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Magnetoresistance Anisotropy of Polycrystalline Cobalt Films:

Geometrical-Size- and Domain-Effects

Woosik Gil,1 Detlef Gorlitz,1 Michael Horisberger,2 and Jurgen Kotzler1

1Institut fur Angewandte Physik und Zentrum fur Mikrostrukturforschung,

Universitat Hamburg, Jungiusstrasse 11, D-20355 Hamburg, Germany2ETH Zurich & Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland

(Dated: 2nd February 2008)

The magnetoresistance (MR) of 10 nm to 200 nm thin polycrystalline Co-films, deposited onglass and insulating Si(100), is studied in fields up to 120 kOe, aligned along the three principaldirections with respect to the current: longitudinal, transverse (in-plane), and polar (out-of-plane).At technical saturation, the anisotropic MR (AMR) in polar fields turns out to be up to twice as largeas in transverse fields, which resembles the yet unexplained geometrical size-effect (GSE), previouslyreported for Ni- and Permalloy films. Upon increasing temperature, the polar and transverse AMR’sare reduced by phonon-mediated sd-scattering, but their ratio, i.e. the GSE remains unchanged.Basing on Potters’s theory [Phys.Rev.B 10, 4626(1974)], we associate the GSE with an anisotropiceffect of the spin-orbit interaction on the sd-scattering of the minority spins due to a film texture.Below magnetic saturation, the magnitudes and signs of all three MR’s depend significantly on thedomain structures depicted by magnetic force microscopy. Based on hysteresis loops and takinginto account the GSE within an effective medium approach, the three MR’s are explained by thedifferent magnetization processes in the domain states. These reveal the importance of in-planeuniaxial anisotropy and out-of-plane texture for the thinnest and thickest films, respectively.

PACS numbers: 73.50.-h; 73.50.Jt; 73.61.-r; 75.47.-m

I. INTRODUCTION

In applied magnetism, the coupling of the magneticmoment to spatial degrees of freedom plays a key role,and this especially applies to modern magneto- or spin-electronics1. Basically, this coupling is provided by thespin-orbit interaction, which in the example of magneto-transport causes the scattering rate of the conductionelectrons, τ−1, to depend on the direction of the lo-

cal magnetization ~M with respect to the current. Inthe archetypal bulk ferromagnets iron, cobalt, nickel,and their alloys, the resistance difference for orienta-

tions of ~M parallel and perpendicular to the current, i.e.the socalled anisotropic magnetoresistance ratio (AMR),∆ρ/ρ ≡

(

ρ‖ − ρ⊥)

/ρ, amounts to some percent. Innanostructured devices like magnetic multilayers, wires,or constrictions in the ballistic regime2, this ratio maybe enhanced to several ten percent.

Basically, the determination of the scattering rate

τ−1( ~M) and of the AMR requires the knowledge of thescattering potential and also of the spin-orbit split bandsat the Fermi-surface ∈F . Some special aspects of theAMR have been evaluated by Smit3, Berger4, Potter5,and Fert and Campbell6, however, the evaluation of

τ−1( ~M) for a realistic case is still lacking, at least to thebest of our knowledge. In this context, we note a recentab initio calculation of the intrinsic anomalous Hall-effectwhich, in contrast to the AMR, depends only to first or-der on the spin-orbit interaction and not on a scatter-ing potential. This quantity was obtained by integrat-ing the k-space Berry-phase over the occupied spin-orbitsplit states of iron7 and was found to agree up to some30 percent with data on iron whiskers8.

The present work is intended to a fairly systematicstudy of the AMR, which is of second-order in the spin-orbit interaction, also in an elemental 3d-ferromagnet.By selecting hcp cobalt with a rather well known band-structure9, some deeper insight into the AMR may be fa-cilitated. By choosing polycrystalline films, we are closerto devices which invariably use polycrystalline materials.We will vary the structural disorder and the temperaturein the films to probe the role of different scattering mech-nisms. These basic properties of the films under study areexamined in Sect.II. Section III is devoted to the AMRin the technically saturated state with main emphasis toa still unexplained phenomenon of the AMR, i.e. the so-called geometrical-size effect (GSE), previously observedin thin Ni10 and Permalloy11 films. Another point ofinterest will be the absolute value of ∆ρ at low tem-peratures: for Ni-alloys, already McGuire and Potter12

pointed out the unsensitivity of ∆ρ against significantvariations of the residual resistivity ρ(0). The influence ofdifferent domain structures, depending on the film thick-nesses, on the magnetoresistance, is investigated in Sec-tion IV and will be discussed by using the results onthe GSE. This low-field regime, where the in-plane AMRswitches at rather small coercive fields (Hc ≈ 10 Oe),may be of interest for applications despite the fact that∆ρ/ρ lies in the range of some percent. The summaryand conclusions are contained in Section V.

II. CHARACTERIZATION OF THE FILMS

By means of DC-magnetron sputtering at an Ar-pressure of 2·10−9 bar, cobalt films of thicknesses 10 nm,

2

10 20 30 40 50 60 70 80 100

0

0

Si

10nm

(004)(103)

(110)

(101)

(100)

(002)in

tens

ity (

a. u

.)

2θ

200nm

Si

188nm

10nm

10 20 30 40 50 60 70 80 100

0

0

Si

10nm

(004)(103)

(110)

(101)

(100)

(002)in

tens

ity (

a. u

.)

2θ

200nm

Si

188nm

10 20 30 40 50 60 70 80 100

0

0

Si

10nm

(004)(103)

(110)

(101)

(100)

(002)in

tens

ity (

a. u

.)

2θ

200nm

Si

10 20 30 40 50 60 70 80 100

0

0

Si

10nm

(004)(103)

(110)

(101)

(100)

(002)in

tens

ity (

a. u

.)

2θ

200nm

Si

188nm

10nm

Figure 1: Fig.1. Wide-angle X-ray diffraction patterns ofCo-films deposited by DC-magnetron sputtering on glass(188 nm) and on oxidized Si (100) (10 nm). The Miller indices(hkℓ) denote the reflections expected for the hcp structure atincident wavelength λCuKα

= 1.54 A.

20 nm, and 188 nm were deposited on Synsil-glass andoxidized Si(100) surfaces and capped by 3 nm thick Al-layers. The thicknesses were measured by a profilome-ter to an accuracy of 0.6 nm and controlled by high-resolution SQUID-magnetometry. X-ray diffraction di-agrams (XRD), as shown in Fig.1, revealed a polycrys-talline hcp-structure with a slight texture of the hexag-onal axis normal to the plane. Surface images recordedby an atomic force microscope (AFM, Q-ScopeTM250,Quesant Instruments Co.) yielded surface roughnessesbetween (1.5 ± 0.3) nm for 10 nm and (3.8 ± 0.5) nmfor 188 nm and indicated the grain sizes to increase from(25 ± 5) nm to (80 ± 5) nm. Within the error margins,these results turned out to be the same for both sub-strates. It is interesting to note that the grain sizes andtheir increase with thickness are consistent with a re-cent report for polycrystalline Co on glass and Si(100)substrates13.

The magnetic properties of all films have been inves-tigated by ferromagnetic resonance (FMR), hysteresisloops, and magnetic force microscopy (MFM). Using ahome-made FMR spectrometer operating at 9.1 GHz, thedirections and magnitudes of small uniaxial anisotropy

fields, ~Hu, in the film planes were determined . On 20 nmCo:Si and 188 nm Co:glass, for example, Hu = 22.3 Oeand 15.3 Oe, respectively, was obtained and the orienta-

tion of ~Hu could be related to the direction of the depo-sition process. Magnetization isotherms were measuredby a SQUID-magnetometer (Quantum Design MPMS2)

along three orthogonal directions of the applied field ~Hat temperatures, which were of interest for the analyses

of the magnetoresistances (MR’s) in Section IV. Therealso MFM images are presented in order to visualizethe domain structure underlying the magnetization pro-cesses, see Fig.7 below. For this purpose, the Q-scope wasequipped with a commercial tip, coated by a 40 nm thinhard Co-alloy (NanosensorsTM), and magnetized perpen-dicularly to the scanning directions. The directions ofthe in-plane magnetization were determined by monitor-ing the domain wall motion induced by a small magneticfield produced by external Helmholtz coils.

The resistances have been measured by an array of

four in-line contacts prepared parallel to ~Hu by ultra-sonic bonding. The driving currents were kept smallenough to produce linear responses and the resultingU/I-ratios were corrected for the sample geometry14 todetermine the resistivities of the films. The sample chipwas mounted to the end of an cold-finger extending fromthe cold-plate of a pulse-tube cooler (PRK, Transmit Co.Giessen, Germany) to the center of a warm-bore super-conducting magnet (130 kOe, Oxford Instruments). APID controller and a heater allowed stable sample tem-peratures between 70 K and 350 K. Measurements of themagnetoresistance in the domain states, i.e. at low mag-netic fields, were performed by means of an electromag-net, by which also the angle between current and fieldcould be varied. More experimental details are given inRef.15. We should mention here, that the structural,magnetic, and transport properties proved to be largelyindependent on the substrate, i.e. synsil-glass or oxi-dized Si(100)15. This feature indicates a dominant effectof the polycrystallinity of the films, i.e. of the depositionprocess. For some practical reasons, we selected threefilms with thicknesses between 10 nm and 188 nm for thepresent study.

The temperature dependence of the zero-field resistiv-ities is depicted in Fig.2 for these three films. The datacan be well described by a sum of three contributions

ρ(T ) = ρ(0) + ρph(T ) + ρm(T ). (1)

According to the inset, the residual resistivity increaseslinearly with the inverse thickness,

ρ(0) = ρb(0)[1 + dc/d].

The characteristic thickness, dc = (18 ± 1) nm cannotbe related to an extra scattering by the film surfaces16

or grain boundaries17 since, the theories predict the 1/d-behavior only for small deviations from the bulk limit,ρ(0) ≥ ρb(0). Hence, the observed increase of ρ(0)indicates scattering by an additional, yet unidentifieddisorder in the thinner films. Using the extrapolatedbulk value, ρb(0) = (11 + 1) µΩcm, the carrier den-sity 5.8 · 1022cm−3 from Hall-data for these films15, andthe free electron model for the conduction electrons inCo18, we find an upper limit for the mean free path,ℓe (0) = ~kF /nee

2ρb (0) ≈ 11 nm. Since this length is

3

0 100 200 3001.0

1.5

2.0

0 2000

20

60

10nm:Si(100)

20nm:Si(100)

188nm:s-glass

ρ(T

)/ρ(

0)

T (K)

ρ(0)

(µΩ

cm)

d-1 (µm)-1

Figure 2: Temperature variation of the zero-field resistance ofthe three Co-films under study. The solid line represents a fitto Equ.1, taking into account the contributions by phonon-mediated sd- and electron-magnon scattering. Inset: lineardependence of the residual resistivity on the inverse thicknessd, including a result for d=5 nm from Ref.5.

significantly smaller than the mean grain sizes observedby AFM, it may be associated with point-defect scatter-ing within the otherwise crystalline grains.

Since the electron-magnon scattering in Co, ρm(T ) =1.5 · 10−5(µΩcm K−2) T 2 19 is small, the temperaturevariation of the resistivities should be dominated byphonons. Indeed, by fitting ρ(T ) to the Bloch-Grueneisenform

ρph(T ) = ρph ·(

T

ΘD

)nΘD/2T∫

0

xn

(sinh x)2

dx

and taking for the Debye temperature ΘD=445 K, wefind an excellent agreement by setting for the exponentn=3, valid for phonon-mediated sd-scattering20,21. Thestrength of this scattering, ρph, becomes smaller in thethinner, more disordered films, however, due to couplingof the phonons to the complicated structure of the d-states, it is difficult to estimate, ρph, even for singlecrystals21.

Finally, it may be interesting to note that the presentresistivities of the 188 nm film are almost identical tothose obtained by Freitas et al.22 on a 300 nm Co filmdeposited by magnetron sputtering on glass. This ap-plies to the residual resistivity, ρ(0) = 14 µΩcm, as wellas to ρ(T ) at room temperature. Significantly smallerρ(0)-values have been detected on diode sputtered22 andepitaxial23 films of similar thickness.

-100 -50 0 50 100

25.7

25.8

26.0a) 20nm, 295K

Hp

Ht

Hl

H (kOe)

Hu

Hp H

t

Hl, I

∆ρp

∆ρtρ

(µΩ

cm)

0 90 180

0.995

1.000 b) 295K Ht=0.6kOe

ϕ I

188nm

20nm

10nm

ρ(ϕ)

/ ρ(

0)

angle ϕ

Figure 3: a) High-field magnetoresistance (MR) of the 20 nmCo-film at room temperature for the three principal direc-tions of the field (see inset), revealing the transverse and polarMR’s, ∆ρt and ∆ρp, and the appearance of a linear negativeMR at M > Ms. The dashed curve through the polar MR-data presents a fit to Equ.2. b) Normalized in-plane MR ofCo-films, ρ(ϕ)/ρ(0), recorded at room temperature as a func-tion of the angle ϕ between current and field above the sat-uration field of the in-plane magnetization. The solid curvesrepresent fits to Equ.2 valid for the anisotropic MR (AMR).

III. HIGH-FIELD MAGNETORESISTANCE

The MR of all films has been studied for three principal

directions of ~H, defined by the directions of the current

(~I|| ~Hu) and the film plane, see inset to Fig.3a. To give anexample, Fig.3a shows the three MR’s of the 20 nm filmat room-temperature. Starting from a common value at

low fields, a negative MR is found for all directions of ~H .While the longitudinal MR, ρ(Hℓ) decreases linearly withfield, the transverse and polar MR’s contain additionalcontributions. Above the saturation fields Hs, where the

films become homogeneously magnetized, ~M(H > Hs) =

Ms~H/H , these additional contributions to the MR also

saturate at values ∆ρt = ρℓ − ρt and ∆ρp = ρℓ − ρp,both indicated in Fig.3a. This contribution results from

4

the spin-orbit induced AMR, since upon rotation of themagnetization either to the transverse or to the polardirection, we realize the angular dependence

ρ(ϕ) = ρ(0) − ∆ρ · sin2ϕ, (2)

where ϕ is the angle between current and the direction

of the magnetization ~M . Such behavior is characteristicof the AMR of polycrystalline samples of cubic or hexag-onal ferromagnets12, and is illustrated by Fig.3b for the

in-plane rotation of ~M in a field H = 0.6 kOe > Hs.Details of the MR during saturation by (weak) in-plane

fields will be discussed in Section IV. Here we lookat the polar MR by increasing Hp in Fig.3a. SQUIDmagnetization data15 reveal Mp(Hp < Ms) = Hp due

to a rotation of ~M against the in-plane demagnetizing

field HN = −Ms, so that the angle between ~M and

current ~I(|| ~Hu) is determined by sin ϕ = Mp/Ms =Hp/Ms. For this case, Equ.2 predicts a parabolic de-crease, ρ(Hp)−ρ(0) = −∆ρp(Hp/Ms)

2, which is depictedin Fig.3a by the dashed curve in full agreement with thedata.

A. Spin-wave contribution

It is evident from Fig.3a that the linear MR, dρ/dH ,

is the same in all directions of ~H. No signature of theclassical Lorentz-MR, which is positive and proportionalto (M + H)2, is realized for H > Hs, even not at roomtemperature. Due to the small mean free path the ab-sence of this effect is plausible, while in epitaxial films itbecomes visible23. The linear MR has been realized be-fore in ρ(Hℓ) on epitaxially grown iron, cobalt, and nickelfilms on MgO and Al2O3

19 and was quantitatively dis-cussed in terms of elastic scattering by thermally excitedmagnons. Roughly spoken, the negative MR can be as-cribed to the suppression of low energy magnons, whichresults from the increase of the magnon gap proportionalto H. The strong thermal increase of dρ/dH is illustratedby Fig.4a for the longitudinal MR to which the AMR doesnot contribute. In Fig.4b, their temperature dependenceis shown for the three films under study and comparedto the result for a 7 nm thin Co-film obtained by Raquetet al.19. These authors fitted their data to a simplifiedmodel for sd-scattering by magnons21,

dρ

dH= A T

(

1 + 2d1T2)

ln

(

T

T0

)

, (3)

where the amplitude A changed only little from 3 to4 pΩcm/K kOe. Since A depends on the sd-exchange,numerical estimates are rather difficult. The coefficientd1 = D1/D0 is determined by the ratio of the mass renor-malization coefficient D1 and the zero-temperature stiff-ness of the spin-waves D0. Independent experimental

0 20 40 60-0.1

0.0

0 100 200 4000.0

0.4

0.8

1.2

1.6

2.0

350K

295K

250K

200K

Hl (kOe)

78K

a) 20nm Co

∆ρ (

µΩcm

)

b) 188nm 20nm 10nm 7nm (Ref.)

-dρ/

dH (

nΩ

cm/k

Oe)

T (K)

Figure 4: a) Longitudinal high-field MR of the 20 nm Co-film at various temperatures between 78 K and 350 K. b)Coefficient of the linear high-field MR (see panel a.) for allfilms vs. temperature. For comparision, the dotted curveshows results for a 7 nm polycrystalline Co-film from Ref.19,while the solid curves are fits to the prediction for scatteringby spin-waves, Equ.4.

data for Co yield d1 = 1.57 · 10−6 K−2 in good agree-ment with calculations, and it was argued19 that d1 mightbe rather insensitive to microstructural details of thefilms. Consequently, we fitted our data to Equ.3 admit-ting (plausible) variations in the amplitude A and founda larger value, d1 = 3 · 10−6K−2. We believe that thedifference is related to the rather strong disorder in thepresent films with a residual resistance ratio (RRR) near2 (see Fig.2), which contrasts to RRR=27 reported byRaquet et al.19 for their thickest films. Hence, one maysuspect that the granular disorder in our films gives riseto a stronger thermal renormalization of ’the spin-waveenergies’.

B. Anisotropic Magnetoresistance

At low temperatures, where the spin-wave contribu-tion vanishes, the AMR effect should prevail. This isdemonstrated in Fig.5 by the MR curves of the 20 nmand 188 nm films measured at T=78 K along the threeprincipal directions of the field. The significant differencebetween the MR’s of both films at smaller fields is relatedto the domain structure and will be discussed in Section

5

19.6

19.7

19.8

-60 -40 -20 0 20 40 60

-Ms

Ms

Hp

Ht

a) 20nm, 78K

Hl

∆ρp

∆ρt

ρ (µ

Ωcm

)

Ms

-Ms

H (kOe)

∆ρp

∆ρt

Hp

Ht

Hl

b) 188nm, 78K

16.8

16.9

17.0

ρ (µ

Ωcm

)

Figure 5: High-field MR of a) 20 nm Co- and b) 188 nm Co-films, measured at 78 K for the three principal directions of~H relative to the current. As for the in-plane MR, see Fig.3b,the quadratic decrease of the polar MR’s, ρp(H), observed forM < Ms (dot curves) signalizes the AMR-effect also. Abovetechnical saturation, M(H, T ) ≥ Ms, the saturation valuesof the transverse (∆ρt) and the polar (∆ρp) AMR’s are indi-cated.

IV. Here we focus on the saturated transverse and po-lar AMR’s, ∆ρt and ∆ρp, which differ significantly fromeach other, but do not change very much with thickness(essentially the same observation is made on the 10 nmfilm). This phenomenon is one of our main results: for allthicknesses, the polar AMR turns out to be about twiceas large as the transverse AMR.

At first, a sizable difference between both MR’s, ∆ρp >∆ρt, has been reported by Chen and Marsocci10 forsingle- and poly-crystalline nickel films. They coinedthis feature as ’geometrical size effect’ (GSE) and be-lieved that it may arise from the electronic structure in-side the film material. More recently, this size-effect hasalso been detected on sputter-deposited 4.5 nm to 100 nmthin Permalloy films11 at a low temperature, T=5 K. Thisstudy revealed that by raising the degree of (111)-texturein the film, ∆ρp was increased so that the ratio ∆ρp/∆ρt

tended towards two. An attempt to explain this GSEby assuming an anisotropic scattering rate due to dif-fuse scattering at the film boundaries, however, did notprovide conclusive results11.

In order to explore the AMR and the GSE of our Co-

0.0 0.2 0.4 0.6 1.0-0.5

0.0

0.5

1.5

0.0

0.1

0.2

0.3

∆ρp/ρ

(%

)

ρ(0)/ρ(T)

∆ρp/

∆ρt

188nm20nm10nm

a)

∆ρp

(µΩ

cm)

0 100 200 4000

1

2

T (K)

b)

Figure 6: Temperature dependences of a) the saturation val-ues of the polar magnetoresistivity ∆ρp and b) of the geomet-rical size-effect. All solid curves are guides to the eye. Inset toa): Parker-plot analysis of ∆ρp/ρ; the straight lines throughthe data indicate a common negative phonon-contribution tothe MR.

films in some more detail, the absolute values and thethermal behavior of both ∆ρ′s are summarized in Fig.6.Two remarkable features should be emphasized: (i) de-spite different temperature variations, the MR’s of allfilms can be extrapolated to the same value at T=0,as shown in Fig.6a for the polar direction; (ii) Fig.6bdemonstrates that the GSE, i.e. the ratio of the polarand transverse AMR’s, remains almost independent oftemperature.

At first, we address to the AMR postponing the dis-cussion of the GSE to the following subsection. Athickness-independence of ∆ρ itself rather than of the ra-tio ∆ρ/ρ has been pointed out earlier for Ni0.7Co0.3 andNi0.8Fe0.2 alloys (see Fig.17 of Ref.12). For all presentCo-films, ∆ρp(0) = 0.19 µΩcm follows from Fig.6a, andwe suspect that the origin of this AMR resides in the crys-talline regions, to which we tentatively assigned alreadythe bulk residual resistivity, ρb(0) = 11 µΩcm, in Sect.II.There we determined the mean-free path, ℓe = 11 nm,which turned out to be much smaller than the grain sizesestimated from AFM images15. Therefore, we relate thelow-temperature AMR ∆ρ(T → 0) also to the scatteringwithin the crystalline grains and believe that the extrascattering, which enhances ρ(0) in the thinner films (seeinset to Fig.2), produces a negligible AMR. In fact, aweak AMR is expected for scattering potentials with re-duced symmetry, e.g. associated with phonons12 or corre-

6

lated structural disorder (grain boundaries, dislocations),because in these cases the directional symmetry-breaking

effect by the magnetization ~M via the spin-orbit interac-tion becomes less effective.

Analyzing the effect of temperature, i.e. of phonon-scattering, we employ the widely used Parker-plot24,based on the relation for the AMR ratio,

∆ρ (T )

ρ (T )=

[(

∆ρ

ρ

)

d

−(

∆ρ

ρ

)

T

]

ρ (0)

ρ (T )+

(

∆ρ

ρ

)

T

. (4)

This equation is valid under the two premises: (i) theelectric transport is dominated by one spin-channel, i.e.the majority channel in Co 18, and (ii) Matthiessen’s ruleapplies for the thermal and defect scattering. The valid-ity of the latter has been demonstrated by the fits ofρ(T )/ρ(0) to Equ.1 indicated in Fig.2. Then a plot of∆ρ(T )/ρ(T ) vs. ρ(0)/ρ(T ) allows to separate the ther-mal contribution to the AMR, (∆ρ/ρ)T , from the defectone, (∆ρ/ρ)d. In fact, the extrapolation of the ’high-temperature’ data, shown by the inset to Fig.6a, is con-sistent with a common intercept at (∆ρ/ρ)T = -0.40 %.Such negative contribution to the AMR has been real-ized early on crystalline Permalloy25 and, more recently,also on polycrystalline Co-films22. It was associated withphonon-scattering rather than with magnon contribu-tions. At lower temperatures, our data break away fromthe straight lines, which in the thickest film occurs ata rather high temperature, where ρ(0)/ρ(T ) ≈ 0.6. Thisfeature indicates a change of the dominant defect type forscattering and has also been observed by Freitas et al.22

on various Co-films with different ρ(0)′s, i.e. differentdegrees of disorder.

C. Geometrical Size Effect

As a guide for discussing the GSE, we refer to Potter’swork5, who calculated the AMR’s produced by majorityand minority spins for polycrystalline cubic ferromag-nets. He assumed an isotropic scattering potential, asit may be present in the grains of our films. Calculatingthe sd-scattering rates, Potter considered the effect of thespin-orbit interaction on localized 3d-states, but ignoredthe influence on the band-structure. Therefore, we ex-pect only a more or less qualitatively correct guidanceby infering the AMR-ratio from Ref.5:

∆ρ

ρ=

1 + r

2 + r

3√

3

64

(

KSO

∈d

)2

− r

560

(

KSO

2 ∈ex

)2

. (5)

Here KSO ≈ 0.1 eV measures the spin-orbit coupling en-

ergy HSO = KSO~L · ~S. The positive contribution to

Equ.5 arises from the longitudinal part of HSO mixingtwo 3d-orbitals of the minority bands separated by ∈d

near the Fermi-surface ∈F . The negative term is dueto the nondiagonal part of HSO, which admixes some ofthe exchange-split majority states to the minority band.The parameter r = τ−1

sd /τ−1ss accounts for the different

scattering rates of the conduction electrons into the 4s-and 3d-states and is mainly determined by the densityof states of the 3d-bands at ∈F . Because the exchangesplitting 2 ∈ex is significantly larger than ∈d, the nega-tive majority spin contribution to the AMR may be smallrelative to the positive one. Taking r ≈ 10 from a recentexperiment on Co-films18, ∆ρ/ρ = (3

√3/64)(KSO/ ∈d

)2 follows from Equ.5. Comparing this estimate withour result for the transverse AMR at low temperatures,∆ρt(0)/ρb(0) ∼= 10−2, we obtain for the effective splittingof the two unperturbed 3d-levels, ∈d≈ 3.0 KSO ≈ 0.3 eV.This finding for ∈d becomes smaller if a finite contribu-tion by the majority spins would be considered in Equ.5,but it seems to be reasonable regarding the other simpli-fying assumptions of the theory5. Here we mention theneglect of the effects of the lattice potential and the spin-orbit interaction on the Fermi-surface and on the densityof states at ∈F

5, and also of possible hybridizations be-tween the s- and d-orbitals6.

Nevertheless, we will extend Potter’s results derivedfor an ’isotropic’, i.e. polycrystalline cubic ferromag-net, to films with polar texture. Let us recall that theAMR originates from a symmetry breaking of the 3d-

orbitals by the magnetization ~M via the spin-orbit cou-pling. The resulting anisotropic charge distribution givesrise to the scattering asymmetry of the conduction elec-trons into these 3d-states. On general grounds one mayexpect that a reduction of the symmetry of the ferro-magnet structure weakens the AMR12, loosely spoken,because then the magnetization induced axial anisotropyof the orbitals becomes less effective. Since the texturein the permalloy11 and in our cobalt films, both perpen-dicular to the plane, appear to be strongly correlatedwith the GSE, we assume the mixing parameter in Equ.5,k2

α = 3√

3(KSO/4 ∈α)2, to be different for the in-plane

(α = i) and the polar (α = p) directions of ~M . ThenEqu.5 remains still valid for the in-plane orientations of~M and ignoring again the small contribution by the ma-jority spins, we have

ρℓ − ρt

ρ=

1 + r

2 + r

3√

3

4k2

i . (6)

In order to determine the effect of the film anisotropyon the polar MR, we introduce k2

α directly into Potters5

result for the perpendicular conductivity of the minorityspins, σα

⊥/σ0 = (3√

3/2r) · ln[

r/(1 + 12

rk2α)

]

. For small

spin-orbit perturbations, rk2α ≪ 1, the difference between

the transverse and polar resistivities becomes:

ρt − ρp

ρ=

1 + r

2 + r

3√

3

4

(

k2i − kp

2)

. (7)

By some trivial algebra we obtain for the GSE from

7

Equs.6, 7:

∆ρp

∆ρt= 2 −

k2p

k2i

. (8)

Hence, this simple model can explain the upper limit oftwo of the GSE, which emerges from our data in Fig.6band also from Fig.6 in Ref.11 for Permalloy films. More-over, this model ascribes the GSE to the electronic struc-ture, as it was suspected by Chen and Marsocci10. Con-sequently, the GSE should not depend on the tempera-ture which is fully consistent with our results depicted inFig.6b.

Equation 8 also predicts that the upper limit of twois reached, when the mixing effect due to the polar ori-ented magnetization is small compared to mixing by the

in-plane ~M , i.e. k2p << k2

t . This case seems to be re-alized in our films, see Fig.6b, and also in the Permal-loy films with increased 〈111〉-epitaxy (Fig.6 of Ref.11).These observations indicate that the spin-orbit induced

anisotropy in the 3d-orbitals near ∈F is smallest, if ~M isaligned parallel to the existing axial perturbation builtin by hcp- or 〈111〉-epitaxy. In this case, the 3d-orbitalshave already the axial symmetry so that an induced mag-netization along the epitaxial (polar) direction may haveonly a moderate effect on the scattering probability intothese states. This is in distinct contrast to the in-plane

orientation of ~M which breaks the symmetry of these or-bitals. Therefore, the mechanism proposed here for theGSE qualifies the film anisotropy of the AMR more pre-cisely as structural, rather than as a geometrical effect.

IV. LOW-FIELD MAGNETORESISTANCE

A. Domain Structures

The formation of domains affects the MR’s of the188 nm thick film and of the thinner films, d≤ 20 nm,in quite different ways. The interesting features can al-ready be realized on the large field scale of Fig.5: (i)for d=20 nm Co (and also for 10 nm, not shown) both,the polar and the transverse MR’s approach the field-independent longitudinal MR, ρℓ, whereas the polar andthe longitudinal MR’s of the 188 nm film tend to the field-independent transverse resistance ρt. In order to providesome solid basis for a detailed discussion of these char-acteristic features of the domain MR’s, we examine thedomain structures by magnetic force microscopy (MFM).

The essential difference between the thick (188 nm)and the thinner films can be infered from MFM imagesof the demagnetized states, shown in Fig.7a. The imageshave been recorded in the dynamic mode of the Q-scopewhich is sensitive to the polar gradient of the polar force,i.e. δFp/δxp = Mp δ2Hp/δx2

p. The 20 nm film con-sists of large, some 10 µm wide domains with in-planemagnetizations separated by 180 Neel walls. The do-main magnetizations are oriented parallel to the uniaxial

Figure 7: Stray field images obtained by magnetic force mi-croscopy: a) In the demagnetized states of 188 nm Co on glass(10 × 10µm2) and 20 nm Co on Si(100) (25 × 25µm2). Forthe 20 nm film, only domains with in-plane magnetizationsalong the uniaxial anisotropy field ~Hu are observed, while the188 nm film displays a maze-structure with an out-of-planecomponent of ~M . b) In the remanent states, the 188 nmfilm reveals stripe domains parallel to the previously appliedfields, ~Hℓ and ~Ht. The insets show the Fourier-transformswhich show the mean size and the directions of the stripedomains.

anisotropy field ~Hu as determined by FMR. A slight lon-

gitudinal ripple of ~M about ~Hu is visible, which mostlikely arises from the polycrystallinity of the film.

In constrast, the 188 nm film exhibits a maze con-figuration of stripe domains with sizable polar compo-nents of the domain magnetizations. The Fourier trans-form of the image in Fig.7a yields a mean width ofthe stripes, dD = (205 ± 15) nm, being rather close tothe thickness as expected for weak stripes by magneto-static reasons26. Recently, the same observations werereported for a 195 nm thin polycrystalline Co-film onglass and related to a hexagonal texture perpendicularto film plane13. MFM images depicted on epitaxial Co-films revealed a reorientation of the domain magnetiza-tion from in-plane to polar between 10 nm and 50 nm27,28 which was explained in terms of the perpendicularmagnetocrystalline anisotropy of Co. These results sug-gest that also in our case the hcp texture, realized bythe XRD (Fig.1), generates such a crystalline anisotropy,which in the 188 nm film becomes large enough to pro-

duce a significant polar component of ~M . Let us alsorecall that we supposed this texture already in the dis-cussion of the GSE.

The other interesting property of these weak stripes isseen in Fig.7b. In the remanent states, stripe patterns arefound aligned with the direction of the previously appliedfields Hℓ or Hb. This socalled rotable anisotropy can be

8

-0.10 -0.05 0.00 0.05 0.10-1

0

119.7

19.8

a) 20nm, 78K

t

l

b) Hl

Ht

M/M

s

H (kOe)

ρ (µ

Ωcm

)

Figure 8: a) Low-field MR of 20 nm Co measured at 78 K forin-plane fields applied longitudinally (Hℓ) and transversely(Ht) to the current. The solid curves are fits to the AMReffect (Equs.2,9) using the magnetization curves M(Hℓ) andM(Ht) as shown in panel b). The longitudinal field has beenapplied along to the growth-induced uniaxial anisotropy fieldHu = 20 Oe determined by ferromagnetic resonance15.

attributed to the stiffness of the domain walls against de-formations26 and is probably supported by a pinning ofthe walls by local anisotropies in the granular structure.The rotatable anisotropy suggests also an ’isotropic’ hys-teresis loop, the shape of which should be independent onthe direction of the in-plane field. In fact, we do observethis feature on the 188 nm film, see Fig.9 below, and willrefer to it when discussing the MR in the domain state.

B. Anisotropic Magnetoresistance

We begin with the low-field resistance of the thin films,exemplified by Figs.3a, 5a for d=20 nm: both the trans-verse and the polar MR’s, ρ(Ht → 0) and ρ(Hp → 0),tend to the longitudinal one, ρ(Hℓ). This behavior isreadily explained by the fact that the resistance is mea-

sured along ~Hu, and that at low fields the domain mag-netization is also directed parallel to Hu evidenced byMFM (Fig.7a). The parabolic decrease of ρ in larger po-

lar fields, ∆ρ(Hp < Ms) ∼ −H2p , was already attributed

to the AMR resulting from the rotation of ~M from an in-plane to the polar direction. Also the detailed variationof the in-plane MR’s, shown in Fig.8a, can be explainedby the AMR. Using the hysteresis loops M(Hi) in Fig.8b,and assuming the relations for the angle ϕ in Equ.2,

cosϕ (Hℓ) = M (Hℓ) /Ms, (9a)

-1.0 -0.5 0.0 0.5 1.0-1

0

1

t

l

a) 188nm, 78K

b) Hl

Ht

M/M

s

H (kOe)

16.9

17.0

ρ (µ

Ωcm

)

Figure 9: a) Low-field MR of 188 nm at 78 K in longitudinaland transverse fields. As in Fig.7, the solid curves are fits toEqus.2,9 using the magnetization curves displayed in panelb). Note the inversion of the longitudinal and transverse fieldMR-variations in comparison to the 20 nm film, shown inFig.8a.

sin ϕ (Ht) = M (Ht) /Ms, (9b)

the in-plane MR can be described rather nicely. Thephysical arguments for these agreements are: (i) the lon-

gitudinal magnetization process, M(Hℓ), is due to thenucleation of 180 Neel walls (see Fig.7a) at the coer-cive field Hc = −Hu (determined by FMR), which thenrapidly cross the film leaving the resistance unchanged;(ii) upon reduction of the transverse field, on the other

hand, a longitudinal ripple of ~M about ~Ht appears which

originates from ~Hu (see e.g. Ch.5.5 of Ref.26). Accord-

ingly, the components of ~M parallel and antiparallel to

the current ~I|| ~Hu are growing continuously so that ρ(Ht)increases until the transverse coercive field Ht

c < Hu is

reached. There the magnetization component along ~Hu

changes sign and increases at the expense of the ripple,so that ρ(Ht) back to ρt at larger negative fields.

A rather different behavior is displayed by the 188 nmthick film. Already in Fig.5b we noticed that at low fieldsthe polar and the longitudinal MR’s tended to the trans-verse MR. As a rather unexpected feature, the transverseMR turned out to be nearly independent of the field alsoin the domain regime, ρ(Ht) = ρt. The detailed varia-tion of the in-plane MR’s at low fields is shown in Fig.9arevealing just the opposite to the behavior of the thinfilms (see Fig.8a): the longitudinal MR displays a strongfield dependence, while the transverse MR remains very

9

small. These results are explained also by the AMR ef-fect. The in-plane MR, shown in Fig.9b, is rather nicelyreproduced by the solid curves which have also been cal-culated from Equ.2. Again, the mean angle ϕ between

current and magnetizations ~M( ~H) has been determinedfrom Equ.9 and the hysteresis loops, Fig.9b. As a mat-ter of fact, we emphasize that the shape of these loopsdoes not depend on the direction of the in-plane field (’ro-tatable loops’). This is consistent with the correspondingbehavior of the weak stripe domains depicted by MFM inFig.7b. In contrast to the thin films, d≤20 nm, no effectby the uniaxial in-plane anisotropy field, Hu = 15 Oe,determined by FMR15, is realized. The much larger co-ercive field, Hc ≈ 200 Oe, stems most likely from thepinning of the stripe domain walls by the random poly-crystalline anisotropy in the films.

C. Effective Medium Approach

Aiming at a more detailed description of the MR inthe 188 nm film, again the domain structure has to betaken into account. For this purpose, we use an effectivemedium model, by which Rudiger et al.23 successfullyinterpreted the AMR of epitaxial Co-films. Introducingthe volume fractions vi for different domain species, theAMR is approximated by

∆ρ( ~H) =

3∑

i=1

υi( ~H)∆ρi. (10)

Here the ∆ρi denote the AMR’s of the corresponding do-main with polar, transverse or longitudinal orientations

of ~M relative to current and film plane. By definition is

∆ρℓ = 0 and if, for convenience, ∆ρ( ~H) is normalize tothe transverse MR, Equ.10 takes the form

∆ρ( ~H)

∆ρt= υt( ~H) + gsυp( ~H), (11)

where gs = ∆ρp/∆ρt denotes the GSE-ratio. The sim-plest case, υt = υp = 0 and, hence, ∆ρ = 0 has beenrealized on the thin films at low fields.

The most interesting example is the 188 nm film, where(i) the low-field MR appears to be inverted relative to thethin films and, moreover, (ii) the transverse resistivityremains at the saturation value ∆ρ(Ht) = ∆ρt, even inthe domain state. We will now attempt to relate thesestriking features displayed by Figs.5b, 9 to the domainstructure observed by MFM, see Fig.7. Observation (ii)in connection with Equ.11 implies for the concentrationof polar oriented domains,

υp(Ht) =1

gs[1 − υt(Ht)] . (12)

Below the saturation field, the magnetization M(Ht) andtherefore, υt(Ht), starts to decrease at the expense of a

finite polar component υp, which leads to the nucleationof stripe domains. Upon further reduction, Ht → 0, thehysteresis loops display a normalized remanent magne-tization M(Ht → 0)/Ms = 0.66(2), i.e. volume frac-tion υt(0) ∼= 2/3. For an estimate, we take the maxi-mum GSE, gs = 2, to find from Equ.11 υp(0) = 1/6 and

by using3∑

i=1

υi = 1,the same longitudinal volume fraction

υℓ(0) = 1/6 = υp(0). The agreement of both volumes im-plies that the nucleation of polar domains is accompaniedby the creation of an equal amount of longitudinally ori-ented domain. Considering the square-like cross-sectionof the stripes following from Fig.7b, this result indicatesthat the flux extending from the polar phase is closed bythe longitudinal volume υℓ(−Hc). The rotatable sym-metry of the hysteresis loops implies for the longitudi-nal direction also υp(Hℓ → 0) = υt(Hℓ → 0) = 1/6.For the longitudinal MR Equ.10 predicts then ∆ρ(Hℓ →0)/∆ρt = 1/6+2 ·1/6 = 1/2, which is in close agreementwith the measured value, see Fig.9a.

Finally, upon reduction of Ht to the coercive field,υp(Ht) increases further. The volume fraction of the po-lar domains at −Hc can be estimated from the stripemaze of the demagnetized state, Fig.7a, which suggestsυt(−Hc) = υℓ(−Hc). Then, from Equ.11 and simple al-gebra we obtain υp(−Hc) = 1/3 = υℓ(−Hc) = υℓ(−Hc).Thus the demagnetized state consist of equal volumes forall six possible magnetization directions, which by con-sidering the symmetry of the stripe structure is again aplausible result.

V. SUMMARY AND CONCLUSIONS

The magnetoresistance of polycrystalline Co-films,which were characterized by XRD, FMR, SQUID-magnetometry, AFM, MFM and temperature variableresistivity, has been investigated in fields up to 100 kOedirected along three principal directions. In the satu-rated state, the MR displayed the socalled geometricalsize-effect (GSE), according to which the MR for the po-

lar orientation of ~M is up to twice as large as for the

in-plane ~M . The determination of the GSE was facili-tated by the facts that the spin-wave contributions couldbe easily subtracted and that in the present disorderedfilms the classical Lorentz MR proved to be negligible.Basing on a correlation between the GSE and a texturedetected previously on Permalloy films11 and also on ourCo-films by XRD, we proposed to attribute the GSE toan anisotropic mixing of the 3d-levels near ∈F by thelongitudinal part of the spin-orbit interaction. By ex-tending Potter’s5 prediction for the AMR of the minor-ity spin channel, we obtained a result which is consistentwith the observed upper limit of two for the GSE andalso with the temperature independence of the GSE. Arelation of the GSE to the electronic structure has al-ready been conjectured in the literature10, but not yetbeen worked out. Of course, regarding the simplicity of

10

the proposed extension and the assumption of a simplespherical Fermi-surface for the final 3d-states in Ref.5,which considers only the local aspect of the spin-orbitinteraction, these consistencies may be fortuitous. How-ever, we believe that the central argument for the ap-pearance of the GSE, i.e. the presence of an additionaluniaxial symmetry in polycrystalline films through a tex-ture in thin films remains valid. Hence, a more detailedreasoning for the GSE considering also the effect of thespin-orbit interaction on the band-structure is indicatedin order to check the present rough model.

The MR’s in the domain state were interpreted usingthe saturated AMR’s and the GSE, the hysteresis loops,and MFM images of the domain structure. For thin Co-films, d ≤ 20 nm, where the magnetization remained the

film-plane and became for ~H → 0 parallel to the weakuniaxial anisotropy field, the MR attained the maximum(longitudinal) value at zero magnetization. The MR inthe domain state of the thickest film, d=188 nm, on theother hand, displayed a rather different behavior. As afunction of the transverse field in the film-plane, the re-sistance turned out to be almost constant, whereas uponreducing longitudinal and polar fields the resistances de-creased and increased, respectively, from their different

saturation values to the transverse MR. MFM imagesand hysteresis loops revealed the formation of rotatablestripe domains with square cross-section due to the hcptexture. By means of an effective medium model23, theMR’s could be quantitatively explained in terms of a fluxclosure configuration of the magnetization componentsabout the directions of the stripes. Approaching the co-ercive field, the stripes terminated in a maze configura-tion, and the fractional volumes of all three magnetiza-tion components proved to be equal. In this model, thesurprising field independence of the transverse MR re-sults from the squared corss-section of the stripes withtransverse flux-closure and from a GSE ratio of two. Weshould note that this discussion did not invoke (possiblesmall) contributions to the MR by the Neel- and Bloch-walls in the thin and thick films, respectively. Such effectshave been reported before in epitaxial Co-films23,29,30

with strong hcp crystalline anisotropy and quantitativelydifferent domain dimensions.

The authors are indebted to the late Prof. J. Ap-pel (Hamburg) for his encouraging interest during theearly steps of this work. Discussions with Proff. P. Boni(Munchen), D. Grundler (Hamburg) and M. Morgenstern(Aachen) are gratefully acknowledged.

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