ARTICLE
pubs.acs.org/crystal

Polyhedral Topological Crystals†)

Masakatsu Tsubota,*,† Katsuhiko Inagaki,‡ Toru Matsuura,§ and Satoshi Tanda†,||Department of Applied Physics and Center of Education and Research for Topological Science and Technology,
Hokkaido University, Sapporo 060-8628, Japan‡
Department of Physics, Asahikawa Medical University, Asahikawa 078-8510, Japan§
Creative Research Institute, Research Department, Hokkaido University, Sapporo 001-0021, Japan
ABSTRACT: We have synthesized micrometer-scale polyhedral ring crystals of TaS3 by thechemical vapor transportation method. The polyhedral ring crystals are closed-loop crystals
with several vertices. We investigated the crystallinity and ﬂatness of the facets by theelectron backscatter diﬀraction technique and found that the orientation of the crystal axischanged abruptly at several points. Observing the size of the crystals, we found a new phaseof topological crystals and propose a phase diagram of ring-polyhedral crystals in terms ofradius and thickness. We also propose a mechanism of polygonization in which the verticesof the polyhedral crystals are formed from concentrated dislocations as a result of distancedependent interactions between them.

’ INTRODUCTION
Recently, there has been increasing interest in loop nanostructures and microstructures.1 3 Theoretical and experimental
attention has been paid to crystals of these structures.4 7
Topological crystals were discovered in 20008 and have been
researched since then. First, a NbSe3 ring crystal was synthesized
together with whisker crystals. Next, other topological crystals(M€obius-strip-, ﬁgure-eight-, and Hopf-link-shaped crystals)
were discovered,9,10 and then topological crystals of othercompounds (TaSe3,11 TaS3,12 and NbS313) were discovered.
Crystal topology broke through a new stage of understanding ofphysical phenomena.12,14,15
Topological crystals are conceptually diﬀerent from conventional crystals. A crystal is constructed by the inﬁnite repetition ofatoms, as shown in Figure 1 (left). As a result, a conventional
crystal displays discrete translational and rotational symmetries,
and its crystal lattices can be mapped onto itself by translational
symmetry or by various other symmetry operations. On the otherhand, a topological crystal does not display discrete translationalor rotational symmetries, because the orientation of the crystal
axes is smoothly changing. This also indicates that the distribution of dislocations/disclinations introduced into a topologicalcrystal [as shown in Figure 1 (right)] must be smooth.16,17 Whena topological crystal needs to be bent, geometrical frustration isrelaxed through the inclusion of edge dislocations, and othertypes of dislocations, such as screw and mixed dislocations, do
not need to be considered.18 20 However, the interplay between
the global topology of the crystals and the distribution of
dislocations/disclinations is still unclear.
In this article, we report a new type of topological crystals having
several vertices in the tantalum trisulﬁed (TaS3) system. Using theelectron backscatter diﬀraction technique, we conﬁrmed ther XXXX American Chemical Society

crystallinity and ﬂatness of the facets in ring and polyhedral crystalsof TaS3 and then found that the orientation of the crystal axes on
the polyhedral ring crystals changes abruptly at several points. To
reveal the growth mechanism of the polyhedral crystals, weinvestigated several topological crystals in terms of radius andwidth. From the radius width diagram, we propose a phase
diagram of smooth to polyhedral topological crystals, based onthat predicted by Hayashi et al.21’ EXPERIMENTAL METHODS
The polyhedral ring-crystal samples of TaS3 were synthesized by thechemical vapor transport (CVT) method. We enclosed the starting
materials (a tantalum wire and sulfur powder in a mole ratio of 1:3) in an
evacuated quartz tube. The tube was heated in a furnace with atemperature gradient. The temperature of the furnace was set at560 C on one side and 600 C on the other side. The tube was putin the furnace for several hours, with the starting material on the hightemperature side. The grain crystals of Ta or TaS3 were transported tothe low-temperature side by convection and grew into crystals there.
After several hours, we picked up the tube and quenched its hightemperature side in liquid nitrogen because this prevents sulfur fromadhering to the quartz tube. We obtained conventional whisker(needlelike) crystals, ring/tube crystals, and polyhedral ring crystals.

’ RESULTSWe previously discovered polyhedral topological crystals in
MX3 (M = transition metal, X = chalcogen), which are ring
crystals having several vertices, for the ﬁrst time.22 These crystals
Received:
Revised:A

March 31, 2011
September 9, 2011

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Figure 1. Structure of (left) needle-shaped and (right) ring-shaped
crystals. As a result of expansion and contraction, dislocations aregenerated (^ symbols).

Figure 3. Intensity maps of (a) ring and (b) polyhedral crystals ofEBSD obtained using ﬁeld-emission scanning electron microscopy. The
samples were observed from this perspective.

angles of the vertices are not equal. In the case of Figure 2c, thecrystal is shaped as a nonagon, and in the case of Figure 2d,
the radius of the crystal becomes gradually smaller from 4.5 to
1.0 μm, like a sharpened pencil.
We investigated the orientation of the crystal axes of the ringand polyhedral crystals as a function of position using electronbackscatter diﬀraction (EBSD) patterns to conﬁrm their crystallinity and the ﬂatness of the facets. The spatial resolution of theEBSD patterns is several tens of nanometers, depending on theelectron probe radius. We set up ring (Figure 3a) and polyhedral(Figure 3b) crystals. In Figure 3, the brightness in the map is
proportional to the intensity of the EBSD pattern.24 Note thatthe high-crystallinity area shows a high EBSD intensity. Figure 4shows the crystal misorientation (diﬀerence in the orientation offacets) proﬁle of two crystals, scanned along the green arrow. The
blue line shows the variation in the orientation of the crystal axis
from the initial point of the green circle, and the red line showsthat from an adjacent point. The orientation of the facets changes
smoothly in the ring crystal, whereas signiﬁcant discrete jumps areobserved at distances of 4, 5, 7, and 9 μm in the polyhedral crystal.

’ DISCUSSION
To elucidate the formation mechanism of the polyhedral
structures, we investigated the size distribution of 32 topologicalcrystals of TaS3, including 7 polyhedral crystals and 25 ring/tubecrystals. We measured their center radius R and width W. The
distribution of the topological crystals is plotted in Figure 5. Eachred square represents a polyhedral ring crystal, and each black
circle represents a smooth ring crystal. Based on this plot, we
became aware that the ring crystals with radii smaller than 2 μm
became polyhedral ring crystals. This suggests that the radius
must play an important role in the localization of dislocations.
We compared our results with the R W phase diagram of ring
crystals predicted by Hayashi et al., who applied Ginzburg
Landau theory for superconductivity.21 They predicted threephases for ring crystals. In phase I, the crystals are free of
dislocations. In phase II, dislocations are introduced into thering crystals to relax the bending energy. In phase III, because thedefects are so dense, there is no interplane rigidity. They also
predicted polygonization in phase II. In Figure 5, phase I iscolored purple, phase II is blue and yellow, and phase III is red.
The black area is the forbidden area for which R < W. To calculate
the phase boundaries, we used the parameters of TaS325,26 as γ2

Figure 2. Topological crystals and polyhedral topological crystals of
TaS3. (a) Ring and (b) tube of smooth topological crystals. (c) Nonagonand (d) pencil-shaped polyhedral topological crystals.

were synthesized together with whiskers and ring crystals. Figure 2
shows typical topological crystals and polyhedral topological crystals.
Whisker crystals are usually very thin, with typical thicknesses of
0.1 10 μm and lengths of 10 μm 1 cm, because of the quasi-onedimensional crystal structure, where the basic structure of TaS3 is atriangular prism whose structure is one metallic atom surrounded bysix chalcogen atoms. The longest dimension is parallel to the axis ofsymmetry of the triangular prism. Our samples of TaS3 belong to theorthorhombic system.12 The crystal of TaS3 displays Pmn21 spacegroup symmetry.23 The lattice constants were found to be a = 36.804
Å, b = 15.177 Å, and c = 3.340 Å. The c axis corresponds to the longest
dimension. Ring/tube crystals, as shown in Figure 2a,b, had typical
thicknesses of 0.1 100 μm, widths of 0.1 10 μm, and radii of
1 100 μm. The one-dimensional axis is parallel to the circumference.
Shown in Figure 2c,d are scanning electron microscopy
images of typical polyhedral topological crystals. The dimensionsof the polyhedral topological crystal in Figure 2c are 500 nm inwidth, 1.8 μm in radius, and 1 μm in thickness. The interiorBdx.doi.org/10.1021/cg2004003 |Cryst. Growth Des. XXXX, XXX, 000–000

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Figure 4. Misorientation proﬁles from an initial point (green circle) to green arrows for (left) ring and (right) polyhedral crystals. The blue line shows
the variation in the orientation from the initial point of the green circle, and the red line shows that from an adjacent point. The insets show theorientations of the crystal axis on each point. The area of the same color corresponds to the same orientation of the crystal axis.

nontrivial and is not conﬁrmed. We found that the crystals mostly
became polyhedral crystals for R < 2 μm in the R W diagram. In
consequence, we drew a boundary between the polyhedral andring crystal areas, indicated by the white line between the blue
and yellow areas in Figure 5. In the end, Figure 5 is divided into
four areas; red, purple, blue, and yellow. We discovered experimentally that the vertices are inherent in ring crystals and that thephase transition from ring to polyhedral crystals is caused by thevariation in radius size.
Our results suggest that the main factor in the polygonizationis the radius-dependent interaction between the edge dislocations. Figure 6 shows the relationship between the interaction energy and the position of dislocations. The red symbols(^ symbols) represent positions of edge dislocations, and the black
curves show the interaction energy.27 When two dislocations
move on the same crystal line or plane, a repulsive force is exerted
between them. As a result of the dislocation motion, plastic
deformation occurs at an edge of a bulk crystal to reduce thestress within it. The passage of a dislocation through a bulk crystal
is equivalent to a slip displacement of one part of the bulk crystal.
This is consistent with a bulk crystal, but in the case of topologicalcrystals, the bullk model is not applicable. Topological crystals
with a closed curve do not have an edge, and hence, slipping
dislocations cannot exit from the crystal. When many dislocations are generated, cylinder-shaped walls of dislocations areexpected to be formed as a result of the arrangement of thedislocations on parallel lines.16,17 Here, we considered two
dislocations moving on diﬀerent crystal lines or planes. Whenthe two dislocations are suﬃciently far apart, a repulsive force isexerted; however, when the distance is short, an attractive force isexerted. As a result of the attractive interaction, the edge
dislocations become concentrated on a vertical line. This iscalled a polygonization wall (Figure 7).28,29 The average distance
between dislocations becomes a function of R. Thus, for aconstant number of edge dislocations, when the radius is large,
a ring crystal is formed with a repulsive force, and when the radius
is small, a polyhedral crystal is formed with an attractive force.
Because the polygonization is controlled by two parameters,
our results are diﬀerent from the structures of other multifaceted
crystals obtained by self-assembly30,31 in terms of the growth

Figure 5. Phase diagrams of topological crystals showing the dependence of the radius and width. Red squares indicate polyhedral crystals,
and black circles indicate ring crystals. The black area shows the range inwhich ring-crystal formation is forbidden. The other three areas divided
by two black lines are based on Hayashi et al.’s theory. When the radius issmaller than 2 μm, the crystals become polyhedrons. The boundary
between polyhedral and ring crystals is indicated by a white line.

≈ 10 2, dx = 0.3 nm, and dy = 2 nm for the equations Rc1 = πW2/2γdy and Rc2 = 2πdy2/γdx in ref 21, where γ2 is Young’s modulus
divided by shear modulus, dx is the lattice constant in the x
direction, and dy is the lattice constant in the y direction. Theboundary radius Rc2 dividing phase III and phase II is equal to133.3 nm, and the boundary radius Rlim, which is equal to thesmallest value of Rc1, dividing phase II and phase I is equal to1138 nm. The boundary Rlim corresponds approximately to thering tube boundary of our results. Hence, the tube crystals
correspond to a clear circle without defects. In the case of ring
crystals containing edge dislocations, the arrangement of them isC

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polygonization phase in the width versus radius phase diagram
for ring-shaped crystals.

’ AUTHOR INFORMATION
Corresponding Author*E-mail: tsubota@eng.hokudai.ac.jp. Phone: +81 (11)706 7293.
Fax: +81 (11)706 7293.

’ ACKNOWLEDGMENT
The authors are grateful to T. Toshima, T. Tsuneta, K.
Ichimura, and K. Yamaya for experimental support and to M.
Hayashi and N. Hatakenaka for stimulating discussions. This
research was supported by a Grant-in-Aid for the 21st Century
COE program on “Topological Science and Technology” andthe Japan Society for the Promotion of Science.
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Figure 6. Interaction energy between the edge dislocations on (a) thesame plane and (b) other planes shifting to the vertical (y) direction. The
red symbols (^ symbols) and black lines show the edge dislocations andinteraction energy in the x direction, respectively. At large distances, arepulsive force is exerted, and at short distances, an attractive force isexerted in the case of b.

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’ CONCLUSIONSIn summary, we have discovered polyhedral topologicalcrystals in the TaS3 system. The formation mechanism is related to the edge dislocations arrayed on a vertical line by thedistance-dependent interactions between them. We propose aD

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