Finite-temperature phase diagram of nonmagnetic impurities in high-temperature superconductors using a d = 3 tJ model with quenched disorder Michael Hinczewski1 and A. Nihat Berker1 3 − 1Feza Gu¨rsey Research Institute, TU¨BITAK - Bosphorus University, C¸engelk¨oy 34680, Istanbul, Turkey, 2Department of Physics, Ko¸c University, Sarıyer 34450, Istanbul, Turkey, and 3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A. 0 We study a quenched disordered d = 3 tJ Hamiltonian with static vacancies as a model of 1 0 nonmagneticimpuritiesinhigh-Tc materials. Usingarenormalization-groupapproach,wecalculate 2 theevolutionofthefinite-temperaturephasediagramwithimpurityconcentrationp,andfindseveral features with close experimental parallels: away from half-filling we see the rapid destruction of a n spin-singlet phase (analogous to the superconducting phase in cuprates) which is eliminated for a p & 0.05; in the same region for these dilute impurity concentrations we observe an enhancement J of antiferromagnetism. The antiferromagnetic phase near half-filling is robust against impurity 8 addition, and disappears only for p&0.40. 1 PACS numbers: 74.72.-h, 71.10.Fd, 05.30.Fk, 74.24.Dw ] l e The electronic properties and phase diagram of high- We consider the quenched disordered tJ model - r Tc materials are particularly sensititive to impurities— on a d-dimensional hypercubic lattice, −βH = t substitution of 3d transition elements (Zn, Ni, Co, Fe), {−βH (i,j)} + µimpn , where −βH (i,j) = s ij 0 i i i 0 at. porlaontehse[r1m].eTtahles (inAtle,rGplaa)y, bfoertwtheeenCduisaotrodmers,osftrtohnegCaunOti2- −Pth iσ(c†iσcjσ+c†jσciσ)P−J(Si·Sj−ninj/4)+µ(ni+nj) is the standard tJ model pair Hamiltonian. The static m ferromagnetic correlations in the parent compound, and P impurities at each site i occur with probability p via - doped charge carriers, offers a window onto the nature µimp = −∞ and do not occur with probability 1 − p d of both the superconducting phase and the normal state i n via µimp =0. above T . Doping by nonmagnetic (S = 0) Zn ions pro- i o c To formulate an RG transformation for this system, c vides one representative example: the most pronounced we use the d= 1 Suzuki-Takano decimation [9–19], gen- [ effect is the rapid destruction of the superconducting eralized to d > 1 through the Migdal-Kadanoff method phase [1, 2]; in YBCO the transition temperature is re- 3 [20, 21]. This technique, adapted for quenched random duced at a rate of ∼ 15K/at.% of impurities, so that v bond disorder, has recently elucidated the phase dia- 1 it takes Zn concentrations of only about 6% to entirely grams of the quantum Heisenberg spin-glass in various 7 eliminatesuperconductivity[2]. Thisisincontrasttothe spatial dimensions [18]. In our case the rescaling for the 1 antiferromagnetic phase at half-filling, which requires a d=1 system (with sites i=1,2,3,...) is: 7 far larger Zn concentration (about 40% in LSCO [3]) to 0 completely suppress. The effects in the metallic region Tr e βH 6 even − above T are equally surprising: nuclear magnetic res- /0 onance ecxperiments have found that Zn atoms induce =TrevenePi{−βH0(i,i+1)+µiimpni} mat laoncdalemnhagannecteicamntoifmerernotmsaagtnneetaicrescto-rnreeilgahtiboonrsCfuorsisteevse[r4a],l =TrevenePeiven{−βH0(i−1,i)+µiimpni−βH0(i,i+1)}+Poiddµiimpni even lattice spacings around the impurity [5, 6]. In lightly d- hole-dopedLSCOtherehavebeenobservationsofanini- ≃ Trie−βH0(i−1,i)+µiimpni−βH0(i,i+1) ePoiddµiimpni " # n tial increase in the N´eel temperature with Zn addition, Yi co aanntdifeervreonmiamgnpeutriictyo-ridnedruc[7e,d8]r.eappearance of long-range = evene−β′H0′(i−1,i+1) ePoiddµiimpni : " # v i Y Xi In this work we model the effects of nonmagnetic im- ≃ePeiven{−β′H0′(i−1,i+1)+µiim−p1ni−1} =e−β′H′, (1) puritiesinhigh-T materialsthroughad=3tJ Hamilto- c r a nian with quenched disorder in the form of static vacan- where the traces and sums are over even- or odd- cies. Through a renormalization-group (RG) approach numbered sites i, and −β′H′ is the renormalizedHamil- we obtain the evolution of the global temperature ver- tonian. Anticommutation rules are correctly accounted suschemicalpotentialphasediagramwithdisorder. Our for within segments of three consecutive sites, at all suc- results capture in a single microscopic model some of cessivelengthscalesastheRGtransformationisiterated. the major qualitative features of impurity-doping in real ThealgebraiccontentoftheRGtransformationiscon- materials: the rapid suppression of a spin-singlet phase, tained in the second and third lines of Eq. (1), yielding analogousto the superconducting phase in cuprates; the therenormalizedpairHamiltonian−β′H0′(i′,j′)through gradual reduction of the antiferromagnetic phase near the relation: exp(−β′H0′(i′,j′))=Trkexp(−βH0(i′,k)+ half-filling; and the enhancement of antiferromagnetism µimpn − βH (k,j )). Under the transformation the k k 0 ′ away from half-filling for small impurity concentrations. original system is mapped onto one with a more 2 τ AF 0.4 (a) (b) d D d D 0.3 0.2 t FIG.1: Hierarchicallatticeonwhichthed=3,b=2Migdal- 1/ 0.1 Kadanoff recursion relations are exact. e r u t 0.0 ra (c) (d) e p d D d D m 0.3 general form of the pair Hamiltonian, −βH0(i,j) = Te −tij σ(c†iσcjσ+c†jσciσ)−JijSi·Sj+Vijninj+µij(ni+ 0.2 n ) + ν (n − n ) + G , where the interaction con- j ij i j ij stantPs K ≡ (t ,J ,V ,µ ,ν ) are nonuniform, and 0.1 ij ij ij ij ij ij distributed with a joint quenched probability distribu- 0.0 tion P(K ). This generalized form of the Hamilto- −0.50.0 0.5 1.0 1.5 2.0 0.5 0.6 0.7 0.8 0.9 1.0 ij nian remains closed under further RG transformations. Chemicalpotentialµ/J Electrondensityhnii Through the relation above we can write the interac- tion constants K′i′j′ of the renormalized pair Hamil- FIG. 2: Pure system (p = 0) phase diagram of the isotropic tonian −β H (i,j ) as a function of the interaction d=3tJ model[11,12]forJ/t=0.444: (a)intermsofchem- ′ 0′ ′ ′ constants Ki′,k and Kk,j′ of two consecutive nearest- ical potential µ/J vs. temperature 1/t; (b) electron density neighbor pairs in the unrenormalized system, K′i′j′ = hnii vs. temperature1/t. Panels (c)and (d)show theanalo- R(Ki′k,Kkj′). This function R comes in two gous phase diagrams for the uniaxially anisotropic case [17], varieties, depending on whether or not there is with tz/txy = 0.3, Jz/Jxy = 0.09, Jxy/txy = 0.444. In both cases antiferromagnetic (AF), dense disordered (D), dilute an impurity at site k, which we shall denote as disordered (d), and τ phases are seen. The solid lines rep- R and R respectively. Starting with a sys- 0 imp resent second-order phase transitions, while the dotted lines tem with quenched probability distribution P(Kij), arefirst-orderphasetransitions(withtheunmarkedareasin- the distribution P′(K′i′j′) of the renormalized sys- sidecorrespondingtocoexistenceregionsofthetwophasesat tem is given by the decimation convolution [22]: eitherside). Dashedlinesarenotphasetransitions,butdisor- P′(K′i′j′) = dKi′kdKkj′P(Ki′k)P(Kkj′)[pδ(K′i′j′ − der lines between the dilute disordered and dense disordered Rimp(Ki′k,KkjR′))+(1−p)δ(K′i′j′−R0(Ki′k,Kkj′))]. The phases. initialconditionfortheRGflowisthedistributioncorre- spondingtotheoriginalsystem, P (K )=δ(K −K ), 0 ij ij 0 where K ={t,J,−J/4,µ,0}. defined above for interactions in series, and a “bond- 0 The RG transformation is extended to d > 1 through moving” convolution for interactions in parallel, using the Migdal-Kadanoff [20, 21] procedure. While approxi- the function Rbm(KA,KB) = KA +KB. In order to mate for hypercubic lattices, the recursionrelationsgen- numerically implement the convolution, the probability eratedbythisprocedureareexactonhierarchicallattices distributions are represented by histograms, where each [23–25],andwe shalluse this correspondenceto describe histogram is a set of interaction constants (t,J,V,µ,ν) the RG transformation for the case d = 3, with length and an associated probability. Since the number of his- rescalingfactorb=2. Theassociatedhierarchicallattice tograms that constitute the probability distribution in- is shown in Fig. 1. Its construction proceeds by taking creases rapidly with each RG iteration, a binning pro- each bond in the lattice, replacing it by the connected cedure is used [26]. Furthermore since evaluation of the cluster of bonds in the middle of Fig. 1, and repeating R functions is computationally expensive, and most of this step an infinite number of times. The RG trans- the weightoftheprobabilitydistributionsiscarriedby a formationconsistsof reversingthis constructionprocess, fraction of the histograms, we have added an additional by taking every such cluster of bonds, decimating over step before the decimation convolution to increase effi- the degreesof freedomatthe four inner sites of the clus- ciency: the histograms with the 100 largestprobabilities ter, yielding a renormalized interaction between the two are left unchanged, while the others are collapsed into a edge sites of the cluster. Denoting these edge sites as i, singlehistogramina waythat preservesthe averageand ′ j , and the four inner sites as k ,...,k , this decimation standard deviation of the quenched distribution. Thus ′ 1 4 canbe expressedasKi′j′ = 4n=1R(Ki′kn,Kknj′). Just we evaluate 104 local decimations at each RG transfor- as in the d = 1 case, this decimation will give, after a mation. P single RG transformation, a system with a nonuniform Allthermodynamicpropertiesofthesystem,inpartic- quenched distribution of interaction constants. We can ular the finite-temperature phase diagram,canbe deter- calculatethequenchedprobabilitydistributionP′(Ki′j′) mined from analyzing the RG flows. In the pure (p=0) of the renormalized system through a series of pairwise case, the transformation described above reduces to the convolutions, consisting of the decimation convolution recursionrelationsderivedforthe d=3tJ modelinear- 3 FIG.3: On-siteandnearest-neighborsingletandtripletpair- pair correlations for the d = 3 tJ model, with p = 0, J/t = 0.444. In(a)and(b)thecorrelationsareplottedasafunction ofchemicalpotentialµ/J atconstanttemperature1/t=0.10. In (c) they are plotted as a function of temperature 1/t, at theconstantelectron densityhnii=0.67. Thecorresponding phases are indicated near the top of each plot, with solid anddottedverticallinesmarkingsecond-orderandfirst-order phase boundaries respectively. lier studies [11, 12], and yields the phase diagramshown in Fig. 2(a,b) for J/t = 0.444. Here we summarize the FIG. 4: Calculated phase diagrams of the d = 3 tJ model, observed phases (for details, consult [11, 12]): near half- with J/t = 0.444, for various values of the impurity concen- filling (µ/J → ∞, hnii → 1), there is a transition with tration p, plotted in terms of temperature 1/t vs. chemical decreasing temperature from a densely-filled disordered potential µ/J. The phases depicted in thefigures are: dilute phase (D) to long-range antiferromagnetic order (AF). disordered(d),densedisordered(D),antiferromagnetic(AF), This AF phase persists away from half-filling down to and τ. The inset shows AF transition temperatures for the µ/J ≈ 1.6, or 5% hole doping. For very large hole dop- near-half-filled system (µ/J =100) as a function of p. ings (& 37%) we go over into a dilute disordered phase (d), with narrow first-order coexistence regions between in the same region. We see similar behavior in Fig. 3(c), the d and D phases. At intermediate hole dopings of where the correlations are plotted as a function of tem- 33-37% a novel phase (τ) is found at low temperatures, perature 1/t at a constant electron density hn i = 0.67. flanked by an intricate lamellar structure of AF islands. i As we decreasethe temperature,approachingthe transi- The τ phase is characterized by the formation of tionintotheτ phase,thereisasignificantincreaseinthe nearest-neighbor spin-singlet pairs, as can be under- singletcorrelationsandrapiddecayofthetripletcorrela- stood from correlation functions calculated using the tions. Spin-singletliquids,i.e.,thehole-dopedresonating RG flows. Let us define a singlet pair-pair correla- valence bond (RVB) state, have featured prominently in tion function Tsing =h∆sing†∆sing+∆sing†∆singi, where ij,kl ij kl kl ij theories of high-T superconductivity (for a review see c ∆sijing = √12(ci↓cj↑ − ci↑cj↓), and the analogous triplet Ref. [27]). As we shall see below, the behavior of the τ correlation function Ttrip in terms of ∆trip = c c + phase under impurity-doping is analogous to that of the ij,kl ij i j ↑ ↑ superconducting phase in high-T materials. 1 (c c +c c )+c c . For clusters of three consec- c √2 i↓ j↑ i↑ j↓ i↓ j↓ Though in this study we focus on the isotropic d = 3 utive sites i, j, k in the lattice, Fig. 3 shows the on-site model, there is evidence that general features of the correlations Tsing, Ttrip, and nearest-neighbor correla- ij,ij ij,ij p = 0 phase diagram discussed above persist even in tionsTsing,Ttrip . InFig.3(a)and(b),weseeaconstant the case of spatial anisotropy, with uniform interactions ij,jk ij,jk temperature slice at 1/t = 0.10 as µ/J is varied. There (t , J ) along the xy planes and weaker interactions xy xy is a broad region of chemical potentials away from half- (t , J ) along the z direction. Through a similar RG z z filling, centered at the τ phase, where both the on-site approach, using the more complicated hierarchical lat- and nearest-neighbor singlet correlations are strong, in tice associated with a uniaxially anisotropic cubic lat- contrastto the tripletcorrelations,which aresuppressed tice [28], itwasfoundinparticularthatthe τ phasecon- 4 tinues to exist in roughly the same doping range even case, was replicated in the d = 2 tJ model using a self- for weak interplanar coupling, though as expected the consistent diagrammatic approach[34], and in the d=2 transitiontemperaturessteadilydecreaseasthecoupling Hubbard model with the dynamical cluster approxima- is reduced [17]. A representative phase diagram, with tion [35]. Thus the enhancement of the AF phase away t /t = 0.3, J /J = 0.09, J /t = 0.444, is shown from half-filling, which we find at small impurity con- z xy z xy xy xy in Fig. 2(c,d). Thus the τ phase may be relevant even centrations,isconsistentwithpreviousexperimentaland in the strongly anisotropic regime important for high-T theoretical indications. c materials, which are characterized by weakly interacting On the other hand for larger concentrations of impu- CuO planes. 2 rities,the dilution ofthe spins inthe lattice becomes the InFig.4weshowtheevolutionofourcalculatedphase dominant effect, and eventually all long-range magnetic diagram with increasing impurity concentration p. The orderisdestroyedinthesystem. WeseethisinFig.4(e)- τ phase is rapidly suppressed for p = 0.01 through 0.04 (h), showing phase diagrams for p = 0.10 through 0.40, [Fig. 4(a)-(d)], and is no longer present by p = 0.05. and in the inset which plots the AF transition temper- The rate at which the τ phase disappears is comparable ature as a function of p near half-filling (µ/J = 100). to the reduction of T with nonmagnetic impurities in c In contrast to the τ phase, the AF phase around half- cuprates, where typically concentrations ≈ 2−6% (de- filling is robust against impurity addition, and only dis- pending on dopant) are enough to eliminate supercon- appears for p & 0.40. Qualitatively similar behavior has ductivity [1, 2]. As the area of the τ phase recedes for been seen in the half-filled compound La Cu Zn O , 2 1 z z 4 these small impurity concentrations, the region it for- whereZnconcentrationsofz ≈0.4arerequired−toreduce merly occupied is replaced by a complex lamellar struc- the N´eeltemperature to zero[3], much largerthan those ture of the AF phase. We can understand this enhance- needed to eliminate superconductivity in the hole-doped mentofantiferromagnetismthroughanRVB-likepicture material. oftheτ phase[29]: inthe purecasethenearest-neighbor To summarize, we have applied an RG approach to singlets resonate in all possible arrangements along the the quenched disordered d = 3 tJ model, and found the bonds;whenanimpurityisaddedsomeofthesearrange- evolution of the phase diagram as a function of impu- ments are “pruned”, because the bonds adjacent to the rity concentration. The spin-singlet phase away from impurity can no longer accommodate singlets. This in- half-filling is quickly destroyed through the addition of hibition of singlet fluctuations leads to enhanced anti- small quantities of static vacancies, while antiferromag- ferromagnetic correlations around the vacancy. Such lo- netism in the same region is enhanced. The antiferro- cal AF enhancement near dilute nonmagnetic impurities magnetic phase near half-filling is less sensitive to impu- has been observed through NMR and NQR studies on rity addition, and completely disappears only at larger Zn-doped YBCO [5, 30], and supported theoretically by concentrations. These results all have close parallels in finite-cluster studies of the d=2 Heisenberg [31] and tJ experimentalresultsfromcuprates. The RG methodde- [32,33]models. Moredramatically,inlightlyhole-doped scribedherefordealingwithquencheddisorderinthetJ La Sr Cu Zn O (with x = 0.017) the N´eel tem- 2 x x 1 z z 4 pera−tureactu−allyincreaseswiththeadditionofZnupto Hamiltonian could be generalized to more complex sys- tems: forexamplethe disorderedHubbardmodel, where z =0.05,beforeturningdownwardsagainathigherz [7]. the double-occupationof sites is allowedthrougha finite A similar, though smaller, effect has been found even at electron-electron repulsion. The role of electron correla- larger hole dopings of x = 0.115 and 0.13, with the T N tions and disorder in this system has led to interesting increasing up to z = 0.0075 [8]. In the case of x = 0.13, phase diagram predictions [36–38], which could be fur- there is even no long-range antiferromagnetic order for ther explored with RG techniques. the Cu spins in the Zn-free compound; it appears for z > 0.0025. 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