PHYSICAL REVIEW B 82, 014105 共2010兲

Relaxation of geometrical frustration in NbSe3 topological crystals
T. Tsuneta*
Graduate School of Education, Nara University of Education, Nara 630-8528, JapanK. Yamamoto, N. Ikeda, and Y. Nogami
Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan

T. Matsuura and S. Tanda
Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan
共Received 3 December 2008; revised manuscript received 17 May 2010; published 8 July 2010兲
Here we report the structural study of NbSe3 topological crystals via synchrotron x-ray measurements on
individual microcrystals. Topological crystals are characterized by the shape of a ring or a twisted loop, thereby
representing a translational and orientational order confined in topologically nontrivial, curved geometries. The
crystal lattice of topological crystals is under considerable amount of strain. The volume-averaged strain
asymptotically increases with thickness and curvature of a crystal. We present a structural model in which aring crystal is composed of well-ordered domains in the shape of concentric cylinders. The internal structurereflects periodic accumulation and release of strain during the growth.
DOI: 10.1103/PhysRevB.82.014105

PACS number共s兲: 61.72.Bb, 61.72.Hh, 71.45.Lr

I. INTRODUCTION

Recently, we have demonstrated that transition-metaltrichalcogenide 共MX3, M = Nb or Ta and X = S or Se兲, a groupof fibrous and pliable materials, can grow as a microcrystal
in the shape of a ring, Möbius strip,1 and Hopf link.2 Termed“topological crystals,” they are the solid-state manifestationof topologically nontrivial geometries. Owing to their onedimensional nature, MX3 offer higher structural freedom thanpreviously known two-dimensional materials, e.g.,fullerenes3,4 共sphere兲 and nanotubes5 共cylinder兲.
Topological crystals intrigued many with unexploredphysics in multiconnected geometries.6–9 A particularly interesting feature of MX3 topological crystals is their charge
density wave 共CDW兲,10,11 a macroscopic quantum state.
CDW emerges as a periodic modulation of electron-density
coupled to ionic displacement. Mechanical deformations canaffect its phase order, e.g., in a manner analogous to a superconductor under a magnetic field.12 In fact, a series of experiments and simulations have revealed that CDW confinedin the ring geometry has weakened intermolecular chain ordering. Namely, an enhancement of Ginzburg-Landau
fluctuations,13–15 a lowering of CDW pinning potential,16 andhigher damping time of phason excitations17 in NbSe3 topological crystals are reported. A unified understanding of these
phenomena is still lacking.
However, little is known about how a topological crystal
retains its crystal order against geometrical frustration. Suppose a crystalline order native to the Euclidean R3 is confined in a cylindrical space R2 ⫻ S1. As the flat space cannot
be mapped to the curved space isometrically and conformally at the same time, the translational and rotational orderwould acquire increasing strain depending on the position. In
such a frustrated system, the original order may partiallybreak into a new, unfrustrated structure.18,19 Alternatively, it
may develop a defect substructure characterized by periodic
accumulation and release of strain as in Frank-Kasper
metals20,21 and liquid crystal blue phases.22,23 These orders
1098-0121/2010/82共1兲/014105共6兲

are known to perfectly fit curved space but when confined ina flat space, they develop characteristic disclination networks. NbSe3 topological crystal is rare self-organized
curved crystal that originally fits a flat space. And hence its
structure and defect substructure is an interesting problem. In
this paper, we report the structural study of individual, pristine topological crystals.
II. EXPERIMENT

NbSe3 belongs to space group P21 / m with lattice parameters, a = 10.009, b = 3.4805, c = 15.629 Å, and ␤ = 109.47°.24
The crystal structure is composed of infinitely long, onedimensional molecular chains aligned in the crystallographic
b, which coincides with the circumference of topologicalcrystals. The chains are bonded to each other in an anisotropic manner, forming stronger ionic/covalent bonding in c
and weaker van der Waals bonding in a.25 Consequently,
NbSe3 usually forms a thin whisker crystal.
Our samples fall into three categories: whisker, ring 共more
specifically, cylinder兲, and “figure-eight” strip, which is aloop with a twist of 2␲ on circling around the circumference.
Figures 1共a兲 and 1共b兲 shows typical samples. See Ref. 26 forsample growth conditions. Figure 1共c兲 illustrates the coordi(a )

(b )

(c )
r

c

a *

W
R

b

T

FIG. 1. Typical topological crystal samples: 共a兲 ring and 共b兲
figure-eight strip. Scale bars: 100 ␮m. 共c兲 Crystallographic axes aⴱ,b, and c, and apparent size parameters radius R, thickness T, andwidth W. r, ␪, and z of the cylindrical coordinates are parallel to aⴱ,b, and c, respectively.

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PHYSICAL REVIEW B 82, 014105 共2010兲

TSUNETA et al.

(600)

(020)

RWP=4.3
S=1.5
RI=1.1

Intensity (arb. unit)

Intensity (arb. unit)

10000

5000

thin

thick

32.8

0

33

33.2

36.6

36.8

2θ (deg)40
2θ (deg)60

FIG. 3. Diffraction profiles of 共020兲 and 共600兲 peaks, for thin
T / R = 0.19 共upper兲 and thick T / R = 2.0 共lower兲 rings. Curves show
Gaussian fits.

FIG. 2. A Rietveld analysis on a ring sample with T / R = 0.19.
The marks and the overlapping curve represent observed and calculated intensity, respectively. The difference between them is plotted below. Ticks are expected peak positions. The agreement factors
are included in the figure.

dered sample.24 This strain is presumably caused when thewhisker sample is cut off. This gives an upper limit of theuncertainty of our pseudopowder diffraction method.
Further analyses of diffraction profiles are conducted for
two of the ring samples. Isolated diffraction peaks are measured using a four-circle diffractometer and fitted to Gaussian curves, as shown in Fig. 3. The broadening of peaks ismeasured by half widths at half maximum C. C includes
contributions from two factors, nonuniform lattice strain andfinite size of domains of coherent diffraction. They are separated by the Williamson and Hall’s method30 in which theroot-mean square of strain distributions ⌬␧i 共i = a , b , c兲 andthe mean domain size li are related to C asnate system and three shape parameters: thickness T, radiusR, which is defined at the middle of the thickness, and width
W. Cylindrical coordinate axes r, ␪, and z correspond to crystallographic axes aⴱ, b, and c, respectively.
We performed x-ray diffraction measurements employing
synchrotron radiation at beam line BL02B1 of SPring-8. We
could obtain intensity 1010 photons/ s at the maximum,
which is enough for our samples as small as ⬃10 ␮m⫻ 50 ␮m ⫻ 50 ␮m. The beam is monochromatized to ␭
= 0.9917– 0.9934 Å. A specially designed Gandolfi camera27
is used to simulate powder diffraction photographs. Intensity
is collected using an imaging plate and integrated into intensity vs 2␪ data.
Structural refinement was carried out by the Rietveld
analysis.28 The peak-shape, prefered orientation, absorption,
background, zero-point, scale, lattice, and crystal structuralparameters were refined using RIETAN-2000 共Ref. 29兲 software. The peak shape is assumed to be a pseudo-Voigt function. Also it is assumed that the atomic occupancy is 1 andthe isotropic atom displacement parameter is common
among an atomic species. Figure 2 shows a Rietveld analysis. Agreement factors were typically as good as S = 1.5,
RWP = 4.3, and RI = 1.1 The lattice parameters of a single crystal whisker were determined as a = 10.008共2兲, b = 3.4860共7兲,
c = 15.632共4兲 Å, and ␤ = 109.45共1兲°. Only b has a nonnegligible deviation of 0.16% from the measurement of pow(a)(b)∆εi (%)εi (%)

0.2

0

C cos ␪ K 2⌬␧i sin ␪
= +
,
␭
li
␭

共1兲

where ␪ is the Bragg angle and K is the Scherrer constant.
We set K = 1.0. C cos ␪ / ␭ would be a linear function ofsin ␪ / ␭, whose slope and intersect correspond to 2⌬␧i and l1i ,
respectively.
III. RESULTS

The lattice parameters of a topological crystal turned out
under certain measure of strain. Volume-averaged latticestrain ␧i 共an overline denotes an average兲 is defined withreference to lattice parameters in Ref. 24. Obtained ␧i areplotted in Fig. 4共a兲 共marks兲. Although the data are highly
scattered in b, they have a tendency to contract in a while
10

(c)

0

10

−1

10

−2

li (µm)

20

a*
b
c

0.4

E11 E33

0.2

58 101
52 89

−0.2
0

1

20

1

20

1

2

10

46 77

−3

0

T/R

1

T/R

2

0
0

1

2

T/R

FIG. 4. 共Color online兲 共a兲 Volume-averaged lattice strain ␧i 共i = a , b , c兲 plotted against a shape parameter T / R. 共b兲 Root-mean-squared
strain ⌬␧ and 共c兲 mean size of coherent domains li of two ring samples. Circles and triangles represent ring and figure eight, respectively.
Curves show data calculated for different elastic moduli.
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RELAXATION OF GEOMETRICAL FRUSTRATION IN NbSe…

−1

C cosθ / λ (nm )

(a)

(b)

(c)

{100}−{800}{001}−{006}

0.1

0.02

tained so that the deformation is pure bending, an elastic ringundergoes stress

0.01

{040}

␴b共r兲 = E22

0.008

{020}

0.05

0.01

2

4

0
0

␴a = ␴c = 0,

共2兲

as a functions of r. Corresponding strain is given by
0.006

0
0

r−R
,
R

2

4

6

0

2

␧b共r兲 =

4−1sinθ / λ (nm )FIG. 5. The Williamson-Hall plots for diffraction series 共a兲
兵h00其, 共b兲 兵0k0其, and 共c兲 兵00l其. Filled and open circles represent
peak profiles while dashed and solid lines represent linear fits, ofthe thin and thick samples, respectively.

expanding in b and c, monotonously depending on T / R.
Strain in ring and figure-eight samples is alike. Besides, nolowering of the crystal symmetry is observed. The existenceof 21 screw axis is confirmed by the intensity I共020兲 / I共040兲
⬎ 104 for rings and I共020兲 / I共040兲 ⬎ 103 for figure-eight strips.
According to peak profile analyses, ⌬␧i has weaker T / R
dependence. Figure 5 shows reasonably linear WilliamsonHall plots for selected diffraction series: 共a兲 兵h00其, 共b兲 兵0k0其,and 共c兲 兵00l其. Obtained ⌬␧i are exhibited in Fig. 4共b兲. While
the magnitude of ␧i in each direction is alike, ⌬␧i in ring
samples are highly anisotropic 共⌬␧b : ⌬␧a : ⌬␧c ⬃ 100: 20: 1兲
and largest in the chain direction b. As plotted in Fig. 4共c兲,mean size of coherent domains are estimated at roughly
0.2 ␮m for the interchain directions while it is larger in the
chain direction. These estimations give the lower limit of li
since peak broadening for 0.2– 0.4 ␮m is similar to broadening originating from the measurement apparatus.
IV. DISCUSSION

Let us evaluate lattice strain for the case with an elastic
continuum body. Here we consider a rectangular material,T ⫻ 2␲R ⫻ W in dimensions, bent into a ring. The three principal axes of strain should coincide with the cylindrical coordinate axes of the ring. It is inferred that orthogonal crystallographic axes of NbSe3, b, c, and aⴱ, take on them in an
actual crystal. For the sake of simplicity, hereafter we treat
NbSe3 as an orthorhombic material with these axes as thecoordinate axes.
Obviously, it is unrealistic that a ring crystal remains perfectly elastic. Assuming that the rotational symmetry is re-

ε b

r

r−R
,
R

␧a = − ␯21␧b,␧c = − ␯23␧b .

共3兲

Eii 共i = 1 – 3兲 and ␯ij 共j = 1 – 3, i ⫽ j兲 represent anisotropic
Young’s moduli and Poisson’s ratios, respectively. According
to these equations, stress would be as large as 10 GPa evenfor our thinnest sample 共t / R = 0.19兲, far exceeding yieldstress of usual matters. Moreover, volume-averaged strain inany direction would be zero in contrast to actual data.
Figure 6 describes a growth model of a ring crystal incorporating the generation of dislocations. 共a兲 A beam of NbSe3
crystal is bent into a ring. Here it should be remarked that aring crystal tends to grow inward.26 共b兲 The growth front at
the inner surface is always under contractive stress. However, while the crystal is defect free, it can outwardly expand
so that the strain profile remains symmetric as Eq. 共2兲. 共c兲When local strain 兩␧b兩 reaches to an elastic limit k at the innersurface, the crystal yields. Dislocations with Burger’s vector
关010兴 are introduced to relax local strain down to ␧b = ␣k
共0 ⱕ ␣ ⬍ 1兲. We assume ␣ = 0 共total relaxation兲 for the time
being. Dislocations stay on the same atomic layer because
the putative glide plane b-c is normal to the radius. In effect,
the array of dislocations makes a domain wall between two
defect-free cylindrical shells. 共d兲 The subsequent growth accumulates contractive strain again, and the inner shell expands while the outer shell presses back. 共e兲 The two shells
attain an equilibrium position depending on their thickness.
When the maximum strain at the growth front reaches to kagain, the process repeats itself.
Now let us estimate strain distribution in this model. Total
strain is given by the linear sum
␧i = ␧i⬘ + ␧i⬙ ,共4兲where ␧i⬘ and ␧i⬙ are contributions from bending and interlayer pressing, respectively.
␧i⬘ would not be as simple as the pure bending solution
关Eq. 共3兲兴. While the solution predicts that the ratio of Poisson
strain would be ⌬␧a / ⌬␧c = ␯21 / ␯23 ⬃ 1, it is as large as 14 in
actual samples. It is probably because the pure bending solution implies that a-c cross sections are deformed into aα kkFIG. 6. 共Color online兲 A growth model of a ring crystal. The color density indicates the magnitude of the strain, which is contractive in
the inner half and expansive in the outer half. Small insets in 共b兲–共e兲 are strain profiles as a function of r. The other inset in 共c兲 illustrates
a domain boundary composed of dislocations formed on one of atomic layers 共solid lines兲.
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PHYSICAL REVIEW B 82, 014105 共2010兲

TSUNETA et al.

Dn共r兲 =

t2

= 1

r1
2

r2

冉

冊

2
2
ri,n
ro,n
+ r2
.
2
2
2
r ro,n − ri,n共7兲pn represents the radial stress component at the interface between the nth and the 共n + 1兲th shells, i.e.,

r i,2 ro ,2⬙ 共ri,n兲 = ␴a,n+1⬙ 共ro,n+1兲.
pn = ␴a,n3

共8兲Strain is then obtained asM⬙ 共r兲 =
␧a,nFIG. 7. 共Color online兲 Geometrical configuration of a ring crystal composed of concentric cylindrical domains. ri,n and ro,n are theinner and outer radii with no external force applied, which does notnecessarily coincide with actual radii. rn and tn are the radius ofneutral plane and the neutral thickness, respectively.

bent rectangle. It causes mismatch at the boundary between
cylindrical shells involving 关001兴 dislocations. However, it ispresumed that ␧b⬘ reaches to the elastic limit before ␧c⬘ andthus 关010兴 dislocations dominate over 关001兴. Thereby ␧b⬘abruptly changes at the boundaries while ␧c⬘ should remain
continuous. We adopt the simplest bending deformation that
ensures the continuity of ␧c⬘, namely,⬘ 共r兲 = − 共␯21 + ␯23␯31兲
␧a,n⬘ 共r兲 = 共1 − ␯23␯32兲
␧b,nr − rn,rnr − rn,rn⬘ = 0,
␧c,n

共5兲

where rn represent the radius of the neutral plane of the nthlayer.
Stress due to interlayer pressing is determined by the distribution of the neutral thickness and radius of each layer.
Figure 7 illustrates a ring crystal composed of N cylindricalshells. The size of the nth shell is defined by its outer andinner radii with no external force applied, ro,n and ri,n, respectively. Its neutral plane is then at rn = 共ro,n + ri,n兲 / 2. The
neutral thickness is given by tn = ro,n − ri,n. Stress within thenth shell is expressed as31

⬙ 共r兲 = An共r兲pn−1 + Bn共r兲pn ,␴a,n⬙ 共r兲 = Cn共r兲pn−1 + Dn共r兲pn ,␴b,nand ␴c,n⬙ = 0, where
An共r兲 =

冉
冉
冉

冊
冊
冊

共6兲

Cn共r兲 = −

2
2
ro,n
ri,n
+ r2
,
2
2
2
r ro,n − ri,n共9兲Radial displacement un共r兲 is derived as⬙ 共r兲,
un共r兲 = rn␧b,n

共10兲

a linear combination of pn−1 and pn.
Once the distribution of ro,n and ri,n 共n = 1 , . . . , N兲 is determined, the condition of continuity of the interface position
ri,n + un共ri,n兲 = ro,n+1 + un+1共ro,n+1兲

共11兲

provides linear simultaneous equations that are solved for pn
共n = 1 , 2 , . . . , N − 1兲. Note that no external stress is applied at
the surface, i.e., p0 = pN = 0. Then we can calculate stress andstrain from a full set of pn.
We run a numerical simulation of the growth process.
Starting from a thin elastic layer having a neutral radius r1= 40 ␮m, we increased the thickness of the innermost layerby small steps, e.g., by one atomic layer 共⬃Å兲. An additionof thickness dt increases ri,n and ro,n by dt / 2, respectively.
The growth of a layer stops when 兩␧b⬘ + ␧b⬙兩 reaches to k at the
inner surface and the next 共n + 1兲th layer is formed. Local
strain in the 共n + 1兲th layer is relaxed to zero, i.e., its neutral
plane coincides with the interface such as
rn+1 = ri,n + un共ri,n兲.

共12兲

At the emergence of each layer, we computed the volume
average of layer thickness and strain, and standard deviationof strain by Simpson’s rule. They correspond to la, ␧i, and⌬␧i, respectively.
Our simulation depends on six uncertain elastic moduli Eii
and ␯ij but few of them are arbitrary. The Young’s modulus
along the chain axis of MX3 varies in the range of 100–300
GPa depending on measurement technique: an average of
350 GPa 共Ref. 32兲 for NbSe3, 100 GPa 共Ref. 33兲, and 350
GPa 共Ref. 34兲 for TaS3, and 200 GPa 共Ref. 35兲 for ZrTe3.
Here we adopt E22 = 200 GPa. Three additional moduli aredetermined from the linear isothermal compressibility of
NbSe3,36 Ka = 1.4% / GPa, Kb = 0.13% / GPa, Kc = 0.59% / GPa,and K␤ = −0.17% / GPa, according to

2
2
ro,n
ri,n
− r2
,
2
2
r2 ro,n− ri,n2− r2
r2 ro,n,Bn共r兲 = − i,n22r2 ro,n− ri,n1␯⬙ 共r兲 − 21 ␴b,n
⬙ 共r兲.
␴a,n
E11
E22

Ki =

1
␯ij
+兺
.
Eii j⫽i E jj

共13兲

Further, the moduli must be narrowed down in order to give
physically meaningful Poisson’s ratio. If we assume 0.05
⬍ ␯ij ⬍ 0.45, only remaining arbitrary parameters are E11= 52⫾ 6 and E33 = 2.8E11 − 58⫾ 5 GPa.
The results of calculations are plotted in Fig. 4 for k= 0.44% 共arbitrarily chosen兲 and several combinations of E11
and E33. Better fits are given by more anisotropic combina014105-4

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RELAXATION OF GEOMETRICAL FRUSTRATION IN NbSe…

rn+1 =

ri,n + un共ri,n兲
1 + ␣kS. Tanda, T. Tsuneta, Y. Okajima, K. Inagaki, K. Yamaya, and N.
Hatakenaka, Nature 共London兲 417, 397 共2002兲.
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3 R. Tenne, J. Mater. Res. 21, 2726 共2006兲.
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S. Iijima, Nature 共London兲 354, 56 共1991兲.
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共2003兲.
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024505 共2005兲.
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Synth. Met. 152, 261 共2005兲.

εi (%)

α=0
= 0.6
= 0.9

0.2

0

0

−0.2

−0.2

−1

10

−1

−2

10

−2

10

−3

10
10

−3

10

0

1

2

0

T/R

1

2

T/R

FIG. 8. 共Color online兲 ␧i 共i = a , b兲 and ⌬␧i 共i = a , b , c兲. Curves arecalculations for E11 = 46, E22 = 200, E33 = 77 GPa and 共a兲 ␣ = 0 andvarying k, 共b兲 k = 0.45% and varying ␣. Symbols represent measured values.

layer pressure is the main source of ␧c, ⌬␧c increases. It
affects in the opposite direction to the stark anisotropy in ⌬␧iin the measured data. Again, although quantitative evaluation
is difficult, the extreme case ␣ → 1 would be ruled out.
In summary, we proposed that a ring crystal is composedof well-ordered domains in the shape of concentric cylinders,which are divided by walls of dislocations. This should be
one typical aspect when a translational and orientational order is confined in a curved geometry. Possibly, the enhanced
CDW fluctuations in topological crystals are related to thedomain structure. We left the figure-eight geometry for futurestudies since its beyond the present approach.
ACKNOWLEDGMENTS

This research was supported by the 21 COE program on
Topological Science and Technology by the Ministry of Education, Culture, Sport, Science and Technology of Japan and
the Japan Society for the Promotion of Science. The authors
thank to K. Inagaki and M. Hayashi for fruitful discussions.

Y. Asano, Phys. Rev. B 72, 092508 共2005兲.
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9*tsuneta@nara-edu.ac.jp(b)k = 0.45 %= 0.15= 0.050.2

共14兲

instead of Eq. 共12兲. At the limit as ␣ → 1, every atomic layer
has few dislocations and the relaxation of local strain is incomplete. A ring crystal would have higher interlayer pressure in the inner half, resulting in larger ␧i. In fact, as shownin Fig. 8共b兲, ␣ ⫽ 0 results in greater ␧i. Also since the inter-1(a)∆εi (%)

tions, e.g., E11 = 46, E22 = 200, and E33 = 77 GPa, implyingthat the chain axis is three to five times harder than the others. This magnitude of anisotropy agrees with that of thermal
expansion of NbSe3.24 The results reproduce overall qualitative features, namely, the sign and the order of magnitude of
␧i and the flatness and anisotropy of ⌬␧i. The difference
between experiments and calculations is ascribed to oversimplification as well as the uncertainty of elastic moduli. For
example, a local change in growth conditions or the stochastic nature of grain boundary formation may disturb the balance of interlayer pressing, resulting in scattered ␧b. In any
case, the major contribution of ␧a is ␧a⬙ and it is always
contractive since it originates from interlayer pressure.
Our result clearly suggests the existence of cylindershaped domains bounded by dislocation walls. Dislocation
walls hamper a ring crystal from expanding so that the volume average of strain is relaxed to zero. The observed features, namely, nonzero ␧i and highly anisotropic ⌬␧i, are theresult of this.
An alternative possibility is that dislocations freely penetrate into a ring to negate local bending strain. If that is thecase, however, both ␧i and ⌬␧i would have been far smaller.
In fact, our calculations show that ␧i and ⌬␧i asymptoticallyapproach zero when elastic limit strain tends to zero, or k
→ 0, as in Fig. 8共a兲. Although quantitative evaluation is difficult, k should be no less than ⬃0.1%, corresponding to areasonable magnitude of yield stress, ⬃100 MPa.
At the other extreme, dislocations may be more sparse.
When the growth front yields, not necessarily totally relaxation occurs because of interdislocation repulsion. This isexpressed by 0 ⬍ ␣ ⬍ 1 instead of ␣ = 0 and thus

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17

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