ARTICLE IN PRESS
Physica B 343 (2004) 75–79
Calculation and measurement of the magnetization process in orthoferrites Martin Evanzin*, Hans Hauser, Yuri S. Didosyan Institute of Industrial Electronics and Material Science, Faculty of Electrical Engineering and Information Technology, Vienna University of Technology, Gusshausstrasse 27–29, 1040 Vienna, Austria
Abstract The prediction of non-linear behaviour of ferromagnetic material—represented by the hysteresis loop—is the aim of this work. By means of the energetic model of ferromagnetic hysteresis it is possible to generate hysteresis loops containing the characteristics of yttrium orthoferrite. In this way, produced loops will be compared with the measured hysteresis loops of a single plate of yttrium orthoferrite. r 2003 Published by Elsevier B.V. Keywords: Domain wall velocity; Hysteresis model; Orthoferrites; Transparent ferromagnets
2. The energetic model
The prediction of non-linear ferromagnetic material properties, which can depend also on both direction and history of magnetization, is still a challenge in many ﬁelds of technical science . The presented phenomenological approach has been proposed as a so-called energetic model, which was initially intended for describing the anisotropic behaviour of grain-oriented FeSi steel sheets [2,3]. Here the model is applied to an isotropic case and emphasis is given on the relationship between the model parameters and the physics of the magnetization process.
The energetic model of ferromagnetic hysteresis (EM) has already been applied for different magnetization processes and materials [4–12]. The main idea of the EM is the interpretation of the magnetization process as a process of order, which is determined by an elementary probability function, based on Newton’s probability formula Q K! N pKi P ¼ QN i¼1 i ð1Þ i¼1 Ki ! PN PN with i¼1 pi ¼ 1 and i¼1 Ki ¼ K for statistical events Ki in N dimensions with weight pi : 2.1. Calculation The magnetic state of ferromagnetic materials is calculated by minimizing the total energy density
*Corresponding author. Fax: +43-1-58801-36695. E-mail address: [email protected]
(M. Evanzin). 0921-4526/$ - see front matter r 2003 Published by Elsevier B.V. doi:10.1016/j.physb.2003.08.039
wT ¼ wH þ wM
ARTICLE IN PRESS M. Evanzin et al. / Physica B 343 (2004) 75–79
~ H ~ of the sum of the energy density wH ¼ m0 M ~ ~ the applied ﬁeld H and the magnetization M and the material energy density wM ¼ wd þ wR þ wI : The latter term is divided into the energy density of demagnetizing ﬁelds wd and into contributions described by statistical domain behaviour: The reversible energy density wR and the irreversible energy density wI : There are N easy directions, mainly due to the magnetocrystalline and strain anisotropy for anisotropic materials. The statistical domains are characterized by their volume fraction vi with N X
vi ¼ 1
and the direction ~ n i of their magnetization Ms : The total magnetization of the ferromagnetic material is then ~ ¼ Ms M
density pj ðsj Þ q q pj ¼ exp sj ð8Þ k k R1 under the approximation (with 0 pj dsj E1 eq 51) is assumed to be decreasing with increasing sj (adaptive constant, related to pinning: q). The reversal function ki describes the altered jump probability at a point of ﬁeld reversal (see below). With N Z si X wI ¼ vi ðsj Þws dsj ; ð9Þ i¼1
vi ðsj Þ ¼ pi pj ðsj Þ and si ¼ jvi vi;0 j=pi follows for the derivative of wI with respect to vi (vi;0 is the (old) statistical domain volume at the last ﬁeld reversal)
qvk 1 ki exp q vk vk;0 : ¼k
ki qvI qvi pk
ni and with the cosine ai;H of the angle between ~ ~ ; the component of M ~ with respect to H ~ and H (which is calculated by the isotropic EM) results to MH ¼ Ms
ai;H vi :
Stable points of the magnetization curve are points with minimum total energy density, i.e. a state of vanishing total variation: N X qwT qwT dvi þ d~ n i ¼ 0: dwT ¼ ð6Þ qvi q~ ni i¼1 The domain walls are moving reversibly (e.g. wall bowing) until an individual Barkhausen jump starting position sj is reached. The losses ws of a single domain wall during an irreversible Barkhausen jump ws ¼ kðsi sj Þ
are assumed to increase with the wall displacement si sj (loss coefﬁcient: k). In the case of a particle assembly, si represents a magnetization rotation which is reversible until the switching position sj is reached . As the domains are covering mainly pinning sites in order to reduce the (stray ﬁeld) energy involved with, the probability
With nðsj Þ ¼ pj =2 and si ¼ 2n 1; when 1 M mþ1 þ1 ¼ n¼ 2 Ms 2
is the statistic domain number and m ¼ M=Ms is the reduced magnetization, follows for the derivative of wI with respect to n h i qwI q ¼ 2k 1 k exp 2 jn n0 j ð12Þ k qn under consideration of qwH =qn ¼ 2m0 Ms H and Eq. (11) corresponding to the irreversible ﬁeld q i k h 1 k exp jm m0 j ; ð13Þ HI ¼ m0 Ms k where the (old) values n0 and m0 are the unit magnetization reversal or reduced magnetization, respectively, at the last reversal point. We have to consider that the parameters itself will depend on the applied ﬁeld in some cases. Since the famous experiments of Sixtus and Tonks , it is well known that the velocity of a jumping wall depends on the ﬁeld acting at the wall in the moment before the jump starts. The dynamic behaviour of domain walls is still subject of investigation, but for the following estimations let us consider the dependence of the domain wall
ARTICLE IN PRESS M. Evanzin et al. / Physica B 343 (2004) 75–79
velocity vw as x
vw ¼ f ðH Hcr Þ ;
where Hcr is the critical ﬁeld, starting the jump (in the order of magnitude of the coercivity) and x is determined empirically between 1 and 2. Consider a planar domain structure with thickness d and width l in a material with electrical resistivity r: Then the domain wall velocity (at H > Hcr ) is limited by microscopic eddy currents to rH vw ¼ : ð15Þ 2m0 Ms d The distance of a full Barkhausen jump is then l=2 and the duration is t ¼ l=2vw : Therefore the additional, ﬁeld-dependent energy loss is accordRt ing to ww;H ¼ 0 pw dt; which yields at constant vW the total loss energy ww ¼ k þ m0 Ms Hcr
due to irreversible domain wall movement at H > Hcr ; this term replaces k in Eq. (12) with cr ¼ l=d: The adaptive cr is a ratio of the domain or grain geometry (domain width to thickness in this simple case). In order to provide steadyness at Hcr it is convenient to use HR instead of H in Eq. (16), because HI is limited by 7Hc and jHR j increases directly with jHj above jHc j: The irreversible ﬁeld is then k HI ¼ þ cr H R m0 M s h q i 1 k exp jm m0 j : ð17Þ k The constant cr can be almost zero in many cases (e.g. some amorphous and ferrites: d > l); it will be E1 for grain-oriented materials with dEl and it will be very large in nanocrystalline materials (d5l). The considerations above are similar for the case of assemblies of interacting particles because the switching velocity and the interaction losses are – above a critical ﬁeld – also proportional to the applied ﬁeld. 2.2. Results Eqs. (2), (5), (6) and (17) yield the hysteresis, where the sgnðxÞ and jxj functions provide
the correct four quadrant calculation: M ¼ Ms m and H ¼ Hd þ HR þ HI :
For the initial magnetization, beginning with M ¼ 0; H ¼ 0; we ﬁnd m0 ¼ 0 and k ¼ 1: The function k describes the inﬂuence of the total magnetic state on pj at points of magnetization reversal . At those points – if the walls have already moved reversibly at least – there will be, in general, a larger distance to be covered reversibly in the opposite direction (maximum: twice the wall bowing). Therefore, k (old value k0 ) depends on the wall displacements s ¼ jm m0 j up to this point of ﬁeld reversal (m0 is the starting value of m at the last ﬁeld reversal) with the simpliﬁcation eq 51: q k ¼ 2 k0 exp jm m0 j : ð19Þ k0 This function is the most simple reversal under the conditions of continuity (at a point of ﬁeld reversal qwI =qn has to be the same immediately before and after reversal) and the similarity of w0 with the incremental susceptibility at coercivity. Other possibilities will be discussed later. The calculation always starts with the initial magnetization curve (m0 ¼ 0; k ¼ 1) and m is increased stepwise (the stepwidth determines the desired resolution of the calculation), which gives the corresponding ﬁeld by Eq. (18). At a point of ﬁeld reversal k is calculated by Eq. (19) and m0 is set to the actual value of m at this point. Then m is decreased stepwise until the next reversal point, etc. Fig. 1 shows the calculation with parameters g ¼ 10:18442; h ¼ 0:5499751; k ¼ 16:10642; q ¼ 26:85174 and cr ¼ 4:71: 2.3. Measurement The sample was a single plate of yttrium orthoferrite cut perpendicular to an optical axis. The optical axes of YFeO3 lie in the bc plane and—at a wavelength of 0:63 mm—form angles of 752 with the axis of weak ferromagnetism (the crystallographic c-axis). The speciﬁc Faraday rotation at this wavelength equals—2900 =cm . The plate’s thickness was 120 mm: Perpendicular to the
ARTICLE IN PRESS 78
M. Evanzin et al. / Physica B 343 (2004) 75–79
Fig. 1. Calculated hysteresis loop of YFeO3 :Hm ¼ 4000 A=m; Ms ¼ 8:36 kA=m:
Fig. 2. Measured hysteresis of the sample: Hm ¼ 4000 A=m; Ms ¼ 8:36 kA=m:
Fig. 4. Measured initial magnetization curve and minor loops of YFeO3 : Hm ¼ 3200 A=m; Ms ¼ 8:36 kA=m:
surfaces of the sample the adjustable homogeneous magnetic ﬁeld H was applied. In the direction of this ﬁeld, a chopped red-light laser beam was sent through the sample, passed a polarization ﬁlter and a converging lens and was absorbed by a photocell connected to an oscilloscope. The peak-to-peak value of the recorded alternating voltage is proportional to the instant magnetization of the sample due to the magnetic ﬁeld. Magnetization curves up to ﬁelds of 400 kA=m were measured. The magnetization of the sample reached values up to 8:36 kA=m as shown in Fig. 2. Figs. 3–4 illustrate the corresponding calculation and measurent of the initial magnetization curve and minor loops.
Fig. 3. Initial magnetization curve and minor loops of YFeO3 : Hm ¼ 3200 A=m; Ms ¼ 8:36 kA=m:
 E. Della Torre, Magnetic Hysteresis, IEEE Press, Piscataway, 1999.  H. Hauser, J. Appl. Phys. 75 (1994) 2584.  H. Hauser, J. Appl. Phys. 77 (1995) 2625.  H. Hauser, P.L. Fulmek, J. Magn. Magn. Mater. 133 (1994) 32.  H. Hauser, P.L. Fulmek, J. Magn. Magn. Mater. 155 (1996) 34.  P.L. Fulmek, H. Hauser, J. Magn. Magn. Mater. 157 (1996) 361.  P.L. Fulmek, H. Hauser, J. Magn. Magn. Mater. 183 (1998) 345.  P.L. Fulmek, H. Hauser, J. Phys. IV 8 (1998) 335. .  H. Hauser, R. Grossinger, J. Appl. Phys. 85 (1999) 5133.
ARTICLE IN PRESS M. Evanzin et al. / Physica B 343 (2004) 75–79  P. Andrei, L. Stoleriu, H. Hauser, J. Appl. Phys. 87 (2000) 6555.  H. Hauser, Elektrotech. Inform. 110 (1993) 57. .  R. Grossinger, H. Hauser, M. Dahlgren, J. Fidler, Physica B 275 (2000) 248.  H. Hauser, P.L. Fulmek, J. Magn. Magn. Mater. 155 (1996) 34.
 K.J. Sixtus, L. Tonks, Phys. Rev. 37 (1931) 930; K.J. Sixtus, L. Tonks, Phys. Rev. 42 (1932) 419.  P.L. Fulmek, H. Hauser, J. Magn. Magn. Mater. 160 (1996) 35.  Y.S. Didosyan, J. Magn. Magn. Mater. 133 (1994) 425.