Effects of Magnetic Field Uniformity on the … OF MAGNETIC FIELD UNIFORMITY ON THE MEASUREMENT OF...

QuantumDesign

Effects of Magnetic Field Uniformity on the Measurement of

Superconducting Samples

Mike McElfresh and Shi Li of Purdue University and

Ron Sager of Quantum Design

10307 Pacific Center Court San Diego, CA 92121-3733 USA www.qdusa.com

Tel: 858.481 .4400 800.289.6996 Fax: 858.481.7410 E-Mail: [email protected]

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EFFECTS OF MAGNETIC FIELD UNIFORMITY ON THE MEASUREMENT OF SUPERCONDUCTING SAMPLES

T a b l e o f C o n t e n t s

A. Introduction ............................................................................................................................................... 1

B. Experimental ............................................................................................................................................. 21. The Sample ................................................................................................................................. 22. The Measurements ...................................................................................................................... 33. Mapping the Magnetic Field Profile .......................................................................................... 54. Altering the Magnetic Field Uniformity .................................................................................... 55. Definitions of Fields, Field Sequences, and Units ..................................................................... 5

C. Field Uniformity and Properties of Superconducting Magnets ............................................................... 6

D. Effects of Field Uniformity on Superconductor Measurements .............................................................. 81. Magnetic Properties of Superconductors - Briefly .................................................................... 82. Effects in the Reversible State .................................................................................................... 93. Effects in the Irreversible State ................................................................................................ 11

E. Modeling the SQUID Signals Resulting from Field Non-Uniformities ................................................ 15

F. Effects of Field Non-Uniformities in Large Applied Fields .................................................................. 19

G. Implications of Anomalous Signals for Magnetization Measurements ................................................. 23

Acknowledgements ...................................................................................................................................... 25

References .................................................................................................................................................... 25

Appendix A: Introduction to the Bean Critical State Model ....................................................................... 26

Appendix B: The MPMS Measurement and Data Analysis ........................................................................ 301. The MPMS Measurement ........................................................................................................ 302. Data Analysis ............................................................................................................................ 313. The MPMS Superconducting Magnet ...................................................................................... 344. MPMS System Calibration ....................................................................................................... 35

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A . I n t r o d u c t i o n

Since the discovery of the high-temperaturesuperconductors, much has been learned about strong-pinning effects of magnetic flux in superconductingmaterials. These materials can display extreme magnetichysteresis, and measurements on them can be affecteddramatically if the magnetic history of the sample isaltered or if the sample experiences even small variationsin magnetic field during the measurement. As a result,during the last several years a variety of unexpected andunexplained phenomena have been reported for thesematerials. Sometimes these effects are real properties ofthe materials under investigation; in other cases thephenomena arise from an interaction between themeasurement apparatus and the material itself.1

Researchers presently use a wide variety of both home-built and commercial equipment to characterize and studythe magnetic properties of superconducting materials.While there are many types of magnetometers in use, wewill focus on the subset of magnetometers that aregenerally used to measure the magnetic properties of smallsamples of materials. These “small samplemagnetometers” use a variety of methods to measure themagnetic properties of the sample under study. We canclassify these measurement methods into those that requiremotion of the sample, within or through a pickup coil(e.g., vibrating sample magnetometers, most SQUIDmagnetometers, extraction magnetometers), and those thatmeasure a force or torque, in which case the sample isheld stationary while the field is rotated or a gradient isapplied (e.g., Faraday balance, Gouy method, torquemagnetometers).

In systems in which the field is changed while holdingthe sample stationary, the sample is subjected to a time-varying magnetic field which affects the magnetic historyof the sample. In this case the measurement results forsuperconductors must account for motion of vortices, eddycurrents induced in the sample, and other dynamic effectswhich arise from the interaction of the time-varying fieldand the sample. Conversely, magnetometers which requirethat the sample be physically moved from one position toanother can introduce similar effects because the magneticfield in such instruments is never exactly uniform. Thisstudy presents a systematic analysis that describes someof the effects that such magnetic field non-uniformity canproduce when making magnetic measurements onsuperconducting materials. All of our measurements usea strong-pinning superconducting thin film with the filmperpendicular to the magnetic field because this geometryis one of the most problematic configurations in which tomake a magnetic measurement. This particular geometrywill produce the largest demagnetization effects, thereby

greatly amplifying the magnitude of the field non-uniformities experienced by the sample. This geometryis also appropriate for the high-Tc materials because theirlarge crystalline anisotropy means that samples frequentlyhave a very large effective aspect ratio, even for materialsin bulk (polycrystalline) form.

In all magnetometers that move the sample during themeasurement, the raw data is an output signal having anamplitude which is a function of either time, position, orboth. The sample property to be measured (usually themagnetic moment of the sample) is then computed usingsome type of analytical model which is fit to the raw datafrom the measurement. However, the analysis methodalways makes assumptions about the magnetic behaviorof the sample. For example, nearly all analysis methodsassume that the magnetic moment of the sample:

1) approximates a magnetic dipole moment, and2) the sign and value of this moment do not change

during the measurement.

We will refer to these two assumptions as the constant-dipole model, and we will see that in the case ofsuperconductors the second of these two assumptions isoften not valid.

We want to emphasize here that the raw data produced bysuch a measurement is, in fact, the true magnetic responseof the sample, and this signal contains importantquantitative information about the magnetic nature of thesample. (This assumes, of course, that the measurementapparatus is operating correctly.) As we have stated,however, strong-pinning, superconducting samples do notnecessarily behave as a magnetic dipole of constantmagnitude, and in those cases the analysis algorithmswhich compute the sample moment can fail, producingunpredictable and misleading results. In other words, theindiscriminate use of a constant-dipole model, which isthe normal analysis model provided with commercialmagnetometer systems, can lead to completely erroneousinterpretations of a sample’s magnetic properties. Thispitfall can be largely avoided if one takes the care todetermine the actual measurement conditions and thenmodels the magnetic behavior of the sample under theparticular circumstances experienced during themeasurement. This does require, however, that oneconstruct an appropriate analysis method to extract therelevant information from the output signal, then applythat analysis method to the raw data from the measurement.

It is worth noting that the constant-dipole model can alsofail for non-superconducting samples under certaincircumstances. For example, the constant-dipole modelwill produce erroneous results when the magnetic moment

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QUANTUM DESIGN

of the sample is not purely a dipole moment (but is aquadrupole moment or a sum of multipolar contributions,for example). This can be the case when the magneticmoment of the background is of the same order as themagnetic moment of the sample, and the sample and thebackground signal from the sample holder are displacedwith respect to one another along the axis of the magnet.We will address the issue of non-dipole samples and non-constant dipoles as separate topics.

Because the standard data analysis routines can produceerroneous results we will denote the magnetic momentcomputed using the constant-dipole model by thedesignation m*, while the true magnetic moment of thesample will be denoted by m. For several reasons, themeasurements we describe herein were all performed ona Quantum Design Magnetic Properties MeasurementSystem (MPMS). First, each of the authors had an MPMSavailable in his laboratory. Second, there are an enormousnumber of these systems currently in use, and the largemajority of them are being used to measuresuperconductors. Third, the MPMS is very versatile,offering a great deal of control over the variables we feltwere most important. Of particular importance to us wasthe fact that the MPMS provides access to the true rawdata from the measurement — the digitized values ofSQUID voltage as a function of position — that are storedin the data file when using the All Scans data-file selection.This makes the raw data available in an easily accessibleformat, allowing experienced users to analyze the raw datausing their own analytical methods and algorithms.

Most of the results of this study should be generallyapplicable to any magnetometer system in which thesample is moved (or the fields are changed) during themeasurement. The specific effects will differ dependingon the detection method, detection geometry, output-signalanalysis method, the specific magnet used, and the distancethe sample is moved. All of these parameters will varyfrom one instrument to another, so the work reported herestrongly suggests the need for similar studies on otherspecific models of magnetometers. This is particularlytrue of any magnetometer system which uses an analyticalmodel and data processing algorithms to compute andreport some sample parameter, as is the case with virtuallyall commercial magnetometer systems.

Our goal in performing this work has been to help theusers of many different magnetometers understand morefully some of the anomalous behaviors that have beenreported over the last several years. Our hope is that thisreport will stimulate further investigations, research, andanalytical modeling into the specific effects that combineto produce the observed measurement results. Whenreading this report, we hope the reader will take enough

time to study all of the figures carefully. The specificchoice of figures was made for the purpose ofsystematically building the concepts necessary tounderstand the important issues, but in many of the graphs,different properties are plotted as a function of the position,z, along the axis of the magnet, making it possible toconfuse the various plots under a cursory examination.

B . E x p e r i m e n t a l

This section provides a brief introduction to the methodsused in these studies as well as a brief discussion of someof the considerations taken into account. A much moredetailed description of the operation of the MPMS andreasons behind various considerations can be found inAppendix B and the manuals provided with the MPMS.In the following we have tried to accommodate users ofboth Système Internationale d’Unités, or SI, and cgssystems of units by giving expressions in both forms andusing (SI) and (cgs) to indicate the applicable units.

1. The Sample

The sample used in these studies was a YBa2Cu3O7

(YBCO) thin film, approximately 3000 Å thick, preparedby pulsed laser deposition onto a heated 0.5 mm thick[100] LaAlO3 substrate. The substrate had dimensions of5 mm x 5 mm while the YBCO film was patterned into a 2mm diameter circle. The film had a superconductingtransition temperature Tc = 90 K and the superconductingmaterial had a mass of about 5 x 10-6 g. The magneticfield was applied perpendicular to the plane of the film(along the c axis). These films generally exhibit largevalues of the critical current density, Jc, with valuesJc ≈ 106 A/cm2 at T = 77 K being fairly typical.

When measuring a sample with a large magnetizationvalue, e.g. a superconductor, it is often important toconsider demagnetization effects. For an elliptically-shaped sample, the total H field inside the sample is givenby H = H0 - Hd, where H0 is the applied field produced bythe current in the magnet coil and Hd is thedemagnetization field. The demagnetization field is givenby Hd = -NM, where N is a shape-dependentdemagnetization factor and M is the magnetization of thematerial. For a long thin sample in a parallel field N ≈ 0,while for a short flat sample in a perpendicular field thedemagnetization correction (NM) can be enormous. Inthe case of a superconductor M is negative so that Hd willbe positive, which means that the “demagnetization” fieldactually increases H inside a superconducting sample. Thevalue of N has a range 0 ≤ N ≤ 1 in SI units and 0 ≤ N ≤ 4πin cgs units. It is possible to estimate the demagnetizationcorrection from the initial (virgin) portion of the

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output voltage. Under ideal conditions, the magneticmoment of the sample does not change during transportthrough the detection coil.

Figure B2(a) shows the position of the various loopsmaking up the second-order-gradiometer detection coil.The position z = 0 cm corresponds to the center of the coiland this is positioned at the midpoint along the length ofthe magnet. Figure B2(c) shows the output voltage of theSQUID electronics as a function of position V(z) as asample with a positive, constant magnetic point dipolemoment is moved through the pickup coil. This V(z) datacan be collected on the MPMS by using the All Scans datacollection option. As both an example of an m(z) plot, aswell as to emphasize the position independence of themagnetic moment under ideal conditions, Figure B2(b)shows the m(z) = constant plot that corresponds to the V(z)curve in Figure B2(c). Figure B2(e) shows the outputsignal V(z) for a sample with a constant, negative dipolemoment as shown in Figure B2(d).

The signal analysis programs currently provided with theMPMS are designed to analyze a specific type of response:a magnetic dipole of constant magnitude moving throughthis particular type of detection coil (second-order

magnetization curve, plotting magnetization as a functionof applied field, M(H), for samples cooled below Tc inzero applied field, since a slope of M/H = -1 (SI) = -1/4π(cgs) is expected for the case in which N = 0.

2. The Measurements

A measurement is performed in the MPMS by moving asample through the superconducting detection coils,which, as shown in Figure B1, are located at the center ofthe magnet. The sample moves along the symmetry axisof the detection coil and magnet. As the sample movesthrough the coils the magnetic dipole moment of thesample induces an electric current in the detection coils.The SQUID functions as a highly linear current-to-voltageconvertor, so that variations in the current in the detectioncoil circuit produce corresponding variations in the SQUID

Figure B1(a). The geometrical configuration of the magnet,detection coils and sample chamber.

Figure B1(b). The configuration of the second-ordergradiometer superconducting detection coil. The coil sitsoutside of the sample space within the liquid helium bath.

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gradiometer). The MPMS provides three methods foranalyzing the SQUID output signals. (For further detailsplease refer to Appendix B.) When these signal analysisroutines are used for cases in which the output signaldeviates significantly from a point-dipole response or ifthe magnitude of the magnetic moment varies during themeasurement (i.e., significant deviations from theconstant-dipole model), the analysis is not reliable andthe magnetic moment value reported by the computerprogram will not be a true measure of the moment. All ofthe data analysis routines provided with the MPMS reportthe magnetic moment of the sample in cgs orelectromagnetic units (emu). In many of the casespresented here, the computer fits are often not validmeasurements of the magnetic moment, but simply the

z (cm)

SQUID output signal

Magnetic Moment

SQUID output signal

Magnetic Moment

Second-order Gradiometer coil

-1012

m(z

)

-2-101

m(z

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1.52 cm

2.02 cm

0.1 cm

MagneticDipole

+ + (a)

(b)

-3 -2 -1 0 1 2 3

(d)

-1

0

1

V (

z)

-2

-1

0

V (

z)

(c)

(e)

result of the best computer fit to a measured curve. Wewill therefore refer to the values produced by computerfits to V(z) signals as the apparent magnetic moment(designated m*). When referring to the true (or actual)magnetic moment of the sample, the designation m willbe used, and even though m* values are not true magneticmoment values, the emu unit will be used in both cases.In this report the linear regression method is used toproduce the m* values, unless otherwise specified.

Even when the output signal deviates significantly fromthat of a point dipole or changes during a measurement,the raw data still contain important and fundamentalquantitative information about the magnetic properties ofthe sample. The measured response in these cases is notan error or an artifact, but a full-fledged magneticmeasurement of the sample, but for various reasons, themeasured response may not correspond to a point dipole-type signal that is easily analyzed to determine the truemagnetic moment. Reasons for deviations can include(but are by no means limited to) large background signals,magnetic field non-uniformities and shielding effects ofthe superconducting magnet. As will be shown, thesedeviations from a non-ideal dipole response often do notinterfere, but rather may even enhance, one’s ability todetermine properties like the superconducting transitiontemperature and the vortex-solid phase transitiontemperature. It will also be shown that in those cases whena quantitative determination of the magnetic moment isactually necessary, one can use an appropriate method forbackground subtraction when applicable or an appropriatemodel for the non-dipole response signal.

A common method for identifying the onset of magneticirreversibility is to make measurements along two pathsthat expose the sample to very different temperature and/or magnetic field histories. For superconductors, the mostpopular method has been the zero-field-cooled/ field-cooled (ZFC/FC) measurement. In this case the sampleis first cooled in zero applied field (H = 0) to the lowestmeasuring temperature. After the temperature hasstabilized, a field is applied and data are collected as thesample is warmed to above Tc

(the ZFC part). A secondset of data is collected as the sample is slowly cooledthrough Tc in the same field (designated FCC for field-cooled-cooling). In contrast to this, for a field cooledwarming (FCW) measurement the sample is cooled to thelowest measuring temperature in the applied field and thenmeasured on warming.

For the FCC measurements presented here, the temperaturewas decreased in steps of 0.25 K to be sure that noundershooting of the temperature occurred. Becausetemperature overshooting while warming is less of aproblem in general, steps of various sizes were used for

Figure B2. (a) The position z of the loops of the second-ordergradiometer detection coil. The position z = 0 corresponds tothe center of the coil and this is positioned at the midpoint alongthe length of the magnet. (b) The ideal behavior of a sample(with a positive value of the magnetic dipole moment) as afunction of position, m(z), as it is moved through the detectioncoil. (c) The output voltage of the SQUID electronics as afunction of position V(z) as a magnetic point dipole with thevalue m(z) = constant > 0, as in (b), is moved through thesecond-order gradiometer pickup coil. Under idealcircumstances the sample approximates a point dipole and thevalue of the magnetic moment does not change during themeasurement. (d) The m(z) curve for an ideal sample with anegative value of the magnetic dipole moment. (e) The outputsignal V(z) for a sample with a constant, negative dipole momentshown in (d).

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the ZFC measurements. A scan length of 2.5 cm (±1.25cm) was used in all ZFC and FCC measurements and thesample went through two cycles at each measuring point.

3. Mapping the Magnetic Field Profile

Quantitative information on the small variations inmagnetic field over the measuring region are veryimportant to understanding the effects we will bediscussing here. Detailed measurements of the magneticfield as a function of position z were made under a varietyof conditions using a Hall-effect device mounted on a longprobe oriented to measure the field component along theaxis of the magnet. Measurements were made with thesample space at a temperature of 300 K. The probe wasattached to the MPMS sample transport to take advantageof its precise positioning capabilities. Measurements ofthe magnetic field profile were usually made over a 6 cmscan length (z = ±3 cm). Using the Hall-effect device, themagnetic field could be measured to a precision of 0.01Oe in low fields, and about 1 part in 105 at higher fields.

Figure B3 shows a magnetic field profile H(z) for themagnet after the field has been changed from a setting of+50 kOe to a setting of zero. This particular H(z) profileshows the magnetic field variations from z = -13 to +13cm. Even though the transport current flowing throughthe magnet is zero, due to flux trapping in thesuperconducting wire from which the magnet isconstructed, the magnetic field is not zero except at a fewpoints. For the m* measurements presented in this report,the sample was scanned over 2.5 cm in the most uniformregion at the center of the magnet.

4. Altering the Magnetic Field Uniformity

The MPMS system provides four different methods forchanging the magnetic field in the system. The no-overshoot, oscillate, and hysteresis modes are useful forroutine changes between arbitrary starting and endingvalues, while the magnet reset option provides a specialmethod for eliminating nearly all of the trapped flux fromthe superconducting MPMS magnet by driving the magnetnormal (non-superconducting). More detailed descriptionsof the operation and attributes of each method arepresented in Appendix B.

The specific magnet field profile H(z) depends on severalfactors including the field-change method used and theprevious history of magnetic field settings. The fielduniformity increases in the order of: no-overshoot,oscillate, magnet reset. The actual field profile is fairlyreproducible for the oscillate and reset modes. In contrastto this, the no-overshoot mode can produce field variationswith a large range of values, however, the trends we showhere, particularly those associated with different sequencesof field changes, are fairly reproducible.

5. Definitions of Fields, Field Sequences, and Units

For the purposes of the discussions here, the unit gauss(or G) will be used when referring to either the magneticfield (usually denoted H) or the magnetic flux density(usually denoted B). The relationship between B and H isgiven by B = H + 4πM (cgs), where M is the magnetizationof the sample and H is the field produced by the magnet.The cgs unit for H is oersted (or Oe) while the cgs unit forB is the gauss, however as a unit of measure 1 Oe = 1 G,so in practice gauss is often used for H. The SI unit for His A/m, while tesla or Wb/m2 is used for B, with B = 1tesla = 10,000 G and B = µ0H + µ0M. In addition to units,several different field designations will be used here. Hwill refer to the actual field (which can be experimentallymeasured) produced by the magnet, while Hset will referto the field that is determined — using the appropriatecalibration coefficient — from a measurement of thecurrent flowing from the power supply through the magnet.The difference between the actual field H and Hset is notdue to errors in the calibration, but rather to flux trappingand the hysteretic nature of superconducting magnets.

In addition to the different field designations used here,there will be frequent references to the particular sequenceof magnetic field settings. As will be shown, the particularsequence of field changes has a significant effect on themagnitude and spatial dependence of the magnetic fielduniformity. In all the cases described, the field will be setto zero (Hset = 0) from either Hset = +50 kG or Hset = -50kG, in which case these sequences of field changes will

Figure B3. Even when the transport current in thesuperconducting magnet has been reduced to zero, the trappingof flux in the superconducting windings can result in adistribution of non-zero fields in the magnet. The axialcomponent of the magnetic field is shown as a function ofposition from z = -13 to +13 cm, where the midpoint alongthe magnet’s length is defined as z = 0.

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H (

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z (cm)

Hs s q = +50 kG ⇒ 0 G

No-overshoot Mode

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magnet is generally the region with the smallest fieldcurvature and greatest uniformity. The superconductingmagnet used in the MPMS has additional windings towardthe ends of the magnet to further reduce the curvature ofthe field near its central region. The detection coils arecentered along the length of the magnet to take advantageof the field uniformity near the center. Variations in fielddue to sample motion during a measurement can be furtherreduced by shortening the scan length so the measurementis made in only the most uniform portion of the magnet.

Figure C1 shows the magnetic field as a function ofposition, z, along the axis of a 55-kG solenoidal magnetin an MPMS magnetometer after reducing the transportcurrent of the magnet to zero (Hset = 0) from the fieldsetting Hset = -50 kG (Hssq = -50 kG ⇒ 0 G) using the no-overshoot mode. This H(z) profile was measured directlyusing a Hall-effect device mounted on a probe. As shownin Figure C1, from z = -3 to +3 cm the field changes byabout 5 G and is fairly symmetric about the center. Usinga computer fit, the shape of the H(z) curve in Figure C1can be approximated by the relation H(z) = -2.99 - 0.11 z+ 0.65 z2 G. (Additional information about fielduniformity at longer distances from the center of themagnet can be found in Quantum Design’s TechnicalAdvisories #1 and #6.)

Figure C2 shows a comparison between the Hset = 0profiles obtained using the no-overshoot mode for twodifferent field-change sequences. In one case, the field

Figure B4. Because there will be frequent references to theparticular order of magnetic field settings used prior to startinga given measurement, a convention for writing these fieldchanges is defined here. Hssq refers to the sequence of Hset valuesto which the magnet was set in order to obtain the particularmagnet state used in the measurement. As will be shown, thesequence of field changes effects the spatial dependence of themagnetic field uniformity.

be noted by Hssq = +50 kG ⇒ 0 G or Hssq = -50 kG ⇒ 0 G,respectively. In some cases an additional field, i.e. themeasuring field, will then be set from one of these twoHset = 0 conditions (or remanent states). In the particularcase that the final field setting is Hset = +25 G, the twosequences would be noted as Hssq = +50 kG ⇒ 0 G ⇒ +25G and Hssq = -50 kG ⇒ 0 G ⇒ +25 G. These definitionsare described schematically in Figure B4. In some of thestudies to be described, either Hssq = +50 kG ⇒ 0 G orHssq = -50 kG ⇒ 0 G will be used above thesuperconducting transition temperature Tc, and then themeasuring field will be set at a temperature below Tc.

C . F i e l d U n i f o r m i t y a n d P r o p e r t i e s o fS u p e r c o n d u c t i n g M a g n e t s

When a magnet is used to produce magnetic fields over awide range of values, there is a practical limit on the spatialuniformity of the field that can be produced. Uniformitycan be improved by incorporating a larger magnet likethose used in high-resolution NMR, in which case theuniformity is usually optimized at the single field valueat which the magnet will be operated. Whenmeasurements are to be made over a wide range of fields,the larger magnet will slow operation, increase thephysical size of the system, and, in the case ofsuperconducting magnets, increase the liquid-helium boil-off. The region midway between the ends of a solenoidal

Set Field to +50 kG(No-overshoot Mode)

Set Field to +25 G(No-overshoot Mode)

Set Field to -50 kG(No-overshoot Mode)

Set Field from +50 kG to 0 G(No-overshoot, Oscillate, or Magnet-reset)

Set Field from -50 kG to 0 G(No-overshoot, Oscillate, or Magnet-reset)

Set Field to +25 G(No-overshoot Mode)

Field-Change Sequences

Hssq

= +50 kG ⇒ 0 G ⇒ +25 G Hssq

= -50 kG ⇒ 0 G ⇒ +25 G

Figure C1. The magnetic field as a function of position z alongthe axis of a 55-kG solenoidal magnet, H(z), in an MPMSmagnetometer after reducing the transport current throughthe magnet to zero (remanent profile Hset = 0) from a field ofHset = +50 kG using the no-overshoot mode. This H(z) profilewas measured directly using a Hall-effect device mounted on aprobe. Note that the point midway along the length of themagnet (center) is defined as z = 0.

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- 2

0

2

4

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H(z

) (G

)

z (cm)

Hs e t

= 0.0 G

(No-overshoot ModeH

s s q = -50 kG ⇒ 0 G)

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was set to zero (Hset = 0) starting from a field setting ofHset = +50 kG (Hssq = +50 kG ⇒ 0 G) and, in the othercase, the field was set to zero starting from a field withthe opposite polarity, Hset = -50 kG (Hssq = -50 kG ⇒ 0G). These two remanent profiles are nearly symmetric (amirror reflection) about H = 0. The differences in thefinal state of the magnet (the specific spatial dependenceof the field) and how the state depends on the particularsequence of magnetic field changes is crucial tounderstanding the various anomalies that can occur whenmeasuring superconducting samples. The different H(z)profiles produced by the two different sequences willresult in very different magnetic-field histories when asample is measured under these two different conditions.(Magnetic-field history refers to the sequence of magneticfields to which the sample itself is exposed. This includesthe field changes due to changing the current in themagnet, but it also includes the field changes that occurwhen the sample is moved to different positions in themagnet where the field value is different due to the fieldnon-uniformities.) For samples exhibiting irreversible(hysteretic) properties (e.g., superconductors andferromagnets), the value of the magnetization can be avery strong function of the magnet-field history.

Figure C2 also shows the effect of using the no-overshootmode to set an Hset = +25 G field from each of theseremanent states (Hssq = +50 kG ⇒ 0 G ⇒ +25 G andHssq = -50 kG ⇒ 0 G ⇒ +25 G). The profiles essentiallyretain the same shape and relative amplitude variation,however neither profile is displaced by +25 G from itsHset = 0 position. Instead, the Hset = +25 G profiles arenow nearly symmetric about H = +25 G, with the profile

Figure C2. A comparison between the remanent profilesobtained using the no-overshoot mode. In one case, the fieldwas set to zero (Hset = 0) from a field setting Hset = +50 kG and,in the other case, the field was set to zero from Hset = -50 kG.Also shown is the effect of setting Hset = +25 G using theno-overshoot mode from each of the two remanent states.

associated with an initial setting of Hset = -50 kG displacedby an additional +9 G and the profile associated with theinitial setting of Hset = +50 kG displaced by about -9 G.

Each of the field-change methods available (no-overshoot,oscillate, and magnet reset ) on the MPMS will produce adifferent H(z) profile. Figure C3 shows the remanentprofiles produced when the oscillate mode is used for thesame field-change sequences that were used to producethe curves in Figure C2. By comparing the amplitudes ofthe curves in Figure C2 and C3, it is clear that using theoscillate mode reduces the amplitude of the non-uniformity considerably compared to the no-overshootmode, with H(z) varying by only 0.4 G from z = -3 to 3cm for the oscillate mode compared with an H variationof about 5 G over the same length when using the no-overshoot mode. The shape of the Hssq = +50 kG ⇒ 0 Gcurve in Figure C3 can be approximated by H(z) = 6.23 -0.0024 z + 0.056 z2 G. Note that when using the oscillatemode, the Hset = 0 curves are not symmetric about H = 0.Instead the Hssq = -50 kG ⇒ 0 G curve straddles H = 0while the Hssq = +50 kG ⇒ 0 G curve is displaced by about+6 G from H = 0. Remarkably, when the magnet is nowset to Hset = +25 G (using the no-overshoot mode only forthis last field change), both H(z) profiles fall fairly closeto H = +25 G. However, the shapes of the profiles changesomewhat from the shape observed at Hset = 0, which isparticularly true for the Hssq = -50 kG ⇒ 0 G curve.

The other method that can be used to alter the H(z) profileis the magnet reset option. The remanent profiles producedfor the cases when the field is set to Hset = 0 fromHset = +50 kG and -50 kG using the magnet reset, are shown

Figure C3. A comparison between the remanent profilesobtained using the oscillate mode. In one case, the field was setto zero (Hset = 0) from a field setting Hset = +50 kG and, in theother case, the field was set to zero from Hset = -50 kG. Alsoshown is the effect of setting Hset = +25 G from each of theHset = 0 states using the no-overshoot mode.

7

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QUANTUM DESIGN

to the metallurgical properties of the original materialsand the exact process by which the wire is fabricated. Inany event, we want to warn the reader that one cannotdirectly apply the quantitative results on field profilespresented in this report to any other magnetometer system.While the general trends observed for field changesequences should apply to all MPMS systems, it isimportant to properly characterize the field profilesexperimentally for your own individual system.

D . E f f e c t s o f F i e l d U n i f o r m i t y o nS u p e r c o n d u c t o r M e a s u r e m e n t s

1. Magnetic Properties of Superconductors - Briefly

The magnetic behavior of superconductors is fairlycomplicated. The complicated nature of their behavior isdue in part to the fact that the magnetic behavior of asuperconductor can depend very strongly on the form ofthe material (e.g., polycrystalline or single crystal) as wellas on the density and types of defects present in the specificsample. This also means that the magnetic behavior canbe altered by processes that effect the material’s structure.

When a small magnetic field is applied to asuperconductor, supercurrents induced near the surface ofthe superconductor can completely screen the appliedmagnetic field from the interior of the sample (B = 0inside). This corresponds to the initial portion of themagnetization as a function of applied field, M(H), curvein Figure D1, which has a slope of -1 (SI) [-1/4π (cgsunits)]. This regime is often referred to as the Meissnerstate. For an ideal type-II superconductor the sample’sinterior would remain completely screened up to the lowercritical field H

c1 after which the magnetic field penetrates

into the sample in the form of quantized lines of magneticfield (flux vortices). The amount of flux that haspenetrated into the sample is proportional to the fluxdensity, B. The relationship between B, H, and M is alsoshown in Figure D1. A key point about the ideal M(H)curve shown in Figure D1 is that it is reversible, with Mfollowing the same curve both on increasing anddecreasing H. A superconductor exhibiting this idealbehavior would not be able to support a supercurrent formagnetic field levels above H

c1, because under these

conditions the penetrating magnetic flux lines provide amechanism for the dissipation of the supercurrents.

In order to support a supercurrent above Hc1, the fluxvortices in a superconductor must be immobilized. Whena transport current flows in an ideal superconductor withpenetrating flux lines, the Lorentz force (FL = J x B) actson the vortices which results in motion of the vorticesthrough the sample. The moving vortices result in

in Figure C4. The amplitudes of the curves in Figure C4are very small, with an H(z) variation less than 0.1 G fromz = -3 to +3 cm, which compares with an H(z)-variationof about 5 G when using the no-overshoot mode and 0.4G using the oscillate mode. The H(z) curve can beapproximated by H(z) = -0.501 + 0.020 z + 0.0023 z2 G.The Hset = 0 curves are not symmetric about H = 0. Inthis case the Hssq = +50 kG ⇒ 0 G curve straddles H = 0while the Hssq = -50 kG ⇒ 0 G curve is displaced by about-0.6 G from H = 0. The Earth’s magnetic field is about0.5 G and the system was not magnetically shielded inthese studies. When the magnet is now set to Hset = +25G (using the no-overshoot mode for only this last fieldchange), both H(z) profiles are now fairly close to H =+23 G and the shapes of the profiles change somewhatfrom the shape observed at Hset = 0. In both cases theH(z)-variation from z = -3 to +3 cm is still smaller than ineither the no-overshoot or oscillate modes, varying by lessthan about 0.1 G.

We want to emphasize that the field profiling [H(z)]measurements and the experimental data presented in thisreport were collected from a single MPMS instrument.In particular, it is not generally true that different magnetson different instruments will all behave in the same way,even if the magnet is the same model as the one used tocollect the data described in this report. Experimentalmeasurements at Quantum Design have shown that theremanent fields in the MPMS magnets vary significantlyfrom one instrument to another. We have no explanationfor this behavior, but we speculate that the detailed natureof the remanent fields in any particular magnet probablydepends strongly on the microscopic structure of itssuperconducting wire, which can ultimately be traced back

Figure C4. A comparison between the remanent profilesobtained using the magnet-reset option. In one case, the fieldwas quenched to zero (Hset = 0) from a field setting Hset = +50kG and, in the other case, the field was quenched to zero fromHset = -50 kG. Also shown is the effect of setting Hset = +25 Gusing the no-overshoot mode from each of the remanent states.

8

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transition. This means that the M(H) loop will behysteretic, following a different M(H) curve for increasingH than for decreasing H. The hysteresis will increase asthe pinning strength increases and there will be acorresponding increase in Jc. This effect is the basis ofthe Bean Critical State model which can be used to relatethe Jc of a sample (a transport property) to the sample’smagnetization using a geometrical factor (see AppendixA for a brief description of the Bean model).2 It isimportant to realize, however, that irreversibility resultsin a history dependence for properties like magnetization,so that the sequence of magnetic field changes andtemperature changes can both effect the value of themagnetization measured for an irreversible sample. Anaccurate (or even relevant) measurement of manyproperties will require the proper preparation (a particularsequence of field and temperature changes) to place thesample in the state where a meaningful measurement canbe made.

2. Effects in the Reversible State

As a sample is moved along the axis of the magnet duringa measurement in a magnetometer, the field H at thesample will change as a function of position due to thenon-uniformity of the magnetic field. A very importantconsequence of this changing local field is that this willproduce a position dependence for the true magneticmoment of the sample, m(z), since the sample’s momentdepends on the applied field in which the sample actuallyresides. Superconductors in the Meissner state have a verylarge, negative volume susceptibility M/H = χ = -1/4π (cgsunits) ~ -0.08 emu/cm3/G. For a sample of asuperconducting material having a density of 5 gram/cm3

and a mass of 1 milligram, a field change of about 5 Gwould correspond to a magnetic moment change of about8 x 10-5 emu. For a thin superconducting sample in aperpendicular field, the demagnetization effects wouldproduce much larger magnetic fields leading tocorrespondingly larger changes of the magnetic momentwith position.

For an ideal superconductor in the Meissner state, thesuperconductor acts like a perfect, reversible diamagnet.Thus, the magnetic moment m is proportional to thenegative of the applied field. To completely screen outthe applied field from the interior of the superconductor,shielding currents near the surface of the superconductorincrease in proportion to the applied field. Thus, smallpositive increases in the magnetic field at the sample willresult in the magnetic moment of the Meissner-statesuperconductor becoming more negative. Figure D2shows the measured H(z) profile for a field setting Hset = 0which was set using the no-overshoot mode from a settingHset = -50 kG. The local H experienced by the sample

Figure D1. The ideal M(H) curve for a reversible type-IIsuperconductor. Below the lower critical field, Hc1, thesuperconductor completely screens out the applied magneticfield H (or Ha). Since B = H + 4πM (cgs), then M = -H/4πfor H < Hc1. Just above Hc1 some of the applied magnetic fieldpenetrates into the superconductor in the form of flux vortices.The amount of flux penetration is proportional to B and, as theapplied field continues to increase, a larger fraction of theapplied field penetrates into the superconductor. At the uppercritical field Hc2, the applied field fully penetrates (B ≈ H) andthe sample goes normal.

a

dH

c1H

c2H

c2H

H

B

M= 0

b a

c d

B = 0

Beq

M

M (H)

H c1

H

T< T , T= constantc

c

- M

B

H

B

- M

b

H

-4πM = H

- M

dissipation giving the superconductor a resistance. Aneffective mechanism for immobilizing flux vortices is tointroduce defects into the material which can serve as sitesfor “pinning” the flux vortices. This pinning is usuallythe result of a flux vortex having a lower energy at a defect.With a sufficiently large density of defects present in thesuperconductor, a supercurrent can be supported up to thecritical current Jc, which is where the Lorentz forceassociated with the applied current just exceeds the pinningforce. Flux motion is only one of the mechanisms thatcan limit Jc. Clearly the dissipation mechanism that resultsin the smallest Jc (the one that produces dissipation at thesmallest value of J) is the one that will determine Jc forthe sample.

One important consequence of flux pinning is that theM(H) curve will now be irreversible below the vortex-solid

9

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QUANTUM DESIGN

increases after the sample passes through the center ofthe magnet. Thus, for a superconductor in the Meissnerstate, m(z) is not constant during a measurement, but ratherwill have a component that varies proportionally with thefield variations it experiences during the measurement.

As shown in Figure D3, in the case of a superconductingsample, the effects of field non-uniformity on the SQUIDoutput signal can be surprisingly significant. The opencircles in Figure D3 have been calculated to show theexpected SQUID output signal, V(z), for a superconductingsample with the position-dependent magnetic moment,m(z), shown in Figure D2. The m(z) curve was displacedby an additive constant so that m(z = -3) = 0 at the startingpoint of the scan. (For comparison, the crosses in FigureD3 show the SQUID output signal V(z) for the position-independent constant-dipole model where m(z) = constant,but m(z) is NOT zero.) A crucial observation here is that,since the magnetic field was originally set to zero (Hset =0), there should have been no signal at all from the sample,so in this particular case the entire SQUID output signalis an artifact of the non-uniform field caused by fluxtrapped in the magnet. Note that this spurious outputsignal is very similar in shape to the ideal constant-dipolesignal, including the symmetry about z = 0, but the extremaare now located closer to z = ±1.47 cm than they are to thez = ±1.55 cm positions associated with the constant-dipolemodel. It is neither accurate nor useful to consider thisartifact signal as a measurement of the sample’s magneticmoment. Even though the analysis algorithm will returna value for the magnetic moment when it operates on thisartifact signal, this “apparent magnetic moment” value ismerely a best fit to the SQUID output signal which isproduced by local field changes during the motion of thesample. In fact, for a superconductor in the Meissner state,m(z) should be constant and equal zero when H = 0. Themeasured response is a true magnetic measurement of thesample, however, it is not the particular measurementwhich the analysis programs were designed to analyze.

V(z) signals for which the extrema occur closer to z = 0than those of point-dipole signals have been observed foryears when measuring superconductors, particularly nearTc.3,4 The origins of these signals have been very difficultto explain, and models employing contributions fromhigher-order magnetic multipoles has failed to account forthese observations. Recently, however, it was shown thatthese kinds of signals could be produced when themagnetic moment has a spatial dependence,5,6 as shownin Figure D3.

It was shown in Section C (see Figure C2) that the H(z)profiles for the Hssq = +50 kG ⇒ 0 G and Hssq = -50 kG ⇒0 G sequences were essentially mirror images of oneanother about H = 0. Since the magnetic moment of the

- 4

- 2

0

2

4

- 4

- 2

0

2

4

- 3 - 2 - 1 0 1 2 3

H(z

) m(z)

z (cm)

H(z)

m(z)

Figure D2. The measured H(z) profile for a field set to Hset = 0using the no-overshoot mode from Hset = -50 kG (Hssq = -50 kG⇒ 0 G). m(z) is a calculation of the magnetic moment of areversible superconductor as it experiences this particular H(z)dependence. m is not constant during a measurement, but rathervaries proportionally with the local H value as the sample movesalong z. In this calculation the second-order polynomial fit tothe H(z) profile in Figure C1 was used.

Figure D3. The SQUID output signal V(z) for a dipole movingthrough the detection coil for the ideal situation in which themagnitude of the moment is unchanged during the measurement,i.e. m(z) = negative constant. Also, the SQUID output signalV(z) for a superconductor sample in the Meissner state havingthe m(z) dependence shown in Figure C1.

initially increases as the sample moves from its startingposition, plateaus near the center of the magnet (z = 0),and then decreases. A calculation of the effect of thisparticular non-uniform field profile (non-uniformity) onthe position dependence of the magnetic moment, m(z),of a reversible superconductor is also shown in Figure D2,where it can be seen that m(z) initially decreases then

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separates a higher-temperature phase in which Jc is alwayszero, and flux vortices cannot be pinned, from a lower-temperature phase in which Jc > 0 and vortices can bepinned. Whether or not one should regard the Jc = 0 phaseas superconducting remains a controversial question(mostly of semantics) because the Jc = 0 phase cannotsupport a supercurrent without dissipation (resistance) yetthere appears to be a significant density of Cooper pairspresent in this phase. It can be difficult to reconcile thatthe flux vortices causing the dissipation are entitiesassociated with superconductors and local supercurrents,yet the word “superconductor” suggests a macroscopiczero resistance state.

A common method for identifying the onset ofirreversibility is to make measurements along two differentpaths that expose the sample to very different temperatureand/or field histories.7 As mentioned earlier, forsuperconductors the most popular method has been thezero-field-cooled/ field-cooled (ZFC/FC) measurement. Inthis case the sample is first cooled in zero applied field tothe lowest measuring temperature (ZFC). Aftertemperature stabilization, a field is applied and data arecollected as the sample is warmed to above Tc. Anotherset of data is then collected as the sample is slowly cooledin the same field from above Tc, which is designated FCCfor field-cooled-cooling.

Figure D5(a) shows the ZFC- and FCC-m*(T) curvesmeasured for the YBCO film in which the magnetic fieldis applied perpendicular to the film, parallel to thecrystallographic c-axis. [Figure D5(b) is an expanded viewof the data over a narrow temperature range near Tc.] Thebehavior observed here is not particular to HTS materials.These films are generally regarded as strong pinning andhave very high Jc values below the vortex-glass transition.The scan length was 2.5 cm (z = ±1.25 cm) in all casesand the apparent moment values, m* , were determinedusing the linear regression routine to analyze the SQUID-output V(z) curves. For the ZFC curves the field was setto Hset = 0 at a temperature above Tc

from either Hset = +50kG or Hset = -50 kG using either the no-overshoot, oscillate,or reset modes. The sample was then cooled to the lowestmeasuring temperature (T = 5 K) where the field was setto Hset = +25 G using the no-overshoot mode. FCC curveswere measured immediately after the corresponding ZFCcurve. Figure D5 shows both sets of ZFC- and FCC-m*(T)curves measured in Hset = +25 G for the two different field-change sequences.

From Figure C2 it follows that the two sequences Hssq =+50 kG ⇒ 0 G and Hssq = -50 kG ⇒ 0 G will generateH(z) profiles that are mirror images of one another.Similarly, there is a remarkable mirror symmetry betweenthe two FCC-m*(T) curves of Figure D5, including even

superconductor is proportional to the field, H, in which itresides, it follows that a pair of H(z) curves that have mirrorsymmetry about H = 0 will produce a pair of m(z) curvesthat have mirror symmetry about m = 0. This can be seenby comparing the curves with open circles in Figures D3and D4, which are clearly mirror images about m = 0. Thisexample is meant to help to reinforce the conclusion thatthe V(z) output signal in these cases is merely a particularmagnetic response resulting from transporting asuperconducting sample through a non-uniform magneticfield, rather than a measurement of a dipole moment ofconstant amplitude. Even though they are essentiallyartifacts, computer fits to the V(z) curves shown in FiguresD3 and D4 would produce moment values of the samemagnitude but opposite in sign. Specifically, a negativeapparent moment m* would result for the V(z) curve inFigure D3 and a positive m* for the curve in Figure D4.

3. Effects in the Irreversible State

The onset of irreversibilities dramatically changes themagnetic properties of a superconductor. Prior to thediscovery of high temperature superconductors (HTS) itwas usually assumed that the onset of irreversibilitiescoincided with the upper critical field, Hc2, which is theboundary that separates the superconducting and normal(non-superconducting) states of a type-II superconductor.In HTS materials observations of a large separation in field(or temperature) between the onset of irreversibilities, andthe apparent onset of the strong diamagnetic responseassociated with superconductivity, led to the identificationof a previously unexpected phase transition, now knownas the vortex-solid phase transition. This phase transition

- 3

- 2

- 1

0

1

2

- 3 - 2 - 1 0 1 2 3

m= -1 = constantm ~ -H(z)(No-overshoot ModeH

s s q = +50 kG ⇒ 0 G)

V(z

)

z (cm)

Figure D4. The SQUID output signal, V(z), for a superconductorsample in the Meissner state calculated using the H(z) profileresulting from the field-change sequence Hssq = +50 kG ⇒ 0 G.The SQUID output signal for the ideal case (m = negativeconstant) is plotted for comparison.

11

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QUANTUM DESIGN

No-overshoot Mode

0

T= 92.5 K

2

4

T= 90.0 K

0

2

4

6

T= 89.5 K

-2

0

2

T= 88.5 K

-1

0T = 92.5 K

-2

-1

0

1 T = 90.0 K

-4

-2

0

2

4 T = 89.5 K

-4

-2

0

2

4 T = 88.5 K

FCCH

ssq = +50 kG ⇒ 0 G ⇒ +25 G

FCCH

ssq = -50 kG ⇒ 0 G ⇒ +25 G

-2

0

T= 88.0 K

-4

-2

0T= 86.0 K

-8

-6

-4

-20

T= 84.0 K

-8

-6

-4

-2

0T= 78.0 K

0

2

4

6

8

-2

0

2

4 T = 88.0 K

0

2

4T = 86.0 K

0

2

4

6

8

T = 84.0 K

T = 78.0 K

0 1.25-1.25z (cm)

0 1.25-1.25

z (cm)

Figure D5. (a) ZFC- and FCC-m*(T) curves measured atHset = +25 G. Hset = 0 was set above Tc using the field-changesequences Hssq = +50 kG ⇒ 0 G and Hssq = -50 kG ⇒ 0 G in theno-overshoot mode, while the measuring field Hset = +25 G wasapplied at T = 5 K in the no-overshoot mode after zero-fieldcooling. (b) Expanded view of data near Tc.

the reversal near Tc which is clearly seen in Figure D5(b).These curves are nearly perfect reflections of each otheracross the line m* = 0. This symmetry is also evident inthe V(z) plots shown in Figure D6, with the symmetry inboth V(z) and m*(z) observed down to the lowesttemperatures. Since the only difference between these twoFCC-m*(T) curves is the field-change sequence used toprepare the field (which has been shown to produce themirror-image H(z) profiles), we must conclude that thesymmetry in the m*(T) curves arises from the effects ofthe non-uniform field experienced by the sample as it istransported during measurement. to Figure D5(b), whenHset = 0 is set from Hset = +50 kG, the FCC-m*(T) curveinitially turns positive as the temperature is lowered belowTc, reaches a maximum and then becomes more negativeeventually reaching a plateau at a negative value of m*.

Figure D6(a). The SQUID output signals, V(z), near Tc for theFCC data shown in Figure D5.

12

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Support for two competing effects can be found in theV(z) data of Figure D6(a). At all the temperatures studiedthe FCC-V(z) curves are related by a simple symmetryoperation (rotation about V = constant). At the lowesttemperatures the V(z) curves exhibit constant-dipolebehavior while at higher temperatures (T > 78 K) it isevident that the extrema that are normally found atz = ±1.25 cm are observed to be much closer to z = 0. Thelower-temperature data are consistent with constant-dipolebehavior in which the true magnetic moment of the sampledoes not change much during the measurement process,while the higher-temperature “compressed-peak” behaviornear Tc is more consistent with a moment that is changingsignificantly during the measurement.

At lower temperatures the two ZFC-m*(T) curves shownin Figure D5(a) are separated by an amount similar to theseparation of the two FCC-m*(T) curves and then fortemperatures very close to Tc they become mirror imagesabout m* = 0. Very close to Tc the ZFC curves followtheir corresponding FCC curves breaking away near thepeaks in the FCC curves. Figure D6(b) shows the SQUIDresponse V(z) curves for the data collected under ZFCconditions. At the lowest temperatures it can be seen thatthe V(z) curves are very similar to the ideal SQUIDresponse signal, which is consistent with production of awell-established critical state at the lowest temperature.Near Tc, however, the ZFC-V(z) signals become quiteanomalous. For the V(z) data collected using Hssq = +50kG ⇒ 0 G, it can be seen that the maxima located beyondz = ±1.25 cm in the ZFC-V(z) curves move closer toz = ±1.1 cm as the temperature increases from T = 86 Kto 88.5 K. In contrast to this, when Hset = 0 is set usingHssq = -50 kG ⇒ 0 G, a new maximum starts to push innear the center (z = 0) of the V(z) curve as the temperaturemoves above T = 87 K and this new maximum becomesquite clear by T = 88.5 K. Hence, the ZFC-V(z) curvesmeasured using the two different field-change sequencesare quite different in the temperature range from about T= 86 K to 89 K. At T = 89 K and above, the curvesassociated with the different field-change sequencesbecome related by a simple symmetry operation, i.e.,reflection of the V(z) data across a line V = constant. Thissymmetry continues to be observed in the V(z) curvesto temperatures above Tc. An examination of the twoZFC-m*(T) curves, at temperatures of 89 K and above,shows them to be symmetric about m* = 0.

Since we know that the two field-change sequencesproduce H(z) profiles that are mirror images of oneanother, the ability to relate the associated V(z) responsesby a simple symmetry operation is fairly strong evidencethat the magnetic response observed in the two ZFC-m*(T)curves near Tc is dominated by the effects of the non-uniform field experienced by the sample as it is transported

Figures D6(b). The SQUID output signals, V(z), fortemperatures below Tc for the ZFC data shown in Figure D5.

-3

-2

-1

0 T= 86.0 K

-2

-1

0 T= 87.5 K

-2

0T= 88.0 K

-2

0

T= 88.5 K

ZFC ZFC

No-overshoot Mode

-1

0T = 86.0 K

-2

-1

0 T = 88.0 K

-2

0 T = 87.5 K

-4

-2

0 T = 88.5 K

Hssq = +50 kG ⇒ 0 G ⇒ +25 G Hssq = -50 kG ⇒ 0 G ⇒ +25 G

T= 89.0 K

0

2

T= 90.0 K

2

4 T= 92.5 K

0 1.25-1.25

z (cm)

0

2

4

-4

-2

0

2 T = 89.0 K

-2

0 T = 90.0 K

-2

0 T = 92.5 K

0 1.25-1.25

z (cm)

The peak in the FCC-m*(T) curve strongly suggests thepresence of two competing effects, both of which areproduced by the transport through the field non-uniformities. One of these effects dominates near Tc

causing the initial upturn, while the second effect increaseswith decreasing temperature, eventually dominating thebehavior.

13

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QUANTUM DESIGN

during measurement. The differences in the two ZFC-m*(T) curves (around 86 K to 89 K) probably result fromthe competition of two properties of the superconductor,one property being the pinning of flux, with its associatedflux gradients, that is generated by the application of afield only after cooling to low temperature, while thesecond property is associated with effects arising fromtransport through the non-uniform field. Near Tc thecomponent of the magnetic moment associated with thetrapped flux finally has a magnitude similar to that of theapparent moment associated with the effects produced bythe field non-uniformities. Thus, the m* values observednear Tc are not the actual magnetic moment m values ofthe sample, but rather the best computer fit of the idealSQUID response to the observed V(z) output. However,at lower temperatures the results should be an accurate

Hssq = +50 kG ⇒ 0 G ⇒ +25 G

3

3.5

4T = 90.0 K

3.5

4

4.5

T = 89.75 K

3.5

4

T = 89.5 K

3.5

4

4.5

T = 89.0 K

-2

-1.5

-1T = 90.0 K

-0.5

0

T = 89.75 K

0.5

1T = 89.50 K

-1

-0.5

0

T = 89.0 K

-1.25 0.0 1.25

z (cm) z (cm)

Oscillate Mode

FCC Hssq = -50 kG ⇒ 0 G ⇒ +25 G

FCC

-1.25 0.0 1.25

-1 10- 2

-8 10- 3

-6 10- 3

-4 10- 3

-2 10- 3

0 100

2 10- 3

0 2 0 4 0 6 0 8 0 100

ZFC (Hs s q

= +50 kG ⇒ 0 G ⇒ +25 G)

FCC (Hs s q

= +50 kG ⇒ 0 G ⇒ +25 G)

ZFC (Hs s q

= -50 kG ⇒ 0 G ⇒ +25 G)

FCC (Hs s q

= -50 kG ⇒ 0 G ⇒ +25 G)

m*

(em

u)

T (K)

Oscillate Mode

Hs e t

= 25 G

(a)

-4 10- 5

-2 10- 5

0 100

2 10- 5

7 5 8 0 8 5 9 0 9 5

ZFC (Hs s q

= +50 kG ⇒ 0 G ⇒ +25 G)

FCC (Hs s q

= +50 kG ⇒ 0 G ⇒ +25 G)

ZFC (Hs s q

= -50 kG ⇒ 0 G ⇒ +25 G)

FCC (Hs s q

= -50 kG ⇒ 0 G ⇒ +25 G)

m*

(em

u)

T (K)

Oscillate ModeH

s e t = 25 G

( b )

Figure D7. (a) ZFC- and FCC-m*(T) curves measured atHset = +25 G. Hset = 0 was set above Tc using the field-changesequences Hssq = +50 kG ⇒ 0 G and Hssq = -50 kG ⇒ 0 G in theoscillate mode, while the measuring field Hset = +25 G wasapplied at T = 5 K in the no-overshoot mode after zero-fieldcooling. (b) Expanded view of data near Tc.


-1.25 0.0 +1.25

-3

-2

-1

0 T = 89.0 K

-3

-2

-1

0 T = 89.5 K

-2

-1.5

-1

-0.5 T = 89.75 K

z (cm)

-4

-2

0

2 T = 89.0 K

-2

-1

0

1

2T = 89.5 K

0.6

0.8

1

1.2T = 89.75 K

1.4

1.6

1.8

T = 90.0 K -1.8

-1.6

-1.4

-1.2T = 90.0 K

-1.25 0.0 +1.25z (cm)

Oscillate Mode

ZFC H = +50 kG ⇒ 0 G ⇒ +25 Gssq H = -50 kG ⇒ 0 G ⇒ +25 Gssq

ZFC

Figure D8(b). The SQUID output signals, V(z), near Tc for theZFC data shown in Figure D7.

14

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○


effects of these differences are reflected in the m*(T)curves in Figure D7(b), which no longer exhibit the nearlyperfect symmetry about m* = 0 that was observed for theno-overshoot mode. As is also evident in Figure D7(b),as a result of using the oscillate mode, the maximum inthe ZFC- and FCC-m*(T) curves is located somewhatcloser to Tc (89.7 K as compared to 89.1 K for themeasurements made using the no-overshoot mode.) Notethat the magnitude of m* for the pair of extrema in theFCC-m*(T) curves is about 1 x 10-5 emu when the oscillatemode is used, compared to m* values of nearly 1 x 10-4

emu for the same sample using the no-overshoot mode.This is entirely consistent with our previous observationthat the field uniformity following an oscillate modechange is about a factor of 10 more uniform than whenusing the no-overshoot mode. Thus, the oscillate modecan produce a nearly ten-fold reduction in the magnitudeof the apparent moments associated with field non-uniformities. Figures D8(a) and D8(b) show the FCC-and ZFC-V(z) curves for several temperatures near Tc forthe data shown in Figure D7(b).

It was also shown in Section C that the field non-uniformities could be reduced even further by using themagnet-reset option. Figure D9(a) shows the ZFC- andFCC-m*(T) curves measured at Hset = 25 G, for the twofield-change sequences Hssq = +50 kG ⇒ 0 G and Hssq =-50 kG ⇒ 0 G set using the reset option. [Figure D9(b)shows the same data in an expanded view in a narrowtemperature range near Tc.] As shown in Figure C4, bothHset = +25 G curves have the same shape and amplitudevariation, and the curves vary by less than 0.1 G betweenz = -2.5 cm and 2.5 cm, with larger (~ 0.1 G) deviationsfor |z| > 2.5 cm. The similarity of the H(z) curves isreflected in the m*(T) curves in Figure D9(b), which againshow the high degree of symmetry about m* = 0 similarto that observed for the no-overshoot mode. It is alsoevident that the maxima in the ZFC- and FCC-m*(T)curves are now located very close to Tc. The m* valuesfor the extremum in the FCC-m*(T) curves are furtherreduced, having an apparent magnitude of about 5 x 10-6

emu, compared to an apparent magnitude of about1 x 10-5 emu for the oscillate mode and 1 x 10-4 emu forthe no-overshoot mode. The sharp change in both ZFCand FCC curves so close to Tc argues for an onset of strongflux pinning very close to Tc. Figures D10(a) and D10(b)show the V(z) curves for several temperatures near Tc forthe data shown in Figure D9(b).

E . M o d e l i n g t h e S Q U I D S i g n a l sR e s u l t i n g f r o m F i e l d N o n - U n i f o r m i t i e s

In Section D it was shown that the low-field FCC-m*(T)data appear to be a result of the particular magnetic

-1 10- 2

-8 10- 3

-6 10- 3

-4 10- 3

-2 10- 3

0 100

2 10- 3

0 2 0 4 0 6 0 8 0 100

ZFC (Hs s q

= +50 kG ⇒ 0 G ⇒ +25 G)

FCC (Hs s q

= +50 kG ⇒ 0 G ⇒ +25 G)

ZFC (Hs s q

= -50 kG ⇒ 0 G ⇒ +25 G)

FCC (Hs s q

= -50 kG ⇒ 0 G ⇒ +25 G)

m*

(em

u)

T (K)

Magnet-reset ModeH

s e t = 25 G

(a)

Figure D9. (a) ZFC- and FCC-m*(T) curves measured atHset = +25 G. Hset = 0 was set above Tc using the field-changesequences Hssq = +50 kG ⇒ 0 G and Hssq = -50 kG ⇒ 0 G usingthe magnet-reset option to set Hset = 0. The measuring fieldHset = +25 G was applied at T = 5 K in the no-overshoot modeafter zero-field cooling. (b) Expanded view of data near Tc.

measurement of the magnetic moment of the sample withinthe error associated with the contribution from the non-uniformities.

It was shown in Section C that by using the oscillate modeto set Hset = 0 it was possible to reduce the H(z) curvature.Figure D7(a) shows the ZFC- and FCC-m*(T) curvesmeasured at Hset = +25 G when the oscillate modewas used above Tc for the two field-change sequencesHssq = +50 kG ⇒ 0 G and Hssq = -50 kG ⇒ 0 G prior tosetting Hset = +25 G at T = 5 K. [Figure D7(b) shows thesame data in an expanded view in a narrow temperaturerange near Tc.] Figure C3 showed that when Hset = +25 Gwas set from these two conditions, the mirror symmetryobserved about H = +25 for the no-overshoot mode wasnot observed. In particular there is a difference in theamplitude of the variations for the two H(z) curves. The

-4 10- 5

-2 10- 5

0 100

2 10- 5

7 5 8 0 8 5 9 0 9 5

ZFC (Hs s q

= +50 kG ⇒ 0 G ⇒ +25 G)

FCC (Hs s q

= +50 kG ⇒ 0 G ⇒ +25 G)

ZFC (Hs s q

= -50 kG ⇒ 0 G ⇒ +25 G)

FCC (Hs s q

= -50 kG ⇒ 0 G ⇒ +25 G)

m*

(em

u)

T (K)

Magnet-reset ModeH

s e t = 25 G

( b )

15

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

QUANTUM DESIGN

behavior of a superconductor moving through a non-uniform magnetic field, rather than a measurement of aconstant magnetic moment. By merely setting themeasuring field (Hset = +25 G) of the magnet using twodifferent field-change sequences, two different FCC-m*(T)curves that are mirror images about m* = 0 were produced.The two sequences differ only in that one case involvessetting Hset = 0 G from Hset = +50 kG, while in the secondcase Hset = 0 G was set from Hset = -50 kG. It was shownin Section C that these two field-change sequencesgenerate H(z) profiles which are mirror images aboutH = +25 G (see Figure C2). Thus, the mirror symmetryof the FCC-m*(T) curves about m* = 0 is not at allsurprising when the effects of the field non-uniformitieson the V(z) SQUID signals are considered.

As discussed in Section D, near Tc the ZFC-m*(T)behavior also appears to be dominated by a magneticresponse associated with the field non-uniformities. It isinteresting to see how the systematic reduction in the fieldnon-uniformity, by using the oscillate mode and then themagnet-reset option, leads to a reduction in thetemperature range below Tc for which artifacts appear todominate the ZFC behavior.

By considering accepted models of superconductorbehavior, and using the experimentally measured H(z)profiles, it should be possible to account for the observedmagnetic responses.8,9 The very large differences betweenZFC- and FCC-m*(T) curves observed under the mostuniform field conditions (using the magnet reset option)suggest that strong-pinning effects completely dominatevery close to Tc. This indicates that the Bean Critical Statemodel should be applicable to the observed behavior, atleast in the cases where the field non-uniformity is largeenough to produce significant flux gradient penetrationand where Jc is large enough to produce significantmagnetic moments.

Using the Bean Critical State model, calculations of theexpected SQUID output signals, V(z), and the positiondependence of the true magnetic moment m(z), that aregenerated as the superconductor is transported through anon-uniform field, were made for a wide range of Jc values(see Appendix A for more details). (Reference 9 is a morerealistic calculation that incorporates demagnetizationeffects.) Examples of the results of calculations for large,intermediate, and small Jc values (relative to the appliedfield) are shown in Figure E1. A second-order-polynomialfit to the measured H(z) profiles shown in Figure C2 wasused for the calculations in Figure E1, with the H(z) curvedisplaced using an additive constant so that m(z) had theinitial value m = 0 at the point where transport throughthe detection coil and the non-uniform field would begin(z = -3). The H(z) curve models the transport of the sample

3.2

3.3

3.4T = 91.0 K

2.8

3

3.2T = 90.5 K

3

3.2

3.4T = 90.0 K

3

3.2

3.4T = 89.0 K

0.9

1

1.1

T = 91.0 K

1

1.2

1.4

1.6

T = 90.0 K

0.8

1

1.2

1.4

T = 90.5 K

1.6

1.8

2

T = 89.0 K

z (cm) z (cm)

Magnet-reset Mode

H ssq = +50 kG ⇒ 0 G ⇒ +25 G

-1.25 0.0 +1.25 -1.25 0.0 +1.25

H ssq = -50 kG ⇒ 0 G ⇒ +25 GFCC FCC


Figure D10(b). The SQUID output signals, V(z), near Tc for theZFC data shown in Figure D9.

-1.5-1

-0.50

0.5T = 89.0 K

2.3

2.35

2.4T = 90.0 K

3.3

3.4

3.5

T = 90.5 K

1.7

1.8

1.9

T = 91.0 K

T = 89.0 K

0.5

1

1.5T = 90.0 K

-2

-1

0

0.7

0.8

0.9T = 90.5K

0.3

0.35

0.4

T = 91.0 K

-1.25 +1.250.0

Hssq = +50 kG ⇒ 0 G ⇒ +25 G

Magnet-reset Mode

-1.25 +1.250.0

z (cm)z (cm)

Hssq = -50 kG ⇒ 0 G ⇒ +25 G

ZFC ZFC

16

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○


from H = 0 through positive field values. It can be seen inFigure E1(a) that, for the case of small Jc values, the smallfield increase that occurs shortly after the sample beginsthe transport process is capable of driving the sample intothe critical state [i.e., m(z) becomes a constant after a smallfield change]. In this case, as the sample passes the center(z = 0) the field decreases and then m(z) increases, withm(z) actually attaining a positive value by the end of itstravel. The convoluted shape of the associated SQUIDoutput signal, V(z), is also shown in Figure E1(a). Thebehavior of m(z) and V(z) are shown in Figure E1(b) forintermediate Jc values and in Figure E1(c) for large Jc

values. For the intermediate Jc value chosen here, thecritical state is reached closer to the center of the travelwhile in the large Jc case, the field non-uniformities arenot large enough for the sample to reach the critical stateand m changes continuously during transport. In all casesfor the H(z) profile used, the sample has a positive momentby the end of its travel (at z = 3.0 cm).

To account for the observed behavior it is useful toconsider the effects expected when a strong-pinningsuperconductor is cycled through a small m(H) loop. Inthe present case, the changes in H occur when the sampleis moved through the non-uniform field during ameasurement. The M(H) behavior for a strong-pinningsuperconductor is shown schematically in Figure E2.Cycling a strong-pinning superconductor up and down infield from H = 0 [through half an M(H) loop] leaves thesuperconductor in a remanent state, and cycling to positivefield values will leave the superconductor in a remanentstate with a positive magnetic moment (or magnetization).This effect was pointed out in the Bean-model calculationsshown in Figure E1, where an excursion to positive fieldsdue to the particular H(z) profile produces a positivemagnetic moment which is evident as the positive m valueat the end of the travel. Very close to Tc, Jc is very smalland a significant remanent magnetization is not likely tobe produced. However as Jc increases with decreasingtemperature, cycling the sample in the non-uniform fieldshould produce a significant remanent magnetization,particularly when the field non-uniformities are large.

The calculations shown in Figure E1 and the schematic ofthe Bean model in Figure E2, suggest that after onemeasuring cycle the sample should actually have aremanent magnetization prior to the beginning of thesample’s next cycle. Within the Bean model there is alimit on the size of this remanent magnetization. Whenthe sample begins moving to positive (increasing) fields— as when Hssq = -50 kG ⇒ 0 G is used but with a positiveremanent magnetization (at z = -3.0 cm) — the truemagnetic moment will gradually decrease, cross throughm = 0, and become negative, reaching a minimum at theposition of largest positive field value (usually at the

0

0.8

1.6

-0.4

0

0.4

- 3 - 2 - 1 0 1 2 3

m (

emu) V

(V)

z (cm)

Bean Model: High Field/Small JC

Hs e t

= 0.0 G

(No-overshoot ModeH

s s q = -50 kG ⇒ 0 G)

V(z)

m(z)

(a)

- 2

- 1

0

1

2

- 4

- 2

0

2

4

- 3 - 2 - 1 0 1 2 3

m (

emu) V

(V)

z (cm)

Bean Model: Low Field/High Jc

H = 0.0 G( No-overshoot ModeH

s s q = -50 kG ⇒ 0 G)

(c)

Figure E1. Calculations of m(z) and V(z) based on the BeanCritical State model in which the value m = 0 was used at thestart of the measuring cycle (z = -3 cm). The experimentallymeasured H(z), following the Hssq = 50 kG ⇒ 0 G sequence inthe no-overshoot mode, was used in the calculation. (a) A verysmall Jc value was used in which case the critical state isattained with only a small field change (close to z = -3). (b) Anintermediate value of Jc in which case the critical state is reachedafter more than 1 cm of travel. (c) A large Jc value in which casethe critical state is not attained over the full field excursionexperienced during the measurement.

- 1

0

1

2

3

- 2

- 1

0

1

2

- 3 - 2 - 1 0 1 2 3

m (

emu) V

(V)

z (cm)

Bean Model: Intermediate Field/Intermediate Jc

Hset

= 0.0 G

( No-overshoot ModeH

s s q = -50 kG ⇒ 0 G )

( b )

V(z)

m(z)

17

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

QUANTUM DESIGN

Figure E2. (a) A magnetic moment as a function of appliedmagnetic field curve for a sample exhibiting ideal Bean-Critical-State-model behavior. (b) The effects on magnetic momentresulting from cycling the sample through the non-uniformmagnetic field. In this case the H(z) profile results in the cyclingof the sample to positive field values leaving the sample with apositive remanent magnetization. The sample follows thereverse path as it is returned to the starting position. Data iscollected only on the initial path. (c) In this case the H(z)profile results in the cycling of the sample to negative fieldvalues leaving the sample with a negative remanentmagnetization.

m

H

Strong Pinningm(H) Loop

(a)

H

z∆H

z∆H

H

measure

reset starting position

1 2 3

45

z= 0

∆H

m'

∆H

m'

(b)

(c)12

3

4

5

12

3

4

5

m'rem

m'rem

H

H

- 1

0

1

2

3

-1.5

- 1

-0.5

0

0.5

1

- 3 - 2 - 1 0 1 2 3

m (

emu) V

(V)

z (cm)



s s q = -50 kG ⇒ 0 G)

V(z)

m(z)

(a)

center, z = 0). After passing the field maximum, the truemagnetic moment will become more positive, finallyreturning to its positive starting value. An m(z) curveexhibiting this behavior is shown in Figure E3(a). Acomputer fit to this V(z) signal would return a negativevalue for the magnetic moment. Thus, the “apparent”effect of flux pinning produced by field non-uniformities(when the field excursions initially move the sample tomore positive fields) would be to produce a negativeapparent magnetization. This should be the case no matterhow large Jc becomes. This sequence was used for theFCC-m*(T) data shown in Figures D5, D7, and D9. Thus,it can be seen that in all cases the apparent magnetizationbecomes negative just below Tc, and in the case when thereset option was used, m* stays negative down to the lowest

Figure E3. Bean model calculations similar to those of FigureE1 except that the magnetic moment at the start of a scan(z = -3) is not zero. From (a) to (c) the initial m value increasesshowing that the shape of the V(z) curve evolves from one thatwould produce a negative apparent magnetic moment m* value,(a) when there is a small initial m value, to one that wouldproduce a positive apparent magnetic moment m* value,(c) when there is a larger initial m value. This scenario isnot supported by present Bean model calculations, even thosewhich include demagnetization effects.

2

3

4

5

- 4

- 2

0

2

4

6

- 3 - 2 - 1 0 1 2 3

m (

emu) V

(V)

z (cm)



s s q = -50 kG ⇒ 0 G) V(z)

m(z)

(c)

0

1

2

3

- 2

- 1

0

1

2

- 3 - 2 - 1 0 1 2 3

m (

emu) V

(V)

z (cm)



s s q = -50 kG ⇒ 0 G)

V(z)

m(z)

( b )

18

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○


temperatures. However, the FCC-m* data for the oscillateand no-overshoot modes reverse sign at lower temperatures.

Figure E3(b) and E3(c) show calculations that demonstratewhat could happen if the remanent magnetization couldincrease beyond the expectations of the Bean model. Withincreasing remanent magnetization at the beginning of thesample’s travel, the V(z) signal evolves from what initiallywould produce a negative apparent moment m* using acomputer fit [Figure E3(a)], into a signal that wouldproduce a positive apparent moment m* using a computerfit [Figure E3(c)]. Unfortunately, even though thisscenario (of larger and larger remanent magnetizationvalues) could explain the observations there is no apparentreason why the remanent magnetization should increasebeyond the expectations of the Bean model. Thus, wepresently have no explanation for the reversal in sign ofthe FCC-m*(T) curves that is observed when the no-overshoot or oscillate modes are used prior to setting themeasuring.

At some point below Tc, the effects of field non-uniformities will no longer dominate the ZFC-m*(T) data.The crossover from artifact dominated m* values to morequantitatively-accurate magnetic-moment values occurscloser to Tc as the field uniformity increases. When usingour particular sample in the specific geometry chosen, themagnitude of V(z) generated by field non-uniformitieswhen the no-overshoot mode is used, is comparable to theV(z) signal generated by a sample with a dipole-momentof about 2.5 x 10-4 emu, while for the oscillate mode itcompares with a moment of about 1.5 x 10-4 emu. For themagnet reset option, the V(z) associated with fieldnon-uniformities is comparable to a moment of about0.5 x 10-4 emu. For the ZFC-m*(T) measurement, thesample has an m* magnitude of about 3.5 x 10-2 emu atthe lowest temperatures and this gradually decreases withincreasing temperature. The crossover from accuratemagnetic moment data to artifact-dominated data occursapproximately at the point where the ZFC-m*(T) curvesassociated with the two different field-change sequencessplit. For the no-overshoot data this occurs at atemperature of about 84.2 K, where the apparent magneticmoment m* has a magnitude of about -3 x 10-3 emu.For the oscillate mode the split occurs at T ≈ 89.5 K (andm* ≈ 4 x 10-5 emu) while there is no evidence of a split forthe data collected after using the magnet reset option.

For those cases in which temperature is the parameter ofimportance (e.g., Tc) it is possible to use the appearanceof non-dipole V(z) signals to make very accuratedeterminations of the characteristic temperature assuggested by Suenaga et al.3 We have determined Jc

to theresolution of the 0.25 K temperature steps using theappearance of non-dipole V(z) signals. Remarkably, the

larger the non-uniformity of the field, the larger will bethe non-dipole signals produced, so that when determininga characteristic temperature one may actually takeadvantage of data which might otherwise be regarded as a“problem” or artifact. The V(z) signal is always alegitimate measurement of the magnetic response. Theappropriate analysis of the output signal can providevaluable quantitative magnetic information.

F . E f f e c t s o f F i e l d N o n - U n i f o r m i t i e si n L a r g e A p p l i e d F i e l d s

As the magnetic field is increased, the onset of irreversiblemagnetic behavior at the vortex-solid phase transition Tg

moves to lower temperatures. The combination of ZFC-and FCC-m*(T) measurements has become a standardmethod for identifying the onset of irreversibilities at Tg.This method exploits the history dependence associatedwith irreversible behavior, since as mentioned earlier, theobserved M value depends on the particular sequence offield and temperature changes that brought it to its presentstate. The temperature range over which the ZFC- andFCC-m*(T) curves are separated corresponds to the regionof (strong) pinning, and therefore irreversible behavior,while the temperature range over which the curves aresuperimposed usually corresponds to reversible behavior.

The application of a large field can lead to some additionalcomplications. What had been a very small backgroundcontribution in small fields can now become a significantcontribution to the signal. An extreme example of thiseffect is a thin superconducting film on a relatively thicksubstrate. The very small mass of superconductingmaterial — in the present case there is about 5 micrograms(5 x 10-6 g) of superconductor on a substrate with a massof 34 milligrams (34 x 10-3 g) — means that above Tg thesignal can be strongly dominated by the substrate. If theJc of the film is large enough, the superconductor’smagnetization will dominate at some temperature belowTg, and the larger the value of Jc, the closer in temperatureto Tg i t will dominate. (While a diamagneticsuperconducting response is expected between the vortex-solid transition and Hc2, this signal is usually too small toobserve experimentally for a thin film sample.) The sameconsiderations exist for crystals and bulk samples as forthin films, with the relative magnitudes of the backgroundand sample signal determining how important thebackground contribution will be.

When the contributions of the background and sample areof comparable magnitude, it is often not a simple matterto subtract out the background contribution. A furtherproblem arises when the center-of-mass of the sample andthe center-of-mass of the background are displaced

19

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

QUANTUM DESIGN

spatially (primarily along z). This is a problem for allkinds of magnetometer measurements, not just for SQUIDmagnetometer measurements made with the MPMS, sincemagnetometers are usually designed to determine themagnetic moment of samples that approximate pointdipoles. (This issue was discussed at some length in theQuantum Design Application Note #1.) Another effectwhich can also complicate an otherwise straight-forwardbackground correction is the fact that below Tc thescreening of the superconductor can change the localmagnetic field experienced by the background. Thus,measuring the background above Tg or in a separatemeasurement may not reflect its actual contribution belowthe Tg of the sample.

Figure F1(a) shows ZFC- and FCC-m*(T) curves measuredat Hset = +10 kG over the full temperature range studied.Hset = 0 was set above Tc using the field-change sequenceHssq = +50 kG ⇒ 0 G or Hssq = -50 kG ⇒ 0 G in the oscillatemode, while the measuring field Hset = +10 kG was appliedat T = 5 K using the no-overshoot mode after zero-fieldcooling. On the scale in Figure F1(a) the FCC-m*(T) dataremains very close to zero, with some small featuresevident at higher temperatures. Figure F1(b) shows thesame results with expanded scales. There are clearlydifferences between the data in which the Hssq = +50 kG⇒ 0 G sequence was used in the field preparationcompared to the data in which Hssq = -50 kG ⇒ 0 G wasused. In Figure F1(b) it is much more evident how m*increases just below Tg for both field-change sequences.This reflects an evolution of the H(z) profile at largerfields. Also more evident is the significant diamagneticsignal above Tg which has its dominant contribution fromthe substrate. Since the parameter of interest in the ZFC/FCC measurement is a characteristic temperature ratherthan a magnetization value, it is possible to use the changesin the V(z) signals to make very accurate determinationsof the onset temperature for irreversible behavior.3

Figure F1(c) shows the same data near Tg with scales

further expanded. The vortex-solid transition at Tg isidentified in the figure as the point where the pairs of ZFC-and FCC-m*(T) curves become superimposed. Becausethe signal in this case is so heavily dominated by thesubstrate there is no evidence of the equilibriummagnetization of the superconductor expected attemperatures between Tg and the temperature of the uppercritical field, Tc2. The V(z) signals near Tg are shown inFigure F2 where it can be seen that there is a dramaticchange in the shape of the V(z) signal at Tg, with constant-dipole curves above Tg ≈ 85 K and extremely anomalouscurves below Tg. Figure F3 shows the V(z) signalsassociated with the data in Figure F1 over a larger rangeof temperatures.

Figure F1. ZFC- and FCC-m*(T) curves measured atHset = +10 kG. Hset = 0 was set above Tc using the field-changesequences Hssq = +50 kG ⇒ 0 G and Hssq = -50 kG ⇒ 0 G in theoscillate mode, while the measuring field Hset = +10 kG wasapplied at T = 5 K using the no-overshoot mode after zero-fieldcooling. (a) Measurements over the full temperature range,(b) on expanded x and y scales, and (c) near Tg.

20

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○


Figure F2. The V(z) signals near Tg for the FCC data shown inFigure F1.

The ZFC- and FCC-m*(T) curves measured at Hset = +10kG — when the no-overshoot mode was used during thefield-change sequences Hssq = +50 kG ⇒ 0 G and Hssq =-50 kG ⇒ 0 G — are shown in Figure F4. These curvesare slightly different than those observed for the oscillatemode, and a brief examination of Figure F1(c) and F4(c)show that it is easier to determine the temperature of thetransition, Tg, when the less uniform field is used. InFigure F4(c) there is a clear departure from the foursuperimposed curves at a temperature of 86 K.

The H(z) profiles in Figure F5 show how H(z) evolves asHset increases from +1 kG to +5 kG. The associated FCC-m*(z) curves are shown in Figure F6. It can be seen thatfor the curves in which Hssq = +50 kG ⇒ 0 G was used,the H(z) profiles evolve from having a minimum near z =0, into an asymmetrical curve (with respect to z = 0) witha minimum, and finally into curves with a maximum nearz = 0. On the other hand, the H(z) curves for which Hssq =-50 kG ⇒ 0 G was used, always have a maximum near z =0, thus the H(z) profile with a maximum near z = 0 ischaracteristic of increasing H with a change of about 5kG required to reverse the H(z) profile (for a magnet stateproduced under these particular conditions). The evolutionof the H(z) profiles is reflected in the FCC-m*(z) data inFigure F6 where the curves associated with Hssq = +50 kG⇒ 0 G become more positive and progressively approach

-5

-4

-3

-2

-1 T = 85.5 K

-4

-3

-2

-1

0 T = 85.0 K

-4

-3

-2

-1 T = 84.5 K

-4

-3

-2

-1

0 T = 84.0 K

-3

-2

-1

0

1 T = 83.5 K

-3

-2

-1

0 T = 83.0 K

-3

-2

-1

0

1 T = 82.5 K

-4

-2

0

2 T = 81.5 K

H = -50 kG ⇒ 0 ⇒ +10 kG

Oscillate Mode

FCC

-1.25 +1.250.0 -1.25 +1.250.0

z (cm)z (cm)

ssq

-2

-1

0T = 61.0 K

-3

-2

-1

0 T = 70.0 K

-3

-2

-1

0 T = 75.0 K

-2

-1

0

1

2T = 80.0 K

-2

-1

0

1

2T = 81.5 K

-2

-1

0

1

2T = 82.5 K

-4-3-2-101 T = 85.5 K

-4-3-2-10

1 T = 90.0 K

H ssq = +50 kG ⇒ 0 G ⇒ +10 kGZFC

-1.25 +1.250.0 -1.25 +1.250.0

z (cm)z (cm)

Oscillate Mode

Figure F3(a). Some of the SQUID output signals V(z) associatedwith the ZFC data shown in Figure F1.

-2

-1

0 T = 90.0 K

-2

-1

0T = 85.5 K

-2

0T = 82.5 K

-4

0

4 T = 80.5 K

-4

-2

0 T = 75.0 K

-5

-3

-1T = 70.0 K

-6

-4

-2

0 T = 61.0 K

H ssq = +50 kG ⇒ 0 G ⇒ +10 kGFFC

Oscillate Mode

-1.25 +1.250.0

z (cm)

-1.25 +1.250.0

z (cm)

Figure F3(b). Some of the SQUID output signals V(z) associatedwith the FCC data shown in Figure F1.

21

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

QUANTUM DESIGN

990

1000

1010

Hssq = +50 kG ⇒ 0 G ⇒ +1 kGHssq = -50 kG ⇒ 0 G ⇒ +1 kG

H (

G)

1990

2000

2010

H (

G)

2980

3000

H (

G)

5000

5010

5020

-3 -2 -1 0 1 2 3

H (

G)

z(cm)




the shape of the curves associated with the sequenceHssq = -50 kG ⇒ 0 G as the field increases to +5 kG.

When the V(z) signals deviate significantly from aconstant-dipole response some rather unusual effects canbe observed in the behavior of the apparent magneticmoment. For example, the results in Figure F7 show largediscontinuities in the m*(T) curves when the iterativeregression method (rather than the linear regressionmethod used in this study) was used to analyze the V(z)results. In fact, there is a factor of about 1.6 between thevalues of m* above and below the discontinuities.

Figure F5. The H(z) profiles produced using the no-overshootmode as measured using a Hall-effect device, evolve in shapeas the magnetic field value is increased. For the case whenHssq = -50 kG ⇒ 0 G is used to set the initial field, the fieldretains a maximum at the center (z = 0), whereas whenHssq = +50 kG ⇒ 0 G is used, there is an evolution from aminimum to a maximum as the field increases to Hset = +5 kG.

Figure F4. ZFC- and FCC-m*(T) curves measured atHset = +10 kG. Hset = 0 was set above Tc using the field-changesequences Hssq = +50 kG ⇒ 0 G and Hssq = -50 kG ⇒ 0 G in theno-overshoot mode, while the measuring field Hset = +10 kG wasapplied at T = 5 K in the no-overshoot mode after zero-fieldcooling. a) Measurements over the full temperature range,b) on expanded x and y scales, and c) near Tg.

22

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Figure F7. ZFC- and FCC-m*(T) curves measured atHset = +10 kG. Hset = 0 was set above Tc using the field-changesequence Hssq = +50 kG ⇒ 0 G in the no-overshoot mode, whilethe measuring field was applied at T = 5 K after zero-fieldcooling. The discontinuities are associated with a change in thecomputer analysis method during the measurement, when a largeerror is encountered in the iterative regression algorithm.

Repeating the signal analysis using the linear regressionmethod produces m*(T) curves without discontinuities.The discontinuities are associated with a change in theanalysis method during the measurement. The computersoftware places limits on the errors associated with theiterative regression fit, and when this error is exceededthe system will repeat the signal analysis using the linearregression method. Since the signal is not a dipole-likeresponse, the different algorithms used in the different fitroutines can produce rather different apparent magneticmoments m*. For this reason, in this study the linearregression routine was used to generate all of the m*(T)curves other than those in Figure F7.

0

0.0004

0.0008

0.0012 Hssq= +50 kG ⇒ 0 G ⇒ +1 kG

m*

(em

u)

0

0.0002

0.0004

0.0006

m*

(em

u)

-0.0001

0.0001

0.0003

0.0005

60 70 80 90 100

m*

(em

u)

T (K)

(a)

(b)

(c)

0.0000

Hssq= -50 kG ⇒ 0 G ⇒ +1 kG

Hssq= +50 kG ⇒ 0 G ⇒ +3 kGHssq= -50 kG ⇒ 0 G ⇒ +3 kG

Hssq= +50 kG ⇒ 0 G ⇒ +5 kGHssq= -50 kG ⇒ 0 G ⇒ +5 kG

Figure F6. The evolution of the H(z) profiles shown in Figure F5lead to a related evolution in the FCC-m*(T) behavior near thevortex-solid phase transition at Tg. In 1 kG (a) the FCC-m*(T)curve associated with Hssq = +50 kG ⇒ 0 G takes on morepositive values and by Hset = +5 kG (c) the two FCC-m*(T)curves exhibit nearly the same behavior.

G . I m p l i c a t i o n s o f A n o m a l o u s S i g n a l sf o r M a g n e t i z a t i o n M e a s u r e m e n t s

Most of the problems described here have a very longhistory that dates back far beyond the appearance of theMPMS or any other SQUID magnetometer. When asample and its background do not approximate a constant,point dipole, complications can arise with anymagnetometer. Field uniformity is always a problem thatcan be reduced by shortening the sample travel-lengthduring measurement or by improving the properties of themagnet. The MPMS incorporates a high-uniformitysuperconducting magnet, but in addition there are featuresdesigned to accommodate a reasonably short scan lengthand, as shown, several field-change methods available forimproving the field uniformity.

Since there are limits to how much improvement can, orshould, take place in automated signal analysis, it isimportant for the scientist to understand the potentialpitfalls in the measurements they are making. It is alsoimportant to know exactly what property (quantity) onewishes to measure in order to decide if the data will besuitable. For example, the values of m* reported by theMPMS using computer fit routines, for temperatures veryclose to Tc and Tg, are clearly not accurate determinationsof the true magnetic moment m of the sample. However,in the particular case presented here, the ZFC/FCC methodcan still provide a very accurate determination of thetransition temperatures Tg

and Tc. In fact, some of the“problems” associated with field non-uniformity discussed

23

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

QUANTUM DESIGN

here may actually enhance the accuracy of measuring thesecharacteristic temperatures.

ZFC-m*(T) and m*(H) data can also be used to providean accurate measurement of certain properties includingJc when the signal associated with the trapped flux is muchlarger than the signal associated with field non-uniformities. The FCC-m*(T) measurement, on the otherhand, is very often misunderstood and misused. The FCC-m*(T) measurement has been used to measure thesuperconducting fraction and is occasionally used tomeasure Jc. In neither case is the FCC measurementgenerally reliable due to the effects of flux pinning. Thisis because flux is not readily expelled from a strong-pinning superconductor as it is lowered in temperature ina field smaller than Hc1. For determining Jc, the mostreliable magnetic method is probably an M(H) [m*(H)]measurement. The virgin curve — the initial part of azero-field-cooled M(H) curve — can be useful fordetermining the superconducting fraction, however it mustbe properly corrected for demagnetization effects.

Some of the SQUID output signals, V(z), shown in earliersections look similar to an ideal dipole signal even thoughthe origin of the signal is completely spurious. Computerfits to these signals can return what appear to be reasonablevalues of the magnetic moment and even give small errorvalues for the regression fit. Caution must be exercisedwhen interpreting results of this kind since, as has beenshown, such results are sometimes not dipole responsecurves with slight deviations, but rather are the responsesassociated with a completely different kind of magneticbehavior. Problems will result from the imprudent use ofanalysis methods meant for extracting a magnetic moment,however, with the appropriate choice of analytical methodvaluable magnetic information can be extracted from themeasurements in all cases.

As mentioned earlier, when the contribution of thebackground and sample are of comparable magnitude, andwhen the center-of-mass of the sample and the center-of-mass of the background are spatially displaced, significantdeviations from constant-dipole V(z) signals can result.In some of these cases it is possible to measure the V(z)signals of the background separately at the same fieldsand temperatures at which the V(z) signals of the combinedsample and background are measured. It is preferable forthe background to be located in exactly the same positionin both runs. The V(z) signals from the two runs can thenbe subtracted to produce a set of V(z) curves that representthe response of the sample alone. An application of thismethod is shown in Figure G1.

It must be emphasized that the effects that have beendescribed here are not unique to any specific

-10.0

-5.0

0.0

5.0

10.0

Sample and Background

V (

V)

V (

V)

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0SubtractedFit

-3 -2 -1 0 1 2 3z (cm)

V (

V)

-10.0

-5.0

0.0

5.0

10.0BackgroundAlone

(a)

(b)

(c)

Figure G1. In some cases the sample with background does notapproximate a magnetic dipole and will produce a V(z) responsefor which the signal analysis routines included with the MPMSare not applicable. By collecting the V(z) curves for (a) thesample with background and then (b) the V(z) for the backgroundalone, at each of the fields and temperatures of interest, one canthen subtract corresponding V(z) curves to produce the V(z)curve for the sample alone. Analysis is made easier by locatingthe background in exactly the same position during both runs.

magnetometer, and are not even unique to SQUIDmagnetometer systems. While the measurementsdescribed in this document were all performed on theQuantum Design MPMS SQUID magnetometer system,any magnetometer measurement in which the sampleexperiences variations in the magnetic field during themeasurement process will be subject to similar effects.One might argue that the effects could be eliminated by

24

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○


just providing a more uniform magnetic field. However,in the case of measurements made at very low fields, andbecause of flux trapping effects in superconductingmagnets, this may not be possible within reasonable costand size constraints for any practical system containing ahigh-field superconducting magnet.

A c k n o w l e d g e m e n t s

We would like to thank Minoru Suzuki of NipponTelegraph and Telephone and Jim Thompson of Oak Ridgeand University of Tennessee for very important insights.We would also like to thank Jost Diederichs of QD foroffering critical comments and Lifang Hou for preparingand patterning the YBCO thin film. Many helpfulcomments were provided by Ron Goldfarb of the NationalBureau of Standards and Technology in Boulder, Colorado.Refereed journal publication of some portions of this textare in preparation with John R. Clem of Iowa StateUniversity and Alvaro Sanchez of Universitat Autonomade Barcelona who have made detailed, realistic theoreticalcalculations of the effects of the measuring processesdiscussed here. Some support for this work was providedby the Director for Energy Research, Office of BasicEnergy Sciences through the Midwest SuperconductivityConsortium (MISCON) grant #DE-FG02-90ER45427.

R e f e r e n c e s

1. D. Farrell, in Physical Properties of High TemperatureSuperconductors, IV, edited by Don Ginsberg(World Scientific, 1989), pp. 49-53.

2. C. P. Bean, Rev. Mod. Phys. 36, 31 (1964).

3. M. Suenaga, D. O. Welch, and R. Budhani, Supercond.Sci. Technol. 5, s1 (1991).

4. A. K. Grover, Ravi Kumar, S. K. Malik, andP. Chaddah, Solid State Comm. 77, 723 (1991).

5. F. J. Blunt, A. R. Perry, A. M. Campbell, and R.S. Liu,Physica C 175, 539-544 (1991).

6. Minoru Suzuki, Kazunori Miyahara, Shugo Kubo,Shin-ichi Karimoto, Koji Tsuru, and Keiichi Tanabe.(preprint)

7. M. McElfresh, Fundamentals of Magnetism andMagnetic Measurements featuring Quantum Design’sMagnetic Property Measurement System, (QuantumDesign, San Diego, CA, 1994).

8. S. Libbrecht, E. Osquiguil, and Y. Bruynseraede,Physica C 225, 337 (1994).

9. Shi Li, Lifang Hou, and M. McElfresh, John R. Clem,and Alvaro Sanchez, (preprint).

25

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

QUANTUM DESIGN

A p p e n d i x A : I n t r o d u c t i o n t o t h eB e a n C r i t i c a l S t a t e M o d e l

The Bean Critical State model is used to relate the criticalcurrent density Jc, which is the largest possiblesupercurrent (zero-dissipation or zero-resistance current),to the total magnetization of a type-II superconductor forthe particular case in which Jc is limited by flux pinning.The model relates the magnetization M to Jc by the simpleformula Jc = sM/d, where d is the sample width or diameterand the constant s is a shape dependent constant having avalue s = 10/π for a slab sample and s = 15/π for acylindrical sample. Several assumptions are usually madewhen using the Bean model. For instance, it is usuallyassumed that the lower critical field Hc1 is very small (it

is usually ignored) and a key assumption is that the currentdensity J in the sample can have only the values +Jc, -Jc,or zero. The simplest version of the model treats the caseof an infinitely long sample in a parallel field such thatdemagnetization effects are not important.

The magnetization as a function of applied magnetic fieldM(H) for an ideal type-II superconductor (without fluxpinning) is shown in Figure APP-A1. As the appliedmagnetic field is increased from H = 0, at temperaturesbelow the superconducting transition temperature Tc,initially the superconductor completely screens out themagnetic field (B = 0 within the sample). When H reachesHc1 magnetic flux enters a type-II superconductor in theform of magnetic-flux quanta known as flux vortices. (Onequantum of magnetic flux φ0 = 2.07 x 10-7 oersted-cm2.)As H continues to increase, the vortex-solid transitionboundary is crossed at Hg, at which point Jc = 0, and thenthe upper critical field Hc2

is crossed, above which thesample is no longer superconducting (it becomes normal).The region Hc1 < H < Hc2 is known as the vortex phase. Amagnetic phase diagram is shown in Figure APP-A2. Incontrast to this, a type-I superconductor fully screens upto the critical field, Hc, above which it becomes normal.

H (

kOe)

0

10

20

30

40

50

60

70 75 80 85 90

T (K)

NormalFluxon

Liquid

Fluxon

Glass

Hc1

Hc2

Tg , Hg

Tg , Hg : Vortex Glass Transition (VGT)

c

Jc = 0

Magnetic Phase Diagram For Type II Superconductor

(Single Crystals)

J > 0

ρρρρL = 0

ρρρρL > 0

Figure APP-A1. Magnetization M as a function of applied fieldH for a long cylindrical sample of an ideal type-IIsuperconductor in a parallel magnetic field. The plots in (a)through (e) compare the values of the applied field H, themagnetization M, and the flux density B inside the sample atvarious points on the M(H) curve. For an ideal superconductor,M, B, and H will all be uniform inside the sample.

c1H

c2H

Beq

M a d

M

H

Superconductor Free Space

(a)

H

c2H

B = 0

0 r

BM

(b)

H

B = 0

0 r

B

M

B > 0

B = H

B > 0

rH

z = -∞∞∞∞

z = +∞∞∞∞

(d)(c)

H

0 r

B

M

H, B

(e)

M = 0

0 r

M

H

0 r

B

M

c1H

4 π M= -H

4 π M= B -H

b c

e

Figure APP-A2. Magnetic phase diagram for a type-IIsuperconductor. The lower and upper critical fields, Hc1 and Hc2

respectively, have long been known whereas the vortex-solidtransition boundary Hg(T) is a more recent discovery.

26

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○


ideal superconductor the same curve is followed ondecreasing applied field.

In contrast to the ideal superconductor in which acontinuum of current values can exist, the Bean modelassumes that only the current densities ±Jc and zero arepossible, so that for a non-zero applied field H, the currentdensity Jc always flows to some depth. As H increases,the depth to which this current (Jc) flows also increases.This is shown schematically for the simplest version ofthe Bean model in the series plots in Figure APP-A3, withthe position dependence of the current shown in FigureAPP-A3(a) and the position dependence of the amplitudeof the magnetic flux density, B(r), shown in

-1 -0.5 0 0.5 1r

P

H /HP = 1.0

H /HP = 0.8

H /HP = 0.6H /H

P

= 0.4H /H = 0.2

(b)

-

0

(a)

0.2

0.4

0.6

0.8

1.0

H/Hp =

Jc

Jc

B

Hp

H /HP >1.0

Figure APP-A3. (a) A calculation of the spatial dependence ofthe current and (b) the position dependence of the amplitude ofthe magnetic flux density B as described by the simplest versionof the Bean Critical State model for a long cylindrical sample ina parallel field. Here r = 0 corresponds to the center (on thesymmetry axis) of the sample and r = 1 corresponds to thesample surface. The only possible values of current density are± Jc and zero, with J = Jc flowing to the depth to which the fluxpenetrates and J = 0 where B = 0.

A superconductor with an ideal-shaped M(H) loop likethe one shown in Figure APP-A1 would not be able tosupport a supercurrent in the vortex phase. This is becausethe flux vortices are free to move in the sample and theapplication of a transport current will cause the vorticesto experience a Lorentz force FL = J x φφφφφ00000, where J is thecurrent density. The force will lead to motion of thevortices which will in turn generate a voltage parallel to Jthat results in dissipation of the supercurrent. In order tokeep the vortices from moving, the vortices must bepinned. Pinning is produced by creating spatial variationsin the properties of the superconductor, usually byintroducing particular types of defects. The largest currentdensity that a superconductor can support withoutdissipation is Jc. In the event that Jc is limited by flux-vortex motion, the Jc will be determined by the balancingof the Lorentz and pinning forces. There are other possiblelimits to Jc, such as the depairing limit which is usuallytwo orders of magnitude larger than the largest Jc

associated with flux pinning. Clearly the mechanism thatproduces the smallest value of Jc will determine the Jc of agiven sample. In granular superconductors, Jc is oftenmuch lower than either the pinning-limited Jc or thedepairing-limited Jc, l imited instead by thesuperconducting weak-links at the grain boundaries.

Before discussing the Bean model further, it is useful todiscuss the behavior of an ideal type-II superconductor(with no defects). Shown schematically in Figure APP-A1 is an infinitely long cylindrical sample. The magneticfield is applied parallel to the axis of the cylinder. As thefield is applied supercurrents at the surface are producedto screen the interior of the cylinder in order to maintainB = 0 inside. For the case of an ideal type-IIsuperconductor for H < Hc1, the currents would only existnear the surface in a thin layer in which the magnitude ofthe field decays into the sample with the characteristiclength λ, the magnetic penetration depth. The B(r) profileassociated with two representative values of the appliedfield less than Hc1 are shown in Figs. APP-A1(a) and APP-A1(b), with the applied field H, the magnetization 4πM,and the magnetic flux density B identified in the figures.It can be seen that as H increases (while still below Hc1),B remains at zero since the interior is completely screened,meaning that the screening currents produce amagnetization value, -4πM, that is equal and opposite tothe applied field. When H exceeds Hc1 [Figure APP-A1(c)]flux penetrates into the superconductor, and in the idealcase this flux is uniformly distributed within the sample.The magnitude of the magnetization is now reduced andthe amount of flux penetrating the sample is proportionalto the flux density B. The magnitude of M continues todecrease [Figure APP-A1(d)] until H = B at the uppercritical field Hc2

[Figure APP-A1(e)], which is where thesample becomes normal (non-superconducting). For an

27

○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○

QUANTUM DESIGN

Figure APP-A3(b). Here r = 0 corresponds to the center(on the symmetry axis) of the sample and r = 1 correspondsto the sample surface. It can be seen that the magneticflux density, B(r), decreases linearly from the surface, andfor small applied fields the region near the axis of thesample is fully screened (B = 0). As the applied fieldincreases, the depth to which flux penetrates alsoincreases, and within the depth to which the fluxpenetrates, a current of magnitude Jc flows. At asufficiently high field, labeled Hp (the penetration field),the flux fully penetrates to the center of the sample andthe sample is now in the critical state, with +Jc or -Jc

flowing throughout the sample. As the field continues toincrease, flux continues to penetrate, however the slopedB/dx remains the same. In fact because dB/dx ~ Jc, andJ = ±Jc, within the Bean model the slope is always ±dB/dxinside the superconductor.

Figure APP-A4 shows an M(H) loop for the simplest caseof the Bean model. The B(r) profiles for applied fieldsincreasing from H = 0 to 2Hp, back down to H = 0, andfinally to negative fields are shown in Figures APP-A4(a)through APP-A4(i) with the corresponding positions onthe M(H) curve identified. The effects of lowering theapplied field after raising it to H = 2 Hp

are shown inFigures APP-A4(e) through APP-A4(i). At the sample’ssurface the field inside the superconductor will match thefield outside, while ±Jc continues to flow in the sample.For the B(r) profile to match the applied field at the surface,the profile must reverse slope near the surface and itfollows that the depth of this profile-reversal increases asthe applied field decreases. This continues until thereversal of the profile reaches the center (on the symmetryaxis) after which a further decrease in the applied fielddoes not change the shape of the profile, although Bcontinues to decrease inside the sample. When the appliedfield is reduced to zero this B(r) profile remains [see FigureAPP-A4(h)]. This final state is known as the remanentcritical state.

A magnetometer measures the magnetic moment of theentire sample, and from the magnetic moment, one candetermine the magnetization of the sample. Magnetizationis usually reported as the magnetic moment divided bythe sample size, usually its mass or volume, however, theseare just two of the ways magnetization value may bereported. The applied field is generally regarded as themagnetic field generated by the magnet coil and isunchanged inside the sample. Since B = H + 4πM, M isjust the difference between B and H. Thus, in the plots ofFigure APP-A4, M is proportional to the area between B(r)and a line corresponding to the applied field H. When thearea enclosed by B(r) is above the line corresponding toH, it follows that M will be positive in sign, and when thearea is below H, M will be negative. During the reversal

H

a

b

c d

e

f

g h i

H

M

H

Jc(x) ~ -dB/dx

H

H

H

Hp

2Hp

H

H

H

2Hp

H H=0

H=0

H=0

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

o r

2Hp

2Hp

Hp

Hp

o r o r

o r o r o r

o r o r

Figure APP-A4. M(H) curve for a long cylindrical sample of astrong pinning type-II superconductor in a parallel magneticfield for which the simplest version of the Bean model applies.The plots in (a) through (i) show the position dependence of theflux density B(r) as the applied field increases from zero to twicethe penetration field, Hp, then down to negative applied fields.B(r) profiles at various positions on the M(H) curve areidentified.

of the field profile on decreasing H, there will be regionsboth above and below the line corresponding to H. Thesum of the areas, taking areas above H to be negative andthose below H to be positive, will determine the sign ofM. (Note that the plots in APP-A4 are cross sectionalprofiles and that the total magnetization will actuallycorrespond to a volume rather than an area.)

The shape of the M(H) loop for a strong pinningsuperconductor can be accounted for by using the Beanmodel to follow the area between B and H through ahysteresis loop. When the sample is cooled below Tc inzero applied field (H = 0) the magnetic moment should beequal to zero. In the Bean model it is generally assumed

28

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that Hc1 = 0. Flux penetrates into the superconductor as Hincreases producing a linear B(r) dependence. Since thedifference between the B(r) curve and the applied field isproportional to the magnetization, the magnitude of themagnetization increases as H increases. However, the Mmagnitude increases by a smaller and smaller percentageas more flux penetrates until the field fully penetrates thesample, at which point the M-magnitude no longerincreases with H. Upon decreasing H, the B(r) profilereverses direction near the surface. Since B is going tohave only one value at any point in the sample, this reversalexists to the depth where the B(r) curves meet. Thereversal point continues to move deeper into the sampleas H is decreased until it reaches the center of the sample,after which the profile remains the same down to H = 0.If the applied field is raised to at least twice Hp, the samplewill attain the remanent critical state when the appliedfield is reduced to zero.

29

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ducting input circuit of the SQUID, so that changes in themagnetic flux in the detection coils (caused by the samplemoving through the coils) produce corresponding changesin the current flowing in the superconducting input cur-rent, which is detected by the SQUID. The general de-sign of the system is shown in Figure APP-B1

The detection coil system is wound from a single piece ofsuperconducting wire in the form of three counterwoundcoils configured as a second-order (second-derivative) gra-diometer. In this geometry, shown in Figure APP-B1(b),the upper coil is a single turn wound clockwise, the cen-ter coil comprises two turns wound counterclockwise, andthe bottom coil is a single turn wound clockwise. Thecoils have a 2.02 cm diameter, and the total length of thecoil system is 3.04 cm. The coils are positioned at thecenter of the superconducting magnet outside the samplechamber such that the magnetic field from the samplecouples inductively to the coils as the sample movesthrough them. The second-order gradiometer configura-tion rejects noise caused by fluctuations of the large mag-netic field of the superconducting magnet, and also re-duces noise from nearby magnetic objects in the surround-ing environment.

A p p e n d i x B : T h e M P M S M e a s u r e m e n tA n d D a t a A n a l y s i s

1. The MPMS Measurement

The MPMS measures the local changes in magnetic fluxdensity produced by a sample as it moves through theMPMS superconducting detection coils. The detectioncoils are located at the midpoint of a superconducting so-lenoid (normally referred to as the superconducting mag-net) which can apply a DC magnetic field to the sampleas specified by the user. The detection coils are connectedto the input of a Superconducting QUantum InterferenceDevice (SQUID) located in a magnetic shield some dis-tance below the magnet and detection coils. In this con-figuration the detection coils are part of the supercon-

Figure APP-B1(b). The MPMS superconducting detection coils,located at the center of the MPMS superconducting magnet, arewound in a second-derivative configuration.

Figure APP-B1(a). The sample chamber, which extends throughthe center of the coils, is thermally isolated so the detection coilsand magnet can remain at 4.2 K. Measurements are made bymonitoring the SQUID output while moving the sample upwardthrough the coils.

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The MPMS determines the magnetic moment of a sampleby measuring the output voltage of the SQUID detectionsystem as the sample moves through the coil. Sincechanges in the output voltage of the SQUID are directlyproportional to changes in the total magnetic flux in theSQUID’s input circuit, transporting a point-dipole samplealong the z-axis of the detection coil system produces theposition-dependent output signal, V(z), shown in FigureAPP-B2(a). In practice, a measurement is performed byrecording the output voltage of the SQUID sensor at anumber of discrete, equally spaced positions as the sampleis drawn upward through the detection coils. To insurethat the mechanical motion of the sample transport mecha-nism does not cause vibrational noise in the SQUID de-tection system, the measurements are performed by re-cording the SQUID output while the sample is held sta-tionary at a series of points along the scan length.

A set of voltage readings from the SQUID detector, withthe sample positioned at a series of equally spaced pointsalong the scan length, comprise one complete measure-ment, which we refer to as a single “scan”. The numberof points to be read during each scan can be selected bythe user; we have used 40 points for the data shown inthis report. For more general measurements we recom-mend using 16 points when one is measuring sampleswhich have large moments, and increasing the number ofpoints to 32 for smaller signals. This range of values is acompromise between the need for greater averaging ofsmall signals and the simultaneous need to stay above thelow-frequency noise regime. After each scan has beencompleted, a mathematical algorithm is used to computethe magnetic moment of the sample from the raw data.

2. Data Analysis

All of the analyses discussed in this report were performedusing the Linear Regression algorithm to fit a theoreticalcurve to the raw data, as shown in Figure APP-B2(a).When using this particular algorithm, the calculation alsocomputes a regression factor, which falls within the rangeof 0 to 1, to indicate the quality of the fit between thetheoretical curve and the raw data. (A value of 1.0 meansthat every point fell exactly on the theoretical curve.)When measuring a sample having a very small magneticmoment, multiple scans can be averaged together to com-pute an average value for the moment and a standard de-viation for the set of measurements.

As discussed in the body of this report, effects which arisefrom the non-uniformity of the dc magnetic field can pro-duce a magnetic response which is inconsistent with theregression algorithms used to fit the raw data points. Thecurve shown in Figure APP-B2(a) is the theoretical curve

for a sample which behaves as a simple point-dipole mov-ing along the axis of the detection coil system. For a realsample, the ideal curve will precisely fit the data onlywhen: 1) the sample is positioned on the longitudinalaxis of the coil system, 2) the linear dimensions of thesample are much smaller than the characteristic dimen-sions of the detection coils, 3) the magnetic moment ofthe sample does not change with position over the entirelength of the scan, and 4) the sample is uniformly magne-tized over its entire volume, When these conditions aremet, the magnetic moment, m, of a sample can be accu-rately determined using the analysis methods provided

Figure APP-B2. The regression algorithms compute themagnetic moment by fitting a theoretical curve to the raw datapoints with the amplitude being used as the fitting parameter.The Linear Regression algorithm assumes that the data areperfectly centered along the scan length as in (a), while theIterative Regression fit, shown by the solid line in (b) cancorrectly fit data which are not centered. The dashed line in (b)shows the Linear Regression fit for off center data.

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with the instrument, but when making measurements onreal physical samples both the position and shape of thesample can have subtle effects on the calculated magneticmoment.

The first two items, which arise from the small geometryof the detection coils used in the MPMS, essentially causethe absolute calibration of the instrument to changeslightly as a function of radial position and sample vol-ume. The effect due to the radial position of the sample iseasily eliminated by using a sample holder which keepsthe sample positioned close to the centerline of the samplechamber. Similarly, effects due to the size and shape ofthe sample can be easily corrected by a simple multipli-cative factor which effectively adjusts the absolute cali-bration of the instrument to compensate for the shape andvolume of the sample. The origin of these two effects andtheir correction is described more fully in Section 4 ofthis appendix, which discusses the calibration of theMPMS system.

Item (3), the variation of the magnetic moment of thesample as a function of position, which is the main topicof this report, is an inherent problem in any magnetom-eter in which the sample physically moves from one loca-tion to another during the measurement. It is also worthnoting that even if one arranges to move the sample oververy small distances (of the order of 0.5 mm), small varia-tions in the magnetic field can still produce dramatic ef-fects when measuring some types of samples, such asstrong-pinning superconductors. Finally, Item (4) repre-sents a problem which is inherent to the nature of thesample being measured. Since there is no a priori reasonto assume a particular distribution for the magnetizationin a nonuniformly magnetized sample, there is no generalpurpose theoretical model which would be suitable foranalyzing such data. For this situation, however, the rawvoltage data from the SQUID is available to the user whenhe selects the “All Scans” data file. This option allowsthe user to store all of the raw SQUID readings from themeasurement, then use his own theoretical model to fitthe raw data and perform subsequent analysis

When measuring reasonably well-behaved samples, theMPMS software provides three different mathematical al-gorithms for computing the magnetic moment of thesample. All three algorithms assume that the currents in-duced in the detection coils are those associated with themovement of a point-source magnetic dipole through asecond-order gradiometer detection coil, and all three canproduce errors when the raw data deviate significantlyfrom the ideal point-dipole signal. The three analysis al-gorithms, which are selected in the MPMS “Set Param-eters” menu, are denoted as “Full Scan”, “Linear Regres-sion”, and “Iterative Regression”.

2.1 The Full Scan Algorithm

The Full Scan algorithm effectively integrates the totalarea under the voltage-position curve, V(z), by comput-ing the square root of the sum of the squares of the datapoints, normalized by the total number of data points inthe scan. This algorithm was originally developed for theMPMS because it is less sensitive to changes in samplevolume than the regression algorithms, but it requires arelatively long scan length (5 cm or more) to provide anaccurate measurement of the absolute moment of thesample. When using the Full Scan algorithm, the mag-netic moment is computed using the equation:

where ∆z = L/(n-1), L is the scan length, n is the numberof data points in the scan, Vi is the voltage reading at thei-th point, and C is a calibration factor that includes thecurrent-to-voltage ratio of the SQUID as well as the volt-age-to-moment conversion factor for the Full Scan algo-rithm. The factor ∆z, which is the distance between datapoints, normalizes the calculation for both changes in thescan length and the number of data points collected.However, measurements made with a different scanlength and/or a different number of data points will giveslightly different values for the absolute value of themagnetic moment.

This algorithm also has a significant limitation when usedto measure very small magnetic moments. Since noise inthe system always adds to the moment, the algorithm cannever give a value of zero when there is any noise in themeasurement. Hence, the smallest measurable momentcan be severely limited by the background noise from themagnet, and there is no effective way to improve the mea-surement by additional averaging.

2.2 The Linear Regression Algorithm

The Linear Regression algorithm computes a least-squaresfit of the theoretical voltage-position curve to the mea-sured data set, using the amplitude of the magnetic mo-ment as its free parameter. The algorithm assumes thatthe center peak of the voltage-position curve, V(z), is lo-cated precisely at the center of the measurement scan.When the sample is properly centered in the pickup coils,the central peak of the three-peaked response curve willbe equidistant from both ends of the data set, as shown inFigure APP-B2(a). For the second-derivative detectioncoils employed in the MPMS, one can easily calculate thetheoretical curve, V(z), for the general condition in which

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ture will about 2 mm above center when the MPMSsystem cools down to its lowest temperature. (The totalvertical displacement of the sample over the temperature

range of 2 K to 400 K is approximately 3 mm.)

The present MPMS software system allows the user toresolve this problem in either of two ways. First, as dis-cussed below, the Iterative Regression algorithm can ac-tually compute the position of the sample by allowing thetheoretical curve to move along the direction of the samplemotion to obtain a better fit to the data points. Alterna-tively, one can use the automatic “Sample Tracking” fea-ture of the MPMS which automatically adjusts the posi-tion of the sample as the temperature changes to keep thesample properly centered to within about 0.3 mm over theentire temperature range. When using the Sample Track-ing feature, however, it is essential that the user accuratelycenter the sample at the beginning of the measurementsequence, since the Sample Tracking algorithm assumesthat the sample was correctly positioned at the beginningof the sequence.

The Linear Regression algorithm is the most reliable cal-culation if one is measuring a sample which has a verysmall magnetic moment. Since this algorithm can inher-ently produce a value of zero (unlike the Full Scan algo-rithm), it can more effectively average the noise whenmeasuring small samples. Furthermore, its most signifi-cant limitation, the errors introduced by changes in thesample position, has been virtually eliminated by theSample Tracking capability of the MPMS system, allow-ing the Linear Regression algorithm to provide reliabledata over the full temperature range of the MPMS. Be-cause of the reliability of the Linear Regression algorithmand the Sample Tracking feature, this is now the recom-mended analysis method.

2.3 The Iterative Regression Algorithm

The Iterative Regression algorithm, which can report thecorrect value for the magnetic moment even when the scanis not properly centered, is designed to compensate forcentering errors as large as 1 cm. This theoretical modeluses the same constant-dipole equation as the Linear Re-gression calculation, but allows the position of the theo-retical curve to vary along the axis of the detection coils.

the magnetic moment of the sample, m(z), varies withposition. For this case, the output signal, V(z), is givenby the following expression:

where all the coils have the same radius, a, and the fourcoils are positioned at z = -d, z = -b, z = b, and z = d.The origin of this coordinate system is at the center of thecoil set, and the z axis lies along the axis of the coils.

In the ideal case, one assumes that the sample moves in aperfectly uniform magnetic field so the magnetic momentof the sample is constant as the sample moves through thedetection coils. In this case, the position-dependent mag-netic moment, m(z), in the above equation is just a con-stant, m, and one can easily compute the theoretical curvewhich is shown in Figure APP-B2(a). When the LinearRegression algorithm is used to compute the magneticmoment of the sample, the theoretical equation givenabove is fit to the set of position-voltage readings using aleast-squares calculation with the magnetic moment, m,as its free parameter. The value of m which minimizesthe least-squares calculation is then reported as the mag-netic moment of the sample.

Since the Linear Regression algorithm assumes that thescan is accurately centered with respect to the detectioncoils, the magnetic moment reported for a scan duringwhich the sample is not properly centered will be incor-rect, with an increasing error as the misalignment in-creases. From Figure APP-B2(b), in which the dashedline shows a typical Linear Regression fit to a set of datawhich are off center by 3 mm, it is clear that the calcu-lated magnetic moment (as indicated by the amplitude ofthe theoretical curve) will be too small by a significantamount. From this example, it is also clear that, whenusing the Linear Regression algorithm, the scan must beproperly centered in the detection coils to obtain an accu-rate value for the magnetic moment of the sample.

However, carefully centering the sample at the beginningof a series of measurements will not guarantee accuratecentering during the entire measurement sequence. Be-cause the temperature range of the MPMS is so large, ther-mal expansion and contraction in the sample rod on whichthe sample is mounted causes the sample position tochange when measurements are made over large tempera-ture range. Consequently, a sample which has been prop-erly centered with the sample chamber at room tempera-

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3. The MPMS Superconducting Magnet

The MPMS system employs a superconducting magnetand a precision current supply to generate the large, highlystable dc magnetic fields required for the measurementsystem. When a large stable magnetic field (> 2 tesla) isneeded over a modest volume, a superconducting magnetis normally chosen, but there are numerous issues whichmust be taken into consideration when using a supercon-ducting magnet. For example, after the MPMS supercon-ducting magnet has been charged to fields greater thanabout 200 G, magnetic field lines will become trapped inthe magnet’s superconducting windings, so that when themagnet current is set back to zero, the actual measurablemagnetic field at the center of the magnet may be as largeas 40 G to 50 G. This so-called “trapped flux” is charac-teristic of all superconducting magnets, although this “re-manent field” which is left in the magnet when the mag-net transport current is discharged to zero will dependstrongly on the size and specific design of the magnet.

While the remanent field in the various MPMS magnetsis small compared to the maximum available field, the re-manent field can still introduce a substantial error whenperforming measurements in the MPMS at very low fields.In this case it is important to determine and account forthe effect of the remanent field in the magnet. One im-portant effect discussed in this report is the variation ofthe remanent field as a function of vertical position (z-dependence) over which the sample moves during a mea-surement. The MPMS system provides several mecha-nisms for reducing the remanent field and increasing itsuniformity when the user needs to perform measurementsat low magnetic fields.

Another important feature of superconducting magnets,particularly when employed in a high-sensitivity measure-ment system such as the MPMS, is that after the magneticfield has been changed, the field in the magnet will relaxlogarithmically in time. This process, which occurs be-cause the magnetic flux creeps across the magnet’s super-conducting windings, proceeds even after the magnet hasbeen placed in a persistent condition (with the persistent-current switch in its superconducting state, and the mag-net power supply turned off). While the relaxation effectis relatively small compared to the maximum availablefield, under some conditions — particularly when mea-suring near the limit of the MPMS’s sensitivity after alarge change in the value of the field — this relaxationcan generate a signal in the high-sensitivity SQUID de-tection system which is many times larger than the signalto be measured.

Because the time constant for the magnet relaxation (typi-cally several minutes) is quite long compared to the length

Mathematically this is achieved by substituting z′ + s forz in the above equation, where the variable, s, now be-comes a second free parameter which can be varied toimprove the least squares fit to the data. Since the vari-able s appears in the equation as a displacement along thez-axis, changing the value of s is equivalent to shiftingthe position of the theoretical curve along the axis of thecoil system.

The resulting equation now has two variables and is non-linear in s, so one cannot simply compute both m and sfrom the Linear Regression algorithm. To find the opti-mum values for both m and s, the Iterative Regressionalgorithm first computes a standard Linear Regression cal-culation taking the value of s to be zero. Then, based on aderivative calculation, a new value of s is selected and thelinear regression calculation is repeated. By comparingthe regression factors on subsequent calculations (whichindicates the accuracy of the regression fit) , the algorithmcan determine the values for s and m which minimize theleast-squares calculation, and the resulting value of m isreported as the magnetic moment of the sample. A typi-cal theoretical fit computed by the Iterative Regressionalgorithm for a scan which is 3 mm off center is shown bythe solid line in Figure APP-B2(b).

The Iterative Regression algorithm was developed to com-pensate for incorrect sample position, and the algorithmworks extremely well for measurements which have a largesignal-to-noise ratio so the peaks are well defined. How-ever, when the sample moment is very small the iterativeprocess can be confused by low frequency noise in themeasurement. More specifically, when the peaks in thesample signal are of approximately the same size as thelow frequency magnetic noise from the superconductingmagnet, the iterative regression algorithm will sometimesfit the analytical curve to features in the data which arecharacteristic of the noise in the system rather than thedata. Also, the iterative process may fail to converge. Thiscondition is indicated by a computed sample positionwhich is more than 1cm off center, well beyond the maxi-mum possible position for a properly centered sample.When this condition occurs, the system reverts to usingthe Linear Regression algorithm.

It is clear from the above discussion that each analysisalgorithm has both strengths and weaknesses. However,as we have consistently emphasized in this report, it isimportant to understand that all of the algorithms can giveincorrect values for the magnetic moment when one ismeasuring samples which are not uniformly magnetized,or when the magnetic moment of the sample changes withposition along the scan length.

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of an single sample measurement (about 10 seconds), thebackground relaxation can usually be treated as a linearbackground signal, which is automatically eliminated bythe data analysis routines. However, if a very small sig-nal is to be measured immediately after a magnetic fieldchange, the background drift can be so large that the16-bit dynamic range of the MPMS digitizing system maynot be able to achieve the resolution needed to measurethe signal in the presence of the relaxing background. Toaccommodate this situation, the MPMS system providesa “Pause” function which allows the user to specify thelength of time to wait after a magnetic field change be-fore beginning the next measurement.

The MPMS system provides four different methods forchanging the magnetic field in the system, allowing theuser to tailor the magnet control for particular measure-ments. The “No-Overshoot”, “Oscillate”, and “Hyster-esis” modes are useful for routine changes between arbi-trary starting and ending values, while the magnet resetoption provides a special method for eliminating trappedflux from the superconducting MPMS magnet. When theNo-Overshoot method is used, the magnetic field ischanged monotonically from the initial field to the de-sired field setting. At the beginning of the field change,the field is ramped quickly, but as the field approachesthe requested value, the field changes much more slowlyto avoid overshooting the target value. When using theOscillate method, the magnetic field alternately overshootsand undershoots the requested value with the amplitudeof the overshoot and undershoot decreasing on each cycle.This process minimizes the relaxation in the supercon-ducting magnet following the field change, but is inap-propriate when measuring samples which display mag-netic hysteresis effects. When changing the magnetic fieldusing the No-Overshoot and Oscillate modes, the magnetis put into a persistent condition after the final field isreached, with the magnet switch heater and power supplyturned off while measurements are being performed.

In the Hysteresis mode, the magnet is charged directly tothe final field, and measurements are performed while themagnet current is being maintained by the power supply(with the magnet switch heater turned on). This allowsmeasurements to be made at sequential field settings muchmore rapidly than normal, but the sensitivity of such mea-surements is substantially reduced due to the increasedmagnetic noise from the magnet supply current when op-erating in this mode. This measurement is most suitablefor measuring the hysteresis curves of samples which havefairly large magnetic moments.

The “Magnet Reset” option provides a mechanism bywhich the user can remove magnetic flux trapped in thesuperconducting magnet windings after the magnet has

been charged to high fields. In the 55-kG (5.5-tesla)MPMS magnets, the Magnet Reset operation comprises acontrolled quench in which part of the magnet is drivennormal while the magnet is sustaining a persistent cur-rent. In the 10-kG (1-tesla) MPMS magnet, and the highuniformity model-5S 50-kG (5-tesla) and model-7S 70-kG (7-tesla) magnets, the magnet heater is capable of driv-ing the entire magnet normal without requiring that themagnet be quenched.

When the reset operation is initiated to induce a quenchof the superconducting magnet, a portion of the super-conducting windings is driven normal (non-superconduct-ing) and becomes resistive. Resistive losses in this seg-ment of the magnet then generate heat which causes addi-tional parts of the magnet to become resistive, and thequench becomes a runaway situation in which all of thesuperconducting magnet windings are eventually drivennormal. When this occurs, the persistent current in themagnet quickly decays to zero, and any magnetic flux pre-viously trapped in the superconducting windings is re-leased. The magnet reset option leaves the magnetic fieldin the MPMS magnet in a state similar to that observedwhen the magnet has just been cooled down from roomtemperature and has not yet been charged to fields aboveapproximately 200 G. The field remaining in the magnetfollowing a quench will typically be approximately thesize of the Earth’s field, about 0.5 G, although this valuemay vary if the MPMS system has an external magneticshield around the dewar.

To provide the capability for the user to make measure-ments in very low magnetic fields (less than 10 milligauss),the MPMS systems offer the “Low-Field Option” whichemploys a flux-gate magnetometer to measure and nullthe field left in the magnet after a quench process has beencompleted. The low-field option, which is available onMPMS systems having 10-kG (1-tesla) magnets and thehigh-uniformity model-5S 50-kG (5-tesla) magnet, auto-matically measures the field at the center of the magnet,then adjusts the current in the magnet to null the remain-ing field to less than 0.005 G (0.050 G for the Model 5Smagnet). The user may also measure the field profile inthe magnet along its vertical axis by scanning the flux-gate magnetometer along the axis of the magnet.

4. MPMS System Calibration

All of the data analysis routines in the MPMS report themagnetic moment of the sample in cgs or electromag-netic units. To insure that the reported values accuratelyrepresent the absolute magnetic moment of the sample,the instrument calibration at the factory employs a sampleof known susceptibility in the form of a right circular cyl-

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inder having a dimension of 2.5 mm for both diameterand height. If the shape of your sample deviates signifi-cantly from a right circular cylinder, or if the sample ismuch larger than the standard calibration sample, you maywish to recalibrate your measurements using a referencesample having the same shape and dimensions as the ex-perimental sample.

As noted in Section 2 above, the absolute calibration ofthe instrument changes slightly with the size and shape ofthe sample and also with the radial position of the samplein the sample chamber. These effects result from the smallgeometry of the MPMS detection coils, which was origi-nally chosen for the multiple benefits it provides. Themost important benefits of the small geometry are: (1) itallows a complete scan to be made over a small distance,minimizing the distance which the sample must move dur-ing the measurement, (2) it improves the rejection of noisefrom the dc magnetic field, (3) it optimizes the magneticcoupling between the coils and the sample, allowing mea-surements to be made on samples which have extremelysmall volumes, and (4) it allows the use of a much smallermagnet, minimizing cost, allowing faster charging rates,and reducing helium loss while the magnet is charging.

However, detection coils which use a small geometry aremore sensitive to the radial position of the sample. To bespecific, measurements in which a sample is located atthe outer edge of the sample chamber will give a some-what different result for the magnetic moment, than asample which is located precisely on the centerline of thecoil system. This effect, which results from the r-cubednature of the dipole field, means that a sample which is

Figure APP-B4. Variation of the measured magnetic moment asa function of sample length and diameter for three combinationsof measurement and analysis selections: (a) Full Scan algorithmwith a 6 cm scan, (b) Linear Regression using a 3 cm scan, and(c) Linear Regression using a 4 cm scan. Linear Regressioncorrections also apply to Iterative Regression algorithm.

Figure APP-B3. The variation of the measured magneticmoment as a function of radial position for measurements usinga second-derivative coil system. This dependence occursbecause the mutual inductance between the sample and thedetection coils changes with the radial position of the sample.

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off center will produce a greater net flux in the detectioncoils than a sample located exactly on the longitudinalaxis of the coil system. Figure APP-B3 shows the varia-tion of the moment as a function of radial position for theMPMS for a point-dipole sample. This effect can be ratherlarge (over 10% at the wall of the sample chamber), but itcan be essentially eliminated by using a sample holderthat keeps the sample reasonably close to the centerlineof the sample chamber.

The nonlinearity associated with the size and shape of thesample also arises from the radial dependence of the coilsystem. By the same argument, if the sample to be mea-sured has a large diameter, the outermost parts of thesample will produce a larger relative contribution to thesignal than the center-most portion of the sample. A simi-lar argument applies to long samples. Since the coils arespaced closely together (with a distance of only 1.52 cmbetween adjacent coils), the end of a long sample will beentering the effective measuring volume of one coil be-fore the opposite end of the sample has left the measuringvolume of the adjacent coil. Because of these two ef-fects, the calculated magnetic moment will not increaseexactly as expected with either increasing length or in-creasing diameter.

Figure APP-B4(a) shows the calculated correction as afunction of sample length for the Full Scan algorithm us-ing a 6 cm scan length. Figures APP-B4(b) and APP-B4(c)show the results of similar calculations for the Linear Re-gression algorithm for a 3 cm scan length and 4 cm scanlength respectively. (The correction factors shown in Fig-ure APP-B4(b) and APP-B4(c) can also be used for theIterative Regression algorithm.) From these figures onecan see that long needle-like samples will give values thatare several percent smaller than expected, while thin disk-like samples will give values that can be a few percent toolarge. The correction for intermediate sample shapes canalso be determined from these figures.

This effect is strictly related to the shape and volume ofthe sample, and the appropriate multiplicative factor tocorrect the reported values of magnetic moment can bedetermined from the graphs shown in Figure APP-B4.Please note that these calculations do NOT include anycorrections for the geometrical demagnetizing factor as-sociated with different shapes. Corrections for the properdemagnetizing factor must be included if one wishes tocalculate a theoretical value for the magnetic moment forany specific sample shape.

This volumetric or shape effect can also be corrected byadjusting the instrument calibration factor for a specificshape by recalibrating the MPMS using a piece of mate-rial of the desired size and shape but having known mag-

netic properties. When an unknown sample of similar sizeand shape is measured, the system will then report thecorrect absolute values of magnetic moment. We havefound that Standardized Pd wire from the National Insti-tutes of Standards and Technology (NIST) is especiallyuseful for making reference samples in a variety of geom-etries.

One final note which may be of interest to the user con-cerns the palladium reference samples supplied with theQuantum Design MPMS. These samples are made from99.995%-purity palladium material which has been fullycharacterized through a series of measurements in anMPMS system. The residual iron content in the materialused for the reference samples was specified by the manu-facturer to be 8 parts per million (ppm), but an indepen-dent assay reported a value of 24 ppm iron content. (Co-balt and Nickel content were reported to be 0.1 ppm.)Although the iron content of the palladium is small, it isstill large enough to produce a noticeable saturable mag-netic moment when measuring this material in low mag-netic fields. Consequently, when using one of these ref-erence samples to check the calibration of an MPMS, oneshould account for the presence of the iron impurities toachieve the best value of the instrument calibration.

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