Conductance anisotropy and linear magnetoresistance in La2-xSrxCuO4 thin films
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Conductance anisotropy and linearmagnetoresistance in La − x Sr x CuO thin films M van Zalk , A Brinkman and H Hilgenkamp , Faculty of Science and Technology and MESA+ Institute for Nanotechnology,University of Twente, 7500 AE Enschede, The Netherlands Leiden Institute of Physics, Leiden University, 2300 RA Leiden, The NetherlandsE-mail: [email protected]
Abstract.
We have performed a detailed study of conductance anisotropy andmagnetoresistance (MR) of La − x Sr x CuO (LSCO) thin films (0.10 < x < R xy in zeromagnetic field. It is demonstrated that the sign of R xy depends on the orientation of theLSCO Hall bar with respect to the terrace structure of the substrate. Unit-cell-highsubstrate step edges must therefore be a dominant nucleation source for antiphaseboundaries during film growth. We show that the measurement of R xy is sensitiveenough to detect the cubic-tetragonal phase transition of the STO substrate at 105 K.The MR of LSCO thin films shows for 0.10 < x < ρ/ρ ∝ | ρ xy ( B ) | , with theconstant of proportionality independent of temperature. Such scaling suggests thatthe linear MR originates from current distortions induced by structural or electronicinhomogeneities. The possible role of stripes for both the MR and the conductanceanisotropy is discussed throughout the paper.PACS numbers: 73.50.Jt 74.25.fc 74.72.-h Submitted to:
J. Phys.: Condens. Matter onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films
1. Introduction
Strong electron correlations lead to a wide variety of exceptional phenomena. In high- T c superconductors strong correlations play an important role. For instance, theyinduce the Mott insulating state in the undoped parent compounds, despite the factthat the electronic bands are half filled. A rich phase diagram appears upon theintroduction of charge carriers in these Mott insulators. Under certain circumstancescharge carriers spontaneously order along lines, called stripes, which separate undopedantiferromagnetic (AF) regions [1]. Diffraction experiments have demonstrated (static)charge and spin modulation in La . − x Nd . Sr x CuO (Nd-LSCO) [2] and La − x Ba x CuO (LBCO) [3, 4], leaving no doubt that stripes exist in these compounds. Two conditionsneed to be satisfied for stripe ordering to occur: (1) A doping near x = 1/8,corresponding to a filling factor of 1/2 for Cu sites along the stripe. This conditionrelates stripes to the 1/8 anomaly [5], a strong suppression of superconductivity at thisdoping. (2) A structural phase transition from the low-temperature orthorhombic (LTO)to the low-temperature tetragonal (LTT) phase, which is believed to provide a pinningpotential for stripes, through the specific rotations of oxygen octahedra surrounding theCu atoms [2].In a wider variety of compounds, among which La − x Sr x CuO (LSCO),incommensurate spin ordering is observed but no evidence for charge ordering [6, 7, 8, 9].In LSCO such spin ordering can be observed throughout the doping range x = 0.02–0.25[6]. Peaks in neutron diffraction data (either at zero or finite energy) resemble these dueto stripes and it is therefore reasonable to propose the presence of a fluctuating stripephase when condition (1) and (2) for a static stripe phase are not fulfilled [10].While static stripe ordering in Nd-LSCO and LBCO has a pronounced effect onsuperconducting and transport properties, such as T c , the thermopower and the Hallcoefficient R H [11, 12, 13], the consequences of fluctuating stripes/incommensurate spincorrelations in LSCO and YBCO remain elusive. An interesting hypothesis is that if fluctuating stripes are conducting [14] and fluctuate along some preferential direction, ananisotropy occurs in the macroscopic conductivity of the host material. Ando et al. [15]have investigated conductance anisotropy in LSCO in the lightly hole-doped ( x = 0.02–0.04) region and in underdoped YBCO, finding the lowest resistance in the directionalong the spin stripes. In addition to conductance anisotropy, several other fingerprintsof stripes have been investigated. Anisotropic magnetoresistance (MR) was reportedfor underdoped YBCO and related to stripes [16]. Lavrov et al. [17] have searched fornonlinear current-voltage effects related to stripe motion induced by applied electricfields. Their negative result implies that if charged stripes exist in thin films, theyshould be pinned strongly.In our work we proceed to investigate conductance anisotropy in LSCO thin films(0.10 < x < ◦ resolution. Furthermore, we investigate thetransverse in-plane ( I ⊥ B , B k c ) MR, motivated by the observation of linear transverse onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films (a) a R e s i s t an c e ( k ) Temperature (K)x = 0.10x = 0.12x = 0.25
10 20 3010 -2 -1 (b) Figure 1. (a) Sample structure consisting of 36 LSCO Hall bars (one of them shownin the inset) covering α = 0–175 ◦ with 5 ◦ resolution. Bonding pads and wiring leadsare covered by Ti/Au. The STO [100] axis aligns with the long side of the sample.Hall bar dimensions are shown in the inset (in µ m). (b) R ( T ) curves for three differentLSCO compositions. MR in LSCO single crystals for doping x = 0.12–0.13 by Kimura et al. [18], whichmight well be a signature of a fluctuating stripe phase. We observe a sensitivity of theconductance anisotropy for lattice symmetry and we find indications for inhomogeneityon a small length scale. We carefully consider whether these could be due to the presenceof stripes, discussing alternative explanations as well. In particular, we discuss the roleof structural antiphase boundaries, which will be shown to be nucleated from substrateterrace edges.
2. Experimental details
LSCO thin films (thicknesses d in the range 30–60 nm) were grown by pulsed laserablation from sintered LSCO targets on SrTiO (001) (STO), (La . Sr . )(Al . Ta . )O (100) (LSAT), and NdGaO (110) (NGO) substrates. All STO substrates except onewere chemically etched [19]. NGO and STO substrates were annealed for at least twohours at 950 ◦ C in an oxygen environment, LSAT substrates for 10 hrs at 1050 ◦ C.Atomic force microscopy (AFM) confirmed atomically flat substrate surfaces with unit-cell-height substrate steps. The miscut angle typically was 0.1–0.2 ◦ .Films were deposited in 0.13 mbar oxygen at a temperature of 700 ◦ C. The laserfluence was 1.2 J cm − . The film growth was monitored by reflective high-energy electrondiffraction, which showed intensity oscillations, indicative for layer-by-layer growth. Thethin films were annealed for 15 min at the deposition pressure and temperature, afterwhich the oxygen pressure was increased to 1 atm, in which the films were annealed 15min at 600 ◦ C, 30 min at 450 ◦ C and subsequently cooled down to room temperature. c -Axis oriented epitaxial growth was confirmed by x-ray diffraction. Latticemismatches result in tensile strain values of 3.2%, 2.4%, and 2.0% for STO, LSAT,and NGO, respectively. onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films R xy ()
50 100-4.5-4.2-3.9
Figure 2.
Conductance anisotropy measured by the transverse resistance for an LSCOthin film ( x = 0 .
12) for different orientations α . Arrows indicate anomalies which willbe discussed in Sec. 3.2. Hall bars in various orientations [figure 1(a)] were defined by photolithographyand Ar-ion milling. The STO [100] axis aligns with the long side of the sample.For each experiment, insulating behavior of the substrate was confirmed. Electricalcontacts were made by wire bonding to sputtered Ti/Au contact pads, defined bylift-off. Resistance and Hall measurements were performed in a commercial cryostat(Quantum Design, PPMS) with magnetic fields applied perpendicular to the thin film.Resistance measurements were independent of applied current (typically 1–100 µ A) andHall measurements were linear over the entire magnetic field range ( B = -9 T to +9 T).No significant changes in resistivity ρ or T c were observed as a result of thermal cycling.Figure 1(b) shows R ( T ) plots for samples with different Sr contents. We verifiedthat the target stoichiometry ( x = 0.10, 0.12, and 0.25) was transferred 1:1 to the thinfilm by comparing Hall coefficients obtained for our thin films with bulk values [20]. Forthe compositions x = 0.10 and x = 0.12, the Hall angle ρ/R H showed a T -dependenceover 50–300 K, whereas for x = 0.25 ρ/R H linearly depends on temperature. Thisbehavior is in perfect agreement with reported high-quality single-crystal and thin-filmdata on LSCO [21, 22, 20].
3. Results and Discussion
Conductance anisotropy, a smoking gun for the presence of fluctuating stripes [15], ismost effectively examined by measuring the transverse resistance R xy = U y /I x ( α ) ( x and y orthogonal directions) for B = 0 T, since the angle-dependence of the longitudinalresistance R ( α ) is easily affected by small inhomogeneities in the sample. We find arelatively large signal for R xy for all substrates and doping (figure 2), which cannot beattributed to misalignment of the voltage contacts, given the resolution of the appliedphotolithography technique.One possible explanation for the anisotropy is the stepped substrate surface, which onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films (b) STO (e) LSAT(c) NGO(d) STO (unetched) (g) LSCO on LSAT(f) LSCO on STO (unetched) La, SrSrCuTiO (a)
Figure 3. (a) Schematical representation of a step-edge induced antiphase boundary(dashed line) in LSCO on STO. Substrate (b–e) and LSCO thin film (f,g) surfaces.Films shown in (f) and (g) were grown on substrates in (d) and (e), respectively. Scalebars denote 1 µ m. might induce structural antiphase boundaries in the film. Such antiphase boundarieshave experimentally been observed in YBa Cu O on STO using high-resolution electronmicroscopy [23]. Figure 3(a) shows schematically how an antiphase boundary would looklike for LSCO. The CuO -planes are interrupted at the structural antiphase boundary.Typical surfaces of our substrates as measured by AFM are shown in figures 3(b–e).In (f,g) it can be seen that the film surface reflects the morphology of the substrate.We have determined the step-edge orientation α se of all our substrates from AFM dataobtained before deposition of the LSCO thin films.Figure 4 shows that the sign of R xy can be predicted with high certainty from the onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films -1 STO STO STO NGO NGO STO (unetched) STO LSAT NGO NGO STO R xy () B =
V- I-V+I+ -45 0 45 90 135-10 -10 -10 -10 -1 SE (degrees) Figure 4.
Room temperature transverse resistance ( B = 0 T) versus Hall barorientation with respect to step-edge direction for different substrates and x = 0.10(red, open symbols), x = 0.12 (black, solid), and x = 0.25 (blue crosses). α se variesrandomly between 10 and 140 ◦ . orientation of the Hall bar with respect to the step-edge orientation ( α − α se ) for allour samples. This provides evidence that antiphase boundaries in LSCO thin films aredominantly nucleated from substrate step edges. The large spread in R xy reflects therandomness exhibited by step edges. For x = 0.10–0.12, we estimate an antiphase-boundary resistivity of ρ AB ≈ − Ω cm at room temperature, which is in line withtypical interface resistances involving high- T c cuprates [24, 25]. From a typical criticalcurrent density value ( J c ≈ A cm − ) we estimate an I c R n product of about 1 mV,which is a reasonable value [25]. For x = 0.25, ρ AB is about 10 times smaller, which isin agreement with an expected decrease in thickness of the depletion region [25]. In figure 2, anomalies can be observed in R xy ( T ) at 105 K. These are most clearlyrevealed upon numerical differentiation. Figure 5 shows that discontinuities in d R xy / d T are present for all doping, however only for STO substrates. By defining ∆ ≡ (d R xy / d T ) T ↓ − (d R xy / d T ) T ↑ we demonstrate in figure 5(c) that ∆ depends onthe Hall bar orientation α . The largest ∆ is observed for α = 45 ◦ , whereas for α = 90 ◦ ∆ hardly exceeds the noise level. We do not observe anomalies in the longitudinalresistance.The sudden change in d R xy / d T at 105 K coincides with a cubic-tetragonal phasetransition in STO [27]. The fact that such behavior is only observed for STO substratesproves that it is in fact induced by this structural transition. Noise in d R xy / d T below105 K can then be attributed to rearrangement of domains in the substrate, since the c axis can align along three orthogonal directions. The deviation from the cubic unitcell in the tetragonal phase ( T <
105 K) is small ( c/a = 1.00056 at 56 K [28]). Sincethe LSCO film is epitaxially connected to the substrate, we expect the change of thesubstrate’s lattice to be fully passed on to the LSCO film.The lattice parameter changes associated with the LTO-LTT transition in LBCO onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films -200204060 cubic STO d R xy / d T ( m / K ) tetragonal x = 0.12 70 140 21080100120 (e)(d)(c)(a) R xy ( m ) Temperature (K) x = 0.25135(cid:176) (b) STO [100]0 45 900102030 ( m / K ) (degrees) -50 d R xy / d T ( m / K ) x = 0.25-2000200 d R xy / d T ( m / K ) x = 0.10100 200-50050 d R xy / d T ( m / K ) Temperature (K)
NGO x = 0.1250 100 150 200-20020 d R xy / d T ( m / K ) Temperature (K)LSAT x = 0.12 105 K Figure 5. (a) Numerical derivative of the transverse resistance R xy showing a clearjump ∆ at 105 K, coincident with the cubic-tetragonal transition in the STO substrateat 105 K. (b) These effects are absent on LSAT, providing evidence that ∆ is indeedrelated to the STO phase transition. On LSAT weak instabilities are found in therange 60–80 K predominantly for orientations close to the Cu-Cu direction (the curvesshowing such an instability are plotted by solid symbols). (c) Orientational dependenceof ∆, measured at 105 K for LSCO on STO (d) R xy for x = 0 .
25, in which the anomalycan be observed without differentiation because of the small value of the antiphaseboundary induced background. The arrow denotes the resistance change estimatedfrom the stress developing in the LSCO layer upon the the STO phase transition,using the pressure-dependent resistivity data from Nakamura et al. [26]. (e) As (a)and (b) but for x = 0 .
10 and x = 0 .
25 (both on STO) and for x = 0 .
12 on NGO. Thelatter shows an instability at T = 78 K for α = 45 ◦ . ( a LTT /a LTO = 1.0017 and b LTO /a LTO = 1.0036 [29]) are a few times larger than thestructural changes induced by the STO. Yet, for LBCO these small modificationsrepresent a significant change in the tilting direction of the oxygen octahedra, providingthe necessary pinning potential to stabilize a static stripe phase [2]. Pinning of thefluctuating stripe phase present in LSCO as a result of the induced lattice asymmetry bythe STO phase transition would naturally lead to the observed conductance anisotropychange. There are however a few difficulties with this stripe pinning scenario. First,one might expect a stronger doping dependence, as a static stripe phase appears insingle crystals of Nd-LSCO and LBCO only around x = 1 /
8. Second, the appearance ofstatic stripes in these compounds coincides with discontinuities in transport properties,in particular in R H [11, 13]. We do not observe any peculiarity in R H around 105 K.Transport properties in LSCO and other high- T c compounds are generally sensitiveto applied pressure, pointing toward a delicate dependence of electronic structureon crystal structure [30, 26]. The observed conductance anisotropy anomalies mighttherefore be a manifestation of pressure effects on transport properties. We estimate thestress developing in the LSCO layer due to the strain change at 105 K from the Young’s onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films –10 Pa [26, 31] to be 0.06–0.6 GPa. Using data from Nakamura etal. [26] we estimate for x = 0.25 at 105 K a maximum resistivity change induced by suchstress of 10 − Ω cm, leading to ∆ R xy ≈
20 mΩ for our structure. This value compareswell to the measured ∆ R xy for this doping; see the arrow in figure 5(d). For lower x ,the pressure dependence of LSCO is stronger and ∆ R xy will likely be larger. This isconsistent with our observations, although a quantitative comparison is difficult becauseantiphase boundaries induce a stronger background in R xy . The expected stress effectin the longitudinal resistance (∆ R ≈ R xy ) is smaller than the noise we measure in R ,which explains why we do not observe anomalies in R . Only the differential measurementof R xy is sensitive enough to reveal the STO cubic-tetragonal phase transition througha resistivity measurement.For NGO, no structural phase transitions are reported in the temperature range50–200 K [32]. For LSAT there might be small distortion from cubic symmetry at andbelow 150 K [33]. We do not observe a transition near 150 K in 5(b). Weak fluctuationsfor T >
150 K could be traced to variation in the temperature sweep rate. Both onLSAT and NGO [figures 5(b) and (e), bottom panel] we observe instabilities in d R xy / d T in the temperature range 60–80 K, predominantly for Hall bar orientations close to α = 45 ◦ and 135 ◦ . These instabilities are only observed when cooling down, and not forincreasing temperature, unlike the effects on STO. Although speculative, they could beexplained by a structural transition in the film, rather than in the substrate. Perhaps thehigh-temperature tetragonal (HTT) structure is sufficiently clamped by the substrateto reduce the transition to the low-temperature orthorhombic (LTO) phase to 60–80 K. Magnetoresistive properties of LSCO and high- T c cuprates in general have beeninvestigated widely, both in the superconducting ( T < T c ) regime [34, 35, 36], as inthe normal state [18, 37, 38, 39, 40]. Many studies have focused on the violation ofKohler’s rule [38], anisotropy of MR in relation to stripes [39], and high magnetic fields[40]. Most work has been done with single crystals. Here we show that the low-fieldmagnetoresistance ∆ ρ/ρ of LSCO thin films shows intriguing non-monotonic behavioras function of temperature with a crossover from quadratic to linear MR at 90 K. Suchbehavior [figure 6(a,b)] is observed for all doping values and all substrates that wereused for this research. Literature reports [37] quadratic MR without linear componentfor much thicker LSCO films on LaSrAlO , which puts the LSCO under compressive strain (with a moderate lattice mismatch of 0.5%).The linear MR ( T >
90 K) in our thin films is weakly dependent on x and substratetype, and comparable in magnitude to linear MR reported in single crystals ( x = 0.12–0.13) by Kimura et al. [18]. In both cases, linear MR weakly decreases with increasingtemperature over 90–300 K. The quadratic component ( T <
90 K) in our data issuppressed rapidly between 50 and 85 K. This behavior is similar for single crystals.The crossover that we observe at 90 K, might therefore be interpreted as a sudden onset onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films
01 -5 0 5-5 0 501234 / ()
50 K708095110140300
STO x = 0.12R = 432 B (T) / ()
50 K
STO x = 0.12 (b)(a) (c) B (T) -5 0 501 / () B (T) NGO ( x = 0.12)R = 324 -5 0 5 B (T) LSAT ( x = 0.12)R = 343 / () STO (unetched) x = 0.10R = 79250 K 11085 STO x = 0.25R = 97 Figure 6. (a) Temperature evolution of MR in LSCO on STO ( x = 0.12). Solid linesare parabolic fits to data for 50, 70, and 80 K. At 90 K, a crossover to linear MR isobserved. (b) The onset of linear MR between 85 and 110 K is observed for varioussubstrates and x . The zero-field resistances at 50 K are shown in the graphs. (c) MRfor several Hall bars on two different samples. We do not observe sample-to-samplevariations. -40 0 4001 / () xy ( nm) 34 nm10 ASTO (unetched) x = 0.10 -20 0 2060 nm100 A xy ( nm)LSAT x = 0.12-40 0 4056 nm10 A 110 K 140 K 200 K 250 300 K xy ( nm)STO x = 0.12 -4 0 4 xy ( nm) 56 nm100 ASTO x = 0.25 Figure 7.
The magnetoresistance plotted versus the Hall resistivity. We observescaling as ∆ ρ/ρ ∝ | ρ xy ( B ) | , with the constant of proportionality independent oftemperature. Film thicknesses and measurement currents are specified in the graphs. of a linear term above 90 K in combination with a gradual suppression of quadratic MRwith increasing temperature. Interestingly, the linear MR appearing in single crystals( x = 0.12–0.13) is present down to 50 K, and as a result, MR decreases monotonicallywith temperature.The doping dependence of linear MR observed in single crystals strongly suggestsa relation to the 1 / et al. [18] propose it to result from magneticfield enhanced fluctuations towards the stripe phase. An alternative explanation interms of a van Hove singularity crossing the Fermi energy at x ≈ x =0.05–0.25 [6]. Nevertheless, one would expect singular behavior near x = 1 / onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films B >
70 T is required to meet the energy scale typical for dynamical spincorrelations ( µ B B > µ ≈ /Vs (at 50 K) all our measurements are taken in thelow-field ( µB ≪
1) regime. For Ag δ Se it was shown [44] that the magnetoresistancefollows a modified Kohler’s rule: b ( T )∆ ρ/ρ = f ( ρ xy /ρ ), with B and the carrier density n entering implicitly through ρ xy /d = R xy ( B, n ). In our case we find a surprisinglysimple non-Kohler type scaling: ∆ ρ/ρ ∝ | ρ xy ( B ) | , with the constant of proportionalitybeing independent of temperature; see figure 7. This suggests the linear term in theMR has the same origin as the Hall resistivity. Clearly the mixing of the Hall andlongitudinal resistances would provide a straightforward explanation for this behavior.One might wonder why linear MR for x = 0.25 is slightly larger in magnitudethan linear MR for x = 0.10 and x = 0.12, despite the fact that the antiphase-boundaryresistivity is significantly smaller for x = 0.25. It should be noted that also the resistivityof LSCO itself is much smaller for this doping and we expect the ratio between the two todetermine the strength of linear MR. The inhomogeneity scenario also provides a naturalexplanation for absence of linear MR in much thicker LSCO films [37]: the effects of thestructural antiphase-boundaries might be washed out toward the thin film surface by theintroduction of other types of defects, giving rise to more isotropic disorder. Moreover,the Hall voltage is smaller for thicker films, as it is inversely proportional to the filmthickness.Some questions remain concerning the inhomogeneity scenario. First, it is unclearwhy linear MR vanishes below 90 K. Both the longitudinal and Hall resistivity do notshow apparent changes of behavior around 90 K. Second, we do not observe a strongsample-to-sample variation in the magnitude of linear MR [see figure 6(c)], which mightbe expected if inhomogeneity is the underlying cause. Third, there is no dependence oflinear MR on the Hall bar angle ( α − α se ). The answers to the last two questions mayreside in the exact identification of inhomogeneity in our samples. Perfectly straight andparallel antiphase boundaries, with homogeneous ρ AB might not give rise to linear MR onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films α − α se . The small length scales of such imperfections might provide enoughaveraging to prevent sample-to-sample variations. Numerical calculations will have tocorroborate the proposed scenario. If the mechanism would fail to account for ourobservations, an electronic origin (e.g. stripes) of linear MR will have to be reconsidered.
4. Conclusion
The transverse resistance R xy in zero magnetic field, usually background in a Hallmeasurement, provides valuable information about the microstructure of the materialunder study. We have used it to demonstrate that unit-cell-high substrate stepedges are the dominant source of structural antiphase boundaries in LSCO thin films.The antiphase boundary resistivity was estimated to be ρ AB ≈ − Ω cm (roomtemperature). In addition, we show that for LSCO R xy can reveal structural phasetransitions of the substrate on which the films are grown. Such transitions are usuallydifficult to detect and require advanced spectroscopic analysis equipment.For the detection of stripes, conductance anisotropy is an important observable.We have shown that in LSCO thin films conductance anisotropy is dominantly causedby antiphase boundaries, which mask possible stripe effects. Future experiments in thisdirection will therefore require substrates with an extremely small vicinal angle, andHall bars at sub-micron scale.The silver chalcogenides have recently attracted interest because of their non-saturating linear MR, which make them suitable for use as magnetic field sensor [43, 44].The MR is linear down to surprisingly low magnetic fields in these materials. This hasbeen explained by the presence of disorder, giving rise to the mixing of longitudinaland Hall resistances [46]. Our LSCO thin films show linear low-field MR in the entiredoping range 0.10 < x < ρ/ρ ∝ | ρ xy ( B ) | with a temperature-independent constant of proportionality. Thissuggests the linear MR of LSCO thin films is related to disorder as well. Structuralantiphase boundaries generated from substrate steps are a likely source of disorder.However, linear MR also appears in single crystals of LSCO, although in a narrowerdoping range ( x = 0.12–0.13) [18]. It is unclear why these crystals in particular wouldcontain many antiphase boundaries. If the presence of such defects can be excludedexperimentally, linear MR must have a different origin, at least in single crystals. Inthat case it will be worth reconsidering the role of stripes, which might similarly deflectthe current from the longitudinal direction, causing the mixing of longitudinal and Hallresistances. onductance anisotropy and linear magnetoresistance in La − x Sr x CuO thin films Acknowledgments
We gratefully acknowledge Jan Zaanen for fruitful discussions. This work is financiallysupported by the Dutch Foundation for Fundamental Research on Matter (FOM), theNetherlands Organization for Scientific Research (NWO) through VIDI (A.B.) and VICI(H.H.) grants, and the NanoNed program.
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