For example, a certain geometrical structure of knots, which scientists
call a Hopfion, manifests itself in unexpected corners of the universe, ranging
from particle physics, to biology, to cosmology. Like the Fibonacci spiral and
the golden ratio, the Hopfion pattern unites different scientific fields, and
deeper understanding of its structure and influence will help scientists to
develop transformative technologies.
In a recent theoretical study, scientists from the U.S. Department of
Energy’s (DOE) Argonne National Laboratory, in collaboration with the
University of Picardie in France and the Southern Federal University in Russia,
discovered the presence of the Hopfion structure in nano-sized particles of
ferroelectrics — materials with promising applications in microelectronics and
computing.
The identification of the Hopfion structure in the nanoparticles
contributes to a striking pattern in the architecture of nature across
different scales, and the new insight could inform models of ferroelectric
materials for technological development.
Ferroelectric materials have the unique ability to flip the direction of
their internal electric polarization — the slight, relative shift of positive
and negative charge in opposite directions — when influenced by electric
fields. Ferroelectrics can even expand or contract in the presence of an
electric field, making them useful for technologies where energy is converted
between mechanical and electrical.
In this study, the scientists harnessed fundamental topological concepts
with novel computer simulations to investigate the small-scale behavior of
ferroelectric nanoparticles. They discovered that the polarization of the
nanoparticles takes on the knotted Hopfion structure present in seemingly
disparate realms of the universe.
“The polarization lines intertwining themselves into a Hopfion structure
may give rise to the material’s useful electronic properties, opening new
routes for the design of ferroelectric-based energy storage devices and
information systems,†said Valerii Vinokur, senior scientist and Distinguished
Fellow in Argonne’s Materials Science division. ​“The discovery also highlights
a repeated tendency in many areas of science.â€
What (and where) in the world are Hopfions?
Topology, a subfield of mathematics, is the study of geometric
structures and their properties. A Hopfion topological structure, first
proposed by Austrian mathematician Heinz Hopf in 1931, emerges in a wide range
of physical constructs but is rarely explored in mainstream science. One of its
defining characteristics is that any two lines within the Hopfion structure
must be linked, constituting knots ranging in complexity from a few
interconnected rings to a mathematical rat’s nest.
“The Hopfion is a very abstract mathematical concept,†said Vinokur,
​“but the structure shows up in hydrodynamics, electrodynamics and even in the
packing of DNA and RNA molecules in biological systems and viruses.â€
In hydrodynamics, the Hopfion appears in the trajectories of liquid particles flowing inside of a sphere. With friction neglected, the paths of the incompressible liquid particles are intertwined and connected. Cosmological theories also reflect Hopfion patterns. Some hypotheses suggest that the paths of every particle in the universe interweave themselves in the same Hopfion manner as the liquid particles in a sphere.
Image
depicts some of the polarization lines within a ferroelectric nanoparticle. The
lines intertwine into a Hopfion topological structure. (Image by Yuri Tikhonov,
University of Picardie and Russia’s Southern Federal University, and Anna
Razumnaya, Southern Federal University.)
According to the current study, the polarization structure in a
spherical ferroelectric nanoparticle takes on this same knotted swirl.
Simulating the swirl
The scientists created a computational approach that tamed polarization
lines and enabled them to recognize the emerging Hopfion structures in a
ferroelectric nanoparticle. The simulations, performed by researcher Yuri
Tikhonov from the Southern Federal University and the University of Picardie,
modeled the polarization within nanoparticles between 50 to 100 nanometers in
diameter, a realistic size for ferroelectric nanoparticles in technological
applications.
“When we visualized the polarization, we saw the Hopfion structure
emerge,†said Igor Luk’yanchuck, a scientist from the University of Picardie.
​“We thought, wow, there is a whole world inside of these nanoparticles.â€
The polarization lines revealed by the simulation represent the
directions of displacements between charges within atoms as they vary around
the nanoparticle in a way that maximizes energy efficiency. Because the
nanoparticle is confined to a sphere, the lines travel around it indefinitely,
never terminating on — or escaping from — the surface. This behavior is parallel
to the flow of an ideal fluid about a closed, spherical container.
The link between liquid flow and the electrodynamics displayed in these
nanoparticles bolster a long- theorized parallelism. ​“When Maxwell developed
his famous equations to describe the behavior of electromagnetic waves, he used
the analogy between hydrodynamics and electrodynamics,†said Vinokur.
​“Scientists have since hinted at this relationship, but we demonstrated that
there is a real, quantifiable connection between these concepts that is
characterized by the Hopfion structure.â€
The study’s findings establish the fundamental importance of Hopfions to
the electromagnetic behavior of ferroelectric nanoparticles. The new insight
could result in increased control of the advanced functionalities of these
materials — such as their supercapacitance — for technological applications.
“Scientists often view properties of ferroelectrics as separate concepts that are highly dependent on chemical composition and treatment,†said Luk’yanchuck, ​“but this discovery may help describe many of these phenomena in a unifying, general way.â€
Simulation reveals the Hopfion structure of polarization lines within a
ferroelectric nanoparticle. (Video by Yuri Tikhonov, University of Picardie and
Russia’s Southern Federal University, and Anna Razumnaya, Southern Federal
University.)
Another possible technological advantage of these small-scale
topological structures is in memory for advanced computing. Scientists are
exploring the potential for ferroelectric materials for computational systems.
Traditionally, the flip-able polarization of the materials could enable them to
store information in two separate states, generally referred to as 0 and 1.
However, microelectronics made of ferroelectric nanoparticles might be able to
leverage their Hopfion-shaped polarization to store information in more complex
ways.
“Within one nanoparticle, you may be able to write much more information
because of these topological phenomena,†said Luk’yanchuck. ​“Our theoretical
discovery could be a groundbreaking step in the development of future
neuromorphic computers that store information more organically, like the
synapses in our brains.â€
Future plans
To perform deeper studies into the topological phenomena within
ferroelectrics, the scientists plan to leverage Argonne’s supercomputing
capabilities. The scientists also plan to test the theoretical presence of
Hopfions in ferroelectric nanoparticles using Argonne’s Advanced Photon Source
(APS), a DOE Office of Science User Facility.
“We view these results as a first step,†said Vinokur. ​“Our intention
is to study the electromagnetic behavior of these particles while considering
the existence of Hopfions, as well as to confirm and explore its implications.
For such small particles, this work can only be performed using a synchrotron,
so we are fortunate to be able to use Argonne’s APS.â€