Magneto-optical studies on ferromagnetic stripe domains in K2CuF4

Journal of Magnetism and Magnetic Materials 21 (1980) 143-156 © North-Holland Publishing Company

MAGNETO-OPTICAL STUDIES ON FERROMAGNETIC STRIPE DOMAINS IN K2 CuFa W. KLEEMANN and F.J. SCHAFER Fachbereich 6-Experimentalphysik, Universit~t Paderborn - Gesamthochschule, 4790 Paderborn, Fed. Rep. Germany

Received 9 April 1980

Ferromagnetic stripe domains are observed in K2CuF4 below Tc = 6.19 K by use of Faraday rotation (FR). They are about 4 , m wide and lie parallel to the c-planes. They are due to a very small intraplanar anisotropy field as a result of the minute orthorhombic lattice distortion observed by Hidaka and Walker. A stripe domain model, including closure domains for eo.mplete demagnetization, is in accordance with the experimental observations if the anisotropy fields/~Aut = 1.4 kOe and/./a~ = 0.10e are introduced. Long-range order, presumably induced by interplanar exchange, is accompanied by 3d lsing critical behavior as verified by critical exponents obtained from different magneto-optical measurements (a = 0.07, # = 0.33 and 8 = 4.6). Changes of the domain structure, of the at-plane FR, and of the ae-plane linear birefringence due to a transverse magnetic field H IIa are discussed. A non-vanishing internal field is preceded by a domain instability. The compatibility of the domain structure with neutron scattering results of Hirakawa and Yoshizawa is discussed.

1. Introduction

indicate the appearance of a transverse stiffness of the • spin system leading to locally ordered regions with giant moments and vortex structures ("condensed magnons"), but without LRO below Te. The inferences given in ref. [9] are mainly based on the apparent absence of any Ising-type intraplanar anisotropy, which is believed to induce LRO, too. However, as has been reported by Haegele and Babel [10] and confirmed by Hidaka and Walker [ 11 ], a very small but finite orthorhombic lattice distortion destroys the square planar arrangement of the Cu 2+ ions in K2 CuF4. This leads to a doubling of the D,~h crystallographic unit cell, probably resulting in the space group D ~ [ 11 ], and the appearance of orthothombic twin domains. They originate from two non-equivalent stacking sequences of the CuF2 planes along the perpendicular c-direction. This must be interpreted as a consequence of the antiferrodistortive order of the Jahn-Teller distorted fluorine octahedra surrounding the Cu 2 ÷ ions [12]. Bearing in mind the alternate inplane orbital ordering [12] simple electrostatic arguments are sufficient to explain the orthorhombic lattice distortion of a given stacking sequence. As a consequence o f the orthorhombic lattice symmetry a binary magnetic symmetry axis within the cplane maybe also anticipated. An Ising-like secondary

The magnetic properties of K2 CuF4 have attracted considerable interest in the last few years. Owing to its layer structure this compound is considered to be a nearly ideal two-dimensional Heisenberg (2d H) ferromagnet with S = 21 [1 ], which should exhibit a phase transition into a phase with infinite susceptibility, but without a spontaneous magnetic moment [2,3 ]. However, 3d long-range order (LRO) has been measured below Tc = 6.25 K by means of neutron diffraction [4] and NMR [5] at zero applied magnetic field. This was explained by the existence of a small, but finite interplanar exchange interaction J ' ~ 8 X 10 -~ J, where J / k B ~ 10 K [1,4,5]. The small xy-type intraplanar anisotropy field (H~ut = 1.4 kOe [5]), on the other hand, should not cause a transition into LRO [2]. I~ has been pointed out by Khokhlachev [6] and Pokrovski [7], that a 2d H-system with small planar anisotropy can be treated as a 2d xy-system at sufficiently low temperatures. Hence, Ks CuF4 seems to be a good candidate to show the spin structure anomalies of the 2d xy-system or the planar rotator, respectively [6,7,8]. Indeed, the recent neutron diffraction experiments of Hirakawa and Yoshizawa [9] on K2CuF4 seem to 143

144

IV. Kleemann, F.J. Schafer / Magneto-optical studies

anisotropy field is expected, owing to anisotropic exchange via spin-orbit coupling, and, possibly, owing to anisotropic dipolar interactions. Now it might be inferred that the observed LRO is merely a consequence of the weak anisotropy rather than being caused by interplanar exchange. This phase transition, however, does not necessarily occur at Tc, which characterizes the transition into the Berezinsky phase, but at a somewhat lower temperature TI [7, 13]. Such peculiar behavior was verified by Karimov and Novikov [14] on 2d layers of NiC12 intercalated with graphite. In order to clarify the somewhat controversial situation concerning the true nature of the magnetic phase transition(s) in K2 CuF4 and its zero-field spin structure, we have performed magneto-optical investigations with essentially three aims *" (i) observation of ferromagnetic domains and their correlation with the orthorhombic easy axis, (ii) independent measurements of TI and Tc in order to check the applicability of the slightly anisotropic 2d xy-model to K2 CuF4, (iii) measurements of critical exponents in order to confirm asymptotic 3d Ising rather than 3d H [4,16] behavior. The paramagnetic linear magnetic birefringence in the ac-plane (ac-LMB) has proved to be a good measure of the short-range spin-order [ 15]. It can thus be used to measure the specific heat anomaly, which occurs at T¢ [16]. On the other hand, the ferromagnetic LRO is known to induce optical anisotropy like Faraday rotation (FR), magnetic linear dichroism (MLD), magnetic circular dichroism (MCD) [ 17-20] and LMB via the Cotton-Mouton effect, and may thus be used to measure the spontaneous magnetization of K2 CuF4. Because of its extremely small anisotropy, the domain structure will have a compensated magnetic moment, thus leading to single domains of microscopic size. Hence our experiments were performed using a polarizing microscope, which makes possible the low temperature observation of the domains and the in situ measurements of the magneto-optical properties. We have confined ourselves to the birefringence effects FR and LMB in the visible spectral range. Special * A preliminary report was given at the DPG spring meeting, AG Magnetismus, at Freudenstadt (Verhandl. DPG (VI), 15 (1980) 265).

efforts had to be made to detect the LMB anomaly at To, which was invisible in our former investigation [15]. Since only a very small portion of the magnetic entropy of the essentially 2d H-system is spent at Tc, the anomaly will be extremely small as has been verified on other 2d H-magnets (e.g. K2 NiF4 [21 ], K2MnF, [22], and (CnH2n+INH3)2 CuCI4, n = 1 and 2 [23]).

2. Experimental procedures Single crystal of K2 CuF4 of good optical quality were grown by Cristal-Tec (Grenoble, France). After preliminary experiments on several different plateletshaped samples we performed all the measurements reported in this paper on the same sample. This was to ensure that all the effects could be correlated with one another without ambiguity concerning different critical behavior (e.g. due to chemical impurities or internal stress distribution) and different demagnetization factors. The sample was cut parallel to the crystal axes referring to the conventional K2 NiF4 structure (a, a and c)with faces of the type "ac" (3.4 X 1.9 mm 2 and 1.9 X 0.95 mm 2), and "aa" (3.4 X 0.95 mm 2), respectively. They were polished to optical quality using ~/am diamond abrasive. In situ orientation of the samples at low temperatures was possible by means of a special sample holder in a horizontal He-gas flow cryostat. This was constructed to meet the geometrical requirements of the microscopic set-up containing a polarizing microscope (250 ×), a photoelastic modulator (PEM), a computer controlled Babinet-Soleil compensator, and Glan polarizers [24]. LMB can be resolved to about 10 -8 on 1 mm thick samples, whereas FR can be measured at an accuracy of about 1 mrad. This becomes possible by means of the now classical modulation methods [25], which also allow for an exact separation of FR and LMB occurring simultaneously, for example in ac-plane measurements. Great care was taken to maintain the exact position and orientation of the sample during the otherwise computer controlled temperature or magnetic field sweeps. This proved to be necessary, since almost all of the experiments needed the highest resolution. Visual checks were performed using the sideon eyepiece of the microscope. Sodium vapor light

W. Kleemann, F.J. Schizfer I Magneto~gpticai studies

(589.3 nm) was used throughout the course of the experiments unless otherwise stated. Temperature stability to better than 0.02 K was achieved between 4.7 and 150 K by use of a conventional regulation device, which comprised a controlled heater acting on the exchange gas and an electrodynamic He outlet valve controlling the pumping rate. Isothermal field sweeps were taken at an improved stability of about 5 mK with the help of computer controlled data selection. Transverse magnetic fields up to 700 Oe were produced by a small electromagnet specially designed to fit the cryostat and the microscope.

145

negligible up to about 125 K. This can be concluded from the excellent agreement between our experimental data points with a fit to the high temperature series expansion for the magnetic specific heat of the S = } ferromagnetic square lattice [26]. The corresponding curve is denoted as HTS in fig. 1 and has been fitted by a least squares routine between 40 and 110 K. The fitting parameter J[kB was found to be 9.9 K in good agreement with the mean value obtained from the results of other experiments [1,4,5]. Inserting this value into the spin-wave expression for the magnetic specific heat in the low temperature range [27], (1)

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3. Experimental results 3.1. Zero field ac-LMB

The temperature dependence of the derivative of the ac-LMB, d I Anac I/dT (DLMB), is shown in fig. 1. Very typically for 2d H-systems [15,21,22] a broad peak emerges at about 1.5 Tc (= 9.3 K), and a long tail corresponding to the decaying 2d short-range order extends up to about 20 Te. Remarkably [15], the contribution of the diamagnetic lattice to the DLMB appears to be completely

we obtain a straight line, which can be considered as the regular extrapolation of our experimental data towards T = 0 (curve denoted as SW in fig. 1). The possibility of a quantitative test of eq. (1) unfortunately lay beyond the low temperature limit of our cryostat. We were fortunate, however, in the search for the specific heat anomaly at Tc. We found a rather sharp small peak near 6.2 K being reproducible on different samples, quite clearly discerned from other small peaks and bumps, which appear quite irregularly in the DLMB curves (fig. 1 ; also see the insert).

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146

w. Kieemann, F.Z Schdfer / Magneto-optical studies

The Tc values as derived from the DLMB curves vary by about 0.05 K from one sample to another. That is why we restricted our comparative measurements of Tc and TI to one and the same sample. Fig. 2 is devoted to this problem, where the DLMB peak at Tc = 6.189 K is compared with the zero-field magnetization curve from FR experiments (cf. section 3.2). Both curves have been measured on the same sample with the light beam focused onto exactly the same spot of the surface, the only difference being a 45 ° rotation of the polarizers from one experiment to the other. It may be anticipated (cf. section 3.2) that TI was found to be 6.185 K. Hence we can conclude that within the experimental accuracy both critical temperatures are equal, Tc = T1. This means that the possibility of the very existence of an intermediate temperature range with infinite susceptibility, but vanishing magnetization, seems to be ruled out in K2 CuF4. This might be a consequence of the twofold symmetry of the intraplanar anisotropy axis [7]. It is interesting to analyze the shape of the DLMB anomaly in the critical temperature range. In order to avoid rounding effects due to digital filtering in the calculation of the derivative, we used a l o g - l o g plot of the original data I A n ( T ) - An(Tc) I versus r = I T/Tc - 1 I. This allows the critical exponent ot to be determined from I An(T) - An( Tc)l = AT 1-c~,

3.2. Observation or ferromagnetic stripe domains The ac-LMB as discussed in section 3.1, is measured by use of crossed polarizers, which are oriented under 45 ° with respect to the a and c axis, respectively. Turning the crystal by 45 °, on the other hand, makes the LB invisible and any transmitted light flux should be extinguished. In K2 CuF4, however, we observe optical activity in this geometry at temperatures below Tc. Fig. 3 shows the intensity ! of transmitted white light measured as a function of temperature over an area of about 2 mm 2 of our ac-sample. There is an obvious correlation between I and Ms2, where Ms is the spontaneous magnetization [4] (fig. 2). Since both experiments report different critical temperatures (Te = 6.185 K and Tc = 6.25 K [4], respectively) a comparison using the reduced temperature scale should be more conclusive. This is done

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correlations. Since geometrical factors depending on the shape of the sample are involved, an exact separation of the contributions to the LMB proportional to M a and to (So" $1 ), respectively, will hardly be possible.

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I¢. Kleemann, F.J. Schiller ~Magneto-optical studies in the insert of fig. 3, where 1 lrz (r) is plotted versus Ms (T'). By letting r = r ' the presumed proportionality is excellent confirmed. Assuming the correctness o f this inference, which will be discussed in detail in section 4.4, we have extracted the critical exponent/~ from the intensity curve and obtain/3 = 0.33 ---0.01 within the fitting range r = 10 - 3 ..... 2 X 10 -2. We note that this value agrees exactly with the neutron data [4] in the same temperature range (see also the discussion in section 4.2). An enlarged plot o f I ( T ) i n fig. 2 clearly shows the crossover into a range with smaller 13values at r > 10 - 2 . For r = 3 X 10 - 2 ..... 3 X 10 - I we find/3 =

147

0.24 again in close agreement the neutron value, ~ = 0.22 [4], though it seems to be questionable if this temperature still merits the designation "critical". We should rather like to draw attention to the smearing out o f the curve above To. A similar tail, though larger by a factor o f 5, was observed in the neutron experiment [4]. According to our experience with the sample dependence o f T¢ we argue that the tail is mainly due to a local variation o f Tc within the sample containing inhomogeneous stress fields. The origin o f the optical activity below T c becomes clear upon a microscopic inspection o f the sample. Using monochromatic light, we observe that it con-

Fig. 4. Faraday rotation patterns of ferromagnetic stripe domains in an ac-sample of K2CuF4 at 4.8 K using monochromatic light (546 nm). In all photos the polarizers were crossed and oriented parallel to a and e, respectively, except for (la) and (lb), which show the Faraday contrast at a rotation angle of ±5 ° of the analyzer. The photos (2), (3) and (4) show the subsequent contrast enhancement, alternate stripe broadening and narrowing, and the f'mal domain instability as a consequence of a transverse external field H IIa. Schematic side-on views of the corresponding domain structures are drawn on the right-hand side (see text).

148

I¢. Kleemann, F.J. Schiller/Magneto-optical studies

sists of a rather regular pattern of stripe domains which are parallel to the a-direction having a mean width of about 4 ~m [fig. 4 (i)]. That these stripes become visible by FR due to Ms in the bulk of the sample will be discussed in section 4.3. Since only the sign of the rotation changes between alternate domains, no intensity contrast should be observed provided that the polarizers are perfectly crossed. Hence, the dark lines between adjacent light stripes in fig. 4 (1), which obviously do not exhibit FR, must be identified with the domain walls containing a central part with M s ± k (k is the light propagation vector). This assumption will be verified by wall width estimations in section 4.3. It has to be noted that, despite the magneto-optical equivalence of alternate Faraday domains, slight deviations from this rule are readily found with a closer inspection of fig. 4 (1)or, much more convincingly, of interference patterns using white light. As is well known, the interference colors are very sensitive to slight differences in optical phase retardations. From the study of a large number of color photos of white light interferrograms we conclude that local stress distributions or lattice distortions may easily disturb the regularity of the domain pattern. In some cases superstructure-like modulations of the interference colors were observed. Scratches on the sample surface may cause local deformations of the otherwise exactly parallel stripes. These observations already give a hint at the smallness of the anisotropy field responsible for the linear structure (cf. 4.1 .). By rotating the analyzer with respect to the polarizer, of course, the familiar FR contrast between alternate domains maybe also achieved. This is demonstrated in fig. 4 (la and b), where rotation angles of about -+5° were needed to compensate the FR of one type of domains at X --- 546 nm and for a sample thickness D = 0.95 mm. Application of an external transverse magnetic field parallel to the a-axis gives rise to a certain domain rearrangement, which can be observed to take place within pairs of domains or larger groups. These motions are usually discontinuous and maybe due to domain wall-pinning processes. The major effect, however, is the continuous increase of the domain widths I¢ of one type of domains (next-nearest neighbors) at the expense of their nearest neighbors. This can be seen in fig. 4 ( 2 - 3 ) for two intermediate values of H.

At moderate field strengths (H ~ 100 Oe) some small groups of alternately magnetized domains suddenly become unstable and transform into broad structureless, dark stripes [fig. 4 (4)]. For a further increase of H the entire sample is eventually converted into one dark single domain. As expected for a saturating transverse field, the FR completely disappears in the limit of vanishing M±. It has to be noted that the collapse field depends linearly on the demagnetization factor N as calculated with the assumption of ellipsoidal sample shape [28]. Observing the aa-plane of the sample, no trace of the domain structure becomes visible below Tc. Evidently, for k Uc the magneto-optical effects of the transverse stripe domains either vanish or compensate. However, if the sample is inspected from the other ac-plane, exactly the same domain pattern appears as in the original ac-plane. A similar field dependence is found under the action of a transverse field H IIa. Owing to the larger demagnetization factor for the new direction, the critical field for domain instability now becomes larger at the corresponding rate, We consider this experiment as an unambiguous proof for the oblique orientation of the spontaneous magnetization with respect to the ac-faces of a given sample. Only under this assumption can FR be observed along both orthogonal a-directions. We shall develop a domain model in section 4.1, which will be compatible with this idea. The formation of domain structures in ferromagnets in the critical region has been the subject of several investigations in the last few years. It has been argued in the case of uniaxial orthoferrites [29] that fluctuations seem to be the main source of the nucleation of stripe domains, which are quite similar to those in K2 CuF4. This conjecture, however, is not easily to confirm in our system. Observing the onset of domain formation at T¢ we have noticed a discontinuous appearance and disappearance of small groups of fully developed stripes. This flickering phenomenon might be dueto fluctuations of macroscopic wavelength, though local temperature fluctuations cannot be ruled out completely. However, the local arrangement of the stripes remained exactly the same in repeated phase transitions. In our opinion this gives a hint at the inhomogeneous character of nucleation, which is probably caused by lattice defects and distortions.

W. Kleemann, F.J. Schafer I Magneto-optical studies

149 .......~...~

3. 3. Magnetic field induced LMB and FR

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The field-induced variation of the domain structure as discussed in section 3.2 gives rise to welldefined field dependencies of the LMB and FR as measured on macroscopic sample regions on the acplane. Averaging over at least 100 single domains we obtain curves, which for our sample (N = 0.15) peak at about 100 Oe (fig. 5: curves Anac and FR). Whereas the FR drops to nearly zero at 120 Oe, Anac attains only a minimum and increases linearly at higher field strengths. The minimum of Anac appears at the critical field H e corresponding to the kink of the field-induced aa-LMB, which is also presented in fig. 5 (curve Anaa ). It has to be stressed that all curves in fig. 5 correspond to the same sample at equal N. It will be shown in section 4.2 that He corresponds to the situation characterized by the complete conversion of the sample into a magnetic single domain with still vanishing internal field, Hi = 0. Hence, we shall have to explain in section 4.2 the peculiar curves of fig. 5 within a multidomain model taking into account the gradual conversion into a single domain. The temperature dependence of the fieldinduced LMB, Anaa and Anac, was measured in the range between 4.7 and 6.8 K (figs. 6 and 7). The plot of Anlaaa versus H in fig. 6 reveals nearly perfect linearity at H < H c ( T ) and T < To, where He marks the kinks as mentioned above. In fig. 6 the framed

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region between the T axis and the eye-guiding line connecting the open circles at Hc(T)is denoted by Hi = 0. It refers to the part of the H - T phase diagram of K2 CuF4 with XII = oo. Apart from this region special attention will be paid to the critical isotherm as measured at 6.22 K (fig. 6) in the discussion (section 4.2). Anticipating, at this early stage, the result

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150

I¢. Kleemann, F.J. Schizfer / Magneto-optical studies

that An~rz essentially measures the magnetization M, it may be noted that fig. 6 also contains the M(T) curves as a function of H, which were published in ref. [9]. For quantitative comparison the H axes must be scaled by the ratio of the different N values involved in both experiments. As for Anac(H)we list several interesting features emerging from fig. 7, which will be discussed in section 4.2: (i) small fields give rise to weak minima, which disappear at T/> Tc; (ii) at moderate field strengths (H ~< 100 Oe) relative maxima are found, which gradually decrease and shift towards H = 0 on approaching T c ; (iii) at T > Tc the curves increase monotonously as functions of H, attaining the largest slope at Tc (critical isotherm); (iv) at fields exceeding 200 Oe the function Anac(H ) - Anac(0 ) versus T exhibits a pronounced peak at Tc as can be seen e.g. from the dashed eyeguide line at H = 330 Oe in fig. 7. It may be noted that larger absolute errors .are obviously involved in Anac(H)as compared with Anaa(H ) (fig. 5). This is due to the fact that a slight temperature instability exerts a dramatic influence on Anac(0 ) (cf. fig. 1), whereas Anaa(0 ) virtually vanishes at all temperatures. Taking into account d IAnac I/dT 2 × 10 - s K -1 near Te (fig. 1) and a temperature stabilization of -+5 mK, we obtain 5 (Anac) -+1 × 10 -7, which is of the observed order of magnitude. At this point our above mentioned statement Anaa(0) = 0 has to be discussed again with respect to the small orthorhombic lattice distortion [10,11 ], which will give rise to orthorhombic LB, as well. In fact, crystallographic twin domains, being stripeshaped and about 20 #m wide and lying parallel to [110] axes (bisecting the conventional a axes), have been observed with the help of their LB contrast on aa-samples between crossed polarizers. This natural LB is very small (An < 10-7) and virtually temperature independent. It does not interfere with Anaa(H) since the neutral lines of both effects form an angle of 45 ° within the aa-plane ([ 110] and [ 100], respectively). It must be remarked that a only few of our samples exhibited such twin domains, which moreover comprised only a few per cent of the total sam.pie volume. Hence, it seems to be justified to start an

interpretation of the observed ferromagnetic domain structure under the assumption of a crystallographic single domain.

4. Discussion

4.1. Model or ferromagnetic stripe domains Since the domain structure below T c becomes visible via FR when viewing both ac-faces of a given sample (section 3.2), an oblique direction of the spontaneous magnetization M s can be anticipated. From symmetry considerations we conclude that one of the [110] axes must be the easy axis, which we shall denote as a' henceforth and which has to be identified with one of the orthorhombic crystal axes [10,11]. The newly detected intraplanar anisotropy in K: CuF4 resembles that of Rb2 CrC14, which exhibits a number of comparable properties [30]: (i) easy-plane transparent 2d H-ferromagnet (ii) antiferrodistortive Jahn-Teller displacement of the halogen ions in the basal plane. From neutron scattering experiments [31 ] it has been inferred that the actual spin arrangement is slightly canted owing to 90 ° rotations of the local easy axes of alternate Cr 2 + ions. In this model fourfold anisotropy is expected with the easy axes along (110). A similar type of anisotropy should also be present in Ks CuF4 and recent neutron scattering results already provide a hint at a very small, but finite gap of the magnon spectrum at q = 0 (Eo 10 - s eV [32]). In our opinion, however, the predominating anisotropy must have a twofold symmetry, since the orthorhombic lattice distortion destroys the equivalence o f a ' and b'. Hence, in the context of the following discussion we shall neglect possible canting and treat K2CuF4 as a ferromagnet with a principal hard c-axis and a secondary orthorhombic easy axis a'. Models of this type have been treated in the theory of "strong" stripe domains, which were detected in permalloy [33,34] and in sputtered films of FeB [35]. The following considerations are mainly based on the latter paper. The anisotropy energy per unit volume of K2 CuF4

151

W. Kleemann, F.J. SchiTfer / Magneto-optical studies

5 K (fig. 9), we have to use a stray-field-free domain model, which is schematically outlined in fig. 8. Starting with stripe shaped bulk domains ("b-domains") with M nearly parallel to the easy axis a' and under the angle 0b = 7r/4 + ~ with respect to a, we have to add closure domains ("c-domains") to compensate the stray-field of the sample. Under the condition that M± must be continuous at any surface or domain wall, a finite angle 0c between M and a on the surface will emerge (0c = n/2 - 4). Hence, the closure domain angle a as defined in fig. 8 is determined by

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Mclosure

I

L-w-J Fig. 8. Stripe domain model for K2CuF4 assuming an easy aa-plane, which contains an oblique easy axis a'. The direction of the magnetization in bulk and closure domains is given in the coordinate frame. Other symbols are explained in the text.

is given by FA = - K o cos 2 ~k - KI cos2~,

(4)

where ~k and ~oare the polar and the azimuthal angles, respectively, which define the direction of M with respect to c and to a' within the aa-plane (fig. 8). A superimposed single-ion anisotropy with C4 symmetry would require an additional term - K s cos 2 2~0, which is neglected as explained above. The anisotropy constants Ko < 0 and Kt > 0 correspond to the anisotropy fields/~Aut= 1.4 kOe [5] andHi~ 7~ 1 0 e [1], respectively, which characterize the exchange interaction with xy- and Ising-like symmetry. Henceforth we shall assume/_/i~ = 0 . 1 0 e and use as a trial parameter IKo/KI I =/~AUt/Hi~ = 1.4 X l04.

(5)

Since Hi~ < < M s , where Ms is of the order 800 Oe at 1000

oHc/N

"+ .. .i • }.

•

-M N=°Is

tan ct = sin 0c/sin 0b.

(6)

In accordance with [35] we shall denote the domain wall energies per unit area as ot so (180 ° Bloch walls between b-domains), ol so- (nearly 180 ° Bloch walls between c-domains, accounting for 0e < < 1r/2), and o45+ (oblique walls between b- and c-domains), respectively. In order to calculate the equilibrium configuration, the total energy has to be minimized. For simplicity we neglect the contribution of closure domains at the vertical ac-faces of the crystal, assuming a fiat sample (D < < L = sample length parallel to the horizontal a-axis). The exact treatment of a sample like ours (D/L = 0.28) will not yield fundamental, but only slight quantitative differences to the results obtained below. The expression for the energy per unit volume including anisotropy and wall energies can be found to be: F = (1

W sin Ob '~ ) ~ ~ n O c ] [-K1 cos2(0b -- ~r/4)]

I¢ sin 0b + - (-K2o sin20c - K~/2) 2D sin 0c sin 0b + Olso/W+ (otso- - O t s o ) D sin 0c

+ 2O4s+ (1 + sin2Ob~

o

500

II,

o

sio

sis

61o

T/K Fig. 9. Temperature dependence of the spontaneous magnetization obtained from neutron diffraction (Ms) [4] and from the linear bireffingence data in fig. 6 (He/N) , respectively.

(7)

Since the sample thickness D is large as compared with the domain width I¢(D ~ 1 mm, W ~ 4 #m), we can neglect the last two expressions in favor of Otso/l¢. In a straightforward calculation we obtain from the necessary conditions

(Z)

W,Oc

¢)0 b,0c 0

,0 b = ~ -

152

W. Kleemann, F.J. Schitfer / Magneto-optical studies

the equilibrium parameters 0b = 4 '

0c = sin - I ( - K 1 / 2 K o ) it2 ,

W = (2Doxso)la/(-KoKl)l/4.

(9)

Thus M can be found to lie parallel to a' in the b-domains (~0 = 0), and nearly parallel to a at the surface since IK1/Kol ,<< 1. Using the assumption ( 5 ) w e obtain 0c = 0.4 °, whereas the c-domain angle is calculated as a = 0.6 °, by use of the formula 0l = tan -1 ( - K a / K o ) la ,

(10)

emerging from eqs. (5), (6) and (9). This implies that the c-domain extends deeply into the bulk of the sample. Their total depth as defined in fig. 8 is D'

I¢(Ko) 1/2 =2 -K11 "

(11)

In the case of our standard sample we calculate D' = 0.24 mm. Indeed, a microscopic focusing experiment yields sharp b-domain walls only within the depths of about 1)/4 and 3D/4 from the upper surface. Near both surfaces the visual domain contrast becomes diffuse, which can thus be explained by interference with oblique domain walls. A critical minimum thickness Dc, where c-domains from both surfaces touch and the model becomes invalid, is predicted from eqs. (9) and (11): Dc = 2 O l s o ( - K o ) l a / K 3~ •

(12)

For our sample we expect De = 0.24 ram, if we use the proportionality D' = D In and the above mentioned empirical values for D' and D. Eq. (12) still lacks experimental evidence and will be tested in the near future. A crucial test for the domain model and the parameters involved is the direct calculation of the domain width W and of the wall width W'. Using eq. (9), the approximate relations for 180 ° Bloch walls [361, olso = (2a "2 Kl JS2/a) xa ,

(13)

W'

(14)

= (27r2 jS2/Kxa) In

and the parameters J/kB = 11.36 K [32], S = ½, a = 5 A (mean nn-distance of Cu 2+ ions), KI = gll/aB tao Hi~ / a3 withgll = 2.30 and/_/i~ = 0 . 1 0 e , and

Ko = gSJ°AUt/a 3 with z = 4 and O~AUt/kB = 0.I 1 K [5], we obtain I¢ = 3.99 lam and 1¢' = 0.94 tam. These values agree excellently with W ~ 4 tam and W' ~ 1 #m as obtained from fig. 4(1) at 4.8 K. Since this temperature is far enough from To, the low temperature approximations (13) and (14) seem to be adequate. Moreover, D e = 0.23 mm is calculated from (12), (13) and (5), being close to our extrapolated value. So far all experimental features are in good agreement with our domain model. Satisfying quantitative agreement is achieved with/_/i~ __0 . 1 0 e . Thus, K2 CuF, can be described henceforth as a 2d H-ferromagnet with 1% xy- and 10 -4 % Ising-anisotropy. Since Hi~ is of the order of magnitude of the earth's magnetic field (HE) this becomes important in zero-field experiments. We have observed, that the regularity of virgin domain patterns differs significantly on turning the stripes from the n o r t h - s o u t h into the s o u t h - n o r t h direction. Since HE hits the sample surface at an inclination angle of about 60 ° , this experiments seems to show that HE supports the domain ordering process during nucleation more efficiently, if HE lies nearly parallel to a'. No such differences were observed after shielding the sample with Mu-metal. 4.2. Intraplanar LMB, spontaneous magnetization, and critical exponents Weak external fields applied parallel to an easy axis (or plane) of a ferromagnet gives rise to a magnetization M, which cancels the field within the sample via its demagnetizing field: Hi=H-

NM=O.

(15)

This equation holds, if the shape of the sample is ellipsoidal, and if the Bloch wall motion is reversible. Both conditions are approximately fulfilled for our standard sample. It can be approximated by an elongated rotational ellipsoid with an axis ratio of 2.3, which has a demagnetization factor o f N = 0.15 [28]. No measurable hysteresis was detected during all field sweeps (figs. 5 - 7 ) . This remark does not, however, completely apply to our visual observations of domain patterns. In their virgin state all samples display a somewhat irregular and diffusely bordered pattern, which becomes more regular after a magnetization cycle. Fig. 4(1)corresponds to the virgin state.

W. Kleemann, F.J. Schiller ~Magneto-optical studies However, coarse hysteresis effects were measured only on one sample, which had been distorted mechanically during an attempt to grind a perfect ellipsoid from an as-cleft rectangular sample. This observation may also have some importance in the interpretation of other experiments on thus prepared ellipsoidal samples of K2 CuF4 [9,37]. The condition, eq. (15), gives rise to the initial linearity of our curves A n ~rz versus H in fig. 6, where we rely on the proportionality z~n~ = M 2 ,

(16)

which will be shown [38] to emerge from the singleion anisotropy and from the C o t t o n - M o u t o n effect. The range of validity o f eq. (15) is rather well-defined (open circles in fig. 6) owing to the very low scatter of the experimental data. The projection of the open circles onto the H - T-plane yields He(T), which can be scaled into a magnetization curve HaMs = He/N. As can be seen in fig. 9, satisfying agreement is found with the neutron data [4], which were scaled to Ms(O) = gugBS/Cu2+-ion = 1200 Oe. A preliminary evaluation of the near critical isotherm at 6.22 K in fig. 6 yields the critical exponent 8 = 4.6 -+ 0.1, obtained from the initial slope of z~n~~ o: M ocH 118 .

(17)

The correction of H for the demagnetizing field was performed by use of the initial slope of the adjacent isotherm at 6.02 K defining 1IN. It must be noted that these slopes increase by about 8% when cooling from 6 to 5 K. In our opinion this is due to changes of the resonance frequencies involved in the C o t t o n Mouton dispersion [38] as a consequence of the magnetic LRO. That is why future, more precise determinations of 6 must be based on 1 [N values obtained from isotherms very near to T ; . It is well known that 6 is very sensitive to any uncertainty in T¢ and may vary by 10%, if the supposed T¢ is deliberately shifted by r = 10 -3 [39]. This might explain the somewhat different results, which were obtained from two different scaling relations for the critical exponent 3': 3' = 13(6 - 1),

(18)

and 3' = 2 - 213 - a .

(19)

153

Inserting our magneto-optically determined values 13= 0.33 (FR intensity; fig. 3) and a = 0.07 (Anac; fig. 2) together with/i = 4.6, we obtain 3' = 1.19 and 3' = 1.27, respectively. Both values agree quite well with the result of susceptibility measurements [37], which in the same fitting range as chosen for 13and (r = 10 -3 ..... 2 × 10 -2) can be found to vary between 1.3 and 1.0. Since 8 was obtained at Tc + 0.04 K, its actual value at T¢ is supposed to be somewhat larger [39]. Hence, 3' = 1.27 is considered to be the more reliable value. On the other hand, the susceptibility result 3' = 1 at z = 10 -3 [37] must be taken with some caution, since no account of the now apparent in-plane anisotropy had been taken. It has been argued in former investigations that K2 CuF4 might behave like a 3d H-System very near to Tc [4,16]. However, after the verification of the small Ising-component of the anisotropy, we are now rather inclined to look for 3d Ising critical behavior. Presumably the crossover into the 3d region comes about by the interplanar exchange J', which is by a factor of 103 stronger than the uniaxial exchange. We have the feeling that all of our critical exponents (a = 0.07,13 = 0.33, 3' = 1.27, 8/> 4.6) are quite coherent with the theoretical predictions for d = 3, n = 1 [41]: (a = 0.110,/3 = 0.325, 3' = 1.24, ~ = 4.82), whereas they seem to differ sensitively from the (d = 3, n = 3)exponents: (0q = -0.12,13 = 0.36, 3' --- 1.39, 6 = 4.80). The crucial point seems to be the specific heat exponent a, which is well-defined in the LMB experiment, but has been made dubious in the evaluation of the Cm data [ 16]. However, whereas a cusp-like behavior at Tc with as < 0 is expected for a 3d Hsystem [40], a logarithmi c diyergence is found. It might be argued that the discrepancy could be due to some uncertainty of the correct baseline o f the Cm data, since, contrary to the LMB data, subtraction of the lattice specific heat from the raw data is involved. 4. 3. Field dependence o f the domain structures and its magneto-optical detection In this section a qualitative explanation of the peculiar field dependencies of the LMB and FR in the ac-plane (figs. 5 and 7) will be given. Starting with the domain model (fig. 8) we assume a gradual increase of those domains being favored with respect to the direction o f H [fig. 4 (2 and 3)]. This gives rise to a net FR

W. Kleemann, F.J. Schdfer / Magneto-optical studies

154

of the whole ensemble of domains, peaking at about 90 Oe just before the b-domains become unstable [fig. 4(4)]. This collapse may be due, either to c-domain touching, or to a critical decrease of the unfavored domains such that the 1 btm wide domain walls touch. The new configuration contains large domains with smaller Mu [fig. 4(4)] and, hence, smaller FR. Above He =NMs = 127 Oe the entire sample is converted into a large homogeneously magnetized single domain. The remaining FR merely probes thermally fluctuating Mu -components, which vanish in the limit of saturated magnetization. A quantitative estimate of the FR is given in fig. 4 (la and b), where rotation angles of about +5 ° are measured at 4.9 K and X = 546 nm on a sample thickness D = 0.95 mm. According to an interpolation formula given by Laiho and Levola [18] a value of +32 ° would emerge under the same conditions, provided that M IIk IIc. However, owing to the oblique orientation of M in the b-domains, which are, moreover, reduced to an effective thickness of roughly D/2 by the c-domains, we rather expect a rotation of + 11 ° in our experiment. This is not far from the measured value. The remaining difference possibly reflects the fact that with k IIa a different geometry as quoted in [18] is met in our experiment (see also section 4.5). Concerning the field-induced change of Anac, we have to take into account that, analogously with the intraplanar MLD [19], the intraplanar induced LMB is angular dependent [38]: Anaa ~xcos 2 2j3

(13 = ~o+ lr/4).

(20)

Hence, &naa vanishes for M IIa' and thus in all b-domains. Therefore no contribution of the b-domains to ~nac is expected to first order. Only the weak minimum at about 20 Oe (fig. 5) seems to be caused by a slight rotation of M towards k in the unfavored bdomains. This follows from the conditions of the stray-field-free model leading to constant 0e throughout, but to a larger 0b in the compressed domains [fig. 4(2)]. The prevailing effects on the &nae.curves originate from the c-domains, which contribute according to eq. (20). Beginning at H = 0 with a bias LMB due to the equilibrium c-domain distribution [fig. 4(1 )], this contribution increases proportionally to the fieldinduced increase of the total c-domain volume [fig.

4(2-3)]. The sudden decrease of ~nac following the peak at about 90 Oe is due to the domain collapse, which drastically reduces the c-domain volume. However, a further increase of H rotates the magnetization parallel to a, thus giving rise to a monotonous increase of Anae up to saturating values at H > 300 Oe. The temperature dependence of Anac versus H in fig. 7 shows that the positive peaks gradually shift towards H = 0 on approaching Tc from low temperatures. This follows automatically from the condition that the domain collapse always takes place at H < He. Owing to the lack of domains above Tc the Anaccurves are monotonous and obtain their largest slope at Tc. This follows from the divergence of XII,which is involved via the relation z£nae(/-/) - &nae(0 ) ¢xM 2 .

(21)

This might be used for another determination of the critical exponent from the critical isotherm in fig. 7 [cf. section 4.2 and eq. (17)]. Comparing the temperature ranges T > T¢ and T < Tc, we have to take into account that c-domains become stable below Te. They give rise to a bias LMB at zero field, which is suppressed in the presentation of fig. 7. The total effect will obey the following proportionality: abias

(0)

( - D° I so Ko ) l a Kla~ M~s,

(22)

if we account for the geometry of our model and use eqs. (9), (11) and (21). This may be simplified by use of the well-known proportionality between etau and KIn [361: bias

nac (0) ¢xD 'a(-Ko/Kl)laM~s.

(23)

On the other hand, we can estimate the temperature effect of this LRO-contribution to &nac (cf. discussion in section 3.1) from the loss of amplitude of the curves in fig. 7 at (nearly) saturating field. This is indicated by a dashed line in the plane H = 330 Oe, which falls off rapidly below Tc. A closer inspection reveals an approximate proportionality of ~ bias (0) with M~s. Hence from eq. (23) we obtain D' ~ constant or

Ko cxK1 ,

(24)

a reasonable result [36], which will be useful in the following section.

w. Kleemann, F.J. Schiller / Magneto-optical studies 4. 4. Domain controlled transmitted light intensity In this section we have to verify the correctness of the relation

I cc Ms2,

(25)

which was used to analyze the temperature dependence of the intensity of white light transmitted through an at-sample between crossed polarizers below Tc (figs. 2 and 3). We can start from the wellknown proportionality between the longitudinal magnetization, Ms u, and the rotation angle O, which also depends on the wavelength X and is proportional to the sample thickness D [ 17,18,38 ]. The transmitted light intensity for a given X and D thus becomes I(~,, D) = Io (X) sin2 O(X, D) ~ Io (X) O2(X, D)

¢clo(X)D2 M2s,i

(26)

in the limit of small rotations (O < < 90°). Hence, for a given domain structure (fig. 8), where only b-domains are FR active, relation (25) can hold only, if the effective thickness of the b-domains does not change with temperature. According to the discussion in section 4.3 this seems to be verified [eq. (24)]. Small deviations from relation (25), however, cannot completely be excluded, in spite of the positive test presented in the insert of fig. 3. Final conclusions could be drawn from measurements o f / o n samples with varying D values, since the relative influence of the c-domains could decrease with increasing D.

4.5. Domain structure and neutron scattering We now arrive at the crucial question, whether our observation and interpretation of the domain structure in K: CuF4 are compatible with the recent results of neutron scattering by Hirakawa and Yoshizawa [9]. These authors used a spheroidal sample with the direction of the applied field parallel to a and the scattering vector K parallel to c. Below To, a linear decrease of the magnetic Bragg scattering at (004) is found for H ~ Hp a gradual recovering of the scattering rate takes place. In the authors' [9] opinion this effect is due to a phase of condensed magnons, which is destroyed by an external field at the same rate as

155

sample regions with giant moments (× = .o) are turned into the direction o f / / ( X = 0). This process is finished at Hp. In this context a model of the spin structure is proposed, which resembles that of the Berezinsky phase of the 2d xy-system. At the first sight this model seems to contradict the existence of domains exhibiting a net FR. However, the relatively small total FR measured on single domains (cf. section 4.3) could be a hint at a spontaneous spin structure, which still contains differently magnetized, submicroscopic regions with only preferential order parallel or antiparallel to the easy axis, respectively. In this picture a net spontaneous magnetization arises in a particular domain, which at the same time has an intraplanar order very similar to that proposed [9]. Such a peculiar structure would tend to enhance the elastic neutron intensity, but to decrease the FR. An applied field, on the other hand, would enhance the LRO even at H < H e . This should be observable in the FR of single domains and seems to be the reason for the polarization enhancement of inelastically scattered neutrons [42]. Further FR measurements on ac-samples in a longitudinal field//II a are planned in order to test the above hypothesis.

Acknowledgements We are grateful for useful discussions to V. Wagner and to J. Ferrd, who also kindly provided the samples of K2 CuF4. Thanks are due to J. Blickwedel and H. Obermeier for their keen assistance during the preparation of the experiments. The work was partly supported by the Landesamt for Forschung NRW.

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156

141.Kleemann, F.J. Schafer ~Magneto-optical studies

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