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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 1

Department of Physics, Nano-magnetism Research Center, Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel 2 HGST, a Western Digital company, 3403 Yerba Buena Road, San Jose, California 95315, USA (Received 2 September 2012; published 22 October 2012) AHE in thin films of the itinerant We extract the paramagnetic anomalous Hall effect (AHE) resistivity ρxy ferromagnet SrRuO3 and show that the AHE coefficient Rs scales with the longitudinal resistivity ρxx . We fit AHE assuming two mechanisms: side jumps and Karplus-Luttinger (Berry phase) the resistivity dependence of ρxy mechanism. For the latter, we consider the effect of a finite scattering time with a fractional power-law relation to 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 fundamental transport property of magnetic conductors discovered more than a century ago; nevertheless, it still attracts considerable interest. Being a manifestation of an interplay between spin and charge currents, there is special interest in the AHE in connection with the emerging field of spintronics.2 As some models attribute the AHE to topological properties of the conduction bands, there is interest in the AHE in connection with other intriguing transport phenomena affected by topology such as the quantum Hall effect and transport properties of topological insulators.3,4 However, despite the intense theoretical and experimental effort a comprehensive understanding of this phenomenon is still lacking. AHE , The AHE is manifested as a transverse resistivity ρxy assuming the current flows along the x axis and the transverse voltage is measured along the y axis, and it is linked to the  of the conductor. When the AHE intrinsic magnetization (M) HE , which is given by is present, it adds to the Hall effect ρxy OHE a combination of the ordinary Hall effect (OHE) ρxy and AHE ρxy : HE OHE AHE ρxy = ρxy + ρxy .

(1)

The OHE is proportional to the magnetic field B and is given by OHE ρxy = R0 B⊥ ,

(2)

where R0 is the OHE coefficient and B⊥ is the component of the magnetic field in the z axis. AHE The expression for ρxy depends on the mechanisms responsible for the AHE. Contributions to the AHE due to extrinsic mechanisms are given by AHE ρxy = Rs μ0 M⊥ ,

(3)

where Rs is called the anomalous Hall coefficient and M⊥ is the component of the magnetization in the z direction. Commonly, 2 Rs is given by Rs = aρxx + bρxx where the linear term in ρxx is attributed to the skew scattering mechanism5 and is expected to dominate in the high conductivity regime (σxx > 106 −1 cm−1 ) and the quadratic term is attributed to the side jumps mechanism6 and is expected to dominate in the low conductivity regime (σxx ∼ 104 –106 −1 cm−1 ). 1098-0121/2012/86(14)/144414(5)

In addition to the extrinsic mechanisms, band topological properties give rise to AHE via the Berry phase (KarplusAHE 2  = −ρxx σxy (M), Luttinger) mechanism.7–11 In this case, ρxy  where σxy (M) is determined by the band structure. The Berry phase mechanism is expected to dominate in the same regime as the side jump mechanism. The AHE in the itinerant ferromagnet SrRuO3 (Ref. 12) has attracted considerable interest for its nonmonotonic temperature dependence including a sign change close to its Curie temperature. The mechanisms responsible for this behavior are controversial. While the Berry phase scenario, assuming a temperature-dependent exchange gap, seems to agree qualitatively with the data,8 it has been shown that the AHE vanishes at a particular resistivity and not at a particular magnetization as one might expect.13 Midinfrared measurements suggest the applicability of the Berry phase scenario at energies above 200 meV while the dc limit is dominated by extrinsic scattering mechanisms.14 Furthermore, we have shown recently that the AHE in SrRuO3 scales in the ferromagnetic phase with ρxx .15 The scaling and the nonmonotonic functional form of the AHE were found to be consistent with two contributions to the AHE coefficient: (a) a Berry phase contribution with a temperature-independent band structure while taking into consideration the finite scattering rate and (b) a side jumps contribution. Here we show evidence that the scaling between the AHE and ρxx extends into the paramagnetic phase. AHE HE We extract ρxy from ρxy by subtracting the contribution of the OHE. We then extract M⊥ by using a relation between field-induced magnetization and magnetoresistance and show that the extraction is consistent with the expected mean-field AHE behavior with no fitting parameters. Knowing ρxy and M⊥ we extract Rs and analyze its dependence on temperature and resistivity. We find that the AHE in SrRuO3 is determined by the total resistivity, irrespective of its sources (phonons or magnons). This is demonstrated by showing a scaling of the extracted Rs with ρxx , when ρxx is changed by changing the temperature or by applying a magnetic field. We also find that the data of different samples with different thicknesses collapse on a universal curve when applying the scaling procedure described in Ref. 15. We show that the AHE in both the ferromagnetic phase and the paramagnetic phase is consistent with the scenario proposed in Ref. 15

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©2012 American Physical Society

NOAM HAHAM, JAMES W. REINER, AND LIOR KLEIN

PHYSICAL REVIEW B 86, 144414 (2012)

when taking into account the fractional power-law relation between the scattering rate and longitudinal resistivity observed experimentally.16,17 II. SAMPLES AND EXPERIMENT

Our samples are epitaxial thin films of SrRuO3 grown on slightly miscut (∼2◦ ) substrates of SrTiO3 by reactive electronbeam evaporation. The films are untwinned orthorhombic ˚ b∼ single crystals, with lattice parameters of a ∼ = 5.53 A, = ∼ ˚ and c = 7.85 A. ˚ The Curie temperature Tc of 5.57 A, the films is ∼150 K and they exhibit an intrinsic uniaxial magnetocrystalline anisotropy. Above Tc the easy axis is along the 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 mentioned otherwise. The films are patterned to allow transverse and longitudinal resistivity measurements that are performed with PPMS-9 (Quantum Design). The field-antisymmetric transverse resistivity is extracted by changing the voltage and current leads. We measure the temperature dependence of the longitudinal and transverse resistivity between 165 and 300 K with magnetic fields up to 8 T applied along the easy axis. III. EXPERIMENTAL RESULTS A. Extracting ρ xAHE y

Figure 1(a) shows the temperature dependence of the anHE tisymmetric transverse resistivity (ρxy ) for different magnetic fields (0–8 T) applied along the easy axis in the paramagnetic phase (165–300 K). As we apply a field at 45◦ from the film HE normal (namely, with a component along zˆ ), ρxy has both an OHE contribution and an AHE contribution. We determine the OHE contribution according to the methods described in Refs. 13 and 19 and we find that it corresponds to a carrier density of 1 electron per Ru (R0 ∼ −3.7 × 10−2 μ cm/T), OHE consistent with previous reports.13,19 Subtracting ρxy from HE AHE ρxy we extract ρxy 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 nonmonotonic. This behavior suggests competing effects. As we will see below, while the magnetization decreases with increasing temperature, Rs increases with increasing temperature. For high fields the increase in Rs dominates for some temperature interval near Tc where the magnetization is close to saturation. B. Extracting M⊥

A direct measurement of the magnetization in the paramagnetic phase is rather challenging due to the strong diamagnetic signal of the substrate, which is much larger than the paramagnetic signal of the thin films. Thus, we use magnetoresistance measurements to extract the field-induced magnetization. Figure 2(a) shows the temperature dependence of ρxx for different magnetic fields (0–8 T) applied along the easy axis. We notice a large negative magnetoresistance near Tc which decreases as temperature increases. Figure 2(b) shows the magnetic resistivity defined as ρ(T ,H ) ≡ ρxx (T ,H ) −

HE FIG. 1. (Color online) The Hall effect ρxy (a) and the AHE AHE resistivity ρxy (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 magnetization established previously20 for SrRuO3 is given by ρ(T ,H ) = −aM 2 ,

(4)

where a is a constant. This relation was used to extract the critical exponents of the ferromagnetic phase transition and was shown to apply near Tc by demonstrating the universal scaling relation related to the phase transition.20 We extract for every sample the coefficient a from the resistivity below Tc as described in Ref. 20. As we show below, the field-induced magnetization in the paramagnetic phase determined using Eq. (4) is consistent with the meanfield approximation. Assuming that for T sufficiently above Tc mean-field approximation applies, the magnetization is expected to obey the self-consistent equation  M/Ms = Bs

 gμB SH 3STceff + M/Ms , kB T (S + 1)T

(5)

where Bs is the Brillouin function, Tceff is the effective mean-field Curie temperature, and Ms is the saturation magnetization. Combining Eqs. (5) and (4) we expect

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M MR /Ms = Bs (gμB Heff /kB T ),

(6)

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 Tceff at T = 165 K. (d) M MR /Ms vs gμB Heff /kB T defined in text for T  180 K (circles) and T < 180 K (squares). The solid line is the Brillouin function. Inset: extracted optimal Tceff as a function of T .

where



ρ /Ms , a  3STceff ρ gμB SH + /Ms . gμB Heff /kB T ≡ kB T (S + 1)T a M MR /Ms ≡

For SrRuO3 in the paramagnetic phase S = 1 and Ms ∼ 288 emu/cm3 .21 Tceff was extracted by considering the field dependence of the magnetic resistivity at constant temperatures. According to Eq. (4) we expect a linear relation between √ ρ and M. We calculate the expected field dependence of the magnetization at a specific temperature according to Eq. (5) assuming different values of Tceff and determine the value for which the best linear√fit is obtained for the calculated magnetization Mcalc and ρ. Figure 2(c) shows the calculated magnetization Mcalc normalized by the saturated √ magnetization Ms as a function of ρ assuming different Tceff at T = 165 K. We obtain the best linear fit for Tceff = 156 K. The figure also shows that assuming Tceff = 150 K or Tceff = 160 K leads to a noticeably worse linear fit. For each temperature the optimal Tceff is found, and the temperature dependence of Tceff is presented in the inset of Fig. 2. The error bars are determined by the distribution of the

optimal Tceff for a randomly chosen subset (six of nine points) of the data. We notice that Tceff slightly grows with temperature above Tc . At temperatures higher than 180 K the error bar in Tceff increases significantly as M becomes linear with H for the field range we use. However, we expect Tceff to saturate. Using Tceff = 159.5 K and plotting M MR /Ms vs gμB Heff /kB T yields a good scaling of the data in the whole temperature and field range as expected from Eq. (6); moreover, the scaling function is in good agreement with the expected Brillouin function with no fitting parameters [see Fig. 2(d)]. We note, however, that deviations from the expected Brillouin function start below T = 180 K. Applying this analysis for different samples yields a similar scaling, with the same scaling function. This result supports the use of Eq. (4) in our analysis. Although the scaling was performed assuming a temperature-independent a, a good scaling is obtained if some temperature dependence is assumed. To bound this variation we 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 Curie temperature, and constrain it to be in the range of Tceff values extracted at lower temperatures (see inset of Fig. 2). We find that 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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PHYSICAL REVIEW B 86, 144414 (2012)

For the 50-nm sample the variation is +13% to −15% and for the 90-nm film it is +5% to −20%. Knowing a we can use Eq. (4) to determine the magnetization in the relevant range of temperatures in the paramagnetic phase for given field and temperature:  (7) M = ρ(H,T )/a.

Using Eq. (3) assuming temperature-independent a we extract Rs for different temperatures and fields by dividing AHE by the extracted magnetization M.22 the AHE resistivity ρxy Figure 3(a) shows the temperature dependence of the extracted Rs for different fields. The curves for the different fields differ

significantly, especially near Tc . Figure 3(b) shows Rs as a function of ρxx extracted for different fields and temperatures. We note that Rs scales with ρxx . The scaling implies that Rs is sensitive to the total ρxx irrespective of its sources; for different fields the same resistivity is obtained at different temperatures and thus with different weights for magnon and phonon scattering. The obtained scaling is consistent with previous results in the ferromagnetic phase.15 We considered the resistivity dependence of Rs in both ferromagnetic and paramagnetic phase for all of the samples, and followed the scaling procedure described in Ref. 15; namely, we normalized the resistivity with the resistivity at which the AHE vanishes in the ferromagnetic phase (the normalized resistivity is denoted as ρ ∗ ) and Rs with the absolute value of the negative peak value of Rs in the ferromagnetic phase (the normalized Rs is denoted as Rs∗ ) in order to

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

FIG. 4. (Color online) (a) The AHE coefficient Rs normalized by the absolute value of its negative peak value Rs∗ as a function of resistivity normalized by the resistivity at which Rs vanishes, ρ ∗ . The solid lines are for the extreme Rs values. (b) AHE coefficient Rs vs resistivity 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.

C. Extracting Rs

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PHYSICAL REVIEW B 86, 144414 (2012)

correct for geometry errors. Figure 4(a) shows Rs∗ as a function of ρ ∗ for three samples. We note that the curves of the different samples in both the ferromagnetic and paramagnetic phase collapse on a single curve. We also note that Rs∗ continues smoothly from the paramagnetic phase into the ferromagnetic phase. The solid lines represent the possible variation range of Rs based on the upper and lower bounds for a.

y;15 however, the data in the extended temperature interval appears to be inconsistent with such an assumption. We also checked the sensitivity of the fit to changes in the exponent α. We assume different values of α and fit the data to Eq. (8). For each α we calculate e(α), the total error between the data and the fit defined by the root of sum of squares of the difference between the fit and data points. The inset in Fig. 4 shows the normalized error E(α) ≡ e(α)/e(α = 0.45) vs the assumed exponent α. We note that reasonable fits are obtained for α ∼ 0.37–0.58 for which the error is less than two times its minimum value.

IV. THEORETICAL MODEL

The AHE in the ferromagnetic phase was described assuming a Berry phase contribution (with a temperatureindependent band structure) which considers the effect of finite scattering rate and a side jumps contribution. According to this scenario Rs is expected to satisfy the equation15 2 Rs = ρxx

2

B 2 + Cρxx , + (¯h/τ )2

(8)

where B is a constant associated with Berry phase,  is a characteristic energy gap in the band structure, C is associated with the side jumps contribution, and τ is the scattering time. Considering the common relation 1/τ ∝ ρxx , we can try to fit the data to the equation. However, Eq. (8) does not provide a good description for both the ferromagnetic and paramagnetic phase. A possible source for this failure is the assumed relation between ρxx and the scattering time. We note that several groups confirmed a relation ρ = D/τ α 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) with ˚ and with a reasonable energy gap of side jump y ∼ 0.2 A  ∼ 18 meV. We note that using Eq. (8) to fit the data below Tc while assuming ρ ∝ 1/τ yields higher values for  and 1

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V. CONCLUSIONS

We measured the paramagnetic AHE in SrRuO3 for a wide temperature and field range. We find that the AHE coefficient Rs scales as a function of resistivity, and thus is insensitive to the source of scattering as resistivity is changed by changing the temperature or by applying a magnetic field. The functional form of the AHE is consistent with a combination of the side jumps mechanism and the Karplus-Luttinger (Berry phase) mechanism which considers the effect of finite scattering time that has a fractional power-law relation with ρxx with an exponent α ∼ 0.37–0.58. ACKNOWLEDGMENTS

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

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