Spin fluctuations and excitations in a 2D xy-ferromagnet: CoCl2 in graphite

SvntheticMetals, 34(1989) 505 511

505

SPIN FLUCTUATIONS AND EXCITATIONS IN A 2D XY-FERROMAGNET: CoC12 IN GRAPHITE

D.G. WIESLER 1 and H. ZABEL 2

Department of Physics, University of Illinois, Urbana, 11 61801 (US.A.) S.M. SHAPIRO

Department of Physics, Brookhaven National Laboratory, Upton, N Y 11973 (U.S.A.)

ABSTRACT We have investigated by neutron scattering the spin fluctuations and excitations in the stage 2 CoC12 - graphite intercalation compound. This compound has easy-plane anisotropy and sufficiently weak interplanar interaction to qualify as a test material for KosterlitzThouless-Berezinsky type phase transitions. We have carried out quasi-elastic scattering measurements to determine the temperature variation of the spin correlation length ~ above the two dimensional ordering transition. We have also probed the dependence on wave vector and temperature of the inelastic scattering cross section, consisting of both a central peak, associated with vortex diffusion, and spin waves, which become strongly damped above the transition temperature. INTRODUCTION COC12 intercalated into graphite has attracted a great deal of interest in recent years as a prototype system of a two-dimensional (2D) easy-plane (xy) ferromagnet. Detailed predictions exist of the static and dynamic properties of such a system through the seminal work of Kosterlitz and Whouless [1], Berezinsky [2], and more recently of Kawataba and Binder [3], Huber [4], and Mertens et al. [5]. While much effort has been'devoted to the understanding of the static properties of CoC12 - GIC, only now studies of the dynamic properties start to emerge. The main ideas of the 2D xy Kosterlitz-Thouless-Berezinsky (KTB) phase transition can be summarized as follows: While a rigorous proof by Mermin and Wagner [6] states that no true long range order exists for 2D xy type spin symmetry at any temperature above T = 0, the low temperature phase is characterized by a quasi long range order with an algebraic decay of the correlation function caused by spin rotational fluctuations. In this phase, the fundamental excitations are topological defects, the so-called magnetic vortices. At low temperatures vortices of opposite circulation come in pairs and are strongly bound. The KTB phase transition is signified by the unbinding of these vortices at TKTB. Above TTKB long wave length magnons become unstable and the Brownian motion of free vortices gives rise to a diffusive spin dynamics. tPresent address: IBM Almaden Research Center, San Jose, CA 95120-6099, USA 2Present address: F,xperimentalphysikIV, Ruhr Universit~.tBochum,D 4360Bochum1, FRG

0379-6779/89/$3.50

© Elsevier Sequoia/Printed in The Netherlands

506 Elastic and inelastic experiments on stage 2 CoC12 -GIC have been reported at several places [7-9], showing that the spins exhibit a two step ordering process. Above Tu ~- 9.4K, only short range magnetic order exists, for T~ > T > Tl 2D ferromagnetic correlation is observed, and for T < Tt ~ 8.8K weak antiferromagnetic coupling occurs between the intercalate CoCl~ planes. Here we discuss quasi-elastic and inelastic neutron scattering data obtained for temperatures mainly above T~. An extended version of this paper is in preparation and the experimental details will appear there. CRITICAL MAGNETIC SCATTERING Critical magnetic scattering has been measured by scanning across the magnetic ridge along c* at L = 0.075(2~r/Ic), where I~ is the e-axis repeat distance. The results, which are reproduced in Fig. 1, show that at high temperatures a broad peak emerges and grows as the sample is cooled. Between 9 and 10 K a sharper component develops atop the diffuse reflection, and as the sample is cooled further, the sharp peak continues to gain intensity while the diffuse component dies off. The data have been analyzed by the usual form

5.25 I

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Figure 1: The 2D critical scattering. Nuclear scattering intensity has been subtracted out of these data. The numbers associated with each scan show the temperatures in Kelvin.

50V

S(Q=) = [1 - ~l(Q=~21 , ~ - , , I F(Q) i~ [IL(1 + ( Q ~ ) ) - P + I~6(Q~)],

(1)

where the first factor represents the neutron scattering selection rule assuming the spins are confined to the xy plane and F(Q) is the magnetic form factor. The cross section is composed of two parts: a modified Lorentzian peak of intensity IL raised to the power p, representing the short range order diffuse scattering, and a 6 function at Q~ : o, corresponding to a magnetization or a 2D Bragg rod. Fig. 2 shows the variation of the Lorentzian component IL with T. IL is essentially the 2D susceptibility, showing the same m a x i m u m at T = 8.8K and shoulder at T : 9.7K that are seen in a.c. susceptibility measurements [10]. Also shown in the same figure is the 6 function intensity 16, which behaves as expected for an order parameter, and which is consistent with earlier measurements.

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Figure 2: The intensities of the 5 and the Lorentzian component of the scattering from fits to the critical scattering are plotted as a function of temperature.

From the fit of the diffuse scattering above 7", to eq. (1) we obtain the temperature dependence of the correlation length ~. For a 2D xy system, ~ is expected to diverge in a characteristic fashion given by the expression ~ N exp(b/tU2), where b is a constant and t = T/T¢ - 1. A K T B plot of log(1/~) versus t -1/2 is shown in Fig. 3, which is seen not to agree well with the K T B prediction, because ~ fails to diverge at Tu • More usual power law functions such as ~ = t -~ do not result in better fits [1l]. A similar discrepancy is found for the intensity IL which should diverge in the critical region according to ~+2-, . Better agreement is found for the profile of the quasi-elastic scattering above T~. Mertens et al. [5] have predicted that correlations between spins arranged into free vortices diffusing through the system above TKTB give rise to a quasi-elastic scattering function that is Lorentzian raised to the power p = 3/2. We have found t h a t above 12K p is indeed 1.5 as predicted, and drops to 1.25 as'T~ is approached. The latter value is expected for a 2D spin system with r / = 1/4. These d a t a are therefore consistent with the K T B prediction for ~/ at TKT B and with the notion of vortex diffusion above TKTB. We have reported earlier that our test to observe a power-law cusp, S(Q) ~ q-2+, for the t e m p e r a t u r e region T~ < T < T= remained inconclusive [8].

508

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Figure 3: Plot of log(I/~) vs. t-1/2. The experimental data are compared with the KosterlitzThouless-Berezinsky prediction (dashed line) for b=1.5.

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Figure 4: Spin wave energies at Q= -- 0.072 as a function of T. The d a t a points are compared with ~hree theoretical predictions for spin wave energies close to T K T B *

509 SPIN WAVE RENORMALIZATION The magnon dispersion measured in the low temperature phase is consistent with the Hamiltonian of a 2D xy system and weak interlayer interaction and has been reported before [12]. Here we describe the temperature dependence of the spin wave energies. Nelson and Kosterlitz [13] have predicted that for a vortex unbinding mechanism the spin stiffness constant should exhibit a discontinous jump at TrKB. Our inelastic neutron scattering data show that at large wave numbers Q°, the spin waves damp out and shift to lower energies. However, at lower Q~ the spin waves seem to lose intensity rapidly by approaching Tu without shifting their energies, while intensity centered at zero energy transfer appears to increase. This central peak intenstity is discussed in the next section. Quantitative fits to the inelastic scattering cross section yield accurate energies for the spin waves, which are plotted for small Q~ values in Fig. 4 and are compared to the predictions of Nelson and Kosterlitz. From this figure it follows that the Nelson-Kosterlitz expression for the jump of the spin stiffness constant fits rather well the observed data, while two other predictions for 2D systems by Ohta and Jasnow [14] and by Pokrovskii and Uimin [15] compare less favorably. CENTRAL PEAK SCATTERING The chief characteristic of a KTB transition is the breakup of bound vortex pairs as T is raised above TKrB. As the vortices become free they diffuse about, driven by interactions with the other vortices in the system, much like the Brownian motion of particles observed on liquid surfaces. The vortex dynamics is governed by the Landau-Lifshitz equations of motion, and the vortex autocorrelation gives rise to a characteristic 'central peak', i.e., ~, peak centered at zero energy transfer. According to Mertens et al. [5], the line shape for the in-plane xx correlation of the free vortices can be expressed as $2

s"(Q.,~)

q3~2

= 2¢2 {,.,2 + ~2[1 + (fQ.)2]},

(2)

where ~ = ~r2~/~ and ~ is the mean vortex speed. The Lorentzian squared line shape has a HWHM given by

r,(o°) = [~(vS- 1)]1/2~11 + (~Q.)211/2

(3)

which goes to a constant value for Q, ~ 0. This finite value is characteristic for soliton diffusion and is also predicted for 1D systems. The widths obtained from fits to the central peak with the Lorentzian squared profile axe plotted in Fig. 5. They follow roughly the predicted behavior, i.e., the width increases with increasing Qa and also with increasing T through the temperature dependence of the vortex speed e(T). From £ ( Q , = 0), e(T) can be determined, which according to Mertens et al. [5] should have the values calculated for a square lattice of lattice parameter a and density of free vortices n~: f~(T) =

(2rb)'/22jS2a2h-ln~/2(1 -

T/TKTB) -1/4

(4)

Inserting this expression into Eq. (3) and using the values for ~ as determined by the elastic scattering experiments, we find that F, should increase much more rapidly with temperature than actually observed. In addition, Fz(Q,) is predicted to saturate at a finite value for Qa -4 0 and to cross over to a linear dependence on Q,, the crossover being at Q~ ~ 1/~. The experimental F,(Q,) at 17.5K is shown in Fig. 6. The best fit to the theory is shown as a solid line and has ~ = (8.0 -I- 1.5) × 104 cm/sec and ~ = 8.2 + 0.3 ) t . These values compare well with an independent determination of ~ from the elastic data at the same temperature giving ~ = 5.2 +0.2 )t, and an estimate for ~ using suitable values for J and b yields 4.2 × 104 cm/sec. The fit itself, however, is rather poor, since it deviates substantially from the data at higher Q, values. A quadratic fit to the data points would be more appropriate. At present we have no theoretical justification for this form.

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>~ 1.0 E

02 14 16 18 ~0 Temperature (K)

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Figure 5: Temperature variation of the half-width I', of the inelastic central peak as a function of temperature and for different Qas;" square: Qa = 0, circle: Qa = 0.036, diamond: Q~ = 0.072.

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Figure 6: Dependence of the central peak width l~z on Qa at 17.5K. The solid line is the best fit to the prediction of Mertens et al. [5]. The dashed line shows a fit to quadratic dependence.

CONCLUSIONS We have performed extensive tests through elastic, quasi-elastic and inelastic neutron scattering of spin fluctuations and excitations in the quasi 2D xy ferromagnet of stage 2 COC12 - GIC close to the critical t e m p e r a t u r e for the spin disordering. We have found t h a t some of the experimental results axe in good agreement with theoretical predictions of the 2D xy phase transition, while others axe fulfilled either poorly or not at all. In particular, we have not found any indication for a bound vortex state below Tu. In contrast, for the 2D regime between T~ and T~ the elastic scattering cross section contains a 6 component indicating long range order rather than quasi long range order. Also, the t e m p e r a t u r e dependence of the correlation function above T does not follow the expected K T B form and in particular fails t o d i v e r g e a t T~. The quasi-elastic and inelastic scattering d a t a can be better reconciled with

511

the existence of unbound spin vortices. The central peak in the inelastic spectrum can be described by a Lorentzian squared profile and remains finite with decreasing wave vector. Also the correlation lengths and the vortex speed obtained from the measured widths of the central peak are in rather close agreement with theoretical predictions. Thus, in the disordered state above T~ the present system conforms closer to a spin vortex picture then below T~. This is probably because above T~ the effects of finite size and interplanar interaction are less severe.

ACKNOWLEDGEMENTS We thank A.R. Bishop, F.R. Mertens, and G.M. Wysin for helpful discussions. One of us (DGW) acknowlegdes fellowship support from the Shell Foundation. The work at Illinois was supported by NSF under grant DMR 86-05565. Work at Brookhaven was supported by the Division of Materials Science, U.S. Dept. of Energy, under contract DE AC02-76CH00016.

References [1] J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6, 1181 (1973). [2] V.L. Berezinsky, Soy. Phys. - JETP 34, 610 (1971). [3] C. Kawabata and A.R. Bishop, Z. Phys. B 65, 225 (1986). [4] D.L. Huber, Phys. Rev. B 26, 3758 (1982). [5] F.G. Mertens, A.R. Bishop, G.M. Wysin, and C. Kawabata, Phys. Rev. Lett. 59, 117 (1987); Phys. Rev. B 39, 591 (1989). [6] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966). [7] D.G. Wiesler, M. Suzuki, and H. Zabel, Phys. Rev. B 36, 7051 (1987). [8] D.G. Wiesler and H. Zabel, Phys. Rev. B 36, 7303 (1987); D.G. Wiesler and H. Zabel, J. Appl. Phys. 63, 3554 (1988) [9] D.G. Wiesler, H. Zabel, and S.M. Shapiro, Physics B 156 & 157, 292 (1989). [10] Y. Murakami, M. Matsuura, M. Suzuki, and H. Ikeda, J. Magn. Magn. Mater. 31-34, 1171 (1983).

[11] D.G. Wiesler, Thesis, University of Illinois, 1989. [12] H. Zabel and S.M. Shapiro, Phys. Rev. B 36, 7292 (1987). [13] D.R. Nelson and J.M. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977). [14] T. Ohta and D. Jasnow, Phys. Rev. B 20, 139 (1979). [15] V.L. Pokrovskii and G.V. Uimin, Soy. Phys. - J E T P 38, 847 (1974).