Determination of an onset of superconducting diamagnetism by scaling of the normal-state magnetization
I. L. Landau, K. M. Mogare, B. Trusch, M.Wagner, J. Hulliger
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Determination of an onset of superconducting diamagnetism byscaling of the normal-state magnetization
I. L. Landau, K. M. Mogare, B. Trusch, M. Wagner, J. Hulliger
Department of Chemistry and Biochemistry, University of Berne, Freiestrasse 3, CH-3012-Berne, Switzerland
Abstract
We propose a simple scaling procedure for the normal-state magnetization M n data collected as functions oftemperature T in different magnetic fields H . As a result, the M n ( T ) curves collected in different fields collapseon to a single M sc ( T ) line. In this representation, the onset of superconducting diamagnetism manifests itself bya sharp divergence of the M sc ( T ) curves for different H values. As will be demonstrated, this allows for a reliabledetermination of temperature T onset , at which superconducting diamagnetism become observable. Key words: type-II superconductors, superconducting diamagnetism, normal-state magnetization, superconductingcritical temperature
PACS:
1. Introduction
An onset of superconducting diamagnetism in avicinity of the superconducting critical tempera-ture T c attracts a lot of attention in high- T c super-conductors [1–9]. This attention is partly dictatedby an obvious fundamental interest to the onset ofsuperconductivity in such complex materials. It isalso important from purely practical reasons be-cause it provides the highest value of T c in non-uniform superconducting samples.At the same time, in spite of apparent sim-plicity of the problem, experimental studies arerather complex. Because all high- T c superconduc-tors have a considerable temperature dependenceof the normal-state magnetization M n , extrapo-lation of M n to a region of the superconductingtransition is not obvious and sometimes question-
80 100 120 140 160 1801.92.02.12.22.3 T (K) M ( - e m u ) fit to T > 100 K datafit to T > 93 K data H = 50 kOesample Y Fig. 1. Magnetization M ( T ) data for Y H = 50 kOe. The solid and the dashedlines are fits of Eq. (1) for two sets of experimental datapoints. able (see, for instance, Fig 1 of Ref. [8] and Figs.8 and 9 in Ref. [9]).As an example, Fig. 1 shows magnetization data Preprint submitted to Elsevier Science 31 October 2018 or an oxygen depleted sample of YBa Cu O − x .Detailed data for this sample will be presented anddiscussed below. We have tried to fit experimental M n ( T ) data by the Curie-Weisse law plus sometemperature independent constant M : χ = C/ ( T − Θ) + M . (1) C , Θ and M were used as fit parameters. As maybe seen in Fig. 1, the calculated curves do not fitexperimental data points particularly well. Fur-thermore, the corresponding values of the onsettemperature T onset noticeably depend on the data,which were chosen for approximation. In such andsimilar situations any conclusion about the onsetof superconductivity is not reliable.Here, we shall introduce a simple procedure toscale the normal-state magnetization data col-lected in different magnetic fields. Because theproposed procedure does not involve any specificassumption about sample properties, it is applica-ble to uniform and non-uniform samples, to singlecrystals and ceramics.
2. Samples
In this section we present a brief description ofsamples, which were used to verify the proposedscaling procedures. Two kinds of high- T c materi-als were investigated: YBa Cu O − x (YBCO) andTl Ba Ca Cu O (Tl-2223). All samples were ce-ramics and were not particularly uniform. Fig. 2shows a superconducting transition for an oxygendepleted sample Y T c is defined as is shown inFig. 2(a). Because this sample was kept on air forabout a year, it contained inclusions with a higherlevel of doping and with correspondingly higher T c values. This resulted in about 25 K long tale of thetransition (see Fig. 2(b)). Definitions of the mean-field critical temperature T c and T ( max ) c are shownin Fig.2. Main characteristics of samples are sum-marized in Table 1.
40 50 60 70 80-4-3-2-1070 80 90 100 110 120 130-4-3-2-10 T c = 86 K ( max ) T (K) M ( - e m u ) ZFC H = Oe M ( - e m u ) T c = 61.2 K T (K) (a)(b) sample Y Fig. 2. Magnetization M ( T ) data for Y H = 10 Oe after zero-field-cooling(ZFC). (a) The main part of the transition. The solid lineis a linear fit to the steepest part of the M ( T ) curve. (b)A tale of the transition with y -scale expanded by threeorders of magnitude. The solid line is a linear fit to M ( T )data at T >
90 K. Definitions of T c and T maxc are shownin Figs. 2(a) and 2(b), respectively.Table 1Main parameters of samples. The effective magnetic field h f − p characterizes a relative strength of a ferromagneticcontribution to the sample magnetization (see Eq. (4) forthe definition)sample compound T c T maxc h f − p T onset (K)Y . ± . . ± . . ± . ± . ±
3. Scaling procedure
We consider two modifications of the scaling pro-cedure. A simpler version is applicable to sampleswith purely paramagnetic normal-state magneti-2ation. If the normal-state magnetization includesalso a ferromagnetic contribution, the procedureshould be modified in oder to account for a non-linearity of M ( H ) curves.3.1. Purely paramagnetic case
Because paramagnets may be described by amagnetic field independent magnetic susceptibility χ p , the sample magnetization M = χ p H , i.e., M isproportional to H at any temperature. This meansthat M n ( T ) curves collected in different fields willcollapse onto a single M sc ( T ) curve if M sc ( H ) = M ( H, T ) M ( H, T ) , (2)where T is any temperature well above T c .
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50 kOe5 kOe1 kOe0.3 kOe T (K) M sc = M ( T ) / M ( K ) sample Tl Fig. 3. Magnetization data for Tl M sc = M ( T ) /M (150 K)versus T .Fig. 3 illustrates results of such a scaling. Agree-ment between the data is practically perfect. Whilemagnetic fields differ by more than two orders ofmagnitude, the differences between the M sc ( T )curves do not exceed an experimental scatter.The scaled magnetization curves for lower tem-peratures are shown in Fig. 4. It may be seen thatthe onset temperature T onset ≈
131 K. There issome experimental uncertainty, but there are nosystematic errors, which may distort the conclu-sions.
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50 kOe30 kOe10 kOe1 kOe M sc = M ( T ) / M ( K ) T (K) sample Tl Tonset = (131 ± ± Fig. 4. The scaled magnetization data for the Tl .3.2.
Paramagnetism with a ferromagneticcontribution -1 0 1 2 3-0.20.00.20.40.6 after +4 kOeafter -4 kOe H (kOe) M ( - e m u ) T = K sample Y Fig. 5. Magnetization data for the sample Y T = 150 K in fields − < H < < H <
50 kOe. .Quite often, the normal-state magnetization ofhigh- T c superconductors cannot be described aspurely paramagnetic. A typical example is shownin Fig. 5. There is a quite symmetrical hysteresislope with non-zero magnetizations at H = 0, whichis an unambiguous signature of ferromagnetism.A simple scaling procedure, which was intro-duced in the previous subsection does not work, asmay clearly be see in Fig. 6. Because, the samplemagnetization M ( H ) = χH cannot be describedwith χ independent of field, this failure is expected.3
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50 kOe20 kOe 7 kOe 3 kOe 2 kOe T (K) M sc = M ( T ) / M ( K ) sample Y Fig. 6. M ( T ) /M (110 K) measured in different fields asfunctions of temperature. . H (kOe) M ( - e m u ) T = K H M sample Y M f ( max ) χ p = dM / dH Fig. 7. M at T = 110 K versus field. The solid line isa linear fit to the data for H ≥ χ p and M ( max ) f .If there are several contributions to the samplemagnetization, M may be written as a sum. In ourcase, M = χ p H + M f , (3)where indexes p and f are related to paramagneticand ferromagnetic contributions, respectively. Fig.7 shows M ( H ) data for the same sample Y H ≥ M is a linear function of H . This allows toconclude that all ferromagnetic inclusions are al-ready saturated and in H ≥ M f is equalto its maximum value M ( max ) f . All calculations forthis sample that we present below were made for T = 110 K. χ p ( T ) = dM/dH ≈ . · − emu/kOe and M ( max ) f ( T ) ≈ . · − . In lowerfields, M f depends not just on field, but also onthe magnetic history (see Fig. 4). However, M f ( H )can easily be calculated by employing of Eq. (3).In order to characterize a relative strength of aferromagnetic contribution, we introduce h f − p = M ( max ) f χ p . (4)The value of this effective field is a convenient wayto characterize ferromagnetic contributions to M .The values of h f − p for the investigated samples arepresented in Table 1.As the next step, we assume that M f ( H ) istemperature independent for all H values. Thisis the only realistic way to scale the normal-statemagnetization data without making rather spe-cific assumptions about sample properties, whichmay distort the final result. Although this cannotbe exact, we shall show that in some limited tem-perature range above T c this assumption workssufficiently well to ensure satisfactory scaling of ex-perimental data. Thus, using Eq (3) and the valueof χ p ( T ), we calculate M f ( H, T ) = M − χ p H .Then, experimental M ( T ) data for different fieldsare scaled according to M sc ( T ) = M ( H, T ) − M f ( H, T ) χ p ( T ) H (5)The results of such scaling with T = 110 K arepresented in Fig. 8(a). As may be seen, M sc ( T )curves calculated from experimental M ( T ) datameasured in different fields perfectly match eachother in a rather extended temperature range T > T c . The onset of diamagnetism is shown inFig. 7(b). It can easily be established that 92 K 5) K. For this sample, h f − p = 0 . 14 kOe,i.e., a relative ferromagnetic contribution is ap-proximately 6 times smaller than that for the sam-ple Y h f = p , but rathertemperature independence of M f ( H ), which is the4 00 110 120 130 140 47 kOe20 kOe 7 kOe 2 kOe0.5 kOe T (K) M sc 90 95 100 1050.800.850.900.951.001.051.10 T (K) M sc (a)(b) sample Y T = K T onset = (92.3 ± K Fig. 8. Magnetization data for the sample Y T > T c . (b)In the transition region. The solid lines are guides to theeye. .basis of our scaling approach. For instance, for thesample Tl h f − p = 3 kOe, i.e., considerablyhigher than for all other samples presented here.Nevertheless, as may be seen in Fig. 10, M sc ( T )curves calculated from data measured in differentfields perfectly match each other in a rather ex-tended ranges of temperatures and fields.There were, however, some cases, in which thequality of scaling was not as good as presentedabove. The results for such a sample are shown inFig. 11. While formal characteristics of this sample,including the value of h f − p are quite similar to thatof the sample Y H ≤ M f ( H ) on temperature. We remind that the tem-perature independence of M f ( H ) is the basis of 100 120 140 160 1800.951.001.051.10 50 kOe20 kOe 5 kOe1 kOe0.2 kOe 70 80 90 100 110 1201.001.051.101.15 50 kOe35 kOe20 kOe10 kOe5 kOe0.5 kOe T (K) M sc T (K) M sc (a) (b) sample Y T = K T onset = (88.5 ± K Fig. 9. M sc for the sample Y T > T c . (b) In the transition region. The solid linesare guides to the eye . 110 120 130 140 150 160 170 180 1900.70.80.91.01.11.2 50 kOe15 kOe 5 kOe2 kOe0,5 kOe T (K) M sc sample Tl T = K T onset = (125 ± K Fig. 10. M sc versus T for the sample Tl .this scaling procedure. At the same time, the datafor H ≥ 10 kOe can be scaled quite well (see Fig.11). This is evidence that, while M f depends ontemperature in lower fields, the saturated value ofthe ferromagnetic contribution to the sample mag-netization M ( max ) f is practically temperature inde-pendent. Indeed, as may be seen in Fig. 12, M f ,calculated according to Eq. (3), is independent offield down to H = 10 kOe, while the data point for5 00 110 120 130 140 150 1600.850.900.951.001.05 50 kOe20 kOe10 kOe 5 kOe 2 kOe 1 kOe 90 95 100 1050.961.001.04 50 kOe35 kOe20 kOe10 kOe2 kOe T (K) M sc M sc T (K) (a)(b) sample Y T = K T onset = (92.5 ± K Fig. 11. M sc versus T for the sample Y T > T c .(b) In the transition region. The solid lines are guides tothe eye. . H (kOe) M f ( - e m u ) sample Y T = K M f ( max ) Fig. 12. M f = M − χ p H at T = 110 K as a function of H .The dashed line corresponds to M maxf = 1 . · − emu. . H = 5 kOe is already substantially below M ( max ) f .3.3. Analysis of M sc ( H, T ) data below T onset Because the divergence between the M sc ( T )curves manifests the onset of superconductingdiamagnetism, the difference∆ M sc = M sc ( H ) − M sc (50 kOe ) (6) may serve as some kind of its quantitative charac-teristic. 110 120 130 140 150-0.04-0.020.00 ˘ M sc χ p H ( - e m u ) T (K)sample Tl T = 135 K0.5 kOe 1 kOe 3 kOe10 kOe75 80 85 90 95 100-0.04-0.020.00 ∆ M sc χ p H ( - e m u ) T (K)sample Y T = 125 K (a)(b) Fig. 13. ∆ M sc χ p H versus T . (a) For the sample Y .In order to compare ∆ M sc results for differentvalues of H , it is convenient to use ∆ M sc χ p H (seeEq. (5)). Fig. 13 shows the ∆ M sc χ p H curves forthe sample Y T onset resulting from this representation of the resultsare the same as may be determined from Figs. 10and 11. In this case, the main source of uncer-tainty in T onset is the distance between the neigh-boring data points. We also note a rather dras-tic difference in ∆ M sc ( H ) χ p H dependencies below T onset for these two samples. For the sample Y M sc ( H ) χ p H is an increasing function of H (Fig.13(a)). Contrary to that, ∆ M sc ( H ) χ p H decreaseswith increasing H . Quite likely, this is one of man-ifestations of general differences between Y- andTl-based cuprates. 4. Conclusions The proposed approach allows to scale thenormal-state magnetization M n ( T ) data in a way6hat the M sc ( T ) curves calculated from experi-mental M n ( T ) data collected in different magneticfields collapse onto the same master curve. Be-cause a diamagnetic contribution to M due tosuperconductivity depends on magnetic field ina way, which is quite different from that for thenormal-state magnetization, the onset of super-conductivity leads to a rather pronounced diver-gence between the M sc ( T ) curves correspondingto different magnetic fields, as it may clearly beseen in Figs. 4, 8, 9, 10, 11(b) and 13. This al-lows for an unambiguous determination of T onset corresponding to the onset of superconductingdiamagnetism. Accuracy of T onset is determinedby an experimental scatter of M ( T ) data and bydistances between neighboring M ( T ) data points.We remind that the proposed approach relies onlyon rather general properties of the normal-statemagnetization and does not include any specificassumptions, which cannot be independently ver-ified. We also note that the possibility to checkthe quality of scaling at temperatures well above T oncet serves as some consistency check. If thescaling procedure does not work satisfactory, theresulting data should not be used for importantconclusions.There are two main reasons to have quite consid-erable values of ( T c − T onset ). (i) Thermal fluctu-ations, which are expected to be especially strongin high- T c superconductors and (ii) inclusions ofsmall quantities of phases with higher values of T c .Here, we mainly consider technical aspects of scal-ing of the normal-state magnetization data anddiscussion of a possible nature of the transition at T = T onset in different samples is beyond the scopeof this paper. Acknowledgements This work was supported by the Swiss NCCRMaNEP II under project 4, novel materials. References [1] J. Mosqueira, E. G. Miramontes, C. Torr´on, J. A.Camp´a, I. Rasines, and F. Vidal, Phys. Rev. B , 15272 (1996).[2] M.J. Naughton, Phys. Rev. B , 1605 (2000).[3] A. Lascialfari, A. Rigamont1, L. Romano, P. Tedesco, A.Varlamov, and D. Embriaco, Phys. Rev. B , 144523(2002).[4] I. M. Sutjahja, A. 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