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Liquid Crystal Analogue of Abrikosov Vortex Flow in Superconductors A.Tanaka,1,2 T. Ota,2 and R. Hayakawa2 Solid State Laboratory, Tokai Establishment, Japan Atomic Energy Research Institute, Ibaraki, Japan 1 Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan 2 () We extend the correspondence between the Renn-Lubensky Twist-Grain-Boundary-A phase in chiral liquid crystals and the Abrikosov mixed state in superconductors to dynamical aspects. We findthatforaTGB samplewith freeboundaries,an externalelectric fieldapplied alongthehelical axisinducesauniformtranslationalmotionofthegrainboundarysystem-ananalogueofthewell- known mixed state flux flow. Likewise, an analogue of the mixed state Nernst effect is found. In muchthesamewayinwhichthefluxflowcarriesintercoreelectricfieldsgeneratingJouleheatinan otherwisedissipation-freesystem,thegrainboundaryflowcarriesalongpolarizedcharges,resulting 6 9 in a finiteelectric conductivityin a ferroelectric. 9 1 PACS numbers: 61.30.Cz, 61.30.Jf n a J Theformalanalogyofthesmecticfreeenergywithina consequenceoftheverysymmetryoftheTGB.[7,5]How- 1 Ginzburg-Landauapproach,withthatforthewellknown ever, a detailed study of the full effect of thermal fluc- 3 case of a superconductor has had a long history in the tuation within the GL formalism requires considerable 2 study of liquid crystals. It was first pointed out by computing efforts, and we choose to set a less ambitious v de Gennes that the essential mean field features of the goal. Inthispaper,westudythe fundamentalproperties 4 phase transition between the Nematic (N) and Smectic- ofdislocationmotionintheTGBsystem. Asaresult,we 0 A(SmA)phasescanbederivedbysimplyresortingtothe obtain several new entries to the list of superconductor- 0 correspondence with the Normal-Superconductor transi- liquidcrystalanalogue,assumarisedinTable1. Amajor 1 tion. [1] Corrections to this mean-field picture was stud- feature is the precise liquid crystal analogue of the flux 0 ied by means of a large-N expansion of the GL energy flow motion of the Abrikisov vortex lattice. Although 6 9 functional, in which a runaway RG trajectory was inter- the long wavelength hydrodynamics of the TGBA phase / pretedas anindicationofa fluctuation-drivenfirstorder has been previously worked out, [10] this paper is, to h transition. [2] our knowledge, the first one which has considered the p An important application of this analogy was carried grain boundary itself and its constituent screw disloca- - m out by Renn and Lubensky, who, based on the mean- tion lines as mobile entities. An important implication e field phase diagramof type-II superconductors in an ap- of this motion is that the dislocation cores carry elec- h plied magnetic field, predicted an intermediate phase tric chargesalongwith the flow throughthe ferroelectric c lying between the cholesteric (Ch) and SmA phases. media. Thus the system is predicted to be conductive. : v [3] This new phase, termed the Twist-Grain-Boundary- Togeneratethemotionofthedislocations,wewilltake i A(TGBA)phase,isthecounterpartoftheAbrikosovvor- advantage of the chiral nature of system. Due to the X texlatticephase(alsoknownasthemixedphase),andis lack of chiral symmetry, polar vector fields can couple r composedofaregularnetworkofscrewdislocations. For linearly to axial vector fields. If we consider the torque a the detailed structure of this lattice, readers are reffered ofthe smectic layernormalas the latter and couple it to to ref.[3]. The TGBAphase wasalsoindependently dis- an external polar vector field, this will effectively exert covered by exerimenters [4], and immediately became a a force on the grain boundary. As the external field, we subject of intense research. consider(1)anelectricfieldappliedalongthehelicalaxis So far, theoretical investigations on the TGBA based of TGBA phase and (2) temperature gradient imposed on the superconductor analogy have been restricted to along the same direction. mean-field,staticaspects. (Seehoweverrefs[7–9].) Con- In the SmA phase of such materials, it is well known sidering however the rich variety of mixed state physics thatwhenanelectricfieldisappliedin-planetothesmec- that continues to be revealed [6] through recent (high tic layers, a finite tilting of the molecular axis from the Tc-motivated) studies of fluctuation effects and vortex smectic layer normal is favored. This phenomenon is dynamics, it is natural to ask what new feature arises calledtheelectrocliniceffect.[11]Weshallshowherethat when one seeks to extend the analogy in similar direc- thestressinducedbytheelectrocliniceffectproducesthe tions. AsinhighTcsuperconductors,itisexpectedthat driving force acting on screw dislocations. Our starting thermalfluctuations haveprofoundeffects onthe TGBA pointis the followingfreeenergy,whichisacombination lattice structure;a simple estimate via the Ginzburg cri- of the covariant Chen-Lubensky(CL) model [3] and the teria strongly suggests this vulnerability, and there is an energy related to the electroclinic effect. [11]: additionalintrinsicinstabilitytowardsdisorderwhichisa 1 g F = d3x[r ψ 2 +C ( iq n)ψ 2 + ψ 4 Next we consider the case where a temperature gra- 0 Z | | | ∇− | 2 | | dient is imposed along the pitch. In analogy with the +K1( n)2+ K2(n n k )2+ K3(n n)2 Nernst effect in a superconducting mixed phase, it is 0 2 ∇· 2 ·∇× − 2 ×∇× anticipated that screw dislocations in the TGBA phase ǫ E2 P2 + 0 + η(n ( u δn)) P], (1)a+re influenced by a thermal force generated in the di- 0 2 2χ − × ∇ − · rectionofthetemperaturegradient. Weshallshowbelow thatthisisindeedthecase. Thephysicsofwhatisshown where, n is the Frank director,ψ(x) is the smectic order below may be summarized as follows. The temperature parameter, ψ(x) = ψ eiq0ueiq0n0x with q = 2π/d, d | | · 0 gradient produces the gradient of smectic density ψ , theseparationbetweensmecticlayers,uthesmecticlayer | | where n = ψ 2. This in turn disturbs the equilibrium displacement,n0 is the normalvectorof smectic layer. η of screwsdisl|oca|tions, giving rise to an effective driving is the coupling constant between the polarization P and force acting on the system. n ( u δn), and χ is the electric susceptibility. The 0× ∇ − Following the approach of Koklov and co-workers [13] terms η(n ( u δn)) P and K k n n reflect 0 × ∇ − · 2 0 ·∇× whoevaluatedtheNernstcoefficientforthesuperconduc- the mentioned symmetry of the system. Differentiating F with respect to δn, gives the stress field J δF′ = tor case solely within the Ginzburg-Landau formalism, ≡ −δδn we can derive [5] an expression for the total force acting B′(v −δn) with v = ∇u, and B′ = 2Cq02|ψ |2 −η2χ. on the ensemble of screw dislocations in the case where J is analogous to the supercurrent Js which is the free the temperature gradient produces a steady nonequilib- energydifferentiatedwithrespecttothe vectorpotential riumdistributionofscrewdislocations. Forthecondition A0,.wxe-asexeist,h¿aFtrovm tδhnevsoluδtnioenissodfiv∇id·edJ i=nto0 tawnodpδδaFnrt=s. (tih.ee.effξse2c/tilvbledf≪orc1ewpeitrhuξnsitthaeresamoencttichecogrrraeilnatbioonunlednagrtyhi)s, − − One isthe “vortex”partv δn ,[12]withnonvanishing v curl, and the other is the u−niform part, δne = ηχ(nB0′×E), fx =Λ∇x(∆ns) inducedbytheelectricfieldE. Applyingthevirtualwork ns principle, we find a Peach-Koehler tyoe force x(∆ns) Bd2 ld Bd2λ2 ld ∇ ln + exp . (4) ∼ n (cid:18)2πl (cid:18)ξ (cid:19) 4l2 −λ (cid:19) s d s d 2 f =J dˆz. (2) l e × B = 2Cq2 ψ 2 , and the penetration length λ = acting on each unit length of the dislocation. Here dˆz is 0| | 2 K /B. It should be noted that the drift force f is the Burgers vector. 2 Now we compare this with superconductors. B′(v− Tpprhoepfiorrtsitotnearlmtoofththeegrriagdhitenhtanodf ssmideecotifcedqe.n(4si)tyha∇s(s∆imnpsl)e, δn ) corresponds to the vortex part of the supercurrent, v form of a single dislocation contribution multiplied by a and B δn corresponds to the transport current. In the s−upe′rcoenductor case, the corresponding force f = factorof1/ld,thedensityofscrewdislocationsinalayer. L The second term reflects the global twisted structure of J Φ ˆz is just the Lorentz force acting on the vortex t × 0 the TGBA system. line per unit length. Φ is the flux quantum. Compar- 0 Now that we have obtained the forces induced by the ing f with f , it is clear that the force acting on the l L coupling to external fields, we now turn to the actual single screw dislocation corresponds to Lorentz force in motion of the dislocations. superconductors. We start with the “model A” type equation for the Next we consider the case where the Peach-Koehler order parameter [14]: force acts on the grainboundaries which is just an array of screw dislocations. For the grain boundary located at δF x = 0, the unit normals nL and nR of the SmA slab at γψ∂tψ = +ζ(t). (5) 0 0 −δψ x < 0 and x > 0 , respectively are forced to be rotated ∗ with respectto eachotheraroundthex-axis byanangle ζ(t)isathermalnoiseobeyingthefluctuation-dissipation of2πα. AuniformelectricfieldE=Exâppliedalongthe relation, <ζ(t)ζ(t )>= 2kBTδ(t t). γ =γ +iγ is x-axis will generate the stress “currents” on the left and genericallyacomp′lexkine|tγiψc|coeffi−cie′nt. ψOnceψt′he deψf′′ect raingdhtJeRend=s o−f eηaχcnhR0Sm×AEs,larbesgpievcetnivbelyyJ(LeSe=e−Fηigχ1n).L0 ×TEhe, mnaomveics,lwiftefwoirlcleh(aMveatgonucsonfosirdceer).thTeheeffeecxtisotfenacehyodfrosudcyh- extensionofeq.(2)tothisconfigurationisstraightforward a force and its relevance to the Hall sign anomaly is an and the force ftgb acting on the grain boundary per unit issueofpresentdebate insuperconductors.[15,16]It can area is be shown that a imaginary part of the relaxation time γ in eq. (refptd) can give rise to such a force. How- ηχEd ψ′′ ftgb =− ld cos(πα)xˆ ∼−2ηχEπαxˆ, (3) ever, renormarization-group calculation shows (γψ′′/γψ′ ) term to be an irrelevant perturbation in the vicinity of where the final expressionis valid for small α. This sim- thetransitionpoint,i. e. neark =k .[17]Furthermore, c2 ple superposition is invalidated near k = k where dis- it was concluded in Ref. [18] that the force is absent in c2 locations cores begin to overlap. the single screw dislocation case. Thus we consider it 2 plausible to neglect such forces. and put γ /γ = 0. In (√l l ξ ) using the solution of eq. (6) leads to the ψ′′ ψ′ d b ∼ s this purely dissipative case, the motion of screw disloca- sameformasineq.(7). Asisthecaseofsuperconductors tion takes the form γlvl = fl, where γl is the viscosity k¯0isexpressedintheformoftheexpansionof kc2k−0k0 near coefficient for a screw dislocation and vl is its velocity. kc2. The parameter β¯ [3] (corresponding to Abrikosov’s γl is determined using the Rayleigh functional for eq.(5) betainsuperconductors)isrelatedtotheconfigurationof R[∂tψ,∂tψ∗]= d3xγψ ∂tψ 2, and the balance if force screw dislocation lattice. Differentiating the free energy | | ∂∂vRl = fl. as γl =R γψπln(λ/ξs). [19] Hence the weak chi- with respect to K2k0, we obtain k¯0 = k0 − k2c2κ−22βk¯0 with rality regime, eq.(5) reduces to the stochastic equation for a single defect line, γlddRtl =−δδRFl +Gl(t). The nois pκ2ola=rizCa1qt0iqongcK2u2r.renDtemtoacnodiinncgidtehweidthissRipwathioicnhdisueintdouctehde G (t) satisfies <G(t)G (t)>= 2kBTδ(t t). Later we l l l ′ γl − ′ by the defects dynamics, we are able to express vl as twhiellesxhpoewrimtheanttatlhpeavrealmoceitteyrsvlη,=χ,ηχdγd.lE is obtained from vl = ηγχψdkE0llbdq√0k¯π0 in terms of the material parameters k0, η, χ, and viscosityγ . At k k (l l ξ2) the screw dislocations cannot ψ 0 ∼ c2 b d ∼ s In summary we have investigated the basic electric be treatedasindependent mobile entities. The static so- propertiesofthe dislocationmotioninthe TGBAphase, lutions of the linearized GL equation for ψ in the super- with a special emphasis on the analogy with vortex dy- conductor case near H can be approximated by eigen- c2 namics in superconductors. In actual experimental sit- functions of the lowest Landau level. An analogous pro- uations, rubbing process to fix the direction of directors cedure canbe takenfor the TGBA casewhen q /l 1. 0 0 ≫ of molecules should not be made in order to make the [3] Here we add the time dependence into the solution. predicted effects observable. The present work should ψ(x)= be considered a starting point for further studies of the dynamics in the TGB system, in which the interplay C exp(iqs(x +U (t)) x) Xs s (k0x ⊥k0(xss+Uss(t))·)2 bpeetcwteedentothgeivrme railseflutoctvuaartiioounsainntderdeesfteinctgdpyhnyasimcsic.s is ex- exp − (6) (cid:18)− ξ¯2 (cid:19) It is pleasure to acknowledgemany helpful discussions withProfessorT.Tokihiroattheearlystageofthiswork. Here, x is the position of s-th SmA slab, U (t) s s A. T. acknowledge a fellowship from the Japan Atomic denotes the displacement of the zero of the s-th Energy Research Institute. T. O. was supported in part eigenfuntion in the lowest Landau level. qs(x + s by the Japan Society for the Promotion of Science. U (t)) = q (0,sin(k (x + U (t)),cos(k (x + U⊥ (t))), s 0 0 s s 0 s s n ξ¯2 = 1/(q k ). The dissipation function R per unit vol- 0 0 umeinthiscaseR= γψq0k0vl2√π. Aswewilldiscussnext, lbq0 thevelocityv (nowdefinedastheaveragevelocitesofthe l zeroes)isdecidedbycombiningRandtheelectricenergy dissipation. First we point out that a d.c. polarization current is [1] P. G. de Gennes, Solid State Commun., 14, 997 (1973). directly observable as a direct consequence of the mo- [2] B.I.Halperin, T. C.Lubensky,and S.K.Ma, Phys.Rev. tion of screw dislocations. This is in sharp contrast to Lett., 32, 292 (1974). dislocation-freeferoelectrics,whereonlya.c. currentsare [3] S. R.Renn andT. C. Lubensky,Phys.Rev.A, 38, 2132 allowed to flow. The origin of the charge carried by the (1988). screwdislocationstemsfromthechiralnatureofthesys- [4] J. Goodby et. al, Nature(London) 337, 449 (1988). tem. As can be seen from eq.(1), the deviation of the [5] T. Ota, A.Tanaka and R. Hayakawa, unpublished. director vector from the layer normalgives rise to polar- [6] K. S. Bedell, M. Inui, D. Meltzer, J. R. Schrieffer, and ization. Around a screw dislocation this is obtained as S. Doniach, Phenomenology and Applications of High- P=is−th2πηeλdm2Ko0di(cid:16)fi|exd−λ2RBle|s(cid:17)seelrf+unχcǫt0iEonp,eeru=nit(cloesnφgt,hsi,nwφh,e0r)e, [7] JT.emTopneerar,tuPrehySsu.pReercvo.nBd,u4ct3o,rs82(8A9d(d1i9so9n1)W. esley, 1992). 0 r aKndφ=tan−1(cid:16)xy−−RRllyx(cid:17). Henceatthesurfaceofthecore [8] TA.,C43.,L5u4b4e9ns(k1y9,9T1)..Tokihiro,andS.R.Renn,Phys.Rev. of a screw dislocation we have a local charge accumula- [9] R.D.KamienandT.C.Lubensky,J.Phys.I.France,3, tion of Q = dS P = ηχd. Thus near kc1 we are 2131 (1993). 53, 650 (1996). ∇ · able to relateRthe velocity to the polarization current by [10] Y. Hatwalne, S. Ramasaway and J. Toner, Phys. Rev. summing up the contribution ∂ Q and get Lett., 70, 2090 (1993). t [11] P. S. Meyer, Mol. Cryst. Liquid Cryst., 40, 33 (1977). ηχdv xˆ j = l =ηχv k¯ xˆ . (7) [12] A. R. Day, T. C. Lubensky, and A. J. McKane, Phys. p l 0 1 ldlb Rev. A,27, 1461 (1983). [13] A. Kuklov, A. Bulatov, and J. L. Birman, Phys. Rev. Here,k¯ isthespatialyaveragedtwistofthedirector,for 0 Lett., 72, 2274 (1994). whichthe relationk¯ = d forsmallαisused. Neark 0 lbld c2 3 FIG.1. Thestressfieldnearthegrainboundary. Jv isthe nonvanishing curl stress field around screw dislocations with its separeation ld. JLe and JRe is the electric field induced spatially uniform stress field of left SmAslab and right SmA slab, respectively. FIG. 2. The smectic layers near the grain boundary. d is layer separation of SmA slabs. The angle between electric field induced stress field in left SmA slab JLe along the Sm layer and that of right SmA slab JRe is 2πα. [14] P.C.Hohenberg,B.I.Halperin,D.R.NelsonandE.D. Siggia, Phys. Rev.B, 21, 2373 (1994). [15] P. Ao and D. J. Thouless, Phys. Rev. Lett., 70, 2154 (1993). [16] F. Gaitan, Phys.Rev.B, 51, 9061 (1995). [17] A.T. Dorsey and M. P. A.Fisher, Phys.Rev.Lett., 68, 694 (1992). [18] H.Pleiner, Philosophical Mag., 54, 421 (1986). [19] K.Kawasaki, Physica A, 119, 17 (1983). superconductors liquid crystals js =−δδAF:supercurrent j=−δδFn:stress Lorentz force Peach-Koehler force vortex lattice flow grain boundary flow due tothe electric field Nernst effect grain boundary flow due tothe temperaturegradient fluxflow resistivity conductivity dueto grain boundary flow TABLEI. Dynamical aspects of the analogy between superconductors and liquid crystals. 4