Discussion

The field dependence of the magnetoresistance has been obtained from the raw data using the definition

(1)

As has been detailed before,[13,25] the angular dependence of the giant magnetoresistance is described by

(2)

where (theta₁ - theta₂) is the relative orientation of the moments of the two magnetic films and G is the coefficient of giant magnetoresistance. The term that describes the angular dependence of the anisotropic magnetoresistance (AMR) has been omitted from equation (2). The AMR is the magnetoresistance that is characteristic of the individual films.[27] The magnitude of the AMR can be separately determined at high field where the GMR term doesn't contribute because (theta₁ - theta₂) = 0.[25] The maximum MR for each sample minus the AMR contribution is listed in Table II as MR_peak. The AMR is also listed in Table II for each sandwich.

Since the moments of the two magnetic films are aligned at the magnetic saturation field H_s, MR(H=H_s) = MR(theta₁ = theta₂) = 0, and equations (1) and (2) are consistent. If the films had antiferromagnetic interlayer coupling, then the moments would be antiparallel at H=0, and MR(H=0) = MR(180°) = G. Because these Fe/Ag/Co_xFe_1-x sandwiches display no antiferromagnetic interlayer coupling, (theta₁ - theta₂) at H=0 is determined by magnetocrystalline anisotropy, as described above, and is typically 0° or 45°. The peak in the magnetoresistance as a function of field therefore occurs at some H where the angular separation of the moments reaches a maximum, here called (theta₁ - theta₂)_peak. As equation (2) shows, the magnetoresistance at the peak is

MR_peak = (3)

Equation (3) implies that in order to obtain the coefficient of giant magnetoresistance G, we must divide the experimentally obtained MR_peak through by . As was discussed above, for four of the five samples the value of (theta₁ - theta₂)_peak can be directly deduced from the magnetization curve. In this case, the intrinsic, magnetic-configuration- independent GMR coefficient G can be obtained directly from Eqn. 3. The deduced magnitudes of G and (theta₁ - theta₂)_peak are collected in Table II. The G coefficient for the x = 0.3 sample is not listed because (theta₁ - theta₂)_peak could not be independently determined from its magnetization curve.

Because the hysteresis loops of these sandwiches are dominated by magnetocrystalline anisotropy, the details of the magnetization reversal process change as one changes the applied field orientation. Consequently the MR loops and (theta₁ - theta₂)_peak also vary as the applied field direction is changed. A different value of MR_peak is therefore obtained for each new applied field orientation. Table II contains the MR_peak and (theta₁ - theta₂)_peak values which correspond to the hysteresis loops of Figs. 2 through 4. The G coefficient should be independent of the applied field orientation if the analysis based on Eqn. 1 is consistent. In fact, for each of the 4 samples whose G is given in Table 1, the magnitude of the G determined from analysis of both the hard and easy axis loops is the same to within about 10%.[28] The agreement of these G values can be slightly improved by taking into account the effect of a small unixial anisotropy which is often observed in samples grown on GaAs or ZnSe.[19] The uniaxial anisotropy causes (theta₁ - theta₂)_peak to be different in loops taken along orthogonal <11> directions.[28] This extra analysis adds great complexity without changing the essential conclusions, so it is omitted here.

Because the GMR depends on the relative orientation of the magnetic moments, it is intrinsically a multilayer property. In contrast, the well-known AMR is dependent on the absolute orientation of each film's moment with respect to the measurement current direction, and is therefore intrinsically a single-film property.[27] Since the different AMR contributions of the Fe and alloy layers cannot be separated experimentally, the values collected in Table II represent an average over both layers. MR_peak and the averaged AMR are displayed versus Co fraction in Fig. 5. Like G , the AMR can also be determined several different ways, which should of course all produce the same number. Values of the AMR determined from the high-field measurements and from the orientation dependence of the low-field hysteresis loops are consistent to within a few percent.[28]

Freitas, Berger, and Silvain have previously measured the composition dependence of the AMR in both bulk and thin-film CoFe alloys.[29] In 920Å films at room temperature, they found a gradual increase in the AMR from about 0.5% for Fe to about 1.5% for Co. In Fig. 5 the AMR is larger for the Fe-rich sandwiches than the Co-rich sandwiches, contrary to the previous findings. The small absolute size of the AMR for these sandwiches is attributable to both surface scattering and shunting by the Ag layers. Of course, neither surface scattering nor shunting can explain why the slope of the AMR vs. Co concentration has a sign opposite to that found previously. To explain this result, one may have to consider the effects of composition-dependent epitaxial strain or changes in local atomic order, both of which could potentially have a large effect on the AMR.[27] More detailed studies of the magnetization and structure of the CoFe alloy system may clarify these issues.[16]

The MR_peak vs. composition data in Fig. 5 are not very meaningful because the sinusoidal factor from Eqn. 3 has not been divided out, and the values of (theta₁ - theta₂)_peak are different for each sample and each field orientation. Clearly the composition dependence of the angle- independent giant magnetoresistance coefficient G is of greater interest. G is perhaps best understood as the GMR that each sample would have at the peak of its hysteresis curve if the magnetic moments of its ferromagnetic layers became exactly antiparallel ((theta₁ - theta₂)_peak = 180°). Only the Co/Ag/Fe sandwich truly had antiparallel moments at its MR peak. Thus the giant magnetoresistance coefficient G is the intrinsic composition-dependent property which should be amenable to theoretical calculation. G and Dr are plotted versus alloy composition in Figure 6a). Unfortunately since (theta₁ - theta₂)_peak could not be determined for the x=0.3 sample, data for this sandwich are not plotted. Note that in uncoupled magnetic multilayers, the problems with unknown angular orientation of the moments in the individual layers are greatly compounded, and only the uncorrected values MR_peak can be quoted with confidence. Even though the GMR of trilayer samples is small in size, trilayers offer the advantage that the values of (theta₁ - theta₂)_peak can usually be reliably determined from magnetization curves.

The magnetic moment data in Fig. 6b) comes from the work of Bardos.[30] A correlation between the composition dependence of the magnetic moment and that of the GMR would be an indication that the degree of conduction electron spin polarization was an important factor in determining the size of the GMR in a given ferromagnet. A correlation of this type is expected in most simple theories of the GMR because the inequality of up- and down-spin resistivities is believed to be essential to the phenomenon. 1, 31 Clearly the spin-averaged resistivity and spin-flip scattering lengths of both the ferromagnetic and paramagnetic layers will also play a role in determining the size of the GMR. A comparison of Figs. 6a) and 6b) shows similar trends in the GMR and magnetic moment data, but more experiments are needed before the issue can be settled definitively. The resolution of this question remains an important objective in understanding the origin of the GMR.

The field-induced change in resistivity, Dr, which is the numerator of equation (1), itself contains all the information about the origin of the GMR, since the denominator is just the zero-field resistivity. Therefore the dependence of Dr on alloy composition may be more directly comparable to fundamental theories than the GMR itself. The values of the field-induced change in resistivity in Table I were obtained by multiplying the experimental zero-field resistivities by G. Examination of Table I shows that the variation of the relative size of Dr with composition is similar to that of the magnetoresistance itself. Extensive studies of Dr as a function of temperature and alloy-layer thickness could provide the information necessary to develop a quantitative theory of the magnitude of the GMR for different ferromagnet/paramagnet systems.