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A novel technique to measure magnetisation hysteresis curves in the peak-effect regime of superconductors
Physica C 298 Ž1998. 122–132
A novel technique to measure magnetisation hysteresis curves in the peak-effect regime of superconductors G. Ravikumar a,) , T.V. Chandrasekhar Rao a , P.K. Mishra a , V.C. Sahni a , S.S. Banerjee b, A.K. Grover b, S. Ramakrishnan b, S. Bhattacharya b,c , M.J. Higgins c , E. Yamamoto d , Y. Haga d , M. Hedo e, Y. Inada e, Y. Onuki e a
Technical Physics and Prototype Engineering DiÕision, Bhabha Atomic Research Centre, Mumbai 400 085, India b Tata Institute of Fundamental Research, Mumbai 400 005, India c NEC Research Institute, Princeton, NJ 08540, USA d AdÕanced Science Research Centre, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-11, Japan e Faculty of Science, Osaka UniÕersity, Toyonaka 560, Japan Received 5 December 1997; accepted 22 December 1997
Abstract Magnetisation hysteresis of type II superconductors measured using SQUID-based Magnetic Property Measurement System ŽMPMS. made by Quantum Design is known to be affected by inhomogeneity in the external field. We present a new method, viz., half-scan technique, for measuring magnetisation hysteresis loops using MPMS. This technique circumvents the problems caused by field inhomogeneity. We have used this technique to measure magnetisation hysteresis loops in the peak-effect regime of superconducting CeRu 2 and 2H–NbSe 2 . The magnetisation hysteresis obtained using half-scan technique is found to be significantly larger than that obtained by conventional MPMS measurements. At very low fields where the extent of field inhomogeneity is not significant and in the reversible region, the results of the half-scan technique are comparable to those obtained using conventional technique. Moreover, hysteresis measurements are shown to be independent of scan length used in the measurement, thus reaffirming that the results are unaffected by the field inhomogeneity. q 1998 Elsevier Science B.V. Keywords: Magnetisation hysteresis; Type II superconductors; Peak effect
1. Introduction In recent years, peak effect in type II superconductors, manifesting in the mixed state as a sharp
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Corresponding author.
peak in the critical current density Jc vs. field H Žor temperature T ., has received renewed attention w1–5x. This effect has been observed in a wide variety of materials ranging from conventional superconductors, such as 2H–NbSe 2 and CeRu 2 w1–9x, to the high Tc cuprate superconductors w10,11x. Part of the interest in the phenomenon is due to a recent suggestion that the peak effect is either a direct manifesta-
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tion of the flux lattice melting or closely related to it w1–6,11x. The flux lattice melting in high Tc superconductors is supposed to occur well below the upper critical field and indeed the peak effect in these systems is seen either at or near this transition. A substantial body of data on the peak effect in nearly all these classes of superconductors is obtained by dc magnetisation hysteresis measurements, particularly utilising the SQUID-based Magnetic Property Measurement System ŽMPMS. by Quantum Design. The peak in Jc vs. H manifests as an enhanced hysteresis in a dc magnetisation experiment. The MPMS employs sample movement over a distance Žscan length. of few centimeters across a pickup coil array in the second-derivative configuration. But we know that the magnetic moment of a superconductor itself varies significantly when moved in an inhomogeneous field. We have recently demonstrated in the peak-effect regime of superconducting CeRu 2 , that magnetisation hysteresis loop delivered by the MPMS depends crucially on the scan length used in the measurement, indicating that the field inhomogeneity interferes with the measurement process w12x. Thus the inhomogeneity in the external field imposes a limit on the capability of the magnetometer to resolve magnetisation hysteresis below a certain value. In other words, high sensitivity of the SQUID is partly sacrificed due to the field inhomogeneity. Therefore, it is necessary to devise an alternative approach which takes into account the actual measurement conditions and then model the magnetic behaviour of the sample under the particular circumstances experienced by the sample during the scan. In this paper, we present a new method to obtain magnetisation hysteresis loops of superconductors which are free of the artefacts which arise because of the sample traversing through an inhomogeneous magnetic field. In Section 2 we shall give a brief background to the analysis described in this paper. In Section 3, we present a new technique to record SQUID responses for measurement of hysteresis loops and Section 4 contains our experimental results based on this new technique along with the data obtained using the conventional technique. The experimental results we report were carried out on single crystals of 2H– NbSe 2 and CeRu 2 . Section 5 is devoted to a discussion and summary of the paper.
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2. Background 2.1. SQUID response In MPMS, a magnetic moment measurement basically involves studying the flux linked with the ‘second derivative’ pickup coil array Žsee Fig. 1. for different sample positions. The net flux per unit magnetic moment, emanating from the sample and intercepted by the four turns of the coil array, is given by
f Ž z . s Ž m 0 R 2r2 . y R 2 q Ž z q Z . q2 w R 2 q z 2 x
y3r2
2 y3 r2
y R2 q Ž z y Z .
2 y3 r2
Ž 1. y7
where m 0 s 4p = 10 WrA m, R Žs 0.97 cm. is the radius of the pickup coil, 2 Z Žs 3.038 cm. is the distance between the two outer turns of the pickup coil array and z is the sample distance from the centre of the pickup coil array. This would imply that the SQUID response of a unit dipole should be given by V Ž z . s cf Ž z .
Ž 2.
where c is calibration factor of the instrument. However, experimentally one observes a constant offset a and a linear drift bz in the SQUID response along the scan length. MPMS software allows for these and also provides a facility to compensate for a possible off-centring of the sample. Taking these factors into
Fig. 1. ‘Second derivative’ pickup coil array used in MPMS by Quantum Design and the SQUID response of a point dipole.
G. RaÕikumar et al.r Physica C 298 (1998) 122–132
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consideration the SQUID response for a magnetic moment m can be written as V mod Ž z . s a q bz q mV Ž z y z 0 .
Ž 3.
where z 0 is the correction for off-centring, which can be termed as centre-shift of the sample. In the so-called iteratiÕe regression algorithm, the measured SQUID response of a sample is fitted to Eq. Ž3., to obtain the magnetic moment m. On the other hand z 0 is assumed to be zero in the linear regression algorithm. 2.2. Effect of field inhomogeneity on a conÕentional SQUID measurement The above analysis presumes that the magnetic moment induced in the sample is constant along the scan length. However, this is not the case, because inhomogeneity in the applied magnetic field can cause strong variations in the induced moment. In a technical advisory w13x, Quantum Design provides the field profile of the solenoid employed in the 5.5 Tesla MPMS system Žschematically shown in Fig. 2a., which is of the form X
H Ž z . rH s 1 y a z 4 X
Ž 4.
where H Ž z . is the actual field experienced by a sample along the scan length. H is the field at z s 0 or H X Ž z s 0. s H and a Ž, 1.25 = 10y4 cmy4 . is a measure of the field inhomogeneity. In Ref. w12x, we used this field profile to model the effect of varying scan length on the magnetisation hysteresis loops measured in the peak-effect regime of CeRu 2 . Generally, in a scan of length 2 l, SQUID response is measured at a number of sample positions Žtypically 40. between z s yl and z s l. Eq. Ž4. reveals that the sample encounters an increasing field for yl - z - 0 ŽA to C. and a decreasing field for 0 - z - l ŽC to E. as shown in Fig. 2a. This field excursion causes the magnetic moment of a superconducting sample to vary along the scan path and the manner in which it varies will depend on the magnetic history seen by the superconductor before commencing the scan. For the initial history of the sample, we consider two cases, viz., forward and reverse legs of the hysteresis loop. In the forward Žreverse. case, prior to starting the scan, the sample is subjected to only increasing Ždecreasing. field till
Fig. 2. Schematic plot showing Ža. field distribution along the scan length, Žb. the magnetisation as a function of field Žleft part. at different points along the scan length and also as a function of z Žright part.. Lower part of the figure is for the forward case while upper part corresponds to the reverse case. Points A, C and E denote the position of the sample z sy l, 0 and q l, respectively.
it attains a value H Ž1 y a l 4 .. At this moment, magnetic field at C Žcentre of the magnet. is equal to H. Using Critical State Model w14x, we argue that in the forward case, magnetic moment of the superconductor is unaffected in the first half of the scan, i.e., for yl - z - 0 ŽA to C. as shown in the lower part of Fig. 2b. However, magnetic moment of the sample changes sharply for 0 - z - l ŽC to E.. On the other hand, in the reverse case magnetisation change is sharp, both between A and C and between C and E as shown in the upper part of Fig. 2b. From the above discussion it is clear that, the basic assumption of m being constant Žin Eq. Ž3.. along the scan length, breaks down in the irreversible region of superconductors. Serious errors creep in, if the magnetisation hysteresis being measured is comparable to the magnitude of field inhomogeneity.
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One of the situations, other than the peak effect, where such errors occur is in the measurement of irreversibility field w15x. A way out of this difficulty is discussed in the Section 3.
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magnetisation hysteresis is rather small, which could otherwise be camouflaged by the field inhomogeneity. 4. Experimental results
3. The half-scan technique 3.1. Forward magnetisation curÕe From the arguments in Section 2.2, we know that in a conventional SQUID response measurement over a scan length 2 l on the forward magnetisation curve, magnetic moment of a superconductor remains essentially unchanged only for the sample motion from A to C i.e., for yl - z - 0. On the other hand, between C and E Ž0 - z - l . magnetic moment of a superconductor varies strongly with the additional complication of possible higher-order multipole moments w16,17x. Therefore, on the forward magnetisation curve, from the so-called all-points scan Žraw SQUID response., we should consider only the raw data between A and C. The magnetic moment is then obtained by using Eq. Ž3. to fit the SQUID response between A and C only. 3.2. ReÕerse magnetisation curÕe On the reverse magnetisation curve, we follow the same strategy as was prescribed in Section 3.1, namely, use that half-scan length over which induced moment remains essentially unchanged. On the reverse magnetisation curve, this is accomplished by adjusting the position of the sample initially at C, i.e., z s 0 before subjecting it to a decreasing external field. The SQUID response is then recorded by moving the sample from C to E Ži.e., from z s 0 to z s l . in which, it experiences a further small decrease in external magnetic field. Since in such a scan the sample traverses through a further field decrease, there is little change in the induced moment. The magnetic moment is then obtained by fitting this response to Eq. Ž3.. From the above arguments, it follows that SQUID response at a given field can be recorded only once in half-scan measurement without upsetting the field distribution in the sample. In short, the half-scan technique is guided by the principles of Critical State Model w14x and allows accurate magnetic moment measurement of superconductors in the regime where
We will now demonstrate the use of the method prescribed in Section 3 by actual examples. We have investigated the magnetic behaviour in the peak-effect regime for two different superconducting systems, viz., single crystals of Ža. Cubic Laves phase CeRu 2 ŽTc s 6.3 K. and Žb. Hexagonal 2H–NbSe 2 ŽTc s 6.1 K.. The CeRu 2 single crystal is the same as that used in Ref. w12x and was mounted on the sample holder with field parallel to the cube edge. 2H–NbSe 2 single crystal is in the form of a thin platelet Ž2 = 2 = 0.4 mm3 . and was mounted with the c-axis parallel to the applied field. 4.1. CeRu 2 Field-dependent magnetisation hysteresis measurements were performed using the conventional technique Žas prescribed in MPMS manual. as well as the half-scan technique. Conventional scans are of 4 cm length Žy2 cm - z - 2 cm. with 40 data points and half-scan measurements are of scan length 2 cm Ži.e., y2 cm - z - 0 cm in the forward case and 0 cm - z - 2 cm in the reverse case. with 20 data points. The sample was centred in a field of 400 Oe on the forward magnetisation curve at 4.5 K. The conventional SQUID response Žy2 cm - z - 2 cm. and the corresponding fits to Eq. Ž3., compensated for the offset a and the drift bz ŽEq. Ž3.., is plotted in Fig. 3a at different fields on the forward magnetisation curve recorded at 4.5 K. In Fig. 3b, we show the half-scan SQUID response and the corresponding fits to Eq. Ž3. Žafter subtracting a q bz . for the sample position between z s y2 cm and z s 0 cm, which is appropriate for obtaining the magnetic moment on the forward magnetisation curve. Even though, at a given field on the forward magnetisation curve, SQUID responses measured by either of the techniques, seem to fit the dipolar response, they amount to very different values of magnetic moment. On the reverse magnetisation curve, following the prescription of Section 3.2, the sample was initially positioned at z s 0 ŽC. before it
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Fig. 3. Ža. Conventional SQUID responses Žscan length of 4 cm. of CeRu 2 crystal and the corresponding fits to Eq. Ž3. at different fields on the forward magnetisation curve measured at 4.5 K. Žb. Half-scan SQUID responses for y2 cm - z - 0 at different fields on the forward magnetisation curve of CeRu 2 ŽT s 4.5 K. and the corresponding fits to Eq. Ž3.. Žc. Half-scan SQUID responses for 0 - z - 2 cm at different fields on the reverse magnetisation curve of CeRu 2 ŽT s 4.5 K. and the corresponding fits to Eq. Ž3..
was subjected to a decreasing external magnetic field. The SQUID response was then recorded between z s 0 ŽC. and z s 2 cm ŽE.. The raw data and the corresponding fits to Eq. Ž3. Žafter subtracting a q bz ., are plotted in Fig. 3c. Each SQUID response was measured with a wait time of 500 s after the field stabilised. Similar data were also recorded at 5 K. The hysteresis loops at 4.5 K and 5 K in the peak-effect regime obtained using the half-scan technique are shown in Fig. 4a and b, respectively. For comparison, we also show the forward magnetisation curves obtained by conventional technique. The forward magnetisation values obtained using conventional measurement and half-scan technique are significantly different in the peak-effect region. However, in the reversible region, both techniques give identical results. We also observe a small but nonzero hysteresis below the peak-effect region with half-scan technique.
On both forward and reverse magnetisation curves, in the peak-effect regime, the SQUID response shows centre-shift which is measured by the parameter z 0 Žsee Section 2.1, Eq. Ž3... In the insets of Fig. 4a and b, we show the z 0 values on the forward and reverse magnetisation curves, respectively, as a function of field. We note that the centre-shift z 0 is more prominent on the forward magnetisation curve Žup to 4 mm. than on the reverse magnetisation curve. The value z 0 on the forward magnetisation curve measured by conventional technique also shows a marked peak in the peak-effect regime but somewhat less than that obtained in the half-scan technique. Outside the peak-effect region, z 0 values are identical in conventional and half-scan techniques. Almost similar features are also observed at 5 K, but with the magnitude of z 0 being much smaller than that at 4.5 K. The arguments in Sections 2 and 3, suggest that the hysteresis loops measured using half-scan tech-
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Fig. 4. Hysteresis curves at Ža. 4.5 K and Žb. 5 K of CeRu 2 obtained using half-scan technique Žsquares., Žcircles. forward magnetisation curve obtained by the conventional technique. Inset shows the corresponding values of centre-shift parameter z 0 .
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technique with scan length l s 1.5, 2 and 2.5 cm. It is remarkable that the magnetisation data for these three different scan lengths fall on the same envelope curve. As shown in Ref. w12x, hysteresis loops of CeRu 2 measured using conventional technique Žusing full symmetric scan. are qualitatively different for different scan lengths. 4.2. 2H–NbSe2
Fig. 5. Hysteresis curves measured at 4.5 K, on CeRu 2 single crystal using half-scan technique with different scan lengths as indicated.
nique should be independent of the scan length used in the measurement. In Fig. 5, we show a comparison of the hysteresis loops measured using half-scan
On this sample, we carried out magnetisation measurements at 4.5 K and 5.1 K, using SQUID MPMS both by conventional Žwith a scan length of 2 cm. as well as by the half-scan techniques Žwith 1 cm scan length.. At 4.5 K, the SQUID responses measured by conventional Žfull scan. technique and the corresponding fits to Eq. Ž3. Žiterative regression. for different fields are plotted in Fig. 6a after compensating for the offset a and drift bz. We note that, in the peak-effect region the centre of the SQUID response measured using the conventional symmetric scan
Fig. 6. Ža. Conventional SQUID responses Žy1 cm - z - 1 cm. recorded on 2H–NbSe 2 crystal at 4.5 K on the forward magnetisation curve at different fields. Continuous line shows the fit obtained using iterative regression algorithm ŽEq. Ž3... Žb. Half-scan SQUID responses Žy1 cm - z - 0. and the corresponding fits to Eq. Ž3. Žusing linear regression algorithm. at different fields on the forward magnetisation curve of 2H–NbSe 2 crystal at 4.5 K. Žc. Half-scan SQUID responses Ž0 - z - 1 cm. and the corresponding fits to Eq. Ž3. Žusing linear regression algorithm. at different fields on the reverse magnetisation curve of 2H–NbSe 2 crystal at 4.5 K.
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technique is considerably shifted from z s 0. However SQUID responses measured using the half-scan method for both forward and reverse cases show negligible centre-shift as shown in Fig. 6b and c, which is in contrast to the case in CeRu 2 . The fact that z 0 values are nearly equal to zero further enables us to use the linear regression algorithm to obtain the magnetic moments. In Fig. 7a, we plot the magnetisation hysteresis loops measured at 4.5 K, both by conventional and half-scan methods. In the peak-effect region, the hysteresis measured using half-scan technique is significantly larger than that measured using conventional technique. Moreover, use of half-scan technique results in a nonzero hysteresis below the onset of peak effect. The hysteresis loop measured using half-scan technique and the conventionally measured forward magnetisation curve at 5.1 K are presented in Fig. 7b. In the inset of Fig. 7b, we show the virgin magnetisation curve in the low-field region where the magnitude of field inhomogeneity is relatively insignificant compared to the magnetisation hysteresis. Therefore magnetisation curves obtained using the conventional technique and the half-scan method are comparable.
5. Discussion and summary of the results A new method, viz., half-scan technique, is suggested for measuring magnetisation hysteresis loops of superconductors using MPMS by Quantum Design. This is demonstrated in the peak-effect regime of superconducting CeRu 2 and 2H–NbSe 2 single crystals. In a conventional measurement, a sample moved along the scan length symmetrically about z s 0, experiences a non-monotonic field change, causing a strong variation in the magnetisation of the Žirreversible. superconducting sample. In the halfscan technique, scans are chosen in such a way that the sample experiences field change in the same sense as that before the scan begins. This ensures that the magnetisation of the superconductor remains on the envelope hysteresis loop defined by forward and reverse magnetisation curves. The half-scan technique suggested in this paper is independent of the exact functional form of the magnetic field profile assumed in Eq. Ž4.. It is
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nowhere used explicitly in the analysis leading to our results. The arguments presented in this paper are applicable for any field profile which monotonically decreases in magnitude on either side of z s 0. This method needs to be suitably amended depending on the exact nature of the field inhomogeneity. The basic ideas of this paper can be equally applied to other techniques Žsuch as extraction magnetometry. where sample motion over a distance of few centimeters is involved. The efficacy of the half-scan technique is clearly demonstrated in Fig. 7a, where a conventional hysteresis measurement using a 2 cm scan length Žin which field inhomogeneity is expected to be very small. gives much smaller hysteresis than that obtained using half-scan technique. Remarkably, below the peak-effect regime, the half-scan technique brings out the magnetisation hysteresis which is otherwise not observed in conventional SQUID measurements Žsee Fig. 7a and b.. This hysteresis signifies finite critical current density below the peak regime which is observed both by transport w1–5x and AC susceptibility measurements w6x. However, in the case of CeRu 2 sample, the hysteresis for fields below the onset of peak effect is extremely small. This is again in agreement with the hysteresis measurements on CeRu 2 using the newly developed Reciprocating Sample Option ŽRSO. available in MPMS where no hysteresis was observed below the peak-effect regime even for scan lengths as short as 1 mm w9x. The hysteresis before the onset of peak effect was also observed in a magnetisation measurement using Faraday method based on capacitance technique ŽSakaki Bara et al., unpublished.. This technique does not involve sample motion. For fields well beyond the peak-effect regime, where superconductors are reversible, the half-scan technique produces results identical to that of conventional measurements. On the forward magnetisation curve of CeRu 2 , SQUID responses for different fields as shown in Fig. 3a, look almost similar to an ideal dipolar signal even though the origin of the signal is completely spurious. Computer fits to these signals can return what appear to be reasonable values of the magnetic moment and even give regression values close to unity. One needs to be careful when interpreting results of this kind since such results are sometimes
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Fig. 7. Ža. Magnetisation hysteresis loops of 2H–NbSe 2 crystal measured at 4.5 K using conventional Žcircles. and half-scan Žsquares. techniques. Žb. ŽSquares. Magnetisation hysteresis loop of 2H–NbSe 2 crystal measured at 5.1 K using half-scan technique and Žcircles. forward magnetisation curve measured at 5.1 K using conventional SQUID measurement. The inset shows the virgin magnetisation curve in the low-field regime obtained using conventional and half-scan techniques.
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Fig. 8. ŽCircles. Conventional SQUID response at 20.5 kOe on the forward magnetisation curve of CeRu 2 . Least-squares fit ŽEq. Ž3.. of the SQUID response for y2 cm - z - 0, extrapolated up to z s 2 cm is indicated by the continuous line.
not dipole signals at all Žsee Ref. w18x.. For example we consider the conventional symmetric SQUID response for y2 cm - z - 2 cm at 20.5 kOe measured at 4.5 K on the forward magnetisation curve of CeRu 2 sample Žsee Fig. 3a.. It appears to fit the dipolar response quite well. However, when a q bz corresponding to the half-scan response between z s y2 cm and z s 0 is extrapolated up to z s 2 cm and then subtracted from the raw data from z s y2 cm to z s 2 cm, we obtain a signal which is representative of the true magnetic behaviour of the superconductor between z s y2 cm and z s 2 cm Žsee Fig. 8.. The continuous line in Fig. 8 is the response of a constant dipole moment. The deviation of the raw data from the constant dipole signal Žfor z ) 0. is primarily due to a strong variation of magnetic dipole moment induced in the superconductor by the inhomogeneous field. However this is not sufficient to understand the zero of the sample response Žopen circles. in the region 0 - z - 2 cm, not coinciding with the zero of the dipolar response Žcontinuous line.. This indicates that, for 0 - z - 2 cm, the SQUID response would have contributions from higher-order multipole moments w16,17x besides a position-dependent dipole contribution. The half-scan technique is equally applicable for accurate measurement of irreversibility line in various superconductors. It was recognized by Perry and
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Campbell w15x that the magnetisation hysteresis in various high Tc superconductors measured using MPMS, is often underestimated because of the field inhomogeneity. This leads to grossly underestimating the irreversibility field in these materials. To extract the true irreversibility field, they devised a method in which sample is stationary while a small ac field ripple is applied along with the dc field. The data analysis associated with this technique to obtain magnetisation hysteresis loops is complicated. Halfscan technique can be directly applied to the measurement of irreversibility line in high Tc superconductors. Finally, it is essential that the results obtained by the new half-scan method presented in this paper should be compared with alternative measurement techniques such as the Reciprocating Sample Option ŽRSO. available in the new SQUID magnetometers by Quantum Design, where scan lengths are in the range of 1 mm. In conclusion, a new method, viz., half-scan technique is developed to measure magnetisation hysteresis loops using SQUID MPMS by Quantum Design. We demonstrated this technique by measuring magnetisation hysteresis loops in the peak-effect region of superconducting 2H–NbSe 2 and CeRu 2 single crystals. Magnetisation hysteresis in the peak-effect region obtained by the half-scan technique is found to be much larger than that measured by the conventional technique. Below the onset of peak effect, half-scan technique clearly brings out the hysteresis in magnetisation which is not observed in a conventional SQUID measurement. It was also demonstrated that in the field regime where the superconductor is reversible, both techniques give identical results. Again at very low fields where the extent of field inhomogeneity is rather small and the hysteresis very large, it was shown that the two techniques lead almost to the same results. The remarkable feature of the half-scan technique is that the hysteresis loop obtained is independent of scan length used in the measurement and therefore is unaffected by the field inhomogeneity. Note: It has been pointed out to us that a "centreshift" in a conventionally measured SQUID response in the peak effect region of CeRu 2 was also noticed by Roy et al. w19x, which was attributed by them to
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an inhomogeneous magnetic state. However, origin of the "centre-shift" discussed in this paper is due to inhomogeneous magnetic field of the solenoid.
Acknowledgements G.R.K. thanks K.V. Bhagwat for valuable suggestions.
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A novel technique to measure magnetisation hysteresis curves in the peak-effect regime of superconductors G. Ravikumar a,) , T.V. Chandrasekhar Rao a , P.K. Mishra a , V.C. Sahni a , S.S. Banerjee b, A.K. Grover b, S. Ramakrishnan b, S. Bhattacharya b,c , M.J. Higgins c , E. Yamamoto d , Y. Haga d , M. Hedo e, Y. Inada e, Y. Onuki e a
Technical Physics and Prototype Engineering DiÕision, Bhabha Atomic Research Centre, Mumbai 400 085, India b Tata Institute of Fundamental Research, Mumbai 400 005, India c NEC Research Institute, Princeton, NJ 08540, USA d AdÕanced Science Research Centre, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-11, Japan e Faculty of Science, Osaka UniÕersity, Toyonaka 560, Japan Received 5 December 1997; accepted 22 December 1997
Abstract Magnetisation hysteresis of type II superconductors measured using SQUID-based Magnetic Property Measurement System ŽMPMS. made by Quantum Design is known to be affected by inhomogeneity in the external field. We present a new method, viz., half-scan technique, for measuring magnetisation hysteresis loops using MPMS. This technique circumvents the problems caused by field inhomogeneity. We have used this technique to measure magnetisation hysteresis loops in the peak-effect regime of superconducting CeRu 2 and 2H–NbSe 2 . The magnetisation hysteresis obtained using half-scan technique is found to be significantly larger than that obtained by conventional MPMS measurements. At very low fields where the extent of field inhomogeneity is not significant and in the reversible region, the results of the half-scan technique are comparable to those obtained using conventional technique. Moreover, hysteresis measurements are shown to be independent of scan length used in the measurement, thus reaffirming that the results are unaffected by the field inhomogeneity. q 1998 Elsevier Science B.V. Keywords: Magnetisation hysteresis; Type II superconductors; Peak effect
1. Introduction In recent years, peak effect in type II superconductors, manifesting in the mixed state as a sharp
)
Corresponding author.
peak in the critical current density Jc vs. field H Žor temperature T ., has received renewed attention w1–5x. This effect has been observed in a wide variety of materials ranging from conventional superconductors, such as 2H–NbSe 2 and CeRu 2 w1–9x, to the high Tc cuprate superconductors w10,11x. Part of the interest in the phenomenon is due to a recent suggestion that the peak effect is either a direct manifesta-
0921-4534r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 0 1 8 - 5
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tion of the flux lattice melting or closely related to it w1–6,11x. The flux lattice melting in high Tc superconductors is supposed to occur well below the upper critical field and indeed the peak effect in these systems is seen either at or near this transition. A substantial body of data on the peak effect in nearly all these classes of superconductors is obtained by dc magnetisation hysteresis measurements, particularly utilising the SQUID-based Magnetic Property Measurement System ŽMPMS. by Quantum Design. The peak in Jc vs. H manifests as an enhanced hysteresis in a dc magnetisation experiment. The MPMS employs sample movement over a distance Žscan length. of few centimeters across a pickup coil array in the second-derivative configuration. But we know that the magnetic moment of a superconductor itself varies significantly when moved in an inhomogeneous field. We have recently demonstrated in the peak-effect regime of superconducting CeRu 2 , that magnetisation hysteresis loop delivered by the MPMS depends crucially on the scan length used in the measurement, indicating that the field inhomogeneity interferes with the measurement process w12x. Thus the inhomogeneity in the external field imposes a limit on the capability of the magnetometer to resolve magnetisation hysteresis below a certain value. In other words, high sensitivity of the SQUID is partly sacrificed due to the field inhomogeneity. Therefore, it is necessary to devise an alternative approach which takes into account the actual measurement conditions and then model the magnetic behaviour of the sample under the particular circumstances experienced by the sample during the scan. In this paper, we present a new method to obtain magnetisation hysteresis loops of superconductors which are free of the artefacts which arise because of the sample traversing through an inhomogeneous magnetic field. In Section 2 we shall give a brief background to the analysis described in this paper. In Section 3, we present a new technique to record SQUID responses for measurement of hysteresis loops and Section 4 contains our experimental results based on this new technique along with the data obtained using the conventional technique. The experimental results we report were carried out on single crystals of 2H– NbSe 2 and CeRu 2 . Section 5 is devoted to a discussion and summary of the paper.
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2. Background 2.1. SQUID response In MPMS, a magnetic moment measurement basically involves studying the flux linked with the ‘second derivative’ pickup coil array Žsee Fig. 1. for different sample positions. The net flux per unit magnetic moment, emanating from the sample and intercepted by the four turns of the coil array, is given by
f Ž z . s Ž m 0 R 2r2 . y R 2 q Ž z q Z . q2 w R 2 q z 2 x
y3r2
2 y3 r2
y R2 q Ž z y Z .
2 y3 r2
Ž 1. y7
where m 0 s 4p = 10 WrA m, R Žs 0.97 cm. is the radius of the pickup coil, 2 Z Žs 3.038 cm. is the distance between the two outer turns of the pickup coil array and z is the sample distance from the centre of the pickup coil array. This would imply that the SQUID response of a unit dipole should be given by V Ž z . s cf Ž z .
Ž 2.
where c is calibration factor of the instrument. However, experimentally one observes a constant offset a and a linear drift bz in the SQUID response along the scan length. MPMS software allows for these and also provides a facility to compensate for a possible off-centring of the sample. Taking these factors into
Fig. 1. ‘Second derivative’ pickup coil array used in MPMS by Quantum Design and the SQUID response of a point dipole.
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consideration the SQUID response for a magnetic moment m can be written as V mod Ž z . s a q bz q mV Ž z y z 0 .
Ž 3.
where z 0 is the correction for off-centring, which can be termed as centre-shift of the sample. In the so-called iteratiÕe regression algorithm, the measured SQUID response of a sample is fitted to Eq. Ž3., to obtain the magnetic moment m. On the other hand z 0 is assumed to be zero in the linear regression algorithm. 2.2. Effect of field inhomogeneity on a conÕentional SQUID measurement The above analysis presumes that the magnetic moment induced in the sample is constant along the scan length. However, this is not the case, because inhomogeneity in the applied magnetic field can cause strong variations in the induced moment. In a technical advisory w13x, Quantum Design provides the field profile of the solenoid employed in the 5.5 Tesla MPMS system Žschematically shown in Fig. 2a., which is of the form X
H Ž z . rH s 1 y a z 4 X
Ž 4.
where H Ž z . is the actual field experienced by a sample along the scan length. H is the field at z s 0 or H X Ž z s 0. s H and a Ž, 1.25 = 10y4 cmy4 . is a measure of the field inhomogeneity. In Ref. w12x, we used this field profile to model the effect of varying scan length on the magnetisation hysteresis loops measured in the peak-effect regime of CeRu 2 . Generally, in a scan of length 2 l, SQUID response is measured at a number of sample positions Žtypically 40. between z s yl and z s l. Eq. Ž4. reveals that the sample encounters an increasing field for yl - z - 0 ŽA to C. and a decreasing field for 0 - z - l ŽC to E. as shown in Fig. 2a. This field excursion causes the magnetic moment of a superconducting sample to vary along the scan path and the manner in which it varies will depend on the magnetic history seen by the superconductor before commencing the scan. For the initial history of the sample, we consider two cases, viz., forward and reverse legs of the hysteresis loop. In the forward Žreverse. case, prior to starting the scan, the sample is subjected to only increasing Ždecreasing. field till
Fig. 2. Schematic plot showing Ža. field distribution along the scan length, Žb. the magnetisation as a function of field Žleft part. at different points along the scan length and also as a function of z Žright part.. Lower part of the figure is for the forward case while upper part corresponds to the reverse case. Points A, C and E denote the position of the sample z sy l, 0 and q l, respectively.
it attains a value H Ž1 y a l 4 .. At this moment, magnetic field at C Žcentre of the magnet. is equal to H. Using Critical State Model w14x, we argue that in the forward case, magnetic moment of the superconductor is unaffected in the first half of the scan, i.e., for yl - z - 0 ŽA to C. as shown in the lower part of Fig. 2b. However, magnetic moment of the sample changes sharply for 0 - z - l ŽC to E.. On the other hand, in the reverse case magnetisation change is sharp, both between A and C and between C and E as shown in the upper part of Fig. 2b. From the above discussion it is clear that, the basic assumption of m being constant Žin Eq. Ž3.. along the scan length, breaks down in the irreversible region of superconductors. Serious errors creep in, if the magnetisation hysteresis being measured is comparable to the magnitude of field inhomogeneity.
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One of the situations, other than the peak effect, where such errors occur is in the measurement of irreversibility field w15x. A way out of this difficulty is discussed in the Section 3.
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magnetisation hysteresis is rather small, which could otherwise be camouflaged by the field inhomogeneity. 4. Experimental results
3. The half-scan technique 3.1. Forward magnetisation curÕe From the arguments in Section 2.2, we know that in a conventional SQUID response measurement over a scan length 2 l on the forward magnetisation curve, magnetic moment of a superconductor remains essentially unchanged only for the sample motion from A to C i.e., for yl - z - 0. On the other hand, between C and E Ž0 - z - l . magnetic moment of a superconductor varies strongly with the additional complication of possible higher-order multipole moments w16,17x. Therefore, on the forward magnetisation curve, from the so-called all-points scan Žraw SQUID response., we should consider only the raw data between A and C. The magnetic moment is then obtained by using Eq. Ž3. to fit the SQUID response between A and C only. 3.2. ReÕerse magnetisation curÕe On the reverse magnetisation curve, we follow the same strategy as was prescribed in Section 3.1, namely, use that half-scan length over which induced moment remains essentially unchanged. On the reverse magnetisation curve, this is accomplished by adjusting the position of the sample initially at C, i.e., z s 0 before subjecting it to a decreasing external field. The SQUID response is then recorded by moving the sample from C to E Ži.e., from z s 0 to z s l . in which, it experiences a further small decrease in external magnetic field. Since in such a scan the sample traverses through a further field decrease, there is little change in the induced moment. The magnetic moment is then obtained by fitting this response to Eq. Ž3.. From the above arguments, it follows that SQUID response at a given field can be recorded only once in half-scan measurement without upsetting the field distribution in the sample. In short, the half-scan technique is guided by the principles of Critical State Model w14x and allows accurate magnetic moment measurement of superconductors in the regime where
We will now demonstrate the use of the method prescribed in Section 3 by actual examples. We have investigated the magnetic behaviour in the peak-effect regime for two different superconducting systems, viz., single crystals of Ža. Cubic Laves phase CeRu 2 ŽTc s 6.3 K. and Žb. Hexagonal 2H–NbSe 2 ŽTc s 6.1 K.. The CeRu 2 single crystal is the same as that used in Ref. w12x and was mounted on the sample holder with field parallel to the cube edge. 2H–NbSe 2 single crystal is in the form of a thin platelet Ž2 = 2 = 0.4 mm3 . and was mounted with the c-axis parallel to the applied field. 4.1. CeRu 2 Field-dependent magnetisation hysteresis measurements were performed using the conventional technique Žas prescribed in MPMS manual. as well as the half-scan technique. Conventional scans are of 4 cm length Žy2 cm - z - 2 cm. with 40 data points and half-scan measurements are of scan length 2 cm Ži.e., y2 cm - z - 0 cm in the forward case and 0 cm - z - 2 cm in the reverse case. with 20 data points. The sample was centred in a field of 400 Oe on the forward magnetisation curve at 4.5 K. The conventional SQUID response Žy2 cm - z - 2 cm. and the corresponding fits to Eq. Ž3., compensated for the offset a and the drift bz ŽEq. Ž3.., is plotted in Fig. 3a at different fields on the forward magnetisation curve recorded at 4.5 K. In Fig. 3b, we show the half-scan SQUID response and the corresponding fits to Eq. Ž3. Žafter subtracting a q bz . for the sample position between z s y2 cm and z s 0 cm, which is appropriate for obtaining the magnetic moment on the forward magnetisation curve. Even though, at a given field on the forward magnetisation curve, SQUID responses measured by either of the techniques, seem to fit the dipolar response, they amount to very different values of magnetic moment. On the reverse magnetisation curve, following the prescription of Section 3.2, the sample was initially positioned at z s 0 ŽC. before it
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Fig. 3. Ža. Conventional SQUID responses Žscan length of 4 cm. of CeRu 2 crystal and the corresponding fits to Eq. Ž3. at different fields on the forward magnetisation curve measured at 4.5 K. Žb. Half-scan SQUID responses for y2 cm - z - 0 at different fields on the forward magnetisation curve of CeRu 2 ŽT s 4.5 K. and the corresponding fits to Eq. Ž3.. Žc. Half-scan SQUID responses for 0 - z - 2 cm at different fields on the reverse magnetisation curve of CeRu 2 ŽT s 4.5 K. and the corresponding fits to Eq. Ž3..
was subjected to a decreasing external magnetic field. The SQUID response was then recorded between z s 0 ŽC. and z s 2 cm ŽE.. The raw data and the corresponding fits to Eq. Ž3. Žafter subtracting a q bz ., are plotted in Fig. 3c. Each SQUID response was measured with a wait time of 500 s after the field stabilised. Similar data were also recorded at 5 K. The hysteresis loops at 4.5 K and 5 K in the peak-effect regime obtained using the half-scan technique are shown in Fig. 4a and b, respectively. For comparison, we also show the forward magnetisation curves obtained by conventional technique. The forward magnetisation values obtained using conventional measurement and half-scan technique are significantly different in the peak-effect region. However, in the reversible region, both techniques give identical results. We also observe a small but nonzero hysteresis below the peak-effect region with half-scan technique.
On both forward and reverse magnetisation curves, in the peak-effect regime, the SQUID response shows centre-shift which is measured by the parameter z 0 Žsee Section 2.1, Eq. Ž3... In the insets of Fig. 4a and b, we show the z 0 values on the forward and reverse magnetisation curves, respectively, as a function of field. We note that the centre-shift z 0 is more prominent on the forward magnetisation curve Žup to 4 mm. than on the reverse magnetisation curve. The value z 0 on the forward magnetisation curve measured by conventional technique also shows a marked peak in the peak-effect regime but somewhat less than that obtained in the half-scan technique. Outside the peak-effect region, z 0 values are identical in conventional and half-scan techniques. Almost similar features are also observed at 5 K, but with the magnitude of z 0 being much smaller than that at 4.5 K. The arguments in Sections 2 and 3, suggest that the hysteresis loops measured using half-scan tech-
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Fig. 4. Hysteresis curves at Ža. 4.5 K and Žb. 5 K of CeRu 2 obtained using half-scan technique Žsquares., Žcircles. forward magnetisation curve obtained by the conventional technique. Inset shows the corresponding values of centre-shift parameter z 0 .
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technique with scan length l s 1.5, 2 and 2.5 cm. It is remarkable that the magnetisation data for these three different scan lengths fall on the same envelope curve. As shown in Ref. w12x, hysteresis loops of CeRu 2 measured using conventional technique Žusing full symmetric scan. are qualitatively different for different scan lengths. 4.2. 2H–NbSe2
Fig. 5. Hysteresis curves measured at 4.5 K, on CeRu 2 single crystal using half-scan technique with different scan lengths as indicated.
nique should be independent of the scan length used in the measurement. In Fig. 5, we show a comparison of the hysteresis loops measured using half-scan
On this sample, we carried out magnetisation measurements at 4.5 K and 5.1 K, using SQUID MPMS both by conventional Žwith a scan length of 2 cm. as well as by the half-scan techniques Žwith 1 cm scan length.. At 4.5 K, the SQUID responses measured by conventional Žfull scan. technique and the corresponding fits to Eq. Ž3. Žiterative regression. for different fields are plotted in Fig. 6a after compensating for the offset a and drift bz. We note that, in the peak-effect region the centre of the SQUID response measured using the conventional symmetric scan
Fig. 6. Ža. Conventional SQUID responses Žy1 cm - z - 1 cm. recorded on 2H–NbSe 2 crystal at 4.5 K on the forward magnetisation curve at different fields. Continuous line shows the fit obtained using iterative regression algorithm ŽEq. Ž3... Žb. Half-scan SQUID responses Žy1 cm - z - 0. and the corresponding fits to Eq. Ž3. Žusing linear regression algorithm. at different fields on the forward magnetisation curve of 2H–NbSe 2 crystal at 4.5 K. Žc. Half-scan SQUID responses Ž0 - z - 1 cm. and the corresponding fits to Eq. Ž3. Žusing linear regression algorithm. at different fields on the reverse magnetisation curve of 2H–NbSe 2 crystal at 4.5 K.
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technique is considerably shifted from z s 0. However SQUID responses measured using the half-scan method for both forward and reverse cases show negligible centre-shift as shown in Fig. 6b and c, which is in contrast to the case in CeRu 2 . The fact that z 0 values are nearly equal to zero further enables us to use the linear regression algorithm to obtain the magnetic moments. In Fig. 7a, we plot the magnetisation hysteresis loops measured at 4.5 K, both by conventional and half-scan methods. In the peak-effect region, the hysteresis measured using half-scan technique is significantly larger than that measured using conventional technique. Moreover, use of half-scan technique results in a nonzero hysteresis below the onset of peak effect. The hysteresis loop measured using half-scan technique and the conventionally measured forward magnetisation curve at 5.1 K are presented in Fig. 7b. In the inset of Fig. 7b, we show the virgin magnetisation curve in the low-field region where the magnitude of field inhomogeneity is relatively insignificant compared to the magnetisation hysteresis. Therefore magnetisation curves obtained using the conventional technique and the half-scan method are comparable.
5. Discussion and summary of the results A new method, viz., half-scan technique, is suggested for measuring magnetisation hysteresis loops of superconductors using MPMS by Quantum Design. This is demonstrated in the peak-effect regime of superconducting CeRu 2 and 2H–NbSe 2 single crystals. In a conventional measurement, a sample moved along the scan length symmetrically about z s 0, experiences a non-monotonic field change, causing a strong variation in the magnetisation of the Žirreversible. superconducting sample. In the halfscan technique, scans are chosen in such a way that the sample experiences field change in the same sense as that before the scan begins. This ensures that the magnetisation of the superconductor remains on the envelope hysteresis loop defined by forward and reverse magnetisation curves. The half-scan technique suggested in this paper is independent of the exact functional form of the magnetic field profile assumed in Eq. Ž4.. It is
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nowhere used explicitly in the analysis leading to our results. The arguments presented in this paper are applicable for any field profile which monotonically decreases in magnitude on either side of z s 0. This method needs to be suitably amended depending on the exact nature of the field inhomogeneity. The basic ideas of this paper can be equally applied to other techniques Žsuch as extraction magnetometry. where sample motion over a distance of few centimeters is involved. The efficacy of the half-scan technique is clearly demonstrated in Fig. 7a, where a conventional hysteresis measurement using a 2 cm scan length Žin which field inhomogeneity is expected to be very small. gives much smaller hysteresis than that obtained using half-scan technique. Remarkably, below the peak-effect regime, the half-scan technique brings out the magnetisation hysteresis which is otherwise not observed in conventional SQUID measurements Žsee Fig. 7a and b.. This hysteresis signifies finite critical current density below the peak regime which is observed both by transport w1–5x and AC susceptibility measurements w6x. However, in the case of CeRu 2 sample, the hysteresis for fields below the onset of peak effect is extremely small. This is again in agreement with the hysteresis measurements on CeRu 2 using the newly developed Reciprocating Sample Option ŽRSO. available in MPMS where no hysteresis was observed below the peak-effect regime even for scan lengths as short as 1 mm w9x. The hysteresis before the onset of peak effect was also observed in a magnetisation measurement using Faraday method based on capacitance technique ŽSakaki Bara et al., unpublished.. This technique does not involve sample motion. For fields well beyond the peak-effect regime, where superconductors are reversible, the half-scan technique produces results identical to that of conventional measurements. On the forward magnetisation curve of CeRu 2 , SQUID responses for different fields as shown in Fig. 3a, look almost similar to an ideal dipolar signal even though the origin of the signal is completely spurious. Computer fits to these signals can return what appear to be reasonable values of the magnetic moment and even give regression values close to unity. One needs to be careful when interpreting results of this kind since such results are sometimes
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Fig. 7. Ža. Magnetisation hysteresis loops of 2H–NbSe 2 crystal measured at 4.5 K using conventional Žcircles. and half-scan Žsquares. techniques. Žb. ŽSquares. Magnetisation hysteresis loop of 2H–NbSe 2 crystal measured at 5.1 K using half-scan technique and Žcircles. forward magnetisation curve measured at 5.1 K using conventional SQUID measurement. The inset shows the virgin magnetisation curve in the low-field regime obtained using conventional and half-scan techniques.
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Fig. 8. ŽCircles. Conventional SQUID response at 20.5 kOe on the forward magnetisation curve of CeRu 2 . Least-squares fit ŽEq. Ž3.. of the SQUID response for y2 cm - z - 0, extrapolated up to z s 2 cm is indicated by the continuous line.
not dipole signals at all Žsee Ref. w18x.. For example we consider the conventional symmetric SQUID response for y2 cm - z - 2 cm at 20.5 kOe measured at 4.5 K on the forward magnetisation curve of CeRu 2 sample Žsee Fig. 3a.. It appears to fit the dipolar response quite well. However, when a q bz corresponding to the half-scan response between z s y2 cm and z s 0 is extrapolated up to z s 2 cm and then subtracted from the raw data from z s y2 cm to z s 2 cm, we obtain a signal which is representative of the true magnetic behaviour of the superconductor between z s y2 cm and z s 2 cm Žsee Fig. 8.. The continuous line in Fig. 8 is the response of a constant dipole moment. The deviation of the raw data from the constant dipole signal Žfor z ) 0. is primarily due to a strong variation of magnetic dipole moment induced in the superconductor by the inhomogeneous field. However this is not sufficient to understand the zero of the sample response Žopen circles. in the region 0 - z - 2 cm, not coinciding with the zero of the dipolar response Žcontinuous line.. This indicates that, for 0 - z - 2 cm, the SQUID response would have contributions from higher-order multipole moments w16,17x besides a position-dependent dipole contribution. The half-scan technique is equally applicable for accurate measurement of irreversibility line in various superconductors. It was recognized by Perry and
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Campbell w15x that the magnetisation hysteresis in various high Tc superconductors measured using MPMS, is often underestimated because of the field inhomogeneity. This leads to grossly underestimating the irreversibility field in these materials. To extract the true irreversibility field, they devised a method in which sample is stationary while a small ac field ripple is applied along with the dc field. The data analysis associated with this technique to obtain magnetisation hysteresis loops is complicated. Halfscan technique can be directly applied to the measurement of irreversibility line in high Tc superconductors. Finally, it is essential that the results obtained by the new half-scan method presented in this paper should be compared with alternative measurement techniques such as the Reciprocating Sample Option ŽRSO. available in the new SQUID magnetometers by Quantum Design, where scan lengths are in the range of 1 mm. In conclusion, a new method, viz., half-scan technique is developed to measure magnetisation hysteresis loops using SQUID MPMS by Quantum Design. We demonstrated this technique by measuring magnetisation hysteresis loops in the peak-effect region of superconducting 2H–NbSe 2 and CeRu 2 single crystals. Magnetisation hysteresis in the peak-effect region obtained by the half-scan technique is found to be much larger than that measured by the conventional technique. Below the onset of peak effect, half-scan technique clearly brings out the hysteresis in magnetisation which is not observed in a conventional SQUID measurement. It was also demonstrated that in the field regime where the superconductor is reversible, both techniques give identical results. Again at very low fields where the extent of field inhomogeneity is rather small and the hysteresis very large, it was shown that the two techniques lead almost to the same results. The remarkable feature of the half-scan technique is that the hysteresis loop obtained is independent of scan length used in the measurement and therefore is unaffected by the field inhomogeneity. Note: It has been pointed out to us that a "centreshift" in a conventionally measured SQUID response in the peak effect region of CeRu 2 was also noticed by Roy et al. w19x, which was attributed by them to
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an inhomogeneous magnetic state. However, origin of the "centre-shift" discussed in this paper is due to inhomogeneous magnetic field of the solenoid.
Acknowledgements G.R.K. thanks K.V. Bhagwat for valuable suggestions.
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