Nonmagnetic impurities in two- and three-dimensional Heisenberg antiferromagnets

Tai-Kai NgDepartment of Physics, Hong Kong University of Science and Technology, Clear Water Bay Road, Kowloon, Hong Kong

~Received 30 July 1996!

In this paper we study in a large-S expansion the effects of substituting spins by nonmagnetic impurities intwo- and three-dimensional Heisenberg antiferromagnets in a weak magnetic field. In particular, we demon-strate a mechanism where magnetic moments are induced around nonmagnetic impurities when magnetic fieldis present. As a result, Curie-type behavior in magnetic susceptibility can be observed well below the Ne´eltemperature, in agreement with what is observed in La2Cu12xZnxO4 and Sr(Cu12xZnx)2O3 compounds.@S0163-1829~96!04542-0#

Recently, there has been increasing interests in the effectsof substituting Cu by nonmagnetic ions~Zn! in high-Tc cu-prates, where interesting effects were discovered for com-pounds in the underdoped regime.1 In particular, magnetic-susceptibility measurements,2 and other experiments3 seemto indicate that local magnetic moments are generated as Cuions were replaced by Zn, both in the underdoped high-Tcyttrium and bismuth compounds,2,3 and for anundopedlan-thanum compound.4 Based on resonant-valence band theo-ries of the t-J model, Nagaosa and Ng5 have constructedexplanations on how local magnetic moments can be gener-ated in these compounds in the disordered phase of two-dimensional antiferromagnets. Their theory is believed to beapplicable to the underdoped regime of high-Tc compounds.However the theory is not applicable to the La compoundwhere Curie-type behavior is observed in magnetic suscepti-bility at temperatures well below the Ne´el temperature wherelong-range antiferromagnetic order exists. More recently,similar experiment on the two-leg ladder compoundSr(Cu12xZnx)2O3 also indicates formation of local magneticmoments in the spin-gap phase as Cu ions are replaced by Znions.6 The compound was found to order antiferromagneti-cally at low temperature atx;0.07.7 Surprisingly, Curie be-havior in magnetic susceptibility remains evenbelow theNeel-ordering temperature,7 indicating that formation ofmagnetic moments around nonmagnetic impurities is a gen-eral phenomenon in Ne´el state of quantum antiferromagnets.

In the presence of long-range antiferromagnetic order, thelow-temperature properties of clean antiferromagnets can beunderstood at least qualitatively using a semiclassical theory(1/S expansion!. In this paper we shall generalize this ap-proach to include the effects of nonmagnetic impurities inthe presence of a uniform magnetic field. Notice that in theabsence of magnetic field, a similar study has been per-formed by Bulutet al.,8 where no ‘‘free’’ magnetic momentswere found to be induced around nonmagnetic impurities.We shall show that local magnetic moments can be inducedby nonmagnetic impurities once magnetic field is applied onthe system through a novel mechanism. Notice that strictlyspeaking, thermal fluctuations destroy long-range magneticorder at two dimensions at any finite temperature. Thus ouranalysis at two dimensions~2D! can only be applied to lay-ered systems like lanthanum cuprates where the magneticcoupling between different layers is much weaker than intra-layer coupling.

To begin with, we first consider the zero temperatureproblem of classical spins interacting antiferromagneticallyunder a uniform magnetic fieldB, with a single nonmagneticimpurity replacing spin at sitei 0. We shall consider the Ne´elstate in the absence of magnetic field to be ordered in thezdirection, with the uniform magnetic fieldB applied in the1x direction. The classical ground state in the absence ofmagnetic field has spins all pointing in1z direction for spinsin theA sublattice and spins all pointing in the2z directionfor spins in theB sublattice. In the presence of a magneticfield, the spins tilt to the1x direction to minimize the mag-netic energy. Letu i

A be the angle tilted away from thez axisfor spin on sitei on theA sublattice andu j

B be the corre-sponding angle for spin on sitej on theB sublattice. It iseasy to show that the classical energyEcl is given in the limitwhenuA(B) are small~weakB-field limit! by

Ecl /S252 (

^ iÞ i0 , j &S 12

~u iA1u j

B!2

2 D 2B8 (iÞ i0

u iA

2B8(j

u jB , ~1!

where^ iÞ i 0 , j & are nearest-neighbor sites in the square~cu-bic! lattice excluding contributions from the nonmagneticimpurity at site i 0 and B85gmBB/S. Notice that the onlyeffect of nonmagnetic impurity is to remove the spin at sitei 0 in our theory. We have also set the spin-couplingJ51 inEq. ~1!. Notice that the energy expression forEcl is validonly to orderO(B2).

Next we consider the continuum limit of the energy ex-pression~1!. Introducing symmetric and antisymmetric anglevariables

us~xW !5S@uA~xW !1uB~xW !#,

ua~xW !5S

2@uA~xW !2uB~xW !#, ~2!

we obtain after some straightforward algebra

PHYSICAL REVIEW B 1 NOVEMBER 1996-IVOLUME 54, NUMBER 17

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Ecl5E ddx

aod F ~d!us~xW !22B8Sus~xW !1@¹ua~xW !#2

1u~x,!

va@us~xW !ao•¹ua~xW !2ua~xW !ao•¹us~xW !#G1E0 ,

~3!

where E052NS2d/2, N5number of sites in the system,ao is lattice spacing andao is a radial vector pointing awayfrom i 0. va is the volume enclosed by the sphere~circle in2D! with radiusao . x,5ao2uxW2xW0u. For an impurity lo-cated at theB sublattice,ao→2ao .

The various terms appearing in Eq.~3! can be understoodrather easily. First of all, in the limitao→0, xi→xj in Eq. ~1!and the only contribution toEcl would be the first twousterms in Eq.~3!. It is also clear that in the limitao→0, termsproportional toua or ua

2 do not appear inEcl , and the onlycontribution fromua can appear as (¹ua)

2 only. The appear-ance of a nonmagnetic impurity at sitei 0 breaks the symme-try between theA and B sublattices and introduces localcoupling between theus andua fields. Notice that additionalcoupling betweenua andus fields will appear if we take intoaccount higher-order terms inB in our energy expressionEcl . However, in the absence of impurities, these terms donot break the symmetry between theA andB sublattices orthe symmetry of interchanginguA anduB fields.

Minimizing Ecl with respect to theus field we obtain

us~xW !51

2d SB8S22u~x,!

vaao•¹ua~xW ! D . ~4a!

Putting us back intoEcl and minimizing with respect touafield, we obtain foruxW2xW0u>ao ,

¹2ua~xW !52B8Saodva

dd~ uxW2xW0u2ao!. ~4b!

Notice that in the presence of magnetic field, the nonmag-netic impurity acts as a source term for theua field ofstrength 2B8S in the limit ao→0. As a result,ua(xW );(2B8S)ln(uxW2xW0u) in two dimensions, and the corre-sponding ‘‘electric-field’’ energy cost;*d2x(¹ua)

2 di-verges logarithmically as the size of system goes to infinity.

The divergence of magnetic energy to orderO(B2) indi-cates that the~classical! response of a nonmagnetic impurityto external magnetic field is intrinsicallynonlinear9 and sug-gests that quantum effect may play an important role in de-termining the correct response of nonmagnetic impurities toexternal magnetic field. In the following we shall derive inthe continuum limit, the 1/S ~spin wave! Lagrangian in thepresence of backgroundus andua fields, and shall show thatthe infrared divergence in classical energy can be cured byformation of local magnetic moment around nonmagneticimpurity once quantum effects are considered.

To derive the spin-wave Hamiltonian in the presence ofmagnetic field, we rotate our coordinate system locally oneach site such that the localz axis is always along the ‘‘clas-sical’’ spin direction. In this coordinate system, the Hamil-tonian becomes

H5 (^ iÞ i0 , j &

@cos~u i1u j !~Si~z!Sj

~z!1Si~x!Sj

~x!!1Si~y!Sj

~y!

1sin~u i1u j !~Si~z!Sj

~x!2Si~x!Sj

~z!!#

2gmBB(iÞ i0

~Si~z!sinu i1Si

~x!cosu i !

2gmBB(j

~cosu jSj~x!2sinu jSj

~z!!. ~5!

The Hamiltonian can be rewritten in the Schwinger-bosonrepresentation of spins in the usual way. In the large-S limit,we may write

Zi↑A 5Zi↑

A 5Zj↓B 5Zj↓

B 5A~2S!,

whereZ(Z) isa ’s are spins Schwinger-boson creation~anni-

hilation! operators for sitei on sublatticea. Expanding theHamiltonian to orderO(S), we obtain to orderO(B2),

H5Ecl1H1/S ,

where

H1/S5S (^ iÞ i0 , j &

F S 12~u i1u j !

2

2 D ~ Zi↓A Zi↓

A 1Zj↑B Zj↑

B !

1S 12~u i1u j !

2

4 D ~Zi↓A Zj↑

B 1Zi↓A Zj↑

B !G1gmBB(

iÞ i0~u i !Zi↓

A Zi↓A 1gmBB(

j~u j !Zj↑

B Zj↑B .

~6!

To further analyze our system we again go to the con-tinuum limit. Introducing symmetric and antisymmetric bo-son fields

f~xW !51

A2@Z↓

A~xW !2Z↑B~xW !#,

p~xW !51

A2@Z↓

A~xW !1Z↑B~xW !#, ~7!

and integrating out thep(xW ) field, we obtain in the con-tinuum limit an effective Lagrangian for thef(xW ) field.10,11

Details of the technique can be found in Ref. 10 and we shallnot repeat them here. ForxW away from the impurity site(ao,uxW2xW0u), we obtain in imaginary time

L1/S;E dtE ddx1

4dSF S ]

]t2euaDf1S ]

]t1euaDf

1m2f1f1c2u¹fu2G , ~8!

where e5B8 andm25S2B82/2. c2;4d(JS)2 is the spin-wave velocity. We have setus(xW )5B8S/2d and have ne-glected (¹ua)

2 terms in derivingL1/S . The latter is of higherorder @O(1/S)# compared with the corresponding term in

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Ecl . The most striking feature of the effective Lagrangian isthat the ua field now appears as thet component of aU(1) gauge field coupling to a charge boson fieldf, withthe dynamics of theua field governed byEcl . In particular,in the presence ofB field, the nonmagnetic impurity appearsas electric charge generating a static electric-field coupling tothe charge bosons. Notice that the effective charge of thenonmagnetic impurity changes sign when it is moved fromone sublattice to another, indicating that the ‘‘sign’’ of thecharge is in fact a sublattice index.10,11

The logarithmic divergence in ‘‘electric-field’’ energy inthe presence of nonmagnetic impurity inEcl in two dimen-sions will be removed if bosons of opposite electric chargeare nucleated from vacuum to screen the effective electricfield, forming effective local magnetic moments around theimpurity. The number of bosons nucleated from vacuum is;2S, as can be seen easily by counting the number ofcharges carried by the nonmagnetic impurity. The magnitudeof magnetic moment formed around the impurity is thus;S. The energy cost for nucleating the bosons can be com-puted by solving the corresponding Schro¨dinger equation forcharge bosons moving in scalar potential of external chargeof magnitude 2Se.12 To capture qualitatively the physics, weestimate the energy cost by using a variational wave functionc(rW);Ae2urWu/j1, wherej1 is determined variationally. Mini-mizing the energy, we find thatj1 is of the orderj1;@JSc2/(gmBB)

3#1/2. Notice that j1→` as B→0 orS→`, indicating that the formation of a magnetic momentaround a nonmagnetic impurity is a nonperturbative quantumeffect which cannot be captured by the usual spin-wavetheory. The energy cost for nucleating the bosons is of theorderE1;(2S)@m12SB82ln(j1 /ao)#.

Next we discuss the situation when there is finite concen-tration of impuritiesn randomly distributed in the system,which is the case of experimental interest. In our continuumtheory, the presence of finite concentration of random impu-rities is equivalent to putting finite concentration of chargeswith magnitude 2SB8 and random sign in our effectivecharge boson system. The behavior of the system is verydifferent from the one impurity case, since now electric fieldsoriginating from opposite charges will cancel, with remain-ing electric-field energy of the order;nEe , whereEe;(2SB8)2ln(l/ao) on average,l;n21/d is the average dis-tance between nonmagnetic impurities. ForEe.E1;2Smorequivalently l@jo;aoe

JS/(2gmBB), it is energetically favor-able to nucleate charge bosons from the vacuum to screen theelectric field, and magnetic moments will be formed aroundnonmagnetic impurities. The magnetic moments couple toeach other weakly withJeff;Je2 l /j1 implying that Curie be-havior will be present in magnetic susceptibility down tovery low temperature;Jeff . As the concentration of non-magnetic impurities increases,l decreases untill;jo . Atthis point a transition occurs where it becomes energeticallyunfavorable to nucleate bosons from the vacuum to screenthe electric field, i.e., there will be no magnetic momentsforming around nonmagnetic impurities at zero temperaturewhen l<jo . The boson-non-magnetic impurity bound statesbecome excited states of the spin system.

The boson-non-magnetic impurity bound states can stillbe observed as effective local magnetic moments at finite

temperatureT>m;gmBB if Ee12Sm.E1 ~correspondingroughly toj1, l ). At this energy range, the boson-impuritybound state appears as a stable excited state of the spin sys-tem with excitation energy;(E12Ee)/2S,m. At finitetemperatureT>m, these states will be occupied by thermallyexcited bosons, forming effective free magnetic momentsaround nonmagnetic impurities. It is important to emphasizethat the formation of an excited boson bound state around anonmagnetic impurity is only possible when there is a gapm in the boson excitation spectrum. In the absence of non-magnetic impurity, a direct computation of the spin-wavespectrum indicates that the spin-wave spectrum splits intotwo branches at small momentumq, with one branch carry-ing a gap larger thanm in our effective Lagrangian, and theother branch gapless asq→0. The contribution from thesplitting of the two branches toH1/S cancels to orderO(B2), leading to our effective Lagrangian with only mas-sive bosons. The splitting of the boson spectrum reappears ifwe include terms to orderO(B4) in H1/S , and the bosonbound state around the nonmagnetic impurity will cease tobe a good eigenstate of the effective Lagrangian~the boundstate turns into a resonant state!. However, the decay rate ofthe resonant state;B4 is small compared with the energy ofthe resonant state;B2 in the limit of the weak magneticfield, indicating that the physical picture of the boson boundstate around a nonmagnetic impurity is still a good approxi-mation to the system in the weak-field limit.

As concentration of impurities increases and distances be-tween impurities decreases further, the overlap between bo-son bound state wave functions at different impurity sitesincreases and a boson impurity band is formed~recall thatthe size of the boson bound state wave function is of theorderj1). For l<j1, the bandwidthW of this impurity bandis of the order;@Am21(c/ l )22m# and the effective impu-rity potential strength is of the orderv;2SB82/(1/l )2. Wefind that the screening of effective impurity potential bycharged bosons is weak and can be neglected. Estimating thescattering lifetime t we find that bosons with energyE<El;cl/j1

2 are strongly localized in this impurity bandusing the criteriaEt<1. Notice that the number of localizedboson states;n21 and decreases asn increases in this re-gime. The magnetic susceptibility will show Curie behaviorat temperature aboveTl;El /kB . ForT<Tl , the number ofeffective magnetic moments contributing to magnetic sus-ceptibility decreases as temperature lowers. Roughly speak-ing, the contribution from nonmagnetic impurities to themagnetic susceptibility can be described by a magnetic-fieldand temperature-dependent density of local momentsn(B,T) which has the following properties:~i! n(B,T);0 atthe region gmBB@kBT, ~ii ! n(B,T);n for l>j1, and;(nj1

4)21 for l<j1 at the regionkBT@gmBB1El , and issmoothly interpolating between the two regions. As tempera-tureT→0,n(B,T)→0 and the contribution to magnetic sus-ceptibility from impurities is of the orderdx;4S2nln(l/ao).

Our theory can be extended to three dimensions in astraightforward way. We find that in the weak-field limit, nomagnetic moment is formed even in the limit of one singleimpurity at zero temperature because of the absence of infra-red divergence associated with the Coulomb potential in 3D.However, bound states of bosons can be formed at

54 11 923BRIEF REPORTS

T>gmBB as in the two-dimensional case, and can still resultin Curie-type behavior in magnetic susceptibility forl@j1;(c/gmBB)

3. As concentration of impurities increasesfurther, the boson bound states turn into impurity bands andthe effective number of free magnetic momentsn(B,T) de-creases when temperature is lower than the boson impurityband bandwidth as in two dimensions.

Summarizing, in this paper we have carried out an analy-sis of the effects of replacing spins by nonmagnetic impuri-ties in Heisenberg antiferromagnets in a magnetic field intwo and three dimensions in a semiclassical 1/S expansion.We find within a continuum approximation that magneticmoments will form around nonmagnetic impurities once themagnetic field is put on the system, resulting in Curie-typebehavior in magnetic susceptibility when concentration ofimpurities is not too high, and temperature is not too low.

Notice that our theory predicts that formation of local mag-netic moments in the presence of external magnetic field andnonmagnetic impurities is a general behavior of orderedquantum antiferromagnets and is not restricted to particularmaterials. The theory explains the observation of Curie-typebehavior in magnetic susceptibility in La2Cu12xZnxO4 andSr(Cu12xZnx)2O3 compounds at temperatures well belowNeel temperature. In particular, the vanishing of local mo-ments atT<gmBB is a theoretical prediction which can betested experimentally.

I thank G. Aeppli, M. Ma, and M. Azuma for very usefuldiscussions and the hospitality of Lucent Technologies, BellLabs. where part of this work was done. The work is par-tially supported by Hong Kong UGC Grant No. UST636/94P.

1Y. Fukuzumiet al., Phys. Rev. Lett.76, 694 ~1996!.2G. Xiao et al., Phys. Rev. B35, 8782~1987!.3H. Alloul et al., Phys. Rev. Lett.67, 3140 ~1991!; K. Kakuraiet al., Phys. Rev. B48, 3485~1993!.

4S-W. Cheonget al., Phys. Rev. B44, 9739 ~1991!; K. Uchi-nokuraet al., Physica B205, 234 ~1995!.

5N. Nagaosa and T.K. Ng, Phys. Rev. B51, 15 588~1995!.6M. Azumaet al., Phys. Rev. Lett.73, 3463~1994!.7M. Azumaet al. ~unpublished!.8N. Bulut et al., Phys. Rev. Lett.62, 2192 ~1989!; see also N.

Nagaosaet al., J. Phys. Soc. Jpn.58, 9781~1989!.9It can be shown that the logarithmic divergence in classical linearresponse is removed once higher-order terms inus and ua areincluded in the energy expression. The resulting classical energyis of orderB2ln(SJ/mBB).

10N. Read and S. Sachdev, Phys. Rev. B42, 4568~1990!.11T.K. Ng, Phys. Rev. B52, 9491~1995!.12The Schro¨dinger equation for charged boson in the effective sca-

lar potential of a external point charge is@E2V(r )#2c(rW)

5(c2¹21m2)c(rW), whereV(r );2SB82ln(r/ao).

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