ARTICLE IN PRESS

Journal of Crystal Growth 297 (2006) 157–160
www.elsevier.com/locate/jcrysgro

Topologically linked crystals
Toru Matsuuraa, Masanori Yamanakab, Noriyuki Hatakenakac,Toyoki Matsuyamad, Satoshi Tandaa,aDepartment of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
Department of Physics, College of Science and Technology, Nihon University, Tokyo 101-8308, Japanc
Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, 739-8521, JapandDepartment of Physics, Nara University of Education, Nara 630-8528, Japanb

Received 31 May 2006; received in revised form 21 July 2006; accepted 5 August 2006
Communicated by M. Uwaha
Available online 25 October 2006

Abstract
We discovered a new class of topological crystals, namely linked rings of crystals. Two rings of tantalum triselenide (TaSe3) single
crystals were linked to each other while crystal growing. The topology of the crystal form is called a ‘‘Hopf link’’, which is the simplest
link involving just two component unknots linked together exactly once. The feature of the crystals is not covered by the conventional
crystallography.
r 2006 Elsevier B.V. All rights reserved.
PACS: 02.40. k; 81.10. h
Keywords: A1. Growth models; A1. Low-dimensional structures; A2. Growth from vapor; A2. Single crystal growth; B1. Inorganic compounds; B1.
Sulﬁdes

1. Introduction
Exotic topological crystals, such as ring-shaped crystals,
Möbius strips of crystals, and ﬁgure-of-eight (2p-twistedstrip) crystals, have been successfully created in niobium(Nb) and selenium (Se) compounds (NbSe3) [1,2], despite
their inherent crystal rigidity. These crystals have topological forms consisting of closed strips with one component[3]. And they have opened a new ﬁeld of investigation
related to topological effects in quantum mechanics andnanotechnology [4–11].
Furthermore, we discovered new topological crystals intantalum (Ta) and selenium (Se) compounds (TaSe3),
which are two ring-shaped crystals linked to each otherexactly at once. Due to the link the two rings cannot be
separated without cutting of chemical bonding. The linked
rings of crystals are categorized in a higher class of
topological crystals because the topology involves twoCorresponding author.

E-mail address: tanda@eng.hokudai.ac.jp (S. Tanda).
0022-0248/$ - see front matter r 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.jcrysgro.2006.08.022

components. This topology is called a ‘‘Hopf link’’, which isthe simplest link involving just two component unknots [3].
Hopf links are realized in soft materials such as molecules[12] and DNA [13], which are known as ‘‘catenanes.’’
However, it is very surprising that rigid crystals have these
topological forms. The discovery of Hopf link of crystals
introduces the concept of framed knots [2] in crystallography. In this paper, we report the growing condition of
the crystals and propose a model of formation mechanismof links.
2. Crystal features
Ring-shaped crystals of TaSe3 have been discovered in
1999 [14], following the discovery in NbSe3. TaSe3 crystals
are known to be a one-dimensional inorganic conductorthat exhibits superconductivity below 2 K [15,16]. TaSe3
belongs to the same crystalline family as NbSe3. Thesecrystals are needle-like crystals called whiskers, since theygrow extremely faster along b-axis than along a- and c-axis
due to their one-dimensional crystal structures. Therefore,ARTICLE IN PRESS
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T. Matsuura et al. / Journal of Crystal Growth 297 (2006) 157–160

they can be bent or twisted more easily than threedimensional crystals, resulting in the coexistence of local
crystal structures and global topological forms [1]. As aresult, these crystals possess similar characteristics to soft
materials.
Fig. 1 shows a scanning microscope image of linkedcrystal rings with diameters of 54 and 46 mm. Both rings are
single crystals of TaSe3. The red and blue ring-shaped
crystals are linked exactly once. The linked rings of crystals
have unusual properties associated with framed knots. By
several approaches such as X-ray diffraction, electronbeam diffraction, and electric transport measurement, it
was conﬁrmed that b-axis of ring-shaped crystals of TaSe3
and NbSe3 head along their circumference direction [17].
Therefore, the linking number of the crystals can be deﬁned
by the trajectories of the b-axis. In the case of the Hopf linkof crystals, the linking number is 1. Since the number is a
topological invariant, uncontinuous transformation, suchas cutting of chemical bond, is necessary to separate thetwo rings. Furthermore, if their thickness grows anyhow
the two rings cannot unite into one ring-shaped crystal.

Fig. 1. Scanning electron microscope (SEM) image of linked rings ofcrystal formed of a compound of tantalum and selenium (TaSe3). The blue
and red rings have respective diameters of 46 and 54 mm. The two rings arelinked with each other exactly once.

These features imply that the crystallography must beexpanded within the concept of framed knots.
3. Experimental procedure
The linked rings of crystals have been synthesized bychemical vapor transportation (CVT) method [18,19]. The
synthesis techniques are the same as those used in previous
experiments [1,2] except that they employ an optimized
non-equilibrium condition involving the circulation andsupersaturation of selenium gas in a furnace.
We soaked a mixture of tantalum powder and selenium
granules of 1:3.15 molar ratio, which was put in one of theends of an evacuated quartz tube in an electric furnace thathas a horizontal temperature gradient of 1.5 1C/cm for 3 h.
The length of the tube was 20 cm and the inner diameterwas 17 mm. A total of 5% of excess selenium was added toenhance the circulation and supersaturation of selenium
gas. The highest temperature region in furnace was set at
720 1C. The end of the tube with the mixture was placed at
the highest temperature region. The temperature gradient,temperature near another end was around 700 1C, namely
the growth temperature of TaSe3 crystals. Under thiscondition, the selenium gas circulation carries tantalum tolower temperature region and TaSe3 crystals grow by vapor
phase epitaxy at an optimal temperature region. Furthermore, the selenium gas is condensed to liquid at the lower
temperature region because the gas is supersaturated.
Therefore, many selenium droplets of various diameters
are repeatedly created and annihilated at the lower
temperature region. It is a good condition for producing
topological crystals since the droplets are indispensable for
producing topological crystals [1,2].
A selenium droplet acts as a spool during crystal growth.
The spooling mechanism is a fundamental process ofcrystals having topological forms. A whisker of TaSe3
crystal grows on the surface of a selenium droplet by vapor
and liquid phase epitaxy and encircles it along a great circle
line (Fig. 2A) because of its one-dimensional crystalgrowth. Finally its ends are connected (B), thus yielding aring once the droplet has evaporated (C). This scenario has
been veriﬁed by observation of the NbSe3 crystals spooledaround selenium droplets [1,2]. Therefore, increase of
number of droplets derives increase of production probability of the ring-shaped crystals and other possible
topological forms in crystals.
By the optimization for producing topological crystals of
TaSe3, we obtained many topological crystals more than
one thousand from one tube. The production ratio wasabout 10 times larger than that of NbSe3. Almost all werering-shaped crystals. We found the topologically linked
ring-shaped crystals in them.
4. Discussion
The spooling mechanism can explain the formation
mechanism of the one-component topological crystals such

ARTICLE IN PRESST. Matsuura et al. / Journal of Crystal Growth 297 (2006) 157–160

as ring-shaped crystals, Möbius strip of crystals, andﬁgure-of-eight crystals [1,2]; however, it cannot explain themechanism of the link of crystals. To make a link, thewhiskers on the selenium droplet must under-run anotherwhisker at least once, or crystals must grow on a torus
droplet. Nevertheless, these scenarios are physically unreasonable.

159

In addition to that, the Hopf link topology cannot beproduced by splitting any one component topologicalcrystals since the linked ring-shaped crystals were nottwisted. For example, we obtain Hopf links by splitting themiddle of 2p-twisted. However, this approach provides alink consisting of two 2p-twisted strips because the numberof twists is conserved. Similarly, any np-twisted strips do
not become the Hopf link of untwisted rings by splitting.
Therefore, the Hopf link of crystals is categorized to newclass of topological crystals.
A promising idea is that the linked rings of crystals areformed by a successive spooling mechanism. It seems likelythat similar processes occur successively and produce
linked rings of crystals. After a ring-shaped crystal isproduced by the ﬁrst spooling mechanism, a small selenium
droplet is attached to the ring (Fig. 2D), and then a newring grows on the second droplet (E). The second ring
crystal is linked once when spooling around the second
droplet. Finally, linked rings of crystals are yielded after
the second droplet has evaporated (F). In fact, we often
observed selenium droplets attached to rings (Fig. 3) under
the optimized non-equilibrium conditions. This is strong
evidence for our proposed mechanism. The small droplet
might be the residue of the droplet on which the ﬁrst ring
grew, or it might have become attached to the ﬁrst ring bychance. Our proposed mechanism provides a reasonable
explanation for the different diameters of the two untwisted
seamless rings of linked crystals.
5. Conclusion

Fig. 2. A schematic of the proposed link formation model. Under an
optimized non-equilibrium condition, many selenium droplets of various
diameters are repeatedly created and annihilated. Selenium droplets act asspools for thin crystals. (A) Once a thin crystal has become attached to adroplet, it grows on the droplet surface as a result of interfacial force. (B)The crystal spools around the droplets and ﬁnally the ends of the crystal
are connected. (C) The non-equilibrium condition means that the selenium
droplet should be annihilated leaving a ring-shaped crystal. (D) After that,a selenium droplet is sometimes attached to the ring-shaped crystal. (E)Then another ring-shaped crystal grows on the surface of the second
selenium droplet. While this second crystal grows, it links with the ﬁrst
ring exactly once. (F) Finally, the second droplet is also annihilated
leaving linked rings of crystals.

We have developed our experimental set-up so that it iscapable of forming many droplets in a controlled mannerby using an optimized non-equilibrium condition thatenhances the chance of encountering a target ring. It may
be possible to grow a crystal chain with a desired numberof links, as well as Borromean rings [20] with threecomponents, with various topologies [3] and various similarmaterials [21,22] by controlling the conﬁgurations of theselenium droplets with the help of recently developed
nanotechnology. Our crystals promote the expansion of theFig. 3. SEM images of ring-shaped crystals of TaSe3 with Se condensates (droplets), which seem the starting form of the successive spooling mechanism inFig. 2D. They are the strong evidence for the formation mechanism of linked rings of crystals proposed in this paper.

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ring topology concept called framed knots [3,24] andaccelerate the reorganization of a framework of crystallography including topological molecules [20–23] unifying
the notions of space group, homotopy, and embedding into
curved surfaces [3].
Acknowledgements
This research was supported by the 21 COE program on
Topological Science and Technology from the Ministry ofEducation, Culture, Sport, Science and Technology ofJapan. The research was also supported by the Grant-inAid for Scientiﬁc Research (A) No. 15204029 (S. T., T.
Matsuyama, and N. H.) and the Grant-in-Aid for Young
Scientists (B) No.15740245 (M. Y.) of the Ministry of
Education, Culture, Sports, Science, and Technology,
Japan. One of the authors (T. Matsuura) is grateful tosupport from Research Fellowship of the Japan Society for
the Promotion of Science for Young Scientists.
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