Anomalously large anisotropic magnetoresistancein a perovskite manganiteRun-Wei Lia,b, Huabing Wangb, Xuewen Wangc, X. Z. Yub, Y. Matsuib, Zhao-Hua Chengd, Bao-Gen Shend,E. Ward Plummere,1, and Jiandi Zhange,1

aNingbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, Zhejiang 315201, People’s Republic of China; bNationalInstitute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0047, Japan; cDepartment of Physics, Florida International University, Miami, FL 33199; dBeijingNational Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China;and eDepartment of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803

Contributed by E. Ward Plummer, July 8, 2009 (sent for review November 28, 2008)

The signature of correlated electron materials (CEMs) is the cou-pling between spin, charge, orbital and lattice resulting in exoticfunctionality. This complexity is directly responsible for their tun-ability. We demonstrate here that the broken symmetry, throughcubic to orthorhombic distortion in the lattice structure in aprototype manganite single crystal, La0.69Ca0.31MnO3, leads to ananisotropic magneto-elastic response to an external field, andconsequently to remarkable magneto-transport behavior. Ananomalous anisotropic magnetoresistance (AMR) effect occursclose to the metal-insulator transition (MIT) in the system, showinga direct correlation with the anisotropic field-tuned MIT in thesystem and can be understood by means of a simple pheno-menological model. A small crystalline anisotropy stimulates a‘‘colossal’’ AMR near the MIT phase boundary of the system, thusrevealing the intimate interplay between magneto- and electronic-crystalline couplings.

metal-insulator transition � CEM

The striking transport properties of many CEMs, such asunconventional superconductivity and ‘‘colossal’’ magne-

toresistance (CMR), are believed to be manifested by a coex-istence of a variety of complex interactions that couple simul-taneously active degrees of freedom (1). Consequently, suchcomplex couplings lead to multifunctionalities displayed by thesematerials. The hallmark of these materials is the multitude ofcompeting ground states that can be tuned or manipulated bydoping, structural manipulations, strain induction, or the appli-cation of external stimulus, such as pressure, electric or magneticfields. In doped manganites Ln1-xAxMnO3 where Ln and A arerare- and alkaline-earth cations, respectively, ferromagnetic(FM) metallic, paramagnetic (PM) insulating and antiferromag-netic (AF) charge/orbital ordered states are among these com-peting ground states (2). External fields influence the subtlebalance of the interactions and induce phase transitions betweendifferent ground states (3–5). Examples include the CMR andgiant anisotropic magnetoresistance (AMR) effects in which thetransport has ‘‘colossal’’ response to an external field, thusoffering intriguing opportunities for new technological applica-tions with advanced correlated electron devices (6).

Magnetoresistance (MR) is a property of materials in whichthe resistivity depends on the strength of applied magnetic field.Anisotropic magnetoresistance (7), on the other hand, is whenthe resistivity also depends on the field orientation with respectto either the electrical current or the crystal axes. Whereas thedependence of resistivity on the field direction relative to theelectrical current, known as ordinary MR or Lorentzian MR, isexpected due to the Lorentz force acting on the conductionelectrons; the dependence of resistivity on the field orientationwith respect to the crystal axes is much more unusual. However,this kind of AMR provides detailed information about theunderlying physics such as spin-orbital and magneto-elasticcouplings in a material and it also enables wide range of

technological applications. The most prominent technologicalapplication of giant MR is in the data storage industry. Yet, theadvantage of AMR over conventional MR is that AMR can beapplied for sensor of field direction and dimensional imaging.

AMR effects are small in most of FM metals and alloys (8),typically with a magnitude of few percent. Recently, it wasdiscovered that giant and unusual AMR effects with a magni-tude of �30% exist in several perovskite manganite thin films(9–14). Particularly, the observed AMR exhibits strong andnonmonotonic temperature (T) (11–14) and field (H) depen-dence (9, 14). This kind of behavior differs dramatically fromwhat has been observed in conventional 3d transition metals oralloys, suggesting a different mechanism beyond conventionaltheories of AMR. For example, the AMR amplitude inLa0.7Ca0.3MnO3 films (10, 11, 13) and in the strainedPr0.67Sr0.33MnO3 films (12) shows a giant enhancement near theCurie temperature or metal-insulator transition (MIT) temper-ature (TMI), whereas conventional AMR in 3d transition metalsor alloys depends monotonically on temperature. Furthermore,the sign of the AMR can be tuned from negative to positive byinducing lattice strain to the films through the lattice mismatchbetween the films and substrate, indicating a strong influence oflattice distortion (12). Most studies of the AMR have focused onthin films. Many extrinsic effects, including the substrate-induced strain and the large demagnetization field due to thelarge apical ratio in film geometry, have impeded thoroughinvestigations of the inherent nature of AMR in this class ofCEMs. In particular, it is unclear whether the observed giantAMR effect is truly intrinsic due to magneto-elastic or spin-orbital couplings. If indeed this is the case, then how does oneunderstand the unusual temperature and field dependence ofobserved AMR? Why does the AMR always have its maximumvalue near TMI? Here, we have taken a different approach bysystematically investigating the single crystals of La1-xCaxMnO3(x � 0.31). Our results on magneto transport clearly show thatthe observed anomalous AMR has an intimate relationship withthe crystalline anisotropy in the system. We provide a phenom-enological model to explain the observed phenomena, whichshould be also applicable to other CEMs with strong magneto-elastic coupling.

La1-xCaxMnO3 (x � 0.31) single crystals are orthorhombic,deviating from the cubic perovskite structure via two dissimilarJahn-Teller (J–T) distortions—in-plane (i.e., in ab-plane) rota-tion and out-of-plane (c-axis) tilt of the MnO6 octahedron (15,16). A MIT coincides with a FM-to-PM transition at TMI � 220

Author contributions: R.-W.L., H.W., and J.Z. designed research; R.-W.L., H.W., X.Z.Y., Y.M.,Z.-H.C., and B.-G.S. performed research; R.-W.L., H.W., X.W., X.Z.Y., Y.M., B.-G.S., E.W.P.,and J.Z. analyzed data; and R.-W.L., E.W.P., and J.Z. wrote the paper.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.

1To whom correspondence may be addressed. E-mail: jiandiz@lsu.edu or wplummer@lsu.edu.

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K, characterized by the T-dependence of magnetization M(T)and the resistivity �(T) shown in Fig. 1 A and B, respectively. Themeasured �(T) vs. H curve demonstrates a typical negativemagnetoresistance behavior. However, the observed MR showsa strong dependence on the field orientation, leading to aremarkable AMR effect.

We used a single crystal of a slab shape (shown in the Inset ofFig. 1B) where the c-axis is perpendicular to the slab plane (xyplane). Fig. 1C shows �(T,�) under a H � 1.0 T field rotating inthe yz plane with angle � between the field and c-axis, whichdemonstrates a strong T-dependence of the AMR effect with the

peak value achieved very close to TMI. The AMR amplitude isdefined as R� � (1 � ��/�//) � 100%, where �// � �(� � 0) and�� � �(� � 90°) are the resistivities for a given field being paralleland perpendicular to the c-axis of the crystal, respectively. ForH � 1.0 T, a peak value of R� � 40% is observed near TMI. Inthis case, the measurement contains no contribution of theconventional Lorentzian MR because the field rotates in theplane perpendicular to the current direction. This anomalousT-dependent AMR must be related to the magneto-crystallineanisotropy and closely associated with the coupled MIT andFM-to-PM transition in the crystal.

We systematically investigated the temperature- and field-dependence of AMR and their correlation with the field-dependent MIT. The results are shown in Fig. 2. TMI increasesalmost linearly with field (see Fig. 2 A). A key discovery here isthat the measured R� peak follows almost exactly the field-dependence of TMI, as shown in Fig. 2B. Large AMR emergesonly near TMI (Fig. 2C), whereas no prominent AMR appears ineither the pure FM-metallic state at T � TMI or the purePM-insulating state at T � TMI. Likewise, the nonmonotonicfield-dependence of R� at a given T close to TMI is evident in Fig.2D, such that R�(H and T) exhibits very similar line shapes forvarying field or temperature.

Fig. 3 presents the measured field-dependence of both thenormalized resistivity and TMI for the field along three differentsample orientations. The measured �(H) has similar values forthe H//x and H//y directions but quite different values for the H//zdirection (see Fig. 3A). A small difference in �(H) between theH//x and H//y directions can mainly be attributed to the Lorent-zian MR, whereas the large difference between the H//z andH//x(y) directions gives rise to the observed giant AMR. Alsoshown in the Inset of Fig. 3B, a small but clear shift of �(T) isevident by orienting H//z- from H//x(y)-direction, thus indicatingan anisotropic field-dependent MIT. As presented in Fig. 3B, theTMI(H) curve for the H//z direction deviates from that for theH//x(y) direction. The crystalline c-axis (i.e., the z direction),which is perpendicular to the sample plane, is the hard axis for

Fig. 1. Magnetic and magneto-transport properties of La0.69Ca0.31MnO3. (A)T-dependence of the magnetization with an Inset showing the unit cellstructure as well as the crystalline axes. (B) T-dependence of the resistivitymeasured from a slab of a single crystal under different fields. The Inset showsa schematic arrangement of the measurements from the crystal with a slabsample size: 3.8 � 0.85 � 0.15 mm3. The sample c-axis is normal to the slabplane. (C) Angle- and T-dependences of the normalized resistivity, �(�)/�(�) �0°, measured under a magnetic field rotating about the x-direction (i.e., thecurrent (I) direction).

Fig. 2. Correlation between MIT and AMR of La0.69Ca0.31MnO3. (A) T- and H-(along c-axis) dependence of the resistivity (�) and TMI defined as the peaktemperature in d�/dT � T curve (see the Inset). (B) A contour plot and line plotsof (C) the T- and (D) H-dependence of the measured R� from the slab singlecrystal as the field was applied in the xz plane (thus including Lorentzian MR).The solid lines through the data points are guides to the eye.

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the field-dependent MIT. We conclude that the observed largeAMR simply reflects the uniaxial magneto-crystalline anisot-ropy in the system.

The correlation between AMR and MIT displayed in Fig. 2and Fig. 3, especially the gap of the two TMI(H) curves shown inFig. 3B, provide a crucial key for understanding the anisotropicmagneto-transport properties. The difference in TMI due to fieldorientations (H//c- or H�c-axis) leads to the observed unusualAMR effect. At a given field strength (vertical cut and betweenthe two TMI curves in Fig. 3B), the sample becomes insulatingwhen H//c-axis while remaining metallic when H�c-axis—creating the R� peak as shown in Fig. 2C. On the other hand, ata given T (horizontal cut in Fig. 3B), the sample becomes metallicas H�c-axis but remains insulating as H//c-axis, thus resulting inthe R� peak as shown in Fig. 2D. When the system is far awayfrom the gap region of the two TMI(H) curves, only conventionalAMR with small amplitude should exist. Furthermore, increas-ing the field gradually reduces the R� (see Fig. 2D). Thus, one cananticipate that the AMR effect as well as the anisotropy offield-dependent MIT (the gap in Fig. 3B) should vanish under asufficiently strong field.

Jahn-Teller distortions (17) and double-exchange (DE) inter-actions (18, 19), which are intimately coupled in this class ofdoped oxides, play vital roles in both the transport and themagnetic properties. Simply speaking, Jahn–Teller (J-T) distor-tions tend to promote insulating phase to the system whereas DEinteraction endorses FM metallic state, although the strength ofDE interaction depends on the degrees of J-T distortions thusmaking a close coupling between them. Bending the Mn-O-Mn

bond by enhancing J-T distortion reduces DE interaction so asbandwidth. The subtle competition between them causes theMIT in these doped manganites. However, external field tendsto disturb such a competition by facilitating the metallic state,evident by appearing negative CMR near the TMI and tuning theMIT (16, 20–22). Owning to the strong correlation character inthese materials, lattice also has its response to applied magneticfield by alter J-T distortions, depending on the detailed latticestructure and applied field orientation. One possible latticeresponse is a suppression of J-T distortions by straightening theMn-O-Mn bond thus causes negative CMR by enhancing DEinteractions. Indeed, the giant AMR observed in La1-xCaxMnO3reflects exactly the different lattice response to the external field,depending on the field direction.

If, in a cubic peroviskite structure, lattice should have identicalresponse to external field in all high-symmetry crystalline direc-tions, thus no AMR is generated by magneto-elastic coupling.However, in orthorhombic La1-xCaxMnO3, the structure haslower symmetry by bearing two distinct J-T distortions from acubic perovskite. The lattice has larger response when the fieldis perpendicular to c-axis than parallel to c-axis, thus creatingdistinct effects on the transport properties including MITthrough different couplings with the DE interaction. This dif-ferent response exactly makes c-axis as the hard axis for thefield-dependent MIT, having a small but crucial anisotropy asclearly shown in the two field-dependent TMI curves (see Fig.3B). Consequently, the system exhibits a giant AMR effect nearthe MIT region that is stimulated by the distinct lattice responseto external field because of the asymmetric J-T distortions.

Based upon the above scenario for the nature of intrinsicAMR, we are able to qualitatively explain the dissimilar mag-nitude of AMR effects observed in different manganite systems(9–14). Because the AMR effect depends on the dissimilarresponse of J-T lattice distortions to external field, a system

A B

C D

Fig. 4. Crystal twinning behavior of La0.69Ca0.31MnO3. Lorentzian TEM im-ages with the selected-area electron diffraction patterns of correspondingareas at T � 80 K with different crystalline orientations. (A) The 45° crystaltwinning between a- and b-axis. (B) The 90° crystal twinning in the c-axis. TheT dependence of the anisotropic R� at H � 1.0 T measured from (C) the slab and(D) a square-rod La0.69Ca0.31MnO3 single crystal. The Inset shows a schematicarrangement of the measurements from the square-rod sample size: 7.0 �0.85 � 0.85 mm3. The c-axis is in the plane perpendicular to the currentdirection.

Fig. 3. Field-orientation dependence of magnetoresistance and MIT inLa0.69Ca0.31MnO3. (A) Field strength and orientation dependence of resistivitynormalized to its zero-field value at the temperatures near TMI and (B) TMI

measured from the slab single crystal (see the Inset of Fig. 1B). The Insetpresents the T-dependence of resistivity on warming under different values ofmagnetic field along z(c)-axis (solid circle) and in xy(ab)-plane. The solid linesare guides to the eye. The dashed arrows indicate the horizontal and verticalcuts for a fixed temperature and field, respectively.

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with stronger J-T distortions should have greater AMR effect.This mechanism explains that, in a similar doping level of x � 0.3,La1-xSrxMnO3 (14) has a smaller AMR effect than Pr1-xSrxMnO3(12) and La1-xCaxMnO3 (9–11, 13) simply because of fewer J-Tdistortions.

Because the intrinsic AMR discussed above is closely relatedto crystalline orientation, twinning in a single crystal coulddramatically affect the characterization of AMR properties.Imagine that a single crystal is randomly and completelytwinned, then, the intrinsic AMR related to crystalline anisot-ropy would be averaged out. In such a case, a possible AMRwould only be the result of the demagnetization field effectsbecause of the geometric shape of the slab sample. However, webelieve that this is not the case. First, demagnetization fieldeffects cannot generate such large amplitude of AMR. Moreimportantly, any demagnetization effect cannot result in theunusual field- and temperature-dependence of AMR as ob-served in the system.

To further explore the issue arising from the twinning anddemagnetization field effect, we have investigated the possibletwinning domain structure in the system. Fig. 4 A and B presentstwo Lorentzian TEM images with selected-area electron diffrac-tion patterns of the corresponding domain areas. The imagesreveal magnetic domain structure, thus reflecting crystallinetwinning domain structure (23). Two distinct twinning cases arerevealed. The twinning between the a- and b-axis, or in-planetwinning, is found via a 45° (or 135°) rotation of each other (Fig.4A); whereas the twinning between the a- (or b-) and c-axis, orout-of-plane twinning, is found through a 90° rotation of eachother (Fig. 4B). It is expected that in-plane twinning is morecommon than out-of-plane twinning in orthorhombic crystals.

Both the small MR difference between the H//x and H//ydirections (Fig. 3A) and relatively small R� as the field rotates inthe xy plane (Fig. 4C) indicate that the twinning existing in theslab sample is mainly the in-plane type.

To corroborate this uniaxial anisotropy scenario, we haverepeated the AMR measurements on a square-rod sample that,because of the increased size of the sample in the z direction, isanticipated to display more out-of-plane twinning and smallerdemagnetization field than the slab sample. The results areshown in Fig. 4D. Compared with the results obtained from theslab sample, the R� values, as the field rotates in the xz (redcurve) and yz plane (green curve), are substantially suppressedin the square-rod sample. Furthermore, the order in the mag-nitude of R� curves (green and blue) between the field rotatingin the yz and xy planes is also switched. These results are exactlyas expected because of the substantial out-of-plane twinningexisting in the square-rod sample—portions of the crystal do-main have their hard-axis (c-axis) oriented in the y direction (orx direction). Such a mixture of hard-axis orientation shouldsuppress the intrinsic uniaxial anisotropy. Because there is acomponent of hard axis in the y direction, the R� in the xy planeis enhanced, whereas that in the yz plane is reduced, resulting inthe order switch of R� curves.

The observed results can be quantitatively explained by aphenomenological model with a uniaxial AMR caused by thecrystalline anisotropy. We start with the description of the modeland then explain the data from both slab and square-rod

Fig. 5. Phenomenological uniaxial anisotropy model for the AMR ofLa0.69Ca0.31MnO3. (A) Calculated normalized resistivity at a given temperatureas a function of magnetic field strength and orientation with respect to thecrystal c-axis (�), based upon the uniaxial anisotropic model (see the Inset forthe uniaxial resistivity ellipsoid). (B) Calculated (solid lines) �-dependence ofnormalized resistivity for three values of fields (corresponding to the threesheets in A) rotating in the yz plane, compared with the experimental results(symbols) measured from the slab sample at T � 225 K. (C and D) Calculatedangle dependence of normalized resistivity for a field of H � 1.0 T rotating inthree perpendicular planes (from y to x, z to y, and z to x direction), comparedwith the experimental results (symbols) measured at T � 225 K from the slab(see Fig. 1) and square-rod (see Fig. 4.) sample, respectively.

Fig. 6. The magnetization of La0.69Ca0.31MnO3 as a function of magneticfield for the two distinct field directions measured at different temperature,H//z(c)- and H�z(c)-axis, respectively.

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samples. Assuming the resistivities for a given field parallel andperpendicular to the c-axis of the crystal are �// and ��, respec-tively, a uniaxial resistivity ellipsoid (a prolate if �// � ��) can beconstructed through the equation

����

�//�2

cos2 � � � ���

��� 2

sin2� � 1, [1]

as is shown in Fig. 5A, where the short-to-long axial ratiorepresenting the uniaxial resistivity ratio is � � ��/�//. Thenormalized resistivity �(�)/�// can be expressed as

���

�//� ���2 cos2� � sin2��1/2. [2]

AMR is completely determined by the ratio �. If the anisotropyis relatively weak, i.e., � 1, the normalized resistivity can beapproximately expressed as

���

�//� cos2� � �sin2� [3]

that is described as a symmetric sinusoidal function of the angle �.In conventional materials, uniaxial AMR is relatively small

(R� � 10%), weakly dependent on field and temperature, andcan be described well with Eq. 3. However, as shown in Fig. 2 Cand D, the R� in La1-xCaxMnO3 shows very strong T- andH-dependence, i.e., � � �(T,H). Furthermore, as shown in Fig.5B, the �(�)/� (0) curve deviates greatly from the symmetricsinusoidal function depicted by Eq. 3. Thus, we used Eq. 2 toreproduce the measured data. For T close to TMI, a normalized�(H,�)/�(H,� � 0) or �(H//,H�)/�(H,� � 0) is calculated andshown in Fig. 5A as a curved 3D MR sheet, where H// � Hcos�and H� � Hsin�, respectively. The only fitting parameter hereis �(H) at a given T.* For a given H, the cross section betweenthe MR sheet and a particular vertical H sheet reveals anormalized �(�)/�(0). Fig. 5B presents the calculated and mea-sured �(�)/�(0) from the slab sample for the three H values atT � 225 K and as H rotates in the yz plane. We obtain � � 0.598for H � 1.0 T and T � 225 K.

As shown in Fig. 5B and Fig. 1C, the line shape of �(�)/�(0)clearly shows a peak-to-valley asymmetry with a larger valleywidth, in contrast with the symmetric line shape given by Eq. 3.The excellent agreement between theoretical and experimentalresults in Fig. 5B along with an evidently large asymmetric lineshape provides the justification for Eq. 2. It is noteworthy thatfilms of Pr2/3Sr1/3MnO3 with large AMR also exhibit similarasymmetry �(�)/�(0) shape (12).

The phenomenological model can reproduce all of the exper-imental AMR data for field rotation in different sample planesfor both slab and square-rod samples. Fig. 5C shows boththeoretical and experimental results with excellent agreementfor the slab sample. For the square-rod sample (see Fig. 4D), themeasured R� is much smaller than that of the slab sample, mainlybecause of the substantial c-axis mixture in the yz plane (see Insetin Fig. 4D). Therefore, we propose a two-component model (seeMethods). Each component satisfies Eq. 2 and has the same �value as the slab sample. The only additional parameter intro-duced here is the twinning portion parameter (�): the amount ofthe minor c-axis component (� � 0.5). This single parameter �is fitted with the �(�)/�(0) data from the square-rod sample, andthe results are shown in Fig. 5D. The agreement between thetheoretical results and experimental data are very impressive,considering that we neglect other contributions such as theLorentzian MR. As shown in the Inset of Fig. 5D, a quitereasonable value of � � 0.40 is obtained, and it remainsreasonably constant at different temperatures, further confirm-ing the validity of the model.

MethodsExperimental. La0.69Ca0.31MnO3 single crystals were grown using the travelingsolvent optical floating zone technique. Samples of both the slab and square-rod shapes for the measurements were cut from the same single-crystal rodsuch that the x direction was almost along the crystal rod growing direction.There is a slightly offset in the doping concentration such that the measure TMI

� 220 K is lower than the published value of TMI � 250 K. The crystalorientation of the samples was carefully examined by X-ray diffraction. TheT-dependence of the magnetization was measured by a superconductingquantum interference device magnetometer (SQUID, Quantum Design). TheMR measurements were carried out using a Physical Property MeasuringSystem (PPMS, Quantum Design) equipped with a motorized sample rotator.The standard four-probe method was used for the resistivity measurements.The angular dependence of the resistivity �(�) was measured at a fixedmagnetic field and temperature by varying the sample orientation relative tothe field. To avoid Joule heating effects, a measuring electrical current wasused below 100 �A in all of the electrical measurements. To ensure that theobserved AMR is not just simply caused by a magnetization anisotropy, wehave measured the magnetization as a function of magnetic field for the twodistinct field directions, H//z(c)- and H�z(c)-axis, respectively. The result isshown in Fig. 6 for the measured magnetization curves at different temper-atures. The magnetization anisotropy with hard magnetic axis in c-axis direc-tion exists in the low field case; the saturation field is only �0.5 T in the FMphase. Therefore, the existence of magnetization anisotropy does not affectour argument for the cause of the observed anomalous AMR.

The magnetic domain structures were imaged by means of Lorentziantransmission electron microscopy (TEM) (Hitachi HF-3000L) equipped with acold-field emission gun and operated at an acceleration voltage of 300 kV. Themicroscope was equipped with a custom-made objective lens, making themagnetic field almost zero at the sample position. Electron-transparent thinsamples were sliced from the same crystal for the AMR measurements andprepared by mechanical polishing and subsequent argon-ion thinning with anacceleration voltage of 4 kV at room temperature. In Lorentzian TEM obser-vations, we used the Foucault mode (Fig. 4 A and B). In this mode, the incidentbeam is split by Lorentzian forces with different directions at the diffractionplane. In the case of the 180° magnetic domains, antiparallel magnetic do-mains produce two spots at the diffraction plane. When only one of the spotsis selected by the objective aperture to form an in-focused image, the imagewith white/black band-shape areas is observed in the in-focused plane. Thesewhite/black areas correspond to antiparallel magnetic domains, and domainwalls are imaged as a boundary between the white and black areas.

Two-Component Model. To include the effect of twinning existing in thesquare-rod sample, we realized that the sample has a mixture of twinningdomains with their c-axis perpendicular to each other (see Fig. 7) and, as aresult, is no longer uniaxial. We considered a simple two-component scenar-io—one has the c-axis along the y direction with a twinning portion � (minorcomponent), whereas the other is along the z direction with a twinningportion (1 � �) (major component), and each component has the same prolateellipsoid dependency as we used for the slab sample. The resistivity-fieldrelation for the minor component, with its c-axis in the y direction, is thengiven by a prolate in the y direction:

*For simplification, here we neglect the contribution of Lorentzian MR and sample shape-induced demagnetization effects. There will be more parameters if these factors areconsidered.

Fig. 7. Schematic view of the relative orientation relation between magneticfield (H), measured current (I), and crystal axis of square-rod sample existingtwo c-axis components due to twinning domains.

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� �

���2

cos2� � � �

��� 2

sin2�cos2� � � �

� //� 2

sin2�sin2� � 1.

When mixed with the major component (Equation (1)), while preservingthe ellipsoid shape, we got the superposition of two perpendicular prolateellipsoids:

� � ���, � � �cos2�

�h2 �

sin2�sin2�

� l2 �

sin2�cos2�

��2 � �1/2

where

�h2 �

�//2��

2

�1 ���2 � ��//

2 ,

and

�l2 �

�//2��

2

���2 � �1 ��//

2 .

�h and �l are the resistivities as the H field is along the major and minorcomponent’s c-axis, respectively, whereas �� is the resistivity when the H fieldis perpendicular to both the major and minor c-axis (or yz plane). This two-component model can account for the loss of uniaxial property in the square-rod sample. In addition, we can obtain the parameter �, the portion of theminor component, by fitting the results from AMR measurements to theproposed two-component model. The result is shown in the Inset of Fig. 5D.The value found is �0.4 for the temperature range around the maximum AMRsignals, which shows that this two-component model can adequately accountfor the experimental results of the square-rod sample.

ACKNOWLEDGMENTS. We thank Hongjun Liu and Hao Sha for their technicalassistance. We acknowledge financial support from the Chinese Academy ofSciences, National Natural Science Foundation of China, State Key Project ofFundamental Research of China, the Zhejiang Natural Science Foundation(China), MEXT of Japan, and the U.S. National Science Foundation throughgrant DMR-0346826 (USA).

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