PHYSICAL REVIEW B 83, 174113 (2011)

Topology-change surgery for crystals
T. Matsuura,1 T. Tsuneta,2 G. Kumagai,3 M. Tsubota,3 T. Matsuyama,2 and S. Tanda3,4
1

Research Department, Creative Research Institute, Hokkaido University, Sapporo, Hokkaido 001-0021, Japan
2Graduate School of Education, Nara University of Education, Nara 630-8528, Japan
3Department of Applied Physics, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan
4
Center of Education and Research for Topological Science and Technology, Hokkaido University, Sapporo, Hokkaido 060-8628, Japan(Received 3 March 2011; published 31 May 2011)
We performed the topology-change surgery for ring-shaped crystals of tantalum triselenide (TaSe3 ) toinvestigate the interplay between the closed-ring topology and elasticity/plasticity of the crystals. We cut theTaSe3 rings using a focused ion beam and observed that the curvatures of the open rings increased from theirinitial curvatures. We found that a change in the radius is proportional to inverse square of the thickness of thecrystals, which corresponds to an inhomogeneous distribution of edge dislocations. From the distribution, wesuggest the existence of cylindrical domain walls in the ring-shaped crystals as a result of the crystal topology.
DOI: 10.1103/PhysRevB.83.174113

PACS number(s): 02.40.Pc, 61.72.Hh, 61.72.LkThe surgery method in Mathematics, which changes thetopology of a space by cut and glue, is a powerful tool toextract topological properties, as Perelman solved the Poincaré
conjecture.1,2 Even in nature, the topology-change surgery
is an important approach since the interplay between global
topology and local symmetry would play a fundamental role incosmology, condensed matter physics, and biology.3–5 By thediscovery of rings, Möbius rings and Hopf links of crystals oftransition metal-trichalcogenide compound systems (NbSe3 ,NbS3 , TaSe3 , and TaS3 ),6–9 we obtained experimental systems
for the interplay between the closed-ring topology and localorder parameters. Using the topological crystals, effects ofthe closed-ring topology on charge density waves10–17 andsuperconductivity18 have been investigated.
As well as the ordered phases in electron systems, the crystal orders of the topological crystals have rich problems of theinterplay between the global topology. The crystal structures of
the topological crystals must be different from normal crystals
due to a topological charge, namely, a global disclination.19 As
shown in Fig. 1(a) (left), the global disclination is obtained as
the total rotation angle of a crystal axis (b axis) of a ring-shaped
crystal along a closed curve on the crystal, where it is 2π
for the ring-shaped crystals, and this value is topologicallyrobust against continuous deformations. Due to the global
disclination of power 2π , the crystal lattice cannot be mapped
onto that of perfect crystals globally. Moreover, the topologicalcrystals must be frustrated geometrically by the deformationenergy from bending or twist, and the deformation energy
would induce topological defects into a topological crystal
for relaxation. Hayashi et al. pointed out theoretically thatthe geometrical frustrations (curvature) of the crystals induce
topological defects (edge dislocations), as analogous withvortices in type II superconductors.20 The x-ray diffraction
measurements for ring-shaped crystals of NbSe3 confirmedthat the averaged bending strain in a ring-shaped crystal isrelaxed by defects.21 However, the distributions of strain anddefects have never investigated experimentally since the x-ray
diffraction measurement has less spatial resolution for thesmall crystals due to the spot size of x ray. In principle, theconventional methods with the plane waves of x-ray/electronbeam diffraction are inappropriate for investigation of the1098-0121/2011/83(17)/174113(5)curved crystals since the directions of the crystal lattices aredepend on the position.
In this paper, we performed the topology-change surgery
experiment with the ring-shaped crystals of tantalum triselenide (TaSe3 ) to reveal the interplay between the global
topology and the local crystal properties (elasticity/plasticity).
The crystal structure of TaSe3 is orthorhombic with lattice constants a = 10.402 Å, b = 3.495 Å, c = 9.829 Å,and β = 106.26◦ ,22 and TaSe3 crystals are known to besuperconductors.23 The b axis is parallel to the circumferenceof the ring-shaped crystals similar to the ring-shaped crystalsof NbSe3 .6,7,24 When a ring-shaped crystal is cut as shown in
Fig. 1(a) (right), the global disclination becomes zero. Thus,
the topology-change surgery from closed-ring to open-ring
crystals means eliminating the global disclination of power 2πfrom the crystals similar to that the vortex in a superconductor
ring is eliminated when the ring was cut. As the results, wediscovered that the changes of the curvatures are proportional
to inverse square of the thickness, which corresponds to thenonlinearity of the defect creation. The analysis based on theelastic theory suggests that the line defects (edge dislocations)form a “topological domain wall.”
To cut the ring-shaped crystals, we used the irradiationof a focused gallium ion beam with a spot size of less than
1 μm (acceleration voltage: 30 kV).14,25 Figures 1(b) and1(c) show the scanning ion beam microscopy (SIM) imagesof the topology-change surgery experiment for typical thinand thick ring-shaped crystals, respectively, where each left
panel shows the original shapes of the ring-shaped crystals
and each right panel shows the final shapes after the topologychange surgery. We observed that the thin ring-shaped crystals
opened elastically after the surgery; however, no open-ring
crystals became to be straight (zero curvature), while theglobal disclination became zero. These results indicate thatthe bending stress is partially relaxed by the defects.
To evaluate the elasticity of the crystals, we introduced
the dimensionless parameter (Rf − R0 )/R0 , where we defined
the initial and the final radiuses of curvatures (R0 and Rf ) as
average values for the outer and inner radiuses, and thickness
T as differential of the outer and inner radiuses. Note thatthe shapes of the open-ring crystals could be fitted by circular174113-1©2011 American Physical Society

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PHYSICAL REVIEW B 83, 174113 (2011)

FIG. 1. (Color online) (a) Schematics of topology-change surgery for a ring-shaped crystal. The global disclination of power ω = C dθb
is changed by surgery from 2π (for the closed-ring crystal in the left panel) to zero (for the open-ring crystal in the right), where θb is relative
angle of b axis of TaSe3 crystals (indicated by the arrows). (b) and (c) Scanning ion beam microscopy (SIM) images of typical thin and thickring-shaped crystals of TaSe3 before topology-change surgery (left), and after surgery (right). The thickness T , original radius of ring-shaped
crystals R0 , and final radius of open-ring crystals after topology change Rf are 1.583, 42.45, and 128.4 μm for the thin ring-shaped crystal(b) and 6.00, 12.05, and 12.35 μm for the thick ring-shaped crystal (c), respectively. (d) Ratio of curvatures (Rf − R0 )/R0 as a function ofnormalized thickness T /R0 . The log-log plot is shown in the inset. The solid line indicates the fitting curve given by A(T /R0 )α , where we use
α = −2 and A = 0.0015.

arcs, elliptic arcs, or cycloid/trochoid curves; however, weconsider all of them to be circular arcs with a mean radius ofcurvature Rf as the zeroth approximation in this paper. If aring-shaped crystal were made from a perfect crystal (withoutdefects), Rf should be infinity, since all bending stress could
be elastically relaxed by the topology-change surgery fromclosed ring to open ring. On the other hand, if the bending
stress in the ring-shaped crystal were completely plasticallyrelaxed by defects, the curvature would not be changed by thetopology change; in this case, (Rf − R0 )/R0 = 0. Hence, thevalue of (Rf − R0 )/R0 represents the number of the defects
inside the ring-shaped crystal.
Figure 1(d) shows the experimental data of (Rf − R0 )/R0
as a function of the normalized thickness T /R0 . The colors of
the symbols correspond to the ranges of the initial radius R0 ,and we found that (Rf − R0 )/R0 did not depend on R0 . The
ring-shaped crystals with small T /R0 values (T /R0 < 0.05)
were elastically opened by the topology-change surgery, andfor some ring-shaped crystals, the final radiuses of curvatures
Rf were more than double R0 , and all the crystals were inthe (Rf − R0 )/R0 < 6 range. We note that the distributionof (Rf − R0 )/R0 did not depend on the width of the crystal.
Thus, we will not discuss the width. From the log-log plot of(Rf − R0 )/R0 versus T /R0 shown in the inset of Fig. 1(d), we
found the following relation:
 T α .
Rf = R0 1 + AR0(1)A fitting line with α = −2, and A = 0.0015 is plotted inFig. 1(d) and in its inset. The second term of Eq. (1) indicatesthe plasticity of the ring-shaped crystals. As discussed later, Aand α involve information about the yield strain and strain
distribution, respectively, reflecting the anisotropic crystal
structure of TaSe3 .
We investigated Eq. (1) using a model of ring-shaped
crystals including defects based on the elastic theory.26 Here,
we here assumed that Rf /R0 depends solely on strain for theb axis (parallel to the circumference), while the stress andstrain in the orthorhombic system of TaSe3 crystals should bestrictly expressed by tensors. This assumption is applicableto the ring-shaped crystals since recent results from the x-ray
diffraction measurement for the ring-shaped crystals of NbSe3
provided corroboration that the strain along the b axis ismuch larger than those for other axes.21 Hence, in this paper,
we considered only the strain along the b axis. If there areedge dislocations in the bent material, the free length of thecircumference (parallel to b) is written as 2π R0 (1 − d (x)),
where d (x) represents the change in the lattice number along
the circumference caused by the edge dislocations. The strainwith an arbitrary radius of curvature R is expressed as
r (x,R) ∼x+ d (x),R(2)

where we assumed d (x)  1. Note that, if the density ofdislocations is a constant value [d (x) = −γ x/R0 (1  γ 0)], r (x,R) is constant, and then α in Eq. (1) is 0. The bending
energy density is written as f (x,R) = Y2 r (x,R)2 , where the174113-2

TOPOLOGY-CHANGE SURGERY FOR CRYSTALSPHYSICAL REVIEW B 83, 174113 (2011)

FIG. 2. (Color online) (a) Schematics of strain r (x,R) and bending energy f (x,R) ∝ r (x,R)2 for a bent perfect crystal. The bendingstrain  with a radius of curvature R is expressed as  = Rx , where x is the distance from a center neutral axis along the radial direction.
The bending stress is assumed to be proportional
to strain , such as σ = Y , where Y is Young’s modulus. The free energy density of thebending strain is expressed as f (x,R) = σ ()d, which is shown as a red curve. (b) Model of a bent crystal with edge dislocations. Thestrain r is relaxed by the edge dislocations. Because of the large anisotropy of the crystal, the core sizes of the edge dislocations are strongly
elongated to b axis direction, where y is about 10 times larger than x . (c) Strain r (x,R0 ) [Eq. (5) with A = 0.0015], bending energyof edge dislocations nd (x). The bottom panel shows nd (x) calculated for R0 = 30 μm. nd (x) reaches its maximum valuef (x,R0 ), and density
√
at |x|/R0 = τ/2 ≡ ( 52 − 54 )A ∼ 0.575 A, and then decreases to a constant value.

stress is assumed as σr (x,R) = Y r (x,R). The total elastic
energy of the open-ring crystal isF (R) =2πR0 (1−d (x))

W/2

T /2

f (x,R)dxdydz−W/2−T /20= π Y R0 W

T /2−T /2

r (x,R)2 (1 − d (x))dx.

(3)

Thus, we obtained the condition of the final radius of curvatureof the open-ring crystals by dFdR(R) |R=Rf = 0, as follows:
112=− 3RfT

T /2−T /2xd (x)dx.

(4)Here, we note that Rf does not depend on Y . d (x) in thering-shaped crystals is obtained from Eq. (4) by using Eq. (1)with α = −2:
 R05+ Rx0A12x(5)d (x) = −   2
2 ,A R2x0 + 1
where we assumed that d (x) is an odd function. The strain
distribution r (x,R) for R = R0 is given by Eqs. (2) and (5).
As shown in Fig. 2(c), r (x,R0 ) deviates from x/R0 when|x|/R√
its maximum value at |x|/R0 =0 increases, and exhibits√
0.341 A. rmax (= 0.183 A) with A = 0.0015 is estimated at
0.71%, the value of which is of the same order as the x-ray
diffraction measurement results for the ring-shaped crystals√of NbSe3 .21 For |x|/R0  0.341 A, the strain r (x) slowly
approaches zero as |x|/R0 is increased.
We found that the distribution of the edge√dislocationsnd (x) has two peaks at x/R0 = ±τ/2 ∼ ±0.575 A, as shownin Fig. 2(c), which suggests that the dislocations gather
on ring with radius
R = R0 ± τ/2. Here, we calculated∗dnd (x) = − 2πRb0 a dx
d (x), where a ∗ = a sin β and b are thelattice constants along the radial and circumference directions,respectively. The peaks indicate that elasticity is dominant
when τ > T /R0 and plasticity is dominant when τ < T /R0 .
If edge dislocations gather on a circle in the ring-shapedcrystals, the cores of the dislocations might overlap eachother and form a cylindrical domain wall as anticipated in
our previous paper,21 similar to the cylindrical vortices in thering-shaped crystals of TaSe3 .18 The cylindrical domain walls
can relax the bending stress homogeneously conserving theglobal disclination. If the strain  = τ/2 is relaxed to zero bya cylindrical domain wall, one edge dislocation is induced for
0 τ
approximately every 45 lattices along the b axis (Nd = 2πR).
b 2The value corresponds fairly well with the order of core
size of the edge dislocations along the b axis ( y = a ∗ /γ ,where γ ∼ 10−1 is the parameter representing anisotropy ofthe crystal structure20 ). Thus, the domain size τ is determinedby the circumference length of the ring-shaped crystals. This
must be the interplay between the global topology and the local
crystal orders.
Furthermore, the cylindrical domain walls would be introduced periodically in thick ring-shaped crystals of T /R0 > 2τ .

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PHYSICAL REVIEW B 83, 174113 (2011)

FIG. 3. (Color online) (a) SIM image of a thick ring-shaped crystal of TaSe3 . On the surface, the concentric lines with a period ofT ∼ 0.8 μm are observed (indicated by the red curves). R0 = 11 μm and T /R0 ∼ 0.072, which is in the order of τ . On the lines, theselenium droplets are also observed. (b) Frequency distribution of randomly selected ring-shaped crystals as a function of T /R0 . The blue
broken curve is a guide for the eyes showing the periodic peaks of the frequency distribution. The black open circles indicate the sort numberof each ring-shaped crystal numbered in order starting with the smallest T /R0 value, and show the periodic step structures. (c) Model ofcylindrical domain walls in the ring-shaped crystal and strain distribution inside crystal.

As shown Fig. 3(a), we found the concentric rings on thesurface of some thick ring-shaped crystals (indicated by redcurves), and on which the selenium droplets congregated.
The period ( T /R0 ) of the rings is in the order of τ . We
also found the frequency (number) distribution of normalized
thickness, T /R0 , of randomly selected ring-shaped crystals
oscillating in the period of τ/2, as shown in the histogramof Fig. 3(b). The number of the rings of a value of T /R0
would increase if the growing speed of thickness becomesslow at the thickness. Hence, the oscillation implies that thecylindrical domain walls induce periodically as a functionof T /R0 . When T /R0 becomes large, the growing speed ofthickness T for the ring-shaped crystals decreases due to anincrease in the bending stress energy [Fig. 3(c), left panel], andif a cylindrical domain wall is introduced, as shown in Fig. 3(c)for T /R0 > τ , the stress is relaxed, and then the growing
speed recovers. These two results suggest that introducing thecylindrical domain walls is the main mechanism of relaxation
for the bending strain in thick ring-shaped crystals. We notethat for T /R0 > τ , (Rf − R0 )/R0 would oscillate with a periodof τ due to the introduction of the domain walls; however,
the amplitude of the oscillation would be smaller than theexperimental noise in Fig. 1(d). The defect density between
the two domain walls should be small, the crystal structures
would be clean as well as single crystals. These features of

the crystals are consistent with the results of the previous
experiments in which we have observed clear charge density
waves and superconductivity in the ring-shaped crystals aswell as single crystals.10–15,18
In summary, we demonstrated that the topology-change
surgery gives us valuable information in crystals. By cutting off
the ring-shaped crystals of TaSe3 , we found a phenomenological relation between the radius of curvature after the topology
change and thickness. From the relation, we extracted thecrystal domain size (τ ) of the ring-shaped crystals. These
results strongly suggest that the cylindrical domain walls
are induced in the ring-shaped crystals, due to the interplay
between the closed-ring topology and the crystal structures.
The topology-change surgery is a significant methodology toextract topological effects, and would be applicable to other
topological materials, such as multiwall nanotubes27,28 andbiomolecules.5
We thank M. Oda, H. Orihara, M. Hayashi, Y. Nogami,
K. Yamamoto, T. Shibayama, K. Yamaya, K. Inagaki,
T. Toshima, and H. Okawa for their fruitful discussions andsupports. This research was supported by the Center of Education and Research for Topological Science and Technology,
Hokkaido University, and the Japan Society for the Promotion
Science.

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1

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M. Dehn, Acta Math. 69, 135.
G. Perelman, e-print arXiv:math/0303109v1.
3
J. P. Luminet, J. R. Weeks, A. Riazuelo, R. Lehoucq, and J. P. Uzan,
Nature 425, 593 (2003).
4
G. E. Volovik, Trans. R. Soc. A 366, 2935 (2008).
5
Y. Timsit and P. Várnai, PLoS ONE 5, e9326 (2010).
6
S. Tanda, T. Tsuneta, A. Y. Okajima, K. Inagaki, K. Yamaya, andN. Hatakenaka, Nature 417, 397 (2002).
7T. Tsuneta and S. Tanda, J. Cryst. Growth 264, 223 (2004).
8S. Tanda, H. Kawamoto, M. Shiobara, Y. Okajima, and K. Yamaya,
J. Phys. IV 9, 379 (1999).
9T. Matsuura, M. Yamanaka, N. Hatakenaka, T. Matsuyama, and S.
Tanda, J. Cryst. Growth 297, 157 (2006).
10Y. Okajima, H. Kawamoto, M. Shiobara, K. Matsuda, S. Tanda, andK. Yamaya, Physica B 284, 1659 (2000).
11T. Tsuneta, S. Tanda, K. Inagaki, Y. Okajima, and K. Yamaya,Physica B 329–333, 1544 (2003).
12T. Matsuura, K. Inagaki, and S. Tanda, Phys. Rev. B 79, 014304(2009).
13M. Tsubota, K. Inagaki, and S. Tanda, e-print arXiv:0906.5206.
14
T. Matsuura, T. Tsuneta, K. Inagaki, and S. Tanda, Phys. Rev. B 73,165118 (2006).
15
K. Shimatake, Y. Toda, and S. Tanda, Phys. Rev. B 73, 153403(2006).
16
T. Nogawa and K. Nemoto, Phys. Rev. B 73, 184504 (2006).
2

17

M. Hayashi, H. Ebisawa, and K. Kuboki, Phys. Rev. B 76, 014303(2007).
18
G. Kumagai, T. Matsuura, K. Ichimura, K. Yamaya, K. Inagaki, andS. Tanda, Phys. Rev. B 81, 184506 (2010).
19
A. E. Romanov and V. I. Vladimirov, “Disclinations in crystrallinesolids,”in Dilocations in Solids, edited by F. R. N. Nabarro (Elsevier
Science Publishers B.V., Amsterdam, 1992), Vol. 9, Chap. 47.
20
M. Hayashi, H. Ebisawa, and K. Kuboki, Europhys. Lett. 76, 264(2006).
21
T. Tsuneta, K. Yamamoto, N. Ikeda, Y. Nogami, T. Matsuura, andS. Tanda, Phys. Rev. B 82, 014105 (2010).
22
E. Bjerkelund and A. Kjekshus, Acta Chem. Scand. 19, 701 (1966).
23
T. Sambongi, M. Yamamoto, K. Tsutsumi, Y. Shiozaki, K. Yamaya,and Y. Abe, J. Phys. Soc. Jpn. 42, 1421 (1977).
24
J. L. Hodeau, M. Marezio, C. Roucau, R. Ayroles, A. Meerschaut,
J. Rouxel, and P. Monceau, J. Phys. C 11, 4117 (1978).
25
T. Matsuura, S. Tanda, K. Asada, Y. Sakai, T. Tsuneta, K. Inagaki,and K. Yamaya, Physica B 329, 1550 (2003).
26
L. D. Landau, L. P. Pitavskii, E. M. Lifshitz, and A. M. Kosevich, Theory of Elasticity, 3rd ed., Vol. 7 (Theoretical Physics)(Butterworth-Heinemann, United Kingdom, 1986).
27
D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda,
A. Dimiev, B. K. Price, and J. M. Tour, Nature 458, 872 (2009).
28
L. Jiao, L. Zhang, X. Wang, G. Diankov, and H. Dai, Nature 458,877 (2009).

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