spin-wave theory for anisotropic heisenberg antiferromagnets

4
PHYSICAL REVIE& 8 VOLUME 44, NUMBER 1 1 JULY 1991-I Spin-wave theory for anisotropic Heisenberg antiferromagnets C. M. Soukoulis, Sreela Datta, and Young Hee Lee* Ames Laboratory and Department of Physics, Iowa State University, Ames, Iowa 50011 (Received 2 April 1991) The anisotropic Heisenberg antiferromagnet (AF), which is defined as a three-dimensional simple- cubic lattice with in-plane antiferromagnetic interaction JII and interplane coupling J& =yJII and is be- lieved to describe the magnetic properties of the cupric oxide materials, is studied using low- temperature spin-wave theory. The dependence of the T 0 staggered magnetization, ground-state en- ergy, transverse susceptibility, spin-wave velocity, and the Neel temperature T& on the anisotropy pa- rameter y (0» y~ 1) are obtained. These resu1ts are found to be in satisfactory agreement with ex- isting experiments on cupric oxide materials. The apparent difference between the muon-spin reso- nance and neutron-scattering results for the ordered moment in the AF state is well explained. The discovery of the copper-oxide superconductors ' has produced great interest in the Heisenberg antiferromag- netic (AF) model. This is because the antiferromagnetic order of Cu + spins has been observed in La2Cu04 and YBaqCu306, which become superconducting when doped. While it has been proposed that magnetism may play an important role in high-temperature superconductivity, the study of magnetic ordering in these families of materials is interesting in its own right. These systems are highly an- isotropic with a very strong coupling between Cu spins in the plane and very weak coupling between planes. Thus, these systems have interesting magnetic properties which may be representative of a wider class of quasi-two- dimensional antiferromagnets. For this reason, we decid- ed to carry out a detailed study of the magnetism of a classical Heisenberg model with interactions in the range appropriate to cupric oxides. It is now well established from numerical work (quan- tum Monte Carlo, Green-function Monte Carlo, series analysis ) and different variational methods that the ground state of the two-dimensional (2D) S = Heisen- berg AF does indeed have long-range Neel order. All of its ground-state properties as well as the long-wavelength excitations are well described by straightforward spin- wave theory. ' Spin-wave theory is based upon a large-S expansion, starting from a ground state with Neel order. Why spin-wave theory is so accurate even for 5 = 2 is still an open and interesting question. Due to these successes, we decided to use the regular spin-wave theory to study the ground-state properties of the anisotropic Heisenberg antiferromagnet on a simple cubic lattice. In order to in- vestigate the dependence of the Neel temperature, T~, on the anisotropy parameter y we generalized the Green's function formalism ' with its random-phase decoupling to the anisotropic Heisenberg antiferromagnet. This random-phase approximation gives a better estimate of Ttv compared with Monte Carlo-simulation results'' than mean-field-theory and low-T spin-wave theory results. One of the most interesting results of this study is that it is possible to have large changes in T~ as the anisotropy y changes, while the staggered magnetization, i.e. , the or- dered moment of the AF state, remains constant. This is consistent with muon-spin rotation experiments' on La2Cu04 ~ and not with the neutron-scattering experi- ments. ' Details of the calculation and discussion of the results will be presented in the remainder of the paper. We have studied the Heisenberg antiferromagnetic model on a simple cubic lattice with in-plane antiferro- magnetic interaction Jll and interplane antiferromagnetic interaction J&, which is much weaker than Jll. The Ham- iltonian for this model is 0 = Jtgs" s J gs's &0= 2 ~l Jill»'(I+&/» The staggered magnetization is (s;& =(s ) '/s), the perpendicular susceptibility at T=O is x (0) = [I (1/2s) (x+ k')1, Wg pg (2) (3) and the spin-wave excitation energy is e(k) =42~ I JiilS(I+V2S)ka, where the first sum is over four nearest neighbors in the plane and the second sum is over the nearest-neighbor in- terplane interactions. The ratio of the two interactions, y = J~/Jt, is inversely proportional to the strength of the anisotropy of the system and should determine the transi- tion temperature T~ of the model. When y=0, we have a 2D Heisenberg AF with no interplane interactions and T~ =0. When y =1, we recover a simple cubic Heisen- berg AF with equal interactions for which T~ =1. 35 for classical spins'' S; with magnitude equal to one. We will use the Boltzmann constant ktt =1 and l J~tl = l. To obtain, within the spin-wave theory, the ground- state properties, as well as the spin-wave excitation ener- gy, for the anisotropic Heisenberg AF model given by Eq. (1), we follow the standard two-sublattice approxima- tion. ' Using the Holstein-PrimakoA' transformation in the two sublattices, the Fourier transform of Eq. (1) can be easily diagonalized. Following the notation in Calla- way's book, we have the following results. The ground- state energy is given by 1991 The American Physical Society

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Page 1: Spin-wave theory for anisotropic Heisenberg antiferromagnets

PHYSICAL REVIE& 8 VOLUME 44, NUMBER 1 1 JULY 1991-I

Spin-wave theory for anisotropic Heisenberg antiferromagnets

C. M. Soukoulis, Sreela Datta, and Young Hee Lee*Ames Laboratory and Department of Physics, Iowa State University, Ames, Iowa 50011

(Received 2 April 1991)

The anisotropic Heisenberg antiferromagnet (AF), which is defined as a three-dimensional simple-cubic lattice with in-plane antiferromagnetic interaction JII and interplane coupling J& =yJII and is be-lieved to describe the magnetic properties of the cupric oxide materials, is studied using low-

temperature spin-wave theory. The dependence of the T 0 staggered magnetization, ground-state en-

ergy, transverse susceptibility, spin-wave velocity, and the Neel temperature T& on the anisotropy pa-rameter y (0» y~ 1) are obtained. These resu1ts are found to be in satisfactory agreement with ex-

isting experiments on cupric oxide materials. The apparent difference between the muon-spin reso-nance and neutron-scattering results for the ordered moment in the AF state is well explained.

The discovery of the copper-oxide superconductors ' hasproduced great interest in the Heisenberg antiferromag-netic (AF) model. This is because the antiferromagneticorder of Cu + spins has been observed in La2Cu04 andYBaqCu306, which become superconducting when doped.While it has been proposed that magnetism may play animportant role in high-temperature superconductivity, thestudy of magnetic ordering in these families of materials isinteresting in its own right. These systems are highly an-isotropic with a very strong coupling between Cu spins inthe plane and very weak coupling between planes. Thus,these systems have interesting magnetic properties whichmay be representative of a wider class of quasi-two-dimensional antiferromagnets. For this reason, we decid-ed to carry out a detailed study of the magnetism of aclassical Heisenberg model with interactions in the rangeappropriate to cupric oxides.

It is now well established from numerical work (quan-tum Monte Carlo, Green-function Monte Carlo, seriesanalysis ) and different variational methods that theground state of the two-dimensional (2D) S = — Heisen-berg AF does indeed have long-range Neel order. All ofits ground-state properties as well as the long-wavelengthexcitations are well described by straightforward spin-wave theory. ' Spin-wave theory is based upon a large-Sexpansion, starting from a ground state with Neel order.Why spin-wave theory is so accurate even for 5=

2 is stillan open and interesting question. Due to these successes,we decided to use the regular spin-wave theory to studythe ground-state properties of the anisotropic Heisenbergantiferromagnet on a simple cubic lattice. In order to in-vestigate the dependence of the Neel temperature, T~, onthe anisotropy parameter y we generalized the Green'sfunction formalism ' with its random-phase decouplingto the anisotropic Heisenberg antiferromagnet. Thisrandom-phase approximation gives a better estimate ofTtv compared with Monte Carlo-simulation results'' thanmean-field-theory and low-T spin-wave theory results.One of the most interesting results of this study is that it ispossible to have large changes in T~ as the anisotropy ychanges, while the staggered magnetization, i.e., the or-dered moment of the AF state, remains constant. Thisis consistent with muon-spin rotation experiments' on

La2Cu04 —~ and not with the neutron-scattering experi-ments. ' Details of the calculation and discussion of theresults will be presented in the remainder of the paper.

We have studied the Heisenberg antiferromagneticmodel on a simple cubic lattice with in-plane antiferro-magnetic interaction Jll and interplane antiferromagneticinteraction J&, which is much weaker than Jll. The Ham-iltonian for this model is

0 = —Jtgs" s —J gs's

&0= —2 ~l Jill»'(I+&/»

The staggered magnetization is

(s;& =(s —) '/s),the perpendicular susceptibility at T=O is

x (0) = [I —(1/2s) (x+k')1,Wg pg

(2)

(3)

and the spin-wave excitation energy is

e(k) =42~I JiilS(I+V2S)ka,

where the first sum is over four nearest neighbors in theplane and the second sum is over the nearest-neighbor in-terplane interactions. The ratio of the two interactions,y =J~/Jt, is inversely proportional to the strength of theanisotropy of the system and should determine the transi-tion temperature T~ of the model. When y=0, we have a2D Heisenberg AF with no interplane interactions andT~ =0. When y =1, we recover a simple cubic Heisen-berg AF with equal interactions for which T~ =1.35 forclassical spins'' S; with magnitude equal to one. We willuse the Boltzmann constant ktt =1 and l J~tl = l.

To obtain, within the spin-wave theory, the ground-state properties, as well as the spin-wave excitation ener-gy, for the anisotropic Heisenberg AF model given by Eq.(1), we follow the standard two-sublattice approxima-tion. ' Using the Holstein-PrimakoA' transformation inthe two sublattices, the Fourier transform of Eq. (1) canbe easily diagonalized. Following the notation in Calla-way's book, we have the following results. The ground-state energy is given by

1991 The American Physical Society

Page 2: Spin-wave theory for anisotropic Heisenberg antiferromagnets

SPIN-WAVE THEORY FOR ANISOTROPIC HEISENBERG. . . 447

where

~-I ——g(1 y)—) '"a,N I

—lb,2 1

(1 y2) 1/2

yq- (2cosk„a+ 2 costa+ 2ycosk, a)/zc

and z =4+2y. The values of X and X' have been calculat-ed numerically and our results for y=O (square lattice)and y=1 (simple cubic lattice) agree with those of An-derson and Kubo.

To understand the dependence of the Neel temperatureT~ on the anisotropy parameter y, one must develop atheory that will give T+ =0, when the interplane couplingJ& =0, i.e., when the model given by Eq. (1) becomes a2D Heisenberg antiferromagnet. The simplest of themean-field approximations is that in which the Geld spinfeels is just determined from the sum of its interactionswith its neighbors. For the classical Heisenberg modelwith spins of unit magnitude, Ttv = —,

'(4~Ji ~

+2)J&()= —,' (Jt[(4+2y), which obviously is wrong be-

cause for y=0, it gives a finite T~. This simple mean-field approach is not good, since it does not correctly takeinto account the strong Auctuations which destroy long-range order in the limit of J& 0. We have, therefore,calculated the Neel temperature T~ from the theory offerromagnetism which is based on the Green functionmethod. ' Bogolyubov and Tyablikov were the first toapply the Green function method to ferromagnetism forthe spin- 2 case and they have evoked the random-phasedecoupling scheme to reduce the Green functions to thelowest order. We have followed ' the random-phasedecoupling method to calculate the Ttv for the generalspin S for the anisotropic Heisenberg model. Within thisapproximation, the result for T~ can be written as

3G(0) '

where G(0) =gt, 1/z(1 —yk), z =4+2y, and yt, is givenby Eq. (6). G(0) is the Green function' of the periodicanisotropic tight-binding system at the bottom of theband. G (0) can be calculated analytically for y =0(square lattice) and numerically for all the other values ofthe anisotropy y. A simpler expression for G(0) can beobtained' by integrating over the two variables k, and k~and yielding G(0) =(I 2/~ )fodk, tK(t), where t =2/(2+ y

—ycosk, a) and K is the complete elliptic integral ofthe first kind. In the limit of small anisotropy parametery~ 0, we obtain G(0) =(I/4tt) ln(32/y), which gives ananalytic expression for the dependence of T~ versus y. Inthis weak limit, we have

correlations in the planes exactly, and the interplane in-teractions within the mean-field approximation. The rela-tion between Tz vs y is very important and in Fig. 1 wecompare the random-phase approximation results [Eq.(7)] for TIv with the Monte Carlo simulations results onthe classical anisotropic Heisenberg system. Notice thatthe agreement is extremely good for all the anisotropies y.Ttv increases very rapidly as y increases for small y, butfor y) 0.1, Ttv is nearly linear with y. In particular, thesmall y dependence of Ttv is described very well by the ex-pression given in Eq. (8). However, our Monte Carlotechniques cannot handle very small y, since at thesesmaller interplane interactions the magnetic correlationlength is much larger than the sample size [most of thesimulation runs" were on sizes 1V -18000 (30x 30 X 20)].This is a very interesting result. It shows that the transi-tion from a paramagnetic Ttv =0 for y=O to an antiferro-magnetic T~~O takes place abruptly. Notice that evenfor very small anisotropic coupling, y=0.001, T~ =0.404,of the same order of magnitude as the y 1 value of Ttv,which is roughly equal to 1.36. This is simply due to thefact that the two-dimensional character (Ttv =0) is veryunstable, being approached (as y~0) very slowly. Inother words, even a very small out-of-plane coupling isenough to destroy the two-dimensional character of thetransition. In Fig. 1, we also plot the results for T~ versus

y which we have obtained" using the modified spin-wavetheory of Takahashi. ' While the agreement with theMonte Carlo results is good close to y =1, for all the othercases the spin wave T& are always lower than our MonteCarlo simulations. In Takahashi's approach, T~ is like aBose-Einstein condensation temperature below whichlong-range order results. The random-phase approxima-

1.2

1.0

0.8

0.6

0.4

0.2

[Jill/ln4z 323 y

(8) 0.0I

0.2I

0.4I

0.6I

0.8I

1.0

Similar results were obtained' for the dependence of themobility edge on anisotropy parameter for a disorderedanisotropic tight-binding model. Equation (8) can also bewritten as (y/32) exp(4'~ J~~ ~/3T~) =1, which is similar toa renormalized mean-field theory" that treats the spin

FIG. 1. Neel temperature T~ as a function of the anisotropyparameter y=J&/J&. Results from the Monte Carlo (MC)simulations (solid triangles), from the random-phase approxi-mation (RPA) (solid circles), and the low-temperature spin-wave theory (open circles) are presented.

Page 3: Spin-wave theory for anisotropic Heisenberg antiferromagnets

448 C. M. SOUKOULIS, SREELA DATTA, AND YOUNG HEE LEE

tion ' scheme to reduce the Green's functions to thelowest order takes into account the hierarchy of coupledequations. As expected, the naive mean-field theory re-sults (not shown in Fig. 1) overestimate T~ for all thevalues of y (e.g., Trv =1.33 for y 0 and T~ =2.00 fory= 1). We have also compared" the results of the renor-malized mean-field theory with the Monte Carlo simula-tions, not shown in Fig. 1 and have found" that T~ ob-tained within this renormalized mean-field theory is muchhigher than the Monte Carlo" for y~ 10

In Fig. 2, we plot the spin-wave results for the staggeredmagnetization (So) and the T=O perpendicular suscepti-bility x&(0) versus the Neel temperature Trv for the caseof S= 2, in order to be able to compare our results withexperiments on the cupric oxide materials. Notice thatthe ordered moment (So) in the AF state remains thesame, close to its y=O value of 0.303, as T~ increasesfrom its zero value, all the way to T~=0.3. As T~ in-creases further, the system is more three-dimensional andits (So) increases almost linearly from Trv =0.4 toT~ =0.9 and then (So) saturates to its three-dimensionalvalue of 0.4216. The behavior of x&(0) vs T~ followsthat of (So). It has been known' that the magnetic prop-erties of the AF compound LazCu04 —~ depend sensitivelyon small diA'erences in the oxygen content y=O to 0.03.The input of 0, equivalent to the substitution of Sr +

for La +, removes electrons and creates holes in the sys-tern, and therefore, suppresses the three-dimensional AFordering. Neutron scattering experiments ' on La2Cu-04 —~ have revealed that the T=O ordering moment of theantiferromagnet decreases by as much as a factor of 3 asthe Neel temperature Tjv decreases. However, muon-spinrotation experiments' on the same samples show that theordered Cu moment remains unchanged close to =0.6pg/Cu despite the large difference in the ordering temper-ature Trv (from 300 to 15 K). A similar behavior hasbeen observed in YBazCu306~„(Ref. 18) and Sr2CuOz-C12 (Ref. 19) samples. In this respect, muon-spin rotationand neutron measurements appear inconsistent. Ourtheoretical results from Fig. 2 show that the T=0 ordered

1.32—I ! t

It 'I I

1.28

moment (gag(So)) remains constant, closed to 0.60pg/Cu, while T~ can change a lot. These results are con-sistent with the muon-spin rotation experiments. %ithinour spin-wave theory, the introduction of oxygen defectsdoes not change the T=O moment but changes mostly JIIand possibly J~ which then drastically changes Trv, sinceT~ is a very sensitive function of the anisotropy y (seeFig. 1). To reconcile with the neutron-scattering results,one has to argue that either the long-range order is notover the whole sample, and that is the reason for thereduction of the T=O moment, or that the magnetic or-dering becomes more and more short ranged with increas-ing oxygen content (i.e., with increasing number of holes)resulting ' in the decrease of Tz. Alternatively, if theT=0 magnetic moment of 0.34 p~/Cu seen in theneutron-scattering experiments for low values of T~ isuniform throughout the volume of the crystal, then itsreduction to such low values, much lower than the spin-wave value, is due to as yet an unidentified mechanism.From Fig. 2, we have that x~(0) remains constant equalto its two-dimensional value of 0.449 Ng pq/2z I J~~ I, whileTz drastically changes. Some preliminary results ' sup-port this picture, i.e., that Tz can change drastically whilex~(0) remains constant. However, more careful experi-ments on single crystals have to measure x&(T), (So), thespin-wave excitation energy, and Tz to check the predic-tions of our anisotropic spin-wave theory. In Fig. 3, weplot the ground-state energy F.o and the spin-wave veloci-ty U, versus the anisotropy y for the 5= 2 case. %'e havenormalized Eo by the Neel energy EN«~ = —N!J~~IzS /2and the U, by v 2z

I J~~ISa. Notice that the behavior of Eo

0.45— I —0.91.26 —1.16

0.40 Eo1.24

(So )

0.35

0.50

—0.7

—0.6

(0)

1.22

1.20

—1.12 ys

—1.10

0.25 —0.51.18— —1.08

I I i I i I i I i I i I I 040.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

TN

1.16—

0.0I

0.2!

O.o( I i I i I

0.6 0.8 1.0

1.06

FIG. 2. The T=O staggered spin (S$) and perpendicularmagnetic susceptibility x&(0) vs the Neel temperature Tz foran anisotropic Heisenberg antiferromagnet with S= 2 . Thex&(0) is normalized by Ng pa/2zI J~~I.

FIG. 3. The ground-state energy Fo and the spin-wave veloci-ty v., vs the anisotropy parameter y. Fo is normalized byEra i= —,

'NzI Jt|IS and v, by ~2z IJiiISa.

Page 4: Spin-wave theory for anisotropic Heisenberg antiferromagnets

SPIN-WAVE THEORY FOR ANISOTROPIC HEISENBERG. . . 449

and v, vs y is quite similar. For y=0, we have the 2D re-sults of Eo =+ 1.3159 EN„~ and v, =1.158&2!Jt!a. As yincreases, Eo and U, decreases and monotically reach their3D results for the simple cubic lattice with y =1.

Within the anisotropic Heisenberg AF model, the onlyparameter that enters the theory is the anisotropy param-eter y. For LazCu04, we have that Ttv/JR~=300/1000=0.300, which gives a value of y= 3 x 10 . The T=0staggered moment is (So) =0.304, while x&(0) =0.45&g ptt/2z! JI~ I

=2.11 x 10 cm /mole using =2 and thespin-wave velocity Av, =de(k)/dk =1.158 2!J~~!a=0.54eVA using a =3.8 A. and J~~ =1000 K. For YBazCu306,we have that Ttv/Jt =450/1200=0. 375, which gives thaty=4X 10, and therefore, (So) =0.31, x~(0) =1.80X 10 cm /mole, and A v, =0.65 eVA. using a =3.85 A.For Sr2Cu02Clz, we have that Tjv/J~~ =250/900=0. 278,which gives y = 1 x 10, and therefore, (So)=0.303,while x&(0)=2.35X10 cm /mole, and Av, =0.50 Ausing a =3.8 A. and Jt =900 K. Notice that the most an-isotropic system is the SrzCuOzC12 since the distance be-tween the Cu02 planes is the largest, while the couplingbetween the planes in YBa2Cu306 is stronger than inLazCu04. The above predictions for x&(0) and Av, arein qualitative agreement with the experiment. ' ' How-ever, more experiments are needed to check the predic-

tions of this theory.In conclusion, we have calculated within the traditional

spin-wave theory the ground-state properties of the aniso-tropic Heisenberg antiferromagnet on a simple cubic lat-tice, this model is believed to describe the magnetic prop-erties of the cupric oxide materials. We have also calcu-lated the Neel temperature Ttv versus the anisotropy pa-rameter y. While the dependence of the T=O orderedmoment versus T~ is in agreement with muon experi-ments, more detailed experiments on single crystals haveto measure x~(T), (So), v„and Ttv in order to be able tocheck the predictions of the anisotropic spin-wave theory.

We would like to thank G. S. Grest, D. C. Johnston, C.Stassis, and B. N. Harmon for useful discussions. C. M.Soukoulis is grateful for the hospitality of the Institute ofMaterials Sciences at the "Demokritos" National Centrefor Scientific Research where part of this work was done.This work was partially supported by European EconomicCommunity (ESPRIT-3041 and Science-ST2J-0312-C).Ames Laboratory is operated for the U.S. Department ofEnergy by Iowa State University under Contract No. W-7405-ENG-82. This work was supported by the Directorof Energy Research and the Office of Basic Energy Sci-ences of U.S. DOE.

Permanent Address: Department of Physics, Jeonbug NationalUniversity, Jeonju, Jeonbug, 560-756, Republic of Korea.

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