PHYSICAL REVIEW B, VOLUME 65, 054408 Spin-dependent electrical transport in ion-beam sputter deposited Fe-Cr multilayers A. K. Majumdar* Department of Physics, Indian Institute of Technology, Kanpur-208016, India and Department of Physics, University of Florida, Gainesville, Florida 32611 A. F. Hebard Department of Physics, University of Florida, Gainesville, Florida 32611 Avinash Singh Department of Physics, Indian Institute of Technology, Kanpur-208016, India D. Temple MCNC, Electronics Technologies Division, Research Triangle Park, North Carolina 27709 Received 19 August 2001; published 3 January 2002 The temperature dependence of the electrical resistivity and magnetoresistance of Xe-ion-beam-sputtered Fe-Cr multilayers has been investigated. The electrical resistivity between 5 and 300 K in the fully ferromag- netic state, obtained by applying a field beyond the saturation field (Hsat) necessary for the antiferromagnetic- AF- ferromagnetic FM field-induced transition, shows evidence of spin-disorder resistivity as in crystalline Fe and an s-d scattering contribution as in 3d metals and alloys . The sublattice magnetization m(T) in these multilayers has been calculated in terms of the planar and interlayer exchange energies. The additional spin- dependent scattering (T) (T,H 0)AF (T,H Hsat FM in the AF state over a wide range of tempera- ture is found to be proportional to the sublattice magnetization, both (T) and m(T) reducing along with the antiferromagnetic fraction. At intermediate fields, the spin-dependent part of the electrical resistivity s(T) fits well to the power law s(T) b cT where c is a constant and b and are functions of H. At low fields 2 and the intercept b decreases with H much the same way as the decrease of (T) with T. A phase diagram T vs Hsat is obtained for the field-induced AF-to-FM transition. Comparisons are made between the present investigation and similar studies using dc-magnetron-sputtered and molecular-beam-epitaxy-grown Fe-Cr multilayers. DOI: 10.1103/PhysRevB.65.054408 PACS number s : 75.50.Bb, 75.50.Ee, 72.15.Eb, 72.15.Gd I. INTRODUCTION rect exchange interaction between the Fe layers through the oscillatory RKKY interaction mediated by the conduction As one of the very few lattice-matched transition-metal electrons. The above antiferromagnetic coupling between Fe pairs one of which is ferromagnetic FM , Fe-Cr multilayers layers was established by means of light scattering from spin offer excellent opportunities for investigating the exchange waves.1 As the external field increases, the spins in different coupling of Fe layers through an antiferromagnetic Cr spacer Fe layers align in the direction of the field, producing a com- layer, giving rise to the so-called giant magnetoresistance pletely ferromagnetic alignment beyond a saturation field GMR . Applications as magnetic-field sensors, especially in Hsat , reducing the resistance. Thus we have a negative mag- reading information, sensing position and speed of moving netoresistance MR . parts, etc., have triggered intense research activity in these Magnetoresistance is defined in textbook fashion by multilayers. GMR sensors are not only very sensitive, but they can be made very small in size. For practical purposes, H,T 0,T not only does one need large GMR, but also small saturation MR 100%. 1 0,T fields. The aim of the present work is to study the temperature It is found in dc-magnetron-sputtered Fe-Cr superlattices2 dependence of electrical resistivity and magnetoresistance in that GMR oscillates as a function of Cr spacer thickness with GMR multilayer stacks prepared by ion-beam sputter depo- three gradually decreasing peaks centered at 11, 27, and 42 Å sition. Typical multilayers reported here comprise 30 repeat Cr for Fe thickness of 32 Å. Also, the first antiferromagnetic layers of Fe 20 Å /Cr 10 Å that have been deposited by region occurs between 6 and 11 Å Cr for Fe thickness lying ion-beam sputter deposition onto Si substrates with xenon between 15 and 40 Å. We have therefore chosen the Cr ions at 900 V and a beam current of 20 mA. thickness around 10 Å corresponding to the strongest peak of GMR in multilayers can be understood in terms of some the antiferromagnetic coupling between the Fe layers and simple ideas as follows. In zero magnetic field the ferromag- hence the highest GMR . netic Fe layers are coupled antiferromagnetically through the The basic qualitative features of GMR can be understood Cr spacer layer, giving rise to a high electrical resistance. even in terms of bulk scattering only if the mean free path This antiferromagnetic AF coupling is ascribed to the indi- mfp of electrons within the layers is much larger than the 0163-1829/2002/65 5 /054408 8 /$20.00 65 054408-1 ©2002 The American Physical Society MAJUMDAR, HEBARD, SINGH, AND TEMPLE PHYSICAL REVIEW B 65 054408 layer thickness. If the mfp of the electrons is larger than the sistance defined as M(T) (T,H 0) (T,H Hsat) Cr spacer thickness, the electrons can feel the relative orien- was found by Mattson et al.2 to follow the equation tation of the magnetization of the successive layers. How- for T 100 K: ever, this interplay between the successive magnetic layers disappears and the GMR vanishes if the mfp is less than the M T M T 0 M T M T 0 aT2, 3 Cr layer thickness. GMR is attributed to the spin-dependent conduction prop- where a is a constant of proportionality. This behavior was erties of ferromagnetic metals. In a ferromagnetic metal or explained in terms of thermal excitation of magnons whose alloy the electrical conduction takes place through indepen- occupation number (n) T2 at low temperatures for aniso- dent channels by spin-up called majority and spin-down tropic materials9 and assuming M(T) n. called minority electrons. This is the two-current model of In Fe(12 Å)/Cr(12 Å) 10 multilayers, prepared by Fert and Campbell3 whose physical basis is the dominance of MBE on MgO 100 substrates, several interesting observa- the spin-conserving scattering and the weakness of the spin- tions were made by Aliev et al.8 Among them are the follow- flip collision, at least at low temperatures. In this picture all ing. electrons of a given spin up or down with s or d or hybrid- a The isothermal magnetoresistance, defined as (0) ized character are grouped together to form majority up or (H) , is proportional to H in the parallel magnetic field minority down bands. If one takes into account the details in the plane of the multilayers orientation and to H2 in the of the band structure of Fe a weak ferromagnet and the perpendicular case. simple Drude conductivity formula for each band, it is easily b A T-Hsat phase diagram was obtained, which clearly shown4 that the majority band has a much higher conductiv- indicated the transition between AF and FM states. ity than the minority band. As a result, in the ferromagnetic c As opposed to the work of Mattson et al.2 which alignment brought about by the saturation field (H looked at the difference between the resistivity in the ideally sat), there is hardly any scattering for the majority-band electrons since antiferromagnetic (H 0) and ferromagnetic (H Hsat) they remain majority in all the Fe layers. On the other hand, alignments , here8 the spin-dependent part of the electrical the minority-band electrons get scattered within every Fe resistivity, defined as s(T) (T,H) (T,H Hsat) for layer. Hence there is a short-circuiting effect, so to say, and fields H 0 where one has both ferromagnetic and antifer- the resistance ( romagnetic fractions , is found to vary in a wide range of FM) drops in the ferromagnetic alignment. However, in the antiferromagnetic configuration zero field temperature below 100 K as both the majority and minority-band electrons are scattered in successive layers, and there is no short-circuiting effect. s T s T 0 s T b cT , 4 Therefore, the resistance ( AF) remains relatively high. A where b very simple calculation based on the two-current model s(T 0) and the temperature exponent are functions of the magnetic field H, and c is a constant of shows that the GMR at low temperatures is given by con- proportionality. The constant was found8 to be 1.7 for sidering only bulk scattering H 0, 2.0 for H 0.5Hsat , and 1 for H Hsat . This is in contrast to the value of 2 for H 0 purely antiferromag- H H netic in the work of Mattson et al.2 Eq. 3 above . GMR sat 0 FM AF 2 / 1 , d 0 s(T) was found to vary linearly with temperature AF / 1 2 from 20 mK to about 1.5 K ``which could be due to electron scattering on critical thermal spin fluctuations.'' 8 All the work summarized above is mostly on MBE-grown where and are the resistivities of the minority and Fe-Cr multilayers and those made by dc magnetron sputter- majority carriers, respectively. It turns out that interface scat- ing. Although the present work is on Fe-Cr multilayers pre- tering from imperfect interfaces, defects, and impurities in pared by ion-beam sputter deposition, we do not believe that Fe-Cr multilayers is also spin dependent and gives rise to the the underlying physics depends in any significant way on GMR. As a matter of fact, it is the imbalance between the differences in these deposition techniques. resistivities of the two bands which is responsible for the GMR for bulk, interface, and spin-flip scattering. The subject II. EXPERIMENT of GMR has been reviewed very well in a recent book.5 Considerable work has been reported on the electrical Fe-Cr multilayers were prepared by the ion-beam sputter- transport and magnetic properties of Fe-Cr multilayers pre- ing technique and characterized by transmission electron mi- pared by sputtering2,6 or molecular-beam epitaxy MBE .6­8 croscopy TEM , atomic force microscopy AFM , Auger The temperature dependence of the electrical resistivity in electron spectroscopy AES , and x-ray photoelectron spec- Fe(30 Å)/Cr(10­ 50 Å) 10 multilayers, prepared by troscopy XPS , as well as resistivity, magnetic hysteresis sputtering and MBE, was interpreted by Almeida et al.6 in loop, and magnetotransport measurements. The film deposi- terms of phonon-assisted s-s and s-d scattering in the tem- tion procedure and film properties, including chemical com- perature range of 15­300 K and in a saturation magnetic position, surface morphology, resistivity, saturation magneti- field of 7.5 kOe. zation, coercive field, and magnetoresistance ratio, are In the antiferromagnetic Fe-Cr superlattice, made by dc discussed in detail in Ref. 10. Values of GMR ratios of the magnetron sputtering, the temperature-dependent magnetore- films are comparable to values measured for polycrystalline 054408-2 SPIN-DEPENDENT ELECTRICAL TRANSPORT IN ION- . . . PHYSICAL REVIEW B 65 054408 Fe-Cr films deposited by the more conventional rf sputtering technique.11 We have deposited the following Fe-Cr multilayer combinations: Si/Cr 50 Å / Fe 20 Å /Cr t Å ...] 30/Cr 50 t Å ..., where t was varied from 8 to 14 Å; this range surrounds the first antiferromagnetic maximum in the Fe-Cr multilayer sys- tem. The deposition rates varied from 5 to 30 Å/min depend- ing on the primary ion-beam energy, the type of ions, and the target material. The films were deposited at room tempera- ture. The effects of variations of the primary ion-beam en- ergy and the type of ions on GMR values were examined; the investigated primary ion energy range was 700­1200 eV for Ar ions and 900­1200 eV for Xe ions. It was demonstrated that the GMR ratio is greater for films deposited using Xe ions than for films deposited using Ar ions, and that for both types of ions the GMR ratio increases as the primary ion- FIG. 1. Magnetoresistance vs external field H kOe oriented beam energy decreases. In this investigation we report the parallel to the layers for a Xe-ion-beam-sputtered Fe-Cr multilayer work on Fe-Cr multilayers of typical structure sample sample 2 at 10 and 300 K. The MR saturates around 13 Fe(20 Å)/Cr(10 Å) 30 layers grown on Si substrates us- kOe (Hsat) and has a typical GMR of 21% at 10 K. ing Xe ions at 900 V and a beam current of 20 mA. The temperature dependence of the resistance between 5 terial has both antiferromagnetic and ferromagnetic fractions. and 300 K of the Fe-Cr multilayers was measured in zero as Instead, for H Hsat , the alignment of the spins of each Fe well as in some applied magnetic fields using the standard layer is parallel to the direction of the external field, giving four-probe dc technique and a magnetic field 0­5.5 T pro- rise to a fully ferromagnetic state. It is well known in crys- vided by a Quantum Design superconducting quantum inter- talline bulk 3d metals and alloys that the electron-phonon ference device SQUID magnetometer MPMS . Both the scattering contribution to the electrical resistivity from sd transport current and applied field were in the plane of the dominates over ss due to the overlap of the s and d bands at film with the current parallel to the field. For measurements the Fermi level. Specifically, for Fe the density of states of in magnetic fields perpendicular to the film plane we used a the 3d majority band at the Fermi level is rather large com- Quantum Design physical property measurement system pared to those of the s bands. The resistivity sd is given by PPMS . We used the same MPMS to measure the magneti- the Bloch-Wilson formula.12 The ``spin-disorder resistivity'' zation of the Fe-Cr multilayers as a function of external coming from the electron-magnon spin-wave contribution fields at temperatures down to 2 K. scattering is well described13 by a relatively small term vary- ing as T2 in ferromagnets like Fe, Co, and Ni. Putting all III. RESULTS AND DISCUSSION these contributions together along with the residual resistiv- ity 0 one can write, assuming Mathiessen's rule Figure 1 shows the magnetoresistance versus external field H kOe for a typical Xe-ion-sputtered Fe-Cr multilayer sample at 10 and 300 K. The MR becomes constant at a saturation field Hsat vertical arrow around 13 kOe with typical values of 21% at 10 K. These values compare favor- ably with 30% and 40% obtained in dc-magnetron-sputtered and MBE-grown samples, respectively. Hysteresis in the MR was negligible as the magnetic field was swept from (0 20 0 20 0) kOe. A. Temperature dependence of the electrical resistivity Figure 2 shows the electrical resistivity versus tem- perature T for sample 1 at several values of the external magnetic field from 0 to 12 kOe. The saturation field (Hsat) has a weak temperature dependence; namely, it decreases with increasing temperature. This is clear from the (T) curves, say, at 10 and 12 kOe. They are closer to each other at higher temperatures. FIG. 2. Electrical resistivity vs temperature T for sample 1 To interpret the temperature dependence of the electrical at several fields between 0 and 12 kOe. The curves are closer to resistivity, (T), at an intermediate field between H 0 and each other at higher temperatures, indicating that the saturation field H Hsat is not simple. In this magnetic field region the ma- Hsat decreases with increasing temperature. 054408-3 MAJUMDAR, HEBARD, SINGH, AND TEMPLE PHYSICAL REVIEW B 65 054408 FIG. 3. Electrical resistivity vs temperature T data points FIG. 4. The deviation of the actual data from the best-fit values from 5 to 300 K for samples 1, 2, and 3 at their respective Hsat . The is plotted as a function of temperature T for sample 2 for fits with solid lines are the excellent least-squares fitted curves for fits to Eq. and without the magnetic T2 term. The deviation is much less 0.1 5 which includes lattice and magnetic scattering contributions. cm in 40 cm and more random for the fits with the mag- netic term than that without it. D /T z3dz T,H H 3 sat 0 A T B. Temperature dependence of the magnetoresistance D 0 ez 1 1 e z At H 0 these multilayers are ideally in an antiferromag- BT2, 5 netic state where the neighboring ferromagnetic Fe layers are where the second term is the Bloch-Wilson contribution all antiferromagnetically coupled, resulting in a higher resis- sd and the third term is the small electron-magnon contribution tivity. Actually, there may be pinholes through the Cr spacer increasing with temperature due to thermal excitation of layer directly coupling the Fe layers ferromagnetically in- magnons. stead. Let us define an antiferromagnetic fraction AFF as Taking the Debye temperature D 420 K for Fe-Cr AFF H 1 M H /M multilayers,6 we have fitted the data for five samples at their s 100%, 6 respective Hsat to Eq. 5 using a three-parameter least- where Ms is the magnetization measured at H Hsat and squares fit program which also evaluated the integral numeri- M(H) is the magnetization when the field is reduced from cally at each iteration. Excellent fits were obtained for all the saturation to H, all at 5 K. Our M(H) measurements on these samples with correlation coefficients of 0.999 995 and values samples show that the AFF is typically 80% at H 0. As H is of the normalized 2 consistent with the number of degrees increased, the Fe layers gradually turn their magnetization in of freedom and error estimates. Figure 3 shows vs T data the direction of the external field, reducing the AFF and points for three samples from 5 to 300 K at their respective hence the resistivity. Finally, the AFF reduces to zero fully Hsat . The solid lines are the best-fit curves to Eq. 5 . It is ferromagnetic alignment and the resistivity and hence the found that the value of the coefficient of the magnetic scat- GMR saturate at H Hsat . tering term B averaged over all five samples is (4 1) We define (T) (T,H 0)AF (T,H Hsat)FM as 10 5 cm K 2 compared to 1.5 10 5 cm K 2 in the difference in resistivity at a given temperature T between bulk ferromagnets Fe, Co, Ni . This higher value of B may the AF (H 0) and the FM (H Hsat) states, both assumed be related to the fact that the resistivity of these Fe-Cr mul- ideal. This (T) is primarily due to the additional spin- tilayers at 300 K is about 5 times larger than that of bulk dependent scattering both bulk and interface in the antifer- iron. romagnetic state. It is assumed here that the residual resis- The fits of the data of Fig. 3 to Eq. 5 without the mag- tivity and the interband s-d scattering dominant for 3d netic (BT2) term are distinctly inferior to those with the metals and alloys do not depend strongly on magnetic fields. magnetic term. The values of 2 are typically 6 times larger Figure 5 plots (T) vs T data stars for samples 1 and 3. and the correlation coefficients poorer for the fits without the Thus the additional spin-dependent scattering resistivity in magnetic term. The deviation of the actual data from the the AF state decreases with increasing temperature. Just like best-fit values residuals is plotted in Fig. 4 as a function of the magnetic field aligns the spins in different Fe layers re- temperature for sample 2 for both the fits. The deviation is ducing the AFF gradually bringing ferromagnetic order in much less 0.1 cm in 40 cm and more random for its place and produces a negative magnetoresistance, here the fits with the magnetic term than without it. Addition of a temperature reduces the antiferromagnetic order potentially Bloch-Gru¨neissen BG term ( ss), which has a T5 depen- bringing down the AFF and hence (T). It is seen from dence, or replacing the Bloch-Wilson term by the BG term Fig. 5 that varies as T2 at low temperatures and is makes the fit much worse. roughly linear at higher temperatures. 054408-4 SPIN-DEPENDENT ELECTRICAL TRANSPORT IN ION- . . . PHYSICAL REVIEW B 65 054408 FIG. 6. s(T) (T,H) (T,Hsat) vs temperature T data points for sample 1 at H 0, 3, 4, 6, and 8 kOe. The solid lines are the least-squares fitted curves for fits to Eq. 4 . FIG. 5. (T) (T,H 0)AF (T,H Hsat)FM vs tempera- ture T data stars for samples 1 and 3. This additional spin- thick Fe films sandwiched between Cr spacer layers. The dependent resistivity in the AF state decreases with temperature. value for Jz is, however, satisfying, close to the recently varies as T2 at low temperatures and roughly linearly at higher identified glass temperature, Tg 140 K, of an antiferromag- temperatures. The solid lines are the least-squares fitted curves for netic glassy phase that coexists with GMR in similar fits to Eq. 7 . multilayer films. Irreversibilities in this glassy phase have been shown to arise from the same interlayer coupling that Singh et al.9 had worked out the reduction in antiferro- drives the antiparallel alignments in GMR.14 magnetic order due to thermal excitation of spin waves in Consistent with the above model are experimental data highly anisotropic antiferromagnets with weak interlayer taken by us not shown and others7 in which the saturation coupling between the antiferromagnetic planes. This theory fields are studied as a function of spacer layer thickness. For has been extended in the present case where each Fe layer is three-ion-beam sputter-deposited ten-layer samples of ferromagnetic, but coupled antiferromagnetically to the Fe(20 Å)/Cr(dCr) with different Cr spacer layer thickness neighboring Fe layers due to the RKKY interaction. In terms (dCr) we have observed that as dCr increases from 8 to 12 Å, of the planar and interlayer exchange energies Jp and Jz , the saturation field (Hsat) decreases from 10 to 5 kOe. As Jz respectively, the sublattice magnetization m(T) at tempera- decreases with increasing dCr , smaller external fields are ture T is given in the Appendix Eq. A6 . Assuming necessary to break the antiferromagnetic coupling between (T) m(T), we have the relation the Fe layers. T 1 T /2 C. Temperature dependence of the magnetoresistance 1 dq , 0 2 J z ln 1 in intermediate fields 0ÏHÏHsat... p 0 1 e Jz /T 1 cos2qz 1/2 7 Following the work of Aliev et al.,8 we have fitted our data for samples 1, 2, and 3 to Eq. 4 with s(T) where m(0) 1. The expression on the right differs only (T,H) (T,Hsat). The data points along with the least- insignificantly from the corresponding expression for squares fit curves are shown in Fig. 6 for sample 1 for H m(T)/m(0) obtained earlier9 for the anisotropic antiferro- 0, 3, 4, 6, and 8 kOe. Excellent fits are obtained for all the magnet cos2 qz instead of cos qz , where it was shown to fall samples with values of 2 consistent with the experimental off as T2 at low temperatures (T Jz), crossing over to an error, correlation coefficients R2 0.999, and small errors in approximately linear (T ln T) fall off at high temperatures the fitting parameters b, c, and . The fits are, however, (T Jz). We note that this expression is relatively unchanged better for smaller fields. Figure 7 shows of Eq. 4 vs when ferromagnetic domains are included. H/Hsat for all the three samples. The solid lines are just We numerically evaluated the integral in Eq. 7 and used guides to the eye. The shape of the curve is rather similar to a three-parameter least-squares fit program to fit the data of the results obtained by Aliev et al.8 summarized at the end Fig. 5. The resulting best-fit curves, shown by solid lines in of the Introduction and Fig. 3 of Ref. 8 for MBE-grown Fig. 5, yield values of 2 consistent with the experimental samples. However, we find some differences, like in our errors and a correlation coefficient of 0.9999. Estimates of Jp work being typically 2 for H/Hsat 1/3, becoming 1 for and Jz are found from the above fits. They are (230 H/Hsat 2/3, and decreasing at still higher fields. This im- 20) K and (70 20) K, respectively. The value for Jp is plies that s vs T curves Fig. 6 are quadratic in lower fields well below the Curie temperature 1040 K for bulk iron, but and linear around H/Hsat 2/3 instead of 1 as in the work could be closer to the unknown Curie temperature of 20-Å- of Aliev et al.8 054408-5 MAJUMDAR, HEBARD, SINGH, AND TEMPLE PHYSICAL REVIEW B 65 054408 FIG. 7. of Eq. 4 vs H/Hsat for all the three samples. The FIG. 9. Magnetoresistance vs external field H kOe at low solid lines are just guides to the eye. fields for sample 4 Ar-ion sputtered for the parallel orientation H in the film plane . The temperatures are between 5 and 205 K every It is found from Fig. 6 that the intercept b decreases with 20 K and 300 K. Clearly, the MR H2 at lower fields. increasing applied fields. This is simply due to the fact that the antiferromagnetic fraction decreases with increasing MR H2 in contrast to the findings of Aliev et al.8 Fig. 1 a field. If we plot b as a function of our measured values of of Ref. 8 who found that the MR is linear in H for their AFF % for samples 1 and 3, we find, as shown in Fig. 8, MBE-grown samples. We found no linear region in the MR that b increases with the AFF in a monotonic fashion both vs H curve even at higher fields until the saturation field decreasing with H . This is a logical conclusion since as H 0, the AFF attains its maximum value giving the highest (Hsat) of 2­3 kOe was reached. As a matter of fact, sample 4 resistivity in the ideally AF ground state. It is to be noted that Fig. 9 reflects an S-shaped curve, having points of inflec- tion. However, for the perpendicular orientation H perpen- the decrease of b with H and the decrease of with T Fig. dicular to the film plane we find that the MR again goes as 5 have a common origin. It is the decrease of the AFF brought about by H and T, respectively. H2, in agreement with the findings of Aliev et al.8 The observed H2 dependence at low external fields is, in fact, expected from simple energy considerations, as argued D. T vs Hsat phase diagram for the AF-FM transition below. The antiferromagnetic ground state of the multilayer Figure 9 shows the low-field magnetoresistance of sample is characterized by the sublattice magnetization m (mA 4 argon-ion sputtered vs external field H for the parallel mB)/2, which takes into account the antiparallel orienta- orientation H in the film plane . It is amply clear that the tion of the spin polarization in alternating Fe layers A and B. The direction of m is arbitrary in the ideal isotropic situation. When a small in-plane magnetic field is applied, the sublat- tice magnetization m aligns itself perpendicular to the direc- tion of the field. This is the lowest-energy configuration as it allows for energy gain in all Fe layers due to twisting of spins in the field direction. If the twist angle is , assumed small, then the energy gain is mH sin mH . The twisting also costs energy Jzm2(1 cos 2 ) 2Jzm2 2 due to loss of antiferromagnetic exchange energy at the layer interfaces. Minimizing the net energy change yields the optimum twist angle (H) H/4Jm as proportional to the field. Now the reduction in the sublattice magnetization or the antiferromag- netic fraction, and therefore the decrease in resistivity, due to this twist is m 1 cos (H) , which goes as H2 for low fields. Figure 10 shows the T vs Hsat phase diagram for sample 4 argon-ion sputtered, same as that of Fig. 9 and sample 1 xenon-ion sputtered . Both are in parallel orientations. Here Hsat(T) is the field at which the MR becomes field indepen- FIG. 8. b of Eq. 4 vs AFF % for samples 1 and 3. b is found dent: i.e., the field-induced AF-to-FM transition is complete. to increase monotonically with the AFF both increasing with H . The values of Hsat(T 0) are 3 kOe for sample 4 and 054408-6 SPIN-DEPENDENT ELECTRICAL TRANSPORT IN ION- . . . PHYSICAL REVIEW B 65 054408 trast to our fits Eq. 7 over a much wider temperature range. From our data at intermediate fields (0 H Hsat), the spin-dependent part of the electrical resistivity, defined as s(T) (T,H) (T,H Hsat), fits very well to Eq. 4 . We find that is typically 2 for H/Hsat 1/3, becoming 1 for H/Hsat 2/3, and then decreasing further at still higher fields. The decrease of the intercept b with increasing H and that of (T) with increasing T Eq. 7 are due to the decrease of the antiferromagnetic fraction Eq. 6 with in- creasing H and T, respectively. Very similar conclusions were reached by Aliev et al.8 in MBE-grown Fe-Cr multilay- ers.Finally, we have also obtained the T-vs-Hsat phase dia- gram for the field-induced AF-to-FM transition. ACKNOWLEDGMENTS FIG. 10. T vs Hsat phase diagram for samples 4 Ar-ion sput- One of us A.K.M. acknowledges the Physics Depart- tered and 1 Xe-ion sputtered . Here Hsat is the field at which the MR becomes field independent; i.e., the field-induced AF-to-FM ment, University of Florida, Gainesville for local hospitality transition is complete. The values of H and experimental facilities. In addition, discussions with S. sat(T 0) are 3 kOe for sample 4 and 11.5 kOe for sample 1. B. Arnason and S. Hershfield are gratefully acknowledged. This research was supported by a DOD/AFOSR MURI grant 11.5 kOe for sample 1. Figure 10 is very similar to Fig. No. F49620-96-1-0026 . 1 b of Ref. 8 on MBE-grown samples for the parallel ori- entation. APPENDIX To obtain the magnon energies in the multilayer system, the following simplified Hubbard model is considered on a IV. CONCLUSIONS three-dimensional lattice consisting of a stack of layers in the The temperature dependence of the electrical resistivity z direction: and magnetoresistance has been studied in ion-beam- sputtered Fe-Cr multilayers. Typical in-plane negative giant H p k ck ck tz ci ci , ci , magnetoresistance is 21% at 10 K saturating at around 1 T. k i Here each Fe layer is ferromagnetic but coupled antiferro- magnetically in zero field to the neighboring Fe layers due to U nn ni . A1 the RKKY interaction. This gives rise to a high resistance. i An external magnetic field aligns the spins in different Fe Here the planar band energy p(kx ,ky) together with the layers producing a ferromagnetic alignment beyond Hsat correlation term describes the ferromagnetic layers, while the which reduces the electrical resistance. The electrical resis- interlayer hopping term tz , which connects sites i to nearest- tivity in the fully ferromagnetic state (H Hsat) between 5 neighbor sites i in the neighboring layers, represents the and 300 K has been interpreted as the sum of a residual AF exchange coupling between layers. We divide the resistivity, electron-phonon s-d scattering, and spin-disorder multilayer system into two sublattices with alternating A and resistivity Eq. 5 . The latter has the same order of magni- B layers, and consider a ground state in which the A layers tude as in crystalline Fe. have spin polarization n n m in the z direction, We have calculated the sublattice magnetization m(T) of i i while B layers have spin polarization n n m in the these Fe-Cr multilayers in terms of the planar and interlayer i i z direction. The sublattice magnetization m, a dimension- exchange energies Eq. A6 of the Appendix . The addi- less quantity, measures the AF order parameter in the tional spin-dependent scattering in the antiferromagnetic multilayer system. state at H 0, defined by (T) (T,H 0)AF (T,H In this two-sublattice basis and in the Hartree-Fock HF Hsat)FM , is obtained from the experimental data by assum- approximation, the Hamiltonian reduces to ing that the residual resistivity and the electron-phonon scat- tering are roughly independent of the field. The decrease in (T) with increasing temperature from 5 to 300 K is ex- H a ak k bk p k z k b , plained as arising from the reduction in the antiferromagnetic k z k p k k order due to the thermal excitation of spin waves, i.e., A2 (T) m(T). Mattson et al.,2 on the other hand, in dc- where ak and bk are the Fourier transforms of the electronic magnetron-sputtered Fe-Cr superlattice, found (T) de- annihilation operator ci , defined on the two sublattices A and creasing as T2 at temperature below 100 K Eq. 3 in con- B, respectively. Here 2 mU and the sublattice magnetiza- 054408-7 MAJUMDAR, HEBARD, SINGH, AND TEMPLE PHYSICAL REVIEW B 65 054408 tion m is determined self-consistently. For simplicity we con- which has the right limiting behavior yielding the antiferro- sider the strong correlation limit in which at the HF level magnetic magnon energy Jz 1 cos2 qz as Jp 0 and the m 1. The interlayer band energy 2 z(k) 2tz cos kz mixes ferromagnetic magnon energy Jpqp as Jz 0. A completely the two ferromagnetic bands, and hence the quasiparticle different starting point in terms of a Heisenberg spin model band energies E(k) p(k) 2 z(k)2 have a mixed for the multilayer system, with planar and interlayer ex- character with features of both the ferromagnetic15 and anti- change energies Jp and Jz , would yield the same result. ferromagnetic ground states.16 As the planar ferromagnetic Going over now to the thermal excitation of magnons in band energy p(k) appears on the diagonal, the eigenvectors the multilayer system, the change m(T) m(T) m(0) in of the Hamiltonian matrix in Eq. A2 are unchanged from the sublattice magnetization at finite temperature T is ob- the AF case.16 tained by considering both the advanced and retarded modes Evaluation of the magnon propagator (q ), involv- in the spin-fluctuation propagator with appropriate Bose ing transverse spin operators (S ,S ) and representing weights. After subtracting out the zero-temperature quantum transverse spin fluctuations about the Hartree-Fock ordered fluctuation part the reduction in the sublattice magnetization state, has been described earlier in the random phase ap- is obtained as proximation RPA for both ferromagnetic15 and antiferromagnetic16 ground states. For the multilayer system 2 the magnon propagator is obtained as 2 q dq J 2 m T pdq p z z J pq p 0 2 2 2 q e q 1 . 2 Jz cos qz A5 q Jz Jpqp 1 J 2 2 z cos qz Jz Jpqp q 2 , A3 Here the upper limit of integration for the qp integral has been taken as for convenience, which is valid at tempera- for small planar momentum q 2 p (qx ,qy). Here Jz 2tz / tures low compared to Jp , as the high-energy modes have 4t2z/U is the exchange energy characterizing the antiferro- exponentially small weight. Integration over the planar mo- magnetic coupling between layers, and Jp , the magnitude of mentum qp finally yields which depends on details of the planar band energy p(k), plays the role of the planar exchange energy. The magnon energy 1 T /2 q is given by m T m 0 2 J dqz ln 1 . p 0 1 e Jz /T 1 cos2 qz 1/2 2 2 2 q J pq p Jz 2 Jz cos2 qz , A4 A6 *Author to whom correspondence should be addressed. Electronic Phys. Rev. Lett. 61, 2472 1988 . address: akm@iitk.ac.in 8 F. G. Aliev, V. V. Moshchalkov, and Y. Bruynseraede, Phys. Rev. 1 P. Gru¨nberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sow- Lett. 81, 5884 1998 . ers, Phys. Rev. Lett. 57, 2442 1986 . 9 A. Singh, Z. Tes anovic´, H. Tang, G. Xiao, C. L. Chien, and J. C. 2 J. E. Mattson, Mary E. Brubaker, C. H. Sowers, M. Conover, Z. Walker, Phys. Rev. Lett. 64, 2571 1990 . Qiu, and S. D. Bader, Phys. Rev. B 44, 9378 1991 . 10 J. M. Lannon, Jr., D. Temple, G. E. McGuire, C. C. Pace, and A. 3 A. Fert and I. A. Campbell, J. Phys. F. Met. 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