Infrared anomalous Hall effect in SrRuO 3 : Evidence for crossover to intrinsic behavior
M.-H. Kim, G. Acbas, M.-H. Yang, M. Eginligil, P. Khalifah, I. Ohkubo, H. Christen, D. Mandrus, Z. Fang, J. Cerne
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Infrared anomalous Hall effect in SrRuO : Evidence for crossover to intrinsic behavior M.-H. Kim, G. Acbas, M.-H. Yang, M. Eginligil, P. Khalifah, I. Ohkubo, H. Christen, D. Mandrus, Z. Fang, and J. Cerne Department of Physics, University at Buffalo, The State University of New York, Buffalo, NY 14260, USA Department of Chemistry, University at Stony Brook,The State University of New York, Stony Brook, NY 11794, USA Department of Applied Chemistry, University of Tokyo, Tokyo, Japan Oak Ridge National Laboratory, Condensed Matter Sciences Division, Oak Ridge, TN 37831, USA and Institute of Physics, Chinese Academy of Science, Beijing, 100080,China.
The origin of the Hall effect in many itinerant ferromagnets is still not resolved, with an anomalouscontribution from the sample magnetization that can exhibit extrinsic or intrinsic behavior. We report the first mid-infared (MIR) measurements of the complex Hall ( θ H ), Faraday ( θ F ), andKerr ( θ K ) angles, as well as the Hall conductivity ( σ xy ) in a SrRuO film in the 115-1400 meVenergy range. The magnetic field, temperature, and frequency dependence of the Hall effect isexplored. The MIR magneto-optical response shows very strong frequency dependence, includingsign changes. Below 200 meV, the MIR θ H ( T ) changes sign between 120 and 150 K, as is observed indc Hall measurements. Above 200 meV, the temperature dependence of θ H is similar to that of the dcmagnetization and the measurements are in good agreement with predictions from a band calculationfor the intrinsic anomalous Hall effect (AHE). The temperature and frequency dependence of themeasured Hall effect suggests that whereas the behavior above 200 meV is consistent with an intrinsicAHE, the extrinsic AHE plays an important role in the lower energy response.
I. INTRODUCTION Ca x Sr − x RuO compounds exhibit unusual properties,such as metamagnetism, quantum criticality, non-Fermiliquid behavior and an anomalous Hall effect (AHE) thatcontinue to challenge the condensed matter community.The Hall effect in SrRuO consists of two parts, the ordi-nary Hall effect (OHE) due to a magnetic field B produc-ing a Lorentz force on moving carriers and the AHE dueto the sample’s magnetization M . The Hall resistivity ρ H is given by: ρ H = ρ yx = R B + ρ AHE yx ( M ) (1)The ordinary Hall coefficient R is related to the carrierdensity. The anomalous Hall resistivity ρ AHE yx ( M ) can bedivided into two categories: 1) the extrinsic AHE arisingfrom impurity scattering and 2) the intrinsic AHE dueto the band structure. The anomalous Hall resistivityis expressed as ρ AHE yx ( M ) = R s ( ρ xx )4 πM + σ Iyx ( M ) ρ xx , (2)where R s is the extrinsic AHE coefficient as a functionof the longitudinal resistivity ρ xx and σ Iyx is the intrinsicAHE transverse conductivity. The coefficient R s containstwo terms, one proportional to ρ xx and one that variesas ρ xx : R s ( ρ xx ) = aρ xx + bρ xx , (3)where a and b are coefficients for two types of scatteringprocesses. The first term is due to asymmetric (skew)scattering from impurities in the presence of magneticorder and is considered to be a classical extrinsic ef-fect. The second term is called “side-jump scattering” from impurities, which has a quantum mechanical ori-gin. Note that although the side-jump scattering scalesas ρ xx just as the intrinsic AHE, and some consider themto be equivalent, the latter is simply proportional to M while the former depends on M through the func-tion σ Ixy ( M ). While the dc AHE is observed in manyitinerant ferromagnetic materials ranging from ruthen-ates to colossal magnetoresistance oxides to diluted mag-netic semiconductors, the degree to which its origin isintrinsic or extrinsic is still not resolved in manycases. The dc AHE measurements by Ref. 3 have foundthat the sign change of θ H ( T ) occurs in SrRuO when ρ xx = 107 µ Ω cm. They considered only the extrinsicAHE and found this critical resistivity when R s = 0,where the sign changes.There have been extensive dc measurements toprobe the temperature and magnetic field dependenceof the Hall effect in SrRuO . There has also been alow temperature study based on Kerr measurements toprobe the frequency dependence of the Hall conductiv-ity above 200 meV. To the best of our knowledge thiswork represents the first systematic study of the temper-ature, magnetic field, and frequency dependence of theHall response in SrRuO in the 0.1 - 1.4 eV energy range.This range is particularly interesting since the only cal-culation to model the frequency dependence of the Hallconductivity σ xy of SrRuO predicts strong spectral fea-tures for σ xy in this range. This work provides the firstexperimental test of this model below 200 meV.Conventional dc Hall effect measurements in novel elec-tronic materials such as high temperature superconduct-ing cuprates (HTSC), diluted nagnetic semiconductors,and ruthenate perovskite materials have been essential inrevealing the unusual character of these systems. Threemajor issues motivate the study of the MIR AHE inthese materials. The first and most general argumentis that since the understanding of dc AHE may be un-resolved, studying the frequency dependence of the dy-namic AHE can provide new information on the micro-scopic causes that are responsible for the AHE. Secondly,MIR measurements would probe more effectively theenergy scales of the system (e.g., the plasma frequency,the cyclotron frequency and the carrier relaxation rates)and provide greater insight into the intrinsic electronicstructure of a wide range of materials. The MIRrange is highly appropriate since the typical band en-ergy scale for the AHE in diluted magnetic semiconduc-tors,
SrRuO , and other materials is in theMIR. It is not surprising that the most striking spectralfeatures in calculations of the AHE are in the MIR range.Unlike conventional MIR spectroscopy that measures thelongitudinal conductivity σ xx , which is related to the sumof the sample’s response to left and right circularly polar-ized, MIR Hall measurements probe the Hall conductiv-ity σ xy , which is related to the difference in the sample’sresponse to left and right circularly polarized light, andtherefore is more sensitive to asymmetries such as spin-splitting etc. Finally, the dc Hall effect can be dominatedby impurity scattering or grain boundary effects. This isespecially important in new materials which often con-tain many impurities and defects. By probing the Halleffect at higher frequencies, the contribution from extrin-sic scattering can be minimized.Magneto-polarimetry measurements can be used to ex-tend Hall effect measurements into the infrared frequencyrange (10 Hz). These measurements are sensitive tothe complex Faraday θ F and Kerr θ K angles, which areclosely related to the complex Hall angle θ H . Since θ H (and θ F ) obeys a sum rule it is very useful to be ableto integrate θ H to higher frequencies to verify whether(and where) the Hall angle sum rule saturates or whetherthere is more relevant physics at even higher frequencies.Finally, since the high frequency behavior of θ H is con-strained by the general requirements of response func-tions, a simple, model-independent asymptotic form for θ H becomes more accurate at higher frequencies.In this paper, we report measurements of the MIRcomplex Faraday, Kerr, and Hall angles in SrRuO filmsin the 0.1 - 1.4 eV energy range. The transmitted andreflected magneto-optical responses in the MIR are qual-itatively similar to results from dc Hall and dc magneti-zation measurements. The measurements are in goodagreement with predictions from a band calculation above 200 meV. The deviations at lower energy are prob-ably due to a stronger contribution from the extrinsicAHE. II. EXPERIMENTAL SYSTEM
The Faraday and Kerr angles are measured using asensitive polarization modulation technique in themid and near-infrared (MNIR) spectral range (115 - 1400 meV) for a SrRuO sample grown by pulsed laser depo-sition at Oak Ridge National Laboratory as described inRef. 29. Several light sources such as various gas lasers,semiconductor lasers, and a custom-modified double passprism monochromator with a Xe light source allow us toperform the measurement in a wide probe energy range.For details of the experimental technique see Refs. 25and 26. The complex θ F and θ K angles are measuredin the MNIR spectral range as a function of magneticfield up to 2 T and temperature from 10 K to 300 K.The small background Faraday signal from the cryostatwindows and the film substrate has been subtracted fromthe data. The complex conductivities σ xx and σ xy andthe complex Hall angle θ H are determined directly fromthe measured complex θ F and θ K using the analysis tech-niques in Ref. 25.The dc longitudinal and Hall resistivities of SrRuO were measured simultaneously using a four-probe vander Pauw geometry at UB’s Magneto-Transport Facil-ity as well as at Oak Ridge National Laboratory using aPhysical Property Measurement System (PPMS, Quan-tum Design) in a six-terminal configuration and 1 mAcurrents. Moreover, the film magnetization was mea-sured by a SQUID magnetometer (Magnetic PropertyMeasurement System; MPMS, Quantum Design) using acustom-modified sample holder, which enables measure-ments of the magnetization perpendicular to the film sur-face, which is the same configuration as our Faraday andKerr measurements.
III. RESULTS
The Faraday and Kerr angles on the SrRuO weremeasured at various temperatures and a wide energyrange to probe the anomalous Hall effect. The continu-ous broadband measurements allow us to extend θ F and θ K measurements to up to 1.4 eV. Although the inten-sity of broadband light is significantly weaker than that oflasers, and therefore the sensitivity is reduced, θ F and θ K are large and can be readily measured in this range. Theanalysis techniques to obtain the complex longitudinal( σ xx ) and transverse ( σ xy ) conductivities and the MIRHall angles ( θ H ) are based on the techniques in Ref. 25.Figure 1 shows temperature dependence of θ F and θ K in SrRuO as a function of applied magnetic field H at aphoton energy of 117 meV. Both θ F and θ K exhibit fer-romagnetic hysteresis until the magnetization disappearsat the Curie temperature T c ≃ θ F ) as a function of H at various temperatures. Notethat the sign of the slope above 130 K is opposite of thatobserved at lower temperatures. Linear fits to the dataat 150 and 160 K indicate non-monotonic behavior of theslope. The strongest change in Re( θ F ) occurs below 1.5T, where the ferromagnetic hysteresis loops close.The Figure 1b) shows θ F and θ K as a function of H at 117 meV and 10 K. The magnitude of Im( θ K ),which is not shown in Fig. 1b), is an order of magnitude FIG. 1. Temperature dependence of θ F and θ K from aSrRuO as a function of applied magnetic field H at a probeenergy of 117 meV. The main panel a) shows the temperaturedependence of the hysteresis loops for Re( θ F ). Linear fits tothe data at 150 and 160 K are also shown to help distinguishthe data from other temperatures. The vertical line indicates H =1.5T, where the hysteresis loops close at low tempera-tures. Inset b) shows Re( θ F ), Im( θ F ), and Re( θ K ) (which ismultiplied by a factor of 5) as a function of H at 117 meVand 10 K. Inset c) shows the hysteresis loops for Im( θ F ) at117 meV and two temperatures, 80 K and 140 K, at whichthe sign change near H = 0 of Im( θ F ) is clearly seen. smaller than that of Re( θ K ). Typically, for metallic filmslike SrRuO in the MIR, the magneto-optic response intransmission ( θ F ) is larger than that obtained in reflec-tion ( θ K ), although the transmitted intensity of light canbe as small as 0.01 %, because θ F ∝ σ xy /σ xx while θ K ∝ σ xy / ( σ xx ) where σ xx ≫ σ xy in metallic films. For thicker and more metallic films, θ F is much more sen-sitive to σ xy than θ K in the MIR. Therefore, measuringmagneto-optic signals in both transmission and reflectionwith a higher sensitivity provides the measurements re-ported here with a distinct advantage over the techniqueused in Refs. 9 and 24, where only the reflected signal wasmeasured. Moreover, the stronger θ F signals are criticalto explore the MIR Hall effect at higher temperatures,where the magneto-optical response is weaker.It is not clear whether the sign change in Re( θ F ) isdue to a sign change in the AHE component or the OHEcomponent with opposite sign (from opposite charge car-riers) which starts dominating at higher temperatures inFig. 1a). However, the AHE nature of the sign change isclearly seen in the Fig. 1c), which shows Im( θ F ) at 80 Kand 140 K at a probe energy of 117 meV. Here the sharpstep near H ≃ θ F ) isquite linear in H at the sign change, which occurs in tem-peratures between 130 K and 160 K where the magneti-zation is smaller due to the proximity to T c . On the other FIG. 2. Temperature dependence of a) θ H ( T ) and b) θ H ( T ) /M ( T ) at 1.5 T at 117 meV and dc. The magnetization M ( T ) is shown in Fig. 3. hand, unlike Re( θ F ), the sign change in Im( θ F ) appearsbetween 80 K and 140 K far T c . The opposite signs for thehysteresis loop steps in Im( θ F ) indicate the sign changeis related to the magnetization, and hence due to the signchange in the AHE coefficient R s . The sign change in theferromagnetic step near H = 0 is not clearly seen in dcHall measurements, which demonstrates another advan-tage of magneto-optical measurements in probing AHEmaterials.Figure 2 shows the Hall angle as a function of tem-perature at an applied magnetic field of 1.5 T and at aprobe energy of 117 meV and 0 meV (dc). Since thehysteresis loops close below µ | H | =1.5 T, this value ofmagnetic field is chosen to characterize the strength ofthe magneto-optical response. It is difficult to separatethe OHE from the AHE above 1.5 T, because both Hallsignals are linear in H . Reference 32 has reported thatthe magnetization in SrRuO does not saturate even atmagnetic fields of 40 T and found that the anomalousHall signal can be linear in H up to 40 T. Althoughthe separation of AHE from OHE has been done in dcmeasurements, it is more challenging to make this sep-aration for MIR data. However, there are two regionswhere the OHE signal can be readily separated from theAHE signal. First of all, at lower temperatures and lowmagnetic fields, where magneto-optical response is fer-romagnetic, θ H is dominated by the AHE. For highertemperatures, it is still hard to separate AHE from OHEbecause the ferromagnetic response is weaker as the tem- FIG. 3. Temperature dependence of a) θ H ( T ) and M ( T ) andb) θ H ( T ) /M ( T ) at 1 T above 200 meV. θ H ( T ) is very similarto the magnetization M ( T ) at 1 T. The horizontal lines in b)are guide lines, which show θ H ( T ) ∝ M ( T ). perature increases towards T c . Second of all, the OHE issuppressed at higher frequency. The Drude model pre-dicts that at as the probe frequency ω increases past thecharacteristic scattering frequency γ H , Re( θ H ) ∝ ω − and Im( θ H ) ∝ ω − . If one assumes that the linear partof the Hall angle ( ∝ θ F ) only comes from the OHE dueto free carriers in Fig. 1, one can use the extended Drudemodel to calculate the Hall frequency ω H , which is re-lated to the carriers’ effective mass, and the Hall scat-tering rate γ H . Furthermore, one can estimate themaximum contribution of the OHE to the overall Hall re-sponse. θ F and θ K produce σ xx , σ xy and θ H = σ xy /σ xx .Slope of the linear behavior of the θ F (and therefore θ H )in Fig. 1 at fields above 1.5 T is nearly constant from 10K to 80 K. The slope translates into ω H = − .
31 cm − /T( − .
039 meV/T) and γ H = 1102 cm − (137 meV). As-suming that the carrier scattering is isotropic, one canuse ω H to obtain a carrier effective mass of 2 . m e , where m e is the bare electron mass. Since the Hall scatteringrate γ H is close to ω = 117 meV, one can expects thatRe( θ H ) ≈ Im( θ H ) ≈ ω H / (2 γ H ) when ω ≈ γ H . The valueof OHE given by the linear part of the hysteresis is lessthan 3% of the total Hall response. Likewise, the OHEis much smaller at the higher frequencies over 117 meVdue to ω ≫ γ H .Figure 2a) shows the sign change of Re( θ H ) and Im( θ H ) at 117 meV near 130 K, where the Hall signchange is observed in dc measurements (dashed line).The magnitude and temperature behavior of θ H at 117meV and dc are very similar. The sign change of Im( θ H )appears at a slightly lower temperature, deeper in theferromagnetic phase, than Re( θ H ). It may explain whythe sign change of Im( θ H ) is more clearly seen in Fig. 1.The magnitudes of both Re( θ H ) and Im( θ H ) are slightlysmaller than that of dc. The dc Hall angle exhibits aminimum value near 40 K, but at 117 meV both Re( θ H )and Im( θ H ) increase monotonically with temperature.Im( θ H ) at 117 meV rises more steeply than Re( θ H ),changes sign earlier, and reaches a maximum value near140 K. Since the MIR Hall angles mostly come from theAHE, the θ H is associated with the magnetization M .Figure 2b) shows the Hall angles divided by the mea-sured M , which is shown in Fig. 3a) (solid line). Thesign change in Re( θ H ) /M and Im( θ H ) /M is clearly seen.The sign change in Im( θ H ) occurs near 120 K and inRe( θ H ) near 150 K. These are symmetrically separatedfrom the dc Hall sign change temperature.Figure 3 shows the temperature dependence of the dcmagnetization and θ H at higher energies of 224, 366,and 676 meV, which is obtained from the formula of[ θ H (+1 T ) − θ H ( − T )] /
2. In this case, θ H ( T ) no longerchanges sign as it does at 117 meV and dc measurements,instead θ H ( T ) approaches zero gradually as shown inFig. 3a). As the probe energy increases, θ H ( T ) behavesmore like the magnetization M . The solid line in Fig. 3a)represents the measured magnetization, which is satu-rated at lower temperature to 0.8 µ B / Ru. At an energy of676 meV, M ( T ) and θ H ( T ) exhibit the same dependenceover nearly the entire range as shown in Fig 3a). Typ-ically, visible and near-infrared Faraday and Kerr mea-surements are used to determine the magnetization ofmaterials. Figure 3b) plots θ H ( T ) /M ( T ) at higher en-ergies of 224, 366, and 676 meV at 1 T. The horizontalguide lines show behavior where θ H is proportional to M . At a probe energy 676 meV the value of θ H /M is aconstant at all temperatures, but at 366 meV this is thecase only at lower temperatures.Figure 4 shows the anomalous Hall part of θ F and θ K at 10 K and 0 T with the sample fully magnetized out ofplane as a function of probe energy. As seen before, theOHE contribution is small, especially at higher energiesat H ≈ H = 0 T and OHE is proportional to H , evensmall OHE cannot contribute to this Hall signal. Both θ F and θ K display a strong energy dependence and changesign mostly at lower energies, but monotonically increaseor remain fairly constant at higher energies as shown inFig. 4. The energy dependence of Re( θ F ) and Im( θ K )is similar. The sign changes are observed at low ener-gies near 250 meV for Re( θ F ) and 130 meV for Im( θ K ).Both signs change from negative to positive as energy in-creases. Note that the signs are defined and determinedin Ref. 25. Likewise, Im( θ F ) and Re( θ K ) exhibit similarfeatures. However, both sign changes appear near 800 FIG. 4. Energy dependence of the AHE a) θ F and b) θ K withthe sample fully magnetized perpendicular to the plane at 0 Tand 10 K. Since the measurement is at H = 0 T, the OHE aswell as background signals from the substrate and windows,which are linear in H , do not contribute to the signal. At theenergy range from 0.4 - 0.7 eV, the intensity of transmittedlight is so weak to measure θ F . meV. The sign of Im( θ F ) also changes from negative topositive, but the sign of Re( θ K ) behaves oppositely. Thedc values of θ F and θ K in Fig. 4 are determined in Ref. 25connect smoothly with the MIR data as E → σ xx and b) transverse (AHE)conductivity σ xy as a function of probe energy. The sym-bols are obtained from θ F and θ K in Fig. 4 using theanalysis techniques in Ref. 25. One advantage of deter-mining both σ xx and σ xy from the same set of θ F and θ K measurements is that the behavior of σ xx , which isfairly well known in this energy range, can be providea consistency test for σ xy , which is not well known. Asexperimentally seen in Ref. 25, the complex σ xx from θ F and θ K is in good agreement over the entire energy rangewith σ xx obtained from reflectance measurements on adifferent SrRuO film, which has a factor of 3 smallerdc resistivity. On the other hand, theoretical predictionsfor the AHE have been limited at finite energies, althoughseveral different models are used to explain the dc AHEin SrRuO . Here we compare the measured MIRconductivities with predictions from a Berry-phase calcu-lation for SrRuO . The solid lines (the real part of σ xx and σ xy ) and dashed lines (the imaginary part of σ xx and σ xy ) are from a Berry phase calculation of the intrinsic FIG. 5. a) The longitudinal conductivity σ xx and b) trans-verse (AHE) conductivity σ xy b) as a function of probe energy.The thin lines are from a Berry phase calculation of the in-trinsic AHE by Z. Fang. Note that this calculation neglectsintraband transitions. AHE by Z. Fang and coworkers. The calculated complex σ xx in Fig. 5a) clearly deviates from the measured σ xx .This is not surprising since the model neglects intrabandtransitions, which play a dominant role at lower energiesin a metal. For energies over 200 meV, the calculationagrees qualitatively and in some energy regions quantita-tively with the measured value. So unlike the calculation,there is no sign change of the measured Im( σ xx ) in ourenergy range.Since the goal of this calculation was to provide in-sights into the behavior of σ xy , it is encouraging that theagreement between the calculated and measured valuesof σ xy is significantly better, as can be seen in Fig. 5b),which shows the anomalous Hall response. The Fangmodel predicts a sign change from electron-like at low en-ergy to hole-like at higher energies in both Re( σ xy ) andIm( σ xy ). The measured Re( σ xy ) changes from electron-like for energies below 150 meV to hole-like at higherenergies. The measured Im( σ xy ) appears to make thesame change above 400 meV. Both Re( σ xy ) and Im( σ xy )extrapolate smoothly to their dc values. Unlike conven-tional metals like gold, which can be modeled using asingle band in the infrared and where Drude behavior (in-traband transition) are responsible for the infrared Halleffect, in SrRuO both intraband and interband transi-tions contribute, so the “electron-like” description is onlymeant to indicate signs, not the microscopic origin of the FIG. 6. The complex a) Hall angle θ H and b) Hall resistivity ρ H at 10 K and and 0 T with the sample fully magnetized outof plane as a function of probe energy. Since the measurementis at H = 0 T, the signals are solely due to the AHE. Hall effect. Of course, in the dc limit the OHE is solelydue to intraband (Drude) behavior. Above 300 meV, thecalculated Re( σ xy ) and Im( σ xy ) values run roughly par-allel to the measured ones. The crossing of Re( σ xy ) andIm( σ xy ) is observed in both the calculated and measuredvalues near 1 eV and 0.8 eV, respectively. Below 300meV, Re( σ xy ) exhibits a peak near 200 meV in both thecalculation and measurements. Another sharp peak-likestructure is predicted in Re( σ xy ) near 50 meV. However,the measured Re( σ xy ) has already dipped below zero at120 meV, in strong contrast to the theoretical upturn at100 meV, and is heading towards the dc value smoothly.For Im( σ xy ), the calculation and the measurements showsthe zero-crossing near 300 meV and 500 meV, respec-tively. As with the calculated Re( σ xy ), the calculatedIm( σ xy ) also exhibits sharp peaks near 130 and 15 meV,but the slope and the large negative offset of the mea-sured Im( σ xy ) suggests that it will continue monotoni-cally towards its dc value of zero. It is interesting to notethat the over the entire measured energy range, the cal-culated σ xy generally agrees better with the data thanthe calculated σ xx . These measurements strongly sup-port the validity of the Berry phase model for describingthe anomalous Hall response of SrRuO above 200 meV,but below 200 meV the measurements do not follow the-oretical predictions.Figure 6 shows the measured complex a) Hall angle θ H and b) Hall resistivity ρ H at 10 K and 0 T (sample fully magnetized out of plane). For low energies, Re( θ H )is nearly constant in the 0 - 120 meV range, whereasIm( θ H ) increases linearly in the same range. The lowenergy MIR θ H results extrapolate smoothly to the dcvalues, suggesting that there are no additional featuresin the 0 - 120 meV range. Above 100 meV, Re( θ H ) in-creases more rapidly with increasing energy, changes signnear 200 meV, and increases monotonically over most ofmeasurement range. Im( θ H ) reaches a minimum valuenear 200 meV, changes sign near 800 meV, and becomesmore positive as energy increases. The energy depen-dence of ρ H in Fig. 6b) is nearly constant below 0.8 eV,but above this energy the value of ρ H increases an orderof magnitude. Above 0.8 eV Re( ρ H ) has peak near 1.3eV, where Im( ρ H ) changes sign. Additionally, Im( ρ H )has a minimum value near 1 eV. IV. DISCUSSION
Exploring the Hall effect in SrRuO as a function ofmagnetic field H , temperature T , and frequency ω (orMNIR energy) can provide new insights into the mate-rial as well as the AHE in general. We have seen theMNIR θ H (or σ xy ) is consistent with the predictions froman intrinsic AHE calculation for energy greater than 200meV. On the other hand, the frequency, temperature andmagnetic field dependence below 200 meV may be moreconsistent with expectations from the extrinsic AHE. Inthis discussion, we explore the evidence for a crossoverbetween intrinsic and extrinsic behaviors in the MIR en-ergy range. Since the extended Drude model (EDM) wassuccessfully applied to the IR σ xx in SrRuO , we willcompare EDM predictions for σ xx and σ xy with our mea-surements in order to disentangle OHE and AHE contri-butions. Secondly, we apply the extrinsic AHE model tothe temperature dependent θ H at 117 meV to test thismodel at and near dc.The EDM is more important at lower energy, because itmodels the intraband transitions which are dominant atand near dc. Note that the EDM considers the frequencydependent relaxation time τ ∗ ( ω ) and plasma frequency ω ∗ p ( ω ), which are renormalized by the mass enhancementfactor m ∗ ( ω ). The longitudinal conductivity σ xx is de-fined in EDM as σ xx = ω ∗ p ( ω ) π ( γ ∗ ( ω ) − iω ) , (4)where renormalized scattering rate is γ ∗ ( ω ) = τ ∗− ( ω ) = γ ( ω ) /m ∗ ( ω ), and frequency-dependent plasma frequency ω ∗ p ( ω ) = ω p /m ∗ ( ω ). The bare plasma frequency ω p ofthe free electron gas can be calculated by the carrierdensity. This model was used in SrRuO to probe pos-sible non-Fermi-liquid behavior exhibited in reflectancemeasurements. It was reported that Re( σ xx ) falls like ω − . and γ ∗ ( ω ) increases linearly with ω in the energyrange between 0 to 1000 cm − (124 meV) at low temper-ature. The renormalized scattering rate is given by theformula γ ∗ ( ω ) = ω [Re( σ xx ) / Im( σ xx )], which is derivedfrom Eq. (4), and using data in Fig. 6a). The scatteringrate γ ∗ ( ω ) obtained from our σ xx increases fairly linearlywith ω in the energy range of 116 - 600 meV (900 - 4800cm − ) and 10 K. Furthermore, in the same energy rangethe conductivity Re( σ xx ) drops like ω − . . Our resultsfor σ xx based on θ F and θ K measurements are consistentwith that in Ref. 28.The EDM allows one to extract the carrier density n ofthe electron gas. One way to obtain n is from the OHE.The Hall scattering parameters are extracted from thelinear in H behavior of θ H at low temperatures, assumingthat this slope is solely due to the OHE. At 117 meVwe obtain ω H = − .
31 cm − / T, γ H = 1102 cm − , anda carrier effective mass of 2.99 m e . Likewise, the MIRHall coefficient R H can be determined from the linearbehavior of θ H ( H ). The coefficient R H is given by: R H ( ω ) = ρ yx ( ω ) B = θ H ( ω ) σ xx ( ω ) B , (5)In a simple Drude model where γ is constant, R H ( ω )is purely real and constant. Considering only Re( R H ) at117 meV, one can estimate n = 1 . × electrons/cm from using the formula R H = − ne , where e is the elec-tron charge. This value is nearly within a factor of twoof that obtained in dc measurements by Khalifah andcoworkers, where n = 2 . × electrons/cm (or n = 1 . × electrons/cm from Ref. 31). This discrep-ancy is probably due to the fact that the linear behavior isnot completely due to the OHE, but also the AHE. If 50%of the net linear Hall angle signal at 117 meV between1.5 and 2 T resulted from the AHE, leading to a factor oftwo decrease in the contribution from the OHE, the car-rier density obtained from the slope of θ H at 117 meV, at10 K, and above 1.5 T would be 2 . × electrons/cm .The carrier density n = 1 . × electrons/cm cor-responds to the plasma frequency ω p = 30517 cm − ,while n = 2 . × electrons/cm corresponds to ω p =47315 cm − . According to Ref. 28, the mass enhance-ment factor m ∗ ( ω ) approaches 1 as the energy exceeds1000 cm − . It means that ω ∗ p ( ω ) ≈ ω p is satisfied from1000 cm − up to 4800 cm − , where γ ∗ ( ω ) ∝ ω . For theEDM σ xx , the appropriate partial sum is ω ∗ p ( ω c ) π = Z ω c π Re [ σ xx ( ω )] dω, (6)where ω c is a cutoff frequency. To stay consistent withthe analysis in Ref. 28, we fit the measured Re( σ xx )to the formula Aω − . , where A is a fitting parameter.If the partial sum is performed to ω c = 1000 cm − ,then ω ∗ p = 32636 cm − , which corresponds to a car-rier density n = 1 . × electrons/cm . For ω c =4800 cm − , ω ∗ p = 48307 cm − , which corresponds to n = 2 . × electrons/cm . These values of carrierdensity are qualitatively in agreement with the valuesobtained from the OHE and dc measurements. The scat-tering parameters determined from the EDM reveal that intraband transitions play an important role in σ xx below100 meV.As with the MIR σ xy ( T ), the behavior of θ H can bedivided into two distinct regions at 200 meV. Below 200meV, the MIR θ H ( T ) is similar to the dc θ H ( T ) as inFigs. 2 and 3, but above 200 meV it looks like the dc mag-netization M ( T ). The agreement in σ xy ( ω ) between themeasurement and the strictly interband, intrinsic, Berryphase model at energies below 200 meV is not as goodas at higher energies as shown in Fig. 5b). Moreover,the EDM analysis suggests that intraband transitions areimportant to σ xy and even more important to σ xx at en-ergies below 100 meV. One possible reason for discrep-ancy between measurement and theory at low energies isthat the lifetime broadening used in the calculation is toosmall for the SrRuO film measured here, which exhibitsmuch broader features. Another possibility is that theintrinsic model does not include intraband transitions,which play an important role near dc.Since much of the dc AHE is commonly framed usingEqs. (1) and (3), it is interesting to study the tempera-ture and frequency dependence of θ H in terms of theseequations. The simplest generalization to finite frequencyof the extrinsic AHE θ H is to simply use the frequencydependent resistivity ρ xx ( ω ) in Eqs. (1) and (3) at H = B = 0 T: θ H ( ω ) = ρ H ( ω ) ρ xx ( ω ) = R s ( ω ) ρ xx ( ω ) 4 πM = ( a + bρ xx ( ω ))4 πM. (7)Although this generalization is not rigorously correct, itcan provide insight into the lower energy AHE. If oneconsiders only the extrinsic AHE at or near dc, the signchange appears to be due to a change in sign of the AHEcoefficient R s . In Ref. 3, the dc AHE always vanishes atthe same value of ρ ∗ xx = 107 µ Ω cm, where θ H ( ρ ∗ xx ) = 0,which suggests that the extrinsic AHE is dominant atdc. In our dc measurements, the sign change occurs atapproximately ρ ∗ xx = 173 µ Ω cm. If one applies Eq. (7)using ρ xx ( ω ) to the sign change at 117 meV, the signchange of Re( θ H ) appears at ρ ∗ xx = 400 µ Ω cm and thatof Im( θ H ) occurs at ρ ∗ xx = 383 µ Ω cm. Typically, thelongitudinal resistivity at 117 meV is two or three timeslarger than the value at dc because ρ xx ∝ γ ( ω ) ∝ ω atlower energies. According to Eq. (7), the sign changeoccurs when ρ ∗ xx = − a/b . The parameters a (skew scat-tering) and b (side-jump scattering) are derived from mi-croscopic origins. The analysis at 117 meV revealsthat a and b are energy-dependent. Therefore, from ρ ∗ xx ( ω ) = − a ( ω ) /b ( ω ) and since ρ ∗ xx grows with ω , itcould be interpreted that the skew scattering term a ( ω )grows with respect to the side-jump scattering term b ( ω )as energy increases for 0 to 117 meV. Another point ofview is that the skew scattering is more sensitive to theprobe energy than the side-jump scattering below 117meV. Since the OHE is zero at H = 0 T or negligiblysmall even at H ≈ σ xy ; 2)the temperature dependence of the MIR Hall effect losesits dc character; and 3) the Drude-like behavior in σ xx significantly decreases. Therefore, the upper limit for theimpurity scattering rate appears to be near 1600 cm − (200 meV). This confirms the importance of the extrinsicAHE in dc and low frequcny measurements, but alsoshows the transition to more intrinsic behavior at higherfrequencies. Our measurements suggest that by increas-ing the probe energy beyond 200 meV, the AHE makesa transition from an extrinsic to an intrinsic character.We find that the Berry phase model accurately describesthe MIR AHE of SrRuO above 200 meV. ACKNOWLEDGMENTS
We thank K. Takahashi, A.J. Millis, N.P. Ong, andJ. Sinova for helpful discussions. We also wish tothank B.D. McCombe for the use of UB’s Magneto-Transport Facility. This work was supported by ResearchCorporation Cottrell Scholar Award, NSF-CAREER-DMR0449899, and an instrumentation award from theUniversity at Buffalo, College of Arts and Sciences. J. Smit, Physica , 877 (1955). J. Smit, Phys. Rev. B , 2349 (1973). Y. Kats, I. Genish, L. Klein, J.W. Reiner, and M. R.Beasley, Phys. Rev. B , 180407(R)1-4 (2004). P. Khalifah, I. Ohkubo, B.C. Sales, H. Christen, D. Man-drus, and J. Cerne, Phys. Rev. B R. Karplus and J. M. Luttinger, Phys. Rev. , 11541160(1954). L. Berger, Phys. Rev. B , 4559 (1970). W.-L. Lee, S. Watauchi, V. L. Miller, R. J. Cava, and N.P. Ong, Science , 1647 (2004). T. Jungwirth, Q. Niu, and A. H. MacDonald, Phys. Rev.Lett. , 207208/207201-207204 (2002). Z. Fang, N. Nagaosa, K. S. Takahashi, A. Asamitsu, R.Mathieu, T. Ogasawara, H. Yamada, M. Kawasaki, Y.Tokura, and K. Terakura, Science , 92-95 (2003). N. Nagaosa, Jairo Sinova, S. Onoda, A. H. MacDonald,and P. Ong, accepted for publication to Review of ModernPhysics (2009); arXiv:0904.4154. W. L. Lee, S. Watauchi, V.L.Miller, R.J.Cava, andN.P.Ong, Science , 1647-1649 (2003). J.M. Harris, Y.F. Yan, and N.P. Ong, Phys. Rev. B ,14293 (1992). H.D. Drew, S. Wu, and H.-T.S. Lihn, J. Phys.: Condens.Matter , 10037 (1996). L.D Landau, Sov. Phys. JETP , 920 (1957); , 101 (1957);and , 70 (1959). For example, see S.G. Kaplan, S. Wu, H.-T.S. Lihn, H.D.Drew, Q. Li, D.B. Fenner, J.M. Phillips, and S.Y. Hou,Phys. Rev. Lett. , 696 (1996). Y. Tokura et al., J. Phys. Soc. Jpn , 3931 (1994), K.Chabara et al., Appl. Phys. Lett. , 1990 (1993), R. v.Helmholt et al., Phys. Rev. Lett. , 2331 (1993), and S.Jin et al., Science , 413 (1994), S.G. Kaplan, M. Qui- jada, H.D. Drew, D.B. Tanner, G.C. Xiong, R. Ramesh,C. Kwan, and T. Venkatesan, Phys. Rev. Letters , 2081(1996). J. Sinova, T. Jungwirth, J. Kucera, and A. H. MacDonald,Phys. Rev. B , 235203/235201-235212 (2003). G. Acbas, M.-H. Kim, M. Cukr, V. Nov´ak, M.A. Scarpulla,O.D. Dubon, T.Jungwirth, Jairo Sinova, and J. Cerne,Phys. Rev. Lett. S. Broderick, L. Degiorgi, H. R. Ott, J. L. Sarrao, and Z.Fisk, Eur. Phys. J. B , 3-6 (2002). Yu.P. Sukhorukov, E. A. Ganshina, B. I. Belevtsev, N.N. Loshkareva, A. N. Vinogradov, K. D. D. Rathnayaka,A. Parasiris, and D. G. Naugle, J. Appl. Phys. , 4403(2002). J. ˇCerne, D.C. Schmadel, M. Grayson, G.S. Jenkins, J. R.Simpson, and H. D. Drew, Phys. Rev. B , 8133 (2000). H.D. Drew and P Coleman, Phys. Rev. Lett. , 1572(1997). J. Cerne, D. C. Schmadel, L. Rigal, and H. D. Drew, Rev.Sci. Instr. , 4755-4767 (2003). M.-H. Kim, G. Acbas, M.-H. Yang, I. Ohkubo, H. Christen,D. Mandrus, M. A. Scarpulla, O. D. Dubon, Z. Schlesinger,P. Khalifah, and J. Cerne, Phys. Rev. B , 214416 (2007). M.-H. Kim, V. Kurz, G. Acbas, C. T. Ellis, J. Cerne,arXiv:0907.3128. J. ˇCerne, M. Grayson, D.C. Schmadel, G.S. Jenkins, H. D.Drew, R. Hughes, J.S. Preston, and P.J. Kung, Phys. Rev.Lett. , 3418 (2000). P. Kostic, Y. Okada, Z. Schlesinger, J. W. Reiner, L. Klein,A. Kapitulnik, T. H. Geballe, and M. R. Beasley, Phys.Rev. Lett. , 2498 (1998). P. Khalifah, I. Ohkubo, H. M. Christen, and D. G. Man- drus, Phys. Rev. B , 134426 (2004). I. Ohkubo, H. M. Christen, S. Sathyamurthy, H. Y. Zhai,C. M. Rouleau, D. G. Mandrus, and D. H. Lowndes, Appl.Surf. Sci. , 35 (2004). M. Izumi, K. Nakazawa, Y. Bando, Y. Yoneda, and H.Terauchi, J. Phys. Soc. Jpn. , 3893 (1997) T. Kitama, K. Yoshimura, K. Kosuge, H. Mitamura, andT. Goto, J. Phys. Soc. Jpn. , 3372 (1999). L. Klein, J. S. Dodge, C. H. Ahn, G. J. Snyder, T. H.Geballe, M. R. Beasley, and A. Kapitulnik, Phys. Rev.Lett.77