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Nonlinear waves of magnetization in iron borate
Journal of Magnetism and Magnetic Materials 103 (1992) 325-334 North-Holland
Ad"
Nonlinear waves of magnetization in iron borate M.V. C h e t k i n a n d V.V. Lykov Department of Physics, Moscow State University, Moscow 119899, USSR Received 8 May 1991
A new type of magnetoelasticwaves in an easy-planeferromagneticwith the transverse corrugation type of self-organization of a structure whose period depends on the external magnetic field, has been observed and magneto-optically visualized. The dynamicsof the domain wall in iron borate and its behavior in the magnetic and intense sound wave field have been studied. The possibility of a dynamic orientational transition under both magnetic and sound field with the destruction of the domain wall upon reaching the critical velocitydepending on the external pressure has been shown.
1. Introduction
2. Experimental technique
Waves of magnetization in weak ferromagnetics (WFM) have received widespread attention in recent years, because, on the one hand, WFM are promising materials for modern microelectronics, primarily in data storage and processing systems, and, on the other hand, the propagation of these waves is associated with a great number of physical, especially nonlinear, phenomena [1,2]. The presence of a magnetoelastic bond comparable with the anisotropy energy in easy-plane antiferromagnetics (EPAFM) is responsible for the nonlinear phenomena there. The elastic waves in E P A F M become nonlinear even at low amplitudes and their propagation may be accompanied by dynamic orientation phase transitions (OPT). Iron borate FeBO 3 is a transparent easy-plane WFM with a strong magnetoelastic bond against the background of a low, practically vanishing anisotropy field in the plane, less than 0.01 Oe [3]. FeBO 3 is very suitable to study various nonlinear magnetoelastic interactions using, primarily, magneto-optic methods. This paper is devoted to the study of dynamics (section 3) of domain walls, their behavior in a strong acoustic field (section 4) and to the investigation of the observed slow nonlinear magnetoelastic wave with self-organization of the transverse corrugation type (section 5).
The investigation of nonlinear waves of magnetization in iron borate was performed in samples prepared as plates with typical dimensions 3-5 mm and 20-100 ~ m thick. A single N6el domain wall (DW) could be produced by the gradient magnetic field of = 70 O e / c m and the external contracting pressure applied in the sample plane, whose value was controlled and varied from 0 to 2 x 109 d y n / c m 2. The D W movement was realized either under the action of the impulse driving magnetic field H with the time-rise of 6 - 8 ns or under the action of a sound wave. If the pressure, the driving magnetic and gradient field are uniform, then a strictly two-domain structure is realized in the sample and there are no domain nuclei at the sample ends. The observation was performed using the Faraday effect which is strongly influenced by the birefrigence in the plate tilted at an angle of several degrees with respect to the horizontal axis (to the vertical one for the study of magnetoelastic waves too) in order to ensure a projection of the magnetization in the direction of the light propagation. Due to the birefrigence effect, the polarization plane rotation was about 1 ° in the domain contrast and less than 0.5 o in the D W contrast and during the propagation of the magnetoelastic wave.
M.V. Chetkin, V.V. Lykoc / Nonlinear wat,es of magnetization in iron borate
326
pie ( p = 0). The choice of the frequency was due to the relaxation time of the magnetic subsystem (--- 10 -8 s) so that it could trace the changes in the elastic subsystem. At the transducer the voltage in the impulse was as high as 1.6 kV, and the acoustic wave phase was rigidly not worse than 1-2 ns synchronized with the light impulse. To estimate the amplitude of deformation in the acoustic wave passing through the iron borate plate, the calibration was made on the basis of the reciprocity principle using two transducers which had in turn been calibrated by means of the electroinduction technique [5]. The method described allowed us to estimate the amplitude in the thin iron borate plate and to directly measure the velocity of the quasilongitudinal sound in iron borate at the given frequency. The value of the velocity was found to be (6.0_+ 0.2) k m / s . The deformation amplitude was as high as = 3 × 10 -s.
The recording was performed using either the high speed photography, when studying the magneto-acoustic interactions, or the double high speed photography, when studying the DW dynamics. In double high speed photography the sample was consequently illuminated by two light impulses of some nanoseconds optical delay. In both cases the oxazin-dye laser with the wavelength of 535 nm pumped by the T E A nitrogen laser, with a light impulse duration of 0.3 ns was used [4]. The image of a moving DW was registered by the double photography technique both in the domain contrast and in the DW contrast. The technique allowed the simultaneous recording of at least three shades of the colour grey, thus making it possible to control the D W structure in the real time scale with a simultaneous variation of its velocity. The sound wave was produced by a piezotransducer with a resonance frequency of 3.4 MHz, attached with phenil salicylate to a glass rod whose other end was pressed to the sample end. The rod provided an acoustic delay of 3-5 txs to separate the electromagnetic induction arising when the transducer was excited from ~he acoustic signal because of the sample being sensitive to the former. A thin layer of oil between the sample and the rod improved the acoustic contact and so it was possible to avoid pressing the sam-
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3. The domain wall dynamics in FeBO 3
The domain wall dynamics in iron borate has been studied earlier elsewhere [6-8]. It has been shown that the nature of the DW dynamics in FeBO 3 resembles that in yttrium orthoferrite [9]. But it has been noted that sometimes the DW disappeared [8], although the reasons and the
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Fig. 1. Dependence of the domain wall velocity on the magnetic field amplitude in FeBO 3. Contracting p r e s s u r e P = 2 0 x 10 s (O); 10 × 108 (-); 3 x 109 ( x ); 0.1 x 108 ( A ) d y n / c m 2.
M.H Chetkin, H E Lykov / Nonlinear waves of magnetization in iron borate c o n d i t i o n s u n d e r w h i c h this h a p p e n s r e m a i n e d unknown. Our study of the dependence of the DW velocity o n t h e d r i v i n g m a g n e t i c f i e l d a m p l i t u d e has
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s h o w n t h a t t h e D W s t a t i o n a r y m o v e m e n t is possible o n l y at a v e l o c i t y less t h a n a c e r t a i n v e l o c i t y V 1 d e p e n d e n t o n t h e e x t e r n a l p r e s s u r e [12]. I f t h e D W m o v e m e n t is s t a t i o n a r y a n d t h e p r e s s u r e is
Fig. 2. Double high speed photographs of a moving domain walls in iron borate. The schematic magnetization direction structure of DW and domain contrast in this technique are shown. At the left side of figure the stationary moving DW is shown (Vow < V1). At the right side the moment of the DW disintegration is shown (Vow = Vl): under illumination of the first beam there is no disintegration, under illumination of the second beam the disintegration has already occurred. Optical delay was 11 ns~ The dark band on the photograph is a distance which DW pass during optical delay. The grey area is the 90 ° domain arisen after 180 o DW disintegration. ,
328
M.V. Chetkin, V.V. Lykov / Nonlinear waves of magnetization in iron borate
high enough, then on the dependence V(H) (fig. 1) one can observe constant D W velocities regions at of 4.4, 4.0, 2.7, 1.8 k m / s coinciding with the velocities of the transverse, surface and Lamb waves, respectively, and resulting from the interaction of DW with these waves. The nature of the latter should be investigated. On reaching the velocity V1 (in the field H1) , the D W disintegrates into two 90 o DWs (to be exact, not 180 ° ones), and the width of the resulting domain is determined by the difference between the driving and gradient field. In other words, when the DW reaches the velocity V1 in the region where H > H1, the dynamic spin-reorientation phase transition (OPT) with the magnetization direction there close to 90 ° takes place, the DW being a nucleus for a new phase. Fig. 2 represents a double high speed photography of a moving domain wall in iron borate. The schematic magnetization direction structure of DW and domain contrast in this technique are shown. At the left side the stationary moving DW is shown (VDw < 1/1). At the right side the moment of the O P T formation and D W ~lisintegration is shown (Vow = V1): under illumination of the first beam there is no disintegration, under illumination of the second beam the disintegration has already occurred. Optical delay was 11 ns. By varying the optical delay time it is possible to estimate the time needed for the O P T to be formed which seems to less than 1 ns (in the experiment the minimum optical delay time was 3 RS).
The dependence of the D W velocity Vl and mobility t~ on the external contraction pressure are given in fig. 3. At low pressures the mobility may be exceedingly high, reaching 200 × 103 c m / s Oe.
The p h e n o m e n o n observed can be explained qualitatively in the following way. The stability of the domain structure in the sample is determined by the external contracting pressure creating the uniaxial anisotropy in the sample plane or effective magnetic field Hp. If Hp has reached the
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the mechanical stress along the x-axis in the sample plane, at P > 0 it contracts the crystal, and at P < 0 it stretches the crystal, C66, B66 a r e the elastic and magnetoelastic constants, M 0 is the magnetization. Thus, for FeBO 3 at P = 1 × 108 d y n / c m 2 and at different values of B66 [3], H = 1 - 2 0 e ; taking into account the magnetoelastic renormalization C66, Hp* = 1 . 5 - 3 . 0 0 e . The antiferromagnetizm vector m changes its direction from mllx to rally, i.e. 180 o D W are formed. A the given pressure P > 0 a similar O P T can be performed in the reverse order in the field H. In our case the O P T takes place in the region where H < Hp*. Thus the field H 1 corresponds to the field Hp*, their values are apparently close. To specify the picture, one should take into account the dynamics process, i.e., the dependence V(H). This question was investigated in ref. [11], where the authors came to the conclusion that a magnetoelastic gap had to exist in spectrum of the D W velocity. The presence of the gap is determined by a strong magnetoelastic interaction and its width determined as the difference (S t - V1), where S t is a transverse sound velocity. Near S t the enhancement of the magnetoelastic energy may be so great that the effective constants of the induced anisotropy change their
329
M.V. Chetkin, V.V. Lykov / Nonlinear waves of magnetization in iron borate
signs, i.e. O P T takes place. As a result the D W disintegrates. Hence the gap value depends on the applied pressure. Fig. 3 gives the theoretical dependence V I ( P ) plotted in accordance with ref. [10].
4. The domain wall in the field of elastic waves in FeBO 3
The D W can be moved under the action of not only magnetic field but effective field of sound wave too This section discusses the D W interaction in thin iron borate plates with the sound wave propagating in the basal plane. Small vibrations (of a few micron) of D W in the sound wave field have been observed earlier in Y I G [11]. In the given study both the gigantic (up to 200 Ixm) vibrations of D W and D W disintegration have been observed for the first time in the intense sound wave field. The investigation was performed on samples with the typical dimensions 3 × 5 x 0.06 mm. But only constant gradient magnetic field and a small contracting pressure for stabilized domain structure may be used. D W moved under sound wave formed by the longitudinal piezotransducer using the technique described in section 2. High speed photography was used too. U n d e r the action of the sound wave upon the DW, we observe the D W vibrations at the sound frequency relative to the equilibrium position. The D W movement was stationary, and the D W structure was stable only when the deformation in the sound wave was small. When the sound amplitude has reached some value u 0 dependent on the external contracting pressure, the dynamic O P T is observed, i.e. the 180 ° D W disintegrates into two DWs, producing a domain with the magnetization direction close to 90 °. The D W serves as a nucleus of a new 90 o phase. Fig. 4 represents the photographs of D W in the sound wave field in iron borate at large sound amplitudes. The dependence of the 90 ° domain width on the sound wave amplitude is shown on fig. 5. With a further growth of the sound amplitude, the D W movement becomes essentially nonstationary, which does not allow an unambiguous
Fig. 4. Photograph of the 90 ° domain arisen after 180 o DW disintegration by sound wave in iron borate. The initial static wide of DW was 2 i~m.
correlation between the sound amplitude and the 90 o domain width. The increasing pressure results in decreasing amplitude of the D W vibrations and in the increase of the sound amplitude threshold value at which the D W disintegration
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pressure added to the external one in the sound wave increases the velocity value V 1. Increasing external pressure results in the increasing of the velocity 1/1 and, hence, of the threshold field HI and u 0 and also in diminishing mobility and vibration amplitude of the DW.
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Fig. 6. Time dependence of the position ~7 of the D W when it moves in the sound wave field. T h e time-scale of disintegration of the 180 ° D W is touch on the picture.
begins. At p > 109 d y n / c m 2 no D W disintegration has been observed. If the sound amplitude u < u0, then the shape of the curves of the D W vibrations corresponds exactly to the acoustic impulse shape. Figure 6 gives the temporal dependence of the position ~7 of the D W when it moves in the sound wave field. The time-scale of disintegration of the 180 o D W is shown on the picture. The physics of this p h e n o m e n o n appears to be as follows. The action of sound upon the D W is equivalent to the action of a certain effective magnetic field Heff in which the vibrational movement of the D W takes place, since Herf -e x p ( - i w l ) , 00 being the wave frequency. The order of Herf value can be estimated by formula (1). For u = 30 ,~, Heff -- 1 0 e , which coincides in the order of magnitude with the static field forming the domain structure. The D W vibration amplitude increases with increasing lieu. Once Heff has reached a certain critical value H 1 which is determined by the external pressure, the O P T takes place with the disintegration of the 180 ° DW, just as in the case of the D W movement under the action of the magnetic field. Similarly, to give a more complete explanation, it is necessary to take into account the dynamic of process, i.e. VDw(H). From fig. 6 one can see that the D W disintegration begins upon reaching the velocity V 1 in the straight direction. When the D W moves in the reverse direction, no disintegration occurs, since in this case the contracting
When a sound propagates through an iron borate plate, the elastic deformation in the wave may prove to be sufficient to change the sign of the anisotropy constant in the plate, which results in a dynamic O P T induced by the wave. In this case the process of the sound wave propagation becomes essentially nonlinear [1]. This section describes a slow nonlinear magneto-acoustic wave which is unstable with respect to the transverse perturbation of the wave front and has a selforganization structure of the transverse corrugation type [12]. Earlier such types of waves have not been observed and treated theoretically. The investigation was performed on iron borate samples in the form of plates using the technique described above but with some modifications. The sample was in a monodomain state created by the external magnetic field of -- 1 0 e directed perpendicularly to the sound wave vector k, hence, at first the magnetization directed perpendicularly to k too. With a small voltage applied to the transducer, the uniformity of the polarization plane rotation was maintained throughout the sample. When the sound wave amplitude was ~ 10 A, we observed the magnetoelastic wave. It will be noted that the amplitude is about one order more then when disintegration of the 180 ° D W is observed. The sample was divided into a series of alternating light and dark bands with the period Apt--500 Ixm and with diffuse and smeared boundaries that were perpendicular to the wave vector k (the longitudinal structure). The velocity of the propagation of the wave V - (1.8 + 0.2) k m / s and it could be determined by two different methods: 1) by measuring Air and recalculating the velocity in terms of the known frequency; and 2) by direct m e a s u r e m e n t of the distance traversed by the wave in a given o
M.V. Chetkin, V..V. Lykov / Nonlinear waves of magnetization in iron borate
331
interval of time, which can be achieved by using the high speed photography. Both methods yield the same value for the wave velocity. Within the accuracy of measurement this velocity was independent of the sample thickness ranging from 30 to 85 Ixm, the sound amplitude and the magnetic fields. With increasing voltage at the transducer and with the sound wave amplitude of = 20 A, additional domain structures emerged in alternating bands, the magnetization direction being along parallel to the wave vector (transverse corrugation). The boundaries between dynamic domains became well defined (fig. 7a). The structures had characteristic bending periodic fronts (fig. 7b). The magnetic domains moved as a whole at the same velocity (1.8 + 0.2) km/s. If the sound divergence is not large, the structures have a nearly o
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regular arrangement (fig. 8). If there is a divergence in the wave, the structures "scatter" too (fig. 9). A further increase of the sound amplitude does not result in any significant change in
332
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tude terminates these processes and has no appreciable effect on the dynamic domain and on the character of their propagation. The application of the magnetic field H in the direction of the wave propagation (H[]k) leads to the enlargement of dark regions and to the reduction of light ones in the transverse domain structure; the reversal of the field direction gives the opposite effects, the corrugation period A± being practically the same (fig. 10). In the magnetic field H_I_k the corrugation period is reduced (figs. 8 and 9). The dependence A± ( H ) in the field H _1_k is shown in fig. 11. The period is
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t0 20 Fig. 11. The dependence of the transverse corrugation period )t ± of magnetoelastic wave on the magnetic field H _L k.
M.V. Chetkin, 14.I4. Lykov / Nonlinear waves of magnetization in iron borate
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gradually reduced with increasing field, the minimum recorded value A± = 80 l~m. Concurrently with the reduction of the corrugation period in the field H ± k, the domain length in the longitudinal direction with respect to the vector direction can be reduced (fig. 9). Thus, in the dark regions of the domain structure the magnetization is close to the direction of the wave propagation, in the light regions the magnetization has the opposite direction (fig. 12). The increase of the magnetic field, both parallel and perpendicular to k, above 40 Oe resulted in a complete disappearance of any structures in the sample, i.e. as a result, the crystal was uniformly magnetized. The external pressure increased the amplitude of the structure formation onset sound. The closure of the magnetic flux at the wave front proceeds through the domain that is in front of the structure. Yet, it is not clear how a magnetic flux closes behind the structure. Perhaps, other layers of the sample take part in this process, namely, the layers that are parallel to the basal plane and are separated by Bloch domain wails. It is also possible that in the nonequilibrium structure there is no flux closure behind the structure.
333
The physics of the phenomenon observed appears to be as follows. The values of the anisotropy constant in the basal plane of FeBO 3 is small and in practice this value is determined by the external pressure only. A low ( < 1 Oe) magnetic field H shifts the DWs and the sample becomes a monodomain and M I I H ( M - magnetization). At first, k ± M. In the wave local contraction region the deformation is directed so that it enhances the magnitude of the induced anisotropy and no changes in the domain structure occur. Conversely, in the region of stretching, the deformation in the wave is directed oppositely to the existing anisotropy. If the sound amplitude is high enough, then in this region the sign of the anisotropy constant is changed because of the value of the anisotropy constant being determined by the sound wave amplitude, i.e. the dynamic OPT takes place. As a result, the magnetic moments in the sound wave stretching region turn out to be aligned along the direction of the wave propagation, and in the contraction region the magnetic moments are aligned perpendicularly to the direction of the wave propagation as well as in the wave propagating in EPAFM examined in ref. [13]. The sound wave propagating near OPT is essentially nonlinear, even at small amplitudes. At large sound amplitudes it becomes unstable with respect to the transverse perturbation of its front (fig. 7b). The principal physical reason for this instability seems to be the same as in the case of the soliton propagation in EPAFM, namely: a strong dispersion of sound in the sample [1]. Therefore, with a small modulation in the transverse coordinate, the regions with a smaller amplitude will outstrip those with a larger amplitude. As a result, the instability of the self-focusing type is created [14]. The perturbation evolution and the formation of the ultimate structure must be affected by the demagnetization field. In this was the self-organization of the dynamic domain structure proceeds. It is worth noting that the velocity of the nonlinear magnetoelastic wave is exceedingly small; it is 5 times smaller than that of the longitudinal sound in an unlimited sample [3] and, as our measurements have shown, it is at least 3
334
M.V. Chetkin, V.V. Lykoe / Nonlinear waves of magnet&ation in iron borate
times smaller than that of the sound in a thin iron borate plate ( = 6.1 km / s ) at the above-mentioned frequency. This wave is likely to originate due to the Lamb bending waves. In this case, however, it is necessary to account for the absence of any structure in the second layer of the sample at another half-wave of the sound. To conclude this section it should be noted that the observation and investigation of nonlinear magnetoelastic waves was made possible as a result of the magneto-optical visualization of the elastic wave in FeBO 3 plate performed in the experiment, which is undoubtedly of practical importance as well.
constant sign and consequently dynamic orientational phase transitions take place.
Acknowledgements T h e authors are grateful to V.N. Seleznev and M.B. Strugatskii who kindly supplied the crystals to be studied and to A.K. Zvezdin, V.G. Shavrov, V.L. Preobrazhenskii and A.F. Kabychenkov for valuable and fruitful discussions of the results obtained.
References 6. Conclusion All the phenomena in FeBO3, described above, are determined by a strong magnetoelastic bond against the background of a low, practically vanishing anisotropy field in the plane. When a DW is moved by both external magnetic field and sound wave there is a certain critical velocity V1 dependent on the external pressure which determines the DW dynamic. On reaching the velocity V~ the DW disintegrates into two 90 ° DWs (to be exact, not 180 ° ones). In other words, when the DW reaches the velocity I/'1, the dynamic spin-reorientation phase transition (OPT) takes place. The DW stationary movement is possible only at a velocity less than a certain velocity V~. Note, in both case the DW was a nucleus for a new phase. But, when the very intensive acoustic wave propagates through the sample, the nucleus is not required for origin of dynamic domain structure. Here, we have the self-organization of magnetic structure on magnetoelastic wave. In all these cases a change of the anisotropy
[1] V.I. Ozhogin and V.L. Preobrazhenskii, Sov. Phys. Usp. 155 (1988) 593. [2] E.A. Turov and V.G. Shavrov, Soy. Phys. Usp. 140 (1983) 439. [3] D. Diehl, W. Jantz, J. Nalang and W. Wettling, Current Topics Mater. Sei. 1 (1988) 1. [4] M.V. Chetkin, A.K. Zvezdin, A.F. Popkov, S.V. Gomonov, V.B. Smirnov and Yu.N. Kyrbatova, JETP (USSR) 96 (1988) 269. [5] A.E. Kolesnikov, Akusticheskie Izmerenija, Leningrad (1983) 255. [6] G.BI Scott, J. Phys. D 7 (1974) 1574. [7] P.D. Kim and D.Dh. Hvan, Soy. Fiz. Tverd. Tela 24 (]982) 230O. [8] M.V. Chetkin and V.D. Tereschenko, Soy. Krystallogr. 33 (1988) 1311. [9] M.V. Chetkin, V.V. Lykov and V.D. Tereschenko Soy. Fiz. Tverd. Tela 32 (1990) 939. [10] A.K. Zvezdin, V.V. Kostyuchenko and A.A. Mykhin, preprint, Institute of Phys., Acad. Sci., Moscow (1983) 209. [11] V.K. Vlasko-Vlasov, V.I. Nikitenko and O.A. Tikhomirov, J. Magn. Magn. Mater. 75 (1988) 383. [12] M.V. Chetkin and V.V. Lykov, Soy. JETP Lett. 52 (1990) 863. [13] A.P. Kabychenkov and V.G. Shavrov, Acta, Phys. Pol. 4 (1988) 531. [14] S.K. Turitzin and G.E. Falkovich, Soy. JETP 1 (1985) 258.
Ad"
Nonlinear waves of magnetization in iron borate M.V. C h e t k i n a n d V.V. Lykov Department of Physics, Moscow State University, Moscow 119899, USSR Received 8 May 1991
A new type of magnetoelasticwaves in an easy-planeferromagneticwith the transverse corrugation type of self-organization of a structure whose period depends on the external magnetic field, has been observed and magneto-optically visualized. The dynamicsof the domain wall in iron borate and its behavior in the magnetic and intense sound wave field have been studied. The possibility of a dynamic orientational transition under both magnetic and sound field with the destruction of the domain wall upon reaching the critical velocitydepending on the external pressure has been shown.
1. Introduction
2. Experimental technique
Waves of magnetization in weak ferromagnetics (WFM) have received widespread attention in recent years, because, on the one hand, WFM are promising materials for modern microelectronics, primarily in data storage and processing systems, and, on the other hand, the propagation of these waves is associated with a great number of physical, especially nonlinear, phenomena [1,2]. The presence of a magnetoelastic bond comparable with the anisotropy energy in easy-plane antiferromagnetics (EPAFM) is responsible for the nonlinear phenomena there. The elastic waves in E P A F M become nonlinear even at low amplitudes and their propagation may be accompanied by dynamic orientation phase transitions (OPT). Iron borate FeBO 3 is a transparent easy-plane WFM with a strong magnetoelastic bond against the background of a low, practically vanishing anisotropy field in the plane, less than 0.01 Oe [3]. FeBO 3 is very suitable to study various nonlinear magnetoelastic interactions using, primarily, magneto-optic methods. This paper is devoted to the study of dynamics (section 3) of domain walls, their behavior in a strong acoustic field (section 4) and to the investigation of the observed slow nonlinear magnetoelastic wave with self-organization of the transverse corrugation type (section 5).
The investigation of nonlinear waves of magnetization in iron borate was performed in samples prepared as plates with typical dimensions 3-5 mm and 20-100 ~ m thick. A single N6el domain wall (DW) could be produced by the gradient magnetic field of = 70 O e / c m and the external contracting pressure applied in the sample plane, whose value was controlled and varied from 0 to 2 x 109 d y n / c m 2. The D W movement was realized either under the action of the impulse driving magnetic field H with the time-rise of 6 - 8 ns or under the action of a sound wave. If the pressure, the driving magnetic and gradient field are uniform, then a strictly two-domain structure is realized in the sample and there are no domain nuclei at the sample ends. The observation was performed using the Faraday effect which is strongly influenced by the birefrigence in the plate tilted at an angle of several degrees with respect to the horizontal axis (to the vertical one for the study of magnetoelastic waves too) in order to ensure a projection of the magnetization in the direction of the light propagation. Due to the birefrigence effect, the polarization plane rotation was about 1 ° in the domain contrast and less than 0.5 o in the D W contrast and during the propagation of the magnetoelastic wave.
M.V. Chetkin, V.V. Lykoc / Nonlinear wat,es of magnetization in iron borate
326
pie ( p = 0). The choice of the frequency was due to the relaxation time of the magnetic subsystem (--- 10 -8 s) so that it could trace the changes in the elastic subsystem. At the transducer the voltage in the impulse was as high as 1.6 kV, and the acoustic wave phase was rigidly not worse than 1-2 ns synchronized with the light impulse. To estimate the amplitude of deformation in the acoustic wave passing through the iron borate plate, the calibration was made on the basis of the reciprocity principle using two transducers which had in turn been calibrated by means of the electroinduction technique [5]. The method described allowed us to estimate the amplitude in the thin iron borate plate and to directly measure the velocity of the quasilongitudinal sound in iron borate at the given frequency. The value of the velocity was found to be (6.0_+ 0.2) k m / s . The deformation amplitude was as high as = 3 × 10 -s.
The recording was performed using either the high speed photography, when studying the magneto-acoustic interactions, or the double high speed photography, when studying the DW dynamics. In double high speed photography the sample was consequently illuminated by two light impulses of some nanoseconds optical delay. In both cases the oxazin-dye laser with the wavelength of 535 nm pumped by the T E A nitrogen laser, with a light impulse duration of 0.3 ns was used [4]. The image of a moving DW was registered by the double photography technique both in the domain contrast and in the DW contrast. The technique allowed the simultaneous recording of at least three shades of the colour grey, thus making it possible to control the D W structure in the real time scale with a simultaneous variation of its velocity. The sound wave was produced by a piezotransducer with a resonance frequency of 3.4 MHz, attached with phenil salicylate to a glass rod whose other end was pressed to the sample end. The rod provided an acoustic delay of 3-5 txs to separate the electromagnetic induction arising when the transducer was excited from ~he acoustic signal because of the sample being sensitive to the former. A thin layer of oil between the sample and the rod improved the acoustic contact and so it was possible to avoid pressing the sam-
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3. The domain wall dynamics in FeBO 3
The domain wall dynamics in iron borate has been studied earlier elsewhere [6-8]. It has been shown that the nature of the DW dynamics in FeBO 3 resembles that in yttrium orthoferrite [9]. But it has been noted that sometimes the DW disappeared [8], although the reasons and the
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Fig. 1. Dependence of the domain wall velocity on the magnetic field amplitude in FeBO 3. Contracting p r e s s u r e P = 2 0 x 10 s (O); 10 × 108 (-); 3 x 109 ( x ); 0.1 x 108 ( A ) d y n / c m 2.
M.H Chetkin, H E Lykov / Nonlinear waves of magnetization in iron borate c o n d i t i o n s u n d e r w h i c h this h a p p e n s r e m a i n e d unknown. Our study of the dependence of the DW velocity o n t h e d r i v i n g m a g n e t i c f i e l d a m p l i t u d e has
327
s h o w n t h a t t h e D W s t a t i o n a r y m o v e m e n t is possible o n l y at a v e l o c i t y less t h a n a c e r t a i n v e l o c i t y V 1 d e p e n d e n t o n t h e e x t e r n a l p r e s s u r e [12]. I f t h e D W m o v e m e n t is s t a t i o n a r y a n d t h e p r e s s u r e is
Fig. 2. Double high speed photographs of a moving domain walls in iron borate. The schematic magnetization direction structure of DW and domain contrast in this technique are shown. At the left side of figure the stationary moving DW is shown (Vow < V1). At the right side the moment of the DW disintegration is shown (Vow = Vl): under illumination of the first beam there is no disintegration, under illumination of the second beam the disintegration has already occurred. Optical delay was 11 ns~ The dark band on the photograph is a distance which DW pass during optical delay. The grey area is the 90 ° domain arisen after 180 o DW disintegration. ,
328
M.V. Chetkin, V.V. Lykov / Nonlinear waves of magnetization in iron borate
high enough, then on the dependence V(H) (fig. 1) one can observe constant D W velocities regions at of 4.4, 4.0, 2.7, 1.8 k m / s coinciding with the velocities of the transverse, surface and Lamb waves, respectively, and resulting from the interaction of DW with these waves. The nature of the latter should be investigated. On reaching the velocity V1 (in the field H1) , the D W disintegrates into two 90 o DWs (to be exact, not 180 ° ones), and the width of the resulting domain is determined by the difference between the driving and gradient field. In other words, when the DW reaches the velocity V1 in the region where H > H1, the dynamic spin-reorientation phase transition (OPT) with the magnetization direction there close to 90 ° takes place, the DW being a nucleus for a new phase. Fig. 2 represents a double high speed photography of a moving domain wall in iron borate. The schematic magnetization direction structure of DW and domain contrast in this technique are shown. At the left side the stationary moving DW is shown (VDw < 1/1). At the right side the moment of the O P T formation and D W ~lisintegration is shown (Vow = V1): under illumination of the first beam there is no disintegration, under illumination of the second beam the disintegration has already occurred. Optical delay was 11 ns. By varying the optical delay time it is possible to estimate the time needed for the O P T to be formed which seems to less than 1 ns (in the experiment the minimum optical delay time was 3 RS).
The dependence of the D W velocity Vl and mobility t~ on the external contraction pressure are given in fig. 3. At low pressures the mobility may be exceedingly high, reaching 200 × 103 c m / s Oe.
The p h e n o m e n o n observed can be explained qualitatively in the following way. The stability of the domain structure in the sample is determined by the external contracting pressure creating the uniaxial anisotropy in the sample plane or effective magnetic field Hp. If Hp has reached the
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the mechanical stress along the x-axis in the sample plane, at P > 0 it contracts the crystal, and at P < 0 it stretches the crystal, C66, B66 a r e the elastic and magnetoelastic constants, M 0 is the magnetization. Thus, for FeBO 3 at P = 1 × 108 d y n / c m 2 and at different values of B66 [3], H = 1 - 2 0 e ; taking into account the magnetoelastic renormalization C66, Hp* = 1 . 5 - 3 . 0 0 e . The antiferromagnetizm vector m changes its direction from mllx to rally, i.e. 180 o D W are formed. A the given pressure P > 0 a similar O P T can be performed in the reverse order in the field H. In our case the O P T takes place in the region where H < Hp*. Thus the field H 1 corresponds to the field Hp*, their values are apparently close. To specify the picture, one should take into account the dynamics process, i.e., the dependence V(H). This question was investigated in ref. [11], where the authors came to the conclusion that a magnetoelastic gap had to exist in spectrum of the D W velocity. The presence of the gap is determined by a strong magnetoelastic interaction and its width determined as the difference (S t - V1), where S t is a transverse sound velocity. Near S t the enhancement of the magnetoelastic energy may be so great that the effective constants of the induced anisotropy change their
329
M.V. Chetkin, V.V. Lykov / Nonlinear waves of magnetization in iron borate
signs, i.e. O P T takes place. As a result the D W disintegrates. Hence the gap value depends on the applied pressure. Fig. 3 gives the theoretical dependence V I ( P ) plotted in accordance with ref. [10].
4. The domain wall in the field of elastic waves in FeBO 3
The D W can be moved under the action of not only magnetic field but effective field of sound wave too This section discusses the D W interaction in thin iron borate plates with the sound wave propagating in the basal plane. Small vibrations (of a few micron) of D W in the sound wave field have been observed earlier in Y I G [11]. In the given study both the gigantic (up to 200 Ixm) vibrations of D W and D W disintegration have been observed for the first time in the intense sound wave field. The investigation was performed on samples with the typical dimensions 3 × 5 x 0.06 mm. But only constant gradient magnetic field and a small contracting pressure for stabilized domain structure may be used. D W moved under sound wave formed by the longitudinal piezotransducer using the technique described in section 2. High speed photography was used too. U n d e r the action of the sound wave upon the DW, we observe the D W vibrations at the sound frequency relative to the equilibrium position. The D W movement was stationary, and the D W structure was stable only when the deformation in the sound wave was small. When the sound amplitude has reached some value u 0 dependent on the external contracting pressure, the dynamic O P T is observed, i.e. the 180 ° D W disintegrates into two DWs, producing a domain with the magnetization direction close to 90 °. The D W serves as a nucleus of a new 90 o phase. Fig. 4 represents the photographs of D W in the sound wave field in iron borate at large sound amplitudes. The dependence of the 90 ° domain width on the sound wave amplitude is shown on fig. 5. With a further growth of the sound amplitude, the D W movement becomes essentially nonstationary, which does not allow an unambiguous
Fig. 4. Photograph of the 90 ° domain arisen after 180 o DW disintegration by sound wave in iron borate. The initial static wide of DW was 2 i~m.
correlation between the sound amplitude and the 90 o domain width. The increasing pressure results in decreasing amplitude of the D W vibrations and in the increase of the sound amplitude threshold value at which the D W disintegration
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pressure added to the external one in the sound wave increases the velocity value V 1. Increasing external pressure results in the increasing of the velocity 1/1 and, hence, of the threshold field HI and u 0 and also in diminishing mobility and vibration amplitude of the DW.
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5. N o n l i n e a r m a g n e t o e l a s t i c waves in iron borate
Fig. 6. Time dependence of the position ~7 of the D W when it moves in the sound wave field. T h e time-scale of disintegration of the 180 ° D W is touch on the picture.
begins. At p > 109 d y n / c m 2 no D W disintegration has been observed. If the sound amplitude u < u0, then the shape of the curves of the D W vibrations corresponds exactly to the acoustic impulse shape. Figure 6 gives the temporal dependence of the position ~7 of the D W when it moves in the sound wave field. The time-scale of disintegration of the 180 o D W is shown on the picture. The physics of this p h e n o m e n o n appears to be as follows. The action of sound upon the D W is equivalent to the action of a certain effective magnetic field Heff in which the vibrational movement of the D W takes place, since Herf -e x p ( - i w l ) , 00 being the wave frequency. The order of Herf value can be estimated by formula (1). For u = 30 ,~, Heff -- 1 0 e , which coincides in the order of magnitude with the static field forming the domain structure. The D W vibration amplitude increases with increasing lieu. Once Heff has reached a certain critical value H 1 which is determined by the external pressure, the O P T takes place with the disintegration of the 180 ° DW, just as in the case of the D W movement under the action of the magnetic field. Similarly, to give a more complete explanation, it is necessary to take into account the dynamic of process, i.e. VDw(H). From fig. 6 one can see that the D W disintegration begins upon reaching the velocity V 1 in the straight direction. When the D W moves in the reverse direction, no disintegration occurs, since in this case the contracting
When a sound propagates through an iron borate plate, the elastic deformation in the wave may prove to be sufficient to change the sign of the anisotropy constant in the plate, which results in a dynamic O P T induced by the wave. In this case the process of the sound wave propagation becomes essentially nonlinear [1]. This section describes a slow nonlinear magneto-acoustic wave which is unstable with respect to the transverse perturbation of the wave front and has a selforganization structure of the transverse corrugation type [12]. Earlier such types of waves have not been observed and treated theoretically. The investigation was performed on iron borate samples in the form of plates using the technique described above but with some modifications. The sample was in a monodomain state created by the external magnetic field of -- 1 0 e directed perpendicularly to the sound wave vector k, hence, at first the magnetization directed perpendicularly to k too. With a small voltage applied to the transducer, the uniformity of the polarization plane rotation was maintained throughout the sample. When the sound wave amplitude was ~ 10 A, we observed the magnetoelastic wave. It will be noted that the amplitude is about one order more then when disintegration of the 180 ° D W is observed. The sample was divided into a series of alternating light and dark bands with the period Apt--500 Ixm and with diffuse and smeared boundaries that were perpendicular to the wave vector k (the longitudinal structure). The velocity of the propagation of the wave V - (1.8 + 0.2) k m / s and it could be determined by two different methods: 1) by measuring Air and recalculating the velocity in terms of the known frequency; and 2) by direct m e a s u r e m e n t of the distance traversed by the wave in a given o
M.V. Chetkin, V..V. Lykov / Nonlinear waves of magnetization in iron borate
331
interval of time, which can be achieved by using the high speed photography. Both methods yield the same value for the wave velocity. Within the accuracy of measurement this velocity was independent of the sample thickness ranging from 30 to 85 Ixm, the sound amplitude and the magnetic fields. With increasing voltage at the transducer and with the sound wave amplitude of = 20 A, additional domain structures emerged in alternating bands, the magnetization direction being along parallel to the wave vector (transverse corrugation). The boundaries between dynamic domains became well defined (fig. 7a). The structures had characteristic bending periodic fronts (fig. 7b). The magnetic domains moved as a whole at the same velocity (1.8 + 0.2) km/s. If the sound divergence is not large, the structures have a nearly o
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regular arrangement (fig. 8). If there is a divergence in the wave, the structures "scatter" too (fig. 9). A further increase of the sound amplitude does not result in any significant change in
332
M. 1/.. Chetkin, 14.V.. Lykov / Nonlinear waves o f magnetization in iron borate
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tude terminates these processes and has no appreciable effect on the dynamic domain and on the character of their propagation. The application of the magnetic field H in the direction of the wave propagation (H[]k) leads to the enlargement of dark regions and to the reduction of light ones in the transverse domain structure; the reversal of the field direction gives the opposite effects, the corrugation period A± being practically the same (fig. 10). In the magnetic field H_I_k the corrugation period is reduced (figs. 8 and 9). The dependence A± ( H ) in the field H _1_k is shown in fig. 11. The period is
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t0 20 Fig. 11. The dependence of the transverse corrugation period )t ± of magnetoelastic wave on the magnetic field H _L k.
M.V. Chetkin, 14.I4. Lykov / Nonlinear waves of magnetization in iron borate
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gradually reduced with increasing field, the minimum recorded value A± = 80 l~m. Concurrently with the reduction of the corrugation period in the field H ± k, the domain length in the longitudinal direction with respect to the vector direction can be reduced (fig. 9). Thus, in the dark regions of the domain structure the magnetization is close to the direction of the wave propagation, in the light regions the magnetization has the opposite direction (fig. 12). The increase of the magnetic field, both parallel and perpendicular to k, above 40 Oe resulted in a complete disappearance of any structures in the sample, i.e. as a result, the crystal was uniformly magnetized. The external pressure increased the amplitude of the structure formation onset sound. The closure of the magnetic flux at the wave front proceeds through the domain that is in front of the structure. Yet, it is not clear how a magnetic flux closes behind the structure. Perhaps, other layers of the sample take part in this process, namely, the layers that are parallel to the basal plane and are separated by Bloch domain wails. It is also possible that in the nonequilibrium structure there is no flux closure behind the structure.
333
The physics of the phenomenon observed appears to be as follows. The values of the anisotropy constant in the basal plane of FeBO 3 is small and in practice this value is determined by the external pressure only. A low ( < 1 Oe) magnetic field H shifts the DWs and the sample becomes a monodomain and M I I H ( M - magnetization). At first, k ± M. In the wave local contraction region the deformation is directed so that it enhances the magnitude of the induced anisotropy and no changes in the domain structure occur. Conversely, in the region of stretching, the deformation in the wave is directed oppositely to the existing anisotropy. If the sound amplitude is high enough, then in this region the sign of the anisotropy constant is changed because of the value of the anisotropy constant being determined by the sound wave amplitude, i.e. the dynamic OPT takes place. As a result, the magnetic moments in the sound wave stretching region turn out to be aligned along the direction of the wave propagation, and in the contraction region the magnetic moments are aligned perpendicularly to the direction of the wave propagation as well as in the wave propagating in EPAFM examined in ref. [13]. The sound wave propagating near OPT is essentially nonlinear, even at small amplitudes. At large sound amplitudes it becomes unstable with respect to the transverse perturbation of its front (fig. 7b). The principal physical reason for this instability seems to be the same as in the case of the soliton propagation in EPAFM, namely: a strong dispersion of sound in the sample [1]. Therefore, with a small modulation in the transverse coordinate, the regions with a smaller amplitude will outstrip those with a larger amplitude. As a result, the instability of the self-focusing type is created [14]. The perturbation evolution and the formation of the ultimate structure must be affected by the demagnetization field. In this was the self-organization of the dynamic domain structure proceeds. It is worth noting that the velocity of the nonlinear magnetoelastic wave is exceedingly small; it is 5 times smaller than that of the longitudinal sound in an unlimited sample [3] and, as our measurements have shown, it is at least 3
334
M.V. Chetkin, V.V. Lykoe / Nonlinear waves of magnet&ation in iron borate
times smaller than that of the sound in a thin iron borate plate ( = 6.1 km / s ) at the above-mentioned frequency. This wave is likely to originate due to the Lamb bending waves. In this case, however, it is necessary to account for the absence of any structure in the second layer of the sample at another half-wave of the sound. To conclude this section it should be noted that the observation and investigation of nonlinear magnetoelastic waves was made possible as a result of the magneto-optical visualization of the elastic wave in FeBO 3 plate performed in the experiment, which is undoubtedly of practical importance as well.
constant sign and consequently dynamic orientational phase transitions take place.
Acknowledgements T h e authors are grateful to V.N. Seleznev and M.B. Strugatskii who kindly supplied the crystals to be studied and to A.K. Zvezdin, V.G. Shavrov, V.L. Preobrazhenskii and A.F. Kabychenkov for valuable and fruitful discussions of the results obtained.
References 6. Conclusion All the phenomena in FeBO3, described above, are determined by a strong magnetoelastic bond against the background of a low, practically vanishing anisotropy field in the plane. When a DW is moved by both external magnetic field and sound wave there is a certain critical velocity V1 dependent on the external pressure which determines the DW dynamic. On reaching the velocity V~ the DW disintegrates into two 90 ° DWs (to be exact, not 180 ° ones). In other words, when the DW reaches the velocity I/'1, the dynamic spin-reorientation phase transition (OPT) takes place. The DW stationary movement is possible only at a velocity less than a certain velocity V~. Note, in both case the DW was a nucleus for a new phase. But, when the very intensive acoustic wave propagates through the sample, the nucleus is not required for origin of dynamic domain structure. Here, we have the self-organization of magnetic structure on magnetoelastic wave. In all these cases a change of the anisotropy
[1] V.I. Ozhogin and V.L. Preobrazhenskii, Sov. Phys. Usp. 155 (1988) 593. [2] E.A. Turov and V.G. Shavrov, Soy. Phys. Usp. 140 (1983) 439. [3] D. Diehl, W. Jantz, J. Nalang and W. Wettling, Current Topics Mater. Sei. 1 (1988) 1. [4] M.V. Chetkin, A.K. Zvezdin, A.F. Popkov, S.V. Gomonov, V.B. Smirnov and Yu.N. Kyrbatova, JETP (USSR) 96 (1988) 269. [5] A.E. Kolesnikov, Akusticheskie Izmerenija, Leningrad (1983) 255. [6] G.BI Scott, J. Phys. D 7 (1974) 1574. [7] P.D. Kim and D.Dh. Hvan, Soy. Fiz. Tverd. Tela 24 (]982) 230O. [8] M.V. Chetkin and V.D. Tereschenko, Soy. Krystallogr. 33 (1988) 1311. [9] M.V. Chetkin, V.V. Lykov and V.D. Tereschenko Soy. Fiz. Tverd. Tela 32 (1990) 939. [10] A.K. Zvezdin, V.V. Kostyuchenko and A.A. Mykhin, preprint, Institute of Phys., Acad. Sci., Moscow (1983) 209. [11] V.K. Vlasko-Vlasov, V.I. Nikitenko and O.A. Tikhomirov, J. Magn. Magn. Mater. 75 (1988) 383. [12] M.V. Chetkin and V.V. Lykov, Soy. JETP Lett. 52 (1990) 863. [13] A.P. Kabychenkov and V.G. Shavrov, Acta, Phys. Pol. 4 (1988) 531. [14] S.K. Turitzin and G.E. Falkovich, Soy. JETP 1 (1985) 258.