PHYSICAL REVIEW B VOLUME 48, NUMBER 10 1 SEPTEMBER 1993-II

Magnetic excitations of a doped two-dimensional antiferromagnet

A. ShermanInstitute of Physics, Estonian Academy of Sciences, Riia lpga, EE2$00 Tartu, Estonia

M. SchreiberInstitut fiir Physikalische Chemic, Johannes Gute-nberg Univ-ersitat,

Jakob-Welder Weg -I1, D 550-99 Mainz, Federal Republic of Germany(Received 22 March 1993)

Magnetic excitations of the two-dimensional (2D) t Jmo-del are considered in the presence of asmall concentration of holes c. The spin-wave approximation used implies long-range antiferromag-netic ordering from the beginning. Migdal s theorem is shown to be valid for the model considered.The energy spectrum of the magnons is determined with the help of the one-pole approximationfor the hole Green's function. If the concentration of mobile holes is larger than a critical value anadditional branch of overdamped magnons arises near the I and M points of the Brillouin zone.This is connected with the generation of electron-hole pairs (the Stoner excitations) by magnons.The appearance of such excitations means the destruction of the long-range antiferromagnetic order.For parameters presumably realized in cuprate perovskites this happens for several percent of holesper site. The relation between the critical concentration and the hole concentration destroying the3D long-range ordering in La2 Sr Cu04 is discussed. The arising short-range order is character-ized by the instantaneous spin correlation length t' c, in coincidence with the experimentallyobserved dependence in this crystal.

I. INTRODUCTION

Magnetic properties of cuprate perovskites have at-tracted considerable attention due to their presumableconnection with high-T superconductivity. Some ofthese properties are rather unusual and still have no satis-factory explanation. One of them is the extreme sensitiv-ity of the long-range antiferromagnetic order to any de-viation &om the ideal crystal. In La2 Sr Cu04 alreadyat x 0.02 the three-dimensional (3D) long-range orderis destroyed and the short-range order with an instan-taneous spin correlation length ( —a/~x in the CuOzplanes arises~ (a is the distance between copper sitesin the planes). Sometimes the explanation of this ex-perimental fact is based on arguments of Ref. 3 aboutcompeting antiferromagnetic and ferromagnetic interac-tions generated by a hole localized on an oxygen orbital.Such interactions can be described in the framework ofthe extended Hubbard model in the limit of strong re-pulsion. However, it is known that in this limit the low-energy dynamics of the model can be described by the t-Jmodel. For the latter model, calculations ' show that forreasonable sets of parameters perturbations produced bya solitary hole in the spin subsystem are too small to ex-plain the extremely small dopant concentrations whichare sufFicient for the destruction of the ordering.

In the present paper an attempt is undertaken to anal-yse these questions on the basis of the 2D t-J model in thepresence of a finite concentration of mobile holes. Themodel is supposed to be a good candidate for the descrip-tion of transport and magnetic properties of the Cu02planes in the cuprate perovskites. To make this prob-lem tractable we use the spin-wave approximation. '

This approximation has been shown to give results whichare in quantitative agreement with exact calculations onsmall lattices in the cases of one and two holes. ' Forthe considered model vertex corrections will be shown tobe small (in analogy to Migdal's theorem for the electron-phonon interaction in metals~z); this makes the systemof Dyson's equations self-consistent. Under the suppo-sition that the homogeneous Fermi-liquid description isvalid and by making use of the one-pole approximationfor the hole Green's function the energy spectrum of themagnetic excitations is determined. Above some criticalconcentration of holes the usual magnon branch (slightlyrenormalized by the hole-magnon interaction) is supple-mented by a new branch of magnetic excitations existingonly in the vicinity of the 1 and M points of the Bril-louin zone. The radii of these regions are approximatelyequal to the Fermi momentum. The imaginary part ofthe f'requency of the new excitations is considerably largerthan the real part; i.e. , the excitations are overdampedmagnons. This fact and the position of the new branchin the Brillouin zone mean that with the appearance ofsuch excitations the long-range antiferromagnetic order,implied &om the very beginning in the spin-wave approx-imation, is destroyed. The correlation length ( of thearising short-range order is connected with the hole con-centration per site c (which equals to the concentrationx of Sr for compositions of interest s) and the intersitedistance a by the relation ( a/~c cited above. The ap-pearance of these overdamped excitations is attributed toan interaction between magnetic excitations of diferentnature: the magnons of localized spins and the Stonerexcitations of mobile holes. For parameters presum-ably realized in La2 Sr Cu04 the obtained results allow

0163-1829/93/48(10)/7492I, '7)/$06. 00 7492 1993 The American Physical Society

MAGNETIC EXCITATIONS OF A DOPED 2D ANTIFERROMAGNET 7493

us to explain the extremely low concentrations of holesdestroying the long-range ordering in this material.

II. DYSON'S EQUATIONS

As mentioned above, our investigation is based on thet-J Hamiltonian in the spin-wave approximation. ' ' A2D square lattice is considered. After the unitary trans-formation which takes into account transversal spin Huc-tuations this Hamiltonian can be represented in the fol-lowing form (for details of the derivation of the Hamilto-nian see Ref. 7):

'R = —P, ) hkthk + ) Cukbktbk

k k

+ ) (gkk'hkhk —k'lk' + H c )Ikk'

where hk is the fermion creation operator of a hole withwave vector k in the classical Neel state ~JV) whichis the reference state for the boson spin-wave opera-tor bk, i.e., bk~JV) = 0. With good accuracy thesetwo operators can be considered as independent for thestates of interest. In Eq. (1), p is the chemical po-tential of holes, wk ——21+1—pk is the unperturbedmagnon frequency with the superexchange constant Jand» = [cos(k a) + cos(k„a)]/2. The interaction con-

I

G(1, t —t') = —i(7 hk(t) hkt(t')),

D(k, t —t') = —i(7 bk(t)bkt(t')),(2)

for the holes and magnons, respectively. Here 7 is thetime ordering operator. From Eq. (1) it is clear that withsome modi6cations which are due to the more complexinteraction Dyson's equations for these Green's functionsare the same as those for the electron-phonon system ina metal

stant gkk ———4t(» k uk +»vk )/~N comprises theeffective hopping constant t and the number of sites N,uk = cosh(nk), vk = —sinh(~k), o.k = ln[(1+»)/(1—»)]/4.

For rather general conditions the effective t-J Hamil-tonian can be obtained from the Hamiltonian of the ex-tended Hubbard model which is widely accepted togive a realistic description of Cu02 planes of cuprateperovskites. In this case the t-J Hamiltonian and itsspin-wave counterpart contain terms describing the staticattraction between holes. These terms can lead to thesuperconducting transition. However, for the descriptionof normal state properties these terms are unessential,and therefore they are omitted in Eq. (1).

The temperature (in units of energy) is supposed to bemuch smaller than p, and J. This allows us to use thezero-temperature Green's functions

D(k(u) = [(u —(u„—II(k~)]G(k(u) = [~+ p, —Z(k(u)]

OO /

II(k ) = —) gk kI'(k' ';k )G(k' ')G(k' —k, ' — ),27r

k I —oo

OO /

Z(k(u) = i ) [gkk I'(k~; k'~')D(k'&u')—OO

+ gk „, „,I'(k —k', ~ —ur'; —k', —~')D( —k', —~')]G(k —k', ~ —~'),

where the vertex part I'(k'~', k~) is the sum of all di-agrams which connect one magnon and two hole lines.The bare vertex gk k and the diagram in Fig. 1 are theterms of the two lowest orders in t in this sum.

In Sec. IV the vertex correction in Fig. 1 will be shownto be much smaller than the lowest order term gk k forwave vectors and frequencies of interest. The same issupposed to be also true with respect to higher ordercorrections. This allows us to substitute I'(k'~', k~) bythe bare vertex gk k. In this way Eqs. (3) become self-consistent.

The energy spectrum of the one-hole states has beenconsidered by different methods in a number of papers(see, e.g. , Refs. 6, 7, 9—11, and 17). The obtained resultscan be summarized as follows: The spectrum consists ofbranches of single-particle excitations characterized bydifferent values of the z component 8, of the total spin (inthe spin-wave approximation) and a hole-magnon contin-uum of scattering states. For parameters presumably re-alized in cuprate perovskites the lowest single-particlebranch corresponds to S, = +1/2 and has four equiva-

I

lent minima at ko = (+vr/~2a, 0) and (0, kvr/~2a). Weuse the system of coordinates with axes along the di-agonals of the plane; in this system the effective masstensor of the hole band is diagonal. In the vicinity ofthe minima the band energy ek can be represented in theform uk+a ——(k~ —k ) /2m' + k„/2m2 for the first two

minima and sk q = k /2m2 + (k„—k„) /2m' for thelatter two minima. In these regions the band is stronglyanisotropic, " m2/mq = 5 —7.

For small hole concentrations the overall character of

FIG. 1. Lowest order vertex correction. Solid and dashedlines denote hole and magnon Green's functions, respectively.

7494 A. SHERMAN AND M. SCHREIBER 48

the hole spectrum is supposed to be the same as for thesolitary hole (we shall discuss this supposition in moredetail below). We assume additionally that the magneticproperties which we want to consider are mainly deter-mined by the lowest hole band. Consequently, the holeGreen's function in Eq. (3) can be approximated by theone-pole expression III. THE MAGNON SPECTRUM

G(k(u) = Z(d —sk + p + z'g sgn(sk —p)

where g ~ +0 and the quasiparticle residue reads

(4) By making use of the approximations discussed in theprevious section, after the integration over w' the magnonself-energy can be represented in the form

n(k' + k/2) —n(k' —k/2)"+k~'k s(k' + k/2) —s(k' —k/2) —(u

+ iver sgn(u) Z ) gk, +k&2 k[n(k'+ k/2) —n(k' —k/2)]8(s(k'+ k/2) —s(k' —k/2) —u), (6)

where n(k) = 0(y, —sk) is the occupation number, 0(z) isthe Heavyside step function, and the symbol P denotesCauchy's principal value of the integration over k' intowhich the summation is transformed. It can be seen fromEqs. (3) and (6) that the self-energy is connected with thegeneration process of electron-hole pairs corresponding tothe bubble diagram.

It will be shown below that dramatic changes of themagnon spectrum are limited to the regions of the Bril-louin zone around the I' [kr = (0, 0)] and M [kM(~2m/a, 0)] points. The sizes of these regions are shownbelow to be of the order of the Fermi momentum k~vt'2pmi. Outside of these regions the spectrum remainspractically unchanged. The formulas given below corre-

I

spond to the region around the I point. Respective for-mulas for the M point can be obtained by substitutingk —k~ for k.

For small k vectors near the I' point the occupationnumbers limit the range of integration over k' in Eq. (6)to vicinities of the minima of the hole band. There theefFective mass approximation can be used. For these re-gions the interaction constant g~+~y2 ~ can be approx-imated by (t/~N) g2/—(ka)2[+(k —k„) 6 (k + k„)]awhere the signs correspond to the four equivalent minimaof the hole band. With these simplifications the imagi-nary part of the self-energy, which describes the magnondamping, can be transformed to the following form:

Imll(k~) = —2sgn((u) ) ka+ o. * "a ek [0(~ +(u)Q~ +(u —0(~ —~)g~ —u)], (7)

where B = (7r/a) /2+m, im2 is of the order of the holebandwidth, ~ = 2p —ek~/2 —w /2&kk /2mi + k„/2m2, and ek i ——k /2m2 + k„/2mi.

Let us analyze the damping as given by Eq. (7). Forthis purpose one should find the quasiparticle residue (5).Using approximation (4) for small hole concentrationsone finds from Eq. (3)

vicinity of the hole band minima in (8), one finds

JBZ ~'w

)7rt2

for these parameters, where m —0.4—0.7. With this valuewe obtain from Eq. (7) the rough estimate for the damp-ing

2

g(ko & ) Z ) ~kok(Sko —Sk& —k —~k)k

Im II(ku) —Jkakyar

where k is the wave vector of one of the hole band min-irna. The usage of the one-pole approximation (4) inEq. (8) gives an upper bound for the value of Z (see thecorresponding discussion in Ref. 10). For parametersof La2Cu04 the effective value of J/t can be estimatedto fall into the range 0.1—0.3 where the hole bandwidth isB 3J. ' This bandwidth is only slightly larger thanthe limiting magnon frequency. By making use of theefFective mass approximation for the integration in the

In accordance with Eq. (7), the following conditionshave to be fulfilled to have a finite damping:

ekcr 2i/Pekcr + ~k + eka + 2i/pekcr)

where ~~ is the magnon frequency. These inequalitiesgive conditions for the Landau damping due to the gen-eration of electron-hole pairs by magnons. For magnonswith wave vectors k & k~ these conditions can be reduced

48 MAGNETIC EXCITATION &OF A DOPED 2D ANTIFERROMAGNET 7495

to the inequality

whereu=u &kiis the velocity of the s

unperturbed me spin waves. For

th th t bd gcourse, even iif long-wavelength magnons

do not deca y, there can be a Rnite dawavelength high-f

ni e amping for short-—requency un ertur

ever, for small holep ur ed magnons. How-

10 ~

a o e concentrations this da1 lli omparison with the r

q. . It will be shown irac ion eads to the a

Ktrations. Moreov 't

n e u lied even for sor small hole concen-reover, it turns out that ine u

energy (6) can bee real part of

e represented in the formo the magnon self-

~2~Z't' . () kayo-cr=+1

Re II(k(u) =l

with 8k = (AJ/ 2/ /pe~~ and q~ = e 44p. The function

~(sk qk )

X(s, q) = g

2x q

q +1(1 —q )cos ss P+ s + 2isi cosP cos — + 1

m/2 dP s — 1——q + 2~s~q) coscos P s — 1——q —2~s~q) cos' P

'

( — ') '4+ '—s + s —2~s~cosg q co

O cos

cos + 8 — q cos

with cos 4 = gl-—q is shown in FiFig. 2. We note that

It follows from Eq. (3) that the renof '

d te ermined by th e equationon

up(v= 2pm

(15)

This coincides with thEq. (11). The m

e condition for a na nonzero dampingmagnons exist in th e region

)

1—= v 2Ja + —Re II(ku). (14) k up1

F v(16)

The difference in thetensor introduces

n he components of to ~e effective massces an unessential

lf h f h dIn this case e

ur er iscussion we ut++1 CQ, —1) 8 = u V

=k/ th hol Io t

th t to ofth

I tt h th h he s ape shown in Fi . 2 w'e

g. Pnge o scale for the 2 axi

c ive

ith ti t (9) th be absolute value oth q & 1 is a roxi

ere ore, an intersection o tve oc-

1 llma

ean an appearance of a slow

Let us deriverive a condition for the existns. ance the plateau

th g Pq and "the hei ht"p= a —2 2vraZ t ~B th

magnons will appear ife }ow-frequency

FIGIG. 2. Function Z(s )h 1 f h

s, q, Eq. j.3 wh'

e magnon self-ener'ona o

q ~ 0 and s ~ l.e -energy. X(s, q) diverges when

7496 A. SHERMAN AND M. SCHREIBER 48

O

0 1

FIG. 3. Graphical solution of Eq. (14) for renormalizedmagnon frequencies in the case q = 0.2. The dotted lineis the velocity of unperturbed magnons.

near the I point and in the region of an analogous radiusnear the M point. The damping (10) is much larger thanthe real part of the magnon frequency uok [see Eqs. (15)and (9)]. Thus, when the low-frequency magnons existthey are overdamped.

The overdamping of the low-frequency branch meansthat the occupation numbers of the respective magnonmodes become indefinite. This contradicts the initial as-sumption of the spin-wave approximation about smallvalues of these numbers. Thus, the appearance of suchexcitations means that the spin-wave approximation be-comes inapplicable to regions of Cu02 planes which arelarger than the characteristic length

(17)

i.e. , the inverse of the limiting wave vector of the over-damped branch [see Eq. (16)]. The relaxational excita-tions of this branch can be interpreted as a movementof the axes of the staggered magnetization in regions ofsize (17). This statement can be reformulated in the fol-lowing form: The spin correlation function decays withdistance as (si so) exp( —~1~/(), where si' denotes thez component of the spin at the position 1. Thus, ( isthe instantaneous spin correlation length. As mentionedabove, the dependence ((x) described by Eq. (17) hasbeen observed in La2 Sr~ Cu04.

From Eq. (9) and condition (15) it is easy to estimatethe critical concentration of holes which corresponds tothe appearance of the overdamped magnons and to thedestruction of the long-range antiferromagnetic order.For the parameters mentioned above it appears to beequal to several percent. This is close to the hole con-centration at which the 3D long-range antiferromagneticorder in La2 Sr Cu04 is destroyed. ' Since we are deal-ing with a 2D model, this comparison needs some addi-tional comments. In the previous considerations the long-range antiferromagnetic ordering has been implied fromthe very beginning. However, it is known that such or-dering is impossible for any nonzero temperature in the2D case. In the real crystal the ordering is possible dueto weak 3D corrections. Nevertheless we suppose thatour consideration is applicable even to the ideal 2D caseat finite but low temperatures (the temperature T has

to be much smaller than p/kg and +2Jak~/kii wherek~ is the Boltzmann constant). Indeed, the correlationlength of the temperature induced spin fluctuations is ex-ponentially large, (' exp( J/kiiT) for J —O. leV (cf.Ref. 18) and low temperatures. This correlation lengthexceeds considerably the correlation length (17) of thespin fIuctuations generated by the holes. Because of thisdifFerence in scale our result that an additional branchof overdamped magnons appears at some hole concen-tration remains unchanged. But the above observationthat for this hole concentration the infinite instantaneousspin correlation length decreases to the value ( given byEq. (17) has to be changed: Now the exponentially largecorrelation length (' decreases to (. The small value ofthe interplane exchange constant means that large 2Dregions of ordered spins play the crucial role in establish-ing the 3D long-range ordering. Thus, the critical valueof the hole concentration, which corresponds to the dras-tic decrease of the 2D correlation length, can be identi-fied with the hole concentration destroying the 3D long-range antiferromagnetic ordering and can therefore becompared with the respective experimentally determinedconcentration of Sr in La2 Sr Cu04.

It follows from estimation (9) that the value of uo andthe real part of the excitation frequency ~i, may be nega-tive [as mentioned, Z(s, q) = 2(—s, q); the intersection ofthe surface I and the plane vs occurs at a negative valueof s in this case]. However, the imaginary part of the fre-quency remains much larger than ~Re cui,

~

and thereforethis case is equivalent to the case Rewk ) 0. The largedamping does not allow one to interpret the negative signof Remi, as an indication that a superstructure (like thespiral state, etc.) occurs in the spin alignment.

Prom Figs. 2 and 3 it can be seen that, besides thediscussed intersection with the plateau in the region ofsmall 8, the plane v8 crosses the wall surrounding theplateau and, for the small hole concentrations consid-ered, the nearly horizontal part of the surface in the re-gion of large 8. This second plateau of I is positioned at avalue of approximately ~2Jo, i.e. , the velocity of unper-turbed magnons. The intersection with the wall corre-sponds to excitations with negligibly small quasiparticleresidues [see Eq. (5) with II substituted for Z]. Indeed,

FIG. 4. Magnon branches along the line connecting the land M points. Overdamped magnons are indicated by thicklines. Solid and dotted lines are the renormalized and initialbranches of usual magnons, respectively.

MAGNETIC EXCITATIONS OF A DOPED 2D ANTIFERROMAGNET 7497

the wall has a large slope and for small hole concentra-tions dII(ku)/der = 2m(~2Ja/v)dX/ds )) 1. Thereforethese excitations can be neglected. The intersection withthe second plateau yields the branch of usual magnonswhich are only slightly perturbed by a small hole concen-tration. The magnon branches are displayed in Fig. 4.As explained above, a part of the branch of overdampedmagnons exists also near the M point.

It follows &om Eqs. (1) and (6) that the overdampedmagnons appear and the long-range antiferromagneticorder is destroyed, because electron-hole pairs are gen-erated by magnons. In the considered situation themagnons are the excitations of localized spins, whilethe electron-hole pairs are the Stoner excitations ofmobile spins connected with holes. Equations (11)and (15), which give the condition for the appearanceof the branch, coincide with the condition that themagnon branch submerges into the continuum of theStoner excitations where the magnons undergo the I an-dau damping.

The overdamped magnons contribute to hole self-energy (3). Because of the magnon damping, for smallhole concentrations their inHuence is small in contrast tothat of the usual magnon branch. This small contributionis the main change in the equation for the hole energiesin comparison with the case of a solitary hole. Thereforethe corresponding deviations of the hole spectrum &omthe spectrum of a solitary hole can be neglected as doneabove.

IV. MIGDAL'S THEOREM

In the previous sections it was supposed that vertexcorrections could be neglected. Now the validity of thisassumption will be verified by comparing the bare vertex

gk, k with the first correction given by the diagram inFig. 1.

The diagram corresponds to the following formula:

OO f

I' ' (k, k ) = ' ) [gk +g', k'gk +k' —k,k'D(k ) +gg —k, —k'gk„—k'D( k, )j—OO

x gk~+k', kG(k& + k', ~& + ~') G(k& + k' —k, ~, + ~' —u). (18)

In accordance with the previous considerations themagnon momentum k in Eq. (18) can be considered tobe small while the hole momentum k~ is approximatelyequal to one of wave vectors k of the hole band min-ima. By making use of approximation (4) for the holeGreen's function and integrating over u', one can see thatthe contribution from the poles of the magnon Green'sfunction becomes negligibly small after the integrationover ~k'~. The main contribution to correction (18) isdue to the poles of the hole propagators. The occu-pation numbers, which arise in I'( ) after the integra-tion over u', and the small values of k limit the inte-gration over k' to the regions where the wave vectorkq + k' is near the Fermi surface. Out of the four re-gions of this kind, two around k' = kz and k' = k~,give the main and equal contributions. We consider theformer case for which the hole energies in Eq. (18) canbe approximated as sk, +k —p, v(~k'~ —ky) andsg, +k k —p --v(~k'~ —kz) —vk . k'/~k'~.

After these simplifications the integration over ~k'~ inEq. (18) can be easily performed. Contributions of theusual and the overdamped magnon branches should beconsidered separately. For small hole concentrations thevelocity u of the former spin waves is much larger thanthe hole velocity v. This allows us to estimate the con-tribution of the branch as

(i) = ga, m~F+ —&& gx, a.

The contribution of the overdamped magnons is nearlypurely imaginary due to the large magnon damping. The

absolute value of this contribution is approximately givenby

(20)

Thus, both contributions are small in comparison withthe bare vertex. For the electron-phonon system con-sidered by Migdal the smallness of the vertex correc-tions is provided by a small ratio of the limiting &equencyof phonons to the Fermi energy. In our case in which theratio of the analogous energy parameters is the oppositethe smallness is connected with the small Fermi energy.

V. CONCLUDING REMARKS

Our consideration was essentially based on the pictureof mobile holes which is inherent to the metallic phase. Infact, it has been found experimentally that in the dop-ing regime 0.02 ( x & 0.06, where the destruction of thelong-range ordering takes place, the resistance within theCu02 planes is metallic and linear in T over a wide rangeof temperatures above 80 K. However, below this temper-ature logarithmic corrections to the resistivity appear,leading to a crossover to semiconducting behavior. Suchlogarithmic corrections are pertinent to the case of weaklocalization when the wave functions can still be labeledapproximately by wave vectors and the hole-magnon in-teraction divers from the interaction in the metallic phaseonly for extremely small wave vectors. We suppose thatour considerations are applicable also to this case. In sup-

7498 A. SHERMAN AND M. SCHREIBER 48

port of this conjecture the following experimental fact canbe adduced: The dependence ((x) described by Eq. (17)is observed not only in the metallic but also in the semi-conducting phase. Besides, there are experimental indi-cations of a smooth variation of the magnetic propertiesacross the metal-semiconductor boundary. Similarly theexperimental result ' that the critical concentration ofholes for temperatures down to T = 10 K remains prac-tically the same as for T 80 K indicates that the pro-cesses which destruct the long-range order are the samein the metallic and semiconducting phases.

Apart from Ref. 3, several investigations were under-taken to explain the extreme sensitivity of the Neel stateon the doping with holes. Two papers ' are basedon models analogous to the one considered here. Themajor difference to the present one is the following: Inboth studies ' expansions in powers of x @+2 of themagnon self-energy are used. Such an approach does notallow one to describe the self-energy properly in the re-gion k & A;~ where the plateau of I is positioned. Wenote that only in this region the self-energy difFers signif-icantly &om zero. As a result, the branch of overdampedmagnons, which is in our opinion the central result ofthe present paper explaining the destruction of the long-range antiferromagnetic ordering, has not been foundthere. Instead, in Ref. 22 the vanishing spin-wave veloc-ity of long-wavelength excitations has been interpretedas an indication of the destruction of the ordering. Asmentioned above, in our opinion the applicability of theused power expansion to the long-wavelength magnons

is doubtful. In Ref. 23 a damping of long-wavelengthmagnons of the slightly renormalized. usual branch hasbeen considered as the criterion for the destruction ofthe long-range ordering. The analogous criterion has alsobeen used in Ref. 24 for explaining the empirically estab-lished dependence ( = a/~x. We have already pointedout that in La2 Sr Cu04 a damping of these magnonsis possible only for hole concentrations much larger thanthose which destroy the Neel state in this crystal.

In summary, we have shown that for values of param-eters presumably realized in La2 Sr Cu04 doping ofthe 20 t-J model by several percent of holes leads tothe appearance of an additional branch of overdampedmagnons. This means that at this critical concentrationof holes the long-range antiferromagnetic ordering, whichis assumed to exist at lower concentrations, is substi-tuted by the short-range ordering with an instantaneousspin correlation length ( a/~x. These results allowone to explain the extremely small dopant concentrationdestroying the Neel state in La2 Sr CuO4 and the ob-served values of the correlation length. Besides, we havedemonstrated that for the model considered Migdal's the-orem is valid.

ACKNOWLEDGMENTS

This work was supported by the Commission of the Eu-ropean Communities under Grant No. ERB-CIPA-CT-92-0098 and, in part, by the Soros Foundation.

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