PHYSICAL REVIEW B 86, 144414 (2012)

Scaling of the paramagnetic anomalous Hall effect in SrRuO3

Noam Haham,1 James W. Reiner,2 and Lior Klein1

1Department of Physics, Nano-magnetism Research Center, Institute of Nanotechnology and Advanced Materials,Bar-Ilan University, Ramat-Gan 52900, Israel

2HGST, a Western Digital company, 3403 Yerba Buena Road, San Jose, California 95315, USA(Received 2 September 2012; published 22 October 2012)

We extract the paramagnetic anomalous Hall effect (AHE) resistivity ρAHExy in thin films of the itinerant

ferromagnet SrRuO3 and show that the AHE coefficient Rs scales with the longitudinal resistivity ρxx . We fitthe resistivity dependence of ρAHE

xy assuming two mechanisms: side jumps and Karplus-Luttinger (Berry phase)mechanism. For the latter, we consider the effect of a finite scattering time with a fractional power-law relationto ρxx .

DOI: 10.1103/PhysRevB.86.144414 PACS number(s): 75.47.−m, 72.25.Ba, 75.50.Cc, 72.15.Gd

I. INTRODUCTION

The anomalous Hall effect (AHE)1 is a fundamentaltransport property of magnetic conductors discovered morethan a century ago; nevertheless, it still attracts considerableinterest. Being a manifestation of an interplay between spinand charge currents, there is special interest in the AHEin connection with the emerging field of spintronics.2 Assome models attribute the AHE to topological propertiesof the conduction bands, there is interest in the AHE inconnection with other intriguing transport phenomena affectedby topology such as the quantum Hall effect and transportproperties of topological insulators.3,4 However, despite theintense theoretical and experimental effort a comprehensiveunderstanding of this phenomenon is still lacking.

The AHE is manifested as a transverse resistivity ρAHExy ,

assuming the current flows along the x axis and the transversevoltage is measured along the y axis, and it is linked to theintrinsic magnetization ( �M) of the conductor. When the AHEis present, it adds to the Hall effect ρHE

xy , which is given bya combination of the ordinary Hall effect (OHE) ρOHE

xy andρAHE

xy :

ρHExy = ρOHE

xy + ρAHExy . (1)

The OHE is proportional to the magnetic field B and is givenby

ρOHExy = R0B⊥, (2)

where R0 is the OHE coefficient and B⊥ is the component ofthe magnetic field in the z axis.

The expression for ρAHExy depends on the mechanisms

responsible for the AHE. Contributions to the AHE due toextrinsic mechanisms are given by

ρAHExy = Rsμ0M⊥, (3)

where Rs is called the anomalous Hall coefficient and M⊥ is thecomponent of the magnetization in the z direction. Commonly,Rs is given by Rs = aρxx + bρ2

xx where the linear term inρxx is attributed to the skew scattering mechanism5 and isexpected to dominate in the high conductivity regime (σxx >

106 �−1 cm−1) and the quadratic term is attributed to the sidejumps mechanism6 and is expected to dominate in the lowconductivity regime (σxx ∼ 104–106 �−1 cm−1).

In addition to the extrinsic mechanisms, band topologicalproperties give rise to AHE via the Berry phase (Karplus-Luttinger) mechanism.7–11 In this case, ρAHE

xy = −ρ2xxσxy( �M),

where σxy( �M) is determined by the band structure. The Berryphase mechanism is expected to dominate in the same regimeas the side jump mechanism.

The AHE in the itinerant ferromagnet SrRuO3 (Ref. 12)has attracted considerable interest for its nonmonotonictemperature dependence including a sign change close toits Curie temperature. The mechanisms responsible for thisbehavior are controversial. While the Berry phase scenario,assuming a temperature-dependent exchange gap, seems toagree qualitatively with the data,8 it has been shown thatthe AHE vanishes at a particular resistivity and not at aparticular magnetization as one might expect.13 Midinfraredmeasurements suggest the applicability of the Berry phasescenario at energies above 200 meV while the dc limit isdominated by extrinsic scattering mechanisms.14 Furthermore,we have shown recently that the AHE in SrRuO3 scales in theferromagnetic phase with ρxx .15

The scaling and the nonmonotonic functional form of theAHE were found to be consistent with two contributions tothe AHE coefficient: (a) a Berry phase contribution witha temperature-independent band structure while taking intoconsideration the finite scattering rate and (b) a side jumpscontribution. Here we show evidence that the scaling betweenthe AHE and ρxx extends into the paramagnetic phase.

We extract ρAHExy from ρHE

xy by subtracting the contributionof the OHE. We then extract M⊥ by using a relation betweenfield-induced magnetization and magnetoresistance and showthat the extraction is consistent with the expected mean-fieldbehavior with no fitting parameters. Knowing ρAHE

xy and M⊥we extract Rs and analyze its dependence on temperature andresistivity. We find that the AHE in SrRuO3 is determinedby the total resistivity, irrespective of its sources (phononsor magnons). This is demonstrated by showing a scalingof the extracted Rs with ρxx , when ρxx is changed bychanging the temperature or by applying a magnetic field.We also find that the data of different samples with differentthicknesses collapse on a universal curve when applying thescaling procedure described in Ref. 15. We show that theAHE in both the ferromagnetic phase and the paramagneticphase is consistent with the scenario proposed in Ref. 15

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NOAM HAHAM, JAMES W. REINER, AND LIOR KLEIN PHYSICAL REVIEW B 86, 144414 (2012)

when taking into account the fractional power-law relation be-tween the scattering rate and longitudinal resistivity observedexperimentally.16,17

II. SAMPLES AND EXPERIMENT

Our samples are epitaxial thin films of SrRuO3 grown onslightly miscut (∼2◦) substrates of SrTiO3 by reactive electron-beam evaporation. The films are untwinned orthorhombicsingle crystals, with lattice parameters of a ∼= 5.53 A, b ∼=5.57 A, and c ∼= 7.85 A. The Curie temperature Tc ofthe films is ∼150 K and they exhibit an intrinsic uniaxialmagnetocrystalline anisotropy. Above Tc the easy axis is alongthe b axis which is oriented 45◦ relative to the film normal.18

We use several films with thickness d between 27 and 90 nm;the data presented are for a 27-nm film if not mentionedotherwise. The films are patterned to allow transverse andlongitudinal resistivity measurements that are performed withPPMS-9 (Quantum Design). The field-antisymmetric trans-verse resistivity is extracted by changing the voltage andcurrent leads. We measure the temperature dependence ofthe longitudinal and transverse resistivity between 165 and300 K with magnetic fields up to 8 T applied along the easyaxis.

III. EXPERIMENTAL RESULTS

A. Extracting ρAHEx y

Figure 1(a) shows the temperature dependence of the an-tisymmetric transverse resistivity (ρHE

xy ) for different magneticfields (0–8 T) applied along the easy axis in the paramagneticphase (165–300 K). As we apply a field at 45◦ from the filmnormal (namely, with a component along z), ρHE

xy has both anOHE contribution and an AHE contribution. We determinethe OHE contribution according to the methods described inRefs. 13 and 19 and we find that it corresponds to a carrierdensity of 1 electron per Ru (R0 ∼ −3.7 × 10−2 μ� cm/T),consistent with previous reports.13,19 Subtracting ρOHE

xy fromρHE

xy we extract ρAHExy shown in Fig. 1(b). We note that for

low fields the AHE decreases with increasing temperature.However, for high fields (H > 4 T) the behavior is nonmono-tonic. This behavior suggests competing effects. As we willsee below, while the magnetization decreases with increasingtemperature, Rs increases with increasing temperature. Forhigh fields the increase in Rs dominates for some temper-ature interval near Tc where the magnetization is close tosaturation.

B. Extracting M⊥

A direct measurement of the magnetization in the para-magnetic phase is rather challenging due to the strongdiamagnetic signal of the substrate, which is much largerthan the paramagnetic signal of the thin films. Thus, we usemagnetoresistance measurements to extract the field-inducedmagnetization. Figure 2(a) shows the temperature dependenceof ρxx for different magnetic fields (0–8 T) applied along theeasy axis. We notice a large negative magnetoresistance near Tc

which decreases as temperature increases. Figure 2(b) showsthe magnetic resistivity defined as �ρ(T ,H ) ≡ ρxx(T ,H ) −

FIG. 1. (Color online) The Hall effect ρHExy (a) and the AHE

resistivity ρAHExy (b) as a function of temperature for different fields.

ρxx(T ,0) as a function of temperature for different fields.The relation between magnetoresistance and magnetizationestablished previously20 for SrRuO3 is given by

�ρ(T ,H ) = −aM2, (4)

where a is a constant. This relation was used to extract thecritical exponents of the ferromagnetic phase transition andwas shown to apply near Tc by demonstrating the universalscaling relation related to the phase transition.20

We extract for every sample the coefficient a from theresistivity below Tc as described in Ref. 20. As we showbelow, the field-induced magnetization in the paramagneticphase determined using Eq. (4) is consistent with the mean-field approximation. Assuming that for T sufficiently aboveTc mean-field approximation applies, the magnetization isexpected to obey the self-consistent equation

M/Ms = Bs

(gμBSH

kBT+ 3ST eff

c

(S + 1)TM/Ms

), (5)

where Bs is the Brillouin function, T effc is the effective

mean-field Curie temperature, and Ms is the saturationmagnetization. Combining Eqs. (5) and (4) we expect

MMR/Ms = Bs(gμBHeff/kBT ), (6)

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SCALING OF THE PARAMAGNETIC ANOMALOUS HALL . . . PHYSICAL REVIEW B 86, 144414 (2012)

FIG. 2. (Color online) (a) Resistivity ρxx vs temperature for different magnetic fields applied along the easy axis. (b) Magnetic resistivity�ρ, extracted from (a), vs temperature for different fields. (c) Mcalc/Ms defined in the text vs

√�ρ for different T eff

c at T = 165 K.(d) MMR/Ms vs gμBHeff/kBT defined in text for T � 180 K (circles) and T < 180 K (squares). The solid line is the Brillouin function. Inset:extracted optimal T eff

c as a function of T .

where

MMR/Ms ≡√

�ρ

a/Ms,

gμBHeff/kBT ≡ gμBSH

kBT+ 3ST eff

c

(S + 1)T

√�ρ

a/Ms.

For SrRuO3 in the paramagnetic phase S = 1 and Ms ∼288 emu/cm3.21 T eff

c was extracted by considering the fielddependence of the magnetic resistivity at constant tempera-tures. According to Eq. (4) we expect a linear relation between√

�ρ and M . We calculate the expected field dependenceof the magnetization at a specific temperature according toEq. (5) assuming different values of T eff

c and determinethe value for which the best linear fit is obtained for thecalculated magnetization Mcalc and

√�ρ. Figure 2(c) shows

the calculated magnetization Mcalc normalized by the saturatedmagnetization Ms as a function of

√�ρ assuming different

T effc at T = 165 K. We obtain the best linear fit for T eff

c =156 K. The figure also shows that assuming T eff

c = 150 K orT eff

c = 160 K leads to a noticeably worse linear fit.For each temperature the optimal T eff

c is found, and thetemperature dependence of T eff

c is presented in the inset ofFig. 2. The error bars are determined by the distribution of the

optimal T effc for a randomly chosen subset (six of nine points)

of the data. We notice that T effc slightly grows with temperature

above Tc. At temperatures higher than 180 K the error bar inT eff

c increases significantly as M becomes linear with H for thefield range we use. However, we expect T eff

c to saturate. UsingT eff

c = 159.5 K and plotting MMR/Ms vs gμBHeff/kBT yieldsa good scaling of the data in the whole temperature and fieldrange as expected from Eq. (6); moreover, the scaling functionis in good agreement with the expected Brillouin function withno fitting parameters [see Fig. 2(d)]. We note, however, thatdeviations from the expected Brillouin function start belowT = 180 K. Applying this analysis for different samples yieldsa similar scaling, with the same scaling function. This resultsupports the use of Eq. (4) in our analysis.

Although the scaling was performed assuming atemperature-independent a, a good scaling is obtained if sometemperature dependence is assumed. To bound this variationwe extract the magnetic susceptibility for high temperatures(T > 200 K) using Eq. (4) assuming a certain variation.Using the Curie-Weiss law, we extract the effective Curietemperature, and constrain it to be in the range of T eff

c valuesextracted at lower temperatures (see inset of Fig. 2). We findthat we cannot exclude a change in the value of a by +10%to −15% between 150 and 300 K for the 27-nm-thick sample.

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NOAM HAHAM, JAMES W. REINER, AND LIOR KLEIN PHYSICAL REVIEW B 86, 144414 (2012)

For the 50-nm sample the variation is +13% to −15% and forthe 90-nm film it is +5% to −20%. Knowing a we can useEq. (4) to determine the magnetization in the relevant rangeof temperatures in the paramagnetic phase for given field andtemperature:

M =√

�ρ(H,T )/a. (7)

C. Extracting Rs

Using Eq. (3) assuming temperature-independent a weextract Rs for different temperatures and fields by dividingthe AHE resistivity ρAHE

xy by the extracted magnetization M .22

Figure 3(a) shows the temperature dependence of the extractedRs for different fields. The curves for the different fields differ

FIG. 3. (Color online) (a) Extracted AHE coefficient Rs as afunction of temperature for different fields. (b) AHE coefficient Rs ex-tracted at different temperatures and fields as a function of resistivity.

significantly, especially near Tc. Figure 3(b) shows Rs as afunction of ρxx extracted for different fields and temperatures.We note that Rs scales with ρxx . The scaling implies that Rs issensitive to the total ρxx irrespective of its sources; for differentfields the same resistivity is obtained at different temperaturesand thus with different weights for magnon and phonon scat-tering. The obtained scaling is consistent with previous resultsin the ferromagnetic phase.15 We considered the resistivity de-pendence of Rs in both ferromagnetic and paramagnetic phasefor all of the samples, and followed the scaling procedure de-scribed in Ref. 15; namely, we normalized the resistivity withthe resistivity at which the AHE vanishes in the ferromagneticphase (the normalized resistivity is denoted as ρ∗) and Rs withthe absolute value of the negative peak value of Rs in the ferro-magnetic phase (the normalized Rs is denoted as R∗

s ) in order to

FIG. 4. (Color online) (a) The AHE coefficient Rs normalizedby the absolute value of its negative peak value R∗

s as a function ofresistivity normalized by the resistivity at which Rs vanishes, ρ∗. Thesolid lines are for the extreme Rs values. (b) AHE coefficient Rs vsresistivity ρxx for a 50-nm sample; the solid line is a fit to Eq. (8).Inset: fit error E as a function of the assumed exponent α.

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SCALING OF THE PARAMAGNETIC ANOMALOUS HALL . . . PHYSICAL REVIEW B 86, 144414 (2012)

correct for geometry errors. Figure 4(a) shows R∗s as a function

of ρ∗ for three samples. We note that the curves of the differentsamples in both the ferromagnetic and paramagnetic phasecollapse on a single curve. We also note that R∗

s continuessmoothly from the paramagnetic phase into the ferromagneticphase. The solid lines represent the possible variation range ofRs based on the upper and lower bounds for a.

IV. THEORETICAL MODEL

The AHE in the ferromagnetic phase was describedassuming a Berry phase contribution (with a temperature-independent band structure) which considers the effect of finitescattering rate and a side jumps contribution. According to thisscenario Rs is expected to satisfy the equation15

Rs = ρ2xx

B

�2 + (h/τ )2+ Cρ2

xx, (8)

where B is a constant associated with Berry phase, � is acharacteristic energy gap in the band structure, C is associatedwith the side jumps contribution, and τ is the scatteringtime. Considering the common relation 1/τ ∝ ρxx , we cantry to fit the data to the equation. However, Eq. (8) does notprovide a good description for both the ferromagnetic andparamagnetic phase. A possible source for this failure is theassumed relation between ρxx and the scattering time. We notethat several groups confirmed a relation ρ = D/τα where D

is a constant and α ∼ 0.4–0.5.16,17 Using this relation with D

and α ∼ 0.4 we find a very good fit as shown in Fig. 4(b) withside jump �y ∼ 0.2 A and with a reasonable energy gap of� ∼ 18 meV. We note that using Eq. (8) to fit the data belowTc while assuming ρ ∝ 1/τ yields higher values for � and

�y;15 however, the data in the extended temperature intervalappears to be inconsistent with such an assumption.

We also checked the sensitivity of the fit to changes in theexponent α. We assume different values of α and fit the data toEq. (8). For each α we calculate e(α), the total error betweenthe data and the fit defined by the root of sum of squares of thedifference between the fit and data points. The inset in Fig. 4shows the normalized error E(α) ≡ e(α)/e(α = 0.45) vs theassumed exponent α. We note that reasonable fits are obtainedfor α ∼ 0.37–0.58 for which the error is less than two timesits minimum value.

V. CONCLUSIONS

We measured the paramagnetic AHE in SrRuO3 for a widetemperature and field range. We find that the AHE coefficientRs scales as a function of resistivity, and thus is insensitive tothe source of scattering as resistivity is changed by changingthe temperature or by applying a magnetic field. The functionalform of the AHE is consistent with a combination of the sidejumps mechanism and the Karplus-Luttinger (Berry phase)mechanism which considers the effect of finite scattering timethat has a fractional power-law relation with ρxx with anexponent α ∼ 0.37–0.58.

ACKNOWLEDGMENTS

We acknowledge useful discussions with E. Shimshoni.L.K. acknowledges support by the Israel Science Foundationfounded by the Israel Academy of Sciences and Humanities.J.W.R. grew the samples at Stanford University in thelaboratory of M. R. Beasley.

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22The extraction was done only for fields and temperatures whichproduced sufficiently large magnetoresistance.

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