{\mu}(k)x^{\mu})$. The group law is preserved thanks to the
$\star$ product defined by $\mathcal{W}(\hat{e}_{k\oplus l})=e_{p(k)\oplus
q(l)}=:e_{p(k)}\star e_{q(l)}$. In general, the $\star$ product is not
commutative. Because of this, we choose the following action for a
$\kappa$-deformed complex scalar field.

$S=\frac{1}{2}\int_{\mathbb{R}^{4}}d^{4}x[(\partial^{\mu}\phi)^{\dagger}\star(\partial_{\mu}\phi)+(\partial^{\mu}\phi)\star(\partial_{\mu}\phi)^{\dagger}-m^{2}(\phi^{\dagger}\star\phi+\phi\star\phi^{\dagger})].$
(3)

To obtain the equations of motion one usually integrates by parts, but in this
deformed context derivatives do not satisfy the Leibniz rule. In fact, if they
did, one could get the following contradiction.

$\displaystyle i(p\oplus q)_{\mu}e_{p\oplus q}$
$\displaystyle=\partial_{\mu}(e_{p}\star e_{q})=(\partial_{\mu}e_{p})\star
e_{q}+e_{p}\star\partial_{\mu}e_{q}=i(p+q)e_{p\oplus q}.$ (4)

Hence one has to use the appropriate deformations for the Leibniz rules. After
some computations, one can verify that the equations of motion satisfied by
the field are the canonical Klein-Gordon equations, and the field which
satisfies them can be written as

$\phi(x)=\int\frac{d^{3}p}{\sqrt{2\omega_{p}}}\xi(p)a_{\mathbf{p}}e^{-i(\omega_{p}t-\mathbf{p}\mathbf{x})}+\int\frac{d^{3}p^{*}}{\sqrt{2|\omega_{p}^{*}|}}\xi(p)b^{\dagger}_{\mathbf{p}^{*}}e^{i(S(\omega_{p})t-S(\mathbf{p})\mathbf{x})}.$
(5)

The action of the discrete transformations $C,P,T$ can be defined in the usual
way in the deformed context, i.e.
$T\phi(t,\mathbf{x})T^{-1}=\phi(-t,\mathbf{x})$,
$P\phi(t,\mathbf{x})P^{-1}=\phi(t,-\mathbf{x})$, and
$C\phi(t,\mathbf{x})C^{-1}=\phi^{\dagger}(t,\mathbf{x})$. Notice that this is
due to the presence of the antipode in the on-shell field. Furthermore, the
action is manifestly invariant under both CPT and $\kappa$-deformed Lorentz
transformations.

##  II Charges and features of the model

We now need to compute the charges for our model. There are two main ways to
do so. The first is by direct computation using the Noether theorem. However,
the fact that derivatives do not follow the Leibniz rule makes this job a
prohibitively difficult one, except for the case of translation charges. The
second way is to use the covariant phase-space formalism. Although very
direct, this second method needs to be carefully defined in the deformed
context. We chose a hybrid approach, namely we derived the translation charges
from the Noether theorem, and then we built a covariant phase space approach
which was able to reproduce the translation charges, allowing then to compute
the remaining charges. Given a symplectic form $\Omega$, and a symmetry in
spacetime described by the vector field $\xi$, the charge $Q_{\xi}$ can be
defined by $-\delta_{\xi}\lrcorner\,\Omega=\delta Q_{\xi}$. In this context
$\delta_{\xi}$ is a vector field in phase space describing the variation
$\delta_{\xi}A$ of any physical quantity $A$ in phase space under the action
of the symmetry $\xi$ in spacetime, $\delta$ is the exterior derivative in
phase space, and $\lrcorner$ represents a contraction. The translation charges
computed directly from the Noether theorem are the following444There is also a
fifth charge $\mathcal{P}_{4}$, see [1] for further details.:

$\displaystyle\mathcal{P}_{\mu}=\int
d^{3}p\,\alpha(p)[-S(p)_{\mu}a_{\mathbf{p}}^{\dagger}a_{\mathbf{p}}+p_{\mu}b_{\mathbf{p}^{*}}^{\dagger}b_{\mathbf{p}^{*}}].$
(6)

The quantity $\alpha(p)$ is a function of momenta whose explicit expression
does not concern us in this context. To reproduce Eq. (6) in the covariant
phase space formalism, we need to assume the following deformation of the
canonical contraction rule between vector fields and forms555Recall that the
canonical relation is $\delta_{v}\lrcorner\,(A\wedge
B)=(\delta_{v}A)B-A(\delta_{v}B)$.:

$\displaystyle\delta_{v}\lrcorner\,(A\wedge
B)=(\delta_{v}A)B+A(S(\delta_{v})B),$ (7)

where $S(\delta_{v})$ can be appropriately defined [2]. Using this definition,
one can compute all the remaining charges, and the creation/annihilation
operators algebra (which turns out to be the canonical one). For example, the
boost charge $\mathcal{N}_{i}$ is

$\displaystyle{\scriptstyle\mathcal{N}_{i}=-\frac{1}{2}\int
d^{3}p\,\alpha\,\Bigg{\\{}S(\omega_{p})\left[\frac{\partial
a_{\mathbf{p}}^{\dagger}}{\partial
S(\mathbf{p})^{i}}a_{\mathbf{p}}\,-a_{\mathbf{p}}^{\dagger}\,\frac{\partial
a_{\mathbf{p}}}{\partial
S(\mathbf{p})^{i}}\right]+\omega_{p}\left[b_{\mathbf{p}}\frac{\partial
b_{\mathbf{p}}^{\dagger}}{\partial\mathbf{p}^{i}}-\frac{\partial
b_{\mathbf{p}}}{\partial\mathbf{p}^{i}}b_{\mathbf{p}}^{\dagger}\right]\Bigg{\\}}}.$
(8)

One can then verify that the charges satisfy the usual non-deformed Poincaré
algebra. However, due to the effects of $\kappa$-deformation, one can also
verify that, e.g, $[\mathcal{N}_{i},C]\neq 0$, meaning that CPT symmetry is
subtly violated. More explicitly, using the definition

$\displaystyle
C=\int\,d^{3}p\,(b^{\dagger}_{\mathbf{p}}a_{\mathbf{p}}+a^{\dagger}_{\mathbf{p}}b_{\mathbf{p}}),$
(9)

one can show that $[\mathcal{N}_{i},C]$ is given by

$\displaystyle\frac{i}{2}\int
d^{3}p\Bigg{\\{}S(\omega_{p})\left[\frac{\partial a_{\mathbf{p}}}{\partial
S(\mathbf{p})^{i}}b^{\dagger}_{\mathbf{p}}-a_{\mathbf{p}}\frac{\partial
b^{\dagger}_{\mathbf{p}}}{\partial S(\mathbf{p})^{i}}+\frac{\partial
a_{\mathbf{p}}^{\dagger}}{\partial
S(\mathbf{p})^{i}}b_{\mathbf{p}}-a_{\mathbf{p}}^{\dagger}\frac{\partial
b_{\mathbf{p}}}{\partial S(\mathbf{p})^{i}}\right]+$
$\displaystyle\qquad\qquad+\omega_{p}\left[\frac{\partial
b_{\mathbf{p}}^{\dagger}}{\partial\mathbf{p}^{i}}a_{\mathbf{p}}-b_{\mathbf{p}}^{\dagger}\frac{\partial
a_{\mathbf{p}}}{\partial\mathbf{p}^{i}}+\frac{\partial
b_{\mathbf{p}}}{\partial\mathbf{p}^{i}}a^{\dagger}_{\mathbf{p}}-b_{\mathbf{p}}\frac{\partial
a_{\mathbf{p}}^{\dagger}}{\partial\mathbf{p}^{i}}\right]\Bigg{\\}}.$ (10)

Notice that in the limit $\kappa\rightarrow\infty$ one recovers the canonical
result $[\mathcal{N}_{i},C]=0$.

##  III Comments and conclusions

The fact that $[\mathcal{N}_{i},C]\neq 0$ has several important physical
consequences. The most apparent one is a difference in the decay time between
particles and antiparticles. Furthermore, we now have a well defined theory
which will allow us to study in details the propagator and the $n$-point
functions in general. From this point of view, it will be interesting to
tackle the loops in this deformed context. Finally, what has been done for the
case of the complex scalar field will be extended to fields of higher spins,
in order to expand the discussion to more realistic phenomena.

## Acknowledgments

Parts of these works were supported by funds provided by the Polish National
Science Center, the project number 2019/33/B/ST2/00050.

## References

  * [1] M. Arzano, A. Bevilacqua, J. Kowalski-Glikman, G. Rosati, and J. Unger, Phys. Rev. D 103, 106015 (2021)
  * [2] A. Bevilacqua, J. Kowalski-Glikman, and W. Wislicki, Phys. Rev. D 105, 105004 (2022)
  * [3] M. Arzano, and J. Kowalski-Glikman, Deformations of Spacetime Symmetries: Gravity, Group-Valued Momenta, and Non-Commutative Fields, Lecture Notes in Physics, Springer, 2021.