Direct Observation of Nanoscale Solitons

in One-dimensional Charge-density Wave

HAN WOONG YEOM AND TAE-HWAN KIM

CENTER FOR ARTIFICIAL LOW DIMENSIONAL MATERIAIS AT THE INSTITUTE FOR BASIC SCIENCE

POSTECH

A solitary
wave, or a spatially localized wave, can propagate constantly without losing
its shape. After the very first observation of a solitary wave on a canal near
Edinburgh by John Scott Russell (1808-82), there were many debates attempting to
explain this fantastic phenomenon for more than 60 years. Even after a full
theoretical understanding of a solitary wave was presented by Korteweg and de
Vries in 1895, one had to wait another 70 years until scientists fully realized
how important the first discovery was in 1834. Due to the 'rediscovery' of the
solitary wave by Zabusky and Kruskal in 1965, this exotic phenomenon was coined
with the term 'soliton' since they are quasiparticles corresponding
to solitary waves, similar to phonons versus sound.

The exceptional
properties and mathematical elegance of solitons have fascinated scientific
minds. In terms of mathematics, the integrable nonlinear partial differential
equations having soliton solutions attracted most of the attention and led to
other beautiful theories such as the inverse scattering transform. Additionally,
the physics of solitons has been useful to analyze various phenomena such as
Bose-Einstein condensation, conducting polymers, ferroelectric domain walls,
dislocations in crystals, and so on.

In the case of
dislocations, one can easily observe their permanent profiles down to the
atomic scale. However, dislocations are not soliton in the strict sense even
though the soliton concept is still useful to describe dislocations. To get
soliton solutions in highly discrete systems such as crystals, one needs to
consider the continuum limit approximation. In actual crystals where the width
of the dislocation is only of the order of one or two lattice spacings, the
continuum limit approximation is not justified and can only give a rather crude
picture. That's why dislocations do not move spontaneously once the stress that
created them is suppressed. In contrast to dislocations, polyacetylene is
estimated to have a width of 7 unit cells or larger. For such a width, the
continuum limit approximation is valid. But, solitons in polyacetylene were not
observed directly in a similar way as the dislocations. Unlike macroscopic
solitons such as solitary waves in water, microscopic solitons remains to be explored
in real space.

In order to
observe mobile solitons directly at the nanometer scale, much experimental
effort [1] has been made, especially in one-dimensional (1D) charge-density
waves (CDW) like polyacetylene, which have degenerate ground states due to
spontaneous symmetry breaking [2]. A soliton interpolates between two
degenerate CDW states of the system (see the left panel of Fig. 1). In this
case, the soliton is expected to follow a hyperbolic tangential function, which
is a solution of the non-linear Klein-Gordon equation. Furthermore, since the
total energy of the system does not depend on the position of solitons, they
are free to move along CDW wires. However, such a mobile soliton was not
directly observed in a solid-state system until recently
[3].

In
this work, we used scanning tunneling microscopy (STM) to directly observe
nanoscale solitons in real space [3]. Our system is a well-known quasi-1D CDW
system, indium nanowires grown on a silicon surface. Each unit wire consists of
two zigzag indium chains and undergoes metal to insulator transition below 120
K through a periodicity-doubling distortion along a wire. Due to interwire
coupling, this specific system has unique fourfold degeneracy of the CDW state.
Even though this uniqueness leads to various CDW phase defects, we can clearly
distinguish intrinsic solitons from other non-solitonic phase defects as guided
by the mathematical soliton solution. The solitons exhibit a characteristic
variation of the CDW amplitude, a hyperbolic tangential form, with a coherence
length of about 3.6 nm or 9 lattice spacings (see the right panel of Fig. 1 and
Fig. 2), as expected from the electronic structure of the CDW state. This
coherence length justifies the continuum limit approximation and solitons thus
are expected to move freely at constant speed in contrast to dislocations.

Figure 1. (left) Schematic picture of a soliton
(red) interpolating two different ground states. An STM image (right top)
showing a soliton (dashed line) in the CDW wires and its corresponding STM line
profile (black dot, right bottom) with the soliton solution (red lines).

On the other hand, by taking STM line profiles repeatedly along only a
single CDW wire, we clearly take a snapshot of moving solitons (see Fig. 3). In
most cases, we could see only abrupt CDW phase shifts (see the inset of Fig. 3)
because solitons move too fast beyond STM's temporal resolution (~0.6 msec).
Unexpected interaction with a non-solitonic defect (colored as green in Fig. 3)
causes the trapping of [S1] solitons
(indicated by triangles in Fig. 3) for a short time (up to 0.3 sec) after a
soliton translates the non-solitonic defect by half a CDW period. This
interaction will be completed when another soliton transmits over the defect by
moving it again. In total, two successive solitons cause a defect to hop
by one CDW period while they are trapped transiently during the interaction
(Fig. 3). This observation implies that energy exchanges occur between
solitons and a defect [S2] during the
interaction. Thus, it might [S3] require an
extra energy cost for a soliton transmission.

Figure 2. 3D representation of the soliton shown in
Fig. 1. Gray balls represent atoms of a scanning tip.

Figure 3. Sequential STM line profiles
(the oldest one at topmost) taken along a single CDW wire. Vertical arrows and
dashed lines indicate CDW phase shifts and positions of a non-solitonic defect
(green). Color (blue and red) means different CDW states. Very after [S6] the first CDW
phase shift (the 2^{nd} profile from top), solitons connecting two
different CDW states are detected around the non-solitonic defect (indicated by
triangles). They disappear immediately after another CDW phase shift (the 4^{th} profile from top). As a result of the passing solitons, the non-solitonic
defect hops by half a CDW period after each CDW phase shift. (Inset) The arrow
indicates two successive CDW shifts of the same CDW wire in a 2D STM image.
Note that the CDW shift appears abruptly in STM images or line profiles since
solitons move very fast beyond temporal resolution of STM.

For the first time, we directly observed both immobile and mobile
solitons in a 1D CDW system at the nanometer scale since the first theoretical
expectation of soliton excitations in polyacetylene in 1979. This direct
observation of such nanoscale solitons is a critical first step forward in both
measuring and analyzing the dynamic mechanisms of nanoscale solitonic devices
in solid-state physics. On the other hand, discreteness effects against the
continuum limit approximation cannot be completely negligible although they are
very weak due to relatively long coherence length. This non-negligible
discreteness effect would bring new features such as a trapping of the soliton
(shown here) or the radiation of lattice vibrations (causing heat). Further
investigation on the discreteness effect will give us feasible experiential
methods to control solitons in many ways; for example, sending, receiving,
converting to other excitations, and so on.

This work was supported by the National Research Foundation of Korea
(NRF) through the Center for Low Dimensional Electronic Symmetry and the SRC
Center for Topological Matter.

Reference

[1]
H. Morikawa *et al.*, Phys. Rev. B 70, 085412 (2004); S. J. Park *et
al.*, Phys. Rev. Lett. 93, 106402 (2004); G. Lee *et al.*, Phys.
Rev. Lett. 95, 116103 (2005); H. Zhang *et al.*, Phys. Rev. Lett. 106,
026801 (2011).

[2]
H. W. Yeom *et al.*, Phys. Rev. Lett. 82, 4898 (1999).

[3]
T.-H. Kim and H. W. Yeom, Phys. Rev. Lett. 109, 246802 (2012).

**Han Woong Yeom** received his PhD from Tohoku
University in 1996. Afterward he joined the Faculty
of Science of University of Tokyo. He returned to
Korea in 2000 to join the physics faculty of Yonsei
University. In 2010 he became a professor at
POSTECH. He now is directing the Center for
Artifi cial Low Dimensional Materials at the Institute
for Basic Science.

**Tae-Hwan Kim** received his PhD from Seoul National
University in 2005. Afterward he worked as a
postdoctoral research associate at the Center for
Nanophase Materials Sciences at Oak Ridge National
Laboratory, USA. In 2010 he became a collegiate
assistant professor at POSTECH. He now is
participating in the Center for Artifi cial Low
Dimensional Materials at the Institute for Basic Science.