Resistance of multilayers with long length scale interfacial roughnessJason Alicea and Selman Hershfield Citation: Journal of Applied Physics 93, 7930 (2003); doi: 10.1063/1.1555799 View online: http://dx.doi.org/10.1063/1.1555799 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/93/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Influence of interfacial scattering on giant magnetoresistance in Co/Cu ultrathin multilayers J. Appl. Phys. 109, 033910 (2011); 10.1063/1.3544471 Effect of interface intermixing on giant magnetoresistance in NiFe/Cu and Co/NiFe/Co/Cu multilayers J. Appl. Phys. 94, 5881 (2003); 10.1063/1.1615704 Influence of the Ar-ion irradiation on the giant magnetoresistance in Fe/Cr multilayers J. Appl. Phys. 93, 5514 (2003); 10.1063/1.1559640 Interfacial roughness effects on interlayer coupling in spin valves grown on different seed layers J. Appl. Phys. 87, 3023 (2000); 10.1063/1.372390 Electrical resistivity in sputtered Co90Fe10/Ag GMR multilayers J. Appl. Phys. 81, 5793 (1997); 10.1063/1.364670

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Resistance of multilayers with long length scale interfacial roughnessJason Aliceaa) and Selman Hershfieldb)

Department of Physics and National High Magnetic Field Laboratory, University of Florida, Gainesville,Florida 32611-8440

~Presented on 14 November 2002!

The resistance of multilayers with interfacial roughness on a length scale large compared to the layerspacing is obtained using the Boltzmann equation. Both the current-perpendicular-to-plane~CPP!and current-in-plane~CIP! geometries are considered in the limits where the mean-free paths areshort and long compared to the atomic spacing. In the short mean-free path limit, the resistancedecreases in the CPP geometry and increases in the CIP geometry. In the long mean-free path limit,the resistance increases in both configurations due to enhanced surface scattering. The giantmagnetoresistance can either be enhanced or reduced by roughness depending on the sampleparameters. Estimates of the short and long mean-free path effects in Fe/Cr multilayers are obtainedusing experimentally determined parameters. ©2003 American Institute of Physics.@DOI: 10.1063/1.1555799#

I. INTRODUCTION

One of the controversial questions about the giant mag-netoresistance~GMR! is how its magnitude is effected by thepresence of long length scale interface disorder, i.e., interfa-cial roughness on a length scale which is large compared tothe atomic spacing. Several experiments have studied theeffects of this type of disorder in both the geometries wherethe current flows perpendicular to the layers~CPPgeometry!1–4 and in the plane of the layers~CIPgeometry!.5–10 Although current theories are consistent withthe CIP experiments,11,12 there are several unexplained ex-periments in the CPP geometry which yield apparently con-tradictory results.

Chiang et al.2 studied the role of interface disorder inCo/Ag multilayers and found that the CPP magnetoresistancewas reduced by increasing interfacial roughness while theCIP magnetoresistance was enhanced. In a set of experimentson Fe/Cr multilayers, Cyrilleet al.1 found that both the CPPand CIP magnetoresistances were enhanced by interfacialroughness. In the CPP case, the enhancement of the magne-toresistance was due to the zero-field resistivity,rAP , in-creasing with roughness while the high-field resistivity,rP ,remained roughly constant. Still, more recent experiments byZambanoet al.4 see no change in the CPP resistivities or themagnetoresistance of Fe/Cr multilayers with increasingroughness.

The effect of interfacial roughness on the CPP GMR istherefore unclear, with some work pointing to an enhance-ment of the GMR, some a reduction, and others no change atall. In this article, we examine theoretically the effects oflong length scale interface disorder in both the CIP and CPPgeometries using the Boltzmann equation.13,14

II. THEORETICAL FRAMEWORK

The multilayers we studied are shown in Fig. 1. Thelayers are separated by sinusoidal interfaces with amplitudeA and periodj, and within each layer the relaxation time,t i ,is constant. For numerical purposes, our calculations are per-formed in two dimensions, although we nonetheless expectthe results to remain qualitatively the same when generalizedto three dimensions.

The Boltzmann equation we used represents elastics-wave scattering within a current-conserving right-handside:

v•¹r f 2eE•¹pf 52S f 2 f̄

tD , ~1!

where f 5 f (r ,p) is the distribution function,f̄ 5 f̄ (r ,upu) is

a!Present address: Department of Physics, University of California, SantaBarbara, CA 93117; electronic mail: alicea@physics.ucsb.edu

b!Electronic mail: selman@phys.ufl.edu

FIG. 1. Geometry of the multilayers studied. The interfaces are modeled assine waves with amplitudeA and periodj. Layer i has a thicknessDyi anduniform relaxation timet i , and the total thickness of the sample isDy. Thecurrent flows in they direction in the CPP configuration and thex directionin the CIP configuration.

JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 10 15 MAY 2003

79300021-8979/2003/93(10)/7930/3/$20.00 © 2003 American Institute of Physics

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the spherical average off in momentum space, andt5t(r )is the relaxation time. For flat interfaces, the linear responsesolution to this Boltzmann equation can be obtained analyti-cally. Such a solution is not possible when long length scaleinterfacial disorder is present; however, there are two limit-ing cases for which one can obtain an essentially exact linearresponse solution for Eq.~1!: The short and long mean-freepath limits.15 These correspond to the cases where the elec-tronic mean-free paths are much smaller and much largerthan the layer thicknesses, respectively. In the short mean-free path limit, the layers behave as macroscopic pieces ofmetal and can essentially be regarded as resistors in series~CPP geometry! or parallel~CIP geometry!. Here, the resis-tance of the multilayer is governed by classical transportequations, and the Boltzmann equation reduces to a Laplaceequation for the electrochemical potential within each layer.In the long mean-free path limit, variations across the sampleare smeared out, and the multilayer behaves as a homoge-neous, isotropic metal. In this case, the gradient of the elec-trochemical potential can be approximated by its averagevalue, and the Boltzmann equation may be integrateddirectly.15

In order to compare with experiments, it is crucial toinclude surface scattering in these calculations. In the shortmean-free path limit, we model surface scattering by insert-ing a thin layer of a higher-resistivity material between thebulk layers. In the long mean-free path limit, surface scatter-ing is included on the right-hand side of Eq.~1! in addition tothe bulk scattering term, 1/tb , by writing

1

t~r !5

1

tb~r !1G(

iE d,d2~r2Ri~, !!, ~2!

where the integration runs along thei th interface,Ri(,) isthe position of thei th boundary, andG is a parameter char-acterizing the surface scattering rate. Equation~2! can bederived by modeling the interfaces as infinitesimal layers ofa higher-resistivity material.

III. RESULTS

The solution of the Boltzmann equation in the shortmean-free path limit shows that the effective CPP conductiv-ity increases with roughness while the CIP conductivity de-creases. These effects can be understood in terms of the cur-rent. In the CPP geometry, roughness allows the current totraverse a less-resistive, nonlinear path through the samplethat reduces the distance traveled through the low conductiv-ity layers. In the CIP geometry, where the layers behaveessentially as parallel resistors, most of the current short cir-cuits through the low resistance layers. Due to interfacialroughness, some of the current flowing through these layerswill be forced to impinge on more resistive layers near theinterfaces. This reduces the short-circuit effect of the highconductivity layers, leading to a decrease in the effectiveconductivity.

Quantitatively, in both geometries the fractional changein the conductivity due to roughness is proportional to(A/j)2, where the proportionality constant has a differentmagnitude in the CPP and CIP configurations and depends

on the layer conductivities and sample geometry. Estimatesfor these proportionality constants can be obtained by ap-proximating the interfaces as grooved rather thansinusoidal:12,15

dsCIP/CPP

sCIP/CPP'616S sCPP

sCIP21D S A

j D 2

, ~3!

wheresCPPandsCIP are the CPP and CIP conductivities forflat interfaces. Equation~3! only roughly approximates theexact proportionality constants computed numerically, but itis useful for comparing to experiments.15

In the long mean-free path limit, we find that the con-ductivity decreases with roughness due entirely to enhancedsurface scattering. When surface scattering is ignored, theconductivity is independent of roughness. To find how thesurface scattering is effected quantitatively by roughness, weextract the interface conductivity,s* , by treating the bulkand interface resistances as resistors in series. To a goodapproximation, the fractional decrease in the interface con-ductivity is equal to the fractional change in the interfacelength resulting from roughness, independent of the modelparameters. For the sinusoidal boundaries we consider, thefractional change in the interface conductivity is

ds*

s*'2p2S A

j D 2

. ~4!

Note that in the long mean-free path limit the conductivity isisotropic, and the CPP and CIP conductivities are the same.

To get the GMR, the difference of the resistivities in theantiparallel (rAP) and parallel (rP) configurations is com-puted: GMR5(rAP2rP)/rP . In the short mean-free pathlimit, when the resistance of each channel is computed nu-merically, we find that roughness can either enhance or re-duce the CPP GMR depending on the sample parameters,while in the CIP geometry, roughness induces a positiveGMR that vanishes when the interfaces are flat. In the longmean-free path limit, we also find that the GMR can be ei-ther enhanced or reduced by roughness depending on theparameters.

In an actual experiment, the effects in the two limitsdiscussed herein will both be present to some extent, as themean-free paths will be in neither limit, but somewhere inbetween. To see if we can account for the measurements ofCyrille et al.1 and Zambanoet al.,4 we now estimate the sizeof these two effects by computing the high-field and zero-field resistivities in Fe~3 nm!/Cr~1.2 nm! multilayers as afunction of the number of bilayersN using experimentallydetermined parameters. The roughness parametersA andj inour model were determined experimentally by Cyrilleet al.1,3 They found thatA increased within the multilayerwhile j remained constant at 10 nm. Thus, the interfaces intheir samples become more disordered as the sample size isincreased. We used for our calculations the average value of(A/j)2 across the multilayer. The bulk and interface resis-tances have been determined by Zambanoet al.4 We usedtheir measurements for the estimates because the resistancesin the two sets of experiments are roughly the same magni-tude even though the trends are different.

7931J. Appl. Phys., Vol. 93, No. 10, Parts 2 & 3, 15 May 2003 J. Alicea and S. Hershfield

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The results of these estimates in the CPP geometry,along with the corresponding measurements obtained byCyrille et al.,1 are shown in Fig. 2. The resistivity measure-ments obtained by Zambanoet al.4 are similar in magnitudebut do not depend onN. Because the changes in the resistiv-ities predicted in the long and short mean-free path limits areopposite in sign and of roughly the same magnitude, theseestimates show that the resistivities can either increase, de-crease, or remain constant with roughness depending onwhich effect is dominant. If the long mean-free path effect ismore important, one would expect trends similar to thosemeasured by Cyrilleet al.1 in the intermediate mean-freepath case becauserAP increases more thanrP in the long

mean free path limit. If the long and short mean-free patheffects cancel, however, one would expect to see the negli-gible changes in the resistivities observed by Zambanoet al.4

Thus, the effects described here are of the correct size toexplain what is seen experimentally, but a theory which ac-counts for an arbitrary mean-free path is needed in order tomake a detailed quantitative comparison to experiment.

ACKNOWLEDGMENTS

The authors would like to thank Tat-Sang Choy, JackBass, and Ivan Schuller for helpful discussions. This researchwas supported by the DOD/AFOSR Grant No. F49620-96-1-0026, the Center for Condensed Matter Sciences, the Uni-versity Scholars Program at the University of Florida, andthe National Science Foundation through the U.F. PhysicsREU Program.

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FIG. 2. Long and short mean-free path estimates of the parallel and antipar-allel resistivities for Fe~3 nm!/Cr~1.2 nm! multilayers as a function of thenumber of bilayersN. For comparison, measurements@obtained by Cyrilleet al. ~Ref. 1!# are also shown. Since the changes in the resistivities pre-dicted in the two limits are opposite in sign, there can be significant cance-lation of the two effects in samples with intermediate mean-free paths. Thus,depending on the mean-free paths, these results may account for either thetrends@seen by Cyrilleet al. ~Ref. 1!# shown in the figure or the negligiblechanges in the resistivities@observed by Zambanoet al. ~Ref. 4!# in thesame type of samples.

7932 J. Appl. Phys., Vol. 93, No. 10, Parts 2 & 3, 15 May 2003 J. Alicea and S. Hershfield

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