# EnHDC: Ensemble Learning for Brain-Inspired Hyperdimensional Computing

Ruixuan Wang, Dongning Ma, Xun Jiao
{rwang8, dma2<EMAIL_ADDRESS>
Villanova University
Villanova, PA 19085

###### Abstract

Ensemble learning is a classical learning method utilizing a group of weak
learners to form a strong learner, which aims to increase the accuracy of the
model. Recently, brain-inspired hyperdimensional computing (HDC) becomes an
emerging computational paradigm that has achieved success in various domains
such as human activity recognition, voice recognition, and bio-medical signal
classification. HDC mimics the brain cognition and leverages high-dimensional
vectors (e.g., 10000 dimensions) with fully distributed holographic
representation and (pseudo-)randomness. This paper presents the first effort
in exploring ensemble learning in the context of HDC and proposes the first
ensemble HDC model referred to as EnHDC. EnHDC uses a majority voting-based
mechanism to synergistically integrate the prediction outcomes of multiple
base HDC classifiers. To enhance the diversity of base classifiers, we vary
the encoding mechanisms, dimensions, and data width settings among base
classifiers. By applying EnHDC on a wide range of applications, results show
that the EnHDC can achieve on average 3.2% accuracy improvement over a single
HDC classifier. Further, we show that EnHDC with reduced dimensionality, e.g.,
1000 dimensions, can achieve similar or even surpass the accuracy of baseline
HDC with higher dimensionality, e.g., 10000 dimensions. This leads to a 20%
reduction of storage requirement of HDC model, which is key to enabling HDC on
low-power computing platforms.

##  I Introduction

Inspired by how human brain functions, hyperdimensional computing (HDC) is an
emerging computing scheme that leverages the abstract patterns and
mathematical properties of vectors in high dimension spaces [19, 17]. Rather
than processing actual numbers, HDC works with hypervectors (HV), which are
high dimensional (e.g., 10000 dimensions), holographic (not micro-coded)
vectors with i.i.d. (independent and identically distributed) elements [11].
As a novel computing scheme, HDC has shown promising performance for various
applications such as language classification [17], voice recognition [9],
biomedical signal analysis [3] and robotics [18].

Compared with traditional computing schemes such as neural networks, HDC has
several advantages such as smaller model size and low computing cost, making
it a promising computing scheme with low power computing platforms and edge
computing devices [10]. In particular, training of HDC models is free of back-
propagation and can be completed on a sample-basis instead of batches, making
it superior to machine learning algorithms such as neural networks that
requires intensive iterations of back-propagation to establish a model. This
exposes more opportunities for one-shot learning and edge deployment. In
addition, the memory-centricity of HDC grants the advantage of easily
embracing the emerging energy-efficient in-memory computing schemes over other
machine learning algorithms such as neural networks that are highly
computation-centric [12].

Ensemble learning is a machine learning paradigm where multiple models (often
called “weak learners”) are trained to solve the same problem and combined to
get better results. Typically, an ensemble learning system aims to improves
the performance by combining diverse weak learners (base classifiers). In a
practical problem setting, ensemble is useful for the following question:
given a particular classification algorithm, which realization of this
algorithm should be chosen? For example, for a multi-layer perceptron (MLP),
different initializations and weights will render different decision
boundaries. It is worth to note that choosing the best classifier with the
smallest error in training data may not be necessarily the best classifier
overall due to overfitting. Therefore, it remains a question as to which
classifier should be chosen and one can be tempted to choose randomly with the
risk of having a particularly poor model at last. Thus, using an ensemble
model that combines the output from several models, e.g., averaging them, can
reduce the risk of an unfortunate selection of a particularly poor classifier.

For the first time, this paper explores the use of ensemble learning on HDC
models and develop the first ensemble HDC classifier for a wide range of
applications. The benefits of using ensemble learning on HDC includes improved
accuracy and reduced model size. In particular, we make the following
contributions:

  * •

To the best of our knowledge, we propose the first ensemble HDC classifier
called EnHDC. By leveraging the aggregated intelligence of a variety of HDC
classifiers with different random initializations, EnHDC is able to achieve on
average 3.2% accuracy improvement over a single HDC classifier.

  * •

To further enhance the performance of EnHDC, we improve the diversity of base
classifiers by varying a diverse set of parameters for base HDC classifiers
such as number of dimensions, data width, and encoding methods. This further
leads to 1.2% accuracy improvement over basic EnHDC classifier.

  * •

We evaluate the EnHDC on four different practical application domains
including image classification, human activity recognition, speech
recognition, and medical diagnosis. EnHDC enables HDC learning with smaller
number of dimensionality, which leads to a 20% model size reduction with no
accuracy drop.

##  II Related Work

Hyperdimensional Computing Most related works on HDC focus on two major
directions: the application of HDC and the improvement of HDC processing.
Particularly, HDC has been intensively applied into emerging applications such
as robotics by integrating the sensory perceptions experienced by an agent
with its motoric capabilities, which is vital to autonomous learning agents
[15]. In bio-sensing applications such as hand gesture recognition, HDC also
shows 97% accuracy, superior to traditional machine learning algorithms [16].
HDC is also used in building efficient recommendation systems which is almost
14X faster and 7X energy efficient than traditional rating prediction systems
[6]. For optimization of HDC, there are cross-layer in-memory or in-storage
computing platforms designed for HDC to enhance energy efficiency [7].
Computation reuse opportunities are also explored to accelerate performance of
HDC in FPGAs [20]. Software and hardware multi-fold approximation techniques
in encoding are also applied to enhance efficiency [13].

Ensemble Learning Ensemble learning has been applied to various applications
or scenarios for enhanced learning performance. In language translation tasks,
transductive ensemble learning (TEL) is proposed to surpass marginal
improvement on accuracy of traditional ensemble algorithms [21]. In zero-shot
learning scenarios, multi-patch generative adversarial nets (MPGAN) with novel
weighted voting strategies are also proposed for improvement of current
ensemble learning algorithms for better performance [4]. Ensemble learning is
also introduced into transfer learning to mitigate the over-fitting drawbacks
of current transfer strategies [23]. By parallelism and sharing weights
information amongst members in the ensemble classifier, BatchEnsembles
achieves 3X speed-up and 3X memory reduction over traditional ensemble
algorithms [22].

Our Work Some earlier literature indicates having a group of classifiers each
targeting at one feature to build an ensemble can achieve higher
classification accuracy while maintaining lower memory footprint [3]. However,
a systematic analysis of ensemble learning performance with HDC is still
absent. EnHDC, to the best of our knowledge, is the first work that
systematically explores ensemble learning performance under different
configurations such as encoding methods, dimensions of HV and data widths.

##  III Preliminaries

In this section, we introduce the preliminaries regarding hyperdimensional
computing, including backgrounds of HDC and the process of using HDC in
learning tasks.

###  III-A Backgrounds of HDC

Hypervectors Hypervector (HV) is a type of high-dimensional, holographic
vectors with i.i.d. elements [11]. Assume we have an HV of $n$ dimensions
which can be noted as Eq. 1, where $h_{i}$ denotes the elements inside the HV.

In HDC, HVs use their high dimensional space to store different layers of
information, thus can represent values, features and even samples. For
example, in HDC for image classification tasks, HVs can represent a grayscale
value, a pixel or even one image. To establish the dynamic connection between
different layers of information in HV, methodologies of aggregating or combing
information from HVs such as HDC operations, are therefore necessary.

$\vec{H}=\langle h_{1},h_{2},\dots,h_{n}\rangle$ (1)

Operations HVs support three basic operations, addition ($+$), multiplication
($*$) and permutation ($\rho$) as noted in Eq. 2. Additions and
multiplications take two operand HVs as input and perform element-wise
operations that add or multiply each element inside the operand HVs index by
index. Permutations only take one operand HV and perform cyclic shift over the
HV. For all the three operations, the input HVs and the output HVs are in the
same dimension.

$\displaystyle\vec{H_{p}}+\vec{H_{q}}=\langle
h_{p1}+h_{q1},h_{p2}+h_{q2},\dots,h_{pn}+h_{qn}\rangle$ (2)
$\displaystyle\vec{H_{p}}*\vec{H_{q}}=\langle
h_{p1}*h_{q1},h_{p2}*h_{q2},\dots,h_{pn}*h_{qn}\rangle$
$\displaystyle\rho_{1}(\vec{H})=\langle
h_{n},h_{1},h_{2},\dots,h_{n-1}\rangle$

Addition is used to aggregate parallel features that usually belongs to one
modular, while multiplication is used to combine different types of features
together to create new features. Permutation is used to reflect spatial or
temporal changes in the features.

Similarity Check Similarity check is used in HDC for the objective of
measuring the similarity $\delta$ of information between different HVs. There
are different algorithms to measure similarity such as Euclidean distance and
Hamming distance, while in EnHDC, we are using cosine similarity as noted in
Eq. 3. A higher similarity between two HVs indicates that they share more
alike information, or vice versa.

$\delta(\vec{H_{p}},\vec{H_{q}})=\frac{\vec{H_{p}}\cdot\vec{H_{q}}}{||\vec{H_{p}}||\times||\vec{H_{q}}||}=\frac{\sum_{i=1}^{n}h_{pi}\cdot
h_{qi}}{\sqrt{\sum_{i=1}^{n}{h_{pi}}^{2}}\cdot\sqrt{\sum_{i=1}^{n}{h_{qi}}^{2}}}$
(3)

###  III-B HDC in Learning Tasks

HDC in learning tasks features three major phases: Training, Inference and
Retraining; all the phases require a fundamental procedure called Encoding.

Encoding Encoding is the process of mapping input features of one sample to
the high-dimensional space available for HDC training, inference and
retraining, i.e., building representative HVs of this sample from the
fundamental item memory using combinations of HDC operations. Item memory is a
type of specially allocated memory during runtime, which stores the bottom
layer HVs, or building block HVs that are used to add or multiply for
establishing other HVs. To ensure the i.i.d. property, HVs in the item
memories are all randomly initialized.

Process of encoding can be noted as Eq. 4. Assume we have the $m$-dimensional
input features $\vec{F}=\langle f_{1},f_{2},\dots,f_{m}\rangle$ for each
sample, a set of corresponding item memories
$\mathcal{R}=\\{\mathcal{R}_{1},\mathcal{R}_{2},\dots,\mathcal{R}_{m}\\}$ and
the combination of HDC operations $E$ determined by the application, the
encoded HV $\vec{H}$ is obtained by looking up each feature’s corresponding HV
in the item memory and then applying them into the HDC operation combination.
This encoded HV will subsequently represent the input sample in training,
inference and retraining.

$\displaystyle\vec{H}=E(\mathcal{R},\vec{F})=E(\mathcal{R}_{1}[f_{1}],\mathcal{R}_{2}[f_{2}],\dots,\mathcal{R}_{m}[f_{m}])$
(4)

Training Training is the process of aggregating encoded hypervectors sharing
the same label together to build the associative memory. Associative memory
stores the class HVs, each representing a class in the learning problem. At
the beginning of training phase, all the HVs in the associative memory is
initialized as zero. Process of training can be denoted as Eq. 5. In the
learning problem with $k$ classes, assume we have encoded the HVs
$\vec{H^{l}}$ for each training sample where $l$ means the class label,
training process to establish associative memory $\mathcal{A}$ is by adding up
hypervectors representing samples from the same class in the training set
together.

$\displaystyle\mathcal{A}=$
$\displaystyle\\{\vec{A^{1}},\vec{A^{2}},\dots,\vec{A^{k}}\\}$ (5)
$\displaystyle=$
$\displaystyle\\{\sum\vec{H^{1}},\sum\vec{H^{2}},\dots,\sum\vec{H^{k}}\\}$

Figure 1: EnHDC framework

Inference Inference is the process of using the associative memory established
in the training phase to determine the class label of an unknown sample.
Process of inference can be denoted as Eq. 6. First, we encode the unknown
sample into its representing HV $\vec{H_{q}}$ referred to as the query HV.
Then we perform similarity check between the query HV and each class HV inside
the associative memory. As aforementioned, higher similarity means higher
common information shared by the two HVs, further indicating that these two
HVs are likely from the same class. Therefore, the class of HVs in the
associative memory having the highest similarity is determined to be the class
of the query HV, namely the predicted label of the sample.

$\displaystyle
l=argmax(\\{\delta(\vec{H_{q}},\vec{A^{1}}),\delta(\vec{H_{q}},\vec{A^{2}}),\dots,\delta(\vec{H_{q}},\vec{A^{k}})\\})$
(6)

Retraining Retraining is the process of fine-tuning the HDC model by
correcting erroneous information from the associative memory. The process of
retraining can be denoted as Eq. 7. Assume we have a query HV $\vec{H_{q}}$
with class label $l$. If it is predicted wrongly as class $l^{\prime}$, then
we subtract it from the $l^{\prime}$ class HV $\vec{A^{l^{\prime}}}$ in the
associative memory to remove the erroneous information and add it into the
correct $l$ class HV $\vec{A^{l}}$. By doing this update, the similarity
between $\vec{H_{q}}$ and $\vec{A^{l^{\prime}}}$ will be reduced while the
similarity between $\vec{H_{q}}$ and $\vec{A^{l}}$ will be increased to
enhance the accuracy of model.

$\displaystyle\vec{A^{l^{\prime}}}$
$\displaystyle=\vec{A^{l^{\prime}}}-\vec{H_{q}}$ (7)
$\displaystyle\vec{A^{l}}$ $\displaystyle=\vec{A^{l}}+\vec{H_{q}}$

##  IV EnHDC Model

In this section, we describe the process of developing EnHDC and how to
enhance the diversity of the base classifiers in EnHDC to further improve the
performance. Fig. 1 illustrates the overview of EnHDC. First, we separately
train and retrain several different base classifiers using different parameter
configurations. These base classifiers are well-trained and we integrate these
base classifiers into one ensemble classifier. Secondly, we encode the testing
sample into query HV and do the inference on every base classifier. And in
phase three, as we collected all the base inference results in phase 2, we
employ majority voting to get the ensemble inference result of the ensemble
classifier.

###  IV-A Base Classifier Development

In traditional ensemble learning, base classifiers are developed with
different initialization settings. In HDC, base classifiers are developed as
described in Section III-B. Each base classifier has different configurations
and different randomly generated item memories $\mathcal{R}$. Thus, the
training outcome, i.e., the associate memory, will have different class HVs
representing the same class. Therefore, for a given query input, the
classification output may be different.

###  IV-B Diversity Enhancement of Base Classifier

To further enhance the diversity of base classifiers, we propose different
configurations for base classifiers, including different encoding mechanisms,
different dimensions, and different data width for representing the numbers.

Encoding mechanisms: We use two encoding mechanisms: the Record based encoding
and the N-gram based encoding. First of all, the Record Encoding is a general
encoding method which mapping every feature vector $\vec{F}=\langle
f_{1},f_{2},\dots,f_{m}\rangle$, containing $m$ features, into hyper-
dimensional space. The Record encoding method finds the minimum and maximum
feature values and converse the range into $p$ feature levels. In Record
encoding method, we assign a set of random orthogonal bipolar level HVs
$\vec{H_{l}}$ = $\langle h_{l1},h_{l2},\dots,h_{ln}\rangle$ to every feature
levels. Meanwhile, for preserving the position independence of feature values
in the feature vector, Record encoding method also assign one set of HVs to
each feature values, called base HVs $\vec{H_{b}}$ = $\langle
h_{b1},h_{b2},\dots,h_{bn}\rangle$. The Record encoding method use the level
HVs to represent each feature value in the feature vector and Base HVs for the
position relationship of features values. Then Record encoding employed by
linearly combining the level HVs and Base HVs in hyper-dimensional space, the
detail of Record encoding shows as Eq. 8.

$\vec{H}_{Record}=\sum_{i=1}^{m}\vec{H}_{li}*\vec{H}_{bi}$ (8)

In Eq. 8, $\vec{H}_{Record}$ is the non-bipolar encoded HV with D dimensions
containing several integer values. Meanwhile, since base HVs are randomly
generated in hyper-dimensional space, these HVs are almost mutually
orthogonal, which means the cosine similarity between any two base HVs
$\vec{H}_{bi}$ and $\vec{H}_{bj}$, we have $\delta$($\vec{H}_{bi}$,
$\vec{H}_{bj}$) = 0.

We also use the N-gram Encoding method, in which we employ the locality-based
sparse random projection [8] as our N-gram encoding method. In this method,
while we are using the D dimensional HVs, we first extended the length of
feature vector from $N$ to D. For instance, in MNIST dataset, when we try to
encode a feature vector with N = 768 feature values into D = 10000 dimension
HVs, we firstly need to attach 13 duplication following the original feature
vector. Meanwhile, we generate a random bipolar D dimensional local-hashing HV
$\vec{H_{s}}$ = $\langle h_{s1},h_{s2},\dots,h_{sn}\rangle$. To encode the
extended feature vector into the hyper-dimensional space, N-gram Encoding
method deploy an N-gram sliding window and take the dot product of the
extended feature vector and projection vector in this window range. The
detailed encoding progress of N-gram Encoding method shows as Eq. 9. In this
situation, the i-th value of $\vec{H}_{N-gram}$ equals to the dot product of
the w feature values from $f_{i}$ to $f_{i+w-1}$ and w element values from
$h_{i}$ to $h_{i+w-1}$, where $w$ equals to the size of sliding window.

$\displaystyle\vec{H}_{N-gram}=\langle v_{1},v_{2},\dots,v_{n}\rangle$ (9)
$\displaystyle{v}_{i}=f_{i}*h_{i}+f_{i+1}*h_{i+1}+\dots+f_{i+w-1}*h_{i+w-1}$

Dimensions: We use three different dimension settings for base classifiers,
1000, 5000, and 10000. This brings another diversity aspect for the base
classifiers.

Data width: To represent the internal data of the HDC base classifiers, we use
$INT\\_8$ and $INT\\_16$ data width to further enhance the diversity of base
classifiers.

###  IV-C Voting Mechanism

The inference phase has two steps in the EnHDC. First, we map each testing
data into a query HV $\vec{H_{q}}$ using the same encoding method during
training. Then we calculate the cosine similarity of each class HVs with the
query HV $\vec{H_{q}}$ in every base classifier, the inference result is
pointed to the class with the highest cosine similarity. Then we finish the
inference process in each base classifier following the same routine in step
1. And for step 2, we collect all the base inference results in the ensemble
classifier to vote the ensemble inference result. We explored different voting
mechanisms to get a better inference result and we tested two different voting
strategies, soft voting and hard voting. Hard voting is the majority voting.
And for soft voting, since each base classifier gave the inference result by
cosine similarity checking, we can sum up all the related cosine distances and
rank them in order, where the champion will be selected as the final result.
Our test result shows that the hard voting strategy achieved better accuracy
in EnHDC. Therefore, we integrate all the base inference results in ensemble
classifier and use majority voting to get the ensemble inference result of the
corresponding query HV.

(a) MNIST (b) CARDIO (c) ISOLET (d) HAR

Figure 2: EnHDC performance under different HV dimension: D = 1000, 5000,
10000

##  V Experimental Results

(a) MNIST (b) CARDIO (c) ISOLET (d) HAR

Figure 3: EnHDC performance under different ensemble size

###  V-A Experiment Setup

We evaluate EnHDC using four application domains: speech recognition (ISOLET
[5]), human activity recognition (HAR [1]), handwritten digits (MNIST [14]),
and cardiotocography (CARDIO [5]). Detail of these four applications are shown
below:

  * •

MNIST: Handwritten digits Recognition dataset aiming at classifying 10
handwritten digits from 0 to 9. In this application, we use 50000 samples for
training and 10000 samples for testing.

  * •

CARDIO: Cardiotocography dataset aiming at classifying measurements of fetal
heart rate (FHR) and uterine contraction (UC) features into 10 classes (label
consensus by obstetricians). It has a total of 2,126 fetal cardiotocograms
(CTGs) signals. In this application, we use 1913 samples for training and 213
samples for testing.

  * •

HAR: Human activity recognition dataset aiming at recognizing 12 types of
human activities. The dataset is built from 30 subjects performing activities
of daily living (ADL) with 561 features. In this application, we use 7767
samples for training and 3162 samples for testing.

  * •

ISOLET: Speech recognition dataset aiming at recognizing voice audio of the 26
letters of the English alphabet. This dataset contains 150 subjects speaking
the name of each alphabet letter twice. In this application, we use 6238
samples for training and 1559 samples for testing.

###  V-B Accuracy Improvement

Fig. 2 and Fig. 3 present the accuracy comparison between different
configurations of EnHDC. The baseline is one HDC classifier while the EnHDC
employing several different base classifiers. According to our experiment
result, we can observe several important facts.

First as shown in Fig. 2, EnHDC contains 8 and 16 base classifiers with
different encoding methods (Record and N-gram encoding). To evaluate the
performance of EnHDC, we compare 3 models under the same dimensionality
setting across $D=1000,5000,10000$, EnHDC is showing consistent higher
accuracy than baseline models. When EnHDC uses 8 base classifiers, the average
improvement is 3.2% over all applications.

Second, normally HDC requires a high dimensionality such as $D=10000$ to
achieve satisfying performance. However, with EnHDC, we can even achieve
higher accuracy under lower dimensionality than that of baseline model with
higher dimensionality. Actually, across all the applications, EnHDC with 8
classifiers under $D=1000$ presents higher accuracy than baseline model under
$D=10000$. The average improvement is 1.37%.

Third, as shown in Fig. 3, the number of base classifiers have a notable
impact on the accuracy. Without the loss of generality, our experiment
features different ensemble sizes, starting from two base classifiers to
twelve base classifiers. As the green line shown in Fig. 3, the accuracy of
EnHDC increases by adding more classifiers but comes to the vertex when using
eight base classifiers for most applications. After this, the accuracy
improvement is saturated. This is consistent with the ensemble theory in [2],
where the performance of ensemble learning algorithms cannot constantly
increase by adding an infinite amount of base classifiers. The accuracy will
peak during the progress of increasing the number of classifiers, and after
this peak value, the overall accuracy cannot have an obvious improvement.

Last but not least, the diversity enhancement can further improve the accuracy
of EnHDC as shown in Fig. 3. EnHDC_enhanced classifier is the EnHDC classifier
with enhanced diversity by varying encoding mechanisms (Record encoding,
N-gram encoding), dimensions ($D=1000,5000,10000$), and data width ($INT\\_8$,
$INT\\_16$). This figure shows that the EnHDC_enhanced classifier can further
improve the accuracy of EnHDC classifier by 1.2% on average across all
applications, which is 4.4% improvement over baseline HDC model.

###  V-C Memory Reduction

With ensemble learning, we are able to reduce the dimensionality requirement
for HDC classifiers. Typically, HDC is required to have a high dimensionality,
e.g., 10000, to achieve a satisfying performance. For example, for CARDIO
dataset in Fig. 2, the baseline HDC classifier with 10000 dimensions has 1.9%
higher accuracy than the baseline HDC classifier with 1000 dimensions.

However, with ensemble learning, we can see that EnHDC with just 1000
dimensions is able to achieve similar level or even surpass the accuracy of
10000-dimension baseline HDC classifiers across all applications. This can
result in a smaller model size. For example, with 8 base classifiers with
$D=1000$ dimensions, this can reduce 20% model size compared to a baseline
classifier with $D=10000$ dimensions. For MNIST dataset with 10 classes, we
have a baseline HDC model with $D=10000$ and $INT\\_8$ data width whose model
size is $8~{}bits*10000~{}dimensions*10~{}classes$ = 800Kb and EnHDC with base
classifiers with $D=1000$ and $INT\\_8$ whose model size is
$8~{}bits*1000~{}dimensions*10~{}classes*8~{}classifiers$ = 640Kb. The overall
model size reduction is 160Kb for MNIST. Meanwhile, in HAR dataset with 12
classes, we have one 960Kb baseline model with $D=10000$ and EnHDC with eight
96Kb base classifiers with $D=1000$, pointing out the model size reduction for
HAR is 180Kb.

##  VI Conclusion

This paper presents EnHDC, the first ensemble HDC classifier. As an ensemble
learning model, EnHDC employs different base classifiers under different HV
dimensions, different data widths, and different encoding methods. In EnHDC,
base classifiers are individually trained and retrained and EnHDC applies the
majority voting to generate the final inference result. We designed several
experiments to evaluate the performance of EnHDC under different HV dimensions
and different number of classifiers. By evaluating EnHDC on four practical
applications, we show that EnHDC can achieve higher accuracy and can reduce
model size compared to baseline HDC classifiers while providing the same or
even better classification accuracy. Meanwhile, by increasing the diversity of
base classifiers in EnHDC, the classification accuracy has an enhanced
improvement compared to the original EnHDC model. This paper presents the
first effort in using an ensemble learning method in HDC for boosting the
performance and opens the door for this potential research direction. Our
future work will consider using a cascading method and more sophisticated
ensemble learning algorithms such as boosting.

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# Ferroelectric Polarization Reversal vs Pump Spot Shape

Veniamin A. Abalmasov<EMAIL_ADDRESS>Institute of Automation and
Electrometry SB RAS, 630090 Novosibirsk, Russia

(August 27, 2024)

###### Abstract

It has recently been shown that the polarization of a ferroelectric can be
strongly affected by light pumping or even transiently reversed by mid-IR
pumping within a record short subpicosecond timescale. It was then suggested
that the reversal of polarization is most likely prevented by the depolarizing
field that appears during the process. To minimize the effect of the
depolarizing field, I propose to create an elongated domain by choosing a
corresponding pump spot shape. I calculate the polarization dynamics for a
particular experimental setup with mid-IR pumping and show that the
polarization reversal is possible in this case.

The polarization created by elementary dipoles in matter is an inherent
property of ferromagnetic and ferroelectric materials, which is largely used
for data storage [1, 2], but may be even more in demand in emerging
technologies [3, 4, 5, 6]. The most straightforward way to control
polarization by varying the appropriate magnetic and electric fields is
limited in speed to hundreds of picoseconds [7]. However, the use of
ultrashort electromagnetic pulses can reduce this time by tens or even
hundreds of times [8].

To date, ultrafast polarization reversal using femtosecond laser pumping has
been successfully achieved in ferromagnets with an unprecedented switching
time of about 20 ps [9]. At the same time, the situation in ferroelectrics
turned out to be more complicated [10]. The original proposal to switch the
polarization by resonant excitation of the ferroelectric mode [11, 12, 13] has
not yet been fully realized experimentally [14, 15, 16, 17]. It was also shown
that optical pumping with excitation of the electronic subsystem [18, 19, 20,
21] leads only to a temporary decrease in polarization [22, 23, 24, 25].
Following the rapid development and achievements of nonlinear phononics [26,
27, 28, 29], it was proposed to pump the IR-active optical phonon mode, which
is nonlinearly coupled with the ferroelectric soft mode [30]. However, in this
way, only a transient switching of the polarization was achieved in a
monodomain LiNbO3 sample, followed by the polarization recovery to its initial
value on a timescale of about 0.2 ps [31]. After that, it was suggested that
the polarization reversal was prevented by the depolarizing field created by
charges at the domain boundaries and it was proposed to use a metal wire
around the pump spot to screen these charges [32].

Figure 1: Two possible shapes of the pump spot (orange and yellow) and the
corresponding domain with opposite polarization and bound charges at its
boundaries.

The influence of the domain shape on the depolarizing field is well known
[33], and this field, determined by the depolarization factor $N$ for
ellipsoidal domains, $E_{d}=-NP/(\varepsilon_{0}\varepsilon_{\infty})$, where
$\varepsilon_{0}$ is the electric constant and $\varepsilon_{\infty}$ is the
high-frequency relative permittivity, decreases significantly for elongated
domains. Thus, I propose to create an elongated domain by choosing an
appropriate pump spot shape in order to reduce the depolarizing field and make
polarization reversal possible.

Consider the experimental conditions as in [31], where in LiNbO3 (LNO)
monodomain crystal, the mid-IR resonant pumping of an optical phonon mode
nonlinearly coupled to the ferroelectric soft mode led to a transient reversal
of polarization. If the domain created in [31] is approximated by an
ellipsoid, the two long axes of which, $a=c$, are equal to the pump spot size
of about 65 $\mu$m, the $c$-axis being parallel to the polarization, and the
short axis $b$ is equal to the pump penetration depth of about 3.2 $\mu$m (see
Fig. 1), the value of the depolarization factor should be about $N\approx
0.04$ corresponding to the ratio $b/c=0.05$ [34]. This value can be further
reduced by decreasing the ratio $a/c$. For the pump pulse frequency of about
19 THz as in [31] the smallest possible spot size is about the pump wavelength
of 15 $\mu$m. For instance, when the $a$-axis of the spot is halved to $a$ =
30 $\mu$m, and its $c$-axis is doubled to $c=120$ $\mu$m, keeping the fluence
constant and leading to the ratios $b/c=0.025$ and $a/c=0.25$, the
depolarization factor is about $N\approx 0.01$, which reduces the depolarizing
field by four times [34].

The LNO crystal in ferroelectric phase (below 1480 K) belongs to $R3c$
symmetry group with 27 optical phonon modes, four $A_{1}$-modes of which are
polarized along the $c$-axis and have a frequency of about 7.5, 8.1, 10, and
19 THz [35]. The calculated ab-initio using Quantum Espresso [36, 37] IR
activities 44.6, 8.8, 0.41, and 53.25 (D/Å)2/u and frequencies 7.2, 7.7, 9.8,
and 18.4 THz of these modes are close to the experimental values [35]. The
eigenvectors of the first, which is a soft ferroelectric mode, and the last
mode, plotted using XCrySDen [38], are shown in Fig. 2. The corresponding
relative displacements of lithium, niobium and oxygen ions along the $c$-axis
for the two modes, hereinafter called $Q_{P}$ and $Q_{\text{IR}}$, are
$-0.563,-0.074,0.225$ and $-0.056,-0.049,0.104$, respectively. The
ferroelectric soft mode $Q_{P}$ involves the motion of lithium and oxygen ions
while the high-frequency $Q_{\text{IR}}$ is mainly associated with the motion
of oxygen ions in accordance with other ab-initio calculations [39, 40, 41,
42, 43]. However, small displacements of heavy niobium ions are also important
because of their large charge of about 6.44 elementary charges for the
$zz$-tensor component compared to 1.05 and $-2.53$ for lithium and oxygen
ions, respectively (see also [41]).

Figure 2: LNO primitive unit cell with arrows indicating ion motion in the
ferroelectric mode $Q_{P}$ (left) and the highest frequency infrared-active
mode polarized along the c-axis $Q_{\text{IR}}$ (right).

To trace the dependence of the polarization dynamics on the depolarizing
field, I solve a system of two equations of motion for the pumped optical
phonon coordinate $Q_{\text{IR}}$ and the ferroelectric mode $Q_{P}$ of the
form [32]:

$\displaystyle\ddot{Q}+\gamma\dot{Q}+\partial F/\partial Q=0,$ (1)

where $\gamma$ is the damping constant.

The Landau free energy in Eq. (1) can be divided into three parts:

$\displaystyle F=F_{0}+F_{E\text{-ph}}+F_{\text{ph-ph}}.$ (2)

The first part describes free optical phonons:

$\displaystyle
F_{0}=-\frac{\omega^{2}_{P}}{4}Q_{P}^{2}+\frac{c_{P}}{4}Q_{P}^{4}+\frac{\omega_{\text{IR}}^{2}}{2}Q_{\text{IR}}^{2},$
(3)

where $\omega_{P,{\text{IR}}}$ are the frequencies of the corresponding modes
$Q_{P,{\text{IR}}}$. The coefficient $c_{P}$ determines the equilibrium value
$Q_{P}^{e}$ through the equation $\partial F/\partial Q_{P}=0$. In the absence
of electric field and nonlinear phonon interactions this gives
$c_{P}=\omega_{P}^{2}/(2(Q_{P}^{e})^{2})$.

The second part of the thermodynamic potential corresponds to the phonon –
electric field coupling:

$\displaystyle F_{E\text{-ph}}=$ $\displaystyle-
E_{p}(Z^{*}_{Q_{P}}Q_{P}+Z^{*}_{Q_{\text{IR}}}Q_{\text{IR}})$ $\displaystyle-
E_{d}(1-Q_{P}/2Q_{P}^{e}))Z^{*}_{Q_{P}}Q_{P},$ (4)

where $Z^{*}_{Q_{P,{\text{IR}}}}$ are the Born effective charges of the
corresponding modes $Q_{P,{\text{IR}}}$, $E_{d}$ is the depolarizing field
amplitude, $E_{p}$ is the electric field of the pump pulse directed along the
$c$-axis (see Fig. 1).

Figure 3: Ferroelectric mode dynamics for the pump pulse amplitudes of $E_{0}$
= 14, 20, and 25 MV/cm when $E_{d}$ = 2.8 MV/cm (dotted) and $E_{d}$ = 1.4
MV/cm (solid).

The depolarizing field in a thin slab with polarization normal to the plane is
$E_{d}=P_{s}/(\varepsilon_{33}\varepsilon_{0})$, which is about 26.4 MV/cm for
the saturation polarization $P_{s}=0.70$ C/m2 and the dielectric constant
$\varepsilon_{33}$ = 30 in LNO [44]. Assuming that the domain is elliptical,
we must multiply this field by the depolarization factor $N$. We should also
take into account the fact that for a domain with reverse polarization, the
depolarizing field is created by bound charges at the outer boundary of the
domain and the inner boundary of the surrounding crystal, and thus doubles.
Indeed, a sign change in $Q_{P}$ in Eq. (Ferroelectric Polarization Reversal
vs Pump Spot Shape) leads to an electric field $2E_{d}$ when the derivative
$\partial F/\partial Q_{P}$ is considered.

The electric field of the mid-IR pump pulse is $E_{p}(t)=E_{0}\sin(\omega
t)\exp(-4\ln 2\,t^{2}/T^{2})$ with a carrier frequency $\omega$, a Gaussian
envelope with a duration of $T=0.15$ ps and an amplitude $E_{0}$ up to 25
MV/cm as in the experiment [31].

Finally, the leading phonon coupling terms up to the forth order in phonon
amplitudes are [32]

$\displaystyle F_{\text{ph-
ph}}=c_{1}Q_{P}Q_{\text{IR}}^{3}+c_{2}Q_{P}^{2}Q_{\text{IR}}^{2}+c_{3}Q_{P}^{3}Q_{\text{IR}}.$
(5)

The first term in Eq. (5) does not contribute much in the polarization
reversal and was neglected here, while the values of the other coupling
constants were taken $c_{2}=34$ meV/u2Å4 and $c_{3}=17$ meV/u2Å4 as in [32].
Together with the value of the depolarizing field $E_{d}=2.8$ MV/cm [32], this
coupling makes it possible to reproduce well the dynamics of polarization
observed in [31] (see dotted lines in Fig. 3). Thus, the depolarization factor
turns out to be about $N=0.1$, which is twice as high as was estimated above
from geometric considerations, but nevertheless is in reasonable agreement
with all the approximations made. The values of the other parameters used in
the calculation were determined in [32] in accordance with [31]. The phonon
frequencies, dumping constants and Born effective charges of the two modes are
$\omega_{P}=7.5$ THz, $\omega_{\text{IR}}=19$ THz, $\gamma_{P}=1.6\pi$ THz,
$\gamma_{\text{IR}}=2\pi$ THz and $Z^{*}_{Q_{P}}=1.36$ $e/\sqrt{\text{u}}$,
$Z^{*}_{Q_{\text{IR}}}=1.40$ $e/\sqrt{\text{u}}$, respectively; the
ferroelectric phonon equilibrium value is $Q_{P}^{e}=3.1$Å$\sqrt{\text{u}}$.

The polarization dynamics for a half-smaller depolarizing field $E_{d}=1.4$
MV/cm, which should be attained for a squeezed pump spot with a ratio
$a/c=0.25$, as discussed above, shows polarization reversal (see solid lines
in Fig. 3). This reversal becomes possible already for fluences above 60
mJ/cm2.

In experiments with optical pumping [23, 24, 25], the penetration depth of the
pump pulse varies from about 3 nm [25] upto 100 nm [24], depending on the
crystal used. In this case, the depolarization factor should already be very
small, and the depolarizing field is strongly reduced. However, the
depolarizing field will decrease even more if the pump spot is elongated, and
it would be interesting to see how the spot shape affects the polarization
dynamics in this case. In addition, due to the short wavelength of optical
pulses, the ratio $a/c$ can be set very small for the same fluence, which
would provide an almost zero depolarizing field. I also note that the
depolarizing field must be greatly reduced in ferroelectric nanowires [22,
45], where, thus, the polarization reversal by the pump pulse should be
possible. However, special attention should be paid to the size effects in
these samples and their influence on the crystal structure [46, 47]. I note
that a $\pi/2$ polarization rotation induced by a strong single-cycle
terahertz pulse in some domains of a multi-domain ferroelectric thin film of
(Ba0.8Sr0.2)TiO3, where the depolarized field should be reduced due to the
natural shape of the domains, was claimed recently in [15, 16, 17].

In conclusion, I propose to use an elongated pump spot in experiments with
optical, terahertz, or infrared pumping of ferroelectric materials to reduce
the depolarizing field in the irradiated domain. I show that this reduction
should be sufficient to reverse the ferroelectric polarization using mid-IR
pumping in LNO under the conditions of the experiment [31] on a subpicosecond
timescale.

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