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Magnetoresistance of La0.7Ca0.3MnO3 thin film biepitaxial step junctions

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Physica B 336 (2003) 267–274

Magnetoresistance of La

0:7

Ca

0:3

MnO

3

thin film biepitaxial

step junctions

S.F. Chen

a,

*, W.J. Chang

a

, S.J. Liu

a

, J.Y. Juang

a

, J.-Y. Lin

b

, K.H. Wu

a

,

T.M. Uen

a

, Y.S. Gou

a

aDepartment of Electrophysics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan bInstitute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan

Received 27 January 2003; accepted 8 April 2003

Abstract

The biepitaxial La0:7Ca0:3MnO3(LCMO) thin films grown on SrTiO3substrates using a buffer layer of anatase TiO2 were fabricated. The magnetoresistance (MR) of biepitaxial step junction (BSJ) across the boundary layer of biepitaxial LCMO (0 0 1) and LCMO (1 1 0) film was investigated. The temperature and field dependence of MR for BSJ are qualitatively similar to those obtained in other type of artificial grain boundaries with a comparable MR ratio at low temperatures. However, the observed linear current–voltage characteristic across BSJ is in sharp contrary to the commonly reported non-ohmic characteristics. The results are consistent with features predicted by the model of spin-dependent transport across a depressed magnetic ordering and metallic-like junction layer.

r2003 Elsevier Science B.V. All rights reserved. PACS: 72.25.b; 75.30.Vn

Keywords: Magnetoresistance; Biepitaxial step junction; LCMO; Grain boundary

1. Introduction

The observation of colossal magnetoresistance (CMR) in perovskite manganites has led consider-able interest due to its potential applications in magnetic controllable electronic devices. Unfortu-nately, a large magnetic field (several Tesla) is required to achieve over 10 percent of resistance change in manganite materials, making it difficult to use this intrinsic magnetoresistance (MR)

property directly for device designing. Thus, devices, which can achieve a large MR ratio with a low magnetic field such as tunneling magneto-resistance devices [1] or grain boundary (GB) junctions [2–7], are more viable candidates for applications. The tunneling magnetoresistance devices, however, have a disadvantage of involving complex fabrication processes as compared to GB junctions.

In polycrystalline samples, which have GBs in nature, Hwang et al.[8]and Gupta et al.[9]have demonstrated prominent low field MR (LFMR). Their experimental results of increasing LFMR with decreasing grain size imply that the GB

*Corresponding author. Tel.: +88635731943; fax: +88635725230.

E-mail address:[email protected] (S.F. Chen).

0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(03)00261-8

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dominates the MR effect [8–10]. Moreover, the observed almost ohmic current voltage character-istics (IVC) evidently suggest that the LFMR might be caused by a combination of spin-scattering and tunneling processes at GBs [11]. In order to analyze the origins of GB MR, various artificial GB samples have been fabricated. For example, the GBs fabricated on bicrystal sub-strates [2–4], on step-edge substrates [6,12] or in form of biepitaxial films[5]. All these experimental results have demonstrated prominent LFMR and noticeable high field MR (HFMR) below Curie temperature (TC) due to GBs. In addition, the

results of ubiquitously observed non-ohmic IVCs for GBs and the nature of almost spin polarized charge carriers in ferromagnetic state in manganite materials, thus, seem to favor that the inelastic tunneling at GBs is the origin of the observed LFMR [13].

In this study, the samples of La0:7Ca0:3MnO3

(LCMO) biepitaxial films have been fabricated and the MR of the biepitaxial step junction (BSJ) across the boundary layer between two different orientation epitaxial films is investigated. This BSJ demonstrates comparable MR ratio and analo-gous LFMR and HFMR behaviors with the forementioned artificial GBs [2–7]. However, the IVC is evidently linear. The results are better interpreted by the mechanism of spin-dependent transport across a metallic and depressed magnetic ordering junction layer. The BSJ with character-istics of large LFMR, ohmic IVC and relatively easy fabrication process might be attractive in future applications.

2. Samples and experimental

In order to grow LCMO (1 1 0) and LCMO (0 0 1) films on the same substrate to produce a BSJ, an appropriate buffer layer is needed. Because various CMR and superconducting thin films with predominantly (1 1 0) orientation have been successful deposited on the TiO2 buffered

substrates in our previous studies[14–16], the TiO2

has been chosen to be the buffer layer in this study. The biepitaxial LCMO thin film samples were prepared on SrTiO3 (0 0 1) (STO) substrates by

pulsed laser deposition with half area of the substrate being buffered by a layer of anatase TiO2: The vertical structure of the sample is

schematically illustrated inFig. 1(a). The TiN thin

La0.7Ca0.3MnO3 La0.7Ca0.3MnO3 R(001) RBSJ R(110) 120 µm, RSJ 30 µm Anatase TiO2 (001) SrTiO3 (001) substrate (a) (b) (c)

Fig. 1. (a) The vertical structure of biepitaxial films. The TiO2

buffer layer is in anatase phase and 15 nm in thickness. (b) The microscopic image of the patterned biepitaxial films shown in the same lateral structure order with (a). The BSJ can be clearly seen in the image. The bridge width is 30 mm and the distance between two nearest contact pads is 120 mm: The segment resistance of LCMO (0 0 1) and LCMO (110) film not crossing BSJ are indicated by Rð0 0 1Þ and Rð1 1 0Þ; respectively. The measured segment resistance crossing BSJ is indicated by RSJ:

The BSJ resistance thus can be obtained by subtracting the thin film resistance not crossing the BSJ to the resistance crossing the BSJ. (c) The atomic force microscopy image in the vicinity of the BSJ region. The sharp BSJ region can be clearly identified. The left and right sides of BSJ are LCMO (1 1 0) and LCMO (0 0 1) films, respectively.

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film was first deposited on a STO substrate with the deposition conditions of substrate temperature Ts¼ 720C and background pressure of 5 

106 Torr: The half side of TiN film was then removed by wet etching method. The TiO2 buffer

layer having a thickness of about 15 nm was then obtained by oxidizing the sample at 1 atm pure oxygen and 800C for 30 min: This TiO2 buffer

layer was observed to be anatase phase, instead of rutile phase (see below). The reason could be due to the high oxygen pressure used in oxidizing the TiN film. The LCMO film having a thickness of about 70 nm was deposited on this TiO2 buffered

substrate. The substrate temperature and oxygen pressure (PO2) during deposition were kept at

720C and 0:35 Torr; respectively. The sample was then in situ annealed at the same temperature with PO2 ¼ 600 Torr for 30 min and then cooled to

room temperature at a rate of 15C=min: The LCMO films obtained thus possess two distinct film orientations simultaneously grown on the same STO substrate. A BSJ is formed across the boundary of two regions. It is noted that the LCMO layer was intentionally made thicker than the TiO2 buffer layer to make BSJ transport the

dominant factor of resistance, instead of step-edge transport.

In order to investigate the BSJ resistance, the sample was patterned by wet etching into 30 mm-wide which symmetrically crosses the BSJ with 4 contact pads at each side of the BSJ for four-point probe measurement. The microscopic image of the patterned sample and bridge dimensions are shown in Fig. 1(b). The sample surface and structure were characterized by atomic force microscopy and X-ray diffraction (XRD). Fig. 1(c) shows the atomic force microscopy image in the vicinity of the BSJ. The sharp changes in height clearly identify the BSJ region. Fig. 2 shows the typical y  2y XRD patterns for each side of the LCMO film. As in evident fromFig. 2(a) and (b), the LCMO film directly grown on STO (0 0 1) substrate is (0 0 1)-oriented while that grown on TiO2buffered region shows predominantly (1 1

0)-oriented structure. These XRD scans indicate both of the biepitaxial film and buffer layer are purely oriented. The full width at half maximum (FWHM) for the LCMO (0 0 1) and LCMO

(1 1 0) peaks are about 0:19 and 0:23: The

FWHM for TiO2 (0 0 4) peak is about 0:54: All

these FWHMs were estimated by Gaussian fit. The reason why the LCMO film grown on an anatase ð0 0 lÞ TiO2 buffer layer results in (1 1 0) preferred

orientation, instead of ð0 0 lÞ orientation, is not clear at present.

All transport and magnetic properties were measured using Quantum Design Physical Prop-erty Measurement System. The BSJ resistance ðRBSJÞ is obtained by subtracting the film

resis-tance measured with no crossing of BSJ from that measured across BSJ.

3. Results and discussion

Fig. 3(a) shows the temperature dependent resistance ðRðT ÞÞ obtained from the segments of bridge located within the (0 0 1) and (1 1 0) regions together with the one across the BSJ, indicated in Fig. 1(b) as Rð0 0 1Þ; Rð1 1 0Þ; and RSJ; respectively.

As is seen clearly, the insulator-metal transition

0 100 200 300 400 500 600 (a) S T O (001) LC M O (001) LC MO (002) S T O (002) Intensity (cps) 15 20 25 30 35 40 45 50 55 0 100 200 300 (b) LC M O (110) A natas e T iO 2 (004) S T O (002) S T O (001) Intensity (cps) 2θ(degrees)

Fig. 2. The X-ray diffraction pattern for (a) the LCMO (0 0 1) and (b) LCMO (1 1 0) films, separately. The purely LCMO ð0 0 lÞ and LCMO ð1 1 0Þ peaks for the respective sides of the biepitaxial films indicate good orientation for each sides.

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temperature ðTIMÞ for Rð0 0 1Þ peaks around

265 K; while that for Rð1 1 0Þ and RSJ is about

60 K lower with a very broad transition. Despite of the dramatic differences displayed in TIM; it is

remarkable to note that the temperature depen-dent magnetization MðTÞ measured over the entire bridge displays only one single ferromagnetic transition with a TC around 260 K; as shown in

Fig. 3(b). Although, it is impossible to measure MðT Þ on the respective segment of the film, there have been evidences from other manganite films deposited on various substrates display a wide range of TIM’s while TC remained nearly constant

[17,18]. It is suggestive that while magnetization may have reflected the intrinsic property, the transport properties appear to be extremely sensitive to the crystalline microstructures. In addition, there are two more peculiar features to be noted in the Rð1 1 0Þ and RSJ: Firstly, there is a

dip in RðT Þ around 240 K: This resistance dip has

been consistently observed to appear at tempera-tures near the TC: However, it is also dependent of

measuring history and the strength of the applied field (see below). Mathieu et al.[5]has suggested that it might be due to local magnetic ordering associated with a distributed ferromagnetic transi-tion within grain boundaries. The XRD results shown in Fig. 2 seem to support this extrinsic effect scenario, in that the LCMO (1 1 0) region exhibits relatively poor crystallinity. The second feature is the slightly upturn in RðTÞ at very low temperatures. This might be, again, due to the relatively poor crystalline ordering-induced weak-localization over the regions covered by segments giving rise to Rð1 1 0Þ and RSJ: However, whether

both are due to purely extrinsic effects or are something intrinsic to crystalline orientations might need further studies to clarify. Finally, although the similarity of Rð1 1 0Þ and RSJ may

lead one to ascribe a single prevailing mechanism to the results, we note that RSJof is more than

two-times higher than Rð1 1 0Þ; which in turn is about another order of magnitude higher than Rð0 0 1Þ: To simplify the discussion, we will focus on magneto-transport behaviors merely attributed to the BSJ in the following. We thus define RBSJ  RSJ 0:5  ½Rð1 1 0Þ þ Rð0 0 1Þ as the resistance arising from the BSJ.

Fig. 4(a) shows the RBSJ and magnetization as a

function of applied field at 75 K in low field regime. Here the scale of low field is usually taken as the field just below the saturation field. In the present case, it is taken as 0:1 T: It is evident that RBSJ peaks at the coercive field ðHcÞ of the

magnetization hysteresis loop with HcE150 Oe

and shows a large LFMR with DR=R0E17%;

where DR  RBSJðHÞ  R0and R0 RBSJðHcÞ: This

closely association between resistance change and magnetic ordering leads the finding that if one normalizes the resistance change ðDRðHÞÞ to the resistance at Hcand plots it as a function of global

magnetization ðMgÞ normalized to the saturation

magnetization ðMsÞ; the data fit quite well with

DRðHÞ=RðHcÞp  ðMg=MsÞ2: ð1Þ

These features are typical for most grain boundary junctions [10,19] and are attributed to either magnetic inhomogeneity-induced scattering [10]

0.2 0.3 0.4 0.5 0.6 (a) Re sistivity ( Ω cm) 0.0 0.5 1.0 1.5 2.0 RSJ R(110) x 2 R(001) Re sistivity (10 -2 Ω cm) 0 50 100 150 200 250 300 0 1 2 3 4 5 (b) Temperature (K) M agnetic m o m ent (10 2 emu /c m 3 )

Fig. 3. (a) The resistance plotted as a function of temperature for the LCMO (001) film (denoted as Rð0 0 1Þ), the LCMO (110) film (denoted as Rð1 1 0Þ) and the segment crossing BSJ region (denoted as RSJ). (b) Temperature dependence of the magnetic

moment of the biepitaxial films. The ferromagnetic transition occurs at 260 K:

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or intergrain spin-polarized tunneling [8]. In both cases, the alignment of the domains associated with the grains in magnetic field around Hc gives

rise to the steep decrease in resistance and predict the same magnetization dependence. Nonetheless, the tunneling model also predicts that there should be no further MR decrease once the magnetization orientation of adjacent grains becoming parallel [6,8,13,20]. However, the ubiquitous HFMR ob-served on various types of grain boundary junc-tions often deviates from the prediction significantly [6,13,20]. As shown in Fig. 4(a), typically, the steep decrease of LFMR crosses over to a much slower one, instead of remaining

field independent. Although there have been cumulative evidences indicating that linear HFMR [6,13]or linear high field magnetoconductance[21] can be expected and the slope should be propor-tional to the grain boundary susceptibility wGBdue

to inelastic tunneling between ferromagnetic grains via one or two localized states, however, as depicted by the inset of Fig. 4(b), we found that dR=dH in the high field regime is weakly propor-tional to H2 plus even higher order terms in H

rather than expected linear dependence in H: We have redrawn the Fig. 4(b) in terms of magneto-conductance, however, similar non-linear high field slope has obtained. Alternatively, Evetts et al. [22] proposed that the thermally activated carrier transport within a defective region adjacent to a grain boundary should give rise to a linear HFMR and the magneto-response may be sub-stantially influenced by the grain boundary mag-netization. In this scenario, the grain boundary layer is not necessary insulating and the magne-toresistance is in effect the response of a highly disordered mesoscale region with a depressed TC

and magnetization. Although this model is capable of explaining most of the features observed in biepitaxial films, it also predicts a linear HFMR in H: It appears that the ubiquitously observed monotonically decreasing HFMR, though is cer-tainly associated with grain boundary effect, still needs further clarifications.

The characteristics of our BSJ were further identified by measuring the IVCs as functions of temperature and magnetic field. As shown in Fig. 5(a) and its inset, the IVCs across the BSJ segment of the bridge display strictly ohmic behavior over the wide range of temperatures and magnetic fields studied. This is indicative that the current BSJ is indeed metallic in nature, in contrast to most other artificial GBs where various extents of non-ohmic IVCs have been observed and spin-polarized tunneling has been concluded [4,6,12]. Although the slight temperature and magnetic field dependence of the BSJ resistance is consistent with that argued by Evetts et al.[22] Nonetheless, we note that the H2-dependence of

HFMR remains unexplained by various models cited above. Finally, to further delineate the markedly different behaviors exhibited by LFMR

-0.3 -0.2 -0.1 0.0 0.1 0.20 0.3 -6 -4 -2 0 2 4 6 Magnetic moment (10 2 emu/c m 3 ) (a) 0.8 0.9 1.0 RBSJ (H)/R BSJ (H C ) -3 -2 -1 0 1 2 3 0.6 0.7 0.8 0.9 1.0 RBSJ (H)/R BSJ (H c ) (b) Magnetic field (T) 1 2 3 -0.08 -0.07 -0.06 Field (T) dR BSJ (H)/ RBSJ (H c )d H

Fig. 4. (a) Magnetic moment of the biepitaxial films plotted as a function of magnetic field at 75 K and the ratio of the field-dependent BSJ resistance RBSJðHÞ to the coercive-field value

RBSJðHcÞ in the low-field regime. The coercive field is about

150 Oe at 75 K: The coincidence of the RBSJmaximum and the

coercive field indicates that the RBSJis strongly correlated to the

magnetic order in films. (b) The RBSJðHÞ=RBSJðHcÞ in high-field

regime. The inset shows dRBSJðHÞ=RBSJðHcÞ dH at magnetic

field higher than 1 T: The almost linear first derivative indicates the weak H2dependence of MR in high field regime.

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and HFMR, we plot the RðT Þ curves in various fields together with MR ratio ðDR=RÞ as a func-tion temperature in low and high field regimes. Here the BSJMRLF and BSJMRHFare defined as

½RBSJðH ¼ 0 T; T Þ  RBSJðH ¼ 0:1 T; TÞ=RBSJðH ¼

0 T; T Þ and ½RBSJðH ¼ 1 T; TÞ  RBSJðH ¼ 3 T; T Þ=

RBSJðH ¼ 1 T; TÞ; respectively. Here, we choose

0:1 T to be the field dividing the low-field and high-field regime, because it is the saturation field for magnetization (Fig. 4(a)). As can be seen in Fig. 5(b), the TIM appears to be rather insensitive

to the applied magnetic field, however, this is commonly observed in bicrystal GB junctions and epitaxial manganite films [17,20]. Also noticed is that the RðT Þ behavior remains essentially the same as the field is increased, except for the monotonically decreasing resistance with increas-ing field. This is indicative of that whatever mechanisms are prevailing for RBSJðT Þ the applied

fields do not alter them. We note that the RðTÞ dips near TC seen in Fig. 3(a) appear to be

suppressed at high fields, as well.

The BSJMRLF and BSJMRHF thus reflect the

effects of the applied field on RBSJðT Þ alone. As

described previously, the RBSJ under discussion actually consists of only the resistance resulting from the presence of the step junction. Since the present RSJis apparently not insulating, our results

thus agree with that the junction is indeed consisted of some mesoscale and metallic regions with depressed TC; as proposed by Evetts et al.

[22], instead of insulating regions concluded from the commonly observed results of non-linear IVCs. Furthermore, The BSJMRLF curve shown inFig.

5(b) between 50 and 230 K can be fitted well by an expression of ðBSJMRLFÞ1=2¼ C  ð1  T =TnÞ

with C ¼ 0:535 and Tn

¼ 290:3 K; as shown in Fig. 5(c). To further discuss this fitting equation, we assume the magnetic moment almost saturates at 0:1 T below TC; as shown inFig. 4(a). Hence,

the BSJMRLF approximately describes the

satu-rated LFMR. According to Eq. (1), the tempera-ture dependent LFMR should similarly describe the temperature dependence of ðMgÞ2 if Ms is

taken as the saturation magnetization at T ¼ 0 K: Thus, if we take the square root of LFMR shown inFig. 5(c), the magnetization curve shown inFig. 3(b) would be expected rather than the linear

0.00 0.05 0.10 0.15 0.20 0.25 RBSJ (a) (b) ∆ R/R 50 100 150 200 250 300 (c) BSJMR HF BSJMRLF H = 0 T 0.1 T 1 T 3 T Resistance (k Ω ) 0 50 100 150 200 250 300 0.1 0.2 0.3 0.4 0.5 (BSJ MR LF ) 1/2 Temperature (K) -2 -1 0 2 -10 -5 0 5 10 T = 150 K 75 K 10 K Current ( µ A) Voltage (V) -2 -1 0 1 2 -10 -5 0 5 10 Cu rren t ( µ A) Voltage (V) 1

Fig. 5. (a) The IVC’s measured for the segment of RSJ at

various temperatures and zero magnetic field. The inset shows IVC’s measured at 10 K with zero magnetic field (solid line) and H ¼ 1 T (dashed line). All IVC’s show clear linear current– voltage relationship. (b) Temperature dependence of RBSJ in

various magnetic fields (right-hand side axis) and the LFMR ðBSJMRLFÞ and HFMR ðBSJMRHFÞ of the BSJ (left-hand

side axis). The BSJMRLF and BSJMRHF are defined as

½RBSJðH ¼ 0 T; TÞ  ½RBSJðH ¼ 0:1 T; TÞ=½RBSJðH ¼ 0 T; TÞ and

½RBSJðH ¼ 1 T; TÞ  RBSJðH ¼ 3 T; TÞ=RBSJðH ¼ 1 T; TÞ;

re-spectively. (c) The square root of the BSJMRLF plotted as a

function of temperature (circles). The BSJMRLF can be fitted

well by an equation of (BSJMRLFÞ1=2¼ C  ð1  T=TnÞ with

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relationship seen in Fig. 5(c). Alternatively, if we take Msto be MsðTÞ the saturation magnetization

at the measuring temperature, then the ½MgðTÞ=MsðT Þ2 at H ¼ 0:1 T would be always

approximately unity below TC: In this scenario,

the square root of LFMR describes the tempera-ture dependence of proportional constant in equation (1). Eq. (1) then becomes ðBSJMRLFÞ1=2

¼ C0 ð1  T=Tn

Þ  ½MgðT Þ=MsðTÞ: It is

sugges-tive that there may be another mechanism hidden in the proportional constant that determines the temperature dependence of LFMR besides the magnetic ordering. Although the linear relation-ship has been consistently found on various GB samples [22], the origin is not yet clear and is worthwhile for further investigations.

The BSJMRHF curve shown in Fig. 5(b)

illus-trates dramatically increasing at TC and keeps

almost constant below TC: The result is similar to

commonly observed temperature independent HFMR on other types of GBs, however, there is no consistently explanation so far[5,8,13]although HFMR has been attributed to arise from barrier materials in GBs. It is suggestive that the HFMR could be intimately associated with the magnetic order in LCMO films and could be interpreted as follows. The scattering resistance due to the fluctuation of mesoscale junction regions is sup-pressed by higher field near TC: Thus, a significant

difference in MR appears between 1 and 3 T of applied fields. As the temperature moves further away from TC; fluctuation reduced and BSJMRHF

reflects purely the difference in MBSJ at different

applied fields and is rather insensitive to tempera-ture. We note that the RðTÞ dips near TCat several

fields may also be explained by magnetization fluctuations and may be turned off by high applied field. In addition, the similar behaviors displayed by the LCMO (1 1 0) region may also arise from similar effects due to its granular features resulting from large lattice mismatches between the buffered-TiO2layer and the LCMO film grown on it.

4. Summary

In summary, we have fabricated biepitaxial LCMO films using TiO2-buffered STO substrates.

By intentionally patterning the films into bridges with a step junction connecting regions with different film orientations, we were able to separate the magneto-transport properties from junction region and intra-film regions. A MR value comparable to other structures (about 20%) at low temperatures was obtained. The detailed temperature and field dependence measurements revealed that the magneto-transport of carriers probably is dominated by spin-dependent trans-port through a metallic-like GB regions rather than spin-polarized tunneling through insulating-like GB regions. In addition, our results are consistent with most of the results previously obtained from similar GBs. We note, however, the detailed mechanisms responsible for the observed LFMRðTÞ and HFMRðHÞ remain to be explored.

Acknowledgements

This work was supported by the National Science Council of Taiwan, ROC under Grant Nos: NSC 91-2112-M-009-041 and NSC 91-2212-M-009-046.

References

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[14] S.J. Liu, J.Y. Juang, K.H. Wu, T.M. Uen, Y.S. Gou, J.Y. Lin, Appl. Phys. Lett. 80 (2002) 4202.

[15] S.J. Liu, S.F. Chen, J.Y. Juang, J.-Y. Lin, K.H. Wu, T.M. Uen, Y.S. Gou, Jpn. J. Appl. Phys. 42 (2003) L287. [16] P.I. Lin, C.W. Liu, C.C. Hsieh, K.H. Wu, J.Y. Juang,

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[20] R. Gross, L. Alff, B. Buchner, B.H. Freitag, C. Hofener, J. Klein, Y. Lu, W. Mader, J.B. Philipp, M.S.R. Rao, P. Reutler, S. Ritter, S. Thienhaus, S. Uhlenbruck, B. Wiedenhorst, J. Magn. Magn. Mater. 211 (2000) 150.

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[22] J.E. Evetts, M.G. Blamire, N.D. Mathur, S.P. Isaac, B.-S. Teo, L.F. Cohen, J.L. Macmanus-Driscoll, Phil. Trans. R. Soc. Lond. A 356 (1998) 1593.

數據

Fig. 1. (a) The vertical structure of biepitaxial films. The TiO 2
Fig. 3 (a) shows the temperature dependent resistance ðRðT ÞÞ obtained from the segments of bridge located within the (0 0 1) and (1 1 0) regions together with the one across the BSJ, indicated in Fig
Fig. 3 (b). Although, it is impossible to measure MðT Þ on the respective segment of the film, there have been evidences from other manganite films deposited on various substrates display a wide range of T IM ’s while T C remained nearly constant
Fig. 4. (a) Magnetic moment of the biepitaxial films plotted as a function of magnetic field at 75 K and the ratio of the  field-dependent BSJ resistance R BSJ ðHÞ to the coercive-field value
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It is interesting that almost every numbers share a same value in terms of the geometric mean of the coefficients of the continued fraction expansion, and that K 0 itself is

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The min-max and the max-min k-split problem are defined similarly except that the objectives are to minimize the maximum subgraph, and to maximize the minimum subgraph respectively..