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Antiferromagnetic domain walls in lightly doped layered cuprates Baruch Horovitz Department of Physics and Ilze Katz center for nanotechnology, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel 4 RecentESRdatashowsrotation oftheantiferromagnetic (AF)easy axisinlightlydopedlayered 0 cuprates upon lowering the temperature. We account for the ESR data and show that it has 0 significantimplicationsonspinandchargeorderingaccordingtothefollowingscenario: Inthehigh 2 temperaturephaseAFdomainwallscoincidewith(110)twinboundariesofanorthorhombicphase. n Amagneticfieldleadstoannihilationofneighboringdomainwallsresultinginantiphaseboundaries. a Thelatterarespincarriers,formferromagneticlinesandmaybecomechargedinthedopedsystem. J However,holeorderingatlowtemperaturesfavorsthe(100) orientation,inducingaπ/4rotation in 8 theAFeasy axis. The latter phase has twin boundaries and AFdomain walls in (100) planes. 2 PACSnumbers: 75.25.+z,75.60.Ch,75.80.+q,61.72.Mm ] n o The effect of holes on antiferromagnetism is a cen- boundaries and DW’s. c tral issue in high temperature superconductivity. In AnAFpolarizedinthe[100]axisfavorsanorthorhom- - r a remarkable recent experiment the role of doping in bic phase since dipoles interact differently for and p →← u Yet1−axl.C1auxsiBnga2aCnuE3OSR6 (txec≈hn0iq.0u0e8.)TwhaesdsattuadisehdowbystJháantotshsye ↑ra↓thpearirssm.aTllhiinstmhaegcnueptroaetleass,tilceacdoiunpglitnogaischeaxnpgeectiendlattotibcee s . antiferromagnetic(AF)polarizationrotatesfrom[100]at constantoforder310−5 10−6. InprincipleESRanalysis at hightemperaturesto[110]around 40K. Acorrespond- candetecttheresulting−changeincrystalfield4,however, m ≈ inganomalywasalsoseenbyµSRinY1−xCaxBa2Cu3O6 the effect seems too weak in the case AF YBa2Cu3O6. - as well as in La2−xSrxCuO4 compounds2. In the un- Yet, this coupling can be sufficiently strong in samples d dopedcompound(x=0)the polarizationremainsinthe of size 1mm so that twin boundaries are necessary to n ≈ [100] directions down to low temperatures. relieve macroscopic strains. o c The ESR data1 also shows that AF domains are We proceedto present evidence for steps (i) , (iii) and [ present and that their relative intensity is controlled by then derive step (ii). We claim that an orthorhombic magneticfields. Inthehightemperaturephasewithfield structure is in fact supported by the mere observation 2 in the [100] direction, the [010] polarized domains are of AF domains. The presence of AF domains and the v 3 preferred since the canting of AF spins has a higher sus- associated domain walls5 are well known to result from 1 ceptibility. This defines a ”depinning” field at which the either entropy effects (when temperature is fairly close 5 unpreferred domains are diminished. E.g. at high tem- to the Neel temperature), or from disorder or from a 1 peratures and above a field of 1T 80% of the domains magnetoelasticcoupling. TheESRexperiment1 hasused 0 ≈ are polarized perpendicular to the field. The ESR data highpurity samples and temperature was wellbelow the 4 shows,curiously,thatthe depinning fieldofthe [110]AF Neeltemperature. HencethepresenceofDW’sisastrong 0 / of Y1−xCaxBa2Cu3O6 (with field in the [1¯10] direction) indicationforthepresenceofthemagnetoelasiccoupling, at is higher thanthat ofthe [100]AFin YBa2Cu3O6 (field supporting step (i) of the scenario above. m in the [010] direction), both at low temperatures. Step (iii) is consistent with data on the compounds6 - In the present work we suggest that the transition in La1.875Ba0.125−xSrxCuO4. These compounds maintain d the AF polarization correlates with a lattice distortion fixed carrier density while allowing for both tetragonal n and with condensation of the holes into a charge density andorthorhombicphases. Inthe tetragonalphase anin- o wave (CDW). Our reasoning involves a scenario for spin commensurateCDW(aswellasanincommensuratespin c andchargeorderingwiththe followingsteps: (i)At high density wave) with wavevector in the [100] direction is : v temperatures and weak magnetic fields the [100] polar- present. For systems near the orthorhombic boundary i X ized AF leads to an orthorhombic structure; the AF do- (less orthorhombicphase)the CDWwavevectordeviates mainwalls(DW’s)coincidethenwithtwinboundariesat slightly from the [100] axis and finally the CDW disap- r a (110) planes. (ii) Strong magnetic fields favor one of the pears in the orthorhombic phase. AFdomainssothatDW’s annihilate. We showthatthis We propose then that the orthorhombic structure in- process results in antiphase boundaries (APB’s) which duced by the [100] polarized AF structure is unstable consist of ferromagnetic lines. (iii) At low temperatures when holes are added and form a [100] CDW. The AF hole ordering can occur by binding states localized at polarizationthen rotates to the [110] direction which, as the APB’s. However, hole ordering favors the (100) ori- shownbelow,isweaklyorthorhombicandhas(100)twin entation, i.e. a CDW with maxima on (100) planes and boundaries. We proceed now to analyze an effective free an (incommensurate) wavevector in the [100] direction. energyandstudyvariousdomainwalls. Inparticularstep This CDW favorsan AF polarization in a diagonal[110] (ii) is shown, i.e. the collapse of two neighboring mag- direction which is weakly orthorhombic with (100) twin netic DW’s (e.g. by applying a magnetic field) leads to 2 We proceed now to study the T-O transformation induced by a [100] polarized AF as well as the possible AF domain walls coexisting with TB’s. An AF domain wall is defined by a localized rotation of the polarization angle θ (e.g. relative to the [100] axis) by π/2, ± i.e. neighboring domains are polarized along [100] and [010], respectively. The coupled AF and strain structure is illustratedinFig. 1. Note thatantiparallelspins favor shorter bonds, therefore a TB induces a π/2 rotation in the AF polarization, as shown in the figure. We claim that neighboring DW’s have the same rotation angle π/2 (or π/2), rather than opposite π/2 and − π/2. To show that the latter has no solution note first − that the macroscopicstrainwhich imposes the TB array does not affect the local differential equation for θ(s), instead it determines the average TB spacing9 being an integration constant. Hence the existence of a static so- lutionθ(s)canbe deducedfromdynamicalstabilitycon- siderations of the AF structure by itself. In particular DW’s with opposite π/2 and π/2 rotations are unsta- − FIG. 1: Twin boundary with spin polarizations (arrows) ex- ble since by approaching each other they can annihilate hibiting an AF domain wall. The dashed line is in a (110) andformthe lowerenergy groundstate; incontrast,two plane and is the s = 0 line defining a TB position. The dis- π/2 DW’s when approaching each other form a higher placement u(s) is parallel to this line and yields the strain energy antiphase boundary (see below), hence they are ǫ = ∂u/(√2∂s) interpolating between the two orthorhombic stable and a static solution is possible. variants. To illustrate this idea more explicitly we consider a slowlyvaryingmagnetizationwithamagnetoelasticcou- plingenergyoftheform aǫcos(2θ);inparticulara[110] − polarization(θ =π/4)doesnotcoupletothisstrain(the antiphaseboundaries whichcanbe detectedby a variety π/4 state has a weaker coupling to a different strain, as of experiments. Consider first the elasticity part, i.e. the tetragonal- considered below). F1 in Eq. (2) is replaced now by a quadratic term +1bǫ2 since the T-O transition is driven orthorhombic (T-O) transition and formation of twin 2 bythemagnetoelasticcoupling,hencethefullfreeenergy boundaries (TB’s). The orthorhombic distortion, for is weakstrain,correspondstoatetragonallatticedisplaced inthe[110]directionwithadisplacementuthatislinear 2 2 1 1 ∂ǫ 1 ∂θ in the coordinate s in the [1¯10] axis, i.e. u s (see F = aǫcos(2θ)+ bǫ2+ c + d (2) ∼ ± − 2 2 (cid:18)∂s(cid:19) 2 (cid:18)∂s(cid:19) Fig. 1). The signs correspond to the two variants of ± theorthorhombicphase. Thefreeenergy,intermsofthe where the constants b,c,d are positive and the last term strain ǫ=∂u/(√2∂s), has then the form7 represents the spin stiffness. (A crystal field cos(4θ) ∼ from spin orbit coupling is neglected; in fact it must be 2 1 ∂ǫ small to ensure that θ = 0 is a ground state.) The two 0 = 1(ǫ)+ c (1) F F 2 (cid:18)∂s(cid:19) orthorhombicvariants correspondto θ =0,π/2 with the strain ǫ= a/b. where 1(ǫ) has a double minima corresponding to the Domain±walls are solutions of the minimum condition F two variants. A static solution interpolating smoothly for Eq. (2), between the two variants is then possible7, and is de- scribed by a function ǫTB(s). This function has neces- ∂2ǫ acos(2θ)+bǫ c = 0 sarily an s value where ǫTB(s) = 0 which defines a twin − − ∂2s boundary location, hence the TB is on a (110) plane. ∂2θ In general, a TB orientation is uniquely determined by −2aǫsin(2θ)+d∂2s = 0. (3) the type of structural transformation, i.e. smooth inter- polation between variants across other planes leads to A TB which interpolates between the variants ǫ= a/b diverging elastic energies8. The formulation of Eq. (1) necessarily leads to rotation of the AF polarizatio±n by is a simple demonstration that for the T-O transition π/2, as in Fig. 1. We show next that a periodic TB TB’s are on (110) planes. A macroscopic strain which array for which θ(s) is a monotonic function is a valid maintains the overall sample shape, e.g. by presence of solution. Consider a segment of such a TB array where 9 a parent (tetragonal) phase , imposes a TB array and θ = 0 at s = s0, becomes θ = π/4 at s = 0 and finally − determines its periodicity. rotates to θ = π/2 at s = s0; in the same interval the 3 ↑ ↓ ↑ ↓ րւ −→←−−→←−ցտ ↓ ↑ ↓ ↑ ? ? ? ? ↑ ↓ · ↓ ↑ ↓ ↑ ↑ ↓ ↑ ↓ ↑ ↓ ↓ ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓ · ↓ ↑ ↓ ↑ ↓ ↑ ↓ · ↓ ↑ FIG. 2: The 1st line shows three AF variants along [100], separatedbytwoDW’s(withthediagonalpolarizations). The 2ndlineshowstheresultofannihilationofthetwoDW’s–the spins marked by ? continue either left or right domains and FIG. 3: An APB along a (11) line, centered on the dots, an antiphase boundary results somewhere in the ? marked showing a ferromagnetic line. Note that the ferromagnetic range. polarization is parallel to that of theAF. strainisantisymmetric,interpolatingfromǫneara/b>0 DWaswellasaTB,haveamuchlargerwidthandthere- (assuming a>0) to ǫ near a/b<0. Integration of Eq. forelowerformationenergies. Thehighenergycostofan − (3) yields APB,relativetothatoftwoDW’s,providesthestability s of the DW array with same sign DW’s. ∂θ(s) ∂θ(s) 2a ∂s − ∂s |−s0 = d Z−s0ǫ(s)sin[2θ(s)]ds. (4) waDsWde’ms ocannstrbaeteedlimbyinathteedexbpyearimstreonntgofmJaágnnoestsiyc fieteldal.a1s, In the interval s0 < s < s0 sin(2θ(s)) > 0 is symmet- leading to a single variant. The process can generate ricwhileǫisant−isymmetricstartingfrompositivevalues, APB’s assuming that they are sufficiently apart and do hence the right hand side of Eq. (4) is positive and ap- not annihilate each other. APB’s carry spin 1/2 per Cu proaches zero at s0. At the next TB, ǫ is antisymmetric along the boundary, i.e. a ferromagnetic line along [110] startingfromnegativevalueswhile sin(2θ(s))is symmet- as shown in Fig. 3. These lines are AF in the [001] di- ric and negative, hence again the right hand side of Eq. rection, yet, the interlayer coupling is weak so that it is (4) is positive. We conclude that ∂θ/∂s > 0 is a con- relatively easy to flip magnetization by a field parallel sistent solution. Furthermore, in the limit c d the to the AF polarization. Hence, when the magnetic field gradients are dominated by the spin stiffness w≪hile the overcomes the interlayer coupling a nonlinear longitudi- strain follows the local AF angle, ǫ (a/b)cos(2θ) and nal susceptibility χk appears due to aligned ferromag- Eq. (3) becomes (a2/b)sin(4θ)+d≈∂2θ/∂2s = 0. The netic lines. Notethatby rotatingthe APB ofFig. 3into latter is the sine-G−ordonequation with well known peri- a (100) plane its lines (within the layers) are AF rather odic domain structure satisfying ∂θ/∂s>0. then ferromagnetic. Aπ/2DWisthereforefollowedbythesamesign+π/2 Consider next an alternative AF phase with polariza- DW. This result is, in fact, a robust consequence of the tion along [110],stabilized by terms beyond those in Eq. energy argument above requiring DW’s stability, hence (2), e.g. CDW ordering (see below). The coupling of it is valid even with additional terms in Eq. (2). A re- nearest neighbors along [100] or [010] is now identical markable consequence of this analysis is that when two andorthorhombicityis not induced, i.e. the firsttermin neighboring TB’s annihilate the result is not a uniform (2) vanishedat θ =π/4. However,secondnearestneigh- ground state, but rather a change of θ(s) by π which is bors form either , or , pairs which are nonequiv- ↑ ↑ → → an antiphase boundary (APB), i.e. all spins on one side alent. Hence a tetragonal unit cell which is formed by ofthe boundaryarereversedrelativetothe groundstate the vectors [110],[1¯10], [001]of the originallattice tends orientations. The annihilation process of DW’s is shown to contractorexpandalong[110]and[1¯10],respectively. in Fig. 2. The 1st line shows the spin arrangement of This unit cell is rotated by π/4 relative to the previous three variants separated by two DW’s with the polariza- one, hence the axis s′ along which an orthorhombic dis- tion angle increasing monotonically. In the 2nd line the tortion alternates is now in the [100] direction, therefore DW’s are eliminated, i.e. the spins are rotated so as to twin boundaries that interpolate between θ = π/4 and be aligned with either the left or the right domain. The θ = 3π/4 are in (100) planes of the original tetragonal diagonal spins join their nearest domain, but the hori- lattice. The detailed description of these TB’s is more zontal ones can choose either domain (indicated by a ? involved than the previous ones since near θ π/2 the ≈ mark), with any choice resulting in an APB somewhere previous strain will couple. However, following the sta- in the ? marked region. Note that the APB carries an bility argument above, we expect again a π/2 DW to additional spin 1/2 relative to the AF ground state. be followed by the same sign +π/2 DW, hence a strong APB’sinvolveamajorrearrangementoftheelectronic magnetic field in a [110] direction would coalesce these coordinates and can result in a localized state, as shown TB’s and lead to (100) APB’s. by a mean field study10, analogous to studies on soli- The final ingredient in our model is that the added tonsinpolyacetylene. AnAPBinvolvesthereforeatomic holesformaCDWwithwavevectoralong[100]asclearly scales and its width is of that order. In contrast, an AF seen in the La1.875Ba0.125−xSrxCuO4 compounds6. In- 4 TABLEI: ThetableliststopologicalstructuresintheAForderinginlightlydopedCuprates,aswellaspertinentexperimental signatures at various magnetic fields. High temperature corresponds to the [100] polarized AF phase and low temperature corresponds to the [110] polarized AFwith a [100] CDW phase. high temperature low temperature strong [100] field (110) neutral APB (100) DW and (100) charged APB ferromagnetic lines, nonlinear χ k strong [110] field (110) DW (100) charged APB higher TCDW nofield (110) DW (100) DW and (100) charged APB stronger depinningthan for theundoped case deed the charged APB in the mean field calculation10 serving APB’s and probing our scenario: (i) APB’s are is more strongly bound when the APB is along a (100) expectedtohaveintragapstates10,henceopticalabsorp- plane. These(100)APB’sformaperiodicarrayofstripes tion should show new lines at high magnetic fields. (ii) equivalent to a CDW with wavevectorin the [010]direc- The ferromagnetic nature of (110) APB lines (Fig. 3) tion. In a [100] polarized AF these (100) APB’s would results in a nonlinear longitudinal susceptibility χ . (iii) k cross (110) DW’s or (110) APB’s; the [100] CDW there- By adding holes the APB intragapstate can be charged, forefavorsa rotated[110]polarizedAFsothatcrossings leading to a spinless charge carrier. These charges affect with DW’s are avoided. Furthermore, in a strong [110] theopticalabsorptionin(i)aswellasthesusceptibilityin field the [110] polarized AF has the appropriate ingredi- (ii). (iv) The CDW onset is facilitated by a strong mag- entsforformingstripes,namely(100)APB’s,facilitating netic field in the [110] direction, hence TCDW is higher the CDW formation. We expect therefore that the tran- in this case. sition temperature TCDW into a CDW be higher with In conclusion, our scenario accounts for the unusual a field in this [110] direction. The predictions for DW’s ESR data1 and predicts a variety of topological struc- andAPB’saresummarizedintableI,aswellaspertinent tures (table I). We propose that experiments based on experimental signatures. these observations can supplement the ESR data as well We shownowthatourmodelaccountsalsoforthe ob- as clarify a central issue in high temperature supercon- servation that depinning field in the [110] polarized AF ductivity – the nature of doping in AF layeredcuprates. Y1−xCaxBa2Cu3O6 is larger than that of the [100] po- 1 larized AF in YBa2Cu3O6, both at low temperatures . Acknowledgements: I thank Prof. Andra´s Jánossy Depinning involves annihilation of DW in pairs leading for stimulating discussions and for his hospitality, and to a single domain. In Y1−xCaxBa2Cu3O6 some of the the Institute of physics at the Budapest University of DW need to cross a charged APB, a process which has TechnologyandEconomicsforhospitalityduringthisre- a barrier. In YBa2Cu3O6 the process involves TB an- search. I also thank D. Golosov for pointing out Ref. 5 nihilation which is continuously achieved as field is in- and A. 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