Charge ordering and long-range interactions in layered transition ...

C harge ordering and long-range interactions in layered transition m etaloxides B ranko P.Stojkovic1,Z.G .Yu1 ,A .R .B ishop1,A .H .C astro N eto2 and N iels G r nbech-Jensen1

arXiv:cond-mat/9805367v1 27 May 1998

1

T heoreticalD ivision and C enter for N onlinear Studies,Los A lam os N ationalLaboratory, Los A lam os,N ew M exico 87545 2 D epartm ent ofPhysics,U niversity ofC alifornia,R iverside,C A 92521 (July 23,2013) W e study the com petition betw een long-range and short-range interactions am ong holes w ithin the spin density w ave picture of layered transition m etal oxides. W e focus on the problem of charge ordering and the charge phase diagram . W e show that the m ain interactions are the long-range C oulom b interaction and a dipolar short-range interaction generated by the short-range antiferrom agnetic uctuations. W e nd four di erent phases depending on the strength of the dipolar interaction and the density of holes: W igner crystal, diagonal stripes, horizontal-vertical stripes (loops) and a glassy-clum ped phase. W e discuss the e ect of tem perature, disorder and lattice e ects on these phases. PA C S num bers: 71.10.H f,71.20.B e,71.10.-w

R ecently therehasbeen m uch interestin thechargeordering and dom ain wallform ation atm esoscopicscalesin doped transition m etaloxides [1]. A popular exam ple of such orderingsare stripes,i.e.,lineararraysofholesseparated by an antiferrom agnetically (A F) ordered background. T he form ation ofdom ain walls and stripes has been discussed in term softhe proxim ity to phaseseparation [2].M acroscopicphaseseparation hasbeen observed in La2C uO 4+ [3],and stripeshave been observed experim entally in La2 x Srx N iO 4+ y in m any di erent experim ents including direct high-resolution electron di raction [4]. M agnetic susceptibility m easurem ents [5], nuclear quadrupole resonance [6]and m uon spin resonance [7] indicate form ation of dom ains in La2 x Srx C uO 4. Stripeshavealso been seen in La1 x C ax M nO 3 forspeci c com m ensurate values of doping [8]. A direct evidence for stripe form ation was observed in neutron scattering in La1:6 x N d0:4 Srx C uO 4 [9]. M oreover,recent neutron scattering experim ents are not inconsistent w ith stripe phases in other high tem perature superconductors such as Y B a2C u3O 7 [10,11]. O n the theoreticalside,stripes have been proposed as a result ofa com petition between short-range attractive interaction between holesfrom the breaking ofA F bonds and the long-range C oulom b interaction [12]. Indirect support for this picture has been given in term s of the m apping ofthe problem into e ective spin m odels [13]. Striped phaseshave been obtained w ithin m ean eld approaches to the short-range H ubbard or t J m odels w hich are only able to generate insulating states [14]. N um ericalm ethods in these m odels have not been able to con rm this picture except for recent D ensity M atrix R enorm alization G roup sim ulations (D M R G ) [15]. In this paper we present a num ericalapproach to the problem ofholes m oving in an A F insulator in the presence oflong-range C oulom b forces. T he ability to handle long-range C oulom b interactionsat nite density has been enhanced recently in the area ofm olecularphysics: assum ing a com putationalcellofarbitrary geom etry and

cyclic boundary conditions it is possible to sum interactions of particles w ith all of their im ages residing in cells obtained by translation from the originalcom putationalcell[16,17]. O n m aking integraltransform ations, C oulom b interactions are com puted by sum m ing over fast-convergentB esselfunctions w ith greataccuracy. U sing M onte C arlo (M C ) and m olecular dynam ics (M D )m ethods,wesystem atically study the interplay between long-range C oulom b interaction and short-range A F interactions ofdipolar nature w hich we take to have both isotropicand anisotropiccom ponents(depending on the lattice structure). O ur m ain result is sum m arized in the phase diagram ofFig.1.In the absenceofdisorderwe nd four phases depending on the density of holes and the characteristic A F energy scales: a W igner crystal, diagonalstripes,horizontal-verticalstripes(loops)and a glassy-clum ped phase. T he order param eter for charge ordering is the Fourier transform ofthe hole density: (q)=

N 1 X iq e N i= 1

ri

;

(1)

w here ri isthe position ofthe ith hole and N isthe total num ber of holes. A peak in (q) at som e wave-vector q = K indicates ordering. O urstarting pointisthe spin density wave(SD W )pictureofthelayered transition m etaloxidesw hich hasbeen very successfulin describing the insulating A F phase of these system s [18,19]. In this picture the electrons m ove w ith hopping energy t in the self-consistent staggered eld ofits spin. B ecause the translationalsym m etry of the system is broken, the electronic band is split into upper and lower H ubbard bands [20]. T hese are separated by the M ott-H ubbard gap, , and at half lling the lower band is lled and the upper one is em pty. T his picture is consistent w ith the angle resolved photoem ission data in the layered A F insulatorSr2C uO 2C l2 [21]. B y doping the system w ith holes w ith planar density s and at low tem peratures, T (kB T < < ), we 1

focus entirely on the lower band w hich has a m axim um at k = Q =2 = ( 1; 1) =(2a), w here a is the lattice spacing. It can be show n that the holes interact via two di erent m echanism s: a short-range attractive force due to A F bond breaking and a long-range dipolar interaction due to the distortion ofthe A F background [19,22]. It was show n that this dipolar interaction gives rise to spiral distortions of the A F background [22,23]. T he dipole m om ent associated w ith each hole is due to the virtual hopping of holes between neighboring sites and scales w ith the A F m agnetic energy. T he dipolar interaction between two holes w ith dipole m om ents d 1;2 at distance r apart has the form : U dip =

1 (d 1 r2

d2)

2 (d 1 r2

r)(d2

r) ;

particle interaction term s.H owever,atlow densitiesitis reasonableto assum ethattheinteraction ofany two holes is weakly perturbed by other holes,and the totalpotentialenergy can be expressed in term s oftwo-particle energies.T herefore,in ournum ericalcalculationswe study the physics of N holes interacting via V (r) as given in (3). W e assum e a rectangular com putationalbox ofsize L x L y w ith L x ,L y up to 100 unitcellsin a C uO 2 plane. A t the beginning of each sim ulation we place the holes at random and assign to each hole a dipole m om ent of constantsize,butrandom direction.W e nd a m inim um ofthe totalpotentialin this system using three di erent m ethods: M C m ethod,Langevin M D and a hybrid M C M D m ethod [29].A llthree m ethods yield essentially the sam e results. Since the system exhibits severalphases (see Fig.1) for som e values ofthe input param eters,its ground state is not alwayswellde ned and m ay,in fact, depend on the initialand boundary conditions. H ence, in order to rapidly reach a hole con guration w ith the lowestglobalm inim um energy we perform sim ulated annealing from high tem peratures. For B = 0 we nd the W igner crystalw ith sm alldistortionsto be the state oflowestenergy,asexpected [30]. T he sm alldistortion ofthe crystalstructure isdue to the periodicity,w hich introduces a sm allspatialanisotropy into the system due to the rectangularshape ofthe com putationalbox. Increasing A w hile retaining B = 0 reduces the lattice constant ofthe W igner crystaluntila criticalvalue is reached w here holes group together. For A = 0 and nite B the situation is quite di erent. A t sm allB and largerdensitiesthe W ignercrystalisunstableand a new phasew ith diagonalstripesisform ed.T his phaseischaracterized by ferro-dipolarorder(seeFig.2a). T he situation here is very sim ilar to that observed in La2 x Srx N iO 4+ y [4]. A s show n in Fig.2c,at larger values ofB a line stripe is form ed,w hich,w ith increasing density tendsto close into loops,form ing a checkerboard pattern.Im portantly,the loop form ation isaccom panied by dipole orientation along the straightportion ofa loop w ith gradualrotation by =2 ateach corner[31].D ue to the rotation ofdipoles at cornersthe loops interact,and eventually form the checkerboard pattern [32]. T he size ofthe inter-hole distance w ithin a line is determ ined by the ratio of B and the C oulom b energy; the loop sizes are determ ined by the hole density alone. T hese results appear to be consistent w ith the D M R G solution ofthe t-J m odel[15]. IfB is increased further the dipolar interaction becom es dom inant over the average C oulom b interaction;the well-de ned pattern disappears and one observes star shaped clum ps ofholes,w hich can,at sufciently high density,form another geom etric structure (e.g., a W igner crystalof clum ps). W e rem ark that in allphasesa non-vanishing value ofA leads to a decrease in the e ective value of B at w hich the transitions occur (Fig.1);the isotropic term A alone never produces any non-trivialgeom etric phase (e.g.,stripes),even w ith

(2)

w hich isrotationally invariant.Itisalso possible to show using W ard identities that the spin part ofthe problem can be described by a two dim ensional(2D )non-linear m odelin the long wavelength lim it [24]. A t nite T the system ism agnetically disordered and characterized by a nite m agnetic correlation length, [25]. T hus,at nite T the dipolarinteraction between the holes,m ediated by the antiferrom agnet,is actually short-ranged. H owever, besides the A F interactions the holes also feelthe longrange C oulom b interaction. T his is clear ifwe consider that rs = r0=a0 (w here r0 is the m ean inter-particle distanceand a0 istheB ohrradius)isvery largein theunderdoped system s(rs 8).T hus,the interaction energy between theholes,w hich behaveslikee2=(a0rs),iscertainly m ore im portant than the kinetic term ( e2=(a0rs2 )) at low densities. T his im plies that the interaction term s should be treated rstand the kinetic energy asa perturbation.Finally,each holecarriesa spin degreeoffreedom as well,but it is possible to show that the overallspin energy is m inim ized in the spin anti-sym m etric channel, as we assum e here. T hus,in our approach,we are left w ith only the chargechanneland the interaction between two holes,1 and 2,has the form (see Eq.(2)) q2 r= A e r=a B cos(2 ; (3) 1 2 )e r w here q isthe hole charge, isthe angle m ade between r and a xed axisand 1;2 aretheanglesofthedipolesrelativeto the sam e xed axis.A isthe strength oftheshortrange bond-breaking interaction and B isthe strength of the dipolarinteraction,w hich we w illassum e to be independent variables. T he m agnetic correlation length is obtained from neutron scattering m easurem ents [26]. It isalso worth m entioning thatthe 1=2- lled Landau level problem has been m apped into an interacting 2D dipole gas [27]. M ore recently it was show n that the sam e type ofdescription ispossiblefora 2D electron gaseven in the absence ofa m agnetic eld [28]. In general,the m any-body problem ofholes in an A F background is extrem ely com plicated, involving m anyV (r)=

2

stripe form ation. T he loops and diagonalstripes tend to deform to pass very close to the im purities in order to m axim ize the attractive energy. H owever,the dipole interaction is su cient to retain the m ain orientation. T his leads us to conjecture thatw ith the addition ofthe kinetic energy the holes can m ove in string segm ents in an orientation given basically by the phase diagram of the clean system . T hese string segm ents are kept together by the dipolar interaction (i.e., string tension). T he stripe m otion would then be caused by m esoscopic therm al or quantum tunneling of the nite strings between the m inim a of the overallpotential. T his would lead to non-lineare ectsin the low tem perature eld dependent conductivity [35]and unusualT dependence of the conductivity [33]. W e have also perform ed sim ulations in the presence of a realistic underlying periodic lattice and have found thatthiscreatesslightdistortions in thephases,pinning loopsm orestrongly [33].In particular,the peaks in (q) sharpen in som e ofthese phases. Finally,at nite T m elting ofthe phases occurs because ofthe sm allenergy scalesand largeentropy in long-range C oulom b tails. In sum m ary, using a novel num erical technique, we have studied the com petition between long-range and short-range interactions and its im pact on hole ordering in layered transition m etal oxides. Em ploying the SD W pictureofthesesystem s,wehavestudied theshortrange attractive force and the dipolar force generated by the short-range A F uctuations together w ith longrange C oulom b forces for a 2D layer. W e have found a rich phase diagram for the clean system w hich includes a W igner solid,stripes,loops and a glassy phase. T his phase diagram is consistent w ith severaldi erent experim entalm easurem ents. W e have also found this system to be rather sensitive to the presence ofcharged im purities. H owever,the stripe phases survive as nite stripe segm entsw hich we believe is key to understanding these system s. Finally we have also investigated the e ects of a periodic lattice,w hich leadsto m inordistortionsofthe various phases. T hese results are of great im portance for the understanding of the phase diagram of layered transition m etaloxides. W e gratefully acknow ledge valuable discussions w ith A . B alatsky, A . C hernishev, J. G ubernatis, C . H am m el, D . Pines, J. Schm alian, S. W hite and J. Zaanen. A .H .C .N .acknow ledges support from the A lfred P.Sloan foundation.W ork atLosA lam oswassupported by the U .S.D epartm entofEnergy.W ork atthe U niversity ofC alifornia,R iverside,was partially supported by a Los A lam os C U LA R project.

inclusion oflatticee ects.W e nd thatthetransition between the ferro-dipolarand the stripephase is rstorder, w hile othertransitionsappearto be ofsecond order[33]. T he stripe tension ofthe hole patternsw illbe quanti ed elsew here [33]. In the casespresented above we have assum ed uniform dipolar interaction. It is wellknow n that there are orthorhom bic and tetragonaldistortions in practically all transition m etal oxides. In particular static stripe form ation has only been observed in the low tem perature tetragonalphase of La1:6 x N d0:4Srx C uO 4 [9]. In order to study the in uence ofthe anisotropy we assum e that the dipole sizesalong x and y directionshave anisotropy ( = 1 corresponds to the isotropic case). Figure 2c show s our solution for = 0:8: the sym m etry is broken and a stripe superlattice is form ed, w ith a charge ordering vector K = ( =‘)x,w here ‘ is the inter-stripe distance.In the SD W m odelthe Fouriertransform ofthe m agnetization S(q)= hSz (q)i is slaved to (1) such that a peak atK in (1)leadsto a peak atQ K in S(q)[33]. T husourresultsyield a neutron peak at( =a =‘; =a). A ssum ing tw inning, this would im ply neutron peaks at ( =a =‘; =a =‘)in agreem entw ith experim ent[11]. T hesam eisobtained in thecheckerboard phase(seeFigs. 2cand 2d).Ifoneincludesthekineticenergy [33],instead of static stripe form ation one would obtain dynam ical stripes like those believed to exist in La2Srx C uO 4. In thiscase the Ferm isurface ofthe system should be m odi ed by the superlattice form ation [34]. O ur results are som ew hat sensitive to the applied boundary conditions: rst,the exact size ofthe checkerboards depends on its com m ensuration w ith the com putationalbox, w hich, in turn depends on the density. O n increasing ofthe size ofthe com putationalbox,the checkerboard pattern show n in Fig.2c acquires point or line defects[33].T hisleadsto the reduction in the higher order peaks observed in Fig.2d w ith no change in their wave-num bers.Second,in a nite system w ith appropriate charge background,the holes do not form geom etric phases,although they stillform stripes[33]. H owever,in this case even a very sm allanisotropy ( 0:95) again leads to stripe form ation as in Fig.2e [33]. W e have also studied im purity e ects (from defects or charged counter-ions). For exam ple,we place the sam e num berofim puritiesasholes,random ly in a plane a distanced = 6A abovetheplaneto sim ulatethesituation in, e.g.,Sr doped cuprates and consider the unscreened attractive C oulom b interaction between im purity and hole. T hechargepattern produced isvery sensitiveto im purity doping (see Fig.2f). T he W ignercrystalbecom es glassy w ith no obvious sign of charge ordering. T his happens because the attractive C oulom b energy between im purities and holes scales like e2=d w hile the average interp hole C oulom b energy behaves like e2 s. T hus w hen 2 s < 1=d the holes are pinned by im purities. M ost strikingly, all other phases are unstable towards nite 3

10 [1] J.M .Tranquada,cond-m at/9802043. [2] Phase separation in cuprate superconductors, ed. by K . A .M uller and G .B enedek (W orld Scienti c,Singapore, 1993). [3] P.C . H am m el et al., in Proceedings of the C onference A nharm onic Properties ofH igh T c C uprates (1994). [4] J.M .Tranquada et al.,Phys.R ev.Lett.70,445 (1993). [5] J.H .C ho et al.,Phys.R ev.Lett.70,222 (1993). [6] F.C .C hou et al.,Phys.R ev.Lett.71,2323 (1993) [7] F.B orsa et al.,Phys.R ev.B 52,7334 (1995). [8] S.M oriet al.,N ature 392,473 (1998). [9] J.M .Tranquada et al.,N ature,375,561,(1995). [10] J.M .Tranquada et al.,cond-m at/9702117. [11] G .A eppliet al.,Science 278,1432 (1997). [12] V . J. Em ery and S. A . K ivelson, Physica C 209, 597 (1993). [13] U .Low et al.,Phys.R ev.Lett.72,1918 (1994). [14] H . J. Schulz, J.Phys. France 50, 2833 (1989); J. Zaanen and J. G unnarsson, Phys. R ev. B 40, 7391 (1989); A .R .B ishop etal.,Europhysics Letters,14 ,157 (1991). [15] S.R .W hite and D .J.Scalapino,cond-m at/9801274. [16] J.Lekner,Physica (A m sterdam ) 176A ,485 (1991). [17] N .G r nbech-Jensen et al.,M olec.Phys.92,941 (1997). [18] J.C .Slater,Phys.R ev.82,538 (1951). [19] J.R .Schrie er,X .G .W en and S.C .Zhang,Phys.R ev.B 39,11663 (1989). [20] N . F. M ott in M etal-Insulator Transitions (Taylor & Francis,London,1974),pg.141. [21] B .O .W ells et al.,Phys.R ev.Lett.74,964 (1995). [22] B .I.Shraim an and E.D .Siggia,Phys.R ev.B 40,9162 (1989). [23] D . M . Frenkel and W . H anke, Phys. R ev. B 42, 6711 (1990). [24] E.Fradkin in Field T heories of C ondensed M atter System s (A ddison-W esley,R edw ood C ity,1991),pg.48. [25] S.C hakravarty et al.,Phys.R ev.Lett.60,1057 (1988). [26] R .J.B irgeneau et al.,Phys.R ev.B 38,6614 (1988). [27] V .Pasquier and F.D .M .H aldane,cond-m at 9712169. [28] D .-H .Lee,unpublished. [29] J. B onca and J.E. G ubernatis, Phys.R ev.E 53, 6504 (1996). [30] E.P.W igner,Phys.R ev.46,1002 (1934). [31] T his is a highly degenerate con guration w hich can be m apped into a classical six vertex m odel [33]. See, for instance, R .J.B axter in Exactly Solved M odels in StatisticalM echanics (A cadem ic Press, London, 1982), pg. 127. [32] T he sim ulations show the presence oftopologicaldefects in the phase ofthe dipole m om ents,i.e.,the totalphase in a loop m ay be shifted by 2 . [33] D etails ofthis calculation w illbe given elsew here. [34] A .H .C astro N eto and F.G uinea,Phys.R ev.Lett.80, 4040 (1998). [35] J. B ardeen, Phys. R ev. Lett. 45, 1978 (1980); D . S. Fisher,Phys.R ev.B 31,1396 (1985).

Doping (%)

8

Diagonal Stripes

Geometric Phase

6 4 2 Wigner Crystal 0 0

Clumps 2

B (eV)

4

FIG .1. Phase diagram as a function of the hole density and the strength ofthe dipolar interaction,B ,for A = 0.

0.5

(a)

ky (b)

-0.5

0

kx

0

-0.5 0.5 0.5

(c)

(d)

ky 0

-0.5

(e)

0

-0.5

k x 0.5

(f)

FIG .2. G eom etric phases resulting from the com petition ofdipolar and C oulom b interactions: panels (a) and (c)show holes (open circles) w ith their dipole orientations,in a sm all section of the com putational box, for the ferro-dipolar and stripe phases, respectively; panels (b) and (d) show contour plotsofthe hole density in m om entum space (see Eq. (1))for the phases show n in panels (a) and (c). Panel(e) show s the stripe phase obtained w ith dipole anisotropy of0.8,and panel (f) hole positions in the presence ofim purities (crosses).

4