Magnetic and magnetotransport properties of magnetite films with step edges

ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 279 (2004) 331–342

Magnetic and magnetotransport properties of magnetite films with step edges a . M. Ziesea,*, R. Hohne , H.C. Semmelhacka, K.H. Hana, P. Esquinazia, K. Zimmerb a

Department of Superconductivity and Magnetism, University of Leipzig, Linn!estrasse 5, 04103 Leipzig, Germany b Institute for Surface Modification, 04318 Leipzig, Germany Received 28 October 2003; received in revised form 12 December 2003

Abstract The magnetoresistance of step edges in magnetite films was systematically studied. An enhancement of the magnetoresistance by the introduction of step edges was observed, especially in the high-field regime. This was modelled by spin-disorder scattering. The analysis revealed magnetic cluster formation at the step edges. r 2003 Elsevier B.V. All rights reserved. PACS: 72.25.-b; 75.70.-i; 75.47.-m; 85.75.-d Keywords: Magnetite; Magnetization; Magnetoresistance; Spin polarized transport

1. Introduction Extrinsic magnetoresistance effects in half-metallic oxides have been intensively studied in recent years [1]. These are interesting from a fundamental point of view, since they are induced by spinpolarized tunnelling across a natural tunnelling barrier, as well as from an application-oriented perspective, since these effects lead to a strong enhancement of the low-field magnetoresistance. In manganite compounds of the type La0:7 Ca0:3 MnO3 extrinsic magnetoresistance effects are well established [2–6]. At low temperatures a large extrinsic magnetoresistance effect was *Corresponding author. Tel.: 00493419732752; fax: 00493419732769. E-mail address: [email protected] (M. Ziese).

observed which decays much stronger with temperature as is anticipated from the temperature dependence of the saturation magnetization. This was attributed to a reduced interfacial magnetization [7–10] and up to date no remedy to solve this problem of manganite compounds has been found. Comparative studies of the temperature dependence of the extrinsic magnetoresistance in various half-metallic oxides showed that the decay of the low-field magnetoresistance is much weaker in double perovskites of the type Sr2 FeMoO6 than in the manganites and CrO2 [11]. This might be related to different transport mechanisms active in the compounds: double exchange—being strongly affected by local structural disorder—in the manganites and CrO2 versus more conventional band transport in the double perovskites [1]. On the other hand, the absolute value of the low-field

0304-8853/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2004.02.001

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magnetoresistance in double perovskites is small. Judging from the perspective of room temperature applications, the progress in the exploration of grain-boundary magnetoresistance in the aforementioned oxides has been insufficient. Apart from problems with the optimization of interfacial parameters, the disappointing performance is deeply rooted in the rather low Curie temperatures of these oxides. Magnetite (Fe3 O4 ) is a notable exception in the class of half-metallic oxides, since it has a rather high Curie temperature of 858 K: Magnetite crystallizes in the inverse spinel structure with tetrahedral sites (A-sites) occupied by Fe3þ ions and octahedral sites (B-sites) occupied by both Fe2þ and Fe3þ ions. Magnetite is a ferrimagnet with the A- and B-site sublattices coupled antiferromagnetically such that the molar magnetic moment is about 4 mB ; mB being the Bohr magneton [12]. Band structure calculations showed a half-metallic ground state [13]. In a first approximation, the rather high conductivity which is exceptional among ferrites is due to electron transfer between the mixed valent iron ions on the B-sites. The transport properties, however, are complex and are dominated by strong correlation effects and a strong electron–phonon coupling. Above the Verwey transition at about 120 K [14], the transport properties might be described by small polaron hopping in combination with small polaron band-motion [15]. Below the Verwey transition, which is a charge ordering transition, the conductivity is strongly reduced and is due to nearest neighbour or variable range hopping [16]. There have been several attempts to search for grain-boundary magnetoresistance in magnetite samples [17–28], but the effects observed so far have been rather small. Most of the researchers believe that the transport mechanism involved is spin-polarized tunnelling. In this work we present a systematic investigation of the magnetic and magneto-transport properties of magnetite films with grain-boundaries introduced by step edges on the substrate. It will be shown that the magnetoresistance enhancement caused by these grain boundaries is a high-field effect due to spin disorder scattering in the disordered regions. We did not observe any indications of a spin-polarized

tunnelling mechanism in these samples, in contrast to the results obtained on step-edge junctions in La0:7 Ca0:3 MnO3 films [29].

2. Experimental details Magnetite films were grown on both epipolished and patterned MgAl2 O4 ð001Þ substrates by pulsed laser deposition from a stoichiometric, polycrystalline magnetite target. Deposition temperature was 430 C and oxygen partial pressure to achieve optimal magnetic and transport properties was 5  106 mbar: A Lambda–Physik Excimer laser at a wavelength of 248 nm (KrF) operated at a repetition rate of 10 Hz and a pulse energy of 0:6 J was used; the fluence was about 2:5 J=cm2 : Prior to film deposition, three substrates were patterned into arrays of step edges using conventional optical lithography and subsequent chemically assisted ion-beam etching. Step-edge height was between 80 and 120 nm; the separation between step edges was 10 mm: The step-edges covered the whole substrate area. In this work data on three films are presented, namely: two step-edge samples (P1, P2) with (1) 80 nm and (2) 120 nm step height and on a magnetite film deposited on an epi-polished substrate (E1) which is used as a reference sample. All films have a thickness of about 80 nm as determined by a Dektak profilometer. Step direction was [100]. For comparison a La0:7 Ca0:3 MnO3 stepedge array on LaAlO3 is used, see [29]. This was fabricated by PLD at 700 C substrate temperature and 0:15 mbar oxygen partial pressure. Step height was 80 nm; step direction [110], film thickness about 30 nm and Curie temperature 230 K: Structural characterization by X-ray diffractometry was performed with a high resolution diffractometer using Cu Ka1 radiation. Atomic (AFM) and magnetic (MFM) force microscopy measurements were performed using a Nanoscope IIIa with a Dimension 3000 scanning probe microscope (Digital Instruments) with an extender electronics module. The cantilever was coated with a thin magnetic layer (standard MESP-tips were used) and scanned across the surface using lift mode, see also Ref. [30]. In this way, it is possible

ARTICLE IN PRESS M. Ziese et al. / Journal of Magnetism and Magnetic Materials 279 (2004) 331–342

that there might be some effect especially on the magnetotransport properties. The magnetization and magnetotransport measurements described here were performed shortly after film fabrication. Measurements on the same samples after one year of storage in a sample box filled with He showed that both the magnetic and magnetotransport properties changed significantly. This might be further discussed in a forthcoming publication.

3. Experimental results 3.1. Structural characterization X-ray diffractometry showed epitaxial growth with values for the full width at half maximum of 0:5 in both the 2y- and the o-scans. Measurements of the in-plane ajj and out-of-plane a> lattice constants indicate partial strain relaxation with values ajj ¼ 0:827 nm and a> ¼ 0:843 nm and a degree of relaxation R ¼ ðajj  asub Þ=ðabulk  asub Þ ¼ 60%: AFM imaging of the films after deposition on the patterned substrate revealed a clear step-edge structure as shown in Fig. 1. Regular step edges with 120 nm (P2) height are seen in this figure. The step edges are separated by a 10 mm distance. The angle of ascent was determined from 12 images and a value a ¼ 3078 was found. The root-mean-square roughness sq of the film was determined in the plateau regions. It is sq ¼ 0:75 nm identical with the value for the epitaxial film and comparable to the 50
m

80

Height (nm)

to disentangle the long-range magnetic and the short-range topographic information during the same image acquisition. In a first scan, topography was measured and in an additional second scan the phase signal was measured at a constant lift height between 50 and 300 nm; yielding the magnetic information. Measurements of the global magnetization were performed with a superconducting quantum interference device (SQUID) magnetometer (MPMS-7, Quantum Design). The diamagnetic contribution of the substrate was subtracted. The local field distribution was measured with a m-Hall sensor array consisting of eleven Hall sensors arranged on a linear array with a mutual separation of 10 mm [31]. The local field obtained in this way is averaged over the active Hall-sensor area of 10  10 mm2 ; this does not permit to separate magnetization reversal processes at the edges and on the plateaus. However, the averaged response can be compared with the local field distribution in smooth films to assess potential differences in the field reversal process. Resistivity measurements were performed in an Oxford Instruments continuous flow cryostat equipped with a 9 T superconducting solenoid. A conventional four probe technique in DC mode was used in van der Pauw configuration [32]. In this configuration, only the average current direction can be specified; it was chosen, since it enables a 90 rotation of the current direction without the need to make new contacts. A comparison with a linear contact arrangement did not indicate significant differences. Magnetite is not the stable iron-oxide phase at room temperature in air. X-ray photoemission spectroscopy studies on our epitaxial magnetite films have shown that a surface layer of 3–5 nm thickness is oxidized in air; after this layer had been formed, the oxidation rate strongly decreases [33]. The oxidized layer might be g-Fe2 O3 : Since the magnetite films investigated in this work are about 80 nm thick and judging from the behaviour of epitaxial films, the oxidation might not be expected to severely affect the magnetic and magnetotransport properties. On the other hand, it is well known that grain-boundary diffusion processes are considerably faster than bulk diffusion, such

333

scan direction

40 0 α

-40 -80

0
m

50
m

0

10

20

30

40

Distance (µm)

Fig. 1. Left: wide area AFM scan on sample P2 clearly showing the step-edge structure. The profile on the right was recorded along the white line. Step edges of about 120 nm height are visible.

ARTICLE IN PRESS M. Ziese et al. / Journal of Magnetism and Magnetic Materials 279 (2004) 331–342

334

specification of sq ¼ 0:5 nm for the virgin substrate. 3.2. Magnetic characterization In Fig. 2(a), the temperature dependent magnetization measured in an applied field of 0:2 T is plotted. All samples show a clear change of the magnetization at the Verwey transition. The Verwey temperatures determined from the maximum slope of dM=dT are 115:0 K (P1), 116:2 K (P2) and 114:0 K (E1). Since the Verwey temperature sensitively reflects changes in stoichiometry, according to this criterion the quality of the films deposited on the patterned samples is similar to the epitaxial film. The differences in Verwey temperature are not regarded to be significant. In Fig. 2(b), hysteresis loops measured at room

temperature are shown. The coercive fields are 31 mT (P1), 27 mT (P2) and 31 mT (E1). Also these values indicate that there is no degradation of the film properties in case of the step-edge samples. Fig. 3 shows magnetization hysteresis loops measured at 80 K after (a) zero-field cooling and (b) field cooling in 2 T through the Verwey transition as well as data taken at (c) 300 K: In each case the magnetization hysteresis was measured with the magnetic field applied perpendicular to the film and in-plane both parallel and perpendicular to the step edges. In all cases, the out-of-plane hysteresis loop is much broader due to demagnetization. At 300 K; the in-plane hysteresis loops do not depend on the field direction with respect to the step edges. At 80 K after ZFC,

0.4

Temperature T (K) 0

50

100

150

200

250

300

0.0

M / M(130 K)

P1 P2 E1

1.00 0.95 0.90

µ0H = 0.2 T 0.85 1.0

Magnetization µ0M (T)

1.05

(a)

-0.4 0.4

(b) 80 K, FC

0.0

-0.4 0.4

(b)

M / M(2 T)

(a) 80 K, ZFC

(c) 300 K

0.5 0.0

⊥ step edges  step edges ⊥ film

0.0 -0.4

-0.5

-0.4

T = 300 K

-1.0 -0.10

-0.05

0.00

0.05

0.10

Magnetic Field µ0H (T) Fig. 2. (a) Magnetization as a function of temperature from 5 to 300 K measured in an applied field of 0:2 T after zero field cooling to 5 K: (b) Magnetization hysteresis loops recorded at 300 K: For all measurements, the magnetic field was applied parallel to the film.

-0.2

0.0

0.2

0.4

Magnetic Field µ0H (T) Fig. 3. Step-edge sample P1. Magnetization hysteresis loops recorded at 80 K after zero field cooling (ZFC) through the Verwey temperature (a), after field cooling (FC) (b) and at 300 K (c). In each case the magnetic field was applied in-plane parallel and perpendicular to the step edges and perpendicular to the film, respectively. The solid lines were derived from the perpendicular hysteresis loops by applying a shearing correction Hint ¼ H  NM with a demagnetizing factor N ¼ 1:

ARTICLE IN PRESS M. Ziese et al. / Journal of Magnetism and Magnetic Materials 279 (2004) 331–342

Hint ¼ H  NM

ð1Þ

0.2

(a) ZFC

0.0

-0.2 0.2

Bloc (mT)

however, a significant difference between the hysteresis loops for the two field directions can be detected. This vanishes after field cooling to 80 K: From these data, we conclude that the step edges influence the formation of crystallographic domains in the monoclinic phase below the Verwey temperature in such a way that the monoclinic c-axis which is the easy axis is preferentially oriented along the step edges. However, this influence is small, since it is overridden by field cooling in 2 T: A shearing correction

335

(b) FC film

0.0

-0.2

with a demagnetizing factor N ¼ 1 appropriate for a thin film was applied to the out-of-plane hysteresis loops to yield the solid lines in Fig. 3. A comparison with the in-plane hysteresis curves indicates that the surface normal is the hard axis in all cases. The magnetization reversal was studied in more detail with the m-Hall sensor technique. Since both the step-edge array and the m-Hall sensor array have periodicities of 20 mm; every sensor will sample the magnetic signal from both plateau and step-edge regions. The local field hysteresis loops of ten m-Hall sensors were measured; a typical sensor response at 10 K is shown in Fig. 4 after (a) ZFC, (b) FC in 2 T jj step edges and (c) FC in 2 T > film. In all cases, a peak in the out-ofplane component of the local field was observed at the coercive field of 0:1 T: The sharpness of that peak depends on the field cooling procedure. The out-of-plane component of the local field is probably due to domain-wall movements. An extensive analysis of this phenomenon has been made for two epitaxial magnetite films [31]. Since the general shape of the stray-field hysteresis loops and the dependence on the field-cooling procedure are the same for epitaxial films and step edges, we conclude that the magnetization reversal mechanism is not significantly affected by the presence of the step edges. This is reasonable, since the typical domain size was found to be typically 200–400 nm in epitaxial magnetite films [31]. AFM/MFM-scans near a step edge are shown in Fig. 5. The MFM images were recorded at various lift heights of 50, 100, 200 and 300 nm; respectively. The step edge that clearly appears on the

0.2

(c) FC ⊥ film

0.0

-0.2 -0.2

-0.1

0.0

0.1

0.2

Magnetic Field µ0H (T) Fig. 4. Step-edge sample P2. Response of a m-Hall sensor measured at 10 K in an applied field oriented at a small angle with respect to the film surface. The measurements were performed after ZFC (a) and FC in a parallel (b) and perpendicular (c) field of 2 T through the Verwey temperature.

topographic image does not cause any magnetic contrast. The MFM images indicate a domain size of 100–400 nm: Micromagnetic calculations using OOMMF1.2a3 [34] have been performed in order to compare to the MFM data. A schematic picture of the sample structure used in the calculations is shown in Fig. 6; it consists of two slabs of volume ð500 þ dÞ  1000  60 nm3 joined by a slab of volume 60  1000  60 nm3 : Mesh size was 10  10  10 nm3 : The exchange stiffness was chosen as A ¼ 12  1012 J=m both within the bulk of the slabs and across the interfaces. To compare to the room temperature MFM data cubic anisotropy was chosen with K1 ¼ 1:1  104 J=m3 : The saturation magnetization was 5  105 A=m: OOMMF numerically solves for the time evolution of the Landau–Lifshitz–Gilbert equation.

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Fig. 5. Topography (upper right) and MFM images of step-edge sample P2 at 300 K: The image size is 10 mm by 10 mm: The MFM images were recorded at (from left to right) lift heights of 50, 100, 200, 300 nm; respectively. The step edge that can be clearly seen in the topographic image does not appear in the magnetic contrast.

A damping parameter a ¼ 0:5 and a stopping criterion of MS1 dM=dt ¼ 0:01 deg=ns were used; both are default values in OOMMF. The film thickness at the vertical scale of the step edge was varied with values of 20, 40 and 60 nm: The magnetization distribution did not significantly change with d; so only the case d ¼ 60 nm will be discussed here. Fig. 6 shows the magnetization distribution on the upper plateau at 180 nm and on the lower plateau at 60 nm: The arrows indicate the in-plane, the grey shades the out-of-plane magnetization components. As is clear from the large demagnetizing factor, the magnetization on the plateaus lies mainly in-plane. Out-of-plane components appear near the step edges, but these are small. This is in agreement with the MFM data. On the lower plateau, two domains are nucleated. This indicates that typical domain sizes are in the range of 250–500 nm: 3.3. Magnetotransport The resistivity of step-edge array P1 is shown in comparison to the epitaxial film in Fig. 7. Both samples exhibit the same characteristics, namely a small jump at the Verwey temperature and exp½ðT0 =TÞ1=2 -behaviour below the transition. At

low temperatures the resistivity of the step-edge sample is enhanced by a factor of two with this enhancement vanishing at room temperature. This behaviour is in clear contrast to the resistivity enhancement of two orders of magnitude observed in La0:7 Ca0:3 MnO3 step-edge arrays. In magnetite films, the creation of extended defects obviously does not disrupt the conduction paths too seriously. The magnetoresistance was found to be strongly anisotropic with respect to the relative orientation of current and step edges; this is shown in Fig. 8. The magnetic field was applied along the step edges. The anisotropy decays with increasing temperature. At low temperatures in small fields, the magnetoresistance for currents parallel to the step edges is positive. Note that the magnetoresistance slopes are different for the two current directions. The magnetoresistance at a constant field of 3 T is shown in Fig. 10(a) as a function of temperature. The magnetoresistance shown in Fig. 8 contains contributions from both the step edges and the anisotropic magnetoresistance (AMR). AMR is a bulk phenomenon; it has been studied in magnetite films on MgO [35]. In order to assess the contribution from AMR, the magnetoresistance

ARTICLE IN PRESS M. Ziese et al. / Journal of Magnetism and Magnetic Materials 279 (2004) 331–342

337

Resistivity (mΩcm)

Resistivity (mΩcm)

Temperature T (K) 10

7

10

6

10

5

10

4

10

3

10

2

10

1

10

0

10

7

10

6

10

5

10

4

10

3

10

2

10

1

10

0

50

100

150

200

250

300

(a) Fe3O 4 step edge P1 epi-polished E1

(b)

0.06

0.08

0.10

1/T

1/2

(K

0.12 -1/2

0.14

)

Fig. 7. Resistivity of step-edge sample P1 and an epitaxial film plotted as function of (a) temperature T and (b) T 1=2 :

Fig. 6. Top: cross section through the sample structure used for the micromagnetic modelling. Sample length is 1000 nm þ d with d ¼ 60 nm; height is 180 nm and width 1000 nm: The magnetization distributions were calculated on the upper plateau at 180 nm (middle) and at 60 nm (bottom). The arrows indicate the in-plane and the grey scale out-of-plane magnetization components.

of the epitaxial film E1 has been measured for both field orientations with respect to the current. Data taken at 100 and 180 K are shown in Fig. 9. At 100 K; the transverse magnetoresistance is positive at low fields and the AMR defined by AMR ¼ ðrjj  r> Þ=r0

ð2Þ

is negative. At 180 K; the AMR vanishes; this is in agreement with measurements on magnetite films on MgO which showed a sign change of the AMR near 130 K [35]. Note that the high-field magnetoresistance slopes are equal for both configurations. This feature was also seen in the AMRmeasurements on Fe3 O4 =MgO; it proves that the anisotropic magnetoresistance contribution is a low-field effect. The negative AMR determined at 3 T is shown in Fig. 10(a) as a function of magnetic field. The additional magnetoresistance induced by the step edges, called step-edge magnetoresistance (SMR), is obtained by subtracting the negative AMR from the measured magnetoresistance anisotropy, see (W) in Fig. 10(a). This yields the data points (%) in Fig. 10. If the step-edge magnetoresistance would be due to spin-polarized tunnelling across a disordered region near the step edges, one

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338

0.01

0.01 I step edges

180 K both orientations

0.00 0.00 -0.01 -0.01 -0.02

MR

MR

-0.02

-0.03

-0.03

100 K I⊥H I  H

-0.04 I ⊥ step edges -0.05

-0.04

120 K 180 K 295 K

-0.05

-0.06

-0.07 -1

0

1

2

3

Fig. 8. Magnetoresistance of step-edge sample P1 as a function of magnetic field at temperatures of 120, 180 and 295 K: In each case, the magnetoresistance was measured with currents applied perpendicular and parallel to the steps. The magnetic field was applied in the plane along the step edges.

would expect a scaling of the magnetoresistance with the spin-polarization P: Although the relation between spin-polarization P and global magnetization M is not straightforward [36], here we use PpM in a first approximation. Within Julliere’s model [37] ð3Þ

since the step-edge magnetoresistance is only a few percent, one would expect SMRp  M 2 : In Fig. 10(b) the measured SMR is compared to M 2 ; there is no evident correlation between the two quantities. The transport mechanism can be further characterized by measurements of the I–V curves and differential conductivity G: In the case of step-edge arrays in LCMO films, a relation G ¼ G0 þ gx V x

0

1

2

3

Magnetic Field µ0H (T)

Magnetic Field µ0H (T)

SMR ¼ 2P2 =ð1 þ P2 Þ;

-1

ð4Þ

Fig. 9. Magnetoresistance of the epitaxial film as a function of the magnetic field at temperatures 100 and 180 K: In each case the magnetoresistance in the longitudinal IjjH and transverse I>H direction is shown. At 180 K; the magnetoresistance anisotropy vanishes.

with xB1:2 was experimentally found [29]. This can be understood within a model of inelastic tunnelling through a disordered barrier [38]. The I–V curves and differential conductivity of the step-edge array P1 and some epitaxial magnetite films have been measured. Fig. 11 shows (a) I–V characteristics and (b) differential conductance of step-edge array P1 for currents applied perpendicular to the steps in zero field. The I–V curves appear linear at all temperatures investigated, 80 KpTp300 K: The differential conductivity increases for voltages above 0:25 V: The data of the normalized conductance for temperatures above 110 K fall on a universal curve, below 120 K the voltage dependence is weaker. Identical curves were also obtained on epitaxial magnetite films. Therefore, the conductivity increase at high bias voltages is probably due to Joule heating. We

ARTICLE IN PRESS M. Ziese et al. / Journal of Magnetism and Magnetic Materials 279 (2004) 331–342

( R⊥-R ) / R0 (%)

8

impurity spins in the barrier. In Fig. 12, the magnetoresistance of the LCMO and magnetite step-edge arrays are compared. The magnetoresistance was measured at 120 K; corresponding to a reduced temperature T=TC ¼ 0:14; in case of the magnetite sample and at 100 K; T=TC ¼ 0:43; in case of the manganite sample. The LCMO stepedge array shows the typical two-step transition with the sharp low-field decrease and the more gradual high-field magnetoresistance. In contrast, the magnetite step-edge array exhibits only a highfield magnetoresistance. From this comparison, we would conclude that the step-edge magnetoresistance is due to spin-dependent scattering at spin disorder near the step edges.

(a)

step edge P1 -AMR SMR

4

0

-4

(b) -2

SMR (%)

339

-4

∝ -M

-6

2

4. Discussion and conclusions

-8 50

100

150

200

250

300

Temperature T (%) Fig. 10. (a) Measured magnetoresistance of step-edge sample P1 (W), negative anisotropic magnetoresistance of the epitaxial film (3) and step-edge magnetoresistance (%). All data determined at 3 T: (b) Step-edge magnetoresistance compared to the square of the measured magnetization.

conclude that the transport characteristics of the step edges are ohmic. Further insight into the transport mechanism can be obtained from a comparison of magnetoresistance data from step-edge arrays in magnetite and LCMO. The transport mechanism at grain boundaries in LCMO is fairly well established [1]. At the grain boundary an insulating barrier is formed; charge transport across the grain boundary is due to direct tunnelling with a significant contribution from inelastic hopping processes via impurity states. Therefore the magnetoresistance shows a sharp low-field drop from spin-polarized tunnelling between the grains followed by a slow resistance decrease at high magnetic fields. This high-field magnetoresistance is caused by the suppression of spin-dependent scattering at

In the previous section, it was clearly shown that the step-edge magnetoresistance is a high-field effect that cannot be attributed to spin-polarized tunnelling through insulating grain boundaries. The magnetoresistance is likely to arise from spin-dependent scattering at spin-disorder in grain boundaries. A simple model is developed here that qualitatively accounts for the observed magnetoresistance and corroborates the interpretation. Consider two neighbouring spin clusters with ~i as shown in Fig. 13(a). Spin disorder is spins S ~0i with modelled by introducing local fields B random directions; these might be exchange or anisotropy fields. The energy of a spin cluster at site i with magnetic moment mi can then be written as Ei ¼ mi ½B cos Yi þ B0i cosðY0i  Yi Þ :

ð5Þ

Minimizing yields an angle Yim of the spin with ~ with respect to the applied field B tan Yim ¼ sin Y0i =ðb þ cos Y0i Þ;

b ¼ B=B0i :

ð6Þ

Since the transfer matrix element of the electron acquires a factor cosðYij =2Þ; Yij being the relative ~i and S ~j ; the conductivity angle between the spins S of this two-spin system in classical approximation

ARTICLE IN PRESS M. Ziese et al. / Journal of Magnetism and Magnetic Materials 279 (2004) 331–342

340

0.0

(a) step edge P1

0.5

0.0 80 90 100 110 120 130 140 150 160 170 180 190 200 210

-0.5

-1.0

(b)

(dI/dV ) / (dI/dV) 0

-0.1

1.10

MR

Current I (mA)

1.0

Fe3 O 4 120 K

low field MR

-0.2

-0.3

1.05 La0.7Ca0.3MnO 3 100 K -0.4

1.00

-1.0 -2

-1

0

1

2

is given by Z 1 Z 1 spZ d cos Y1 

1 2p

Z

Z

2p

0

dj1

1

d cos Y2 1

dj2 cos2 ðY12 =2Þexp½ðE1 þ E2 Þ=kT

0

ð7Þ with cosðY12 Þ ¼ cosðY1 ÞcosðY2 Þ þ sinðY1 ÞsinðY2 Þ  cosðj1  j2 Þ: Z denotes the partition sum. Integration over the azimuthal angles yields Z 1 Z 1 sp Z 1 d cos Y1 d cos Y2 1

0.5

such that only integrals of the type Z p dYf ðYÞ exp½E=kT

I¼

ð9Þ

1.0

Fig. 12. Magnetoresistance of the step-edge array P1 at 120 K compared to the magnetoresistance of a La0:7 Ca0:3 MnO3 stepedge array at 100 K: The dashed lines represent an extrapolation of the high-field magnetoresistance. The standard definition of the low-field magnetoresistance is indicated.

have to be calculated. In the following, the index i will be dropped, the angles Y0i ; the magnetic ~0i s will be moments mi and the magnitudes of the B assumed equal. Furthermore, the energy E and the function f are expanded to second order (Gaussian approximation) and the integral is extended to ½N; N

1 @2 E EC Em þ ðY  Ym Þ2 2 @Y2m   1 ¼ Em 1  ðY  Ym Þ2 : 2

1

 ð1 þ cos Y1 cos Y2 Þexp½ðE1 þ E2 Þ=kT ; ð8Þ

0

0.0

Magnetic Field µ0H (T)

Voltage U (V) Fig. 11. (a) I–V curves measured on the step-edge array P1 in zero field at the temperatures indicated in the figure. (b) Differential conductivity dI=dV normalized to the value at zero bias voltage.

-0.5

This yields IC expðjEm j=kTÞ    Z N jEm j 2 ðY  Ym Þ exp   f ðYm Þ 2kT N

ð10Þ

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Temperature T (K)

(b)

B0 i

B Si

50

e-

Θi

N

250

300

4 2 µ0 H = 3 T

0

(b) 3



jEm j ðY  Ym Þ2 ðY  Ym Þ2 exp  2kT

 ð11Þ

sffiffiffiffiffiffiffiffiffiffiffiffi  2pkT 1 kT ¼ expðjEm j=kTÞ f ðYm Þ þ f 00 ðYm Þ : jEm j 2 jEm j ð12Þ In lowest order in kT=jEm j one obtains for the conductivity:   3 kT 2 2 sp1 þ cos Ym 1  ð13Þ 2 jEm j

MC (%)

N

200

6

Θ0 i

Fig. 13. Schematics of (a) a charge transfer process between two neighbouring spin clusters and (b) the directions of the spin ~i ; the local anisotropy or exchange field B ~0i and the clusters S ~: applied field B



150

(a)

Si

Sj

1 þ f 00 ðYm Þ 2Z

100

8

MC (%)

(a)

341

step edge P1 T = 180 K

2

1

0 -1

0

1

2

3

Magnetic Field µ0H (T) Fig. 14. Step-edge magnetoconductance of step edge P1 (a) as a function of temperature at 3 T and (b) as a function of magnetic field at 180 K: The dotted lines are fits of the theoretical expressions as discussed in the text.

with Em ¼ m½B cos Ym þ B0 cosðY0  Ym Þ

ð14Þ

and cos2 Ym ¼

b2 þ 2b cos Y0 þ cos2 Y0 : b2 þ 2b cos Y0 þ 1

ð15Þ

The magnetoconductance data sðBÞ=sð0Þ  1 have been analyzed as a function of temperature and magnetic field using Eqs. (13)–(15). Two fitting curves are shown in Fig. 14. In order to reduce the number of fitting parameters the approximation Em ¼ E0 has been made. A fit of the magnetoconductance derived from expression (13) to the data, shown in Fig. 14(a), as a function of temperature yields an energy E0 ¼ 500720 K for fixed values of cos2 Y0 in the range 0.4–0.8; cos2 Ym then lies in the range 0.55–0.98. In case of a single spin, this would correspond to an exchange field of 750 T: In

view of the Curie temperature of magnetite of 858 K; this is a reasonable value. However, the analysis of the field dependence of the magnetoconductance yields random field values of B0 ¼ 471 T and an angle Y0 ¼ 30 75 approximately independent of temperature. A fitting curve to the magnetoconductance data at 180 K is shown in Fig. 14(b). The magnitude of the random field would indicate its nature as an anisotropy field. Accordingly, from the exchange energy of 500 K; this would imply a magnetic moment of the spin cluster of about 190 mB magnetons. From this analysis, the following picture emerges. At the step edges, cluster-formation occurs and the magnetoresistance is induced by spin dependent electron transfer between different clusters. The cluster size was estimated to be about 190 spins. The magnetic moments of the clusters

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are misaligned by comparatively small effective fields of the order of 4 T: These might arise from variations of the local anisotropy constants. The existence of rather large clusters justifies the use of Boltzmann statistics in the calculation of the conductivity.

Acknowledgements This work was supported by the Deutsche Forschungsgemeinschaft under contract No. DFG ES 86/7-1 within the Forschergruppe ‘‘Oxidische Grenzfl.achen’’ and under DFG ES 86/6-1.

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