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AF bilayers
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Journal of Magnetism and Magnetic Materials 285 (2005) 79–87 www.elsevier.com/locate/jmmm
Hysteretic resonance frequencies and magnetization reversal in exchange biased polycrystalline F/AF bilayers D. Spenato, S.P. Pogossian Laboratoire de Magne´tisme de Bretagne, UBO/CNRS/FRE 2697, 6 avenue Le Gorgeu, CS 93837, 29238 BREST Cedex 3, France Received 30 December 2003; received in revised form 9 June 2004; accepted 8 July 2004 Available online 7 August 2004
Abstract Polycrystalline ferromagnetic(F)/antiferromagnetic(AF) exchange biased bilayers of NiFe/MnNi have been grown by RF sputtering. The measured hysteresis loops are shifted and exhibit a strong asymmetry along the easy axis of the F film. The dynamic properties have been investigated through Complex Permeability frequency Spectra measurements in the 100 MHz–6 GHz range. The dispersion relation of the resonance frequency of imaginary part of the permeability shows hysteretic behavior when measured in presence of a static magnetic field applied along the easy axis of the F layer. A model for magnetization reversal in such bilayer based on the exchange energy development into Fourier-like power series is used. We show that it is possible to fit, with a set of parameters, the experimental asymmetric shape of hysteresis loop of the exchange biased bilayer. The same set of fit parameters are used to calculate the dispersion relation. The hysteretic behavior of the experimental resonance frequencies was found to be in agreement with theoretical results. Such a model reconciles irreversible (such as hysteresis loop) and reversible (such as AC permeability) measurements of the exchange bias. r 2004 Published by Elsevier B.V. PACS: 75.60.Ej; 75.30.Gw; 75.30.Et; 76.50.+g; 75.40.Gb Keywords: Exchange bias; Thin magnetic films; Magnetization reversal; Spin dynamics; Ferromagnetic resonance
1. Introduction Despite many theoretical and experimental studies in the last decade on an old [1] phenomenon known as ‘‘exchange biasing’’, a full underCorresponding author. Fax: +33-298-017-395.
E-mail addresses: [email protected] (D. Spenato), [email protected] (S.P. Pogossian). 0304-8853/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.jmmm.2004.07.018
standing of the exchange anisotropy at the interface between antiferromagnets (AF) and ferromagnets (F) is not yet clearly understood. One of the intriguing particularities of the exchange bias field is that different measurement techniques seem to lead to different values of the latter. In the hysteresis loops measurements, the unidirectional exchange anisotropy J E is given by H s M S tF ; where H s (also called ‘‘exchange field’’) is
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the displacement of the hysteresis loop, and M S and tF are respectively the saturation magnetization and the thickness of the F film. Hysteresis loop measurements involve irreversible switching of the magnetization of the F film and introduce magnetic structure modifications in the F film as well as in the AF spin structure, i.e., the AF spins at the interface may rotate with the F magnetization due to strong exchange coupling leading to a domain wall creation in the AF layer [2]. Other techniques called ‘‘reversible’’ have also been used to study exchange biased bilayers since the magnetization of the F film during the measurement is scarcely perturbed. The first reversible technique was based on the measurement of the anisotropic magnetoresistance (AMR) on Co/CoO bilayers as a function of the angle between the exchange field and the rotating external magnetic field [3]. The exchange anisotropy is obtained from the fit of the angular-dependent resistivity curve to a theoretical model based on the assumption of fixed AF moments aligned along the easy direction of F film. This technique gave exchange anisotropy energies two times larger than that obtained from hysteresis loop measurements. The second reversible technique is the AC susceptibility measurements [4] of the exchange anisotropy energy (J AC ). That gives values of the exchange anisotropy energies 10 times larger than the one obtained with hysteresis loop measurements. Brillouin light scattering (BLS) have also been used to measure the unidirectional anisotropy in exchanged biased NiFe/FeMn and Fe/FeF2 bilayers [5]. In the latter case, the measured spin-wave frequencies were fitted to a theoretical model by using free anisotropy energy density with a perpendicular, unidirectional, uniaxial and fourfold symmetries. The authors have found that the shift of the hysteresis loop was 25% smaller than the one measured with BLS. One of the most popular reversible technique used in the study of exchange biased bilayers is the ferromagnetic resonance (FMR). Waksmann [6] and his co-workers used FMR to study the exchange anisotropy in thin epitaxial films of NiFe/MnFeNi bilayers. Since the 1960s, FMR in exchange bias systems have been studied both experimentally[7–17] and theoretically [14,17–20]. Some authors have observed that
the unidirectional exchange anisotropy was about 20% less than the loop shift measured via hysteresis loop [7] or magnetoresistance [14] techniques. The difference was explained by introducing a ‘‘rotatable anisotropy’’ related to the domain configuration in the AF layer [14]. Based on the domain wall model proposed by Mauri et al. [2], Xi et al. [21] have derived expressions relating the exchange anisotropy measured by various techniques like FMR, AC susceptibility and hysteresis loop (HL). The authors have shown that the exchange anisotropy measured with FMR (J FMR ) should be greater than that measured with HL (J HL ), whereas the experimental results show J FMR oJ HL : They attributed it to the granular nature of the films. Measurements performed by Rodriguez-Sua´rez et al. [22] on different kind of exchange biased samples have shown that J FMR oJ AC oJ HL : The authors assumed that unstable AF grains, which couple to the external field via the F magnetization, may exhibit a distribution that is dependent on the applied magnetic field. The technique of measuring complex permeability frequency spectra (CPS) of exchange biased bilayers [23,24] has drawn less attention. It gives exchange field values several times higher than the ones obtained with HL measurements [24]. Thus, interface coupling models that reconcile these measurements are very important since only a few authors have carried out spin dynamic calculations associated with magnetization reversal in the case of strong asymmetry observed in the hysteresis loops measurements. We propose a theoretical model based on the development of the energy functional for the exchange field in Fourier-like powers series that accounts for the asymmetric shape of the measured HL along the easy axis of F film in order to reconcile these measurements. The dispersion relation of the resonance frequency versus the applied magnetic field is calculated for the forward and reverse field direction using the fit parameters of the static hysteresis loop. The resonance frequencies show drops at the spin–flop transitions. The measured resonance frequencies using CPS can be accounted for in the frame of our model. Hence we show that it is possible, within the same model, to reconcile
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the irreversible and reversible magnetic measurements in exchange biased bilayers.
2. Experimental procedure Ni81 Fe19 (52 nm)/Mn46 Ni54 (80 nm) bilayers were grown on Corning glass substrate. After deposition, samples were annealed in a magnetic field of 80 kA/m, aligned with the easy axis of the F film, at 300 1C for 5 h to induce the exchange field. The details of the sample preparation and structure analysis have been published elsewhere [25] and may be summarised as follows: a (1 1 1) texture for the Ni81 Fe19 was favored and the annealed Mn46 Ni54 films deposited on the underlying Ni81 Fe19 film were found to have a (1 1 1) texture with a FCT structure after annealing [25]. The magnetic properties such as the saturation magnetization M s and the coercivity H c were obtained from magnetization loops (VSM) measured at room temperature. The coercivity of the biased NiFe layer was defined by the half of the shifted M–H loop width. The CPS of the bilayers were measured from 30 MHz to 6 GHz using a broad band method based on the measurement of the reflection coefficient S11 of a single turn coil loaded by the film under test with a network analyser [26]. For the used frequencies and because of the small magnetic volume of the samples, a quasi-TEM approximation can be assumed even when inhomogeneous or anisotropic films are involved. Among the deposited samples, we have chosen the ones with low values of the exchange biasing and coercive fields essential for the detection of a signal in the CPS measurements. All the results presented in this paper have been carried out on the same sample. An example of the measurement of the CPS measured on a NiFe/ MnNi bilayer along the direction perpendicular to the F layer easy axis is presented in Fig. 1. The asgrown state of the MnNi is nonmagnetic FCC structure and there is no magnetic interaction between the NiFe and the MnNi layer. We have measured the CPS of the as-grown bilayers along the direction perpendicular to the easy axis of the F film, the results are presented in Fig. 1. This spectrum is characterized by the uniform mode of
Fig. 1. Real (solid line) and imaginary part (dotted line) of the complex permeability spectra of a Ni81 Fe19 ð52 nmÞ= Mn46 Ni54 ð80 nmÞ bilayer: as-grown and annealed at 300 1C for 5 h (H s ¼ 950 A=m).
resonance of a NiFe single layer with the characteristics of spin rotation damping processes: the resonance frequency of m00 at about 700 MHz and the low frequency m0 of about 2200[27]. In order to achieve an antiferromagnetic tetragonal (FCT) state, a high temperature annealing was performed. For the annealed samples, the level of the real part of the permeability m0 ð0Þ at low frequency decreases and the roll off frequency increases (Fig. 1). It can be seen that the imaginary part of the complex permeability m00 shows a lower resonance peak, a higher resonance frequency f res (2.5 GHz) and a wider resonance peak with the increase of the exchange field. The broadening of the high-frequency response may be due to a distribution of local exchange fields because of the polycrystallinity of the samples. It is well known [28,24] that the theoretical value of m0 ð0Þ at low frequency behaves like 1 þ ðM s =H eff ) and, as observed, the enhancement of the effective field due to exchange biasing leads to a reduction of the level of m0 ð0Þ: With respect to the magnetization precession, the exchange bias leads to the enhancement of the resonance frequency.
3. Model The magnetization reversal that occurs in exchange biased bilayers cannot be explained with
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a simple algebraic addition of an exchange field to the applied magnetic field H a because of the asymmetric shape of the hysteresis loop. In a previous paper [29] we have proposed a Fourierlike power series expansion of the exchange energy. In the model it is assumed that the F layer couples with AF-grains (or clusters of grains in which the spins behave coherently; let this be also called ‘‘grain’’ hereafter). The easy axis distribution of different grains of AF layer with respect to the magnetization direction of the F film leads us to the concept of mean easy axis direction of different AF grains. The exchange coupling between different grains, and the way they are influenced by F layer, may result to a deviation of the statistical mean easy axis of AF grains from that of F layer. We admit in our model, a uniform switching of the total magnetization of the F layer. The measurement of the transverse component of the magnetization is a method to investigate magnetization reversal in F/AF bilayers [30]. We have measured both the transverse and parallel magnetization with respect to the direction of the applied magnetic field. The results are presented in Fig. 2 where the angle between the applied magnetic field and the easy axis of the F layer is denoted by a: In this figure, the measured values of the magnetization are normalized with respect to the value of the parallel saturation magnetization M S : It can be observed in Fig. 2(a) that in the case of a magnetic field applied along the easy axis (a ¼ 0) the transverse magnetization may reach 50% of the parallel saturation magnetization. The two maxima of the transverse magnetization correspond to the coercive fields. A strong asymmetry is observed in the transverse magnetization loop (Fig. 2(a)). This fact suggests that, in our samples, the magnetization reversal along the easy axis is complicated and should be considered as a mixture of magnetization rotation and domain-wall propagation. For a ¼ p=2 (see Fig. 2(c)) the transverse magnetization is almost equal to the parallel one, which is consistent with the magnetization reversal mechanism occurring by rotation. For the intermediate angles, the magnetization reversal can be considered nearly rotational (see, for example, Fig. 2(b)). In order to describe the magnetization reversal for all the
(a)
(b)
(c)
Fig. 2. Transverse and parallel magnetization for a Ni81 Fe19 ð52 nmÞ=Mn46 Ni54 ð80 nmÞ bilayer at (a) 0, (b) p=3; (c) p=2 of the applied field referred to the easy axis of the F layer.
angles by a unique model, we have naturally assumed that the magnetization reversal takes place by coherent rotation. Along the easy axis, the magnetization reversal may take place by rotation and the formation of complicated domain structures as shown by several authors[31,32]. By the way, this may explain the small deviation between the experimental and theoretical results as discussed later in this paper. The magnetic configuration of the F/AF bilayer is shown in Fig. 3. The total free magnetic energy per unit area of the exchange coupled F layer and AF layers can be written as follows [29]: E ¼ K F tF ð1 sin2 y cos2 jÞ H a M F tF sin y cosðj aÞ þ
M 2F cos2 ytF þ E e ðj b; yÞ; 2
ð1Þ
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The equilibrium position of the magnetization is determined (ye ; je ) from the total energy minimum: @E ¼ 0; @y
Fig. 3. Vector diagram for an exchange coupled F/AF bilayer submitted to an applied field H a :
where K F is the anisotropy constant of the F layer with its magnetization M F : As shown in Fig. 3, the polar and the azimuthal angles of the magnetization M F are denoted y and j: The first term represents the anisotropy energy per unit area of the F layer. The second term corresponds to the Zeeman energy per unit area of the F layer submitted to an external static field H a applied at an angle a relative to the anisotropy easy axis of the F layer. The third term represents the demagnetizing field energy per unit area due to the shape anisotropy of the thin film. The last term represents the exchange anisotropy energy per unit area that represents the F/AF coupling, where b is the angle of the AF statistical mean easy axis direction with respect to the F easy axis. This angle term may be related to the b-degree domain wall in one of the sublattice of the AF layer as proposed by Mauri et al. [2]. The total reduced energy in terms of effective fields can be written as: E red ¼
E HK ð1 sin2 y cos2 jÞ ¼ tF M F 2 M F cos2 y H a sin y cosðj aÞ þ 2 X n H n ½cosðj bÞ sin y ;
ð2Þ
n
where H K is the anisotropy field and the H n are the coefficients of the power series development.
@E ¼ 0: @j
ð3Þ
Based on the above model, the magnetization curves of the F/AF bilayer are obtained by numerical calculations. We have limited powers series by n ¼ 5 terms, since the fit to the shape of easy and hard axis hysteresis curves was rather good [29]. The fitting to the easy axis hysteresis loop can be improved better by taking into account more terms up to n ¼ 7; but in this case the hysteresis loop calculated for other aa0 directions of the applied magnetic field shows parasitic minor loops. Once the equilibrium position is determined, the resonance frequency is calculated by using the Smit formalism [33] that gives " 2 2 2 # o 1 @ 2 G @2 G @ G ¼ 2 2 ; ð4Þ 2 2 g @y@j M S sin ðye Þ @y @j where G is the free energy density E=tF where E is the total free magnetic energy per unit area of the exchange coupled F layer as used in Eq. (1). The energy second derivatives are calculated at the equilibrium position.
4. Results and discussion In a first step, we have fitted the magnetization loop along the easy axis to our model using Eq. (2). The comparison between theoretical and experimental loops are presented in Fig. 4(a). In that figure, a strong asymmetry between the left and right side of the measured hysteresis loop can be observed. The magnetization reversal on the left side of the loop exhibits a hard-axis-type slope while the reversal for the right-hand side of the loop occurs with a vertical branch of the loop. The model gives values expected either for the hysteresis loop shift (H s ¼ 950 A=m) or for the coercivity (H c ¼ 980 A=m), or for the strong asymmetry in the magnetization reversal.
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(a)
(b)
(c)
Fig. 4. Representative hysteresis loops for a Ni81 Fe19 ð52 nmÞ=Mn46 Ni54 ð80 nmÞ bilayer at (a) 0, (b) p=3; (c) p=2 of the applied field referred to the easy axis of the F layer. The experimental data are shown by scattered empty dots. The solid lines are the theoretical magnetization curves.
The values of fit parameters are b ¼ 0:064; H k ¼ 533 A=m; H 1 ¼ 1661 A=m; H2 ¼ 903 A=m; H 3 ¼ 616 A=m; H 4 ¼ 501 A=m; H 5 ¼ 599 A=m: The first term H k is about two times larger than the anisotropy field of the as-deposited NiFe layer (i.e. 360 A/m). The second term H 1 ; that may be related to the exchange biased field, is about two times larger the hysteresis loop shift. One can observe a distinct kink on the left side of the calculated magnetization loop, observed experimentally by Leighton et al. [34]. They attributed it to a two-stage magnetization reversal. According to Mewes et al. [35], the Stoner–Wolfarth model shows sharp edges which are rounded in the experiment for polycrystalline samples. This may explain the smooth variation of our experimental results. A non-negligible contribution of the domain wall propagation observed experimentally with the transverse magnetization may also
contribute to these small differences between the experimental data and the theoretical curve of the easy axis hysteresis loop. We have used the same set of fit parameters to calculate the magnetization curves for an applied field making different angles with respect to the F easy axis. The results for p=3 and p=2 are presented in Fig. 4(b) and (c) respectively. As observed in Fig. 4(b) and (c), the small deviation of experimental and theoretical curves is attributed to the initial dispersion of the as-deposited NiFe anisotropy revealed by a small coercivity measured along the hard axis. That may come from the aligning field in the sputtering chamber which lacks uniformity and may give rise an anisotropy dispersion in the F film. That dispersion was not taken into account in our model for two reasons. First, the lack of the exact distribution of aligning magnetic field and second, to avoid cumbersome calculations. Energy diagrams are often useful to explain the magnetization reversal in magnetic heterostructures. The formation of energy barriers is a useful way of understanding the process of magnetization reversal. Fig. 5 represents the three-dimensional plot of the reduced energy E red ðH a ; jÞ where j is the angle between the F easy axis and the in-plane magnetization vector, H a is the applied field. As one can see, there are two potential wells at j ¼ 0 and j ¼ p: When the applied field is positive the magnetization is in the j ¼ 0 well, and as the applied field is reduced the magnetization follows a path number 1 to minimize its energy and falls progressively into the j ¼ p well. When the applied field is reversed, the magnetization remains in the j ¼ p well, with the further increase of the applied field the magnetization returns brusquely to the original direction through a different path (number 2). The forward and return paths are schematically shown in Fig. 5 by arrows. That explains the asymmetry of the loops along the easy axis of F layer. We have measured the CPS of the annealed bilayer in presence of a static magnetic increasing and decreasing field applied along the easy axis as in the hysteresis loop measurement. The measured resonance frequencies of the imaginary part of the CPS have been plotted versus the applied field and
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(a)
Fig. 5. Three-dimensional plot of the energy as a function of the applied field and the in-plane magnetization (y ¼ p=2; a ¼ 0). The applied field is along the easy axis of the F layer and the position of the in-plane magnetization is referred by j:
the results are presented in Fig. 6(a). In order to compare the static and dynamic hysteresis loops, the magnetization hysteresis loop measured along the easy axis is shown in Fig. 6(b). For the annealed exchange biased sample, the dispersion relation split into two branches corresponding to the forward and reverse loop of the hysteresis curves, the resonance frequencies curves are hysteretic. This may be associated with the strong asymmetry observed in the hysteresis loop as presented in Fig. 6(b). In the first magnetization reversal (forward), the measured resonance frequency at first decreases and then, increases showing a minimum at 1.83 GHz for an applied field of about 0:9 kA/m. This minimum is not well pronounced. In the reverse loop the frequency decreases and shows a minimum at 1.83 GHz for an applied field of about 0:2 kA/m. At this field the resonance frequency jumps from 1.83 to 1.93 GHz. This latter minimum corresponds roughly to the coercive field in the reverse magnetization direction denoted as H 2 in Fig. 6(b).
(b) Fig. 6. (a) Resonance frequencies of m00 as a function of the applied field of an annealed Ni81 Fe19 ð52 nmÞ=Mn46 Ni54 ð80 nmÞ bilayer during an hysteresis loop sweep. Open circles: measured frequencies for the forward applied field, full triangles: measured frequencies for the reverse field. The solid lines are the calculated frequencies. (b) Open circles: forward branch of the experimental hysteresis loop, full triangles: reverse branch of the experimental hysteresis loop. The calculated hysteresis loop along the easy axis is drawn by a solid line.
This discontinuous jump of the resonance frequencies at H a ¼ H 2 corresponds to the change of the magnetization direction from one stable position aligned with the external field (j ¼ p) for jH a joH 2 to an other one (j ¼ 0) for jH a j4H 2 : In the forward loop, as can be seen in Fig. 6(b), the magnetization does not switch from one stable position (j ¼ 0) to another (j ¼ p) at a fixed value H 1 but rather rotates from one position to another in relatively larger interval of applied field dH a (see Fig. 6(b)). In this field interval, the resonance frequency increases continuously from 1.83 to 2.4 GHz, but these values stay lower than the one for the reverse branch in the same field interval.
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The experimental behavior of the magnetization with the forward field suggests a coherent rotation mechanism resulting to a non-negligible component of the magnetization along the hard axis which itself is along the applied AC field. This may explain the lower value of f res for the forward field in the interval dH a : Such kind of hysteretic behavior of the resonance frequencies have been recently observed in NiFe films with CPS measurements [36]. The authors observed that the jumps of the resonance frequencies appear at fields lower than the coercive field of their samples. This hysteretic behavior of the resonance frequency in exchange biased bilayers has been recently theoretically investigated by Stamps [37]. The author attributes this behavior to different effective fields governing the resonance frequencies for the forward and reverse field directions and obtains hysteretic resonance frequencies by including in energy, terms, that take into account the coupling to both sublattices of the AF and the formation of a planar domain wall. The method for calculating the bilayer resonance frequency is described as follows: for each applied static field along the easy axis, the equilibrium position of the magnetization is determined with the easy axis fit parameters obtained previously and the resonance frequency is calculated with the use of Eq. (4). The comparison between theoretical and experimental resonance frequencies are presented in Fig. 6(a). It can be seen that the theoretical resonance frequency curve fits well the measured one. The theoretical resonance dispersion spectrum shows two distinct abrupt spikes at forward fields as shown in Fig. 6(a) (solid line), one at 1:5 kA/m and the other at 2:4 kA/m. These two spikes correspond to two abrupt jumps in the theoretical forward hysteresis loop (Fig. 6(b), solid line). The experimental forward resonance curve shows a smooth and large minimum that corresponds to a smooth M-reversal, in the same interval (dH a ), as observed in the forward experimental hysteresis loop (Fig. 6(b), open circles). During the reverse loop, a small drop in the theoretical frequency dispersion spectrum appears a 0:2 kA/m and corresponds to the coercive field in the reverse magnetization direction denoted as H 2 in Fig. 6(b)
(full triangles). In many theoretical calculations for the resonance frequency [20,19] the asymmetry in the resonance frequency was not taken into account. Such a behavior of resonance spectrum was predicted by Kim [38] in a theoretical model. The author attributes the first drop to the rotation of the magnetization of the F layer and the second one to the depinning of the partial wall from the interface. We have measured a smooth variation of the resonance frequency for the forward loop instead of two distinct drops as theoretically predicted. That may be attributed to the polycrystallinity of our sample.
5. Conclusion In order to reconcile data on exchange anisotropy measured by irreversible and reversible techniques, a theoretical model based on the representation of the exchange energy by Fourier-like power series (proposed by us previously) has been used. We have shown that it is possible to fit both, the asymmetrical shape of the easy axis hysteresis loop, and the hysteretic frequency dispersion spectra of ferromagnetic resonance to the theoretical model. Our theoretical calculations predict the existence of two branches in the dispersion relation that agrees with the experimental observations. The model reconciles the reversible and irreversible measurements of the exchange bias.
Acknowledgements The authors thank J. Ben Youssef for samples preparation and A. Chevalier for his technical help.
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Hysteretic resonance frequencies and magnetization reversal in exchange biased polycrystalline F/AF bilayers D. Spenato, S.P. Pogossian Laboratoire de Magne´tisme de Bretagne, UBO/CNRS/FRE 2697, 6 avenue Le Gorgeu, CS 93837, 29238 BREST Cedex 3, France Received 30 December 2003; received in revised form 9 June 2004; accepted 8 July 2004 Available online 7 August 2004
Abstract Polycrystalline ferromagnetic(F)/antiferromagnetic(AF) exchange biased bilayers of NiFe/MnNi have been grown by RF sputtering. The measured hysteresis loops are shifted and exhibit a strong asymmetry along the easy axis of the F film. The dynamic properties have been investigated through Complex Permeability frequency Spectra measurements in the 100 MHz–6 GHz range. The dispersion relation of the resonance frequency of imaginary part of the permeability shows hysteretic behavior when measured in presence of a static magnetic field applied along the easy axis of the F layer. A model for magnetization reversal in such bilayer based on the exchange energy development into Fourier-like power series is used. We show that it is possible to fit, with a set of parameters, the experimental asymmetric shape of hysteresis loop of the exchange biased bilayer. The same set of fit parameters are used to calculate the dispersion relation. The hysteretic behavior of the experimental resonance frequencies was found to be in agreement with theoretical results. Such a model reconciles irreversible (such as hysteresis loop) and reversible (such as AC permeability) measurements of the exchange bias. r 2004 Published by Elsevier B.V. PACS: 75.60.Ej; 75.30.Gw; 75.30.Et; 76.50.+g; 75.40.Gb Keywords: Exchange bias; Thin magnetic films; Magnetization reversal; Spin dynamics; Ferromagnetic resonance
1. Introduction Despite many theoretical and experimental studies in the last decade on an old [1] phenomenon known as ‘‘exchange biasing’’, a full underCorresponding author. Fax: +33-298-017-395.
E-mail addresses: [email protected] (D. Spenato), [email protected] (S.P. Pogossian). 0304-8853/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.jmmm.2004.07.018
standing of the exchange anisotropy at the interface between antiferromagnets (AF) and ferromagnets (F) is not yet clearly understood. One of the intriguing particularities of the exchange bias field is that different measurement techniques seem to lead to different values of the latter. In the hysteresis loops measurements, the unidirectional exchange anisotropy J E is given by H s M S tF ; where H s (also called ‘‘exchange field’’) is
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the displacement of the hysteresis loop, and M S and tF are respectively the saturation magnetization and the thickness of the F film. Hysteresis loop measurements involve irreversible switching of the magnetization of the F film and introduce magnetic structure modifications in the F film as well as in the AF spin structure, i.e., the AF spins at the interface may rotate with the F magnetization due to strong exchange coupling leading to a domain wall creation in the AF layer [2]. Other techniques called ‘‘reversible’’ have also been used to study exchange biased bilayers since the magnetization of the F film during the measurement is scarcely perturbed. The first reversible technique was based on the measurement of the anisotropic magnetoresistance (AMR) on Co/CoO bilayers as a function of the angle between the exchange field and the rotating external magnetic field [3]. The exchange anisotropy is obtained from the fit of the angular-dependent resistivity curve to a theoretical model based on the assumption of fixed AF moments aligned along the easy direction of F film. This technique gave exchange anisotropy energies two times larger than that obtained from hysteresis loop measurements. The second reversible technique is the AC susceptibility measurements [4] of the exchange anisotropy energy (J AC ). That gives values of the exchange anisotropy energies 10 times larger than the one obtained with hysteresis loop measurements. Brillouin light scattering (BLS) have also been used to measure the unidirectional anisotropy in exchanged biased NiFe/FeMn and Fe/FeF2 bilayers [5]. In the latter case, the measured spin-wave frequencies were fitted to a theoretical model by using free anisotropy energy density with a perpendicular, unidirectional, uniaxial and fourfold symmetries. The authors have found that the shift of the hysteresis loop was 25% smaller than the one measured with BLS. One of the most popular reversible technique used in the study of exchange biased bilayers is the ferromagnetic resonance (FMR). Waksmann [6] and his co-workers used FMR to study the exchange anisotropy in thin epitaxial films of NiFe/MnFeNi bilayers. Since the 1960s, FMR in exchange bias systems have been studied both experimentally[7–17] and theoretically [14,17–20]. Some authors have observed that
the unidirectional exchange anisotropy was about 20% less than the loop shift measured via hysteresis loop [7] or magnetoresistance [14] techniques. The difference was explained by introducing a ‘‘rotatable anisotropy’’ related to the domain configuration in the AF layer [14]. Based on the domain wall model proposed by Mauri et al. [2], Xi et al. [21] have derived expressions relating the exchange anisotropy measured by various techniques like FMR, AC susceptibility and hysteresis loop (HL). The authors have shown that the exchange anisotropy measured with FMR (J FMR ) should be greater than that measured with HL (J HL ), whereas the experimental results show J FMR oJ HL : They attributed it to the granular nature of the films. Measurements performed by Rodriguez-Sua´rez et al. [22] on different kind of exchange biased samples have shown that J FMR oJ AC oJ HL : The authors assumed that unstable AF grains, which couple to the external field via the F magnetization, may exhibit a distribution that is dependent on the applied magnetic field. The technique of measuring complex permeability frequency spectra (CPS) of exchange biased bilayers [23,24] has drawn less attention. It gives exchange field values several times higher than the ones obtained with HL measurements [24]. Thus, interface coupling models that reconcile these measurements are very important since only a few authors have carried out spin dynamic calculations associated with magnetization reversal in the case of strong asymmetry observed in the hysteresis loops measurements. We propose a theoretical model based on the development of the energy functional for the exchange field in Fourier-like powers series that accounts for the asymmetric shape of the measured HL along the easy axis of F film in order to reconcile these measurements. The dispersion relation of the resonance frequency versus the applied magnetic field is calculated for the forward and reverse field direction using the fit parameters of the static hysteresis loop. The resonance frequencies show drops at the spin–flop transitions. The measured resonance frequencies using CPS can be accounted for in the frame of our model. Hence we show that it is possible, within the same model, to reconcile
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the irreversible and reversible magnetic measurements in exchange biased bilayers.
2. Experimental procedure Ni81 Fe19 (52 nm)/Mn46 Ni54 (80 nm) bilayers were grown on Corning glass substrate. After deposition, samples were annealed in a magnetic field of 80 kA/m, aligned with the easy axis of the F film, at 300 1C for 5 h to induce the exchange field. The details of the sample preparation and structure analysis have been published elsewhere [25] and may be summarised as follows: a (1 1 1) texture for the Ni81 Fe19 was favored and the annealed Mn46 Ni54 films deposited on the underlying Ni81 Fe19 film were found to have a (1 1 1) texture with a FCT structure after annealing [25]. The magnetic properties such as the saturation magnetization M s and the coercivity H c were obtained from magnetization loops (VSM) measured at room temperature. The coercivity of the biased NiFe layer was defined by the half of the shifted M–H loop width. The CPS of the bilayers were measured from 30 MHz to 6 GHz using a broad band method based on the measurement of the reflection coefficient S11 of a single turn coil loaded by the film under test with a network analyser [26]. For the used frequencies and because of the small magnetic volume of the samples, a quasi-TEM approximation can be assumed even when inhomogeneous or anisotropic films are involved. Among the deposited samples, we have chosen the ones with low values of the exchange biasing and coercive fields essential for the detection of a signal in the CPS measurements. All the results presented in this paper have been carried out on the same sample. An example of the measurement of the CPS measured on a NiFe/ MnNi bilayer along the direction perpendicular to the F layer easy axis is presented in Fig. 1. The asgrown state of the MnNi is nonmagnetic FCC structure and there is no magnetic interaction between the NiFe and the MnNi layer. We have measured the CPS of the as-grown bilayers along the direction perpendicular to the easy axis of the F film, the results are presented in Fig. 1. This spectrum is characterized by the uniform mode of
Fig. 1. Real (solid line) and imaginary part (dotted line) of the complex permeability spectra of a Ni81 Fe19 ð52 nmÞ= Mn46 Ni54 ð80 nmÞ bilayer: as-grown and annealed at 300 1C for 5 h (H s ¼ 950 A=m).
resonance of a NiFe single layer with the characteristics of spin rotation damping processes: the resonance frequency of m00 at about 700 MHz and the low frequency m0 of about 2200[27]. In order to achieve an antiferromagnetic tetragonal (FCT) state, a high temperature annealing was performed. For the annealed samples, the level of the real part of the permeability m0 ð0Þ at low frequency decreases and the roll off frequency increases (Fig. 1). It can be seen that the imaginary part of the complex permeability m00 shows a lower resonance peak, a higher resonance frequency f res (2.5 GHz) and a wider resonance peak with the increase of the exchange field. The broadening of the high-frequency response may be due to a distribution of local exchange fields because of the polycrystallinity of the samples. It is well known [28,24] that the theoretical value of m0 ð0Þ at low frequency behaves like 1 þ ðM s =H eff ) and, as observed, the enhancement of the effective field due to exchange biasing leads to a reduction of the level of m0 ð0Þ: With respect to the magnetization precession, the exchange bias leads to the enhancement of the resonance frequency.
3. Model The magnetization reversal that occurs in exchange biased bilayers cannot be explained with
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a simple algebraic addition of an exchange field to the applied magnetic field H a because of the asymmetric shape of the hysteresis loop. In a previous paper [29] we have proposed a Fourierlike power series expansion of the exchange energy. In the model it is assumed that the F layer couples with AF-grains (or clusters of grains in which the spins behave coherently; let this be also called ‘‘grain’’ hereafter). The easy axis distribution of different grains of AF layer with respect to the magnetization direction of the F film leads us to the concept of mean easy axis direction of different AF grains. The exchange coupling between different grains, and the way they are influenced by F layer, may result to a deviation of the statistical mean easy axis of AF grains from that of F layer. We admit in our model, a uniform switching of the total magnetization of the F layer. The measurement of the transverse component of the magnetization is a method to investigate magnetization reversal in F/AF bilayers [30]. We have measured both the transverse and parallel magnetization with respect to the direction of the applied magnetic field. The results are presented in Fig. 2 where the angle between the applied magnetic field and the easy axis of the F layer is denoted by a: In this figure, the measured values of the magnetization are normalized with respect to the value of the parallel saturation magnetization M S : It can be observed in Fig. 2(a) that in the case of a magnetic field applied along the easy axis (a ¼ 0) the transverse magnetization may reach 50% of the parallel saturation magnetization. The two maxima of the transverse magnetization correspond to the coercive fields. A strong asymmetry is observed in the transverse magnetization loop (Fig. 2(a)). This fact suggests that, in our samples, the magnetization reversal along the easy axis is complicated and should be considered as a mixture of magnetization rotation and domain-wall propagation. For a ¼ p=2 (see Fig. 2(c)) the transverse magnetization is almost equal to the parallel one, which is consistent with the magnetization reversal mechanism occurring by rotation. For the intermediate angles, the magnetization reversal can be considered nearly rotational (see, for example, Fig. 2(b)). In order to describe the magnetization reversal for all the
(a)
(b)
(c)
Fig. 2. Transverse and parallel magnetization for a Ni81 Fe19 ð52 nmÞ=Mn46 Ni54 ð80 nmÞ bilayer at (a) 0, (b) p=3; (c) p=2 of the applied field referred to the easy axis of the F layer.
angles by a unique model, we have naturally assumed that the magnetization reversal takes place by coherent rotation. Along the easy axis, the magnetization reversal may take place by rotation and the formation of complicated domain structures as shown by several authors[31,32]. By the way, this may explain the small deviation between the experimental and theoretical results as discussed later in this paper. The magnetic configuration of the F/AF bilayer is shown in Fig. 3. The total free magnetic energy per unit area of the exchange coupled F layer and AF layers can be written as follows [29]: E ¼ K F tF ð1 sin2 y cos2 jÞ H a M F tF sin y cosðj aÞ þ
M 2F cos2 ytF þ E e ðj b; yÞ; 2
ð1Þ
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The equilibrium position of the magnetization is determined (ye ; je ) from the total energy minimum: @E ¼ 0; @y
Fig. 3. Vector diagram for an exchange coupled F/AF bilayer submitted to an applied field H a :
where K F is the anisotropy constant of the F layer with its magnetization M F : As shown in Fig. 3, the polar and the azimuthal angles of the magnetization M F are denoted y and j: The first term represents the anisotropy energy per unit area of the F layer. The second term corresponds to the Zeeman energy per unit area of the F layer submitted to an external static field H a applied at an angle a relative to the anisotropy easy axis of the F layer. The third term represents the demagnetizing field energy per unit area due to the shape anisotropy of the thin film. The last term represents the exchange anisotropy energy per unit area that represents the F/AF coupling, where b is the angle of the AF statistical mean easy axis direction with respect to the F easy axis. This angle term may be related to the b-degree domain wall in one of the sublattice of the AF layer as proposed by Mauri et al. [2]. The total reduced energy in terms of effective fields can be written as: E red ¼
E HK ð1 sin2 y cos2 jÞ ¼ tF M F 2 M F cos2 y H a sin y cosðj aÞ þ 2 X n H n ½cosðj bÞ sin y ;
ð2Þ
n
where H K is the anisotropy field and the H n are the coefficients of the power series development.
@E ¼ 0: @j
ð3Þ
Based on the above model, the magnetization curves of the F/AF bilayer are obtained by numerical calculations. We have limited powers series by n ¼ 5 terms, since the fit to the shape of easy and hard axis hysteresis curves was rather good [29]. The fitting to the easy axis hysteresis loop can be improved better by taking into account more terms up to n ¼ 7; but in this case the hysteresis loop calculated for other aa0 directions of the applied magnetic field shows parasitic minor loops. Once the equilibrium position is determined, the resonance frequency is calculated by using the Smit formalism [33] that gives " 2 2 2 # o 1 @ 2 G @2 G @ G ¼ 2 2 ; ð4Þ 2 2 g @y@j M S sin ðye Þ @y @j where G is the free energy density E=tF where E is the total free magnetic energy per unit area of the exchange coupled F layer as used in Eq. (1). The energy second derivatives are calculated at the equilibrium position.
4. Results and discussion In a first step, we have fitted the magnetization loop along the easy axis to our model using Eq. (2). The comparison between theoretical and experimental loops are presented in Fig. 4(a). In that figure, a strong asymmetry between the left and right side of the measured hysteresis loop can be observed. The magnetization reversal on the left side of the loop exhibits a hard-axis-type slope while the reversal for the right-hand side of the loop occurs with a vertical branch of the loop. The model gives values expected either for the hysteresis loop shift (H s ¼ 950 A=m) or for the coercivity (H c ¼ 980 A=m), or for the strong asymmetry in the magnetization reversal.
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(a)
(b)
(c)
Fig. 4. Representative hysteresis loops for a Ni81 Fe19 ð52 nmÞ=Mn46 Ni54 ð80 nmÞ bilayer at (a) 0, (b) p=3; (c) p=2 of the applied field referred to the easy axis of the F layer. The experimental data are shown by scattered empty dots. The solid lines are the theoretical magnetization curves.
The values of fit parameters are b ¼ 0:064; H k ¼ 533 A=m; H 1 ¼ 1661 A=m; H2 ¼ 903 A=m; H 3 ¼ 616 A=m; H 4 ¼ 501 A=m; H 5 ¼ 599 A=m: The first term H k is about two times larger than the anisotropy field of the as-deposited NiFe layer (i.e. 360 A/m). The second term H 1 ; that may be related to the exchange biased field, is about two times larger the hysteresis loop shift. One can observe a distinct kink on the left side of the calculated magnetization loop, observed experimentally by Leighton et al. [34]. They attributed it to a two-stage magnetization reversal. According to Mewes et al. [35], the Stoner–Wolfarth model shows sharp edges which are rounded in the experiment for polycrystalline samples. This may explain the smooth variation of our experimental results. A non-negligible contribution of the domain wall propagation observed experimentally with the transverse magnetization may also
contribute to these small differences between the experimental data and the theoretical curve of the easy axis hysteresis loop. We have used the same set of fit parameters to calculate the magnetization curves for an applied field making different angles with respect to the F easy axis. The results for p=3 and p=2 are presented in Fig. 4(b) and (c) respectively. As observed in Fig. 4(b) and (c), the small deviation of experimental and theoretical curves is attributed to the initial dispersion of the as-deposited NiFe anisotropy revealed by a small coercivity measured along the hard axis. That may come from the aligning field in the sputtering chamber which lacks uniformity and may give rise an anisotropy dispersion in the F film. That dispersion was not taken into account in our model for two reasons. First, the lack of the exact distribution of aligning magnetic field and second, to avoid cumbersome calculations. Energy diagrams are often useful to explain the magnetization reversal in magnetic heterostructures. The formation of energy barriers is a useful way of understanding the process of magnetization reversal. Fig. 5 represents the three-dimensional plot of the reduced energy E red ðH a ; jÞ where j is the angle between the F easy axis and the in-plane magnetization vector, H a is the applied field. As one can see, there are two potential wells at j ¼ 0 and j ¼ p: When the applied field is positive the magnetization is in the j ¼ 0 well, and as the applied field is reduced the magnetization follows a path number 1 to minimize its energy and falls progressively into the j ¼ p well. When the applied field is reversed, the magnetization remains in the j ¼ p well, with the further increase of the applied field the magnetization returns brusquely to the original direction through a different path (number 2). The forward and return paths are schematically shown in Fig. 5 by arrows. That explains the asymmetry of the loops along the easy axis of F layer. We have measured the CPS of the annealed bilayer in presence of a static magnetic increasing and decreasing field applied along the easy axis as in the hysteresis loop measurement. The measured resonance frequencies of the imaginary part of the CPS have been plotted versus the applied field and
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(a)
Fig. 5. Three-dimensional plot of the energy as a function of the applied field and the in-plane magnetization (y ¼ p=2; a ¼ 0). The applied field is along the easy axis of the F layer and the position of the in-plane magnetization is referred by j:
the results are presented in Fig. 6(a). In order to compare the static and dynamic hysteresis loops, the magnetization hysteresis loop measured along the easy axis is shown in Fig. 6(b). For the annealed exchange biased sample, the dispersion relation split into two branches corresponding to the forward and reverse loop of the hysteresis curves, the resonance frequencies curves are hysteretic. This may be associated with the strong asymmetry observed in the hysteresis loop as presented in Fig. 6(b). In the first magnetization reversal (forward), the measured resonance frequency at first decreases and then, increases showing a minimum at 1.83 GHz for an applied field of about 0:9 kA/m. This minimum is not well pronounced. In the reverse loop the frequency decreases and shows a minimum at 1.83 GHz for an applied field of about 0:2 kA/m. At this field the resonance frequency jumps from 1.83 to 1.93 GHz. This latter minimum corresponds roughly to the coercive field in the reverse magnetization direction denoted as H 2 in Fig. 6(b).
(b) Fig. 6. (a) Resonance frequencies of m00 as a function of the applied field of an annealed Ni81 Fe19 ð52 nmÞ=Mn46 Ni54 ð80 nmÞ bilayer during an hysteresis loop sweep. Open circles: measured frequencies for the forward applied field, full triangles: measured frequencies for the reverse field. The solid lines are the calculated frequencies. (b) Open circles: forward branch of the experimental hysteresis loop, full triangles: reverse branch of the experimental hysteresis loop. The calculated hysteresis loop along the easy axis is drawn by a solid line.
This discontinuous jump of the resonance frequencies at H a ¼ H 2 corresponds to the change of the magnetization direction from one stable position aligned with the external field (j ¼ p) for jH a joH 2 to an other one (j ¼ 0) for jH a j4H 2 : In the forward loop, as can be seen in Fig. 6(b), the magnetization does not switch from one stable position (j ¼ 0) to another (j ¼ p) at a fixed value H 1 but rather rotates from one position to another in relatively larger interval of applied field dH a (see Fig. 6(b)). In this field interval, the resonance frequency increases continuously from 1.83 to 2.4 GHz, but these values stay lower than the one for the reverse branch in the same field interval.
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The experimental behavior of the magnetization with the forward field suggests a coherent rotation mechanism resulting to a non-negligible component of the magnetization along the hard axis which itself is along the applied AC field. This may explain the lower value of f res for the forward field in the interval dH a : Such kind of hysteretic behavior of the resonance frequencies have been recently observed in NiFe films with CPS measurements [36]. The authors observed that the jumps of the resonance frequencies appear at fields lower than the coercive field of their samples. This hysteretic behavior of the resonance frequency in exchange biased bilayers has been recently theoretically investigated by Stamps [37]. The author attributes this behavior to different effective fields governing the resonance frequencies for the forward and reverse field directions and obtains hysteretic resonance frequencies by including in energy, terms, that take into account the coupling to both sublattices of the AF and the formation of a planar domain wall. The method for calculating the bilayer resonance frequency is described as follows: for each applied static field along the easy axis, the equilibrium position of the magnetization is determined with the easy axis fit parameters obtained previously and the resonance frequency is calculated with the use of Eq. (4). The comparison between theoretical and experimental resonance frequencies are presented in Fig. 6(a). It can be seen that the theoretical resonance frequency curve fits well the measured one. The theoretical resonance dispersion spectrum shows two distinct abrupt spikes at forward fields as shown in Fig. 6(a) (solid line), one at 1:5 kA/m and the other at 2:4 kA/m. These two spikes correspond to two abrupt jumps in the theoretical forward hysteresis loop (Fig. 6(b), solid line). The experimental forward resonance curve shows a smooth and large minimum that corresponds to a smooth M-reversal, in the same interval (dH a ), as observed in the forward experimental hysteresis loop (Fig. 6(b), open circles). During the reverse loop, a small drop in the theoretical frequency dispersion spectrum appears a 0:2 kA/m and corresponds to the coercive field in the reverse magnetization direction denoted as H 2 in Fig. 6(b)
(full triangles). In many theoretical calculations for the resonance frequency [20,19] the asymmetry in the resonance frequency was not taken into account. Such a behavior of resonance spectrum was predicted by Kim [38] in a theoretical model. The author attributes the first drop to the rotation of the magnetization of the F layer and the second one to the depinning of the partial wall from the interface. We have measured a smooth variation of the resonance frequency for the forward loop instead of two distinct drops as theoretically predicted. That may be attributed to the polycrystallinity of our sample.
5. Conclusion In order to reconcile data on exchange anisotropy measured by irreversible and reversible techniques, a theoretical model based on the representation of the exchange energy by Fourier-like power series (proposed by us previously) has been used. We have shown that it is possible to fit both, the asymmetrical shape of the easy axis hysteresis loop, and the hysteretic frequency dispersion spectra of ferromagnetic resonance to the theoretical model. Our theoretical calculations predict the existence of two branches in the dispersion relation that agrees with the experimental observations. The model reconciles the reversible and irreversible measurements of the exchange bias.
Acknowledgements The authors thank J. Ben Youssef for samples preparation and A. Chevalier for his technical help.
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