The a.c. susceptibility third harmonic component of NdO1!0.14F0.14FeAs: Aflux dynamic magnetic analysis

Daniele Di Gioacchino a,n, Augusto Marcelli a, Alessandro Puri a,b, Antonio Bianconi b

a INFN–LNF, Via E. Fermi, 40-00044 Frascati, Rome, Italyb University ‘La Sapienza’, Dipartimento di Fisica, P.le A. Moro, 4-00100 Rome, Italy

a r t i c l e i n f o

Keywords:A. SuperconductorD. Magnetic propertiesD. Phase transition

a b s t r a c t

In this contribution we present the analysis of the third harmonic susceptibility data of the newsuperconductor NdFeAs1!0.14F0.14. ‘Cole–Cole’ polar plots respect to the magnetic frequency of theexciting field are presented and discussed. Data show that NdFeAs1!0.14F0.14 exhibit a ‘bulk pinning’with a 3D flux dynamic character. A comparison of the responses of high Tc materials and an evaluationof theoretical critical states is also presented.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The recent discovery of superconductivity in arsenide oxidematerials [1] triggered many different investigations on thesematerials. RFeAsO1!xFx systems (R"rare earth) are characterizedby a high Tc (from 26 to #55 K) and a layered crystal structurewith FeAs layers associated to the superconductivity and RO1.xFxlayers acting as charge reservoirs. In these compounds a complexinterplay among magnetism, layered structure and multiplesurface Fermi sheets occur [2–4]. In addition, low magnetic fieldand high-field transports measurements [5,6] both show anintermediate behavior between that of a low-Tc superconductorand HTSC cuprates. In this contribution we present a flux dynamicinvestigation, to obtain information on the thermal fluctuation ofvortices and the dimensionality of the pinning processes con-nected to the critical current behavior vs. temperature, appliedmagnetic field in the sample. Actually these parameters areimportant for practical applications and the ac multi-harmonicsusceptibility [7], in particular the third harmonic component [8]is the best tool for this kind of analysis. In addition to the a.c. thirdharmonic study of the NdO1!0.14F0.14FeAs as a function of bothtemperature and frequencies in a low applied magnetic field, wecompare the observed behaviors with ideal critical state modelsand with those of other high Tc materials (HTSC) such as MgB2,YBCO and BiSCOO2223.

2. a.c. multi-harmonic susceptibility and flux dynamic in asuperconductor

The dissipative effects in the superconductor mixed flux vortexstate, can be investigated with ’’a.c. magnetic susceptibility’’ (w)experiments [7]. In Fig. 1 we show the real part of the firstharmonic (w0

1) (red curve) vs. temperature for one sample ofNdO1!0.14F0.14FeAs. The curve describes the magnetic behavior ofa characteristic superconductor sample with a pure negativediamagnetic signal. At the same time as the first harmonicimaginary part (w00

1) (blue curve) is correlated to the magneticenergy absorbed by both linear and non-linear processes, and isproportional to the magnetization (M) cycle vs. a.c. applied field(Hac) [9]. The interpretation of the harmonic susceptibility in asuperconductor is based on critical-state models [10–12] wherethe magnetic B profile is correlated, by Ampere’s law, to thecurrent induced in the sample that is equal to the critical value Ic.Indeed, in a one-dimensional sample with a thickness 2d, thewidth D inside a sample where is induced a supercurrent Ic, isD"Bac/Ic, where Bac"m0Hac. As a consequence, when D"d asupercurrent flows through the entire sample volume. In this casethe internal local field Bp can be written as Bn" Icd and the firstand higher harmonic susceptibilities depends only by oneparameter: d"Bp/Bac [13]. In other words when d"1, thepenetration field reaches the center of the sample.

In particular the low frequency complex multi-harmonicsusceptibility (wn " w0

n$ iw00

n) can be used to identify the non-linear processes that sustain the superconducting critical currentdensity and the linear normal losses that limit the superconduct-ing properties. The method offers great advantages because, whileskin electronic losses are linear processes and the susceptibilitymeasures only the first harmonic component, a phase transition,in particular a superconducting transition, is a non-linear

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0022-3697/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.jpcs.2010.03.004

n Corresponding author.E-mail address: daniele.digioacchino@lnf.infn.it (D.D. Gioacchino).

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processes characterized by non-harmonic contributions. In thislatter case, high harmonics (wn41) together with the first compo-nent are simultaneously presents in the susceptibility behaviors.

High harmonic components can be regarded as the Fouriercoefficients [13] of the steady magnetization cycles in thepresence of an external oscillating magnetic field. These coeffi-cients are affected by the magnetic flux entering and leaving thesample. The hysteretic behavior and the high harmonic suscept-ibility components describe the interaction between the fluxvortex array and extrinsic properties such as the pinning due tothe disorder present in the material. However, the analysis iscomplex due to the large number of parameters involved: thecoherence length (x) that describes the core of the weak pinningpotential in a disordered state; the inter-vortex distance (a0)leading to the pinned vortex bundle formation; l defining the fluxextension range, the elastic properties of the flux lattice definedby the Cii elastic modules such as flux tilt (C44), shear (C66),compression (C11) [14], the mean thermal displacement ou4 ofthe flux vortex and the dimensionality (nD with n"1,2,3) of theflux-pinning interaction due to the material layered structure. Thecorrelations between these scale lengths determine the presenceof different flux pinning motion regimes respect to magnetic field,temperature and electric current with a specific vortex mobilitydescribed by the characteristic ’time relation’ or ‘windowfrequency’ occurring in a superconductor, e.g., single vortexes,small bundle vortexes, large bundle vortexes, weak or strongpinning interactions.

In this scenario, decreasing the temperature the vortex systemmoves in a non-equilibrium critical state that slowly decaystoward the equilibrium towards a final flux glass state [15,16].

The susceptibility components wn and in particular the thirdharmonic susceptibility probes the superconducting electrictransport processes in different regions of a ‘phenomenologicalspace’ [16] whose axis are the magnetic field, the temperature andthe current (Fig. 2). Different ‘Hac frequencies’ induce changes inthe current level and may set different ‘time windows’ for eachprocess. In particular, the high harmonic component of themagnetic susceptibility is related to the I–V superconductorcharacteristic. The latter is strongly non-linear and, as showedin Fig. 3, the ‘I–V curvature’ describes the dynamic behavior of fluxlattice-pinning interactions around the critical current value Ic.The I–V curvature and high harmonics (wn41) are proportionalquantities. When the frequency increases, in a superconductorgrows the potential V actually probing changes in the ‘I–V

curvature’ associated to the different flux dynamics induced inthe sample [17].

3. Experimental set-up

The superconducting NdF0.14FeAsO1!0.14 bulk sample has beenprepared by high pressure synthesis from Nd, As, Fe, Fe2O3, FeF3powders (the purity of all starting chemicals was better than99.99%). The components were mixed together according to thenominal stoichiometric ratio of NdO1!0.14F0.14FeAs, thengrounded thoroughly and pressed into small pellets. The latterwere sealed in boron nitride crucibles and sintered in a high-pressure synthesis apparatus under a pressure of 6 GPa and at thetemperature of 1300 1C for 2 h [18]. The sample dimensions are1.5%1.5%1.0 mm3. The a.c. high harmonics susceptibility re-sponse of the sample has been measured in a susceptometerbased on pick-up double coils surrounded by a driven coil(Fig. 4) [19]. The sample was set on a sapphire holder, whichfits in the pick-up coil bridge. Two coils connected in series androlled up in opposition realize the bridge. One is strongly coupled(pick-up coil) while the second is weakly (balance coil) coupled tothe sample. The sample is cooled in the ‘zero magnetic fieldcooling (ZFC) set-up’ in a thermally controlled He gas flowcryostat. The range of the ac driving magnetic field frequencywas 107 Hzo fo1070 Hz with amplitude of 1.3 Gauss. A

Fig. 1. a.c. susceptibility measurements vs. temperature of the NdFeAsO1!0.14F0.14sample. The real part of the first harmonic (red curve) and the imaginary part (bluecurve). (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Fig. 2. A phenomenological phase diagram magnetic field–temperature–currentfor an anisotropic HTSC material. The axis of the current is proportional to the time[16].

Fig. 3. The I–V superconductor characteristic. Different potential V are showed forthree different frequencies (f1of2o f3).

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multi-harmonic lock-in amplifier measured the Fourier compo-nents of the induced signal. The sample was oriented parallel tothe ac magnetic field, and during the acquisition the temperaturehas been measured by a CGR thermometer. Susceptibility data arethe result of the subtractions between signals collected with andwithout the sample vs. frequency and temperature.

4. NdO1!0.14F0.14FeAs third harmonic measurements anddiscussion

In Fig. 5 we show the real and imaginary parts (w03,w

00

3) of thethird harmonic of the NdF0.14FeAsO1!0.14 sample vs. temperatureand frequency. Real parts describe the oscillating signals aroundthe zero value. All the odd high harmonic susceptibilitycomponents are polynomial functions with maxima and minimaof the ratio d, [13] a parameter proportional to the penetrationdepth of the a.c. magnetic applied field into the sample andconnected with the critical current via critical state models. Fordo1 odd high harmonic real part vanish. In addition, in this puresinusoidal excitation case, the even harmonic componentsdisappear [13]. The behaviors allow identifying the onsettemperature of the sample, i.e., the critical temperature Tcat 46.5 K.

In a typical ZFC procedure, after cooling a magnetic field isapplied to the sample and the superconductor switches in a flux-pinning critical state. The state decays with time, via an ‘activatedtemperature creep’ process, in a final stable flux-pinning ‘glassstate’ pattern (see Fig. 2) [16]. The frequency dependence of thenon-linear magnetic response describes this behavior. For thissample, at low magnetic field and around Tc, the ‘critical state’decays in a ‘glass state’ with a faster characteristic creep time raterespect to the explored experimental time window range(9.3–0.93 msec), corresponding to an applied field frequency inthe range 107–1070 Hz. For this reason, the w0

3,w00

3 temperatureonsets are characterized by very weak frequency dependence.

The frequency dependence of the third harmonic susceptibilitycomponents vs. temperature is illustrated in Fig. 5 and describesthe change of the effective flux-diffusivity connected with theflux-pinning interaction. This variation is due to the change ofthe potential V induced by the frequency, as described in thesuperconducting I–V characteristic shown in Fig. 3.

To clarify the role of the w3 frequency dependence, we mayanalyze the polar plot (Cole–Cole) of w03 vs. w00

3 components[20–22]. These lens shaped closed curves shown in Fig. 6, addressboth a change of the initial w phase and of the area of the closed

Fig. 4. A schematic layout of the a.c. susceptibility measurements.

Fig. 5. The third harmonic components (w01!3 ,w

00

1!3) of the a.c. magnetic suscept-ibility curves vs. temperature and frequency for the NdFeAsO1!0.4F0.4 sample.(Top: real part; bottom: imaginary part.)

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curves as a function of the applied magnetic field frequency. Inthis figure we show the presence of a ‘pure critical state with ahomogeneous pinning bulk’ almost independent by the frequencywith a ‘lens’ shape [20,21] (red curve) and a ‘‘pure critical statewith a surface pinning barrier’ with a cardioid profile (violetcurve) [21].

The first indication of the ‘lens shape’ closed curves is that a‘flux bulk pinning’ is operative in the NdF0.14FeAsO1!0.14 sample.The second is the rotation of the polar plots in a counter-clockwise mode if compared to the ideal bulk pinning criticalstate [13,20,23]. This corresponds to a change of the ’phase lag’ Ywith the frequency excitation, associated to a temporal shiftbetween the magnetic excitation signal (Bac) on the surface and tothe magnetic response induced at some distance below thesurface (Bp). The shift is correlated to an effective non-linearmagnetic diffusivity due to the flux-pinning interaction [20]. The[ frequency behavior shown in Fig. 6 is correlated also to thedifferent I–V curvatures [17] observed in the real case and probedvs. frequency. It depends by the flux motion, the pinninginteraction and by its dimensionality inside the sample [17].

All critical state models describe magnetic configurationindependent by the frequency. However, due to the ‘bulk pinning’,in the critical state the w0

3!w00

3 Cole–Cole plot is characterized bythe phase Y"901 up to d"1 because for dr1 the w0

3 amplitude isequal to zero (see Fig. 6—red curve) [13] while in the ‘surfacepinning barrier’ this occurs only for d"1 [21].

Data of the NdFeAsO1!0.14F0.14 show a phase change with aclockwise rotation increasing the frequency in the range231oYo641 when the frequency is in the range 107–707 Hz.On the contrary, polar plots are independent in the 707–1070 Hzrange although atY"641 the value is different by the ideal one inthe critical state (YBulk pinning"901) probably due to the finitethickness of the sample [23]. Actually, at high frequencies, themagnetic flux profile inside the NdFeAs1!0.4F0.4 sample issubstantially similar to the critical state and at these frequencies.The potential V induced in the sample determines around the Icvalue, a similar curvature in the I–V characteristic. In this caseexperiments probe an I–V characteristic region with a very steepslope, e.g., f2 or f3 in Fig. 3.

It is useful for the analysis of the flux dynamic in theNdF0.14FeAsO1!0.14, the comparison of the third harmonic polarbehavior, measured with the same experimental set-up used forothers HTSC materials characterized by a different flux-dimen-sionality, pinning and critical temperature.

In Fig. 7 we compare the w03!w

00

3 Cole–Cole plots of three HTSCmaterials at different frequencies. Here the temperature isconsidered an implicit variable.

The MgB2 sample (panel A) has a Tc"38.2 K and it ischaracterized by a w3 polar plots with a clockwise rotation vs.frequency in the angular range 431oYo741 and in the interval107–1070 Hz. The frequency rotation of the plots is similar to thatof the NdFeAsO1!0.14F0.14 with a clockwise rotation. However, athigh frequency is not present a frequency independent region.Other differences with the NdFeAs1!0.4F0.4 plots are connectedwith the closed curve contours that have a cardioids-like shape.However, increasing the frequency, the ‘cardioid’ position of theMgB2 departs from the ideal position of the ‘surface barrier fluxpinning critical state’ shown in Fig. 6 (violet line) where theYbarrier surface pinning"01. In this case, a flux dynamic dominated bya three dimensional (3D) Kim–Anderson ‘creep activation’ in abulk collective pinning scenario it is also possible [8]. Theclaiming is based on the numerical solution of the non-linearmagnetic diffusion equation for the magnetic flux motion in asuperconductor of thickness d with sinusoidal boundary condi-tions such as the ‘set-up’ of a.c. susceptibility experiments

@B@time

"@@x

Ur&B,J'm0

! "@B@x

# $&1'

in which the diffusion coefficient r(J), is described by the Kim–Anderson creep resistivity expression

rcreep " 2rffJc&t'J

Ue!Up&t'=KTUsinhJUp&t'Jc&t'KT

# $&2'

where rff is the flux flow resistivity, t"T/Tc is the reducedtemperature and

J"1m0d

U@B@ !x

&3'

is the current density induced in the sample where !x " x=d is adimensionless variable.

The temperature dependence of the pinning potential and thecritical current in the collective pinning model used are [8]

Up "U0&1!t4' &4'

Jc " Jc0&1!t2'3=2 % &1$t2'!1 &5'

rcreep " rffJc&B,T'

J% e&!Up&K ,T'=mKT'%(&Jc &B,T'=J'm!1)

The results analyzed using the Cole–Cole w3 plot are showed inFig. 8. We may recognize a ‘cardioid-like’ feature and a frequencyclockwise rotation. Although the polar plot in the figure shows adifferent behavior respect to the MgB2, for this numerical case [8]we used the YBCO parameters with a thickness d"1 cm largerthan that of the MgB2 sample.

In a YBCO sample with a Tc"92 K the polar plots of the w3 inFig. 7 (panel B) also show an increasing clockwise rotation vs.frequency in the range 301oYo581 and 35–1070 Hz. All closedcurves have an ellipsoidal (lens) characteristic and for this sampleboth the frequency rotation and the plot look similar to those ofthe NdF0.14FeAsO1!0.14. Also in this case an independent behaviorof the Y angle at high frequency values is not observed in theexplored frequency range. The flux dimensionality in the YBCOlayered superconductor has been strongly debated but at presentit has been substantially established that a 3D flux pinningbehavior occurs for a low amplitude of magnetic field [24].

Fig. 6. NdFeAsO1!0.14F0.14 polar plots (Cole–Cole), i.e., w03 vs. w00

3 vs. frequency(temperature is an implicit variable). The ideal ‘lens shape bulk pinning’ (red) andthe ‘cardioid shape surface barrier’ (violet) critical states are also showed forcomparison. (For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article.)

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The w3 of a BiSCCO 2223 tape with Tc"110 K gives the polarplots of Fig. 7 (panel C) with a counter-clockwise rotation vs.frequency. The polar plots in the range 107–307 Hz have theangular value Y"631 independent by the frequency, while in the307–1070 Hz interval the angular value ranges between631oYo421.

The magnetic flux profile inside the BiSCCO 2223 at lowfrequencies is quite similar to that in the critical state. Thefrequency rotation exhibited by the NdFeAsO1!0.14F0.14 sampledisplays the opposite behavior. Another difference in the closedcurve profiles, are contours between cardioid/lens features. Thesebehaviors address for the BiSCCO material a clear 2D fluxdimensionality [25]. Indeed, respect to the YBCO system, in thislayered oxide superconductor material the CuO metallic planesare strongly separated by buffer layers. As a consequence a largeanisotropy is present and an array of ‘pancake flux vortices’ on the

CuO layer joined by very weak Josephson junctions is a suitabledescription of the observed behavior. Considering also the high Tcof the Bi2223 sample, for temperatures around Tc, Josephsonjunctions are broken and only separate superconducting layersremain. Only pinning in the CuO plane is active, and an uncoupledlayered material with a clear 2D flux dynamic behavior exists. Inthis case, only at low frequencies a critical state independent bythe frequency occurs. When the frequency increases in a weak 2Dflux-pinning material, the potential V induced in the sample andthe thermal fluctuations are able to melt the pinning vortexlattice. As a consequence the superconductor critical statechanges in a liquid state [26].

To investigate the 2D behavior, we considered the numericalsolutions of the non-linear diffusion equation (Eq. (1)) with aresistivity coefficient described by the collective pinning [27,28]in the glass model [17,29] with a flux creep dynamic regimes

rcreep " rffJc&B,T'

J% e&!Up&K ,T'=mKT'%(&Jc &B,T'=J'm!1) &6'

In this 2D bulk pinning framework, considering the elasticresponse of the flux vortex lattice in a superconductor [30] onlythe shearing C66 elastic moduli [31] is active while the tilt C44elastic moduli [17,28] is not yet effective

Up&B,T'px&B,T'a0

! "2

% C66&B,T' % d &7'

Jp&B,T'pC66&B,T' % x&B,T' &8'

The magnetic behaviors of these parameters are [8,17,28]

C66&B,T'pB2c2&t' % (b&1!b2' % &1!0:58b$0:29b2') &9'

x&B,T'p%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%&1$t2'=&1!t2'

q=

%%%%%%%%%%%%%%%%%%%%%1=&2!2b'

q&10'

a0 "F0=B; b" B=Bc2&T' &11'

The results summarized in Fig. 9 confirm that in a 2D fluxpinning bulk state, increasing the frequency the w3 polar plotshave a counter-clock rotation behavior.

Fig. 7. MgB2 (A), YBCO (B) and Bi2223 (C) w polar plots for different frequencies (the temperature is an implicit variable).

Fig. 8. Polar plots (Cole–Cole) w03 vs. w00

3 results from 0.4 to 400 Hz (thetemperature is an implicit variable) based on the solutions of the non-linearmagnetic diffusion equation (Eq. (1)) with the creep resistivity (Eq. (2)) and acollective pinning model (Eqs. (3) and (4)). See Fig. 9 in Ref. [8]

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Moreover for the BiSCCO 2223 sample shown in Fig. 7 (panel C)if we consider that the rotation changes the low frequency ‘lensshape’ into a high frequency ‘cardioid-like’ feature as a functionof the frequency, a shift in the flux dynamic occurs. At highfrequency, the superconducting response is induced near the tapesurface and a ‘surface barrier’ pinning is presents. If the ‘surface’pinning is stronger respect to the 2D bulk pinning, the polar plotarea increases as a function of the frequency.

To compare different superconductors and to clarify differ-ences or similarity, we plot in Fig. 10 the initial Y angle values vs.frequency for different HTSC materials. NdFeAsO1!0.14F0.14, YBCOand MgB2 exhibit a similar frequency behavior suggesting thepresence of a 3D flux-pinning behavior in contrast to the opposite2D Bi2223 flux pinning response. Moreover it is also evident thatdifferent slopes and the presence or absence of a plateau in thecurves vs. frequency address the presence of different collectiveweak pinning distributions inside these materials. In particular,while for the YBCO and MgB2 samples the frequency independentcritical state at 1070 Hz is not yet achieved, at the same frequency

YBCO the Y value is 581 while for the MgB2 we find Y"741. Forthe latter, the ideal critical state (Y"901) is compatible at highfrequency addressing the better flux pinning characteristic of theMgB2 respect to the NdFeAsO1!0.14F0.14 sample that exhibits aplateau already starting at low frequencies.

In the NdFeAsO1!0.14F0.14 in a ‘real’ critical state an indepen-dent frequency behavior is achieved. This suggests that thismaterial with good 3D pinning characteristics is capable tosustain high critical currents. The different frequency slopesshowed in Fig.10 are correlated to the different thermal activa-tions and to the flux pinning dimensionality of the super-conductors.

Finally, from the polar plot shapes in Fig. 7 both YBCO andNdFeAsO1!0.14F0.14 samples show similar pinning structures.The w3 polar plots have ellipsoidal shapes and their differentfrequency behaviors depend by the different weight of thetemperature activation respect to the pinning potential in theflux creep motion. Actually, for the YBCO sample the criticaltemperature is around 90 K, while it is only 45 K for theNdFeAsO1!0.14F0.14 sample.

5. Conclusions

The third harmonic, Cole–Cole polar plots as a function of thefrequency of the magnetic excitation may explore the complexflux-pinning dynamic response of a superconductor. In thiscontribution we demonstrate that a 3D weak collective bulkpinning is active in the NdFeAsO1!0.14F0.14. In this framework, it isclear that the amplitude of the Y angle and the shape of the w3polar plot define, respect to the experimental set-up considered,the flux-pinning interaction of a superconductor.

Acknowledgment

A special thank is due to F. Tabacchioni (INAF) and to the entiretechnical staff of the DAFNE-Light laboratory for their continuoustechnical support.

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