The hard-axis magnetization curve for a sandwich with 13Å of Cr and a total of 108Å of Fe is shown in Fig. 1. Also shown is a theoretical magnetization curve calculated by minimizing the total energy of the system as a function of the orientation of the two Fe moments, M1 and M2. The energy density expression is[5]
E = (1/2)(E1 + E2) + J cos(theta1 - theta2), (1)
where Ei = K1(a1i2 a2i2 + a2ii2 a3i2 + a3i2 a1i2) + Ku cos2(thetai - pi/4) - MiH (2)
with alphaji the direction cosines of Mi, K1 the usual cubic anisotropy, Ku an in-plane anisotropy which is commonly found in very thin Fe films on GaAs or ZnSe, and thetai(H) the azimuthal angle between Mi and the easy [100] direction. Here J is a phenomenological parameter introduced to describe the antiferromagnetic coupling between the two Fe films. The theoretical magnetization curve found from the energy density expression is in good agreement with the experimental data, as Fig. 1 shows.
The transverse magnetoresistance (H perpendicular to I) trace of the same tCr = 13Å sample is shown in Fig. 2. For all of the Fe-Cr-Fe samples studied, the steps in the magnetoresistance data occur at the same applied field as the steps in the magnetization curve. In order to understand the dependence of the magnetoresistance on the orientation of the Fe moments, two contributions have to be considered. First, there is the intrinsic magnetoresistance of the individual Fe layers, which depends on the angle theta between the applied field and the current direction.[8] Secondly, there is an additional part of the magnetoresistance which is due to the suppression of the antiferromagnetic alignment. One might expect this second contribution to the magnetoresistance to be a function of the angle, DeltaPhi = (phi1 - phi2), between the moments of the two Fe films.[6]
The effect of the intrinsic anisotropy of the Fe layers on the magnetoresistance can be gauged by examining Dr/r(phi) at a high field where the antialignment has been suppressed. Fig. 3 shows the magnetoresistance of the tCr = 13Å sandwich as a function of the angle between a 6 kOe applied field and the current direction. The data have been fit to the standard form[8]
Dr/r (phi) = [Dr/r]long + {[Dr/r]trans - [Dr/r]long} sin2(q) (3)
where [Dr/r]long is the longitudinal magnetoresistance (H || I), and [Dr/r]trans is the transverse magnetoresistance (H perpendicular to I). The fit results in the approximately field-independent anisotropy parameter {[Dr/r]trans - [Dr/r]long} = -3.8 X 10-3. This anisotropy is slightly larger than that that of pure Fe,[8] a finding which is in agreement with previous work on Fe-Cr-Fe sandwiches.[4]
In order to analyze the second contribution to the magnetoresistance, which is due to the suppression of the zero-field antialignment, one must assume a form for the DeltaPhi-dependence of Dr/r. Once a form has been selected, one can invert the Dr/r(H) data to obtain an "experimental" DeltaPhi (H) curve. Fig. 4 is a plot of DeltaPhi vs. H obtained from the data of Fig. 2 by assuming that the magnetoresistance has the form
Dr/r (DeltaPhi) = [Dr/r]sat * cos2(DeltaPhi/2) (4)
where [Dr/r]sat is the value of the transverse magnetoresistance at magnetic saturation. The form of Eqn. 4 is suggested by the magnetization curve, which indicates that DeltaPhi is 0o at saturation and near 180° at zero field. The angular difference DeltaPhi calculated from Eqs. 1 and 2 is also shown in Fig. 4. The generally good agreement between the calculated and experimental curves in Fig. 4 indicates that the magnetoresistance does in fact vary approximately as cos2(DeltaPhi/2). Camley and Barnas[6] previously pointed out that the transmission coefficient of an electron propagating between the two Fe layers will go as cos2(DeltaPhi/2) when the angle between the two Fe moments is DeltaPhi.