VOLUME 73, NUMBER 6 PHYSICAL REVIE% LETTERS 8 AUGUsT 1994

Weak Ferromagnetism and Tricriticality in Pure La2Cu04

Tineke Thio' and Amnon Aharony"-

'NEC Research Institute, Princeton, New Jersey 08540Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics and Astronomy,

Tel Aviv University, Tel Aviv 69978, Israeland Center for Materials Science and Engineering, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139(Received 18 March 1994)

New data for the field dependent magnetization M(H) of pure antiferromagnetic LaqCu04 are fittedby a theory which treats the interlayer coupling using mean-field theory (MFT), and includes theDzyaloshinskii-Moriya coupling. The fits yield a novel measure of the two-dimensional (2D) staggeredsusceptibility y», which deviates below 360 K from the 2D Heisenberg model. The fits also implythat the weak-ferromagnetic transition below the Neel point T& is weakly first order, with a tricriticalpoint close to TN. The MFT nicely reproduces the order parameter and the cusp in y = dM/dH.

PACS numbers: 75.30.Kz, 74.72.Dn, 75.40.Cx, 75.50.Ee

La2Cu04, the insulating parent compound of the high-temperature cuprate superconductors, can be regarded asa model system for the S = 1/2 two-dimensional (2D)square-lattice Heisenberg antiferromagnet (AF) (2DHA)[1], which is expected to have a Neel temperature Ttt =0. Indeed, at temperatures above -370 K the in-planecorrelation length $2D measured via neutron scattering [2]basically agrees with theories of the 2DHA [3,4], and isquite large. However, in practice this system exhibits3D long range order at a surprisingly high temperature,T~ = 325 K. This transition has been attributed to botha small interlayer coupling J& and a small in-planeanisotropy [2,5 —8]. Indeed, both the high-T correlationlength, $2o [2], and low Torder param-eter, M [8], wereconsistently fitted following this scenario. However, theneutron data for $2o became unreliable below 370 K, andfits for the staggered moment Mt using a generalizedSchwinger boson theory [8] were good only at low T.One therefore needs better data and better theories nearTN. The situation became even more intriguing in viewof a recent Letter by MacLaughlin et al. [9], whichquestioned the order of the AF transition at T&, andclaimed that it is "abrupt but continuous. "

La2Cu04 has a great advantage over other cuprates,due to a Dzyaloshinskii-Moriya (DM) term [10], whichresults from the orthorhombic distortion. This term leadsto a sharp peak in the magnetic susceptibility at T~, andto a weak ferromagnetic moment perpendicular to eachCu02 plane [5]. At zero magnetic field (H = 0) thesemoments order antiferromagnetically, since J& ~ 0. A fi-nite H J Cu02 induces a transition to overt weak ferro-magnetism (WF). In Ref. [5] we treated J& in mean-fieldtheory (MFT), and obtained reasonable quantitative fitsto the zero-field susceptibility g(T). However, these fits

required the 2D staggered susceptibility g2o (appropriatewhen J& = 0) as input.

3.0

Eo 2.5

E 2.0

1.5

0.5200 250 300 350 400

Temperature (K)

FIG. 1. Magnetic susceptibility g = dM/dH for 0 i CUO2.The solid line is the MFT fit [Eq. (7)].

In this Letter we present new magnetization data M(H),and generalize the MFT to allow for H & 0 and for afirst-order AF transition. The MFT gives excellent fitsto M(H) for 0 & H ( 5.5 T and 300 & T & 400 K, andyields a novel measure of gzD(T), in a T range where

neutron scattering gave no results. In this range, gzo(T)grows faster than the 2DHA theory prediction, probablyindicating a crossover to 2D XF behavior. The fits showthat the AF transition at T~ is indeed very close to atricritical point (TCP) [9]. The MFT also reproducesMt (T) from neutron scattering [2,8], and g(T).

Single crystals of pure La2Cu04, grown from top-seeded solution [11],were annealed at p = 10 5 torr and800'C for 30 min. The magnetization M(H) was mea-sured in a Quantum Design SQUID magnetometer, withH J Cu02 (H ~~ b in orthorhombic notation). The "zero-field' magnetic susceptibility g(T) = dM/dH, obtainedfrom M(H) at 0.5 ( H ( 1.0 T and shown in Fig. 1,shows a sharp peak at T~ = 323 K, indicating that thesamples are thoroughly reduced [12]. M(H) is linear forT & 350 K (Fig. 2); as T approaches Ttt, the data showsnegative curvature. Just below T~, M(H) is sigmoidal;

894 0031-9007/94/73 (6)/894(4) $06.001994 The American Physical Society

VOLUME 73, NUMBER 6 PHYSICAL REVIEW LETTERS 8 AvovsT 1994

~ 0.03

EOP

~ 0.02—EOEO

a) 0.01CA

0 ~ I

0 1 2 3 4 5Magnetic field {T)

FIG. 2. M(H) for H i Cu02. data (symbols) and MFT fit(solid lines) at temperatures indicated.

12

FQF'tpFDMpFPM)~Fi=1

Here,

F; = —(&2D) '(M; )~ + 4A(M; )4 + 6B(M; ) (2)

represents the free energy of the AF ordering in the ithCu02 plane. The AF unit cell of La2Cu04 containstwo Cu02 planes, so i = 1,2. Equation (2) representsa 2DHA, with gzD(T) diverging exponentially at T = 0[3]. The sixth-order term was added to allow first-ordertransitions. Small anisotropies [6] force the staggeredmagnetization M; ii c (in the plane); M; denotes itsc component.

The DM [5,10] interaction is allowed by the orthorhom-bic distortion of the lattice, which involves a small ro-tation of the Cu06 octahedra around the orthorhombica axis. It is of the form FDM = D (S„; X SB;), whereS~; and S~; are the sublattice magnetizations within theith Cu02 plane [M; = (S4; —Sii;)/2 and the ferromag-t

netic moment per spin is M; = (S„; + Sii;)/2 in eachplane]. The pseudovector D ~i a is represented by anti-symmetric elements Jb' in the nearest-neighbor exchangetensor, with J~' = ~D~ [5]. Given the directions of Dand M;, M; is generated along the b axis, and the DMt

term in Eq. (1) can be written as F; = —CM; M;, whereC = zJ"'(z = 4 is the coordination number in the plane)and M; is the b component of M; (H ii b). The ferro-magnetic term is F," = zoo 'M; —HM;, where go =1/(2zJ~~), and Jjv~ is the nearest-neighbor isotropic ex-change. Finally, the interplane coupling is represented byFjIIt =

2 Jg Ml M2 Note that J& is defined per turbo bondst t

(there are two neighboring layers). Minimizing F, we find

M, = gp(H + CM; ) andt

(g2o) 'Mi + A(Mi ) + B(Mi ) + JgM2 = CMi,

(g2o) M2 + A(M2 ) + B(M2 ) + JiMi CM2t -l t t3 tS (3)

below 320 K the sigmoid develops into a jump, indicativeof the first-order nature of the WF transition.

Our MVI' starts from the free energy, defined perCu spin:

Defining M- = 2(M, ~ M2), M+ and M are thet l t t t

components of the staggered moment in the AF unit cellwith propagation vector r li a and r ii c, respectively [6].At H = 0, Mt becomes nonzero below TN. As we dis-

cuss below, a finite H generates a nonzero M+. Abovet

the WF transition Mt vanishes, and the ordering is domi-

nated by M+. For convenience, we define I = LCM+,t

m = gpCMt, a = A(gpC) 2, and b = B(AC) . Sub-

stituting also M; = go(H + CM; ) into Eq. (3) yieldsMt = 0 (above the WF transition) or

[42o) ' —C'go —Ji] + a[(m')' + 3m']

+b[(mt) + 10(mt) m + 5m"] = 0 (4)

(below the transition), and

[(g2D)' —C gp + Ji] + a[m + 3(mt)2]

+ b[m + 10m (m~) + 5(mt) ] = (Cgo) H/m. (5)

Equation (4) shows that nonzero values of either J~ orof C suffice to generate nonzero solutions for Mt at finite

T, when H = 0. Equation (5) shows that CypH acts as a

staggered field, generating nonzero M+ for all H 4 0. Tocompare with experiments, we note that the net ferromag-netic moment per spin is M =

2 (M~ + M2) = gp(H +CM+). To account for the diamagnetic core suscepti-

bility g«re we replace this by M = goffsetH + goCM+ =go ffget H + I, and treat g,«, = go + g««as a fittingparameter. Solving Eq. (4) for mt, and substituting intoEq. (5), thus yields H(M), with g,ff„„Cgp, Ji, a, b, and

(X2D)' —C'go appearing as parameters which should be

fitted by our data. We assume g,«„„Cgo, and J& to beindependent of T. Our fits showed weak T dependencein b, and we therefore set it also at a constant value.

(g2D) C gp and a were fitted separately for each T.The fits were done such that the total mean-squared devia-tions of the data to the fits at all temperatures was mini-mized simultaneously.

The MFT gives excellent fits to the M(H) data(solid lines in Fig. 2); at low T the first-order charac-ter of the transition is reflected in the MFT by M(H)becoming multivalued in H. The MF T fits yield

ff t (7.5 ~ 0.2) X 10 cin /mole; subtracting—9.9 X 10 cm3/mole, [13] we find

1.74 X 10 cm /mole which corresponds to J~~ =110 ~ 3 meV, in agreement with Ref. [5] and tobe compared with J~~ = 128 6 me V from Ramanscattering [14] and with J~tv = 135 ~ 5 meV from thespin-wave dispersion [15]. We further find J& =1.6 ~ 0.1 p,eV, in agreement with Ref. [5]. Finally,b = (7 ~ 3) X 10 3 eV/emu4.

The MFT fits give Cgp = J"'/2Jtv~ =(1.88 4- 0.06) X 10; with Jlv~ = 135 meV we findJb' = 0.51 + 0.03 meV and C2g = 3.8 + 0.6 p,eV.Neutron scattering measurements on thoroughly reducedLa2Cu04 indicate [7] J '(T = 0) = 1.8 ~ 0.4 meV. The

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VOLUME 73, NUMBER 6 PH YS ICAL REVIEW LETTERS 8 AUGUsT 1994

discrepancy can be explained in part by the T dependenceof the orthorhombic distortion @ [12,16], which is theorigin of J"'. Taking into account that @ (which vanishesat TTo = 550 K for pure La2Cu04) at T = 0 is largerthan at T = Tz by roughly [12] a factor of 2, we estimatefrom our MFT fits J~'(T = 0) = 1.0 ~ 0.2 meV. Inoxygen-doped La2Cu04+Y (T& = 240 K) a similar, albeitsmaller, discrepancy between Jb'(T = 0) = 1.1 meV and

[5] J '(T = Tz) = 0.7 meV can be explained quantita-tively by P(T) increasing [16] from Tz to T = 0 by afactor =1.5. In the temperature range of our data, J"'varies by roughly 30%. However, it is only within 20 Kof T~ that the exact value of J~' is critical to the MFTfits: above T& the MFT is dominated by g2D, and at lowT it is dominated by terms containing mt.

Figure 3 shows our new values of (gzD) ', found fromthe MFT fits (closed circles), using C2go = 3.8 p, eV. Tocompare our data with earlier work, we start with theformulation of Chakravarty, Halperin, and Nelson (CHN)[3]. They related gzo to the 2DHA correlation length $2o,via the structure factor S(0),

S(0) C, kT 1

kT C~2 (2m. p )' (1 + kT/2~p )'

and suggested the values C, = 4.3 and C~ = 0.5. With

J&N = 135 meV [14,16] one expects a spin stiffness con-stant 2m p, = 1.15J&& = 0.155 eV [17]. Using $2o(T)from neutron scattering [2], Eq. (6) gives gzD(T), andthe results turn out to be in good agreement with ours,without adjustable parameters. However, more recentwork [18,19] showed that S(0) is better fitted by S(0) =C($2o/a), where C is a constant and a is the lattice con-stant. In view of this, we used this modified expressionand fitted C to obtain g2o(T) from the neutron scattering

data for $2o (we found C = 0.35). These results appearas open squares in Fig. 3.

The currently best theoretical expression for gqo(T)comes from the theory of Hasenfratz and Niedermayer(HN) [4), with 2np, .= 155 meV. Using this expressionin Eq. (6) gives the dot-dashed line in Fig. 3. Using themodified expression gives the solid line in the same fig-ure. The neutron scattering data shown in the figure havea somewhat stronger temperature dependence than the HNprediction; however, recent neutron scattering data [19],taken to higher temperatures, follow the HN predictionvery well, and confirm the use of the modified expressionfor S(0). The low Tvalu-es of (g2D)

' from our MFT fits,as well as from the neutron scattering, systematically falla little below the HN prediction. The MFT fit is insensi-

tive to the value of g2D below T = 315 K; we thereforet

do not attribute significance to the apparent saturation ofg2o at the lowest T. The deviation of g2D(T) from thet ~ ~ t

HN prediction for T & 360 K could indicate a crossoverfrom 2D Heisenberg to 2D XY behavior. Indeed, our datain this range can be fitted by a Kosterlitz-Thouless form

[20], g2o(T) = C exp[2.625 [TxT/ (T —TxT)] ), with

C = 210 eV ' and TxT = 295.9 K (dotted line in Fig. 3).However, this fit is not very stable, and the data are alsoconsistent with a simple exponential in 1/T (dashed line),as described below.

The fourth-order parameter a(T) [solid circles in

Fig. 4(a)] is positive at high T, and changes sign closeto T~ (indicated by the arrow in the figure), indicatingthat the transition becomes first order at low T. TheMFT fits also yield the 3D rTiagnetic order parameter(mt)2 at H = 0, determined from a(T) and g2o(T)through Eq. (4); it is plotted in Fig. 4(b) (solid circles).In the same figure we show (mt)2 measured by neutron

10

10

10

2

2

tLI 0E

)ttI

O

4

(a)I I I I I I I

I I I I I I I

10

10 I I

0.001 5 0.002 0.0025 0.003 0.00351/T (1/K)

8

E 6

o 4

2E

0

b)

neutron scatt.I I I I I I I

100 200 300 400FIG. 3. Temperature dependence of g2n(T), from MFT fits(solid circles), and from $2n(T) using a revised Eq. (6) and datafrom Ref. [2] (open squares). The various lines are describedin the text.

Temperature (K)

FIG. 4. (a) a(T) and (b) [m~(T)]2 from neutron scattering [2](open squares) and from M(H) MFT (closed circles).

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VOLUME 73, NUMBER 6 PH YS ICAL REVIEW LETTERS 8 AUGUsT 1994

scattering [2], normalized to the MFT "data" at

T = 300 K (open squares). The two sets of data show

excellent agreement with each other, with only oneadjustable parameter, the amplitude of (mt)z. This in

turn allows us to invert Eq. (4) to find a(T) from (mt)2

[open squares in Fig. 4(a)].With this knowledge of a(T), (mt), and g2o(T), we

can predict Y (T) = dM/dH for H 0 by differentiating

Eq. (4) and taking m = 0:

g(T) = g, tt«, + (Cgo/2)

IX X2D—CXp+ J

(7)

We use values for g, ff g J"'/J~tt, J&, and b fromthe MFT fits, above. Instead of using the data val-ues of a(T), we fit these values between 200 and

350 K to a quadratic T dependence [solid line in

Fig. 4(a)] a(T) = ao + a~T + azT with ao = 3.39 X10 eVemu, a) = —7.91 X 10 eVemu K ', and

a2 = 2.13 X 10 eVemu K . Similarly, the valuesof m t (T) in Fig. 4(b) are well represented by theempirical fit [2] to (m1)~ = IO~T —Tz~t', with Io =4.41 X 10 emu, T)y = 323.3 +. 0.5 K, and p =0.54 0.02 [solid line in Fig. 4(b)]. At this point,

we could either use the measured values of gzo(T)from Fig. 3 and calculate g(T) or use the mea-

sured values of g to determine gzo (T). Choosingthe latter procedure, we assumed the form gzo (T) =t

y exp(E~/kT), and fitted Eq. (7) to our g(T) data,allowing F~ to vary; X was determined by requiring

that t = [g2o(TN)] —C go —J~ = 0 [Eq. (4)]. Thesolid line in Fig. 1 shows that the fit agrees withthe data very well for 200 & T & 400 K; we find

E~ = 0.35 ~ 0.03 eV, and (g )' = 1.5(+os) eV. In-

deed, this simple approximate form, shown as the dashedline in Fig. 3, is consistent with the data for gzo(T), aswell as with the HN results in that temperature range(solid line in Fig. 3 ).

In MFT, a TCP occurs at H = 0 if t and a(T)vanish simultaneously. The results of Fig. 4(a) are notsufficiently accurate to indicate the sign of a(T& =323.3 K), but it is clear that TN (where t = 0) is veryclose to a TCP. The small value of a near T~ explainsthe sharp increase of Mt below T~, observed both hereand in Ref. [9]. The implication of this result is that forall H ) 0 and T & TN, the WF transition is first order.Usually, one needs to vary an additional parameter (e.g.,H) to reach a TCP [21]. The close vicinity of the TCP toTN is thus unusual. One source of a negative contributionto a(T), which might yield a fluctuation driven first ordertransition, comes from fluctuations in the "nonordering"parameter M+ [21]. Additional fluctuations which yieldsimilar negative contributions and turn a(T) negative

very close to T~ would arise from all the other Fouriercomponents of Mt in the b direction, all of which have

very strong correlations [22].ln conclusion, our generalized MFT, expanded to sixth

order, gives excellent fits to our magnetization datain pure La2Cu04, and yields J& = 1.6 ~ 0.1 p,eV and

J~~ = 110 meV, in agreement with previous measure-

ments. We further find Jb'(T = 0) = 1.0 ~ 0.2 meV,smaller than that found from neutron scattering (1.8 ~0.4 meV). The MFT allows an extension to T = 315 Kof the measurement of gzo(T). The new data may indi-

cate a crossover to 2D XF behavior. The MFT fits indi-

cate that the H = 0 Neel transition is very close to be-

ing tricritical, and the WF transition is first order for a11

T ~T~.We gratefully acknowledge stimulating discussions

with M.A. Kastner, R.J. Birgeneau, A.B. Harris,B. Keimer, M. Greven, and A. Cassanho. The work atMIT was supported in part by the Center for MaterialsScience and Engineering at MIT under NSF Grant No.DMR90-22933 and by the U.S.-Israel Binational ScienceFoundation.

[I] R.J. Birgeneau and G. Shirane, in Physical Propertiesof High Temperature Superconductors, edited by D. M.Ginsberg (World Scientific, Singapore, 1989).

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[10] I. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958);T. Moriya, Phys. Rev. 120, 91 (1960).

[11] P. J. Picone et al. , J. Cryst. Growth $5, 576 (1987).[12] J. Saylor et al. , Phys. Rev. B 40, 6854 (1989).[13] P. W. Selwood, Magnetochemistry (Interscience, New

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Singh et al. , Phys. Rev. Lett. 62, 2736 (1989).[15] G. Aeppli et al. , Phys. Rev. Lett. 62, 2052 (1989).[16] D. I. Vaknin et al. , Phys. Rev. Lett. 5$, 2802 (1987).[17] R. R. P Singh and D. . A. Huse, Phys. Rev. B 40, 7247

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