Magnetization and magnetoresistance of Ni/nanoporous-GaN composites
Yonatan Calahorra, Josh Dawson, Yana Grishchenko, Saptarsi Ghosh, Abhiram Gundimeda, Bogdan F. Spiridon, Rachel A. Oliver, Sohini Kar-Narayan
MMagnetization and magnetoresistance ofNi/nanoporous-GaN composites
Yonatan Calahorra , Josh Dawson Yana Grishchenko Saptarsi Ghosh Abhiram Gundimeda Bogdan F. Spiridon Rachel A. Oliver Sohini Kar-Narayan Department of Materials Science and Metallurgy, University of Cambridge, CB30FS, Cambrdige, UK Department of Materials Science and Engineering, Technion - IIT, Haifa, 3200003,IsraelE-mail: [email protected]
Abstract.
Multifunctional semiconductors widen the application scope of existingsemiconductor devices. Here we report on the ferromagnetic/semiconductor couplingobtained through nickel and porous-GaN composites. We realised nickel-infiltratedand nickel-coated porous-GaN structures, and examined the subsequent magnetic andmagnetotransport properties. We found that the magnetization of the porous-GaN/Nicomposites strongly depended on the amount of deposited nickel and evolves fromrelatively isotropic and remanent (up to 90% remanent-to-saturation magnetization)response to an anisotropic response characteristic of thin-films. The magnetoresistanceof nickel sputter-coated porous-GaN structures was measured at 300 and 200 K.The temperature dependant measurements suggested that transport in the GaNlayers is dominating the current. Nonetheless, depending on sample pore size,the magnetoresistance displayed high (12-fold) out-of-plane/in-plane anisotropy, andsignificant hysteresis. These results are uncharacteristic of GaN transport and pointtowards magneto-piezo-resistive coupling between the nickel and the porous GaN.These results encourage deeper investigation of magnetic nanostructure propertytuning and of magnetic property coupling to GaN and similar materials. a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b agnetization and magnetoresistance of Ni/nanoporous-GaN composites Keywords : GaN; nanoporous; magnetic materials; magnetoelectric; magnetoresistanceSubmitted to:
JPhys Materialsagnetization and magnetoresistance of Ni/nanoporous-GaN composites
1. Introduction
The coupling between mechanical, electrical and semiconducting properties offered bypiezoelectric semiconductors makes them interesting multifunctional materials, andcan enhance and add a further dimension to semiconductor applications [1–3]. Thepiezotronic effect, where the potential barrier height of a semiconductor junction changeswith mechanical excitation, is a widely studied manifestation of this coupling [4–8]. Thechange of barrier height results in a change in the current output (I-V) characteristics.If the junction is light emitting, the mechanical excitation can modulate the emission:the piezophototronic effect [9]. GaN technology is dominant in light emission applica-tions, with electrical polarization playing a hindering role in GaN optoelectronics [10].Furthermore, polarization is fundamental to GaN high electron mobility transistors(HEMTs) [11] and the piezotronic effect can affect HEMT operation [12, 13], extendingthe basic principle of polarization discontinuity that enables GaN HEMTs. Both aspectsmake the electromechanical properties of GaN attractive research topics.Another class of multifunctional materials is magnetoelectric (ME) materials. Mag-netoelectric materials increasingly take the form of piezoelectric-magnetostrictive com-posites, in a practical expansion of the fundamental concept of a multiferroic [14]. Thecomposite is characterised by intimate contact interfaces between constituents, whichcan take various topologies. The magnetization and applied electric field (or polarizationand applied magnetic field), are thus mechanically coupled. ME composites offer excit-ing possibilities in voltage-controlled magnetism (attractive for spintronics [15]), as wellas magnetic-to-electric transduction for sensing and power transfer [16] and miniaturisedantennae [17]. The nature of ME composites naturally directs attention to high perform-ing piezoelectric materials such as ceramics, and ME literature is predominantly focusedon them. However, the multifunctionality of piezoelectric semiconductor is appealing,and inspires work on ME-semiconductor material composites, where semiconductor de-vice operation can be controlled by external magnetic excitation. For example, thechange in the device resistance under magnetic field - magnetoresistance (MR).III-N material based ME applications are mostly focused on AlN [17–20], as AlNis the best piezoelectric among III-Ns. Correspondingly, GaN is less explored for theseapplications [21]. However, there is generally no semiconducting aspect to these demon-strations per se, largely stemming from the large band gap of AlN, and the traditionalME mindset, inherited from the ceramic materials world. GaN can mitigate that, andoffer true ME-semiconductor applications, while still benefiting from relatively highpiezoelectricity (compared to non-nitride III-Vs [22]), wide range of semiconducting real-world applications and silicon compatibility. We are currently aware of no more than ahandful of truly III-N based ME-semiconductor applications, focused on ME mediatedmagneto-optic coupling [23, 24]. These demonstrations, and the ‘non-semiconducting’ones mentioned above, usually consider laminate structures (2-2 topology); alternatively,there is potential in more complex ME composite topologies, for example matrix em-bedded particles (0-3) or nanowire structures (1-1) [25]. The composite point of view agnetization and magnetoresistance of Ni/nanoporous-GaN composites ∼
40% porous GaN (involume) exhibited a 3-fold enhancement of the piezoelectric charge coefficient ( d ) [30].A similar approach has been considered for porous InP filled with magnetic material,following a reported enhanced piezoelectric response [31, 32]; these reports however didnot study the ME property coupling. Overall, this study demonstrates a non-laminatenickel/GaN composite, the interplay between synthesis process and magnetic propertiesand the subsequent magnetic control of electronic properties.
2. Materials and Methods
N-doped GaN (2 · - 2.3 · cm -3 ; 1 µ m) was grown on low-dislocation GaNpseudo-substrate [33] on sapphire by metal-organic vapor phase epitaxy (MOVPE) andporosified, as described previously [34, 35]. Electrochemical etching was carried out ina two-electrode configuration. The sample was immersed in oxalic acid (0.25M) withthe nitride surface of the doped layer exposed and a DC potential of 10-14 V wasapplied between the sample and a platinum counter electrode. In our process, porositydegree follows the voltage, the depth follows time and the process as a whole dependsupon doping (through the influence on conductivity) [35]. The chosen doping level was1 . · cm -3 and the etching voltage was either 10 or 12 V, see further discussion inSupporting Information S1. In order to reuse the electrochemical sample configuration,the etch was done such that a part of the n-doped layer was intact, with the purpose ofacting as a working electrode for subsequent nickel electrodeposition (Figure 1). Two methods were employed: electrodeposition to realise Ni-infiltrated GaN, andsputtering to realise Ni-coated GaN. Electrodeposition was carried out in a threeelectrode configuration, as previously reported [25], using a counter Pt electrode, andAg/AgCl reference electrode and the sample as a working electrode. See complete detailsin Supporting information S2. Calibration has shown that for the selected workingpotential (-1 V), the deposition rate was linear with time. Following deposition, the agnetization and magnetoresistance of Ni/nanoporous-GaN composites
Goodfellow
Limited). The experimental conditions usedwere 30 W, 2.15 Pa. A permanent marker pen was used as a lift-off resist on thesample edges and non-etched top: it prevented deposition on the covered areas, anddissolved in acetone. Three depositions were carried out resulting in films of 87 ±
8, 185 ±
13, 190 ±
21 nm, as measured by
Dektak profilometer. The nominal thickness of thelast deposition was 300 nm, and the cause for the significant deviation of the sputterthickness from nominal value is unclear.Figure 1 shows a schematic of the samples and fabrication concepts used in thiswork: electrochemical etching of conductive GaN followed by electrodeposition of nickel(using the same apparatus), or sputtering of nickel on top of GaN. Table 1 lists thesamples which underwent vibrating sample magnetometry analysis.Figure 1: Sample preparation schematic. A highly doped GaN layer waselectrochemically etched to form porous structures. The same configuration was usedto electrodeposit nickel within the pores. Some of the samples were sputter coated withnickel. The inset shows two electrodeposited samples (top) and one sputtered sample(bottom).
Scanning electron microscopy (SEM) and energy dispersive X-ray (EDX) spectroscopywere carried out, at room temperature (300 K), on an
FEI Nova NanoSEM operatedat 5 kV by imaging secondary electrons. The SEM is equipped with a silicon DriftDetector Energy Dispersive X-ray spectrometer. SEM was also performed using a
Hitachi TM3030 desktop tabletop instrument at 12 kV. agnetization and magnetoresistance of Ni/nanoporous-GaN composites Vibrating sample magnetometry (VSM) was carried out using a PrincetonMeasurements Corporation electromagnet with an
Applied Magnetics LaboratoryInc.
Precision Bipolar Magnet Controller and a Princeton Measurements CorporationMicroMag Controller. The maximal DC H field applied was about 1 kOe, significantlysmaller than the limit of uniform field (about 10 kOe). The probe was vibrated at 83Hz with an amplitude of 1.0 mm, and a continuous magnetic field sweep was used witha pause of 300 ms after each step increment. The maximum applied magnetic field was10 kOe (1 T), the field increment of the sweep was 5 Oe, the magnetic moment rangewas 500 memu, and the averaging time of each measurement was between 0.1-1 s. Thethe empty holder was measured and exhibited a very low diamagnetic response. Nodemagnetization correction factor was used.Room temperature magnetotransport measurements were carried out in a PhysicalProperty Measurement System (PPMS) DynaCool (
Quantum Design ) to extract MR.The field was applied both in-plane and out-of-plane (relative to the sample), with amaximum value of 100 kOe (10 T). The resistance between sputtered nickel and thenominally conductive edge of the sample was monitored by measuring the AC voltageresponse of the device to an AC current source. See further details in SupportingInformation S3.
3. Results and discussion
Table 1 lists the samples used in this study which underwent VSM characterization,classified by etching voltage (10 or 12 V) and nickel deposition method (sputtering orelectrodeposition). Control samples of nickel electrodeposited on ITO coated polymer(Polyethylene terephthalate) are also shown. This sample distribution allows compar-ing the effects of deposited layer thickness and pore size on the structural and magneticproperties of the deposited nickel: relative remanence, M R /M S , coercive field, H C , andthe anisotropy between out-of-plane (OOP) saturation and in-plain (IP) saturation, M S,OOP /M S,IP . M R is the remanent magnetization and M S the saturation magnetiza-tion.Figure 2 shows cross sectional SEM images of 10 and 12 V etched samples with nickeldeposited for 100 seconds, and a 12 V etched sample with nickel deposited for 500 sec-onds. The 10 V sample grew a coat of nickel already at 100 seconds, while the 12 Vdid not, even after 500 seconds. The over-grown samples had a metallic finish com-pared to an opaque finish in the infiltrated samples. This directed all electrodepositionexperiments towards 12 V etched samples - to properly study infiltration effect. Theunderlying reason is the increased degree of etching with higher voltage, thereby provid-ing more room for infiltration. This is further shown in Supporting Information FigureS1, with a significant increase in surface pore size for higher etching voltage. agnetization and magnetoresistance of Ni/nanoporous-GaN composites M R ,relative to the saturation magnetization, M S , of a single in-plane (IP) or out-of-plane(OOP) hysteresis loop, the coercive field, H C , and the saturation magnetization ratiobetween OOP and IP measurements for the maximal field applied. Double rows showIP and OOP measurement values. Sample Etch [V] Duration/thickness M R M S H C [Oe] M S,OOP M S,IP
NotesSP1 10V 87 nm 0.71 101 0.7 a,b0.76 76SP2 IP
10 V 185 nm 0.52 175 - a,cSP3 IP
10 V 190 nm 0.42 119 - a,cSP4 IP
12V 185 nm 0.34 118 - a,cED1 12V 150 s 0.84 330 0.86 b0.9 214ED2 12V 200 s 0.80 69 0.82 -0.84 147ED3 12V 500 s 0.43 139 0.63 -0.38 163ED4 10V 1800 s 0.46 70 - d- -ED5
ITO - 1800 s/ 0.65 88 - d200 nm - -ED6
ITO - 1800 s/ 0.65 60 0.34 e650 nm 0.52 126 a - M S taken at 500 Oe without full saturation. b - OOP coercivity lower. c - OOP response not measured. d - no switching in OOP response e - Electrodeposited using higher working potential, yielding faster deposition Figure 3 shows the normalised magnetization curves obtained by VSM of severalelectrodeposited samples. Some appear a few times to allow comparison of in-planeand out-of-plane magnetization. Two results are striking: i) short electrodepositiontimes (ED1, ED2) result in prominent magnetic remanence - as demonstrated by high M R /M S values in table 1; ii) these samples also exhibit low in-plane/out-of-planeanisotropy, as evident by high M S,OOP /M ,IP values in table 1. Samples with longernickel electrodeposition times, exhibit characteristics approaching those of thin films.In particular, a thick electrodeposited layer on top 10 V etched GaN (ED4) shows nosaturation in the hard-axis at 1000 Oe, and a relatively low coercive field in-plane.Noticeably, the 500 second electrodeposited sample (ED3), where no top nickel layerwas observed, appears to be an intermediate: where the in-plane curve is similar to the agnetization and magnetoresistance of Ni/nanoporous-GaN composites ; d) Cross-sectional SEM of a 12 V etched, 500 second electrodepositedsample. The the dashed yellow lines indicate the boundary between etched and non-etched GaN.out-of-plane, as well as to the in-plane curve of the thicker layer, with coercive fields atintermediate values as well.Two control electrodepositions (ED5 & ED6, which were part of a larger series ofcalibration and control electrodepositions - see Supporting Information Figure S2.1) areshown in Figure 3f. Both samples were electrodeposited for 1800 seconds, while differentworking potentials were used: -1 V for ED5 and -1.5 V for ED6. This translated tomore than a 3 fold increase in the ED6 thickness, as electrodeposited layer thicknesswas found to be linear in time but not with voltage (Supporting Information FigureS2.1). The comparison shows that ED5 demonstrates similar characteristics to ED4,which could be expected as both samples underwent the same electrodeposition process.Furthermore, it shows that ED6 presents similar in-plane characteristics, however withincreased remanence in the hard-axis.We carried out SEM-EDX spectroscopy of the cross-section shown in Figure 2b,to better understand the initial stages of the infiltrated structure, as shown in Figure 4.Electrodeposition lasted 100 seconds and the sample shown was etched using 12 V,similar to most of the electrodeposited samples. It is evident that unlike the expectedresult of wire-like formation of nickel from the bottom-up, nickel growth nucleatesthroughout the porous volume. This is most likely a result of maintained conductivityin the porous layer - resulting in multiple conductive pathways for electrodeposition. agnetization and magnetoresistance of Ni/nanoporous-GaN composites i.e. non-porous) electrodeposition ofnickel also demonstrates multiple nucleation points, which exhibit a smeared hysteresiscurve, becoming abrupt for the thicker films [37]. This is unlike the observations here -and therefore does not explain well the magnetization of the electrodeposited Ni at theearly stages, indicating further complexity in nucleation site interactions and structure.Ferromagnetic nanoparticles demonstrate competition between single-domainattributes of large coercivity, decreasing with increasing size [38] and a low coercivitylimit, which increases with increasing size [39], where the extreme case of smallparticles is manifested as superparamagnetism [40]. However, our results here showincreased coercivity in the samples of the shortest electrodeposition, as well as significantcoercive field asymmetry. Both findings could indicate an interaction of nickel with asurrounding surface layer of antiferromagnetic NiO (exchange bias [41]), formed after agnetization and magnetoresistance of Ni/nanoporous-GaN composites Figure 5 shows the magnetization of nickel thin films sputtered onto porous GaN. Thesecurves show a thickness evolution of the characteristic in-plane magnetization from arelatively sharp and remanent curve for the thinnest layer, to unsaturated diamond-likecurves for the thicker layers. There was no significant difference found between thecharacteristics of nickel deposited on larger pores (SP4 compared to SP2 and SP3) -probably due to most of the layer forming above the surface. The unsaturated curve agnetization and magnetoresistance of Ni/nanoporous-GaN composites
The motivation for this work was realizing GaN based ME composites. Nonetheless, weexperienced an issue of maintained conductivity in the porous samples (the top layernickel and remaining conductive GaN where electrically connected, through the porouslayer) - this screens piezoelectricity in the porous GaN layer, and subsequently hindersmagnetoelectricity. We attribute this to the highly doped sample chosen for theseexperiments (1.85 · cm -3 ), compared to the lower value used in our previous work agnetization and magnetoresistance of Ni/nanoporous-GaN composites · cm -3 ) [30]), where the porous layerbecame non-conductive. The maintained conductivity has led the ‘device’ part of thework to focus on MR instead of ME coupling. Table 2 summarises the magnetotransportmeasurements of SP3 and SP5: two Ni-sputtered samples.Table 2: The MR properties of two samples sputtered with 190 nm nickel. The resistancevalue, R , is shown with an estimated accuracy of ± .
05 Ω, unless stated otherwise. The
M R value is the resistance change at the maximal field compared to initial, zero-fieldresistance. The anisotropic MR,
AM R = ( R OOP − R IP ) /R IP , is calculated at themaximal field. Sample Etch [V] Temp.[K] R | H =0 [Ω] M R max [%]
AM R | H max [%] NotesSP3 10V 300 62.34 1.77 0.75 a200 203.28 -0.59 0.39 b,c300 63.06 0.76 0.72 aSP5 12V 300 285.65 - - d200 381.5 ± ± a - Maximal MR found OOP. b - Maximal (absolute value) MR found IP. c - OOP MR not monotonic, making AMR non-trivial. d - IP and OOP responses very similar. Figure 6 shows the in-plane (solid lines) and out-of-plane (dashed lines) MR of twosamples, measured between a top sputtered nickel layer (190 nm) and a side contact tothe conductive GaN layer. The two samples differed in the etching process of 10 (SP3)and 12 V (SP5), corresponding to smaller and larger pores. We note that the appliedfield maximum here (10 T) is significantly larger than the normal range of observedsaturation in MR [50–52], as well as magnetostriction [50, 53, 54], of nickel. This allowsexamining both the low and high field characteristics.Figure 6a and Figure 6d show the MR of the two samples measured initially at 300 K(RT). Notably, the responses were markedly different, indicating a profound effect of theporous structure on the magnetoresistive properties of the device: SP3 demonstrated apositive response, while SP5 a negative response. There are contradicting reports aboutthe sign of the parallel (field and current direction) MR of nickel [51, 52], however, allreports show a decreasing trend with field, even when there is an initial rise to yield apositive value which is - unlike the result in Figure 6a. This is an indication that theMR of SP3 was not dominated by nickel transport, and that possibly, the transport ofSP5 was significantly affected by nickel MR. Navarro-Quezada and coworkers studiedthe high-field (up to 6 T) MR of phase separated (Ga,Fe)N particles on the surface ofGaN [55]. Their measurements on a control GaN sample (no iron) showed a similar agnetization and magnetoresistance of Ni/nanoporous-GaN composites µ A initially to 10 mA amplitude). We cannotexclude that joule heating (6 orders of magnitude higher power) altered the device, e.g. ,by annealing of the Ni/GaN interface [56, 57] (see Supporting Information Section S3for more SP5 results). Therefore we continue discussing SP3 alone, which was measuredconsistently by applying an AC current with an amplitude of 50 µ A.Figure 6b shows the MR of SP3 measured at 200 K. The zero-field resistanceincreased roughly four-fold in the transition from RT to 200 K. This would beexpected for a sample where conductance is primarily semiconducting and not metallic.Furthermore, the low field MR was negative, as expected for nickel, indicating thatthere are parallel current paths in this device. Interestingly, the out-of-plane and in-plane responses were different apart from the low-field negative MR: the out-of-planeresponse maintained the positive parabolic characteristic, while the in-plane responsedemonstrated a negative parabolic relation.Figure 6c shows the MR of SP3 measured again at RT. The zero-field resistancereturns to the previous RT value at the end of the first field cycle, as do the general trendsof the out-of-plane and in-plane responses. The AMR value is lower compared to thefirst RT measurement since the initial increase in resistance was no longer exhibited.Furthermore, the AMR value represents the relative difference at maximal field, anddoes not reflect the considerable difference between the in-plane and out-of-plane MR.The in-plane MR is considerably low, 0.063%, and the out-of-plane/in-plane MR is 12.Such an anisotropic response has not been observed by Navarro-Quezada et al. in theirmeasurements of GaN without iron - indicating that the nickel layer influenced the MRof our device.Moreover, the curves in the second RT measurement were less symmetric. Thisis further evident in the low-field region of the out-of-plane response (Figure 6f). Theblack arrows show the field sweep direction, and the larger blue arrow points to theestimated curve minimum - noticeably away from zero-field. To further examine thispoint we fitted the positive part of the negative field sweep and the negative part of thepositive field sweep to distinct parabolas, as shown in Figure 7. This was driven by theassumption that the minimal resistance value of the two curves would be different if theresistance (while still dominated by the GaN, as established earlier) was indeed coupledto the magnetization. The full results of the fitting process are shown in SupportingInformation S3. The results obtained were: -11500 Oe for the negative sweep and-500 Oe (estimated error of 10%). This finding suggests that the magnetization ofthe nickel layer plays a part in the MR, verifying our assumption, thereby suggesting agnetization and magnetoresistance of Ni/nanoporous-GaN composites
4. Summary
In this research we present the nature of the magnetic properties of Ni/porous-GaNcomposites prepared either by electrodeposition of nickel inside the pores or sputteringonto the porous surface. Our results indicate a rich response, intricately dependenton the deposition method, amount of nickel deposited and the porosity (degree, and agnetization and magnetoresistance of Ni/nanoporous-GaN composites
Data and Supporting InformationAcknowledgments
The authors thank Dr. Nadia Stelmashenko for assistance and guidance in carrying outsputtering deposition and VSM measurements, and Dr. Cheng Liu for assistance andguidance in magnetotransport measurements. Y.C., R.O. and S.K-N. are grateful forsupport from Henry Royce Institute - Cambridge Equipment grant EP/P024947/1 andthe Centre of Advanced Materials for Integrated Energy Systems ”CAM-IES” grant agnetization and magnetoresistance of Ni/nanoporous-GaN composites
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Yonatan CalahorraJosh DawsonYana GrishchenkoSaptarsi GhoshAbhiram GundimedaBogdan F. SpiridonRachel A. OliverSohini Kar-Narayan
Supplementary Information Section S1. GaN growth and Etching
Figure S1 shows cross-sectional SEM images of about 1 µ m porous GaN etched atdifferent voltages, of differently doped samples. Etching is stronger for higher dopingand higher voltages, and the resulting effect is visible to the naked eye, in light reflectanceproperties.Figure S1. a) Cross-sectional SEM image showing etching efficiency increased withvoltage; b) top-view SEM image showing increased etching with voltage; c) opticalimages of samples similar to the ones reported, doped 1.85 · cm − and etched at 10and 12 V. Taken at different angles, the samples reflect light differently, following thedifferent etching.2 Section S2. Nickel electrodeposition
Calibration depositions were made using ITO coated PET. The relation betweenthickness and deposition voltage was found to be highly non-linear, while it was linearwith deposition time. This also reflected in magnetization, where the magnetization vs.the total charge passed in the process also showed a linear trend.Figure S2.1 Thickness of electrodeposited nickel as vs. voltage (top, for 1800 s), and vs.time (bottom, using -1 V).Figure S2.2 Magnetization vs. electrodeposition charge obtained by integrating over thecurrent.3Figure S2.3 SEM-EDX data obtained from the sample shown in Figure 4 in severallocations. No significant distinctions along the sample cross section were found.4
Section S3. Magnetoresistance measurements
Figure S3.1 shows the sputtered samples under the bonding stereoscope, as capturedby a cellular phone camera. A 4-probe measurement was applied in order to allow verylow AC signals to be applied and measured (as the resistance was very low), howeverthe physical connections are in 2-probe, with the current and voltage leads connectedto the same pads (the nickel layer and the solder on the sample side). The resistancemeasured includes therefore contact resistance, which should be very low, if present atall.Figure S3.1 The 10 V (left) and 12 V (right) etched samples bonded to the PPMSsample holder. Each pad is connected by two bonds as contingency. Figure S3.2 showsthe measured MR of SP5 at 200 K and back at 300 K. Notably, the resistance at200 K increased as well, however only in about 50% (compared to a 3-fold increase inSP3). The resistance in the return to 300 K was not similar to the first measurement- an indication that the device has indeed changed, possibly due to the higher currentapplied. The consistent feature is that there was no obvious distinction between thein-plane and out-of-plane measurements.5Figure S3.2 Magnetoresistance data of SP5 at 200 K (top) and the second measurementat 300 K (bottom). Out-of-plane data is dashed.The fitted curves in the main Figure 7 are given by: R right = 4 . · − H + 9 . · − H + 63 .
02 (1) R left = 4 . · − H + 4 . · − H + 62 .
98 (2)where H is in 10 Oe and R in Ω. The fitting values are obtained by MATLAB’s cftool ,with a 95% confidence. We therefore estimate the error of H | min ( − b/ a ) as 10% (with b and a the 2 nd order polynomial coefficients). The R-square values of both fits were > ..