- Email: [email protected]
Magneto-optic imaging of domain walls in ferrimagnetic garnet films
ARTICLE IN PRESS
Physica B 398 (2007) 476–479 www.elsevier.com/locate/physb
Magneto-optic imaging of domain walls in ferrimagnetic garnet films H. Ferraria,b,, V. Bekerisa,b, T.H. Johansenc a
Physics Department, FCEyN, UBA, Buenos Aires, Argentina b CONICET, Argentina c Department of Physics, The University of Oslo, Norway
Abstract Magneto-optic (MO) imaging is based on Faraday rotation of a linearly polarized incident light beam illuminating a sensitive MO layer (MOL) placed in close contact to the sample. For in-plane magnetized layers of Lu3xBixFe5yGayO12 ferrimagnetic garnet films, zig-zag domain formation occurs whenever the sample stray parallel field component, HJ, changes sign. Considering the anisotropy, exchange and magnetostatic energies in the Ne´el tails, and the contribution of an applied magnetic field, it is possible to describe the zigzag walls that separate domains with opposite in-plane magnetization. The size of the walls decreases with the spatial derivative of HJ. We studied the evolution of these domains as we steadily forced the change in sign of HJ to shorter length scales, from hundreds to a few microns. We describe the samples used to control the change in sign of HJ at the MOL plane, and we analyze the images that evolve from zig-zag walls to much more complex closed domain structures. r 2007 Elsevier B.V. All rights reserved. Keywords: Magneto-optics; Zig-zag domain walls
1. Introduction Magneto-optical (MO) imaging provides a powerful tool for the direct observation of magnetic flux distributions. The method combines relatively high spatial resolution (1 mm) and magnetic sensitivity (0.2 G) with short measuring times (oms) and large imaging areas (up to 10 10 mm2) [1]. In samples where no significant MO Faraday effect has been observed, it is necessary to use magneto-optical layers placed close to the sample surface as the field-sensing element. The visualization technique is based on the Faraday rotation of a polarized light beam illuminating an MO layer (MOL) placed directly on top of the sample surface. The light passes through the layer, is reflected at a thin mirror coating deposited on its back face, and passes a second time through the sensitive film, thus doubling the Faraday rotation angle, yF. For normal light incidence, the rotation angle grows with the magnitude of the local Corresponding author. Physics Department, FCEyN, UBA, Buenos Aires, Argentina. Tel.: +54 11 4576 3300; fax: +54 11 4576 3357. E-mail address: [email protected] (H. Ferrari).
0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.05.015
perpendicular magnetic field. By using a pair of polarizers in an optical microscope, one can directly visualize and quantify the field distribution across the sample area. The main requirements for the magneto-optical active materials are wide temperature operating ranges for which the Faraday rotation coefficient, i.e., the rotation angle per unit magnetic field is large, negligible hysteresis effects, and also absence of disturbing magnetic domains. Active layers with in-plane easy magnetization direction [2] have significantly improved the MO imaging technique [3]. Here, a magnetic field normal to the garnet plane rotates the magnetization vector out of the plane and leads to a Faraday rotation for normal light incidence. The main objective of the present work is to study the evolution of the zig-zag domain walls observed in an MOL. In each domain, in plane MOL magnetization Ms points in the direction of the parallel field component, By, generated by the sample. Any change of sign in By tends to create zigzag domain walls in the MOL, and these domains appear superimposed on the image of the sample’s normal field component. To study these domain walls, it is necessary to control the parallel field component at the MOL plane, and
ARTICLE IN PRESS H. Ferrari et al. / Physica B 398 (2007) 476–479
particularly its change in sign. For this purpose, we have used periodically magnetized audio tapes as samples [4]. The tapes were magnetized using sawtooth audio waveforms of different frequencies, with the well-known tape recording technique [5]. The tape stray magnetic field components have been calculated analytically elsewhere [6], at the MOL location. Images were obtained for tapes magnetized at different frequencies and the evolution of domain walls is investigated. The paper is organized as follows. Section 2 describes the experimental array. The model for the formation of domain walls and the calculated magnetic fields are presented in Section 3. Results and discussions are addressed in Section 4. 2. Experimental The recorded signal is a direct spatial reproduction of the original temporal signal. Sawtooth wave functions were computer generated and recorded on commercial audio tapes. We used 200, 500 and 1000 Hz audiosignals, which, for tapes with linear velocity v ¼ 15/8 in/s, during recording produced samples magnetized with spatial periodicity l ¼ 0.238, 0.095 and 0.048 mm. The MOL we have used in the present work is a Bidoped garnet film, Lu3xBixFe5yGayO12 with x0.5 and y0.7 with in-plane magnetization [7]. The indicator film was deposited to a thickness of 4.5 mm by liquid-phase epitaxial growth on the (1 0 0) oriented gadolinium gallium garnet substrate. A thin (120 nm) Au layer was evaporated onto the film in order to reflect the incident light and thus to provide a double Faraday rotation of the light beam. To image each sample, the pair polarizer analyzer was set slightly out of crossing so that perpendicular fields of opposite polarity were distinguished in the MO image, i.e., a medium gray color represents zero field. This angle was fixed to y0 ¼ 41710% for all samples. The air gap between the sample and the MOL indicator was minimized. In the calculations, the finite distance between the sample surface
477
and the MOL was taken into account, i.e., the magnetic field is calculated at a distance of 1/100l, where l is modulation period. A standard Olympus BX60M microscope was used with 10 magnification, and a Roper Scientific CoolSnap cf camera recorded the images, which were transferred to a computer for processing.
3. Modeling The tape was modeled as an infinite strip of width w/2pxpw/2 and thickness s/2pzps/2 (see inset in Fig. 1) where the in-plane magnetization is My(y). For the purpose of this work, we will take the limit s ¼ 0. To calculate H at the sensitive layer location, we made use of the Maxwell equation for a source with magnetization and without free current: r H ¼ 0 and r H ¼ 4pr M. The tape magnetization M ¼ My(y)y was
Fig. 2. Schematic pffiffiffiffiffiffiffiffiffiffi magnetization at the FGF plane. 2L is the width of the wall, and a sin y is the width of the Ne´el transition region.
Fig. 1. Inset: tape geometry: (a) sawtooth waveform tape magnetization My(y); (b) z-field component; and (c) y-field component generated by the tape at the FGF plane.
ARTICLE IN PRESS 478
H. Ferrari et al. / Physica B 398 (2007) 476–479
assumed periodic with period l, with amplitude m0 and particularly in this work a sawtooth waveform is examined. Calculating in the usual way, the volumetric charge density related to the magnetization variation, and as source of the magnetic potential, fM. The field H is given by H ¼ rfM. One arrives at the results plotted in Fig. 1 for both magnetic field components, Hz (panel b) and Hy (panel c), perpendicular and parallel to the strip, respectively [6]. In the present work, we will refer to the role of Hy generated by the magnetic tape in the formation of domain walls in the MOL. We observe that Hy changes sign at y/l ¼ 7n/2 very sharply (smoothly) for n even (odd) (n integer; see Fig. 1c). As will be shown below, this determines the size of the domain walls (L) at the MOL as Ms changes sign. We turn now to the expression for the domain wall energy in the MOL. When the component of Ms normal to a domain wall is not continuous, the wall becomes magnetically charged. Therefore, zig-zag walls occur rather than straight walls. Fig. 2 shows a zig-zag wall and the direction of Ms. The shape of this type of wall results from the competition between the magnetostatic energy associated with the charge and the anisotropy energy associated with the deviations of Ms from the easy axis. Other contributions are the exchange interactions and in the case of an applied magnetic field, the Zeeman term. The geometrical characteristics of the zig-zag walls are the vertex angle, 2y, and the zig-zag amplitude, 2L. We shall show evidence that L is sensitive to the rate of change in sign of Hy. The expression for the wall energy E per unit length is: E ¼ EN+EM+EA+EZ, where EN stands for Ne´el wall, EM, EA and EZ for magnetostatic, anisotropic and Zeeman energy per unit length, respectively. All terms in Eq. (1) were calculated in Ref. [8] except the Zeeman term that was evaluated assuming that Hy is linear in y, Hy ¼ AHy
inside the wall: pffiffiffiffiffiffiffi 2L 4 4pDM s pA 2 2 2 þ DKLy þ E ¼ 8M s D 3 2 ln y S 3 2 tan y ða Þ a pffiffiffiffiffiffiffiffiffiffi L , AH M s y sin y
ð1Þ
where Ms is the MOL saturation magnetization modulus, 2L is the width of the wall, 2D is the MOL thickness, K is the in-plane uniaxial anisotropy energy volume density, A is the exchange energy per unit length, a is the Ne´el tail distance and S is the tape width. When the energy E is minimized with respect to L, one obtains for L the following expression: L¼
16M 2s D2 . ð4=3ÞDKy þ AH M s tan y ða2 =yÞ 2
(2)
4. Results and discussion Fig. 3a–c shows MO images, for f ¼ 200, 500 and 1000 Hz, where dark vertical lines are the underlying images of the perpendicular field Hz generated by the tape sawtooth waveform. We will not concentrate on these lines here, but on the triangular domains that appear in the garnet itself. These arise from the change in sign in Hy. The size 2L of the domains depends on dHy/dy at points where Hy ¼ 0 (compare with Fig. 1c). In our simplified model, 2L depends on AH, as shown in Eq. (2), and large (small) domains correspond to small (large) AH, i.e., a slow (steep) change in sign. Due to the first term in the denominator in Eq. (2), as frequency is increased, domain walls become closer, and when adjacent walls get in contact, closed domains appear. In Fig. 3d–f, we show the edges of domains of Fig. 3a–c. Interestingly, 2y is almost independent of the rate of change in Hy, and the vertices of the larger domains seem to touch the smaller ones to the left due to some
Fig. 3. Magneto-optic images corresponding to tape-recorded sawtooth waveforms of (a) 200, (b) 500, and (c) 1000 Hz. Domain contours of MOI for (d) 200, (e) 500, and (f) 1000 Hz. Some lines are defects of the MOL.
ARTICLE IN PRESS H. Ferrari et al. / Physica B 398 (2007) 476–479
misalignment, creating a closed structure. Closed domains occur when frequency is high enough to connect adjacent walls. The walls curve to match, as is shown in Fig. 3c and 3f, generating an increase in Zeeman energy outside the closed domains, where the y component of MS points antiparallel to Hy in large areas. Measuring L vs. f will lead, in future work, to determine the relevant garnet constants. References [1] Ch. Jooss, J. Albrecht, H. Kuhn, S. Leonhardt, H. Kronmu¨ller, Rep. Prog. Phys. 65 (2002) 651.
479
[2] V.K. Vlasko-Vlasov, L.A. Dorosinskii, M.V. Indenbom, V.I. Nikitenko, Yu.A. Ossip’yan, A.A. Poluyanskii, Sov. J. Low Temp. Phys. 17 (1991) 72. [3] L.E. Helseth, R.W. Hansen, E.I. Ilyashenko, M. Baziljevich, T.H. Johansen, Phys. Rev. B 64 (2001) 63. [4] H. Ferrari, V. Bekeris, T.H. Johansen, Physica B 354 (2004) 365. [5] D. Wei, H.N. Bertram, F. Jeffers, Magn., IEEE Trans. 30 (1994) 2739. [6] H. Ferrari, M. Thibeault, V. Bekeris, T.H. Johanbsen, J. Magn. Magn. Mater. 313 (2007) 98. [7] T.H. Johansen, M. Baziljevich, H. Bratsberg, Y. Galperin, P.E. Lindelof, Y. Shen, P. Vase, Phys. Rev. B 54 (1996) 16264. [8] M.J. Freiser, IBM J. Res. Dev. 23 (1979) 330.
Physica B 398 (2007) 476–479 www.elsevier.com/locate/physb
Magneto-optic imaging of domain walls in ferrimagnetic garnet films H. Ferraria,b,, V. Bekerisa,b, T.H. Johansenc a
Physics Department, FCEyN, UBA, Buenos Aires, Argentina b CONICET, Argentina c Department of Physics, The University of Oslo, Norway
Abstract Magneto-optic (MO) imaging is based on Faraday rotation of a linearly polarized incident light beam illuminating a sensitive MO layer (MOL) placed in close contact to the sample. For in-plane magnetized layers of Lu3xBixFe5yGayO12 ferrimagnetic garnet films, zig-zag domain formation occurs whenever the sample stray parallel field component, HJ, changes sign. Considering the anisotropy, exchange and magnetostatic energies in the Ne´el tails, and the contribution of an applied magnetic field, it is possible to describe the zigzag walls that separate domains with opposite in-plane magnetization. The size of the walls decreases with the spatial derivative of HJ. We studied the evolution of these domains as we steadily forced the change in sign of HJ to shorter length scales, from hundreds to a few microns. We describe the samples used to control the change in sign of HJ at the MOL plane, and we analyze the images that evolve from zig-zag walls to much more complex closed domain structures. r 2007 Elsevier B.V. All rights reserved. Keywords: Magneto-optics; Zig-zag domain walls
1. Introduction Magneto-optical (MO) imaging provides a powerful tool for the direct observation of magnetic flux distributions. The method combines relatively high spatial resolution (1 mm) and magnetic sensitivity (0.2 G) with short measuring times (oms) and large imaging areas (up to 10 10 mm2) [1]. In samples where no significant MO Faraday effect has been observed, it is necessary to use magneto-optical layers placed close to the sample surface as the field-sensing element. The visualization technique is based on the Faraday rotation of a polarized light beam illuminating an MO layer (MOL) placed directly on top of the sample surface. The light passes through the layer, is reflected at a thin mirror coating deposited on its back face, and passes a second time through the sensitive film, thus doubling the Faraday rotation angle, yF. For normal light incidence, the rotation angle grows with the magnitude of the local Corresponding author. Physics Department, FCEyN, UBA, Buenos Aires, Argentina. Tel.: +54 11 4576 3300; fax: +54 11 4576 3357. E-mail address: [email protected] (H. Ferrari).
0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.05.015
perpendicular magnetic field. By using a pair of polarizers in an optical microscope, one can directly visualize and quantify the field distribution across the sample area. The main requirements for the magneto-optical active materials are wide temperature operating ranges for which the Faraday rotation coefficient, i.e., the rotation angle per unit magnetic field is large, negligible hysteresis effects, and also absence of disturbing magnetic domains. Active layers with in-plane easy magnetization direction [2] have significantly improved the MO imaging technique [3]. Here, a magnetic field normal to the garnet plane rotates the magnetization vector out of the plane and leads to a Faraday rotation for normal light incidence. The main objective of the present work is to study the evolution of the zig-zag domain walls observed in an MOL. In each domain, in plane MOL magnetization Ms points in the direction of the parallel field component, By, generated by the sample. Any change of sign in By tends to create zigzag domain walls in the MOL, and these domains appear superimposed on the image of the sample’s normal field component. To study these domain walls, it is necessary to control the parallel field component at the MOL plane, and
ARTICLE IN PRESS H. Ferrari et al. / Physica B 398 (2007) 476–479
particularly its change in sign. For this purpose, we have used periodically magnetized audio tapes as samples [4]. The tapes were magnetized using sawtooth audio waveforms of different frequencies, with the well-known tape recording technique [5]. The tape stray magnetic field components have been calculated analytically elsewhere [6], at the MOL location. Images were obtained for tapes magnetized at different frequencies and the evolution of domain walls is investigated. The paper is organized as follows. Section 2 describes the experimental array. The model for the formation of domain walls and the calculated magnetic fields are presented in Section 3. Results and discussions are addressed in Section 4. 2. Experimental The recorded signal is a direct spatial reproduction of the original temporal signal. Sawtooth wave functions were computer generated and recorded on commercial audio tapes. We used 200, 500 and 1000 Hz audiosignals, which, for tapes with linear velocity v ¼ 15/8 in/s, during recording produced samples magnetized with spatial periodicity l ¼ 0.238, 0.095 and 0.048 mm. The MOL we have used in the present work is a Bidoped garnet film, Lu3xBixFe5yGayO12 with x0.5 and y0.7 with in-plane magnetization [7]. The indicator film was deposited to a thickness of 4.5 mm by liquid-phase epitaxial growth on the (1 0 0) oriented gadolinium gallium garnet substrate. A thin (120 nm) Au layer was evaporated onto the film in order to reflect the incident light and thus to provide a double Faraday rotation of the light beam. To image each sample, the pair polarizer analyzer was set slightly out of crossing so that perpendicular fields of opposite polarity were distinguished in the MO image, i.e., a medium gray color represents zero field. This angle was fixed to y0 ¼ 41710% for all samples. The air gap between the sample and the MOL indicator was minimized. In the calculations, the finite distance between the sample surface
477
and the MOL was taken into account, i.e., the magnetic field is calculated at a distance of 1/100l, where l is modulation period. A standard Olympus BX60M microscope was used with 10 magnification, and a Roper Scientific CoolSnap cf camera recorded the images, which were transferred to a computer for processing.
3. Modeling The tape was modeled as an infinite strip of width w/2pxpw/2 and thickness s/2pzps/2 (see inset in Fig. 1) where the in-plane magnetization is My(y). For the purpose of this work, we will take the limit s ¼ 0. To calculate H at the sensitive layer location, we made use of the Maxwell equation for a source with magnetization and without free current: r H ¼ 0 and r H ¼ 4pr M. The tape magnetization M ¼ My(y)y was
Fig. 2. Schematic pffiffiffiffiffiffiffiffiffiffi magnetization at the FGF plane. 2L is the width of the wall, and a sin y is the width of the Ne´el transition region.
Fig. 1. Inset: tape geometry: (a) sawtooth waveform tape magnetization My(y); (b) z-field component; and (c) y-field component generated by the tape at the FGF plane.
ARTICLE IN PRESS 478
H. Ferrari et al. / Physica B 398 (2007) 476–479
assumed periodic with period l, with amplitude m0 and particularly in this work a sawtooth waveform is examined. Calculating in the usual way, the volumetric charge density related to the magnetization variation, and as source of the magnetic potential, fM. The field H is given by H ¼ rfM. One arrives at the results plotted in Fig. 1 for both magnetic field components, Hz (panel b) and Hy (panel c), perpendicular and parallel to the strip, respectively [6]. In the present work, we will refer to the role of Hy generated by the magnetic tape in the formation of domain walls in the MOL. We observe that Hy changes sign at y/l ¼ 7n/2 very sharply (smoothly) for n even (odd) (n integer; see Fig. 1c). As will be shown below, this determines the size of the domain walls (L) at the MOL as Ms changes sign. We turn now to the expression for the domain wall energy in the MOL. When the component of Ms normal to a domain wall is not continuous, the wall becomes magnetically charged. Therefore, zig-zag walls occur rather than straight walls. Fig. 2 shows a zig-zag wall and the direction of Ms. The shape of this type of wall results from the competition between the magnetostatic energy associated with the charge and the anisotropy energy associated with the deviations of Ms from the easy axis. Other contributions are the exchange interactions and in the case of an applied magnetic field, the Zeeman term. The geometrical characteristics of the zig-zag walls are the vertex angle, 2y, and the zig-zag amplitude, 2L. We shall show evidence that L is sensitive to the rate of change in sign of Hy. The expression for the wall energy E per unit length is: E ¼ EN+EM+EA+EZ, where EN stands for Ne´el wall, EM, EA and EZ for magnetostatic, anisotropic and Zeeman energy per unit length, respectively. All terms in Eq. (1) were calculated in Ref. [8] except the Zeeman term that was evaluated assuming that Hy is linear in y, Hy ¼ AHy
inside the wall: pffiffiffiffiffiffiffi 2L 4 4pDM s pA 2 2 2 þ DKLy þ E ¼ 8M s D 3 2 ln y S 3 2 tan y ða Þ a pffiffiffiffiffiffiffiffiffiffi L , AH M s y sin y
ð1Þ
where Ms is the MOL saturation magnetization modulus, 2L is the width of the wall, 2D is the MOL thickness, K is the in-plane uniaxial anisotropy energy volume density, A is the exchange energy per unit length, a is the Ne´el tail distance and S is the tape width. When the energy E is minimized with respect to L, one obtains for L the following expression: L¼
16M 2s D2 . ð4=3ÞDKy þ AH M s tan y ða2 =yÞ 2
(2)
4. Results and discussion Fig. 3a–c shows MO images, for f ¼ 200, 500 and 1000 Hz, where dark vertical lines are the underlying images of the perpendicular field Hz generated by the tape sawtooth waveform. We will not concentrate on these lines here, but on the triangular domains that appear in the garnet itself. These arise from the change in sign in Hy. The size 2L of the domains depends on dHy/dy at points where Hy ¼ 0 (compare with Fig. 1c). In our simplified model, 2L depends on AH, as shown in Eq. (2), and large (small) domains correspond to small (large) AH, i.e., a slow (steep) change in sign. Due to the first term in the denominator in Eq. (2), as frequency is increased, domain walls become closer, and when adjacent walls get in contact, closed domains appear. In Fig. 3d–f, we show the edges of domains of Fig. 3a–c. Interestingly, 2y is almost independent of the rate of change in Hy, and the vertices of the larger domains seem to touch the smaller ones to the left due to some
Fig. 3. Magneto-optic images corresponding to tape-recorded sawtooth waveforms of (a) 200, (b) 500, and (c) 1000 Hz. Domain contours of MOI for (d) 200, (e) 500, and (f) 1000 Hz. Some lines are defects of the MOL.
ARTICLE IN PRESS H. Ferrari et al. / Physica B 398 (2007) 476–479
misalignment, creating a closed structure. Closed domains occur when frequency is high enough to connect adjacent walls. The walls curve to match, as is shown in Fig. 3c and 3f, generating an increase in Zeeman energy outside the closed domains, where the y component of MS points antiparallel to Hy in large areas. Measuring L vs. f will lead, in future work, to determine the relevant garnet constants. References [1] Ch. Jooss, J. Albrecht, H. Kuhn, S. Leonhardt, H. Kronmu¨ller, Rep. Prog. Phys. 65 (2002) 651.
479
[2] V.K. Vlasko-Vlasov, L.A. Dorosinskii, M.V. Indenbom, V.I. Nikitenko, Yu.A. Ossip’yan, A.A. Poluyanskii, Sov. J. Low Temp. Phys. 17 (1991) 72. [3] L.E. Helseth, R.W. Hansen, E.I. Ilyashenko, M. Baziljevich, T.H. Johansen, Phys. Rev. B 64 (2001) 63. [4] H. Ferrari, V. Bekeris, T.H. Johansen, Physica B 354 (2004) 365. [5] D. Wei, H.N. Bertram, F. Jeffers, Magn., IEEE Trans. 30 (1994) 2739. [6] H. Ferrari, M. Thibeault, V. Bekeris, T.H. Johanbsen, J. Magn. Magn. Mater. 313 (2007) 98. [7] T.H. Johansen, M. Baziljevich, H. Bratsberg, Y. Galperin, P.E. Lindelof, Y. Shen, P. Vase, Phys. Rev. B 54 (1996) 16264. [8] M.J. Freiser, IBM J. Res. Dev. 23 (1979) 330.