Geometrically Frustrated Crystals: Elastic Theory and Dislocations ∗ Masahiko Hayashi and Hiromichi Ebisawa Graduate School of Information Sciences, Tohoku University, Aramaki Aoba-ku, Sendai 980-8579, Japan and JST-CREST, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan 6 Kazuhiro Kuboki 0 Department of Physics, Kobe University, Kobe 657-8501, Japan 0 (Dated: February 6, 2008) 2 Elastic theory of ring-(or cylinder-)shaped crystals is constructed and the generation of edge n dislocations due to geometrical frustration caused by the bending is studied. The analogy to su- a perconducting(orsuperfluid)vortexstateispointedoutandthephasediagram ofthering-crystal, J which dependson radius and thickness, is discussed. 3 2 PACSnumbers: 61.72.Bb,61.82.Rx,71.45.Lr,74.25.Op ] i c What is the lowest energy state of matter? The an- topological physics. For example, TaS3 and NbSe3 show s swer to this fundamental question has been believed to superconductivity and charge density wave order under - l be the “crystalline state” as far as the system has an appropriate conditions [14], and, then, topology-induced r t infinite extension and quantum fluctuation is negligible. novelstatesareexpected. Severaltheoreticalworkshave m Most elements or compounds freeze into crystals when been elaborated in this direction [15, 16, 17, 18], and . cooled down slowly. Even the electrons form the so- experimental investigations are under way. t a called Wigner crystal if the density is low enough. The m basic character of the crystalline state is the breaking of - translationalsymmetry; the system is divided into small d fundamental cells. Although some exceptions, such as n Penrose lattice or quasicrystals, have been known, this o profound form of existence occupies the most intriguing c [ expositions in the “mineralogical department of natural history museum”[1]. 1 In recentdecades, with the help of developedfine pro- v 4 cessing technology, especially symbolized by the term 0 “nanotechnology”, a new field seems to be added to 5 the studies of crystallography. The topic of this field 1 is the crystallization in nano-scales. A group of mate- 0 rials which is most attracting attention from both basic 6 and applied scientists is evidently Fullerenes and nan- 0 / otubes [2, 3, 4, 5, 6, 7], where spherical or cylindrical t a form of crystals and their subspecies are found to be m stable structures of carbon in nano-scales. Tanda et al. have shown that quasi-one-dimensional crystals, such as - d TaS3orNbSe3,whichareusuallygrownasneedle-shaped FIG.1: Thestructureofthering-(orcylinder-)shapedcrystal n crystals, can be made into rings or cylinders by control- madeofaorthorhombiclattice. Becauseoftheexpansionand o ling the growing conditions[8, 9, 10, 11, 12]. They have contractioncausedbythebending,edgedislocationsarelikely c evensucceededinproducingsingle-crystalMo¨biusstrips. tobegenerated. HereW isthethicknessoftheringandRis v: These crystals with topologically nontrivial shapes are theradius. R is defined at thecenter of thethickness. i X named “topological matters”. Some other cylindrical r crystals were also found recently, such as MoS2[13]. The In order to understand the physical properties of the a expanding varietyofthe topologicalmatters seems to be “topologicalmatters”,theknowledgeaboutcrystalstruc- invoking questions about the nature of crystallization in ture is indispensable. In this paper we study the struc- nano-scales. tures of ring-(or cylinder-)shaped crystals, which we call The crystalstructure largelyaffectsthe physicalprop- simply the “ring crystals” hereafter. Special attention erties of the system and, therefore, these crystals pro- is paid to the formation of crystal defects due to the videexcellentexperimentalenvironmentsininvestigating elastic stress which topological matters undergo because of their nontrivial forms. In case of the ring crystals the edgedislocationsarelikelytobegeneratedbecauseofthe bending, as one can see from Fig. 1. In this paper we ∗Electronicaddress: [email protected] derive, within the linear elastic theory, the condition for 2 theformationofedgedislocationsinaringcrystalwhose define the metric tensor g by (dR~)2 = g dx dx , ij i,j ij i j braedtwiueseanntdhethsiucpkenrecsosnadruecRtinagnd(oWrs,urpeesrpfleucitdiv)evlyo.rtAicneaslaongdy where gij = ∂∂xR~i · ∂∂xR~j (i,j ={1,2}, and xP1 =x, x2 =y). The elements of the metric tensor are given as the dislocations in a ring crystal is invoked. Our results may be useful in understanding physical phenomena in 2 r+v ring crystals. g11 gxx = (1+ux)2 +[vx]2, (2) ≡ R (cid:20) (cid:21) Elastic theory of a ring crystal ′ 2 uy 2 To construct the elastic theory of a ring crystal we g22 ≡gyy =[r +vy] + R(r+v) , (3) start by making a rectangular strip of a crystal into a 1 h i ring as depicted in Fig. 2, which introduces elastic de- g12 ≡gxy = R2(r+v)2(1+ux)uy+vx(r′+vy), (4) formation. We assume that the system is uniform in ′ where r =dr/dy, u =∂u/∂x etc. In this paper, we z-direction (along the axis of the cylinder) and suppress x ··· assume r(y) =R+y ( W/2 y W/2) for simplicity. the z-coordinatehereafter. The freeenergyisconsidered − ≤ ≤ Strain tensor u , which describes the elastic deforma- to be that for unit length in z-direction. The crystal is ij tion of the crystal, is given by u = 1(g δ ), where anisotropicanditis assumedthatthe elastic couplingin ij 2 ij − ij δ is the Kronecker’s delta. Assuming an orthorhombic the direction along A-C and B-D is much stronger than ij system for simplicity, the elastic free energy is given by that along A-B and C-D (see Fig. 2). This situation is realized in most ring crystals grown until now. L/2 W/2 1 1 F = dx dy λ (u2 )+ λ (u2 ) Z−L/2 Z−W/2 (cid:26)2 xxxx xx 2 yyyy yy +λ u u +2λ (u )2 , (5) xxyy xx yy xyxy xy (cid:27) where L = 2πR is the circumference of the ring [19]. We also assume that the principal axes of the crystal are along x- and y-axis, which yields λ = λ /2. xxyy xyxy − From now on we denote λ1 λxxxx, λ2 λyyyy and ≡ ≡ λ3 λxyxy = λxxyy/2. ≡ − Next we determine the free energy of the ring crystal withinthelinearelasticity. Todothisweexpandthefree energy with respect to small parameters, u , u , v , v , x y x y y/R and v/R to the second order, which yields 2 F = d~r λ1 y +u + 1v + λ2v2+ λ3(u +v )2 2 R x R 2 y 2 y x Z (cid:20) (cid:18) (cid:19) λ3 y 1 +u + v v . (6) x y − 2 R R (cid:18) (cid:19) (cid:21) FIG. 2: Making a rectangular strip of a crystal into a ring. Here welimit ourselvesto the crystalswith W R. We Thestrip(a)ismadeintoaring(b)byintroducingtheelastic further neglect v-field here, since λ2 λ3 in m≪ost ring deformation. Thesystemis assumed tobeuniform in z-axis, ≫ crystals, and dominant elastic fluctuation is expected to which is perpendicular to thepage. arisenotfromvbutfromu. Thenweobtainthefollowing free energy, Theelasticfreeenergyoftheringcrystalisconstructed byexamininghowthelinearelementoftheoriginalcrys- F = d~r λ1 u + y 2+ λ3u2 . (7) tal d~r = (dx,dy) is modified by the bending in the ring 2 x R 2 y crystal. Let us suppose that the position in the original Z (cid:20) (cid:16) (cid:17) (cid:21) crystal ~r = (x,y) is moved to R~ = (X,Y) in the ring By redefining the parameters by corbytsatianl.thHeefroelloorwtihnoggorenlaalticoono,rdinate system is used. We γ = λ3, K = λ1γd2x, y¯= y, W = W, rλ1 (2π)2 γ γ x+u x+u φ0γ 2πu(x,y) (X,Y)= (r+v)cos ,(r+v)sin , (1) A= y¯, θ(x,y¯)= , (8) R R −dxR dx (cid:18) (cid:19) F is rewritten as where r = r(y) denotes the radius of the lattice plane whose y-coordinate is y in the original system, and L/2 W/2 K 2π 2 tuh=e cuo(mxp,yle)teanrdingv s=havp(xe,(yi.)e.a,redissmloacalltidoenv-ifarteieonrisnga)r.ouWnde F =Z−L/2dxZ−W/2dy¯ 2 "(cid:18)θx− φ0A(cid:19) +θy2¯#, (9) 3 where dx and φ0 are the lattice constant in x-direction Thecoresizeinx-direction,Λ,isgivenbyΛ=dy/γ (see andthesuperconductingmagneticfluxquantum,respec- Ref. [23] for a similar calculation). If we consider that tively. Thisis nothingbutthe freeenergyofanisotropic the core size in y-direction to be d , the core size in the y superconductor[20]. The superconducting“phase” θ un- scaled system (see Eq. (8)) is given by d /γ in both x- y dergoes the quantization condition d~l θ = 2πn , andy-directionandthecoreisconsideredtobeisotropic. where C is a simple loop and n is thCe to·ta∇l number ovf In case of a type-II superconductor, the order is de- v vortices enclosed in C. The vorticesRcorrespond to the stroyed by a magnetic field when the distance between edge dislocations in the crystal system. Thus the quan- vortices becomes comparable to the vortex core radius, tization of vorticity corresponds to the quantization of and this field strength is called the upper critical field the Burgers vector. Here the vector potential A is pro- Hc2 = φ0/(2πξ2), where ξ is the coherence length. In portional to 1/R, which means that the strength of “ge- the present case, noting the correspondences, ometrical” frustration is proportional to the curvature. H γ d y Entry of the first dislocation , ξ , (12) φ0 ⇔ dxR ⇐⇒ γ Here we estimate the free energy of a single disloca- tion and study the condition for the creation of the first where H is the magnetic field, we obtain the critical dislocationinthesystemasthestrengthoffrustrationis radius Rc2 = 2πd2y/(γdx) corresponding to Hc2. If increased. We should note that the “penetration depth” R<Rc2,thefrustrationduetobendingissostrongthat λisinfiniteinthepresentsystem. Inaλ= (extremely the crystal is full of dislocations, whose cores are almost type-II)superconductor,asinthecaseofro∞tatingsuper- overlapped. In this case, each plane can be almost free fluid, the lower critical field Hc1 depends on the system fromthe elasticcoupling to neighboringplanes. In other size [21]. We encounter a similar situation here. words,theinter-planeelasticityisdestroyedbythedislo- Weassumethatthedislocationislocatedatthecenter cations. If Rc2 <R < Rc1, the density of dislocations is of the system (x=0, y¯=0) and, instead of treating the low enoughand the coresare separatedfrom eachother. boundary condition seriously,we employ a rather simple In this case, the crystal may retain a finite elasticity. cutoff procedure [22]. The phase modulation due to the Phase diagram of ring crystals vortex is put as θ = zˆ ~r/~r2, where zˆ is the unit ∇ − × | | TheR-W phasediagramofringcrystalsisdepictedin vector in z-direction and ~r = (x,y¯). Substituting this Fig. 3. As is simply imagined, R > W/2 is an absolute into Eq. (9), we obtain the dislocation free energy Fd = KπlogW π2KγW2. Here the divergence at y¯ = 0 is requirement for the formation of a ring crystal. The for- d¯y − 2dxR bidden area is hatched in the figure. It should also be cutoff at y¯ = d¯ = d /γ (d is the lattice constant in noted that our approach is not reliable in the region too y y y | | y-direction). The condition for the dislocation creation close to R=W/2, because of the assumption R W. ≫ (namely, Fd <0) is, then, obtained as Wehavedividedthewholeregionintothree“phases”(I III). In the phase I, the crystal is free from disloca- π W2 ∼ tions(the “Meissnerphase” insuperconducting analog). R<Rc1 = . (10) 2γdxln(W/dy) The phase boundary Rc1 is reentrantas a function of W because of the logarithmic contribution in Eq. (10), al- Many dislocations and destruction of elasticity thoughthelowerpartofthecurveismeaninglessbecause W √ed . (We denote the smallest R in the phase I Now we study the strongly frustrated regime where y ≤ many dislocations are created. In this case, the size of by Rlim.) Inthe phaseII, the frustrationis not too large andthesystemcontainsdislocationswellseparatedfrom the dislocation “core” is important. To study this, we eachother (the “mixed state”), andthe crystalmay still discretize again the y¯-coordinate in Eq. (7) as follows, hold rigidity. In the phase III, the frustration is large L Ny λ1 y 2 and the dislocation cores are overlapped, destroying the F =d dx ∂ u + inter-plane rigidity of the crystal (the “normal state”). y x j 2 R Z0 (cid:20)Xj=1 (cid:16) (cid:17) Possibledislocationpatternineachphaseisalsodepicted Ny−1 λ3 dx 2 2π(uj+1 uj) inTFhige.d3i.stribution of dislocations are observed directly cos − . (11) − j=1 4π2 (cid:18)dy(cid:19) (cid:26) dx (cid:27)(cid:21) byscanningtunneling microscopeetc. Phononspectrum X measurement is also a powerful tool to investigate the Because of large anisotropy λ1 λ3, the core is crystalrigidity,sincethere aredrasticchangesinelastic- ≫ strongly elongatedalong x-directionalthough the size in ityamongI III.Thismayaffectthephysicalproperties ∼ y-direction is comparable to the lattice spacing. There- of the system also. Studies on various ring crystals from foreweextracttwocrystalplanesbetweenwhichthecore such point of view may be an interesting topic in the fu- is located and neglect the coupling to other planes. The ture [24, 25]. Actually, Tanda et al. have observed that solution which minimizes F in Eq. (11) can be obtained thecriticaltemperatureofchargedensitywavetransition analytically as uj+1 = uj = (dx/π)arctan(x/Λ) (the in NbSe3 is significantly modified in ring crystals, whose − coreis locatebetween the j-thandthe (j+1)-thplane). relation to crystal defects may be worth investigating. 4 graphite sheets is negligible. This is consistent with the fact that thin nanotubes consist of graphite sheets with differentchiralitiesandthe inter-planecouplingbecomes significant only in thicker tubes (R > 75nm)[27]. How- ever an accurate treatment of different chiralities based on dislocation theory is a future problem. In case of transition metal trichargogenides, we study orthorhombic TaS3, for which the order of λ1/λ2 is es- timated from Young modulus E and shear modulus G in Ref. [28, 29] as G/E = λ3/λ1 = γ2 10−2. Using ≃ dx = 0.3 nm and dy = 2 nm, we obtain, Rc2 = 133.3 nm and Rlim = 1138 nm. Although the phase I may be difficult to realize (e.g., R 30µm and W 40nm), ≃ ≃ crossoverfromIItoIIImaybeexperimentallyaccessible. It is also interesting to investigate tubes with R W ≃ bynumericallyminimizingthefreeenergy,Eq. (5)or(6), FIG.3: Thephasediagramofaringcrystalandthedisloca- instead of using superconducting analog. Such study is tion pattern in each phase(see text). Inthecrystal pictures, now prepared. dislocation cores are shown by parts of arcs. In summary, we have clarified the distribution of dis- locations, in the ring-(or cylinder-)shaped crystals with Now we apply our theory to actual systems. First, we various radiuses and thicknesses. Our results may be study multi-wall carbon nanotubes, whose elastic con- useful for the studies ofthe physicalproperties of“topo- stants are approximately those of graphite. From Ref. logicalmatters”,andfortheirfutureapplicationstoelec- [26], we read λ1 = c11 100, λ2 = c33 4, λ3 = trical and mechanical devices. c44 0.02 (dyn/cm2), an≃d λ3/λ1 = γ2 1≃0−4. Using MH and HE were financially supported by a Grant- ≃ ≃ dx = 0.25 nm and dy = 0.34 nm, we obtain Rc2 = 46.2 In-Aid for Scientific Research from the Ministry of Ed- nm and Rlim = 395 nm. When R < Rc2, the crystal ucation, Science, Sports and Culture, Japan. KK was is full of dislocations, and the elastic coupling between financially supported by the Sumitomo Foundation. [1] N. W. Ashcroft and N. D. Mermin, Solid State Physics (2002). (Saunderscollege, 1976). [16] M. Hayashi, H. Ebisawa, and K. Kuboki, Phys. Rev. B [2] S.Iijima, J. Phys.Chem. p.3466 (1987). 72, 024505 (2005). [3] R.F. Curl and R.E. Smally, Science 242, 1017 (1988). 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