Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 213-216North-Holland

Invited paper

Experiments on Haldane gap in quasi-one-dimensionalantiferromagnets

J.P. Renard, V. Gadet \ L.P. Regnault 2 and M. Verdaguer 1

Institut d'Electronique Fondamentale, Bal. 110, CNRS URA 11, Unicersite Paris Sud, 91405 Orsay Cedex, France

213

We review recent experimental works on several linear chain compounds which approximate the ideal one-dimensionalHeisenberg antiferromagnet with S = 1. They include Ni(CZHgNZhNOZCl04 (NENP) and two other Ni (II) compounds withsimilar structure, AgVPz~ and a new promising Ni (II) chain Ni(NOzh N(CH 3)4 (TMNIN). All these compounds exhibit thelow temperature susceptibility decrease and the absence of 3D long range magnetic order which are characteristic of theHaldane gap. We especially discuss the break of the gap induced by large applied fields in NENP and TMNIN and the effectof copper impurities introduced in the chains of NENP.

1. Introduction

The interest in quantum anti ferromagnetic chainshas been strongly renewed by the Haldane conjecture[1]. Following this conjecture, the low temperature mag­netic behavior of the one-dimensional Heisenberg anti­ferromagnet (ID-HAF) described by the HamiltonianH = LIS/, SI+l is strongly dependent on the spin valueS. For integer S values, the ID-HAF should exhibit asinglet ground state separated from the first tripletexcited state by an energy gap Eo, while the spectrumof excitations is gapless for half-odd integer S values. Itresults from this quantum gap, that the ID-HAF withinteger S remains disordered down to T= 0 K with afinite correlation length ~ [1,2], while non-integer SID-HAF has an ordered Neel state with an infinitecorrelation length at T = 0 K. The Haldane conjectureis now well supported by numerical calculations onfinite rings for exchange coupled S = I spins usingfinite-size scaling techniques [3-5] and Monte Carlomethods [6-8]. These latter calculations have provided areliable value of the energy gap Eo = 0.41 J [6], and ofthe reduced correlation length ~ = 6 [7,8].

The effect of magnetic anisotropy has also beenconsidered. A very important result is the persistence ofthe Haldane gap for not too large single ion anisotropyvalues {3,5].This allows to check the theoretical predict­ions by experiments on real quasi-ID-AF with S = I, inwhich there always exists single ion magnetic anisotropyD(S;)2. Such an anisotropy term splits the first tripletexcited state, into a doublet and a singlet. The first

I Laboratoire de Chimie des Meraux de Transition, CNRSUPA 419, Universite Pierre et Marie Curie, 4 Place Jussieu,75252, Paris Cedex 05, France.

z Centre d'Etudes Nucleaires de Grenoble, DRF/SPh-MDN,85X, 38041, Grenoble Cedex, France.

excited state is the doublet for D > 0 and the singlet forD<O.

An other important deviation from the idealID-HAFis the interchain coupling J', which in a gapless chain,induces three-dimensional (3D) magnetic order at finitetemperature. For weakly coupled S = I, ID-HAF, no3D magnetic order can develop down to 0 K if IJ'/ J Iis smaller than a critical value IJ'/J Ie [9,10] of about0.02 for 4 neighboring chains. The persistence of thequantum disordered state in systems with magneticanisotropy and 3D interactions is highly favorable forexperimental observations of Haldane gap.

2. Experimental evidence for Haldane gap in Ni(NzC2HshN02Cl04 (NENP)

The first attempts for observing the. Haldane gapwere done by neutron scattering experiments in CsNiCI 3

[11,12]. This compound has a very small magnetic ani­sotropy but a relatively large interchain exchange whichinduces 3D-AF long range order at TN = 4.85 K andthus forbids the clear observation of the gap at lowtemperature. A way for reducing IJ'/J I consists inincreasing the chain spacing while keeping a large Jvalue. This is realized in Ni (NzCzHsh(NOz)CI04(NENP) an orthorhombic crystal of NiH chains wellisolated from each other [13]. Numerous experimentsincluding susceptibility, neutron scattering [14-16],specific heat [17] and high field measurements [15,18,19]have been performed on NENP. All of them supportedthe Haldane conjecture.

Direct evidences for gap are given by the susceptibil­ity data of a single crystal and by the spectrum ofmagnetic excitations around k = orr obtained by inelasticneutron scattering (INS). These last experiments showan upper excitation mode at E)};/k= 30 K related tospin fluctuations parallel to the chain axis (b) and a

0304-8853/90/$03.50 <D 1990 - Elsevier Science Publishers B.V. (North-Holland) and Yamada Science Foundation

214 J.P. Renard et al. I Experiments on Haldane gap in quasi-ID antiferromagnets

lower mode at Eo/k = 14 K related to spin fluctuationsperpendicular to b. High resolution INS measurementsreveal a splitting of this lower mode into two compo­nents at about 0.2 meY from each other correspondingto dEo/Eo = 0.18. Both susceptibility and INS datareflect the influence of magnetic anisotropy in NENP.A single-ion anisotropy term '£;D(S/)2 (D:::: 12 K, zllb)splits the excited triplet state into a singlet at EJ}; and adoublet at Eo. this latter one being splitted into twocomponents by a small orthorhombic anisotropy term'£;E[(S;"")2 - (S/)2J(E« D).

The low temperature magnetic susceptibility behav­ior is mainly determined by the lowest gap Eo. Indeedthe experimental data arc roughly consistent with thefollowing temperature dependence:

0.03 r-----.---r----.------,

O'-__-'- ...L-__---' -'

o 50 100 150 200T(K)

Fig. 1. Molar susceptibility of a powder of Ni[(ClI )4N](N02h(TMNIN). The solid line is a fit to high temperature seriesexpansion for S = 1 Heisenberg antiferrornagnet with JI k = 12

K and g = 2.25.x(T) = x(0) + C exp( -Eo/kT), (1)

;"0.Q2EGlCIlClu

\'\ TMNIN

\ -,<,

<,<,

<,

' ..........--

Affleck [20] has pointed out that for purely axialsymmetry, XZ(T) should go to zero at T ..... 0, in con­trast to Xx·J'T). For orthorhombic symmetry,

(2)

which gives, using the INS experimental dataXZ(O)/XX(O) :::: 2 X 10- 2•

This is much smaller than the experimental value0.33. This discrepancy is likely due to an extrinsiccontribution of defects or impurities which cannot beneglected at low temperature because of its T - 1 varia­tion. This spurious contribution is the major difficultyfor the precise determination of the Haldane gap fromsusceptibility measurements.

3. Other quasi-ID AF exhibiting a Haldane gap

Since the first studies of Haldane gap in NENP,several S = 1 quasi-ID AF have been synthesized andexperimentally studied, mainly by susceptibility mea­surements. Their main characteristics are reported intable 1.

Table 1Main characteristics of S = 1 quasi-1D-AF exhibiting a Haldanegap

Compound Jlk IJ'PI DIJ Ealk Ref.(K) (K)

NENP 48 4 xlO-4 0.2 14 [14-16)NINO 52 0.3 10-15 [15,16,211NINAZ 100 30 [16]

145 30.50 [21]AgVP2S6 >400 10- 5 320 [23,24)TMNIN 12 4.5

Two orthorhombic nickel compounds chemically andstructurally related to NENP, Ni(C)H ION2hN02Cl04(NINO) and Ni(C)H ION2h(N)Cl04 (NINAZ) exhibitthe low temperature susceptibility decrease which char­acterizes Haldane gap systems [16,21]. Recent magneticspecific heat measurements in (NINAZ) confirm theexistence of the gap [21J. AgYP2S6 is a basically differ­ent system since the S = 1 ion is v>. This compound isattractive because of its simple structure formed ofzigzaging v> chains in the ab crystallographic plane,very well decoupled from each other [22]. Recent neu­tron and susceptibility measurements [23,24J show verylarge anti ferromagnetic intrachain exchange J> 400 K,direct evidence of the Haldane gap at Eo = 27.5 meYand no 3D magnetic order down to 5 K. Ni«CH3)4N)(N02») (TMN1N) is another promising system becauseof its simple structure similar to that of the archetypal1D-AF, Mn«CH3)4N)CI3 (TMMC) [25] and of a rela­tively small J value which makes the experimentalstudies easier. The powder susceptibility of TMNIN(Fig. 1) is fairly consistent with the predicted one forS = 1,lD-HAF with J /k = 12 K and g = 2.25 above 20K. Below 10 K, the steep susceptibility decrease withoutany apparent impurity contribution reflects the effect ofthe gap.

4. Effect of magnetic field on Haldane gap

There arc still few experimental data [15,18,19J andtheoretical predictions [4,20] on S = 1 anti ferromagneticchains in applied magnetic fields. Quite generally, onecan expect that a Zeeman energy term - gp.nH'£;S/depresses levels of higher total spin and thus reducesthe energy gap . Above a critical field lIe = Eo/gap.n(a = x, y, z), the gap should vanish and the magnetic

J.P. Renard et al. / Experiments on Haldane gap in quasi-Hi antiferromagnets 215

Providing that the field is not too small , one canexpect an exponential dependence of magnetization,

fttZ(H)=aexp[-(Eo-gzJlnH)/kT]. (4)

Our high field data obtained by SQUID magnetome­try (Fig. 3) arc in excelIent agreement with relation (4)

•

80

7

/

•20

NENP

:>E..enOl~

c:;­"0

10,

E

40 60H (kOe)

Fig. 3. Molar magnetic moment of pure NENP versus mag­netic field applied along b and a axis: b axis: T= 2 K (e),T= 4 K (o): a axis: T= 2.5 K (.), T = 4 K (D). The solid

lines are fits to relation (4) with Eo/k = 14 K.

without any adjustable parameter except for a =

0.024g zJlD per atom. For field applied along the a 'axis(perpendicular to z), the field dependence of magnetiza­tion cannot be described by a relation similar to (4).Indeed, in this orientation, the gap has not a lineardecrease with H [20}. In the experimental field range,the magnetization for H parallel to z is larger than forH perpendicular to z, which is consistent with thepredicted smalIer value of the gap for II along z.

From relation (3), with Eo = 14 K and s' = 2.15,the critical field value for which the gap vanishes isHe = 97 kOe in good agreement with pulsed field mea­surements [19). For H perpendicular to z ; the situationis less clear since the pulsed field measurements [18,19]yield /Ie values which are larger than the expectedvalue (20) EO/gJlD = 94 kOe. In neutron scattering ex­periments in high field [15], only the two predictedupper gaps have been observed. The observation of thelower gap is clearly needed to fulIy confirm the theoreti­cal predictions.

5. Effect of Cu impurities in NENP

In a previous paper [16], we have reported pre­liminary susceptibility measurements on NENP singlecrystals in which a smalI proportion x of Ni2+ ions isreplaced by Cu 2 + ions. At low , temperature, a largeimpurity contribution following a Curie law was ob­served. We succeeded to grow NENP single crystalswith CUB concentration x up to 6%. Susceptibility aswell as high field magnetization of these samples showthat each copper ion introduced in the' Ni2+ chainreleases a nearly free nickel spin. This is illustrated infig. 4 where the high field magnetization along b axis,of NENP: 0.06 Cu is compared to the sum of two

(3)

o

8060

T=15K /0__ 0

o -

o 0 ../ T= 1.9 K

o It"-o ~ ,

o .~,~,'

o........., .!.~. ./

o ~....o 20

2.10 'I- TMNIN

E10'..enOlu

.."0E

moment per atom should increa se as the one of agapless 5=1, ID-HAF:m" = (g"Jln)2H/41S. This be­havior has been first observed in a powder of NENP forwhich the experimental mean value of He was about 90kOe [15]. More recently, experiments on single crystalsof NENP [18,19] revealed an anisotropic behavior of/Ie' He being larger for applied field perpendicular tothe b axis than for fields parallel to the b axis .

The field dependence of magnetization measuredwith a homemade high field SQUID magnetometer of apowdered sample of TMNIN is shown in fig. 2. Fortemperatures well above Eo/k, a quasi-linear M(H)dependence is obtained. For temperatures below Eo/k,the M(/I) curve exhibits a net change of slope aroundHe = 30 kOe. With g = 2.25, this leads to Eo/k = 4.5 Kand to a ratio Eo/1 = 0.37 close to the Monte Carloestimation for the 5 = I, ID-HAF (0.41). This result isconsistent with a small value of the magnetic anisotropywith respect to intrachain exchange J in agreement withsusceptibility measurements on a bundle of tiny singlecrystals which do not show significant difference forfield parallel and perpendicular to the chain axis .

In NENP, the single ion anisotropy lowers the en­ergy gap to Eo/k = 14 K. In applied field along z (baxis), the gap has a linear dependence with the field [20]given by:

40H (kOe)

Fig. 2. Molar magnetic moment of TMNIN powder versusmagneti c field at T = 1.9 K and T = 15 K. The solid lines arelinear fits to the low field and high field parts of the curve atT = 1.9 K. The critical field of 30 kOe corresponds to EG / k =

4.5 K.

216 J.P. Renard et 0/.1 Experim ents on Haldane gap in quasi-LD antiferromagnets

6. Conclusion

References

The authors wish to thank P. Beauvillain, C. Chap­pert and P. Veillet for experimental help in high fieldmeasurements and l. Affleck for the preprint of hisrecent theoretical work.

[1) F.D.M. Haldane, Phys. Lett. A 93 (1983) 464; Phys, Rev.Lett , 50 (1983) 1153.

(2] r. Affleck, T. Kennedy, E.H. lieb and H. Tasaki, Phys.Rev. Leu. 59 (1987) 799.

(3) R. Botet and J. Jullien, Phys. Rev. B 27 (1983) 613. R.Botet, J. Jullien and M. Kolb, Phys. Rev. B 29 (1983)5216.

(4) J.B. Parkinson and J. C. Bonner, Phys. Rev. B 32 (1985)4703.

(5) H. Betsuyaku, Phys. Rev. B 36 (1987) 799.(6) M.P. Nightingale and H.W. Blote, Phys. Rev. B 33 (1986)

659.(7) K. Nomura, Phys. Rev. B 40 (1989) 2421.(8) M. Takahashi, Phys. Rev. B 38 (1988) 5188; Phys. Rev.

Leu . 62 (1989) 2313.(9) Y.A. Kosevich and A.V. Chubukov, Sov, Phys. J.E.T.P. 64

(1986) 654.(10) I. Affleck, Phys. Rev. Lett. 62 (1989) 474.(11) WJ.L. Buyers. R.M. Morra . R.L. Armstrong, M.J. Hogan.

P. Gerlach and K. Hirakawa, Phys. Rev. Lett. 56 (1986)371.

(12) M. Steiner. K. Kakurai, J.K. Kjems, D. Petitgrand and R.Pynn, J. Appl. Phys. 61 (1987) 39.53.

113) A. Meyer. A. Gleizes, J.J. Girerd, M. Verdaguer and O.Kahn. Inorg. Chern. 21 (1982) 1729.

(14) J.P. Renard. M. Verdaguer, L.P. Regnault, W.A.C. Erke­lens, J. Rossat-Mignod and W.G. Stirling, Europhys. Lett.3 (1987) 945.

(15) J.P. Renard, M. Verdaguer, L.P. Regnault, W.A.C. Erke­lens, J. Rossat -Mignod, J. Ribas , W.G. Stirling and C.Venier, J. Appl. Phys. 63 (1988) 3538.

(16) J.P. Renard, L.P. Regnault and M. Verdaguer, J. de Phys.49 (1988) C8-1425.

(17) J . Ferre, J.P. Jamet, c.r. Landee, K.A. Reza and J.P.Renard. J. de Phys. 49 (1988) C8-1441.

(18) K. Katsumata, H. Hori, T. Takeuchi, M. Date, A. Yama­gishi and J.P. Renard, Phys. Rev. Lett. 63 (1989) 86.

(19) Y. Ajiro, T. Goto, H. Kikuchi, T. Sakakibara and T.lnami, Phys. Rev. Lett. 63 (1989) 1424.

(20) I. Affleck, preprint.(21) V. Gadet, M. Verdaguer, J .P. Renard, J. Ribas. M. Mont­

fort. X. Solans, C.P. Landee, J.P. Jarnet and A. Dworkin.to be published.

(22) P. Colornbet, S. lee. G. Ouvrard and R. Brec, J. Chern.Res. S (1987) 134. S. Lee. P. Colornbet, G. Ouvrard andR. Bree, Mater. Res. Bun. 21 (1986) 317.

(23) H. Mutka, J.L. Soubeyroux, G. Bourleaux and P. Col­ember, Phys. Rev. B 39 (1989) 4820.

(24) C. Payen, P. Molinie, P. Colombet and G. Fil1ion, J.Magn. Magn. Mal. 84 (1990)95.

(25) L.J. de Jongh and A.R. Miedema, Adv. Phys. 23 (1974) 1.(26) J.P. Boucher. Hyperfine Interactions 49 (1989) 423.

80

00

........

60

x=Oo

NENP: x euT:2K

20

:)

EGltilen<.l

Gl

"0E

40H(kQe)

Fig. 4. Molar magnetic moment of copper doped NENP withx = 0.06 versus magnetic field 'along b at T= 2 K. The fullcurves are the calculated ones as the sum of two Brillouin

functions iot S = 1/2 and for two values of g = 2.15 and 2.2.

The existence of Haldane gap in antiferromagneticchains with S = 1 is now well supported by experiment.The new Ni(II) chain compound TMNIN, appears as apromising system close to the Heisenberg model with arelatively small gap value EGlk = 4.5 K, which makeseasier precise studies in applied field.

In spite of its relative large single-ion anisotropy,NENP appears as a model system, in which severalexperimental techniques provide a reliable value of thegap EGlk = 14 K. A simple picture of the field effecton the gap, at least for fields parallel to the symmetryaxis h. is well supported by experiment. The improvedknowledge of NENP allows more sophisticated experi­ments including the study of impurity effects and thestudy of spin dynamics by nuclear magnetic resonance[26].

Brillouin functions for S = 1 and 1/2 spins in the sameproportion equal to the Cu concentration x = 0.06. Thissimple model allows to fit the experimental data withoutadjustable parameters except for g", At fields above 60kOe, the impurity moment is nearly saturated and themagnetization increase is mainly due to the gap de­crease induced by the field following relation (3).

The magnetization increase is smaller for NENP:0.06 Cu than for pure NENP, which is likely due to anincrease of the gap induced by impurities estimated to(1.5 + 0.5) K from the data at 80 kOe. This increase isconsistent with the calculated gap variation with n infinite chains of n spins [6]. For the mean value of11:::::: (0.06)-1:::::: 17, the gap should increase by about 10%which is close to the experimental result.