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Direct Observation of Nanoscale Solitons...
HAN WOONG YEOM
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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]혻solitons (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.혻 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]혻during the interaction. Thus, it might [S3]혻require 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]혻the first CDW phase shift (the 2nd 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 4th 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]혻 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.

 
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