Origin of the Planar Hall Effect in Nanocrystalline Co60Fe20B20

K.M. Seemann,1,* F. Freimuth,2 H. Zhang,2 S. Blugel,2 Y. Mokrousov,2 D. E. Burgler,1 and C.M. Schneider1

1Peter Grunberg Institute (PGI-6) and Julich-Aachen Research Alliance (JARA-FIT), Forschungszentrum Julich,D-52425 Julich, Germany

2Peter Grunberg Institute (PGI-1), Institute for Advanced Simulation, and Julich-Aachen Research Alliance (JARA-FIT),Forschungszentrum Julich, D-52425 Julich, Germany

(Received 26 October 2010; published 19 August 2011)

An angle dependent analysis of the planar Hall effect (PHE) in nanocrystalline single-domain

Co60Fe20B20 thin films is reported. In a combined experimental and theoretical study we show that the

transverse resistivity of the PHE is entirely driven by anisotropic magnetoresistance (AMR). Our results

for Co60Fe20B20 obtained from first principles theory in conjunction with a Boltzmann transport model

take into account the nanocrystallinity and the presence of 20 at.% boron. The ab initio AMR ratio of

0.12% agrees well with the experimental value of 0.22%. Furthermore, we experimentally demonstrate

that the anomalous Hall effect contributes negligibly in the present case.

DOI: 10.1103/PhysRevLett.107.086603 PACS numbers: 72.25.Ba, 71.70.Ej, 72.15.Gd

Electron transport effects employing the electronic spindegree of freedom lie at the heart of spintronics and itsapplications not only in information technology [1] butmore increasingly also in sensorics for biomedical pur-poses, where the utmost sensitivity for detecting minutemagnetic stray fields is needed. The planar Hall effect(PHE) and the anomalous Hall effect (AHE) as well astunneling magnetoresistance are promising phenomena forrealizing highly sensitive sensors. From a materials pointof view, nanocrystalline Co60Fe20B20 (CoFeB) attractsgrowing attention since it combines large saturation mag-netization with low coercivity [2,3], both favoring highsensitivities.

PHE and AHE are both observed as a voltage transverseto the applied current [4,5] in contrast to the anisotropicmagnetoresistance (AMR), which is measured in the lon-gitudinal geometry. For PHE the magnetization M lies inthe plane spanned by the current density j ¼ jex andthe direction ey of the transverse voltage measurement,

and for AHE the component of M perpendicular to jand ey matters. Although AMR has been known since

Thomson’s—later known as Lord Kelvin—observationsin 1856 [6], PHE was discovered only a century later inpolycrystalline permalloy [7]. More recently, PHE has alsobeen found in crystalline La2=3Fe1=3MnO3 [8] and as a very

large effect at low temperatures in the dilute magneticsemiconductor (Ga,Mn)As [9]. The transverse resistancexy characterizing the PHE and the longitudinal resistivity

xx denoting the AMR are given by [10]

xy ¼ ðk ?Þ sin cos

xx ¼ ?1þ k ?

?cos2

;

(1)

where k (?) is the resistivity along (perpendicular to) thedirection of the in-plane component of M, and is the

angle enclosed by j and M. A transverse voltage ariseswhenever the current is neither perpendicular nor parallelto the magnetization. Even though the AMR is a subtlespin-orbit effect, quantitatively reliable predictions fromfirst principles based on the density functional theory(DFT) can be made [11,12].In this Letter, we elucidate the role of AMR in the PHE

in nanocrystalline Co60Fe20B20 thin films experimentallyand by ab initio calculations. We measure AMR and PHEin longitudinal and transverse four-probe transport experi-ments. The single-domain behavior enforced by an ex-change bias underlayer enables a direct comparison withtheory, and we calculate AMR and PHE signals by thesemiclassical Boltzmann approach using electronic struc-ture calculated from first principles. In order to account forthe nanocrystallinity of our real CoFeB system, an angularaveraging of the conductivity tensor is performed. Morespecifically, we first calculate Co0:75Fe0:25 (CoFe) withoutboron as the magnetically equistoichiometric pendant toCo60Fe20B20 and employ the virtual crystal approximation(VCA) [13] to take the disorder of CoFe into account. Inthe case of CoFeB we use a combined supercell plus VCAapproach. In view of the residual [110] texture that hasbeen found for CoFeB films [14], a possible contribution tothe transverse resistivity by anisotropic Berry-phase AHEis also examined [15]. However, in our measurements it isfound to be negligible. Hence, we unambiguously showthat PHE has the same physical origin as AMR by verifi-cation of the experimentally obtained transport data with afirst principles approach.Our experimental system consists of magnetron-

sputtered CoFeB with nanocrystalline morphology. Wedeposit films of 20 nm thickness in situ at room tempera-ture (RT) on an antiferromagnetic Ir20Mn80 underlayer.The exchange interaction between the antiferromagnetand the ferromagnet induces a unidirectional exchange

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anisotropy with the exchange bias fieldHex oriented alongey—perpendicular to the long axis of the Hall bar. We

measure the resistivities xy and xx in the typical Hall

geometry with a Hall contact size of 25 25 m2 at RTby recording the transverse and longitudinal voltage sig-nals, respectively. The external field H is applied in thefilm plane and swept between70 andþ70 mT, while thesample is rotated around the film normal againstH in stepsof 5. The rotation angle H is measured with respect toHex. The resistivity xx obtained is shown in Fig. 1(a) aftersubtraction of a constant offset and yields the AMR, whichis accompanied by a transverse resistivity xy, the PHE,

displayed in Fig. 1(b) after subtraction of a geometry-related offset.

In our experiments we control the direction H of theexternal field, which does not necessarily coincide with thedirection of M. In particular, at small magnetic fields andin the presence of magnetic anisotropy we have to intro-duce the angle between H and M, which can be derivedanalytically from the Stoner-Wohlfarth model based oncoherent single-domain behavior. Minimizing the free-energy density E ¼ MH cosðÞ MHex cosðH Þcomprising Zeeman and exchange bias contributions, weobtain

¼ arctan

Hex sinH

H þHex cosH

: (2)

In our case Hex is perpendicular to j and we set inEqs. (1) ¼ 90 þ H to obtain expressions thatcan be fitted to the experimental xx and xy data in

order to determine the intrinsic AMR ratio ½ðk ?Þ=?expt ð=Þexpt.Least-squares fits of Eqs. (1) using Eq. (2) to the AMR

and PHE data in Figs. 1(a) and 1(b) are displayed inFigs. 1(c) and 1(d). We find good agreement for all appliedfields H. Note that the total variations of both xx and xy

amount to the same value 5:0 108 cm as expectedfrom Eqs. (1). Therefore, our data confirm the magneticsingle-domain behavior of the CoFeB film. The fits con-sistently yield an exchange bias field Hex ¼ 5:2 mT and aRT AMR ratio of ð=Þexpt ¼ 0:22%.

We also observe a strong temperature dependence of thePHE signal, which we investigate in the temperature rangebetween RT and 10 K for two pronounced field directions:For H ¼ 45 we observe typical AMR hysteresis loops[Fig. 2(a)], and for H ¼ 90 the curves assume thecharacteristic shape of hysteretic Hall loops [Fig. 2(b)].The PHE curves forH ¼ 45 are asymmetric and shiftedbyH on the field axis due to the exchange bias effect. Theshift at RTof 1.7 mT is less thanHex ¼ 5:2 mT determined

Experimental dataAMR: ρxx

(10-8 Ω cm)

Fit

Angle of applied field ΦH (˚)

App

lied

field

H (

mT

)

0

0.00

5.00

3.75

1.25

2.50

20

40

-60

-40

-20

60

0

20

40

-60

-40

-20

60

0 180 3600 180 360

PHE: ρxy

(10-8 Ω cm)

-2.50

2.50

1.25

-1.25

0.00

(a)

(b) (d)

(c)

FIG. 1 (color online). Experimental resistivities xx (a) andxy (b) measured at RT as a function of field magnitude H and

direction H. Corresponding fits using Eqs. (1) are shown in (c)and (d) and yield a RT AMR ratio of ð=Þexpt ¼ 0:22%.

T = 250 K

T = 293 K

T = 200 K

T = 150 K

T = 100 K

T = 50 K

T = 30 K

T = 10 K

T = 250 K

T = 293 K

T = 200 K

T = 150 K

T = 100 K

T = 50 K

T = 30 K

T = 10 K

µ0H (mT)

(b) ΦH = 90°

(a) ΦH = 45°

Tran

sver

se v

olta

ge U

xy

-200 -100 0 100 200

17.8

19.2

23.0

25.5

27.6

30.4

32.1

34.2

8

6

400 100

T (K)200 300

5

10

∆ρxy

(10

-8 Ω

cm

)∆ρ

xy (

10-8 Ω

cm

)

0 100T (K)

200 3000

10

20

30 7

5

6

4

Hc

H)

Tm(

* (mT

)17.8

19.3

22.6

25.3

27.7

29.6

29.8

30.4

µV

µV

µV

µV

µV

µV

µV

µV

µV

µV

µV

µV

µV

µV

µV

µV

FIG. 2 (color online). Temperature dependence of the mea-sured PHE signal Uxy for (a) H ¼ 45 and (b) H ¼ 90. Thecurves are vertically offset for clarity and the Uxy values are

given for each temperature. Insets: Evolution with temperatureof the field shift H and coercivity Hc, respectively, togetherwith the field-dependent amplitude xy around zero field.

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from the transport data in Fig. 1 since the field is applied atan angle with respect to Hex. The curves for H ¼ 90represent hard-axis loops and are completely symmetricwith respect to H ¼ 0. The coercive field Hc of the hyste-retic Hall loops evolves very similar to the exchange biasshift of theH ¼ 45 data [lower (red) curves in the insetsof Fig. 2]. The temperature dependence of the PHE signalvariation around zero applied field xy is also depicted

in the insets of Fig. 2. Interestingly, the field shift Hincreases at lower temperatures, only making it more ob-vious that the PHE signal xy is actually a manifestation of

the AMR. A further point of evidence for the AMR natureof xy is that the field-dependent amplitude xy around

zero field scales negatively with increasing temperature,unlike AHE does [5].

We performed DFT calculations of the electronic struc-ture of CoFe and CoFeB using the full-potential linearizedaugmented plane wave (FLAPW) code FLEUR [16] and thegeneralized gradient approximation. For the theoreticaldescription of disordered bcc CoFe we make use of theVCA [13]. Using a 6.8 hartree plane-wave cutoff and a20 20 20 k mesh, the computed theoretical latticeconstant for Co0:75Fe0:25 is 0.282 nm and the average spinmagnetic moment is 2:02B, in very good agreement withcalculations based on the coherent potential approximation[17]. Within the FLAPW method VCA is only applicableto binary alloys with both atomic species adjacent in theperiodic table. Thus, we employ a combined supercell plusVCA approach for Co60Fe20B20. Experimentally, it hasbeen found that boron predominantly fills interstitial sitesin bcc CoFeB alloys rather than substituting for transitionmetal sites, and that the bcc CoFe lattice does not expandupon addition of boron [18–20]. Hence, we set up a bccsupercell consisting of 4 VCA CoFe atoms into which weincluded one boron atom at an interstitial site. In units ofthe lattice constant specified above, the Cartesian coordi-nates of the VCA atoms are ð0; 0; 0Þ, ð12 ; 12 ; 14Þ, ð0; 0; 12Þ,ð12 ; 12 ; 34Þ. The boron atom is located at ð14 ; 12 ; 0Þ. Since an

angular averaging of the resistivity tensor will be per-formed later, it is sufficient for the purposes of this studyto take only this single configuration into account, insteadof performing a configurational averaging. The calculatedaverage spin magnetic moment per magnetic VCA atom of1:27B is in good agreement with the value of 1:1B,which we measure experimentally for Co60Fe20B20.

In order to obtain the AMR ratio, ð=Þtheor, from firstprinciples, we compute the conductivity tensor , whichwithin the Boltzmann theory is given by

¼ e2

@2VN

Xkn

ðrkknÞ ðrkknÞðkn FÞ; (3)

with N as the number of k points and V as the unit-cellvolume. denotes the relaxation time (which we assume tobe k independent), kn is the single particle energy of bandn at k point k, and F is the Fermi energy. The group

velocity associated with the electron in band n is vkn ¼ð1=@Þrkkn. Spin-orbit interaction causes a dependenceof the band energies kn—and consequently also of theconductivity tensor —on the direction of the magnetiza-tion M leading to AMR. The function in Eq. (3) isapproximated by a Gaussian of width

ðNVÞ1=3 jrknknj. Inorder to compute Eq. (3) computationally efficiently weuse Wannier interpolation [21,22].Equation (3) is evaluated for 26 different magnetization

directions M ¼ ðsin cos’; sin sin’; cosÞ in both CoFe

and CoFeB. The resistivity tensor ðMÞ ¼ 1ðMÞ isexpanded in terms of the directional cosines of the mag-netization [23]. We found that in order to fit our data it issufficient to cut the expansion after the 4th order term. Thelongitudinal resistivity along the unit direction e ( ¼r; ; ’) in spherical coordinates is given by the projection

ðMÞ ¼ eðMÞe . The two resulting M-dependent

AMR ratios are given by ½rðMÞ ðMÞ=rðMÞ and

½rðMÞ ’ðMÞ=rðMÞ. The angular dependence of

the AMR ratios in ordered bcc CoFe displays cubic sym-metry. In the case of the CoFeB supercell, the inclusionof boron violates the cubic symmetry and therefore theresistivity exhibits an additional geometrical anisotropy inaddition to the anisotropy due to the spin-orbit interaction.However, this geometrical anisotropy may be eliminatedeasily by defining a new effective resistivity tensor,which is given by the sum of resistivity tensors rotatedabout the [111] direction. Consider, for example, the ma-trix elements xxðexÞ, yyðeyÞ, and zzðezÞ. From these the

matrix element xxðexÞ ¼ ½xxðexÞ þ yyðeyÞ þ zzðezÞ=3is constructed, and yyðeyÞ ¼ zzðezÞ ¼ xxðexÞ. The

resistivities for the other matrix elements and generalmagnetization directions are obtained analogously.The resulting AMR ratios of the effectively cubic

CoFeB system are shown together with those of CoFe for ¼ 90 and ¼ 45 in Figs. 3(a) and 3(b), respectively.The angular dependence of the AMR ratios is rather strong,as expected for single crystals [24,25]. The main effectof adding boron is a reduction of the AMR ratios whilequalitatively the angular dependence of the AMR ratios issimilar in the CoFe and CoFeB systems. The strong angu-lar dependence clearly shows that calculations for manymagnetic configurations have to be performed in order tocalculate the polycrystalline average reliably.In order to describe a nanocrystalline material in the

absence of texture we average over all possible grainorientations. Then the resistivities k and ? are related

to the angular averages of the projected resistivities

hrðMÞi and hðMÞ þ ’ðMÞi=2, respectively. This pro-cedure yields for VCA in CoFe an AMR ratio ofð=Þtheor ¼ 0:19%. For CoFeB a smaller AMR ratioof 0.12% is obtained, which is in good agreement withthe experimentally determined value of ð=Þexpt ¼0:22% considering the approximations made.

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Since a residual [110] texture has been found in CoFeBfilms [14] that can give rise to a magnetization componentperpendicular to the film, we also examine the possibilitythat the transverse signals shown in Figs. 1 and 2 maypartly be governed by the anomalous Hall effect [26]. Wehave performed a computational assessment of the intrinsicAHE based on the topological Berry curvature of Blochelectrons in single-crystalline bcc Co0:75Fe0:25 and also insingle-crystalline Co60Fe20B20. We obtain—for the exag-gerated case of a fully out-of-plane magnetized sample—anomalous Hall resistivities of AHE

CoFe 2:0 107 cmand AHE

CoFeB 3:0 107 cm, which both are 1 order of

magnitude larger than the measured PHE signal shown inFigs. 1 and 2. The anisotropy in the computed Berry-phaseAHE, dominant over other impurity and disorder drivencontributions to the AHE anisotropy [15], gives a field-dependent amplitude of AHE

CoFe 1:3 108 cm. In a

controlled experiment, we apply a saturating magneticfield normal to the film plane of the Hall bar device, findingthat in the case of full saturation the measured AHE signalmerely yields 1=10 of the PHE signal. This indicates thatthe role of finite disorder present in a CoFeB sample is tosignificantly suppress the intrinsic AHE, and, correspond-ingly, its anisotropy, precluding any significant contribu-tion to the measured field-dependent PHE signal.

In conclusion, we have elucidated in a joint experimen-tal and computational approach that PHE in CoFeBoriginates from AMR without contributions from AHE.Experimentally this is exhibited by an angle and

temperature dependent investigation of the transverse andlongitudinal signal. Using band structure calculations andsemiclassical Boltzmann transport theory to calculateAMR, we obtain good agreement with the experimentalAMR ratio of nanocrystalline CoFeB. The maximal AHEmeasured in CoFeB is significantly smaller than the an-isotropy of the Berry-phase AHE calculated for CoFe.K.M. S. thanks A. T. Hindmarch, C. H. Marrows, B. J.

Hickey, and Ph. Mavropoulos for fruitful discussions.Y.M. acknowledges the HGF-YIG Programme VH-NG-513 for funding and the Julich Supercomputing Centre forcomputational time.

*k.seemann@fz-juelich.de[1] I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76,

323 (2004).[2] S. Ikeda et al., Nature Mater. 9, 721 (2010).[3] A. Weddemann et al., New J. Phys. 11, 113027 (2009).[4] N. Nagaosa, J. Sinova, and S. Onoda, Rev. Mod. Phys. 82,

1539 (2010).[5] K.M. Seemann et al., Phys. Rev. Lett. 104, 076402

(2010).[6] W. Thomson, Proc. R. Soc. London 8, 546 (1856).[7] K. L. Yau and J. T. H. Chang, J. Phys. F 1, 38 (1971).[8] J. Li et al., J. Phys. Condens. Matter 22, 146006

(2010).[9] H. X. Tang et al., Phys. Rev. Lett. 90, 107201 (2003).[10] D. A. Thompson, L. T. Romankiw, and A. F. Mayadas,

IEEE Trans. Magn. 11, 1039 (1975).[11] J. Banhart and H. Ebert, Europhys. Lett. 32, 517 (1995).[12] S. Khmelevskyi et al., Phys. Rev. B 68, 012402 (2003).[13] L. Bellaiche and D. Vanderbilt, Phys. Rev. B 61, 7877

(2000).[14] Y. T. Chen and C. C. Chang, J. Alloys Compd. 498, 113

(2010).[15] E. Roman, Y. Mokrousov, and I. Souza, Phys. Rev. Lett.

103, 097203 (2009).[16] See http://www.flapw.de.[17] M. Lezaic, P. Mavropoulos, and S. Blugel, Appl. Phys.

Lett. 90, 082504 (2007).[18] C. L. Platt, N. K. Minor, and T. J. Klemmer, IEEE Trans.

Magn. 37, 2302 (2001).[19] A. Yang et al., J. Appl. Phys. 103, 07E736 (2008).[20] A. K. Rumaiz et al., Appl. Phys. Lett. 96, 112502 (2010).[21] J. R. Yates et al., Phys. Rev. B 75, 195121 (2007).[22] F. Freimuth et al., Phys. Rev. B 78, 035120 (2008).[23] T. R. McGuire and R. I. Potter, IEEE Trans. Magn. 11,

1018 (1975).[24] R. P. van Gorkom et al., Phys. Rev. B 63, 134432 (2001).[25] Y. Gondo and Z. Funatogawa, J. Phys. Soc. Jpn. 7, 41

(1952).[26] M. Bowen et al., Phys. Rev. B 71, 172401 (2005).

ϕ [°]

0

0.5

1

1.5

2A

MR

[%]

(a) θ=90°

0 15 30 45-0.5

0 15 30 45

ϕ [°]

-0.4

-0.2

0

0.2

0.4 (b) θ=45°

FIG. 3 (color online). Calculated M-dependent AMR ratiosas a function of the azimuthal angle ’ for (a) ¼ 90 and(b) ¼ 45. Co75Fe0:25 is illustrated in dark gray (blue)while Co60Fe20B20 is light gray (red). The solid lines denoteðr ’Þ=r and the dotted curves are ðr Þ=r.

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