PHYSICAL REVIEW B, VOLUME 63, 064407 Interface resistance of disordered magnetic multilayers K. Xia and P. J. Kelly Faculty of Applied Physics and MESA Research Institute, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands G. E. W. Bauer Department of Applied Physics and DIMES, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands I. Turek Institute of Physics of Materials, Academy of Sciences of the Czech Republic, CZ-616 62 Brno, Czech Republic J. Kudrnovsky´ and V. Drchal Institute of Physics, Academy of Sciences of the Czech Republic, CZ-182 21 Prague, Czech Republic Received 9 October 2000; published 19 January 2001 We study the effect of interface disorder on the spin-dependent interface resistances of Co/Cu, Fe/Cr, and Au/Ag multilayers using a newly developed method for calculating transmission matrices from first-principles. The efficient implementation using tight-binding linear-muffin-tin orbitals allows us to model interface disor- der using large lateral supercells whereby specular and diffuse scattering are treated on an equal footing. Without introducing any free parameters, quantitative agreement with experiment is obtained. We predict that disorder reduces the majority-spin interface resistance of Fe/Cr 100 multilayers by a factor 3. DOI: 10.1103/PhysRevB.63.064407 PACS number s : 75.70.Pa, 72.10. d, 71.15.Ap When two layers of magnetic material are separated by a and layer dependence of CPP-GMR remarkably well.3 Be- non-magnetic spacer layer, the electrical resistance of the cause it turns out to be strongly spin dependent and domi- system depends strongly on whether the magnetization direc- nates the magnetoresistance for layer thicknesses which are tions are aligned parallel or antiparallel. This effect is known not too large, the key to understanding CPP magnetoresis- as giant magnetoresistance GMR .1 The huge interest2­4 in tance lies in understanding the origin of the interface resis- the physics of GMR is largely driven by the wide application tance. The methodology which we have developed allows us potential of the effect, which has already been realized in to include specular and diffuse scattering on an equal footing magnetic recording heads. without introducing any arbitrary fitting parameters. GMR can be observed in a number of different measuring Explicit expressions for the interface resistance were de- configurations. The current-in-plane CIP configuration is rived by Schep et al.14 in terms of the transmission matrix T experimentally the simplest and is what is used at present in which describes how the electronic structure mismatch at an applications. However, for gaining a better understanding of A/B interface affects electron transport. In the limit in which the underlying physics, the current-perpendicular-to-the- there is no coherent scattering between adjacent interfaces, plane CPP configuration3,5­9 is preferred because of its presumably due to sufficiently strong bulk scattering, the in- higher symmetry, which should make it easier to understand, terface resistance is given by and because of higher MR ratios. The factors usually considered in theoretical treatments of h 1 1 GMR are the potential steps encountered by electrons pass- R 1 1 A/B , 1 ing from one material to another, impurity scattering in the e2 T 2 NA NB bulk of the layers, and defect scattering at the interfaces.2­4 There has been a great deal of discussion about the relative where T , are the probabilities for eigenstate in material importance of these ingredients and their spin dependence, A to be transmitted through the interface into the eigenstate which cannot be resolved solely on the basis of model cal- in material B, the sum is over the Fermi surface, and culations which include these effects in parametrized form. e2/h NA(B) is the Sharvin conductance of material A B . In Once the question has been suitably posed, however, detailed Ref. 14 transmission matrices and interface resistances were electronic structure calculations can be used to resolve the obtained for ideal Co/Cu interfaces using a first-principles issue quantitatively. For example, the effect of potential FLAPW-based embedding technique. These, and similar re- steps and their microscopic origin could be established in sults obtained by Stiles and Penn,15 demonstrated that a com- this way.10,11 bination of spin-independent bulk scattering and strongly In this paper we wish to address the relative role of specu- spin-dependent specular interface scattering arising from the lar and diffuse interface scattering. This has been studied by spin dependence of the band mismatch can account for the a large number of authors but so far only using simple mod- observed spin dependence of the interface resistances. These els which do not allow for detailed quantitative analysis of results are at odds with the common wisdom that metallic specific materials.12,13 We focus on the interface resistance of heterointerfaces cannot be perfect due to unavoidable rough- the resistor model which describes the observed thickness ness and/or interface alloying. Indeed, for the one case of 0163-1829/2001/63 6 /064407 4 /$15.00 63 064407-1 ©2001 The American Physical Society K. XIA et al. PHYSICAL REVIEW B 63 064407 TABLE I. Results of calculations. makes it very suitable for testing our method without the complicating factor of spin dependence. The interface resis- System Roughness Rmaj(f m2) Rmin(f m2) tance we find for a clean Au/Ag interface based on Eq. 1 is 0.047 f m2 which is very close to the experimental value, Au/Ag 111 clean 0.094 0.050 0.004 f m2. The resistance of a Au/Ag interface Au/Ag 111 rough a 0.118 becomes 0.059 f m2 when the interface contains two lay- Au/Ag 111 exp. 0.100 0.008 ers of Au0.5Ag0.5 alloy. The uncertainty arising from using different alloy configurations within the 6 6 unit cell is less Co/Cu 100 clean 0.33 1.79 than 1%. Cohcp /Cu(111) clean 0.60 2.24 We calculate the interface resistance of Co/Cu interfaces Co/Cu 111 clean 0.39 1.46 for both 100 and 111 orientations. The lattice constant Co/Cu 111 rough a 0.41 1.82 0.03 used for fcc Co/Cu is 3.549 Å. We focus mainly on the Co/Cu 111 exp. 0.26 0.06 1.84 0.14 111 orientation as this is the structure which is predomi- nantly seen in the experimental samples. The interface alloy Fe/Cr 100 clean 2.82 0.50 is again at least two atomic layers thick.8,9 We treat the in- Fe/Cr 100 rough a 0.99 0.50 terface disorder as two layers of CoCu alloy modeled using a an 8 8 lateral supercell. The largest uncertainty between 2 layers 50-50 interface alloy. different configurations of two layers of 50-50 alloy is about Co/Cu 111 interfaces for which direct comparison could be 2.5%, which is much smaller than the experimental error made with experiment, the agreement though reasonable, bar. For interface alloy compositions ranging from 50-50 to was not perfect. We therefore address the following ques- 44-56 the interface resistance does not change within our tions: Why does a calculation for a perfect interface agree as numerical accuracy. With two layers of interface alloy, the well as it does for a sample produced by sputtering? Can calculated transmission probability for the minority spin theory and experiment be brought into even better agreement electrons decreases by about 10% bringing the calculated by taking into account disorder? Is the finding that specular interface resistance into near perfect agreement with experi- interfaces are a reasonable first order approximation generic ment. We find that disorder gives rise to mainly forward or a coincidence found only for the Co/Cu system? scattering of the electrons so that the decrease of the ballistic The FLAPW-based method used in Ref. 14 was compu- component is almost canceled by the increase of the diffu- tationally too demanding to allow interface disorder to be sive part. This is the reason why the calculations for the treated. Starting instead with the more efficient surface defect-free Co/Cu interface14,15 were in reasonable agree- Green's function method16 implemented with a tight-binding ment with experiment. The strong diffuse scattering also ex- linear muffin tin orbital basis,17 we can now calculate the plains why the two-channel resistor model performs so well transmission and reflection matrices needed in the Landauer- down to relatively thin layers in which bulk scattering should Bu¨ttiker formulation of transport theory,18 but now for much not be important. larger systems. In this paper, we present the results of calcu- The resistance of a Cu/Co interface calculated with two lations for Co/Cu, Fe/Cr, and Ag/Au layered systems in layers of 50% interface alloy is 0.41 f m2 for the majority which we model interface disorder by means of large lateral spin and 1.82 0.03 f m2 for the minority spin. The error supercells. The electronic structure is determined self- bar for the minority-spin results from using a finite lateral consistently within the local spin density approximation. To supercell for modeling disorder and configuration averaging. model the interface, we randomly distribute the appropriate The majority spin bands of Cu and Co, similar to the bands concentration of different atoms within lateral supercells19 of Au and Ag, are so well matched that interface disorder has containing as many as 10 10 atoms. For the disordered lay- very little effect on the interface resistance. We observe ers the potentials are determined self-consistently using the Table I that there is near perfect agreement for the minority layer CPA approximation.16 The calculations are carried out spin certainly within the overall uncertainty of the calcula- with a k mesh density equivalent to 3600 k mesh points in tion but that the calculated resistance for the majority spin the two-dimensional Brillouin zone BZ of a 1 1 interface case which was already too large in the absence of disorder unit cell. The numerical error bar resulting from this sam- is even slightly increased by disorder. pling is smaller than 0.2% of the conductance. The interface With two layers of interface alloy, almost 80% of the resistances calculated for Co/Cu 100 and 111 in the clean minority spin conduction results from diffuse scattering. We limit using Eq. 1 agree with those obtained by Schep using can see this diffuse scattering in a different way by calculat- an entirely different code to within about 0.1 f m2 or 5%. ing the conductance for Cu/Co/Cu, Cu/Co/Cu/Co/Cu, and In the presence of defects, the conductance can be ex- Cu/Co/Cu/Co/Cu/Co/Cu in a manner quite analogous to that pressed as the sum of a ballistic part and a diffuse part; the described above for the single Cu/Co interface. To be able to transmission matrix elements between two Bloch states with perform this large calculation we had to use a smaller super- the same k correspond to ballistic scattering, those between cell (6 6) so that the error bar is larger than for the 8 8 two Bloch states with different k to diffuse scattering. The calculation. In these calculations the boundary Cu layers are calculated results are shown in Table I. The Au/Ag interface semiinfinite ``leads'' and the interface disorder is two layers has fcc 111 texture and the interface roughness is estimated of 50-50 alloy. Results are insensitive to the individual layer to be at least two layers thick in the MSU samples.20 This thicknesses, chosen here to be 10 atomic layers. In Fig. 1 the 064407-2 INTERFACE RESISTANCE OF DISORDERED MAGNETIC . . . PHYSICAL REVIEW B 63 064407 FIG. 1. Differential interface resistance as the number of inter- faces increase for a disordered Cu/Co multilayer embedded between Cu leads. differential resistance per interface is shown for all four sys- tems. For the minority spins, the deviation from the result obtained for a single interface is quite small. The strong dif- fuse interface scattering destroys the phase coherent scatter- ing between subsequent interfaces so that Ohm's law holds when the number of interfaces is increased. For the majority spin, however, the interface resistance does not obey Ohm's law but decreases as the number of interfaces increases. It appears to saturate at a value of 0.07 f m2 which is only a third of the experimental value, but consistent with Mathon's calculation for a multilayer with random layer thicknesses.21 The majority spin potentials for Co and Cu are so similar that the scattering from a double alloy layer is insufficient to FIG. 2. Number of propagating channels in the first Brillouin break the coherence which is considerably longer range than zone. a , b , and c are for the majority spin of a clean Fe/Cr for the strongly scattered minority spins. We would have to interface, bulk Fe, and bulk Cr; d , e , and f are for minority spin assume that the majority spin electrons remain coherent for electrons of a clean Co/Cu interface, bulk Co, and bulk Cu, respec- transport through four interfaces in order to obtain a value of tively. The numbers in brackets are the total number of propagating the average interface resistance close to the experimental channels per unit cell. The grayscale interpolates the number of value of 0.26 f m2. Compared to real samples with bulk propagating channels per k point between zero white and four defects and lateral variations in the layer thicknesses, it ap- black . pears that we overestimate the coherence length in the ma- posite to that of bulk Fe. The effect of disorder is to suppress jority spin case.4 the interface asymmetry rather than enhance it. As was the For the 111 orientation we also considered an interface case for the Co/Cu majority spins, interface disorder has only between hcp Co and fcc Cu. For a clean Co(0001) (111) hcp /Cufcc a small effect on the well-matched minority spin channel. interface both majority and minority spin resistances are sub- For the majority spin channel, however, the transmission stantially larger than for the fcc case and even larger than the probability for a clean interface is very low due to a large experimental values Table I . band mismatch. For a disordered interface, the ballistic con- The Fe/Cr interface resistance is computed for the tribution to the conductance can only decrease by a small bcc 100 orientation, which is the low index orientation with amount but the diffuse component increases enormously the largest spin-asymmetry.15 We used a lattice constant of leading to a large net increase in the transmission. 3% Fe 2.87 Å. To model the interface roughness a 6 6 lateral impurities in the first Cr layer or 3% Cr in the first Fe layer supercell was used. The uncertainty from configuration aver- increase the transmission probability by more than 10%. aging is less than 10%. Two interdiffused atom layers suppress the spin asymmetry Whereas for Co/Cu the majority-spin band structures and the MR efficiently-the interface resistances resulting were well matched, for Fe/Cr the situation is reversed and it from two 50-50 interface alloy layers are 0.99 and is the minority-spin electronic structures which match well. 0.50 f m2 for majority and minority spin, respectively. Using Eq. 1 , the interface resistance is 2.82 and Thus, the interface quality is much more critical for a large 0.50 f m2 for majority and minority spin, respectively, so CPP-MR in the Fe/Cr than in the Co/Cu system. that the Fe/Cr interface has a negative spin-asymmetry, op- The qualitative difference between Fe/Cr and Co/Cu can 064407-3 K. XIA et al. PHYSICAL REVIEW B 63 064407 be understood using Figs. 2 a ­2 c and Figs. 2 d ­2 f tem, interface disorder can increase or decrease the interface where we show as a function of k the number of majority- resistance. For some interfaces such as Au/Ag and Co/Cu, spin propagating channels for the Fe/Cr interface, bulk Fe, the band mismatch at an interface is responsible for most of and bulk Cr, respectively, and the number of minority-spin the interface resistance. For other systems such as Fe/Cr, the propagating channels for the Co/Cu interface, bulk Co, and interface resistance can dramatically depend on the interface bulk Cu, all in the first BZ. For the Co/Cu 111 interface perfection. For Fe/Cr interface, the majority-spin interface minority-spin states, the Fermi surfaces of both Co and Cu resistance is reduced by as much as 70% by interface disor- occupy a large part of the 2D BZ so that there are a large der. Interface disorder enhances the spin asymmetry in the number of states with the same k in both materials which Co/Cu system but decreases it for Fe/Cr. In the systems con- can, in principle, propagate in the absence of disorder. Sum- sidered, the diffuse scattering arising from interface disorder ming over all k , however, the transmission probability of breaks the phase coherence in high resistance spin channels, states coming from Cu is only about 60%; the character of but not necessarily for the low resistance spin channels. the bulk states on either side is such that they match poorly. Note added: After submission of the manuscript we were Defect scattering tends to reduce the transmission probability kindly informed by the authors about a manuscript by D. and thus increases the interface resistance of the Co/Cu mi- Bozec et al. Phys. Rev. Lett. 85, 1314 2000 . The theory nority spin channel. On the other hand, we can identify two part of that paper contains an empirical tight-binding study mechanisms by which interface disorder decreases the inter- of the limitations of the two-channel resistor model which face resistance in the Fe/Cr majority-spin channel by a factor agrees with the conclusions we draw from Fig. 1. 3. Majority spin electrons with small k are almost com- pletely reflected at the Fe/Cr specular interface because the This work is part of the research program for the ``Stich- electronic states on both sides of the interface do not match ting voor Fundamenteel Onderzoek der Materie'' FOM , well. Defect scattering is found to increase the transmission which is financially supported by the ``Nederlandse Organi- of these electrons strongly. Furthermore, for k outside of satie voor Wetenschappelijk Onderzoek'' NWO . This this central area, there are no propagating states on the Cr study was supported by the NEDO joint research program side. Propagating modes in Fe with larger k , which are NTDP-98 , the Grant Agency of the Czech Republic 202/ totally reflected at the specular interface, can be scattered 00/0122 , and the Grant Agency of the Academy of Sciences diffusely into the center of the BZ where there are many of the Czech Republic A1010829 . We acknowledge ben- states available in Cr. efits from the TMR Research Network on ``Interface Mag- In summary, we have studied the interface resistance of netism'' under Contract No. FMRX-CT96-0089 DG12- Co/Cu, Fe/Cr, and Au/Ag interfaces. Depending on the sys- MIHT . 1 M.N. Baibich et al., Phys. Rev. Lett. 61, 2472 1988 ; G. Binasch 11 P. Zahn, I. Mertig, M. Richter, and H. Eschrig, Phys. Rev. Lett. et al., Phys. Rev. B 39, 4828 1989 . 75, 2996 1995 . 2 P.M. Levy, Solid State Phys. 47, 367 1994 . 12 A. Brataas and G.E.W. Bauer, Phys. Rev. B 49, 14 684 1994 . 3 M.A.M. Gijs and G.E.W. Bauer, Adv. Phys. 46, 285 1997 ; J.- 13 S. Zhang and P.M. Levy, Phys. Rev. B 57, 5336 1998 . 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