A heuristic, general, field-dependent critical-current model for the magnetic properties of type-II superconductors

PHYSICA Physica C 233 (1994) 165-178

ELSEVIER

A heuristic, general, field-dependent critical-current model for the magnetic properties of type-II superconductors D.

Gugan

H.H. WillsPhysicsLaboratory, Royal Fort, BristolBS8 1TL, UK Received 13 August 1994

Abstract

A field-dependent critical-current model is developed which allows the computation of hysteresis loops by current superposition for a wide variety of forms of magnetisation curve. Some typical loops are illustrated, as are also the real and the imaginary parts of the AC voltage response for the harmonics n = 1, 2, and 3 as a function of AC modulation field and of DC bias field. The fundamental AC response takes a particularly simple form for square-wave rectification, and it is used to explain experimental data on thin discs of YBCO which were previously thought to be anomalous.

1. Introduction

The magnetic behaviour of type-II superconductors is easily found for critical-state models with a constant value o f the critical current, Jc, by using the method o f current superposition. Gugan and Stoppard [1 ], and Gilchrist [2], have recently emphasised that the initial magnetisation as a function o f applied field, M ( H ) , can be written in terms o f reduced variables

m(-M/Ms) =m(h),

( 1)

where m approaches - h at low h, and - 1 at large h. Ms is here the saturation magnetisation, and h = (HI Hs), Ha being the field at which the low-field linear region o f m extrapolates to the value rn = - 1, so that Hs is numerically equal to Ms, and both are proportional to Jc. As was shown in Ref. [ 1 ], the well-known plate and cylinder models o f Bean [ 3,4 ] can be written as

m( h < p ) = -

[ 1 - ( 1 - h / p ) p] ,

(2)

with p = 2 and 3, respectively, while p = 7 and p = describe models which have been advanced for samples in the form of thin films: p = 7 corresponds to the extreme oblate limit o f the treatment by Bhagwat and Chaddah [ 5 ] o f the general ellipsoid, and p = oo gives the exponential approach to saturation discussed by Sun et al. [ 6 ]. The choice of scaling field adopted here, Hs, is different from that normally used, the penetration field H* (see e.g. Bean loc. cit.), but it appears to be a more natural choice (cf. Refs. [ 1 ] and [ 2 ] ), and in fact we can see from Eq. ( 1 ) that for these models the constant p parametrises the sharpness o f the approach to saturation, since H*=pHs. Other models have recently been developed for thin films which take proper account o f the large transverse fields generated by the circulating supercurrents when the field is applied normal to the plane o f the film. For ribbons, Brandt et al. [7,8] obtain

m(h) =-

tanh(h),

(3)

while for circular discs Clem and Sanchez [9] (fol-

0921-4534/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSD10921-4534 (94) 00581-8

D. Gugan / Physica C 233 (1994) 165-178

166

(4)

induced in the sample continues to flow until it is exactly compensated in magnitude as a result of the emf. induced by the progressive reversal of the applied field. Using the principle of current superposition, the magnetisation and the effective field along the branches (b) and (c) of the hysteresis loop in Fig. 1 both scale by a factor of two, and the magnetisation along the three branches is given by

(5)

(a) 0 < h < h l :

ma=m(h), ml =re(h1) ,

(6)

(b) h~>h>h2:

rnb=rnt-2m{(h~-h)/2},

(7)

(c) h2<-h<-h~: m ¢ = m 2 + 2 m { ( h - h 2 ) / 2 } ,

(8)

lowing the important but slightly erroneous work of Mikheenko and Kuzovlev [ 10 ], - see also Ref. [ 11 ] ) obtain

m(h)=-(2/~)a(x)

,

where x -= h ( n / 4 ) , and where

G(x) = [arc cos{ 1/cosh (x) } +sinh(x)/cosh2(x)] .

Both Eqs. (3) and (4) are of the same reduced form as Eq. ( 1 ), and although Brandt et al. (in particular) have emphasised that the difference of flux profile inside a thin film from that in the infinite geometry of the Bean models gives rise to significant differences in the magnetic response, nevertheless, as Gilchrist [2] has illustrated, the resultant magnetisation curves are very similar. It is straightforward to calculate the hysteresis curves and the AC response for critical-state models of the form ofEq. ( 1 ), provided that one can assume a constant (field-independent) value of the critical current, and provided that one can apply the principle of current superposition. This is described at various levels of detail in Refs. [ 1-11 ], but a brief outline is given here as an introduction to the extended treatment to be introduced shortly. In the simplest critical-state magnetisation loop, a superconductor is magnetised from an initial unmagnetised, current-free state and taken through a cycle of applied fields as sketched in Fig. 1. At the positions of field reversal, C) and (~), the supercurrent distribution previously re(h)

Q

1, I

~2

/ ~b

I

hl

h

¢

which can easily be seen to close when the loop is completed. The AC response in zero DC bias field can be found by writing h = hasin 0, where 0 is taken to vary from =/2 to 3~/2 along path (b) etc. The voltage induced in a secondary coil, V, can be written in reduced form as

VI4kMs( =- - v) 3~/2

=-(1/2)

j

{dmb(O)/dO}R(O)dO,

(9)

n/2

where k expresses the electrical coupling between the sample and the detector coil, where a factor of two enters from the identical integral over - n/2_< 0 < ~/ 2, and where R ( 0 ) is the rectification function. For sin-wave rectification at the nth harmonic of the modulation frequency, R ( 0 ) is cos(n0) or sin(n0) for the real and the imaginary parts of the harmonic it response, v~ and vn, respectively, while square-wave rectification at the modulation frequency is often convenient, when R ( 0 ) is sign(cos(0)) or sign(sin(0) ), giving v's and v~' respectively. For many purposes the reduced voltage, v, is the quantity most directly related to experiment, but the AC response is often presented as an effective susceptibility, and this can be calculated from v as

Zn =4v,/nnha;

2~ =v/ha.

(10)

I

,m2 .

.

.

.

For square-wave rectification Eq. (9) can be integrated directly, giving

.

v'~=m(ha), Fig. 1. Schematicinitial magnetisationcurve and hysteresisloop for superposition modelswith field-independent critical current.

v~=m(ha)-2m(hJ2);

(11)

for sin-wave rectification the integration is straightforward in principle but often complicated in prac-

D. Gugan / Physica C 233 (1994) 165-178 tice if analytic results are desired, cf. Ref. [ 1 ]. The derivation so far has been for zero DC bias field, but it is easy to confirm that the voltage response is independent of any steady bias field when Jc is constant, and also that the branches (b) and (c) of Fig. 1 are symmetrical, so that the AC response can contain no even harmonics. For bulk samples it is well-known that Jc does depend on the field, and the consequences of this have been explored in several particular cases: always for the Bean model, though with a variety of specific assumed forms of J¢ ( H ) . The most complete analyses appear to be given by Chen and Goldfarb [ 12 ], and by Bhagwat and Chaddah [ 13 ], who derive the magnetisation curves (though only for zero bias field), while Ji et al. [ 14 ] give a particularly clear discussion of the AC harmonic response within a restricted model, and compare their theoretical results with their experimental observations. For thin films the situation has not been so clear; however, there is now no doubt that in the experiments o f L i u et al. [ 15 ] fields of several T applied normal to an epitaxial film of YBCO reduced the measured saturation m o m e n t at 10 K to about a tenth of its low-field value, and some fall-off was also seen in hysteresis loops measured at relatively low fields of about 50 mT. Similar effects were also seen by Angadi et al. [ 16 ] in similar thin films at 70 K, though at considerably lower fields, as would be expected from an approximately twenty-fold reduction of the critical current density at the higher temperature. AC susceptibility experiments on thin films have hitherto been discussed within the formalism outlined above, and have not appeared to require the consideration of any field dependence of J~, however, results recently published by Stoppard and Gugan [ 16 ] are inexplicable by the constant J~, current-superposition model, and this provided the motivation for the development which is presented below.

2. A generalised superposition model

167

by introducing a normalised critical current, j ( h ) , which depends upon the modulus of h, and where j ( 0 ) = 1, thus

m = j ( h )m{h/j( h ) } .

(12)

The extra range o f b e h a v i o u r that Eq. (12) can allow will be illustrated later, but it is clear that the magnetisation must always lie within the envelope +_j(h). The calculation of the hysteresis loop follows the same stages as before, but it is necessary now to consider how the frozen-in supercurrents (corresponding to m l and m2 of Eqs. ( 6 - 8 ) ) are affected by the applied field since superposition now takes place with a variable induced current density. Referring to the schematic diagram in Fig. 2, and starting again at the unmagnetised state, the magnetisation for the branches (a) and (b) is branch (a), O<_h<_hL:

ma=j(h)m{h/j(h)};

ml =j~m(hL/j~) ;

(13)

branch (b), h~ > h> h2 :

rnb=m~-{J'l + j ( h ) } m { ( h ~ - h ) / 2 j ( h ) } .

(14)

m(hd)~i / i// /

-~

~\\\ \

//

\

|

+ilh)

/

h2

/

I

/ / / h

d

" (

-

\I

h2,j2, m2

hl, jl, ml

\

/

/

lhl

I/

ii

\

/ / \

/

2.1. Hysteresis loops The critical current enters implicitly into Eq. ( 1 ) through the normalising parameters Ms and Hs, so that the generalisation ofEq. ( 1 ) can be written down

Fig. 2. Schematic initial magnetisation curve and hysteresis loop for the generalised superposition model. The dashed lines indicate the bounds +j(h). N.B. The mean field for the loop, hd= (hL+ h2 )/2, is taken to be positive.

D. GuganI PhysicaC 233 (1994) 165-178

168

The term ml in Eq. (14) is a constant since (with the convention that the mean bias field for the loop is positive) the initial screening current distribution, shown schematically in Fig. 3, is never subjected to an e m f w h i c h could increase it, nor does it experience a higher magnitude o f field which would decrease it below the value j~. The superposed, reversed supercurrent induced along branch (b) by the continual decrease of applied field penetrates the sample from the surface (assumed to be on the right in the schematic diagrams o f Fig. 3) and progressively reverses layers o f the initial current density, - j l , so that all the resultant superposed supercurrent flows with density j (h), the e m f in the current-reversed volume always increasing the reversed current to its maxim u m level allowed by the form o f j ( h ) . For branch (c) one can write down an expression which is formally equivalent to Eq. (14), (cf. Eqs. ( 7 ) a n d (8) for the constant Jc case), but there is an extra complication since there may be a field range for which

h2

j(h)

1

. . . . . .

h=O

-,. : m -

hl

~ - - .'7" --..

hl

Ih2l

j2(h) ~

'~,

h=O

h2

-'" . . . . . .

",L\ \~\

.,

t\

fi,'l

I

N~\ \ N I \ I \ \\\',KN\

- - - qy.~,

=

........

.... ....

\.N\

v/l, vr/~;

\,

\l\

\

~ \

....

-i2 -1

. . . . ......

.,

L.,

.... ~ ~....

(I) branch b (hl ~ h2)

,,. " '

.

.

.

.

.

~. .

+j(h)}m{(h-hz)/2j(h)} , mz(h)=m, -~1"~+jz(h)}m{hJjz(h)},

(15) (16)

where ha = (h 1- hE ) / 2 is the amplitude o f the cyclic field. At the end of a complete cycle, h=hl, and j z ( h ) = j ( h ) = j l , and Eqs. (15) and (16) show that the loop is properly closed. The equations so far have been written in terms of the reduced variable h, but for specific applications one must make some assumption as to how the component fields ha and ha are to be combined to produce the magnetisation m(h), and the critical current j ( h ) - for which they may combine differently. In what follows we can see that simple and realistic assumptions inserted into Eqs. ( 13-16) allow one to make a straightforward numerical calculation o f the hysteresis loops and of the AC response for any chosen form of elementary magnetisation curve m (h) and critical current j ( h ). Examples are given in Section 3, below, but before presenting these, it is of interest to consider some analytic results for the AC voltage response.

i\

". N ~ ' \ \ ~ \ K ~

•

mc=m2(h)+~j2(h)

r,.

. t . . \ \ N \

-11

branch (c), h2
2.2. Analytic results for the AC voltage response

".~ \ r-.

\ \l\ \

h > I h2l, which causes the remanent current j2 and the magnetisation associated with it, m2, to be field-dependent for a part of this branch, i.e. j2(h) = j ( h ) if h > I h21 but otherwise J2 (h) =J2. We thus obtain

•

.L ._.L__,

(ii) branch c (h2---D..- h 1 )

Fig. 3. Schematic diagrams of current distribution corresponding to branches (b) and (c) of Fig. 2. In (i), the close cross-hatching indicates the supercurrent distribution at the end of the initial magnetisation process, while lighter cross-hatching indicates the superposed (and reversed) current distribution, j~ +./2, at the end of branch (b); intermediate states, j, +j(h), are represented by the areas under the broken lines. In (ii), the light cross-hatching is sketched for [h2J--hd, so that h2 is negative) the remanent supercurrent distribution corresponding to turning point 2 in Fig. 2, m2, becomes dependent on field: the forward induced supercurrent distribution is determined by j(h), as for branch (b) (see text for further details).

For square-wave rectification functions in Eq. (9), the equations for the magnetisation o f branches (b) and (c), Eqs. (14) and (15), integrate by parts directly, with only a minor complication arising from the condition h > Ih21 for branch (c). In fact this condition is more conveniently expressed as hd > or < ha, depending on the relative sizes of the bias and the cyclic fields. For the real part of the response there is no difference between these two cases, and one obtains v's = ( 1 / 2 ) (j~ +j2)m(ha/j2) •

(17)

For the imaginary part of the square-wave response there prove to be two regimes, depending on whether hd is greater or less than hal2, v" = ( 1 / 2 ) 0"1 + k ) m ( h J k )

D. Gugan / Physica C 233 (1994) 165-178 - (1/2)(2jd+jl +k)m(hJ2jd) ;

(18)

in this expression k=j2 if hd<_hJ2, but otherwise k=jd, where Jd is the reduced current at the mid-point of each branch of the cycle, i.e. at the field hd where the square-wave of the rectification function reverses.

2,2.1. Special cases of Eqs. (17) and (18)

with the detailed experimental results of Liu et al. [15].

2.2.1.3. Low-field expansions of ha, ha These shed light on the behaviour to be expected experimentally when hd is increased from zero at fixed modulation amplitude, ha, and vice versa. To simplify matters it is helpful to writej(h) as a reduction from unity, i.e. j ( h ) = 1 - g ( h ) , a form which is appropriate for low fields. If the limiting field dependence is quadratic we may write

2. 2.1.1. The saturation regime (m = - 1) As m ( h ) in Eqs. (17) and (18) tends to the limit 1, one finds that

Jl -- 1 - g ( h~ ) = 1 - fl( h~ +ha)Z;

v~+jd.

J2 = 1 - g ( h 2 ) = 1 --fl(hd - h a ) 2 ,

-

v's+ -- (Jr +j2)/2;

(19)

For the usual case where the AC response is measured with zero DC bias field, h+=0, we have Jl =j2=j(ha), and j+ has the value of unity independent of ha, so that the voltage ratio, v'ffv~ ( = Z ' f f Z ~ ) = - j ( h a ) , is a direct measure of the reduced critical current. As hd increases from zero the value of v~' decreases while remaining independent of ha. v'~ is more complicated since it involves both jl and j2, and in particular, if ha=hal, then j2= 1 and v'~ passes through a local extremum of magnitude { I +j(2hd )}/2, while on either side of this field the magnitude ofv'~ tends to a constant value determined by the larger of ha and ha (see Fig. 7(a) below).

2.2.1.2. ha>> h a Here we have jl ~J2 ~Jd, and if we introduce the reduced variable x=-(ha/jd), we obtain for the susceptibility Z'~=,m(x)/x;

z~m(x)/x-2m(x/2)/x.

(20)

Eqs. (20) predict exactly the same behaviour as the current-independent forms, provided that we use the variable x; this similarity is easy to understand from the shape of the hysteresis loops in this regime (see Fig. 5 (b) below), and it has also been pointed out on rather different grounds by Shatz et al. [ 18 ], who have made use of the similarity to study Jc ( T, H) in the context of the Bean plate model. From Eq. (20) one can see that as ha becomes small, both Z~', and the differential reverse susceptibility of the quasistatic magnetisation curves, must have the constant value - 1, independent of hd; this conclusion agrees

169

(21)

where fl is independent of field. Substituting f o r j in Eqs. ( 17 ) and (18) and carrying out the small-field expansions allows the change of the voltage response to be obtained in terms of ha, hd, and ft. Using the Bean plate form of m ( h ) one obtains for the real response

v's - V's(hd = 0 ) =--6v's = 2flh~ ha × [ 1 - h a / 2 + h d 8 + f l ( h a - h a ) 2 ( 1 - h a / 2 ) ...] . (22) The particular form of magnetisation model proves to have only a minor effect on terms inside the square bracket on the rhs of Eq. (22), and in what follows only the leading terms in the response will be written out explicitly. For the imaginary response there are two regimes, corresponding to hd greater or less than h J 2 in Eq. ( 18 ),

5v"( hd <-hJ2 )=2flh~hd[ 1 ...l,

(23)

6v~(ho > h J 2 ) = (3fl/4)h3a[1 ...] + (fl/2)h2hd(1 +flh 2) [1 ...].

(24)

The first term on the rhs of Eq. (24) is independent ofho, and arises because the zero-bias-field reference state (which is experimentally convenient for isolating the small effects due to ha) must be obtained from the hd~0 limit of the regime ha
D. Gugan / PhysicaC 233 (1994) 165-178

170

jl=l-otl(hd+ha)l ; J2=l-al

(hd-ha)l ,

(25)

then there is an extra complication associated with the sign change as hd> or < ha. For ~v's obtained from Eq. (17) one finds

~v',(hd
(26)

~v's(hd>ha)=ah2[1 ...]+(ot/2)h2hd[1 ...],

(27)

where the first term on the rhs of Eq. (27) arises from the hd=0 reference state, as explained for Eq. (24). For five' from Eq. ( 18 ) one has the change of regime at ha = hJ2, giving

6v" ( hd <_h,/2 ) = ( 3a/2 )hahd[1 ...]

plate form of the magnetisation function, Eq. (2); using other forms for m (h) makes almost no discernible difference to the features illustrated here. The limits + j ( h ) form an envelope for the hysteresis loops which is shown on the figures. Figs. 4 ( a ) and 4 (b) show hysteresis loops with zero bias field, and with the critical current scaling field hb = 3. For relatively low values of the cyclic field, ha, the computed curves of Fig. 4(a) show a strong similarity to those measured by Liu et al. [ 15 ], cf. their Fig. 5(a), with sign of a field dependence ofjc appearing when ha~ hb/2. The higher fields of Fig. 4(b) show the development towards a regime of a lower, saturated, value of critical current which is generally similar to the data ofLiu et al. [ 15 ], their Fig. 5 (b),

(28) I

I

I •

•

• .

"

|'

"1

-

I .

•

I .

.

.

.

~v"(hd > h J 2 ) =(3ot/4)h~[1 ...]+(a2/4)h2hd[1 ...].

(29)

Unlike ~v~, no further change of regime occurs in fiv~' at hd=ha, since this arises from the term j2, which can be seen to be absent from Eq. ( 18 ) when hd> ha/ 2. These expressions (22-29) show that one can obtain both the characteristic form, and the magnitude of the initial field dependence ofj~ from a study of the variation of~v~ and fiv~' with ha and ha.

I

hd=O

I

,

.

,

.

.

-2

,

.

,

-1

0

1 h

3. S o m e r e s u l t s f r o m t h e g e n e r a l i s e d

superposition I

model

m,

i

i

i

l

I

i

(b)

3. I. Magnetisation curves The appearance of the t e r m s j ( h ) in Eq. (12) as compared with Eq. ( 1 ) can lead to a wide variety of behaviour which is very similar to published magnetisation curves. Figs. 4-6 illustrate a few examples: all have been calculated from Eqs. ( 13-16) with jc given by

j ( h ) = { 1 + B(h/hb)2)/{ 1 + (h/hb)2} ,

hd=O

(30)

where B is a constant, 0_< B_<_l, which determines the lower limit on j ( h ) - as appears to be seen in some experiments- and where hb sets the scale for the variation ofj~ with field. All the results shown in Figs. 46 have been computed with B = 0.2, and with the Bean

L

-20

I -10

I 0

I

20

10 h

Fig. 4. Hysteresisloopscomputedfrom Eqs. (14) and (15), with bias field ha=0. The dotted lines in (a) indicate the assumed form ofj(h), i.e. Eq. (30), with B=0.2 and hb= 3. A Bean plate magnetisation functionhas been assumed.

171

D. Gugan / Physica C233 (1994) 165-178

and also to the data of Angadi et al. [ 16 ], their Figs. 4 and 5, bearing in mind that the maximum fields of the latter correspond to ha ,~ 100. Figs. 5 (a) and 5 (b), also calculated with hb= 3, show how the hysteresis loop alters as the bias field is increased from zero. The loops in Fig. 5(a) clearly show a loss of symmetry indicative of second harmonic content in the AC response, while those in Fig. 5 (b) show the return to an almost symmetrical shape at high fields, close to what would have been seen for simple models with jc = 1, but with the magnetisation appropriately reduced. For all these curves it can be confirmed by inspection that the reverse differential • susceptibility (i.e., d m / d h a at the points of field reversal) has the expected constant value, equal to that of the initial susceptibility. A value of hb = 3 seems to be realistic for compari-

son with the experimental data cited, but one can also investigate by computation cases where hb is much less; e.g. Fig. 6 (a) shows hysteresis loops with hb = 0.3 for ha= 1, and hd=0 and 1, respectively. The shape of the loop in the biased field shows an extreme distortion, which is reminiscent of some of the minor loops seen by Roy et al. [ 19 ], in particular their Fig. 6. Roy et al. discuss their results in terms of a "history effect" which could arise from the complex magnetic structure of their bulk, ceramic samples; however, the present analysis shows that a field-dependent critical-state model can lead to a similar behaviour even in homogenous samples. At lower values of the cyclic field other strange behaviour appears to be possible, as shown in Fig. 6 (b), where the curves corI

I

I

I

I

I

!..-"

I

"...I

I

..

I

;=

'1"

.

(a)

.a. . . . . . . . . . .

ha=l •.,

.

ha=l

I

-1

• ,. I

I

I

I,.'"l

I

0

i

I

I

I

]

I

I

I

".l,'"

I

I

0

I

I

(hd + h)

."

l".,

-5

1

I

.-

-2

-1

I

.

i

I

..

I

(a) d

I

"'

I

0.4

I

(hd + h)

I

I

I

I

I

I

I

m

I

I

I

I

I zero shifted I ~" \

--

0.2

(b)

m

o

41.2

0

hd--0 ..........

.....

..

-0A

ha=2

I

-0.8

I

-0.4

I

I

0

I

I

I

0.4

0.8 h

-1

I

-25

I

I

I

q

0

I

1

I

(hd + h)

I

25

Fig. 5. Hysteresis loops computed from Eqs. (14) and ( 15 ), with bias field, ha, increasing from zero. Other details as for Fig. 4.

Fig. 6. Hysteresis loops computed from Eqs. (14) and ( 15 ). Details as for Fig. 4 except that hb = 0.3. (a) Shows the extreme distortion which can arise under a modest bias field; (b) shows an unusual reversalof the path taken around the loop as h, decreases

from 0.6 to 0.4.

172

D. Gugan / Physica C 233 (1994) 165-178

responding to ha = 0.4 and ha = 0.6 show a change from a clockwise rotation around the loop to the usual anticlockwise sense, with the two branches of the loop intersecting at ha=0.5: however, this behaviour does not appear to have been observed experimentally. All these loops have been generated by current superposition from an initial state of zero magnetisation, which has been assumed to be a current-free state, as corresponds to common experimental conditions. A general loop from an arbitrary starting point (within the limits +_j(h)) could be obtained by an extension of the method, though it does not seem likely that it would shed further light on the macroscopic magnetisation or on the AC response. At the microscopic level, however, there may be large differences: e.g. the initial m = 0 state assumed here has zero supercurrent flow, while the state cyclically demagnetised from saturation to zero has very thin, counter-rotating, critical-supercurrent sheets throughout the sample. Evidently, these two states may well give rise to a different behaviour, but this is beyond the scope of critical-state theories.

1

" , " ~ .... /-~.~ , ~ /, 7 : _ ~

3.2.1. Overall behaviour The real and the imaginary parts of the square-wave voltage response obtained from Eqs. ( 17 ) and ( 18 ) are shown in Fig. 7. The critical current has been assumed to vary according to Eq. (30), as before, and the graphs show this function, and also the AC response for the case of a constant critical current, Jc = 1. For low values of the DC bias field, hd, the field-dependent case follows Eq. (19) in the saturation regime, ha >- 3, and illustrates the very different behaviour of v~ and v~'. For v~' the plateau regions follow j(hd), and when hd >> ha one can confirm by superposition of the logarithmic plots that they scale as (ha/ Jd), as predicted by Eq. (20). The peaks seen in the magnitude ofv's tend to a constant height (cf. Section 2.2.1.1), which for the current dependence of Eq. (30) can be calculated to he ( 1 + B ) / 2 , i.e. a value of 0.6 here. The curves make it clear that relatively large DC bias fields are needed to make appreciable changes to the AC response. Fig. 8 shows numerically calculated values of the sin-wave rectified response at the fundamental frequency, V'l and v'~'. v'l shows the characteristic h a ,/2

[ (a) I

V1

0.3

0.1

I 0.1

1

1

"l . . . . .

.,

10

I /

ha

(b)

"

I hd=o'~

100

I

..

J(ha) ~

Vsn

3

I

"-

__

//

~

o.3 =3~___~2 =10

3.2. AC voltage response

I ' j=l response

j(ha)

, o., I 0.1

".

jl,

,:lr..pon

, II II h, 1

,

,

10

ha

100

Fig. 7. Real (a), and imaginary (b), voltage response for squarewave rectification at the fundamental frequency, v~ and v", for a range of DC bias fields. Dashed lines show the response for the case with field-independent jc, and the dotted lines show the variation o f j (ha); computational details as for Fig. 4.

fall-offat high fields (cf. Ref. [ 1 ] ), but otherwise can be seen to map onto Fig. 7(a) for the square-wave response. This similarity of form is only partly seen for v~' since the high-field behaviour now becomes asymptotically dependent on j (ha), and independent of ha, unlike the behaviour of v~' at high fields shown in Fig. 7 (b). Nevertheless, there is a striking (if expected) similarity between the sin- and the squarewave rectified response, and in Fig. 9 this is shown to extend to detailed features of the harmonic response. Third harmonic response has been numerically cal-

173

D. Gugan I Physica C 233 (1994) 165-I 78 i

I

[

I

~

~

.~./

i j=l response

1

(a)

I

I

I

I

I .

~+

(a) l V3"

-Vl I

10"1

10-1

h d = 0; 2; 5; 10; 30

10-2

i

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~.\J

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I

hd=0 ~ ==2

100

v'! ;

0.1

, [

(b)

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J

10"3

i ha

/ Vl.

10-2

I

i

1

3

lO

I

I

I

|~

/'l

/

/-

/

N/

/

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0.3

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100

Fig. 8. Real (a), and imaginary (b), voltage response for sinwave rectification at the fundamental frequency, v~ and v'(, for a range of DC bias fields, cf. Fig. 7. See the text for comments on the topological similarity between this and Fig. 7.

culated for both sin- a n d square-wave rectification in a field region ( h , ~ 8) where a p p l i c a t i o n o f a small D C field, hd--<3, gives rise to a deep m i n i m u m with a zero-crossing o f the response: again, there are differences to be seen, but the similarities are striking. Scans o f ha at constant h, c o m p u t e d for h a r m o n i c s n = 1, 2, 3 with sin-wave rectification are shown in Fig. 10; for n = 1 the results can be related to those shown in Fig. 8. The a s y m m e t r y o f magnetisation p r o d u c e d by the D C bias field causes the second harmonic to increase as h 2 for low bias fields, and to have a m a x i m u m near to the c o n d i t i o n ha ~ h,, as is con-

0,1

,

i=,

I ,

1

,

3

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,

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Fig. 9. Imaginary pan of the third-harmonic voltage response for (a), sin-wave and (b), square-wave rectification, v~ and v;s; computational details as for Fig. 4. Positive and negative voltages are indicated by solid and dotted lines, while for the field-

independent response curves they are indicated by dashed and dash-dot lines. The similarity between the curves is discussed in the text. sistent with the general shape o f the hysteresis loops. Typical scans showing the h a r m o n i c b e h a v i o u r as ha is swept at constant hd are shown in Fig. l 1, while in Fig. 12 the real part o f the second harmonic, v~, is shown as a function o f h , for various values o f the D C bias field. These diagrams o f the AC voltage response, Figs. 7-12, all for the Bean plate magnetisation function, and all with the critical current given by Eq. (30), give some idea o f the b e h a v i o u r to be expected in real

174

D. Gugan IPhysica C 233 (1994) 165-178 i

i

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i

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ha=3

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Fig. 10. Real (a), and imaginary (b ), harmonic voltage response for sin-wave rectification, v" and v" (n= 1, 2, 3), as a function of DC bias field, hd, at constant modulation field, ha=3. Negative voltages are shown by dotted lines in (b); the asterisk for n = 1 indicates data with sign reversed. materials where there is a field-dependent critical current. Square-wave rectification shows the essential features just as well as does sin-wave rectification, but it appears to be m o r e revealing in some respects since the analytic results (17) and ( 18 ) focus attention o n the critical parts o f the hysteresis loop, the phase angles 0, rt/2, and n. It would be possible to obtain analytic results for the second ( a n d higher) h a r m o n i c s using square-wave rectification, v~s and ~tt 2s say - a n d m a n y phase-sensitive detectors allow this to be m e a s u r e d - but the analytic expressions

t 0.1

1

10

ha

100

Fig. 11. Real (a), and imaginary (b), harmonic voltage response for sin-wave rectification, v" and v~ (n= 1, 2, 3), as a function of modulation field, ha, at constant DC bias field, hd= 10. Negative voltages are shown by dotted lines; the asterisk for n = 1 indicates data with sign reversed. would involve a d d i t i o n a l phase angles where the total effective fields do not have a simple form, so that the resultant equations would not be as perspicuous as are Eqs. ( 1 7 ) and (18).

3.2. 2. Low-fieM asymptotic behaviour Fig. 7 shows that for low modulation fields, ha < 0.3, i.e. low relative magnetisation, the voltage changes systematically with hd, even for large values o f hd. The asymptotic expressions for this convenient experimental regime have been o b t a i n e d analytically for

175

D. Gugan I Physica C 233 (1994) 165-178 i

I

i

i

I

i

I

i

I

reel response

hd=5

he : 0.01; = 0.32

\

V2' (ha2)

I

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10-1 10-2

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imaginary response

.(?.. .~7-

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,

,

10

ha

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\

I

I

I

I0 "2

I

l

I0 "I

I

1

I

I

10

hd

100

100 i

Fig. 12. Real part of the second harmonic response, v~, as a function of h,, for several fixed values of the DC bias field, hd. Negative voltages are shown by dotted lines. x ha2 i

some simplified forms of j ( h ) in Section 2.2.1.3 above, Eqs. (21-29 ). Fig. 13 shows computed values of the change of reduced voltage from the reference state with ha=0, i.e. v=-vs(ha=0)---~v,, which is plotted as (Sv,/h ] ) versus ha; Fig. 13 (a) shows both the real and the imaginary response calculated using j ( h ) from Eq. (30) with B = 0.2 and hb = 3, while Fig. 13 (b) shows the real response calculated from an exponential form o f j ( h )

10"1

(31)

The imaginary response calculated from Eq. (31 ) has closely the same shape as the real response, but the kinks occur at twice the corresponding values of ha, and the magnitudes are smaller by a factor of about 1.5. The low-field expressions (21-29) are valid for values of ha < 1 in both parts of Fig. 13, and the different regimes predicted analytically are clearly seen, in particular, the two regions of response linear in ha, but differing four-fold in magnitude, predicted for 8v~' when the initial change of critical current with field is quadratic (Fig. 13 ( a ) ) . These results indicate that one can obtain both the magnitude and the characteristic form of the initial dependence ofjc on field from a study of 8vs on h, and ha. The absolute size of the effect is not large, but if we consider the

I

I

I

i

I

I

I

I 10"1

J

I 1

~

[ 10

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/

10-2

10-3 real response

(b) 10.4

i 10-3

j ( h ) = B + (1 - B ) exp( - h / 2 h b ) .

I

I 10-2

i

i hd

100

Fig. 13. Change of square-wave rectified voltage response as a function of bias field, ha, for low modulation fields, ha. In (a), the field-dependent current is given by Eq. (30), while in (b), the field-dependent current is given by Eq. (31 ), both with B = 0.2 and hb=3; see the discussion in Section 2.2.1.3.

relative change, the effect should be easily detectable; for instance, if we consider v~' for the quadratic case (25, 26), and use the Bean plate model value of v~' (hd-- 0) = h 2/ 8 [ 1 ], we find for fields up to hd = h,/ 2_<0.3, that the ratio 8vs' (ha)/v" ( h a = 0 ) = 16flha, and one can obtain extra sensitivity by moving into the regime ha> hJ2.

3. 2.3. Experimental data There is a wealth of information on the susceptibility of hard superconductors for which a criticalstate model is appropriate, but little of it is suitable for detailed comparison with the present work, par-

176

D. Gugan / Physica C 233 (1994) 165-178

ticularly for the effects to be expected from the application of a DC bias field. Experiments on thin films in perpendicular fields are probably the most relevant because such samples appear to be well-defined geometrically, and homogeneous in magnetic structure. The initial objective of this work was to find the reason for the anomalous results reported by Stoppard and Gugan [ 17 ] for thin films of YBCO under high modulation fields, where instead of the behaviour expected from the magnetisation models in common use at the time, I v~/v~' I = I x ' s l x ' ~ 1, they observed a systematic deviation from unity at the higher effective fields, with the ratio falling to ~ 0.2. The experimental position is not ideally simple because the measurements were not made at a fixed temperature with a complete sweep of ha, but at a series of constant temperatures each with a limited range of ha; however, the sections of isothermal response proved to superpose rather closely, as explained in Ref. [ 17 ], so that the overall effective range of field sweep was 0.01 < ha-< 100. The high-field resuits for Z's and Z~' are shown in Fig. 14, where the overlapping sets of experimental data are shown, together with theoretical curves assuming a field-independent critical current. The experimental data are shifted to the right for clarity (by about 2.5 × ), but the shift is fairly constant for X~', which implies that the experimental values have nearly the 1/h dependence predicted by Eq. (20) when the bias field is zero and Jd = 1. The near-agreement with theory for g~' gives support to the procedure adopted for combining the results for the different temperatures, and since the positions of the data for Z'~ are completely determined by the scaling of ha adopted for Z~', it is clear that as the effective field increases from unity, the magnitude of Z'~ falls well below the theoretical curve: a natural explanation of this follows from Eq. (20), where X'~ = - j ( h a ) / h a , and the form o f j ( h a ) which can be inferred from these data is shown by the crosses in Fig. 14. This analysis suggests that for this sample the dependence of critical current on (reduced) field between 60 K-77.5 K (i.e. 0.72-0.93 of the critical temperature) is approximately independent of the ten-fold reduction of the low-field critical current which occurs over the same temperature range, see Ref. [ 17 ]. This conclusion contrasts with the work of Shatz et al. [ 18 ], who studied a ceramic sample of YBCO in the high DC bias field regime

7

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77.5K

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I 3

I 4

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x ~

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~x \x~..xm

x

I 6

I I 8 10

I 20

I I 30 40

I 60

[ 80

ha

Fig. 14. Real and imaginaryparts of the square-wavesusceptibility as a function of ha in bias field ha= 0. The solid lines are computed from the Clem and Sanchez equations for thin discs, with Jr constant [9]; the experimental data on a thin film of YBCO come from Stoppard and Gugan [ 17], and are shifted to the right to avoid overlap with the theoretical curves. The dashed line through the crosses shows the form ofj(ha) which can be inferred from these data (see Section 3.2.4 ). hd >> ha, at temperatures close to To: they also find that their data can be fitted to universal curves by suitable scaling; however, their rather complicated data analysis appears to imply that the field dependence of the critical current has some extra, explicit dependence on temperature which becomes quite strong near to

To. The form o f j ( h a ) for ha between about 4 and 20 shown in Fig. 14 implies that j¢ varies as approximately ha 3/4 over this range, but the accuracy of the data is insufficient to lead to any conclusion about the form ofj¢ at low ha, or about whether there is any tendency for j~ to approach a non-zero asymptotic limit at high fields, as seems to be observed in the static magnetisation data already cited [ 15,16 ], and as has been incorporated into Eqs. (30) and (31); however, from the fall-off in j(ha), one can deduce that the scaling field hb is approximately 6 for this sample of YBCO. The theoretical curves plotted in Fig. 14 are given by the thin-disc formulae of Clem and Sanchez [ 9 ],

D. Gugan / Physica C 233 (1994) 165-178

which were expected to be the formulae most closely applicable to the experimental samples. All magnetisation models tend to the same results in the highfield limit; e.g. for the constant jc Clem and Sanchez case, Z~ and Z~' have the same magnitude above ha-~ 6, while for the Bean plate model the exact 1/ha regimes begin at ha=2 and 4 for Z's and Z~', respectively, and even at fields below the saturation regime for m (h) it is fairly difficult to see the effects of different forms of the magnetisation function, which is the reason that the results exemplified in this paper have nearly all been based on the Bean plate model. However, the low-field data reported for the same sample by Stoppard and Gugan [ 17 ] give unequivocal experimental evidence in favour of the recent magnetisation expressions which take proper account of the thinfilm geometry [ 7-11 ]; it is proposed to discuss this elsewhere, this work of Stoppard and Gugan is in preparation. The harmonic response for thin-film samples subjected to both ha and hd has been measured at 77 K by Lain et al. [20] in a DC field of 100 Oe (hd~ 3). When the symmetry of the magnetisation loop is broken, the even harmonics increase rapidly with hd, as has frequently been reported for bulk samples, by e.g. Miiller et al. [ 21 ], Ji et al. [ 14 ], Ishida and Goldfarb [22], and Navarro et al. [23], and the difference in behaviour for n = 2 as compared with the odd harmonics n = 1 and n = 3 is graphically illustrated in Fig. 10. Unfortunately L a m e t al. do not give any data for the second or other even harmonics, while the bias field they applied produced little effect on the 7th harmonic power for any modulation field between about 0.2 and 200 Oe, since of about 30 data points (their Fig. 3 ( c ) ) only about 3 show any shift, and these all decreased by not more than a factor of two - and one can calculate that the result should be similar for all the odd harmonics. Little can be concluded from their results except that hb for their sample was considerably greater than hd, and that probably hb> 10.

4. S u m m a r y and c o n c l u s i o n s

The heuristic model which has been presented allows a simple current-superposition analysis of critical-state behaviour for different forms of m (h) to be

177

extended to include a field dependence of the critical current, which can also take a variety of different forms. The model provides a simple framework for a general analysis whose possible range ofbehaviour is large, and only a few typical cases have been presented here, mostly using the Bean magnetisation function ( p = 2 in Eq. (2)); there is no difficulty in using other forms of m ( h ) , but for the higher-field regime, where the new feature of this model, the field dependence of the critical current, becomes significant, the results do not vary much from the Bean form. The calculated hysteresis loops show obvious similarities with published experimental data, as discussed in Section 3.1, and they also suggest that novel forms of hysteretic behaviour may sometimes be possible, as shown in Fig. 6(b). The AC response follows directly from the magnetisation loop, and all the harmonic behavior can be computed. The present heuristic model does not give new information about the AC response in the critical state, but it does suggest how one can analyse experimental data by a powerful, general method. Square-wave rectification at the fundamental frequency yields analytic results, Eqs. (17) and (18), which are especially revealing since they focus attention on a few critical parts of the magnetisation cycle, rather than averaging around the loop. As has been argued elsewhere [ 1], the essential features of the odd-harmonic response are all contained in the square-wave-rectified, fundamental response, so that the extra information which can be found by applying a DC bias field can be obtained from an analysis of vs(ha, hd), as shown for example in Figs. 13 (a) and (b). Unfortunately, there seem to be no data for thin films which allows this to be carried out in detail, the data of Lam et al. [ 20 ] giving only a lower limit of hb ~ 10 for the scaling field for the field-dependence ofjc (cf. Eq. ( 30 ) ); however, since this is higher than the value hb g 6 found above for the sample of Stoppard and Gugan [ 17 ], it suggests that hb is a parameter which may have a significant sample dependence, a conclusion which could have important consequences for the technical exploitation of thin films in high magnetic fields.

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D. Gugan / Physica C 233 (1994) 165-178

Acknowledgement I am grateful to Oliver Stoppard for access to his detailed experimental results, and for his comments on this paper.

References [ 1 ] D. Gugan and O. Stoppard, Physica C 213 ( 1993 ) 109. [2] J. Gilchrist, Physica C 219 (1994) 67. [3] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [4] C.P. Bean, Rev. Mod. Phys. 36 (1964) 31. [ 5 ] K.V. Bhagwat and P. Chaddah, Physica C 190 (1992) 444. [6] J.Z. Sun, M.J. Scharen, L.C. Bourne and J.R. Schrieffer, Phys. Rev. B 44 ( 1991 ) 5275. [ 7] E.H. Brandt, M.V. Indenbom and A. Forkl, Europhys. Len. 22 (1993) 735. [ 8] E.H. Brandt and M.V. Indenbom, Phys. Rev. B 48 (1993) 12893. [ 9 ] J.R. Clem and A. Sanchez, ( 1993 ) preprint. [ 10] P.N. Mikheenko and Y.E. Kuzovlev, Physica C 204 (1993) 229.

[ 11 ] J. Zhu, J. Mester, J. Lockhart and J. Turneaure, Physica C 212 (1993) 216. [12] D.-X. Chen and R.B. Goldfarb, J. Appl. Phys. 66 (1989) 2489. [13] K.V. Bhagwat and P. Chaddah, Phys. Rev. B 44 (1991) 6950. [ 14] L. Ji, R.H. Sohn, G.C. Spalding, C.J. Lobb and M. Tinkham, Phys. Rev. B 40 (1989) 10936. [ 15 ] C.J. Liu, C. Schlenker, J. Schubert and B. Stritzker, Phys. Rev. B48 (1993) 13911. [16] M.A. Angadi, A.D. Caplin, J.R. Laverty and Z.X. Shen, Physica C 177 ( 1991 ) 479. [ 17] O. Stoppard and D. Gugan, Physica C 217 (1993) 197. [18] S. Shatz, A. Shaulov and Y. Yesherun, Phys. Rev. B 48 (1993) 13871. [19] S.B. Roy, S. Kumar, A.K. Pradhan, P. Chaddah, R. Prasad and N.C. Soni, Physica C 218 (1993) 476. [20] Q.H. Lam, C.D. Jeffries, P. Berdahl, R.E. Russo and R.P. Reade, Phys. Rev. B 46 (1992) 437. [21 ] K.-H. Miiller, J.C. MacFarlane and R. Driver, Physica C 158 (1989) 366. [22] T. Ishida and R.B. Goldfarb, Phys. Rev. B 41 (1990) 8937. [23] R. Navarro, F. Lera, C. Rillo and J. Bartolome, Physica C 167 (1990) 549.