NISTIR 3977

Alternating-Field Susceptometry and

Magnetic Susceptibility of

Superconductors

R. B. Goldfarb

M. Lelental

C. A. Thompson

United States Department of Commerce

National Institute of Standards and Technology

NISTIR 3977

Alternating-Field Susceptometry and

Magnetic Susceptibility of

Superconductors

R. B. Goldfarb

National Institute of Standards and Technology

Boulder, Colorado 80303

M. Lelental

Eastman Kodak Company

Rochester, New York 14650

C. A. Thompson

National Institute of Standards and Technology

Boulder, Colorado 80303

October 1991

Presented at: Office of Naval Research Workshop on

Magnetic Susceptibility of Superconductors and Other Spin Systems,

Berkeley Springs, West Virginia,

20 May 1991

Published in:

“Magnetic Susceptibility of Superconductors and Other Spin Systems,”

R. A. Hein, T. L Francavilla, and D. H. Liebenberg, Editors,

Plenum Press, New York (1992)

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director
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ALTERNATING-FIELD SUSCEPTOMETRY AND

MAGNETIC SUSCEPTIBILITY OF SUPERCONDUCTORS

R. B. Goldfarb

National Institute of Standards and Technology

Boulder, Colorado 80303

M. Lelental

Eastman Kodak Company

Rochester, New York 14650

C. A. Thompson

National Institute of Standards and Technology

Boulder, Colorado 80303

ABSTRACT

This review critically analyzes current practice in the design, calibration, sensitivity

determination, and operation of alternating-field susceptometers, and examines

applications in magnetic susceptibility measurements of superconductors. Critical

parameters of the intrinsic and coupling components of granular superconductors may

be deduced from magnetic susceptibility measurements. The onset of intrinsic

diamagnetism corresponds to the initial decrease in electrical resistivity upon cooling,

but the onset of intergranular coupling coincides with the temperature for zero

resistivity. The lower critical field may be determined by the field at which the

imaginary part of susceptibility increases from zero. Unusual features in the

susceptibility of superconductor films, such as a magnetic moment that is independent

of film thickness and the variation of susceptibility with angle, are related to

demagnetization. Demagnetizing factors of superconductor cylinders are significantly

different from those commonly tabulated for materials with small susceptibilities.

Rules for the susceptibility of mixtures with specific demagnetizing factors are used to

estimate the volume fraction of superconducting grains in sintered materials. Common

misunderstandings of the Meissner effect, magnetic units, and formula conversions are

discussed. There is a comprehensive summary of critical-state formulas for slabs and

cylinders, including new equations for complex susceptibility in large alternating fields.

Limitations on the use of the critical-state model for deducing critical current density are listed and the meaning of the imaginary part of susceptibility is considered.

1
INTRODUCTION

The term “susceptibility” was originated by William Thomson (Lord Kelvin) in his

annotated Reprint of Papers on Electrostatics and Magnetism} He defined “the

magnetic susceptibility of an isotropic substance [as] the intensity of magnetization

acquired by an infinitely thin bar of it placed lengthwise in a uniform field of unit

magnetic force.” The specification of an infinitely thin bar eliminated the need to

consider demagnetizing fields. The stipulation of a field of unit magnetic force defined

susceptibility as the ratio of magnetization M (magnetic moment per unit volume) to

magnetic field strength H. Thomson distinguished between susceptibility^ and

permeability, a term he devised to mean the ratio of magnetic induction B to H.

Magnetization and susceptibility measurements on superconductors detect signals,

usually inductively, that have their origins in circulating persistent shielding currents, in

addition to any magnetic properties of the material. We distinguish between eddy

currents in normal metals, which decay with time, and shielding currents in

superconductors, which do not. Susceptibility may be measured using direct or

alternating magnetic fields, yielding the ac susceptibility or the dc susceptibility. For

either, we define x the differential magnetic susceptibility dM/d//; we do not

necessarily require that d// -*> 0. In dc susceptibility, the zero-field-cooled (ZFC) curve demonstrates flux shielding (flux exclusion) upon warming, and the field-cooled (FC)

curve demonstrates the Meissner effect (flux expulsion) upon cooling. Whether

measured upon warming or cooling, ac susceptibility (with no dc bias field) always

measures shielding.

It is easier to define a superconductor as a material with zero electrical resistivity

than it is to experimentally verify zero resistivity. A four-point measurement of

resistivity involves the selection of current, voltage criterion, correction for

thermoelectric voltages, contact geometry, and the effect of magnetic field, including the

self-field of the current. The magnetic manifestation of zero resistivity is that a material

is a superconductor if it exhibits perfect diamagnetic shielding; that is, its susceptibility

X is exactly - 1 (in SI units, where numerical results must be corrected for any sample

demagnetizing factor). Susceptibility is reminiscent of electrical conductivity cr, both

functions of temperature T\ xi'HIxi^) ~ o’(T)/cr(0). In the normal state, both are small.

In the superconducting state, both are large. Important variables are the magnitude of

the measuring field and the definition of the critical temperature in terms of the

onset, midpoint, or end of the diamagnetic transition.

MEISSNER EFFECT

The Meissner-Ochsenfeld effect^ is the expulsion of magnetic flux upon cooling a superconductor through in a dc magnetic field or upon reducing the magnetic field

through the upper critical field H^2 constant temperature. Type-II superconductors

will not exhibit a pronounced Meissner effect if they have good flux pinning (important

for high critical current density J^) in the mixed state. Thus, while a material that has

a Meissner effect is a superconductor, the converse is not necessarily true. What is

2
sometimes called the “Meissner state” in superconductors is better termed the shielding

state, perfect diamagnetism/

It has long been known that the Meissner effect is incomplete in many materials.

Shoenberg stated that “for many of the element superconductors, it has not yet been

possible to obtain a specimen which shows a complete Meissner effect....”'^ Tantalum,

a type-I superconductor, is a case in point, with a Meissner effect of only 1%.^

Shoenberg attributed the incomplete Meissner effect in tantalum to its “mechanical

state rather than ... chemical impurities.”^ Years later, Alers et al. returned to this: “It is well known that for tantalum the Meissner effect is practically nonexistent because

the metal freezes in all of the existing flux when it becomes superconducting.... Pure

tantalum in bulk ... [is] made by sintering ... small flakes or grains.... Thus from a physical point of view, the metal is not homogeneous, and one can understand that the

Meissner effect might not be realized by a metal of this physical make up.”^ The

similarity between the morphologies of sintered tantalum and sintered Y-Ba-Cu-O

(or any of its analogs) will not be lost on most readers. However, sintered materials are not unique in this respect. Type-II elements and alloys and other inhomogeneous

superconductors, including melt-cast tantalum,"^^ similarly fail to show a significant

Meissner effect. The Meissner effect is also incomplete in single crystals of

YBa2Cu30-7 _5, which suggests intragrain pinning sites.'^'^

It is experimentally found that, in weak fields, the Meissner effect approaches

a value defined by the ZFC susceptibility curve. This is not surprising; it is tautologous that FC upon warming is equivalent to ZFC when the measuring field is zero.

If the FC curve were completely reversible for warming and cooling in the limit of zero

field, the Meissner effect would of necessity approach 100%.

CRITICAL TEMPERATURE

In field -current-density- temperature {H-J-T) space, there is a critical surface,

with axis intercepts J^, and 7^, separating the superconducting and normal states.

Its intersection with the H-T plane may be regarded as versus T or versus H, and

similarly for intersections with the J-T andJ-H planes. For type-II superconductors

the //-axis intercept is the upper critical field H^2

- mixed state lies between H^2

and the lower critical field

Measured as functions of temperature, transitions in resistivity p and susceptibility

X may be used to define T^. Ideally, should be determined at / = 0 and H = 0. However, measurements of p require some J and measurements of x require some H.

These are best kept small, unless the current and field dependences are specifically

required. Electrical resistivity complements susceptibility. Resistivity is a onedimensional measurement. A specimen will show zero resistivity if there is a single

zero-resistance percolation path. A higher-7^ phase can mask the presence of a lower7^ phase. Either of these cases could lead to erroneous conclusions regarding the

microstructure of the specimen under study. Susceptibility is a two-dimensional

measurement in the sense that a surface current sheath is required for full

3
diamagnetism. The interior of the material could remain normal or have a lower T^.

Both measurements deceive because they do not probe the entire sample volume.

“Onset” temperatures (defined as onset upon cooling, even if the measurement is made

upon warming) occur with the first zero-resistance segment and the first zero-resistance

current circuit, for p and x respectively. In low-dimensional systems, fluctuation effects

may obscure the onset of superconductivity.'^^

Granular and Multifilamentary Superconductors

Sintered high-T^ superconductors and composite low-T^ superconductors with

closely spaced filaments exhibit two critical temperatures. One is intrinsic to the

superconductor and the other is characteristic of the coupling between either

grains^"^^””'^ or filaments.'^^ In such materials, the coupling component supports

supercurrents and has its own effective T^, and multifilamentary

niobium-titanium (Nb-Ti) and niobium-stannide (Nb3Sn) superconductors, the

coupling component is the normal-metal matrix and the coupling mechanism is the

proximity effect."^^""^ The situation is less certain in sintered high-7^ compounds,

but lack of stoichiometry at the grain boundaries could give rise to normal metal

barriers'^^”'^^ and proximity-effect coupling.'^'^”'^*^ Another coupling mechanism in

sintered materials is microbridges between grains."^^ Low-dimensional compounds,

such as Nb^Se^, exhibit coupling attributed to superconductor-insulator-superconductor

Josephson junctions.

Because of the large change in shielded volume that occurs at of the coupling

component, there is a striking change in susceptibility. The change in resistivity, in

comparison, is minor because the coupling component forms a small part of the

conduction path. A crushed sintered sample yields isolated grains with only intrinsic

characteristics.'^^’^'^’^'^’'^'^’'^^’'^^ Both intrinsic and coupling critical temperatures are

i.o

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1 1 1 (Bi-Pb)2 Sr2Ca 2Cu30jj 80 A/m (1 Oe)

Coupling

Losses

Intrinsic

1 losses

Intrinsic diamagnetic

shielding

X'

Coupling diamagnetic /

shielding y

1 i L

100 Hz

40 60 80 100

Temperature (K)

120

Fig. 1. Internal ac susceptibility for (Bi-Pb)2Sr2Ca2Cu30^. The measurement field,

80 A*m“^ rms at 100 Hz, is large enough to separate the intrinsic and coupling

components. The real part does not extrapolate to - 1 because the

demagnetizing factor used to reduce the data was approximate.

4
field dependent, the latter more This dependence may be examined with

increasing ac measurement fields or dc bias fields.

For high quality, strongly coupled, sintered superconductors, the two critical

temperatures coincide for small measuring fields.^ ^ The coupling is not depressed

as much with increasing measurement field compared to a poor quality, weakly coupled

sample. A typical measurement"^^ for a sample of high quality, sintered, high-T^

(Bi-Pb)2Sr2Ca2 Cu30^

is presented in Fig. 1. The real {x') and imaginary (/") parts of

internal susceptibility (corrected for demagnetizing factor) are shown as a function of

increasing temperature, with the intrinsic and coupling segments identified. The

measurement field of 80 A-m~^ rms is large enough to separate the two components.

For a small measurement field of 0.8 A’m“^ rms, the critical temperatures overlap.

Figure 2 shows internal ac susceptibility curves for poor quality, weakly coupled,

sintered YBa2Cu307_g measured in 0.8 A*m“^ and 80 A-m~^ rms. Even for the

lower measurement field, the coupling (90.3 K) is considerably below the intrinsic

(91.1 K). There is no intrinsic x" peak for this sample for the fields used.

Identification of Critical Temperature

In resistivity measurements, is the temperature at which a percolation path is established. The corresponding temperature for magnetic susceptibility occurs when a bulk shielding path is established. This occurs at of the coupling component, in

particular at the onset of coupling. The distinction between the intrinsic onset and

coupling onset is pertinent for samples with weak coupling and for measurements made

in moderately large fields.

Fig. 2. Internal ac susceptibility for sintered YBa2Cu307_g with weak coupling

measured in 0.8 A-m“^ and 80 A-m"^ rms at 1000 FIz. Even for the lower

measurement field, the coupling (90.3 K) is measurably below the intrinsic

(91.1 K). The sample was approximately a cylinder, diameter 0.9 mm,

length 5 mm. The real part does not extrapolate to - 1 because the sample

volume used to compute susceptibility was approximate.

5
The critical temperature is sometimes taken as the midpoint of the diamagnetic

transition and the width of the transition is quoted.'^'^ There are several problems

with this. First, there are two transitions. Second, the widths of the transitions are field

dependent. Third, a large part of the transition to full diamagnetism is due to coupling.

Fourth, the complete intrinsic transition is often obscured by the coupling transition.

Therefore, it is more useful to define the critical temperatures as the onset

temperatures, although the precise onset temperatures are uncertain, particularly due

to fluctuation effects.

Figure 3 shows ac susceptibility and ac resistance measured on a bar of

(Bi-Pb)2Sr2Ca2Cu30^. To compare resistivity and susceptibility curves, we first

Fig. 3. Comparison of ac susceptibility and resistance as functions of temperature for

(Bi-Pb)2Sr2Ca2Cu30^. (a) Mass susceptibility for four ac fields (rms values

shown) at 1000 Hz. Intrinsic is 107.6 K and, for the smallest measuring

field, coupling is 103.6 K. (b) Resistance for an ac measuring current of

1 mA rms at 500 Hz. The critical temperatures obtained from susceptibility are labeled.

6
calculate the self-field of the transport current used in the resistivity measurements.

For a sample with circular cross section and uniform current density, the field inside

the sample is H{r) = rII{lTTa^), from Ampere’s circuital law, where r is the radial

coordinate, I is the current, and a is the sample radius. The average field obtained by

integration over the sample cross section (rather than over the radius) is {H) = //(Sira).

For our sample (actually of rectangular cross section with an effective radius a « 0.7

mm), the self-field of the transport current (1 mA) is negligible compared to any of the

measuring fields, and the best com*parison is with the lowest-field susceptibility curve.

Intrinsic is 107.6 K and coupling for the lowest-field measurement is 103.6 K, as seen in Fig. 3(a). These temperatures are identified in the plot of resistance-. Fig. 3(b).

Fig. 4. Comparison of ac susceptibility and resistance as functions of temperature for

YBa2Cu307_g with extremely weak intergranular coupling, (a) Mass susceptibility for three ac fields (rms values shown) at 1000 Hz. Intrinsic is 92.1 K

and, for the smallest measuring field, coupling is 13.9 K. (b) Resistance for

an ac measuring current of 0.1 mA rms at 500 Hz. The critical temperatures

obtained from susceptibility are labeled. The intrinsic critical temperature is unambiguous for this sample, which is semiconducting in the normal state.

7
The positive slope of the resistance curve above indicates a normal conductor and

the intrinsic does not correspond to a distinguishing feature. Coupling coincides

with the zero-resistance temperature.

Figure 4 shows ac susceptibility and ac resistance measured on a bar of sintered

Y-Ba-Cu-O with extremely weak intergranular coupling. The comparison is informative. From Fig. 4(a), intrinsic is 92.1 K but coupling is only 13.9 K and

very field dependent. In Fig. 4(b), there is a peak in the resistance curve at the intrinsic

where the material goes from the semiconducting state to the superconducting state. Resistance is zero at about the coupling T^. In summary, the temperature for zero

resistivity is related to the susceptibility coupling onset temperature, which is determined

by the quality of intergrain coupling. The temperature for the initial drop in resistivity

is related to the intrinsic onset temperature, which is determined by the quality of the

grains. Other experimental studies are consistent with these conjectures."^"^”^^

DEMAGNETIZING FACTORS

Demagnetizing factors are important for the understanding of the susceptibility

of superconductors and especially films. In the equation for magnetic induction,

B = + A/), H is the internal field, equal to the external or applied field

corrected by the demagnetizing field The source of the demagnetizing field is taken

to be magnetic poles on the surface of a magnetized specimen. In ellipsoids, the poles

are distributed in such a way that all fields are uniform. These fields include

and H, and the magnetization M. They are related vectorially by the equation

+ — NA/, where N is the demagnetizing tensor. If is along a principal axis of the ellipsoid, then H = - NM, where N is the scalar

demagnetizing factor. (If is not along a principal axis of the ellipsoid and 0,

A/ is uniform but not coaxial with and the direction and magnitude of M depend on

X-) For ellipsoids of revolution (spheroids) ^ is a function of the aspect ratio y of

the ellipsoid (ratio of the polar axis to the equatorial axis) and is independent of

susceptibility

N = (1 - y^) ^[1 - y(l - y^) ^\os V] (y < 1). (la)

(y = 1), (lb)

/»/ = (y2 _

2)“l[y(y2 _ l)“‘\osh“V ~ 1] (y > 1). (Ic)

Demagnetizing factors for cylinders have been examined in detail by Chen, Brug,

and Goldfarb/^ For cylinders, N is a function of y (ratio of length to diameter) and

also Xf which is assumed to be constant in the sample. With along the cylinder axis,

M and are both nonuniform except in two cases. When x = 0, M is uniform. The

approximation = 0 is used for saturated ferromagnets, diamagnets, and paramagnets.

When X is uniform and equal to The condition x ^ ^ applies to soft

ferromagnetic materials. When x~ - 1, (A/ + H^) is uniform and equal to [That

is, B = + fid + M) = 0.] This applies to superconductors in the shielding state.

There are two types of demagnetizing factors for cylinders. The fluxmetric (or

ballistic) demagnetizing factor A/y is the ratio where ( indicates an

8
Table 1. Longitudinal and transverse magnetometric

demagnetizing factors and for

cylinders with = -1, after Taylor/'^

y

is the ratio of length to diameter.

y ^mx

0 1 0

0.25 0.6764 0.2136

0.5 0.5258 0.2928

1 0.3692 0.3669

2 0.2341 0.4237

4 0.1361 0.4596

00 0 0.5

average over the center plane of the cylinder. The magnetometric demagnetizing factor

is the ratio where ( indicates an average over the volume of the

cylinder. Fluxmetric factors are used when magnetization is measured ballistically, with a short search coil closely wrapped around the center of a long sample. Magnetometric

factors are used with magnetometers that sense the entire sample volume, such as vibrating-sample magnetometers, SQUID magnetometers, and ac susceptometers. Both

Ay and depend on y and xExperimental and theoretical work on demagnetizing factors for cylinders dates

back to the 1880s. However, perhaps the only research ever published until recently

for = -1 was Taylor’s paper on conducting cylinders. He calculated polarizabilities, which we can convert to for several values of y for both longitudinal

and transverse fields. Our reduction of his results is given in Table 1. Note that

^mx ^my *" ^mz ~'^mx ^mz ~ subscripts X, y, z indicate the orthogonal

axes, with the applied field along z. (The sum of the three orthogonal magnetometric

demagnetizing factors for cylinders equals 1 only when x = 0-)

complete range of y and x are given in Ref. 52. Values of for - 1 are different

from those often tabulated for^ = 0. For y = 1, for example, - 0.3116 for = 0. One caveat is that values of for ;i' = - 1 are for superconductors in the shielding

state. Superconductors in the mixed state do not have constant susceptibility, which is one of the basic assumptions in the derivation.

The measurement of susceptibility requires the application of and the

measurement of M. The susceptibility dA//d//^ is characteristic of the sample and may

be termed the external susceptibility Xgxr internal susceptibility x^ characteristic of

the material, is dM/d//. The two susceptibilities are related: x ~ XexJO- ~ ^Xext)

Xext “Ml + ^X)' When ac susceptibility is measured, Xext ^ complex quantity:

Xext = Xat ^X"ext’ internal susceptibility is also complex: x = X' ^X"' When

relating the two quantities x ^nd Xexv imaginary parts are separated, resulting in

X' = ixi„- N(X'J + X’ext^)] / + X’J) - m’ext + 1).

Z" = + X"J) - 2Nx'^ + 1].

9

(2a)

(2b)
This rule appears periodically in the literature.'^'^"-^^ (We should point out that an

interesting artifact occurs in these equations for x' and )(’ in terms of x'exv ^nd N.

When N « 1 and x'ext ~ ^ '^

"ext ~ might occur for films in the normal state, x' and x" diverge, causing severe scatter in x' and x"- Such values of x'ext X"ext uncommon in actual measurements of thick films.)

Volume Fraction of Superconductor Grains

When the ideal (“X-ray”) density of a superconductor is known, and when there are no nonsuperconducting phases present, the volume fraction of superconducting

grains can be estimated from mass and volume measurements. Otherwise, susceptibility curves give some information on the volume fraction. When grains are fully coupled,

the entire volume of a granular sample, including voids and nonsuperconducting phases,

is shielded and;^ = -1. When grains are uncoupled, the inductive susceptibility signal

represents a summation of shielding signals from many grains; voids and

nonsuperconducting phases do not contribute.

Consider a sintered pellet of spherical superconductor grains, each with

susceptibility Xg = ~ occupying a volume fraction 0 in a medium of Xv ~ 0- ^he

measuring magnetic field strength be large enough to decouple the grains. The

demagnetizing factor of each sphere is -j. Let the bulk pellet have a very different

demagnetizing factor (0, for example). If the total internal susceptibility x is based on

the volume of the bulk pellet, including voids, can one deduce 0 from the value of x

(|;^| < 1)? When we first addressed this problem*^ ^ we suggested that, for grains of

unknown geometry, 0 =» \x\.

The susceptibility of mixtures was discussed by Maxwell in his Treatise.^^ For

perfectly conducting spheres in a nonmagnetic medium, the exact relationship is X = -3(f>/(2 + 0), or 0

= -2^/(3 + x)y where x

is the internal susceptibility of the

mixture. One implication is that, for dense pellets (0 1), the effective demagnetizing

factor for a susceptibility measurement is that of the pellet, not that of the grains, even

when the grains are decoupled. (If demagnetizing fields are thought of as arising from

surface magnetic poles, a dense mixture will have pole cancellation except at the surface

of the pellet.) The effective susceptibility of granular superconductors, including the

effect of magnetic penetration depth, has been examined recently."^*^”^

When the magnetic penetration depth A is on the order of the grain size, a

significant fraction of the grain volume does not contribute to the x signal.^'^ Typically,

A is on the order of 0.2 pm in high-T^ superconductors below For

illustration,^ a 100% dense sample composed of uncoupled plates of thickness 10 pm

would have, for // in the plane of the plates, ;i' = -0.7. The reduction in;i' is especially severe near 7^, where A becomes quite large: A(7)/A(0) = [1 -(7/7^)'*]“ where « = 4

in the two-fluid model, but empirically n ** 1 for Y-Ba-Cu-O.^

Therefore, to estimate the volume fraction of superconducting grains in a sintered

sample containing voids and nonsuperconducting inclusions: (1) The grains should be

decoupled by using measuring fields large enough to depress the coupling transition

10
temperature. Thereby, voids and nonsuperconducting phases are not included in the

shielded volume. (2) The grain dimensions should be significantly greater than A so that a large fraction of the grain volume is shielded. Otherwise, A should be included in the

estimation. (3) The %value used to estimate the volume fraction should be taken well

below the intrinsic of the grains. This avoids the increase in A near T^. Chen et al. have precisely modeled xi'H ^^d ;^(//) curves and calculated the volume fraction of

grains using A and the critical-state model for both grains and matrix. They deduced

the existence of grain clusters in some samples based on discrepancies between actual

77 77

volume fractions and those computed from the susceptibility data. '

A destructive way to obtain the volume fraction of superconducting grains is to

crush the sample pellet, collect all the powder, and use the original sample volume to

compute X- For loosely packed powder, the appropriate demagnetizing factor would

be closer to that of a grain, typically approximated as a sphere. This is most effective

when the crushing simply separates the grains from each other. If the grains were finely

pulverized, their size may approach A.

SUSCEPTIBILITY OF SUPERCONDUCTOR HUMS

To easily distinguish the superconducting and normal states, resistivity is best

measured in specimens with at least one thin dimension. Susceptibility, in contrast, is best measured when there is a large sample volume. Susceptibility measurements on

films thus present special problems and require some interpretation. In measuring the

susceptibility of films there are considerations of adequate shielded volume, field

orientation with respect to the film plane, demagnetizing-factor corrections, and film

thickness compared to A.

''The Absurdity of This Result ...”

The magnetic susceptibility of superconductor films measured in perpendicular

fields presents a paradox arising from perfect diamagnetism and a demagnetizing factor

N that approaches 1. For perfectly shielded superconductors, x - -1, so Xext ~

-(1 - As superconductor films get thinner, N ^ 1 andXgxt In the early

days of electromagnetism. Maxwell commented, “If the value of k [susceptibility] could

be negative and equal to 1/(47t) [in CGS units, 1 in SI units] we should have an infinite

value of the magnetization in the case of a magnetizing force acting normally to a flat

plate or disk. The absurdity of this result confirms what we said in Art. 428.”^’^

Magnetization M is the measured magnetic moment m per sample volume V^. As

the superconductor gets thinner (with its cross sectional area constant), X/^t

because - 0, not because m -* Furthermore, as the film gets very thin and

N ^ I, flux immediately penetrates the film for any and the superconductor is no

ye ^ longer in the shielding state. But is not m linearly proportional to or thickness tl

If it were, M would be independent of t. We will show that, for a range of r, m remains

constant for thick films of constant diameter d. Since a susceptometer pick-up coil

voltage V is proportional /n, this means that v is independent of t. The reason for this

singular behavior is that, as / 0, (1 - N) becomes proportional to t.

11
We model a circular superconductor film as an oblate ellipsoid, with major axes equal to the diameter d and minor axis equal to t. The ellipsoid volume

where y is the aspect ratio t/d. For small M = I - -jTry + 2y^. For y < 0.003,

the first two terms alone give N accurate to four significant figures. We use the

linear approximation andj^^^., = ml{V/i^) = -(1 - N)~^ to get: m = - - N) = (independent of and a gauge for superconductor films,

= -2/(^). (3)

For computational convenience, we extrapolate these arguments to a flat cylinder: ~ ^nd m = -^d^H^.

The magnetic moment of a superconductor arises from shielding currents which,

for small applied fields, flow within a penetration depth A of the surface. If we consider

a cylinder of diameter d and height r, the magnetic moment of the current loop is m = -jrrd I, where / is the total shielding current. In terms of a current density/,

m = - jTrdyXt, which is equal to a constant from the previous discussion. (Strictly, the

shielding current in a superconducting cylinder does not flow in a uniformly wide sheath

on its circumference.) What happens when t gets too small? As t decreases J must

increase until it equals the critical current density Any further decrease in t will

result in flux penetration into the superconductor. From m = = -jvd^JXt,

the critical thickness t^ is simply 2H^dl('rrXJ^), which offers a way to determine if A is known. The analysis breaks down for thin films (t - A) for which there is an enhanced

effective

We have experimentally verified some of these points with a series of seven

superconducting Bi-Sr-Ca-Cu-O thick films. They were made by metallo-organic

decomposition and had the usual granular characteristics. Their diameter was 3.22 mm

and they ranged in thickness from 2.8 (± 0.2) pm to 0.37 (± 0.02) pm. Plots of x'ert _ 1 a function of temperature, measured in a field of 0.8 A- m , 1000 Hz, normal to their

surface, were almost flat at low temperatures and indicated good diamagnetic shielding

at 4.2 K for all except the 0.37-^m film, x^t ^*2 ranged from -390 for the 2.8-jLtm

film to -2080 for the 0.55-pm film, (x^f was -2560 for the 0.31-pm film.) These

values are about half those expected from Xext ~ “2/(7ry), which is not unreasonable

considering the approximations involved. The pick-up coil voltages that gave rise to

these x^t ranged from 4.0 to 4.4 pV (almost constant). The 0.37-ju,m film voltage was

smaller, 3.6 pV. The imperfect shielding for the 0.37-^m film suggests that t^ » 0.4 pm.

Dependence on Field Angle

When the field is perpendicular to the superconductor film plane there is more

susceptometer signal than when the field is parallel. Aside from considerations

regarding magnetic penetration depth, the reason is that, for the perpendicular

orientation (N ~ 1), the applied field is enhanced to give a large internal field,

H = HJ(\ - N), and so are the magnetization, M = -HJ(\ - N), and the external

susceptibility, Xext ~ “(^ ” N)~^. (For magnetically soft, ferromagnetic films, in

comparison, x

^ ^ ^^d Xext

12
Misalignment of a superconductor film in a magnetometer or susceptometer

causes errors in the measurement of Xexr example, Gyorgy found that Xext ^

film of dimensions 7.6 mm x 5 mm x 0.13 iim varied from -0.08, for parallel to

the film surface, to -0.17 for H at an angle of 0.5° from the film plane, to -18 for

yo ^

at an angle of 8°. Teshima et al studied the angular dependence of the mixed-state

magnetic hysteresis loop and concluded that the magnetization is perpendicular to the

film plane for any direction of the applied field.*^^

We can examine the field-angle problem analytically. We start by modeling the

film as an isotropic (constant x), oblate ellipsoid. We use the relations

HaX = (4a)

/ a-' + Nj), (4b)

where the subscript ^ denotes the three ellipsoid axes x, y, z. is at an angle 6 with

respect to the normal to the film plane (z axis), so = H^cosd and = H^sind.

For a superconductor in the shielding state, we take Using 2N^ + = 1, we

have

= -H^cosd / (1 - N^), (5a) = -H^sind / (1 - = -IH^sind / (1 + N^). (5b)

But we measure M only in the ,axis of that is, M^cosd + M^ind, so our measured

susceptibility is

x^, = -cos^ J (1 - N^) - 2sm^e / (1 + N^). (6)

For superconductor films {N^ -» 1), a slightly out-of-plane {d < -jw) will cause

to dominate. The same analysis for magnetically soft, ferromagnetic ellipsoids

gives

= H^cosd / (7a)

M, = IH^sind / (1 - N^), (7b)

Xext - cos^0 / + 2sin^0 / (1 “

(8)

For ferromagnetic films, a slightly off-axis (6 > 0) will cause much of the

magnetization to be in plane.

Multiple Phases

When susceptibility is measured as a function of temperature in perpendicular

field, Xext curves appear strikingly different from the x curves obtained after correcting

for demagnetizing factor. Measurements made with the field perpendicular to the film

plane give very large values of external susceptibility. Values of -1500 for x'ext typical. However, when corrected for demagnetizing factor, any large negative value

of x'ext

''^ll convert to « - 1; there is negligible difference in x' between x^ = ~ 1500

13
and x'ext - “15. Furthermore, huge errors in volume will barely affect the x' result.

The obvious way to avoid these difficulties is to measure x

with the field parallel to the

plane of the film. However, this arrangement gives inadequate signal if the sample

volume is insufficient or if the magnetic penetration depth is large relative to the film

thickness. Furthermore, it is sometimes difficult to align films perfectly parallel to the

applied field. If these problems arise, one has to resort to measurements in

perpendicular field in which Xext measured and x is computed.

Superconducting Bi-Sr-Ca-Cu-O thick films, about 1 ixm thick, were prepared

by metallo-organic decomposition on single-crystal (lOO)-oriented MgO substrates.*^^

By adjusting the heat treatment of the films, we varied the relative concentration of the

two superconducting phases, Bi2SrCa2Cu20j. (“2122,” nominal = 85 K) and

Bi2Sr2Ca2Cu30^ (“2223,” nominal = 110 K). Each phase had both intrinsic and

coupling components. The films were highly oriented, composed of platelets with c axes perpendicular to the film plane. The platelets were 10-30 fim wide and 0.2 -0.3 ^m

thick. The films were characterized by X-ray diffraction, electrical resistivity, and ac

susceptibility in parallel and perpendicular fields with the intent of ascertaining the

relative concentration of the 2212 and 2223 phases.

The areas of the low-angle (002) X-ray diffraction peaks give a relative measure

QJ

of the concentration of the two phases, (Diffraction does not distinguish between

intrinsic and coupling components.) For the (002) reflection, 50% of the signal comes

from the top 0.16 /xm of the film. This is important because phase segregation through

the film thickness is likely. Resistivity provides evidence of two phases, but the lowertemperature 2212 phase will not be detected once the concentration of the highertemperature 2223 phase exceeds the percolation threshold. Thus, the relative

magnitudes of the resistivity transitions will give only a hint of the phase fractions.

For susceptibility, which is more useful, Xext ^

noted above, the

demagnetization correction to obtain x' from x'ext nonlinear. Thus x'^^ gives a better

indication of the relative fractions of the different intrinsic phases. The coupling

components should not be included in the determination; this might require

measurements at several fields. (As expected, we found that the coupling component

was tied to the “parent” phase. In single-phase 2212 samples, for example, there was

no 2223 coupling component.) x^ highly nonlinear, is useful for highlighting the

transition temperature of each phase, especially the higher-temperature 2223 phase,

which is often obscured in Xext' strategy for determining the phase fractions, not

suitable for routine measurements, is to scrape the films into a powder and measure its susceptibility, thus eliminating the coupling components and reducing the demagnetizing

factor problem. Perhaps the major utility of susceptibility measurements in granular

superconductors, however, is for characterizing the quality of intergrain coupling by the

field dependence of the coupling transition temperature.

In Table 2 we give values of the percentages of the 2212 and 2223 phases as inferred from resistivity. X-ray diffraction, and susceptibility, for one of the films while

intact and after powdering. The presence of a small amount of Bi2Sr2CuO^ was

ignored except in the X-ray determinations, which do not sum to 100. As far as we

14
Table 2. Volume percentage of 2212 and 2223 phases

in a Bi-Sr-Ca-Cu-O film as inferred

from different measurement methods.

Morphology Method 2212 2223

Film P 5 95

Film X-ray 12 82

Film Xexv 75 25

Film Xexr^ 60 40

Powder X-ray 32 47

Powder Xext 30 70

Powder x' 40 60

know, none of these determinations is “correct.” Our intent is to illustrate the difficulty

of the problem.

UNITS

Magnetic volume susceptibility % is dA//d// in both SI and CGS. In terms of base

units, is a dimensionless quantity. In CGS, however, x is usually expressed as emu,

emu-cm”^, or emu'cm~^-Oe“^ The designation “emu” is merely an indicator that

electromagnetic units are in use; it is not a unit. The unusual x units in CGS arise from

ambiguity in the units for In CGS, H is in Oe (dimensionally and numerically

equivalent to G). Magnetization, when written as is in G. When magnetization

is expressed simply as M (the magnetic moment m per unit volume) its units are erg-G“^*cm“^ (conventionally expressed as emu*cm~^), which are dimensionally but

not numerically equivalent to G. Occasionally, CGS susceptibility is written as Arrx

(dimensionless), which is equal to SI susceptibility (dimensionless).

Some of the symbols used in this paper and their associated SI units are M [A-m“^], J [A-m“^], a [m], H [A-m~^], and W [J-m“^]. To convert equations to

CGS EMU, replace symbols for H by (47t/liq)“’^^//, symbols for J by (47r/jUQ)’'^V, symbols

for M by (47r/Ato) '^^^» symbols for;^ by Attx, and simplify as necessary. The symbols and

their associated CGS EMU are M [emu-cm~^], / [abamp-cm”^], a [cm], H [Oe], and

W [erg-cm~^]. (Note that in CGS Gaussian units, J would be in statamp*cm“^.) We

avoid the use of “practical” or mixed units.

SUSCEPTIBILITY OF BULK SUPERCONDUCTORS, CRITICAL-STATE MODEL

In this section we review magnetic formulas for isotropic superconductors of

px oo several geometries in the Bean critical-state model. Except for some of the high

field susceptibility equations, most can be found in the literature in one form or another. Transport current, dc bias field, lower critical field, and surface barrier are all taken to be zero. is the applied dc field. Hp is the full-penetration field, a function

of the critical current density (considered isotropic) and the cylinder radius (or slab

half-thickness) a. M is the magnetization per unit volume of superconductor, which is equal to the half-width of the hysteresis loop (7 AM) for//^ > Hp. is the maximum

15
field for the hysteresis cycle, that is, the amplitude of the ac field. W is the hysteresis

loss per unit volume per field cycle, x' ^nd x" are the real and imaginary parts of ac

susceptibility. The equations for x' and are derived from Fourier integrals of the

magnetization^^ using the complete equations for magnetization as a function of

field,^^ not the initial curve. By x' and x' we mean the fundamental Fourier

components x{ and x{-

^ a check, we know from first principles^^’^^ that

X" = for any For > Hp, the Bean hysteresis loop is almost

rectangular, and W A point worth emphasizing is that x' ^ Xdc = ^l^aOnly in the linear limit < Hp does x' Xdr Another point, applicable to the

cylinder in transverse field, is that the susceptibilities are based on the applied field,

and are not corrected for demagnetization.

Infinite Slab, Thickness 2a, Parallel

The equations below are for the initial magnetization curve (0 < < Hp)

and the descending portion of the hysteresis loop (which depends on whether H^ < Hp

or H^ > Hp). To get the ascending portion, replace M by -M and H^ by -H^ in

the equations. For example, the ascending magnetization curve for H^ < Hp is M = -H^ - j(H^^ - 2HJI^ - Hj-yHp, based on Eq. (11). To calculate x' and x"

we use COS®- For ^ f^p< we integrate from 0 = 0 to cos ^(1 - 2/r), and

from 6 = cos“Xl ~ 2/r) to tt, where x = H IH^. A useful trigonometric identity is sin[cos ^(1 - 2/r)] = {2/x)(x - 1)

>/2

Hp = J,a (9)

(OS//, VI (10)

1 + - »aWp ^ Ha (11)

?!fip (12)

-1 + (13)

2r/(37T) (14)

-H ^ Hp^- TfX H^-a Ufim - H,f!Hp

-2Hp < //, S‘Hm) (15a)

ifip (-fim ^ ^ "m--2Hp) (15b)

(16)

{(-1 + ^x) cos 1(1 - 2/r)

[-1 -1 4/(3i) - 4/(3J:^)] {x - 1)^'} / w (17)

(6/jc - 4/;c^)/(3ir) (18)

Radius tI, AxuU

-98

(19)

H//Hp - (Os//, S//p (20)

16
H„ <Hp. M= + hJh^

- (21)

w = (22)

X' = — 1 + j: — 5x^/16 (23) = (Ax - 2x^)l(irr) (24)

M = -\Hp A - fiaflHp

+ (H„ - (25a)

M = ^ Ha ~^p) (25b)

W = (26)

x' = {(-1 + X - 5x^/l6) cos~^(l -- llx)

+ [-19/12 + |x + llx - 2/(ic2)] (x - I)''''} / 7T (27) = (Ajx - 2/x^)l(3ir) (28)

Infinite Cylinder, Radius a, Transverse H

This case has not been solved exactly, but a few approaches have been used

successfully. The method of Carr et al. gives analytic equations in the limits of small

and large fields^""^^^ and is mostly numerical in between.'^^'^ The hysteresis loss in SI units is W' = for H„ < H^, and W = for The

method of Zenkevitch et al., presented below, gives reasonable equations for the

full field range.'^^'^“^^ For M we preserved the form of the equations developed in

Ref. 104. In terms of the equations are simply twice those for the cylinder in axial

H. Different functional forms are given by other authors.^^^''^^

m p

= U^hr (29)

M = + (H, - HpfIHp^] (OSH, VI (30)

M = jHp[l + (H„ -H,- 2Hpf/(2Hpf] - + (H„ - Hp] i-Hn, VI (31)

W = //// (32)

Xext

II 1N) + 1 (33)

X”ext = (& - 4r^)/(3ir) (34)

M = ^Hp[l + (H„ -H,- 2Hpf/(2Hpf] -

(Hm -2Up <H, <l"m) (35a)

M = iHp i-Hm -2Hp) (35b)

W = hoHmHp - |mo«/ (36)

Xexi = {(-2 + 2* - ^3c^) cos -1(1 - 2/r)

+ [-19/6 + + 2/r - 4/(3ot2)] (X - 1)*"’} / TT (37)

X^ext = (8/x - 4/x^)/(3ir) (38)

17
In Fig. 5 we plot x' )(' as functions of for the three cases. As we

discuss in Ref. 89, Hp is a good proxy for temperature, and the inverted abscissa in

Fig. 5 is meant to indicate the temperature dependence of susceptibility. Note that,

for the transverse-field case,

^ — 2 asjr -* 0. For this geometry the demagnetizing

factor is j; after correcting for demagnetization, x'

Critical Current Density

Transport is measured directly using electrical techniques.-^^ Alternatively,

several magnetic methods may be used.'^^^ To estimate from magnetic

measurements, one could measure M (at > Hp) and derive what has become

known as “magnetization 7^.” Alternatively, one could measure the reduction in field

A//^ = IHp required to reverse the magnetization in the hysteresis loop.^^^’'^'^"

Other magnetic methods are based on ac susceptibility, usually with a dc

bias field,'^^^"'^'^^ sometimes using harmonic analysis.‘^^’^'^’^“^“'^““ Curve fitting is often

involved. For example, at the peak of x'\ f^rn ~ cylinders'^ and

for slabs'^^'^ (Fig. 5). The dimensions of the sample are used in all these methods.

The equations that relate to the magnetization are based on the assumption

that 7^ is a constant, independent of //^. Apparently begging the question, the equations

are often used to deduce 7^ as a function of This is not necessarily a serious error provided that certain conditions are satisfied: (1) The sample is homogeneous and

isotropic, although the critical-state model has been extended for anisotropic critical

current densities.-^ (2) The sample has dimensions consistent with the model.

However, in the fully penetrated state {H > Hp) in the Bean formalism, the

magnetization is saturated even for finite dimensions. Thus, for example, the infinite

cylinder equation for M can be used for a finite disk in perpendicular field.*^^ (3) The

field at which magnetization M is taken should be large enough {H > Hp) such that 7^

is not a strong function of H^.^^ Specifically, estimates of 7^ from A/ at //^ = 0 are

^p/

Fig. 5. External ac susceptibility for slab and cylinder geometries as functions of

decreasing HpJH^ = x~^ based on the critical-state model.

18
likely to be in error. (4) Flux vortices are well pinned; that is, there is no flux

QQ

creep. In practical type-II superconductors with high/^, or in high-T^ superconductors

at low temperature, this is generally true. (5) There is little contribution from

surface barriers'^“'^“^“^ and reversible magnetization.'^”^ This often applies at low

temperatures, where the hysteresis loop symmetrically spans positive and negative M

at high fields.

All these conditions for use of the critical-state model are seldom satisfied

simultaneously. Furthermore, interpretation of the magnetization loop and application

of the critical-state formulas are different for weakly linked and homogeneous samples.

For weakly linked samples, such as sintered, high-7 superconductors, the magnetization

?5 ^ at high fields gives the intragrain critical current. The average dimension of the grains

should be used to deduce /^. In low fields the grains are coupled. The correct dimension is that of the specimen and the deduced is comparable to transport But as noted above, the critical-state model cannot be used accurately at low fields, and in any

event, the coupled material is not homogeneous. For homogeneous samples, such as

single crystals and samples with contiguous oriented grains, the basic critical-state model

can be used if the field is applied such that shielding currents flow isotropically.

Lower Critical Field

In materials with intrinsic and coupling components, there will be two lower

critical fields. Generally, type-II superconductors corresponds to the field

at which the initial magnetization-versus-field curve deviates from linearity. In

practice, detection of this field value is difficult because the deviation may be subtle,

especially if the critical current density is large. However, if the magnetic field is cycled, an area, indicative of hysteresis loss, will be traced out in the M-H plane when

H > This is effectively the field cycle used in ac susceptibility

measurements, and hysteresis loss will appear as a positive imaginary part of

susceptibility Therefore, when the field amplitude used in a measurement

exceeds positive x" is expected and measurements of x\'^ constant

or x'\^ constant may be used to deduce an upper limit for

Equivalent to the x'(D method, the corresponding feature in x'i^^ is a departure from full diamagnetism.'^^"^’^'^”'^'^^

Susceptibility and its harmonics, measured as functions of H, reveal a distinct

feature at In another method, is included in an expanded critical-state

model and is a function of the remanent (trapped) magnetization.'^'^^ In small

specimens, such as thin films, one should be aware of the enhancement in that

arises when the magnetic penetration depth is on the order of one of the sample

dimensions.”^’^'^'^"^'^^"^'^^

Interpretation of Peak in x"

The critical-state model can account for many of the features in the temperaturedependent x" (and x') without invoking any kind of loss mechanism or irreversibility

other than magnetic hysteresis. The model may even be used to describe the behavior

19
of the intergranular coupling component of sintered superconductors.'^^’*^'^^ The

following interpretation of the peaks in %" versus increasing T is based on the criticalstate approach and can be applied to both the intrinsic and coupling

components.^ Susceptibility is measured in an ac field of constant magnitude.

We divide the temperature scale into three ranges. (1) For T < T^, the ac field causes

shielding currents to flow on the surface of the sample and a line to be traced out in

the M-H plane. There is no hysteresis because J < H < and x" = 0- (2) For

T somewhat below T^, and have decreased and shielding currents have to flow

within the sample. The hysteresis loop in the M-H plane has an area associated with

it, and x

' > 0- TTie losses and x" attain their maximum values after supercurrents and

penetrated flux reach the center of the specimen. (3) As T approaches

approaches 0, and the M-H loop has collapsed. Even though H^^ also approaches 0,

there is no area to the loop and no hysteresis loss; x' - 0- "This interpretation is in

accordance with the expectations of the critical-state model, in which all energy losses are hysteretic and frequency independent.

Other loss mechanisms besides hysteresis may contribute to;^" in superconductors.

These losses may be classified as time-dependent or time-independent.'^'^-^ In

particular, frequency effects may be explained by flux creep and flux flow.-^'^^”^'^'^

Other frequency-dependent contributions to x'* could be eddy currents of normal-state

electrons in a two-fluid model'^’-^'^'^”^'^^ and vortex lattice viscosity and viscous

damping.'^’-^'^*^”-^^^ One physical interpretation of the ac susceptibility of

superconductors is in terms of BCS theory, the generation of supercurrents, and the

establishment of the Meissner state.'^'^'^''^^^ Others have used a superconductor glass

model and scaling with critical exponents.^'^'^''^'^'^ The time- and field-dependent onset

of irreversibility (an “irreversibility plane''), which may occur below the temperature of

the x" onset, may be observed by the generation of odd harmonics of susceptibility

Occasionally the intrinsic x" peak is not apparent.^^'^ There are several cases where this is likely, with regard to granular high-7^ superconductors. (1) In well

coupled materials, a small measuring field will cause the coupling peak to obscure the

intrinsic peak. (2) There may be insufficient total grain volume. (3) Grain sizes may

be on the order of (4) The grains may be superconducting only on their surface;

the interior is normal, perhaps owing to deficient oxygenation, or superconducting only

at a lower temperature. In this state, there would be insufficient superconducting

volume and therefore a low level of losses. (5) In good quality grains,'^'^^ H^^ may be

large just below T^. If H^^ only falls below H^ at no x” peak will be seen. That is, at the temperature that flux penetrates the grain, there is no longer any bulk pinning.

A larger H^ will often elicit a measurable x'"-

Upper Critical Field

A plot of H^2 versus T is the same as a plot of versus H. Following the

distinction of intrinsic and coupling components, there are both intrinsic and coupling

upper critical fields. Susceptibility can be used to deduce H^2 versus T (at high

temperatures) in much the same way as it was used to determine H^^ versus T. At the

onset temperature there is an equivalence between the measuring field and H^2^^

20
SUSCEPTOMETER DESIGN

Construction

Low-frequency ac susceptibility measurements are often made with a coaxial

mutual-inductance coil system consisting of a primary excitation field coil, a secondary

pick-up coil, and a secondary compensation coil (three-coil system). The two secondary

coils, connected to a bridge circuit, have the same dimensions. The midpoint between

them is concentric with the primary coil.'^'^^"'^'^^ An alternative three-coil system,

suitable for short or long samples, or when coil length, eddy currents, or,temperature

gradients are a problem, is with all coils concentric."^^ Here, the pick-up coil is close

to the sample, but the compensation and field coils are not. To ensure balance between

the pick-up and compensation coils, they are wound so that the mutual inductance of

each with respect to the field coil is the same. Another three-coil system uses a largebore field coil with side-by-side secondary coils.'^^^ If the compensation coil, or some

other field compensation source, is omitted (two-coil system),^^* the measured

quantity is ac permeability = fi' + ifi"), which is numerically related to ac

susceptibility; = ixq^I + x), X')y

1^" ~ where ijlq is the permeability

of vacuum. If a single coil is used (one-coil system), one can relate the changes in

inductance and resistance of the coil to ac x- Calibration is readily achieved in any

magnetometric system when the sample is contained within the pick-up coil.

At audio-frequencies, sensitivity is greatest for the susceptometer (three-coil)

configuration. The ac susceptometer relies on inductive coupling between coils. A

large number of turns on the pick-up coil increases the signal-to-noise ratio at low

frequencies but causes capacitive coupling at high frequencies."^

A

typical

instrument with 520 turns of 28 gauge wire (0.32 mm diameter) on the primary and

1340 turns of 38 gauge wire (0.10 mm diameter) on each secondary has capacitive

coupling above about 5 kHz. The usable frequency range can be extended by reducing

the number of tums."^^ Resonance methods, using a single coil^^’"^^'^“"^^ or a twocoil bridge,^ may be used up to radio frequencies. A sample inserted in one of the

coils causes a change in self inductance and phase that is related to the susceptibility.

These methods are quite sensitive but have been often neglected. For metallic samples,

eddy-current signals may present a problem at high frequencies.

Alternating-field susceptibility characterizes the shielding properties of

superconductors, whether measured upon cooling or upon warming after zero-field

cooling. Typically, measurements are made in zero dc field as a function of ac field or

in small ac fields as a function of dc bias field.'^^’'^'^^’"^^*^ The laboratory environment

should not be ignored as a source of dc field, particularly in materials, such as sintered

high-7^ superconductors, that are weakly coupled. For precise measurements it is desirable to surround the susceptometer with a high-permeability magnetic shield not

too close to the coils.

The pick-up and compensation coils may be connected in series opposition or in

parallel to the differential input of a lock-in amplifier. The lock-in amplifier may be

used either as a null detector upon adjustment of standard inductors and resistors

21
(Hartshorn bridge)/ or simply as an off-balance meter. An input band-pass filter

should be used to attenuate harmonics with typical attenuation of more than 60 dB.

In harmonic susceptibility measurements, filtering is used to isolate the harmonic of

interest, as was done in Ref. 89. A constant ac current source (transconductance

amplifier) should be used to drive the alternating-field coil if the temperature of the coil

changes during the measurement. If the pick-up coil and compensation coil are immersed in liquid helium, Johnson (thermal) noise will be reduced and the coil

resistances will remain constant. A disadvantage of this arrangement is that the sample

will be weakly coupled to the pick-up coil if a reentrant Dewar is used to control the

temperature of the sample. We will describe a system in which the coil temperature

changes with that of the sample. To maximize sample coupling, the pick-up coils are wound beneath the field coil. It is generally considered desirable to match the

impedance of the pick-up coils to the input impedance of the lock-in amplifier. In

practice, however, we have found that there is no advantage in terms of noise or

sensitivity, and that the transformer may contribute phase shifts.

Inevitably, there will be mismatch between the pick-up and compensation coils.

This can vary with temperature and over time as the coils contract and expand. Twoposition susceptometers avoid this problem.'^ The sample is measured in each

coil, with its position controlled by a sample rod and piston. The voltage signal

attributable to the sample changes sign but the signal arising from coil imbalance does

not. When the two measurements are subtracted and divided by two, the imbalance

signal is removed. To minimize the imbalance and exploit the dynamic range of the

lock-in amplifier, we use a trimming loop in series with the field coil. Its position is adjusted once to increase the coupling to either secondary coil. Its contribution to the

measurement field is negligible. Sometimes a grounded “coil foil” (a sheet consisting

of thin parallel strands of insulated copper wire) is used between the field coil and pickup coils to reduce their capacitive coupling. We have found that such a shield

contributes little to the susceptometer performance. Another possible use for coil foil

7 77? is for thermal stability.

To help achieve an isothermal environment, we use a sapphire (AI2O3 ) tube as a coil form. Sapphire is a good thermal conductor and poor electrical conductor.

Metal coil forms are not used because they can contribute eddy-current signals.

Metal structural components and heaters are well separated from the coils for the

same reason. The normal heat leak in the Dewar can be used to slowly warm the

sample, or the temperature may be actively controlled. A resistance or semiconductor

thermometer and its connecting wires are thermally anchored to the sapphire. (In

systems designed for use with high dc bias fields, the thermometer should be relatively

insensitive to field. Carbon glass is an example.) Our sample holder is designed in

three identical sections so that, in both the upper and lower positions, each secondary

coil detects the same holder material except for the sample itself. The holder is open

at the sides for sample insertion and removal. The clearance between the sample

holder and the sapphire tube is small. To reduce the possibility of damage to the coils

from sample-rod motion if air becomes trapped and frozen, we use the release

mechanism shown in Fig. 6. The 0-ring releases at about 7 N force.

22
Phase Adjustment

The phase angle \p of the lock-in amplifier used to detect the pick-up coil voltage

must be adjusted to correctly separate the real and imaginary parts of susceptibility.

The adjustment may be done before the measurement or by computation, after the

measurement: x'm = X’m.O + X'mfi ^inV. X"m = X'w.O ~ X’ext.O where

the 0 subscripts indicate the measured susceptibilities before adjustment of the phase

angle.

Here is an opportunity for the experimentalist to use good judgement. The first

guideline is that x" tnust never be negative. Phase adjustment is accomplished when

samples are in low-loss states. Examples are superconductors in the shielding state (low

temperature and low measuring field) and spin glasses'^ in the paramagnetic state. Precise phase adjustment is necessary to observe frequency shifts in the susceptibility curves.

The phase adjustment should be repeated for each sample and each measurement

frequency but not for each measurement field. It is not practical to adjust phase for

each measuring temperature, but, as temperature changes, the resistance of the

susceptometer coils does change. This could cause some phase change in the mutual

inductance bridge. To avoid this, the lock-in amplifier should be referenced to the

voltage drop across a resistor in series with the primary coil.

On the secondary circuit, the input impedance of the lock-in amplifier is large

enough to make any change in the coil resistance insignificant. If desired, however, wire

with a residual resistivity ratio close to 1, such as brass, could be used to wind the

coils instead of copper magnet wire. (Wire with magnetic impurities should be

avoided.) An added advantage would be the suppression of any eddy currents in the

coils themselves. We have found copper wire to be satisfactory. Phase adjustment at

different temperatures may be unavoidable if there are problems with eddy currents in

structural elements.

Free-moving

sample rod

(mechanism locked)

Frozen

sample rod

(mechanism released)

Threaded

sleeve

0-ring

Fig. 6. Release mechanism used to decouple the top of the sample rod from the drive

piston if there is excessive resistance to motion. The device is at room

temperature, above the experiment Dewar.

23
Measurements

With the use of computer controlled instruments, it is tempting to take as much

data as possible in a single measurement session. Multiple measurement frequencies

may be used at each temperature. It is usually not advisable to change magnetic field

strength during the measurement of hysteretic materials because, at high fields,

magnetic flux penetrates a superconductor and remains pinned. When measurements

are continued at reduced fields, the pinned flux might wiggle around and contribute to

the susceptometer signal. For a similar reason, ac measurements are usually best made

upon warming, after cooling in zero field. Upon cooling through 7^, flux exclusion is often incomplete and pinned flux may remain in the sample. The effect is small and

it would be of concern only in precise work.

SUSCEPTOMETER CALIBRATION

Analytical and Numerical Calibrations and Standards

A susceptometer pick-up coil can be calibrated analytically for spherical

samples."^*^^ We use the dipole field of a uniformly magnetized sphere of magnetic

moment m = where M is the magnetization and is the sample volume, and

calculate the total flux <I> through a thin pick-up coil of radius a, length and n

We assume M = Xext^a ~ ^0 sinZir/r, where Xext external

susceptibility in SI units and Hq and / are the amplitude and frequency of the applied

field Finally, we use Faraday’s law, v = d4>/d/, where v is the pick-up coil voltage,

and get

= WK7 0^ + / (« (39)

where we now refer to the rms field and voltage and have ignored any sample voltage

induced in the compensation coil. For a given pick-up coil, we assign a constant a to

the quantity [( 7

^)^ + / («ir/iQ). When measuring harmonic susceptibility, the

harmonic frequency, not the field frequency, is used for

It is also possible to calibrate susceptometers numerically for cylindrical

samples with small susceptibilities » 0) or small demagnetizing factors {N =» 0).

Either will have almost uniform magnetization. The procedure models the sample as a solenoid and requires computation of the mutual inductance L* between the model

solenoid and the susceptometer pick-up coil. Once L* is known for the sample,

we have Xext ~ ^rms ! where is the pick-up coil voltage, € is the

sample length,/ is the frequency, and//^ is the applied field. Usually L* is calculated

numerically, but if the pick-up coil is thin, L* can be calculated analytically.

Other calibration methods use standards. These include materials with known

susceptibility, and magnetically soft ferromagnets Of - ») and superconductors

(;^ = -1) with known demagnetizing factors (such as spheres and cylinders). These

are discussed in Ref. 182. If cylinders are used, accurate values of corresponding

to the standards’ susceptibilities, are necessary.

24
Eddy Current Method

A classical exercise is the calculation of the complex magnetic susceptibility of an

isotropic, conducting sphere in a uniform ac magnetic fie Id/ The apparent

susceptibility is due to eddy currents, not to the magnetic properties of the material,

much like the magnetic susceptibility of a superconductor arises from shielding currents.

The real and imaginary parts of external susceptibility are calculated in terms of the

sphere radius a and the skin depth 5,

X'ext ~ -|(5/fl)[sinh(2fl/5)-sin(2fl/5)] / [cosh(2^/5)-cos(2^/5)] - y, (40a)

^ext ~f

(5/fl)[sinh(2fl/5)+sin(2^/5)] / [cosh(2^/5)-cos(2^z/5)] - ^(5^/a^). (40b)

In the limit of zero resistivity, 5 0, x'ext ~ 7'^ using N = -j for a sphere, -1.

This result provides the basis for another calibration method that uses spheres of normal conductors such as copper. One requirement is knowledge of the

temperature dependence of 5 or, equivalently, resistivity p. Matthiessen’s rule,

expressed in terms of the residual resistivity ratio RRR = p(273 K)/p(4 K), is p(7) = pi(T) + p(273 K)/RRR, where p^ is the intrinsic resistivity. For copper,

p(273 K) = 1.543x10“^ H-m, and values of RRR range from 10 to 2000. Values of

Pl(T) are tabulated.^*^^ From p(T) we calculate 5(T) = \p(T)/{7rfp,)f^^, where / is the

measurement frequency and p, = /xq for a nonmagnetic material. The point is that a copper sphere with known RRR will have predictable curves of x'ext X'

X") as functions of temperature and frequency. [If the ac susceptometer is already

calibrated, this method can be used to measure p{T).] In Fig. 7 we show x'ext ^^id x’^{ as functions of temperature for a copper sphere, a = 3.088 mm, at 10, 100, and 1000

Hz (points). The measurement field was 80 A-m“^ rms, although the susceptibilities are independent of field. The curves shown are the predicted x'ext X'at

Fig. 7. Apparent susceptibility of a copper sphere as a function of temperature at 10,

100, and 1000 Hz based on eddy currents and skin depth. Points are measured

susceptibilities; curves are calculated susceptibilities.

25
eddy-current equations. The curves overlap the data at 10 Hz. The adjustable

parameters were the lock-in-amplifier phase angle xp (for each frequency) and RRR

(adjusted to 150). We have used the 1000-Hz data in lectures to show students an

“onset of diamagnetism” near 300 K: a room-temperature superconductor!

Two related problems are for infinite cylinders in perpendicular and parallel

fields.'^ The infinite-cylinder solutions, together with the demagnetizing

factors in Table 1, suggest that calibrations can be done with finite-sized cylinders

provided 5 is small (that is, p is small and / is large).

SUSCEPTOMETER SENSITIVITY

The equation that describes the response of an ac susceptometer is Xext ~

av!{V f H ), where volume susceptibility in SI units, a is the calibration ^ ^ 7 — 1 — 1 constant (a function of the pick-up coil geometry) [A-m -V ^-s ], v is the pick-up

coil voltage [V], is the sample volume [m^], / is the frequency [s~^], and is the

magnetic field strength [A-m“^]. The sensitivity of an ac susceptometer depends on

a and on the precision in the output of the ac voltmeter, typically a lock-in

amplifier. Usually v is proportional to/, so oLVpl

is the magnetic moment precision

[A-m^]. (There is actually some degradation in voltage precision at low frequencies

such as 10 Hz due to 1// noise.) For one of our susceptometers, a = 2.1546 and vJ f - 2.5x10“^®, giving a moment precision of 5x10”^® A-m^ (5x10“^ emu). For

comparison, commercial vibrating-sample magnetometers are able to measure about

5x 10”^ A-m^ (5x 10“^ emu) and commercial SQUID magnetometers can detect about

10“^^ A-m^ (10~^ emu). In principle, their sensitivity can be improved by increasing

the pick-up coils’ filling factor.-^ Altemating-gradient-force magnetometers

can measure about 10“^^ A-m^ (10“^ emu).^^"^

Our moment precision of 5x10“^® A-m^ means the precision in Xext 5 X

10“ ^^/(F^//). (That is, we can measure the susceptibility of a sample more precisely

if we have a larger sample or use a larger measuring field.) For the favorable case of a sample sphere of diameter 5 mm measured in a field of 800 A-m“\ the susceptibility

precision would be 10“^ (SI). For a 3-mm-diameter sphere measured in 80 A*m“^ the

precision would be only 4x10“"^ (SI). This assumes that both and H are known

exactly. Precision is not the same as accuracy, which depends on instrument calibration.

ACKNOWLEDGMENTS

We had valuable discussions with D.-X. Chen (Universitat Autonoma de

Barcelona), I, B. Goldberg (Rockwell International Corporation), T. Ishida (Ibaraki

University), and J. K. Krause (Lake Shore Cryotronics, Inc.). R. A. Hein (University

of Washington) offered many thoughtful comments and suggestions. A sintered

Y-Ba-Cu-O sample was provided by A. K. Mehrotra. D. L. Ried, R. L. Spomer, and

R. J. Loughran assisted with data acquisition. T. W. Petersen, R. M. Gerrans, and

R. W. Cross prepared the figures for publication. Work at the National Institute of

Standards and Technology was supported by the Electronics and Electrical Engineering

Laboratory. This paper is not subject to copyright.

26
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32
BL-114A

(5-90)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

1 . PUMJCAT10N OR REPORT NUMBER

NISTIR 3977

2. PERFORMINQ OROANtZATION REPORT NUMBER

B92-0009

3. PUBLICATION DATE

October 1991

TrrUE AND SUBITTLl

Alternating-Field Susceptometry and

Magnetic Susceptibility of Superconductors

5. AUTHOR(S)

R. B. Goldfarb, M. Lelental, and C. A. Thompson

PERFORMINO OROANOATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTmiTE OF STAIWAROS ANO TECHNOLOGY

BOULDER, COLORADO B0303-332B

7. CONTRACT/QRANT NUMBER

t. TYPE OF REPORT ANO PERIOD COVERED

9.

SPONSORING ORGANIZATION NAME ANO COMPLETE ADDRESS (STREET, CITY, STATE, ZIP)

10.

SUPPLEMENTARY NOTES

Published in: "Magnetic Susceptibility of Superconductors and Other Spin Systems,”

R. A. Hein, T. L. Francavilla, and D. H. Uebenberg, Editors, Plenum Press, New York (1992).

11.

ABSTRACT (A 200-WORO OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. E DOCUMENT INCLUDES A SIGNIFICANT BIBLIOGRAPHY OR

LITERATURE SURVEY, MENTION IT WRE.)

This review critically analyzes current practice in the design, calibration, sensitivity determination,

and operation of alternating-field susceptometers, and examines applications in magnetic susceptibility

measurements of superconductors. Critical parameters of the intrinsic and coupling components of granular

superconductors may be deduced from magnetic susceptibility measurements. The onset of intrinsic

diamagnetism corresponds to the initial decrease in electrical resistivity upon cooling, but the onset of

intergranular coupling coincides with the temperature for zero resistivity. The lower critical field may be

determined by the field at which the imaginary part of susceptibility increases from zero. Unusual features in the susceptibility of superconductor films, such as a magnetic moment that is independent of film thickness and

the variation of susceptibility with angle, are related to demagnetization. Demagnetizing factors of

superconductor cylinders are significantly different from those commonly tabulated for materials with small

susceptibilities. Rules for the susceptibility of mixtures with specific demagnetizing factors are used to estimate

the volume fraction of superconducting grains in sintered materials. Common misunderstandings of the

Meissner effect, magnetic units, and formula conversions are discussed. There is a comprehensive summary

of critical-state formulas for slabs and cylinders, including new equations for complex susceptibility in large

alternating fields. Limitations on the use of the critical-state model for deducing critical current density are listed

and the meaning of the imaginary part of susceptibility is considered.

12.

KEY WORDS (6 TO 12 ENTRIES; ALFHABET1CAL ORDER; CAFfTAUZE ONLY RRORER NAMES; ANO SEPARATE KEY WORDS BY SEMICOLONS)

calibrations; critical current; critical field; critical-state model; critical temperature; demagnetization;

demagnetizing factors; granular superconductors; intergranular coupling; Meissner effect; susceptibility;

susceptometers; superconductors; thick films.

13.

AVAILABILITY UNLIMITED FOR OFF1CUL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NT1S). ORDER FROM SUPERmTENOENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE,

WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NT1S), SPRINGFIELD, VA 22161.

14. NUMBER OF PRINTED PAGES

36

IS. PRICE

ELECTRONIC FORM
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