- Email: [email protected]
Third harmonic ac susceptibility of superconducting strips and disks
Physica C 334 Ž2000. 107–114 www.elsevier.nlrlocaterphysc
Third harmonic ac susceptibility of superconducting strips and disks M.J. Qin ) , C.K. Ong Centre for Superconducting and Magnetic Materials and Department of Physics, National UniÕersity of Singapore, Lower Kent Ridge Road, Singapore 119260, Singapore Received 20 December 1999; received in revised form 26 January 2000; accepted 7 February 2000
Abstract Third harmonic ac susceptibility of superconducting strips and disks with finite thickness Žcross-sections 2 a = 2 b . in a perpendicular magnetic field Ha s H0 sinŽ v t . has been calculated. The flux creep effects have been taken into account by using a power-law electric field EŽ j . s Ec Ž jrjc . n. Detailed results are present for different thickness b and creep exponents n, which are compared with those derived from the Bean critical state model and from numerically solving the flux creep equation for infinite slab in parallel field. In the limits of bra 4 1 and n ` the results from the parallel situation and the Bean model are reproduced, respectively. q 2000 Elsevier Science B.V. All rights reserved.
™
PACS: 74.60Ge; 74.60Jg; 74.72Fq
1. Introduction Magnetic measurements, such as magnetization hysteresis loops, magnetic relaxation and ac susceptibility have been widely used to study flux dynamics of high temperature superconductors, as a result, critical current density, activation energy as well as UŽ j . relationship have been determined experimentally by this contactless technique w1–7x. Although the magnetic measurement has an advantage of using contacless and non-destructive technology, experimentally, in order to get maximum signal, most magnetic data have been taken on thin films and single crystals Žusually in the form of platelets resembling thin film rectangles or disks. with the ) Corresponding author. Tel.: q65-874-2625; fax: q65-7776126. E-mail address: [email protected] ŽM.J. Qin..
applied field perpendicular to the film plane. Unfortunately, the analysis of the experimental data in this perpendicular field geometry is complicated by the strong demagnetizing effects, resulting in numerous studies on this subject for both fundamental and technological reasons. A conventional method to treat the demagnetizing effects is to use the demagnetizing factor w8–11x. However, a number of authors have pointed out that in the critical state of a superconductor, in which current is uniformly distributed in the bulk, this method is invalid not only quantitatively but qualitatively w12–15x. Because the theoretical model was not available, they used numerical calculation to treat this problem instead. Mikheenko and Kuzovlev w16x presented an exact critical state model for superconducting disks with zero thickness in perpendicular magnetic field. They took into account a current distribution in the
0921-4534r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 0 0 . 0 0 1 8 7 - 8
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
108
vortex-free region of the disk ŽMeissner state., in addition to current density < j < s jc , which is assumed constant and flows in the vortex-penetrated region. Zhu et al. w17x extended this model by taking into account the fact that the current density should not change abruptly from the vortex-penetrated region to the vortex-free region. This model stimulates many other studies on dc and ac response of superconducting disks and strips with zero thickness under a perpendicular field by taking into account the field dependent critical current density w18,19x. Recently, a more realistic situation of finite thickness sample in perpendicular magnetic field was considered by Brandt w20–23x, by time integration of a integral equation, he calculated the magnetic moment, flux and current penetration, hysteresis loop and fundamental ac susceptibility of type-II superconductors of nonzero thickness in a perpendicular applied magnetic field. This method applies to arbitrary cross-section and current–voltage characteristics EŽ j . of conductors and superconductors. Flux creep can be taken into account by assuming a nonlinear current–voltage characteristics, actually he considered the power-law electric field EŽ j . s Ec Ž jrjc . n , which includes the linear Ohmic Ž n s 1. and Bean Ž n `. limits. Brandt w20–23x has presented systematical results for linear and nonlinear ac susceptibility of superconducting disks and cylinders in perpendicular magnetic field. However, to our best knowledge, no systematical results for harmonic ac susceptibilies in this situation have been reported. And the interpretations of harmonic ac susceptibility in superconductors are usually by means of the critical state model w24–28x, characterized by a stepwise current–voltage relationship: E s 0 for j - jc and E A j for j ) jc , which completely neglects the dissipative processes in the subcritical state j - j c, where jc is the critical current density. Analytical and numerical works w1,29,30x have also been presented for ac susceptibility in the flux creep region where the current–voltage relationship is highly nonlinear,
™
E Ž j . s Ecexp y
U Ž j,T , B . T
,
Ž 1.
where Ec is defined by Ec s EŽ j s jc ., UŽ j . is the activation energy dependent on temperature T, mag-
netic field B and current density j. However, the analysis is only for infinite long cylinder or slab in parallel magnetic field. In this paper, we calculate third harmonic ac susceptibility of finite thickness superconducting disks and strips in perpendicular magnetic field using the method developed by Brandt, and compare the results with those derived from the critical state model and those derived by numerically solving the flux creep equation EB
E sy
Et
Ex
Ž BÕ . .
Ž 2.
The thermally activated flux velocity Õ is given by Õ s Õ 0 Ž jrjc .expwyUŽ j .rk B T x, where Õ 0 s u vm , u is the hopping distance, vm is the microscopic attempt frequency and the factor jrjc is introduced to provide a gradual crossover to the viscous flow regime Õ A j at T 4 UŽ j .. The UŽ j . relationship is chosen to be UŽ j . s U0 lnŽ jcrj ., when inserted into Eq. Ž1., it results in the power-law EŽ j . relationship E Ž j . s BÕ 0
j
ž / jc
s q1
s Ec
j
ž / jc
n
,
Ž 3.
with s s U0rk B T, n s s q 1 and Ec s BÕ 0 .
2. Computational method 2.1. Equation of motion for the current density The equations of motion for the current density have been derived by Brandt w20–23x for general strips and disks in perpendicular field. We outline here the results for general strips with cross-section of 2 a = 2 b in the x–y plane, and infinite long along z direction. A time dependent homogeneous magnetic field H a I yˆ is applied. The applied field induces currents in the bulk and at the surface of specimen which flow along z. The current density j s jŽ x, y . zˆ generates a planar magnetic field H which has no z component. We assume that B s m 0 H where B s = = A is the induction and A s AŽ x, y . zˆ the vector potential. As the scalar fields jŽ x, y . and AŽ x, y . obey the 2D Laplace equation m 0 j s y= 2 Ž A q xBa ., where yxBa corresponds to
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
the vector potential of the applied magnetic field. The solution to this 2D Laplace equation for the geometry considered can be written as, a
A Ž r . s ym 0
b
X
X
X
Ž 4.
with r s Ž x, y . and rX s Ž xX , yX . and the integral kernel Q Ž r,r X . s
ln < r I rX <
a
2p
ms4
Ž xy2 q yy2 .Ž xy2 q yy2 . s ln 2 2 4p Ž xq q yy .Ž xq2 q yq2 . 1
a
b
X
X
y1
H0 d x H0 d y Q
Ž 5.
Ž 6.
Here Q y 1Ž r,rX . is the inverse kernel defined by a
X
b 2 X
y1
H0 d x H0 d r Q
Ž r,rX . Q Ž rX ,rY . s d Ž r I rY . . Ž 7 .
According to the induction law = = E s IB˙ s ˙ Combining this with I= = A, we have E s yA. Eq. Ž4., we have, X
b
H0 d x H0
mŽ t . ei m v tdŽ v t . .
™
X
Ž 8.
The equation of motion for the current density can be obtained by inverting Eq. Ž8. as, a
p H0
2p
H0
Ž 11 .
™
Usually, the xm are normalized such that for H0 0 or v ` the ideally diamagnetic susceptibility x Ž0, v . results; this normalization is achieved by dividing all xm , Eq. Ž11., by the magnitude of the initial slope < mX Ž 0 .< s lim H a ™ 0 < EmŽ Ha .rEHa <. In the following, only the third harmonic ac susceptibility x 3 s x 3X y i x 3Y is considered.
d y Q Ž r,r . j˙Ž r ,t . X
q xB˙a Ž t . .
j Ž r ,t . s my1 0
i
2.2. Parallel limit
a
E j Ž r ,t . s m 0
Ž 10 .
For sinusoidal HaŽ t . s H0 sinŽ v t ., one may define the nonlinear complex ac susceptibility xm s xmX y xmY , m s 1, 2, 3, . . . ,
xm Ž H0 , v . s
Ž r,rX .
= A Ž r X . q x X Ba .
b
H0 d xH0 d yj Ž x , y . x.
with x "s x " xX , y "s y s "yX . Formally, Eq. Ž4. may be inverted and written in the form j Ž r . s ymy1 0
j˙Ž r,t .d t. The detailed numerical calculation of the above equations can be found in Refs. w20–23x. In the following, we consider the power law electric field EŽ j . s Ec Ž jrjc . n , as it describes the linear response case for creep exponent n s 1, the Bean critical state for n ` and the flux creep region for 1 - n - `. As the current density is derived from Eq. Ž9., the magnetic moment per unit length of a strip with rectangular cross-section in a perpendicular field Ha I yˆ can be obtained as w20–23x,
™
X
H0 d x H0 d y Q Ž r,r . j Ž r .
y xBa ,
109
X
b
X
X
H0 d x H0 d y Q Ž r,r .
= E j Ž rX ,t . y xX B˙a Ž t . 4 .
Ž 9.
In the general case of nonlinear EŽ j . and arbitrary sweep of BaŽ t ., this integral equation can be numerally time integrated by starting with jŽ r,0. s 0 at time t s 0 and then putting jŽ r,t q d t . s jŽ r,t . q
™
For the case of b `, we numerically solve Eq. Ž2.. The discretion and numerical integration of Eq. Ž2. are carried out by using a simple single step method, as has been discussed in detail in our previous paper w29x. As the profiles for B are derived, the current density j can be obtained by means of ŽEBrE x . s m 0 j, and therefore the magnetic moment and ac susceptibility can be further derived from Eq. Ž10. and Ž11., respectively. For all examples calculated in this work, the values for the parameters used in Eq. Ž2. are: u v m s 1mrs,
Bdrm 0 jc s 1r4p .
110
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
3. Results and discussion The calculated third harmonic ac susceptibility x 3 s x 3X y i x 3Y of a superconducting strip with side ratio bra s 0.03 is plotted in Fig. 1 as a function of
Y
X
Fig. 2. The data of Fig.1 replotted as x 3 vs. x 3 .
the ac field amplitude H0rHp for creep exponent n s 3, 5, 11, 51 Žfrom top to bottom.. Hp is the full penetration field of superconducting strip in perpendicular magnetic field in the Bean limit w31x, Hp s j c
X
Y
Fig. 1. Third harmonic ac susceptibility x 3 s x 3 y i x 3 of a strip with aspect ratio br as 0.03 in a perpendicular ac field with frequency v s Ec rŽ m 0 jc a2 ., as a function of the amplitude H0 r Hp Žthe full penetration field Hp s 0.0861, Eq. Ž12... Plotted X Y are x 3 Žsolid lines. and x 3 Ždotted lines. curves on a semilogarithmic plot for creep exponents ns 3, 5, 11, 51.
b
p
b a2 arctan qln 1q 2 b a b
2a
ž
/
.
Ž 12 .
It can be seen from Fig. 1 that at low or at high H0rHp values, both x 3Y Ždotted lines. and x 3X Žsolid lines. are zero, while in the transition region x 3Y and x 3X oscillate between positive and negative values. A large positive peak can be observed in both x X Ž H0rHp . and x Y Ž H0rHp .. As n is increased, both positive peaks increase, the transition region narrows and shifts to large H0rHp values. This is in agreement with the results of the fundamental ac susceptibility w20–23,32x. For n s 3, the positive peak in x 3Y is lower than that in x 3X , but as n is increased, the peak in x 3Y increases and becomes larger than that in x 3X for n s 51. In order to see the difference between different flux creep exponents n more clearly, we replot the data of Fig. 1 as polar plot of x 3Y vs. x 3X in Fig. 2. For each creep exponent n, a closed curve is observed. As n is increased, the area closed by the curve increases, i.e., both the positive peaks in x Y and x X increase, while the small negative peak in x 3X decreases. There is a region for n s 3, where both x 3X and x 3Y are negative, but this region is not
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
111
observed for larger n, it is not clear yet, whether this is the feature of small n or caused by calculating errors. Shown in Fig. 3 are x 3X and x 3Y of a superconducting strip with side ratio bra s 0.3 as a function of the ac field amplitude H0rHp for creep exponent n s 3, 5, 11, 51 Žfrom top to bottom.. Similar features as in Fig. 1 can be observed. However, note that there are no negative peaks in x 3X curve for all n
X
Y
Fig. 4. Third harmonic ac susceptibility x 3 s x 3 y i x 3 of a strip with aspect ratio br as 0.03, 0.3, 1,3 in a perpendicular ac field with frequency v s Ec rŽ m 0 jc a2 ., as a function of the amplitude H0 r Hp Ž Hp s 0.0861, 0.4237, 0.7206, 0.8958, respectively.. X Y Plotted are x 3 Žsolid lines. and x 3 Ždotted lines. curves on a semilogarithmic plot for creep exponents ns 3.
Fig. 3. Similar as Fig.1 but for br as 0.3.
and the positive peak in x 3Y curve is always smaller than that in x 3X curve. The results for other thickness have also been obtained and shown in Fig. 4, where x 3X and x 3Y as a function of the ac field amplitude H0rHp are plotted for side ratios bra s 0.03, 0.3, 1,
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
112
X
X
Y
Fig. 5. Third harmonic ac susceptibility x 3 s x 3 y i x 3 of a infinite long slab in parallel field, derived from the Bean critical state model Župper panel. and from numerically solving Eq. 2 Žlower panel..
3 Žfrom top to bottom. and creep exponent n s 3. As bra is increased, the positive peaks increase, the transition region narrows and shifts to large H0rHp values.
Y
Y
Fig. 7. Third harmonic ac susceptibility x 3 s x 3 y i x 3 of a strip and a disk with aspect ratio br as 0.03 in a perpendicular ac field with frequency v s Ec rŽ m 0 jc a2 ., as a function of the X amplitude H0 r Hp Ž Hp s 0.0861, 0.1260.. Plotted are x 3 Žsolid Y lines. and x 3 Ždotted lines. curves on a semilogarithmic plot for creep exponents ns 3.
™
In order to compare to the parallel limit, we plot in Fig. 5 x 3Y and x 3X vs. H0rHp for Ž bra `. derived from the Bean critical state model Župper panel. and from numerically solving the flux creep equation Žlower panel., Hp is the full penetration field in the Bean limit Hp s jc a ŽEq. Ž12. with
X
Fig. 6. Polar plot of x 3 vs. x 3 for br as 3, ns 3; br as 3, ns 51 and br as`, ns 3. Also plotted is the curve derived from the Bean critical state model.
Y
X
Fig. 8. The data of Fig. 7 replotted as x 3 vs. x 3 .
™
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
bra `.. For the Bean model, x 3X keeps zero for H0 - Hp , and then a positive peak follows, that is, x 3X is nonnegative. When flux creep effects are taken into account, as can be seen from the lower panel and Fig. 1, before the positive peak, a small negative peak can be observed, and the transition begins at much lower H0rHp values. A direct comparison can be obtained by plotting the polar plot of x 3X vs. x 3X , as is shown in Fig. 6, where x 3Y vs. x 3X for bra s 3a, n s 3, bra s 3, n s 51 and bra s `, n s 3 are plotted, also plotted is the curve derived from the Bean critical state model. It can be seen clearly that even for n s 3 the peaks of the parallel limit Ž bra `. are larger than the peaks of bra s 0.03 for n s 51, which indicates that the sample thickness has a much more prominent effect on ac susceptibility than the flux creep exponent n. From ŽFigs. 1, 3, 4 and 6., we can see that as bra is increase, it goes to the parallel limit, and with both bra and n are increased, it goes to the Bean critical state model, so the calculations are self-consistent. ac susceptibility of disks have also been calculated, the results are shown in Fig. 7, where x 3X and x 3Y of a strip Župper panel. and a disk Žlower panel. with aspect ratio bra s 0.03 are plotted as a function of the amplitude H0rHp Ž Hp s 0.0861, 0.1260. for creep exponents n s 3. The difference is obvious, compared to the results of the strip, the transition region of the disk is smaller and shifts to lower value of H0rHp . When the data of Fig. 7 are shown as polar plot of x 3Y vs. x 3X in Fig. 8, the difference between the peaks for both sample shapes can be clearly seen, which is also observed for other side ratios, not shown here for simplicity. This feature should be contrasted to the case of fundamental ac susceptibility, where the peaks in x 1Y ŽT . curves for different sample shapes are observed to be almost the same w33x.
™
4. Conclusions In summary, third harmonic ac susceptibility of superconducting strips and disks with finite thickness Žcross-sections 2 a = 2 b . in perpendicularly applied magnetic field Ha s H0 sinŽ v t . have been calculated. The flux creep effects have been taken into account by using a power-law electric field EŽ j . s
113
Ec Ž jrjc . n. Detailed results are present for different thickness b and creep exponents n, which are compared with those derived from the Bean critical state model and from numerically solving the flux creep equation for infinite slab in parallel field. In the limits of bra 4 1 and n ` the results from the parallel situation and from the Bean critical state model are reproduced, respectively.
™
Acknowledgements We are grateful to Professor E. H. Brandt for kind offer of the program and for useful discussions.
References w1x G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys 66 Ž1994. 1125. w2x S.Y. Ding, G.Q. Wang, X.X. Yao, H.T. Peng, Q.Y. Peng, S.H. Zhou, Phys. Rev. B 51 Ž1995. 9107. w3x M.J. Qin, X.X. Yao, Physica C 272 Ž1996. 142. w4x A.P. Malozemoff, T.K. Worthington, Y. Yeshurun, F. Holtzberg, P.H. Kes, Phys. Rev. B 38 Ž1988. 3203. w5x F. Gomory, ¨ ¨ T. Takacs, ´ Physica C 217 Ž1993. 297. w6x A. Gupta, P. Esquinazi, H.F. Braun, H.W. Neumuller, Europhys. Lett. 10 Ž1989. 663. w7x H.H.P.M. Emmen, V.A.M. Brabers, W.J.M. de Jonge, Physica C 176 Ž1991. 137. w8x Y. Yeshurun, M.W. McElfresh, A.P. Malozemoff, J. Hagerhorst-Trewhella, J. Mannhart, F. Holtzberg, G.V. Chandrashekhar, Phys. Rev. B 42 Ž1990. 6322. w9x S. Senoussi, S. Hadjoudj, R. Maury, A. Fert, Physica C 165 Ž1990. 364. w10x F.M. Sauerzopf, H.P. Wiesinger, H.W. Weber, Cryogenics 30 Ž1990. 650. w11x M. Wacenovsky, H.W. Weber, O.B. Hyun, D.K. Finnemore, K. Mereiter, Physica C 160 Ž1989. 55. w12x M. Daumling, D.C. Larbalestier, Phys. Rev. B 40 Ž1989. ¨ 9350. w13x D.J. Frankel, J. Appl. Phys. 50 Ž1979. 540. w14x H. Theuss, A. Forkl, H. kronmuller, Physica C 190 Ž1992. ¨ 345. w15x L.W. Conner, A.P. Malozemoff, Phys. Rev. B 43 Ž1991. 402. w16x P.N. Mikheenko, Yu.E. kuzovlev, Physica C 204 Ž1993. 229. w17x J. Zhu, J. Mester, J. Lockhart, J. Turneaure, Physica C 212 Ž1993. . w18x J.R. Clem, A. Sanchez, Phys. Rev. B 50 Ž1994. 9355. w19x J. McDonald, J.R. Clem, Phys Rev. B 53 Ž1996. 8643. w20x E.H. Brandt, Phys. Rev. B 54 Ž1996. 4246. w21x E.H. Brandt, Phys. Rev. B 55 Ž1997. 14513. w22x E.H. Brandt, Phys. Rev. B 58 Ž1998. 6506.
114 w23x w24x w25x w26x w27x
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
E.H. Brandt, Phys. Rev. B 58 Ž1998. 6523. C.P. Bean, Rev. Mod. Phys 2 Ž1964. 31. A.M. Campbell, J. Phys. C 2 Ž1969. 1492. A.M. Campbell, J. Phys. C 4 Ž1971. 3186. L. Ji, R.H. Sohn, G.C. Spalding, C.J. Lobb, M. Tinkham, Phys. Rev. B 40 Ž1989. 10936. w28x T. Ishida, R.B. Goldfarb, Phys. Rev. B 41 Ž1990. 8937.
w29x M.J. Qin, X.X. Yao, Phys. Rev. B 54 Ž1996. 7536. w30x C.J. van der Beek, V.B. Geshkenbein, V.M. Vinokur, Phys. Rev. B 48 Ž1993. 3393. w31x A. Forkl, Phys. Scr. T 49 Ž1993. 148. w32x M.J. Qin, C.K. Ong, Physica C 325 Ž1999. 173. w33x Th. Herzog, H.A. Radovan, P. Ziemann, E.H. Brandt, Phys. Rev. B 56 Ž1997. 2871.
Third harmonic ac susceptibility of superconducting strips and disks M.J. Qin ) , C.K. Ong Centre for Superconducting and Magnetic Materials and Department of Physics, National UniÕersity of Singapore, Lower Kent Ridge Road, Singapore 119260, Singapore Received 20 December 1999; received in revised form 26 January 2000; accepted 7 February 2000
Abstract Third harmonic ac susceptibility of superconducting strips and disks with finite thickness Žcross-sections 2 a = 2 b . in a perpendicular magnetic field Ha s H0 sinŽ v t . has been calculated. The flux creep effects have been taken into account by using a power-law electric field EŽ j . s Ec Ž jrjc . n. Detailed results are present for different thickness b and creep exponents n, which are compared with those derived from the Bean critical state model and from numerically solving the flux creep equation for infinite slab in parallel field. In the limits of bra 4 1 and n ` the results from the parallel situation and the Bean model are reproduced, respectively. q 2000 Elsevier Science B.V. All rights reserved.
™
PACS: 74.60Ge; 74.60Jg; 74.72Fq
1. Introduction Magnetic measurements, such as magnetization hysteresis loops, magnetic relaxation and ac susceptibility have been widely used to study flux dynamics of high temperature superconductors, as a result, critical current density, activation energy as well as UŽ j . relationship have been determined experimentally by this contactless technique w1–7x. Although the magnetic measurement has an advantage of using contacless and non-destructive technology, experimentally, in order to get maximum signal, most magnetic data have been taken on thin films and single crystals Žusually in the form of platelets resembling thin film rectangles or disks. with the ) Corresponding author. Tel.: q65-874-2625; fax: q65-7776126. E-mail address: [email protected] ŽM.J. Qin..
applied field perpendicular to the film plane. Unfortunately, the analysis of the experimental data in this perpendicular field geometry is complicated by the strong demagnetizing effects, resulting in numerous studies on this subject for both fundamental and technological reasons. A conventional method to treat the demagnetizing effects is to use the demagnetizing factor w8–11x. However, a number of authors have pointed out that in the critical state of a superconductor, in which current is uniformly distributed in the bulk, this method is invalid not only quantitatively but qualitatively w12–15x. Because the theoretical model was not available, they used numerical calculation to treat this problem instead. Mikheenko and Kuzovlev w16x presented an exact critical state model for superconducting disks with zero thickness in perpendicular magnetic field. They took into account a current distribution in the
0921-4534r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 0 0 . 0 0 1 8 7 - 8
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
108
vortex-free region of the disk ŽMeissner state., in addition to current density < j < s jc , which is assumed constant and flows in the vortex-penetrated region. Zhu et al. w17x extended this model by taking into account the fact that the current density should not change abruptly from the vortex-penetrated region to the vortex-free region. This model stimulates many other studies on dc and ac response of superconducting disks and strips with zero thickness under a perpendicular field by taking into account the field dependent critical current density w18,19x. Recently, a more realistic situation of finite thickness sample in perpendicular magnetic field was considered by Brandt w20–23x, by time integration of a integral equation, he calculated the magnetic moment, flux and current penetration, hysteresis loop and fundamental ac susceptibility of type-II superconductors of nonzero thickness in a perpendicular applied magnetic field. This method applies to arbitrary cross-section and current–voltage characteristics EŽ j . of conductors and superconductors. Flux creep can be taken into account by assuming a nonlinear current–voltage characteristics, actually he considered the power-law electric field EŽ j . s Ec Ž jrjc . n , which includes the linear Ohmic Ž n s 1. and Bean Ž n `. limits. Brandt w20–23x has presented systematical results for linear and nonlinear ac susceptibility of superconducting disks and cylinders in perpendicular magnetic field. However, to our best knowledge, no systematical results for harmonic ac susceptibilies in this situation have been reported. And the interpretations of harmonic ac susceptibility in superconductors are usually by means of the critical state model w24–28x, characterized by a stepwise current–voltage relationship: E s 0 for j - jc and E A j for j ) jc , which completely neglects the dissipative processes in the subcritical state j - j c, where jc is the critical current density. Analytical and numerical works w1,29,30x have also been presented for ac susceptibility in the flux creep region where the current–voltage relationship is highly nonlinear,
™
E Ž j . s Ecexp y
U Ž j,T , B . T
,
Ž 1.
where Ec is defined by Ec s EŽ j s jc ., UŽ j . is the activation energy dependent on temperature T, mag-
netic field B and current density j. However, the analysis is only for infinite long cylinder or slab in parallel magnetic field. In this paper, we calculate third harmonic ac susceptibility of finite thickness superconducting disks and strips in perpendicular magnetic field using the method developed by Brandt, and compare the results with those derived from the critical state model and those derived by numerically solving the flux creep equation EB
E sy
Et
Ex
Ž BÕ . .
Ž 2.
The thermally activated flux velocity Õ is given by Õ s Õ 0 Ž jrjc .expwyUŽ j .rk B T x, where Õ 0 s u vm , u is the hopping distance, vm is the microscopic attempt frequency and the factor jrjc is introduced to provide a gradual crossover to the viscous flow regime Õ A j at T 4 UŽ j .. The UŽ j . relationship is chosen to be UŽ j . s U0 lnŽ jcrj ., when inserted into Eq. Ž1., it results in the power-law EŽ j . relationship E Ž j . s BÕ 0
j
ž / jc
s q1
s Ec
j
ž / jc
n
,
Ž 3.
with s s U0rk B T, n s s q 1 and Ec s BÕ 0 .
2. Computational method 2.1. Equation of motion for the current density The equations of motion for the current density have been derived by Brandt w20–23x for general strips and disks in perpendicular field. We outline here the results for general strips with cross-section of 2 a = 2 b in the x–y plane, and infinite long along z direction. A time dependent homogeneous magnetic field H a I yˆ is applied. The applied field induces currents in the bulk and at the surface of specimen which flow along z. The current density j s jŽ x, y . zˆ generates a planar magnetic field H which has no z component. We assume that B s m 0 H where B s = = A is the induction and A s AŽ x, y . zˆ the vector potential. As the scalar fields jŽ x, y . and AŽ x, y . obey the 2D Laplace equation m 0 j s y= 2 Ž A q xBa ., where yxBa corresponds to
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
the vector potential of the applied magnetic field. The solution to this 2D Laplace equation for the geometry considered can be written as, a
A Ž r . s ym 0
b
X
X
X
Ž 4.
with r s Ž x, y . and rX s Ž xX , yX . and the integral kernel Q Ž r,r X . s
ln < r I rX <
a
2p
ms4
Ž xy2 q yy2 .Ž xy2 q yy2 . s ln 2 2 4p Ž xq q yy .Ž xq2 q yq2 . 1
a
b
X
X
y1
H0 d x H0 d y Q
Ž 5.
Ž 6.
Here Q y 1Ž r,rX . is the inverse kernel defined by a
X
b 2 X
y1
H0 d x H0 d r Q
Ž r,rX . Q Ž rX ,rY . s d Ž r I rY . . Ž 7 .
According to the induction law = = E s IB˙ s ˙ Combining this with I= = A, we have E s yA. Eq. Ž4., we have, X
b
H0 d x H0
mŽ t . ei m v tdŽ v t . .
™
X
Ž 8.
The equation of motion for the current density can be obtained by inverting Eq. Ž8. as, a
p H0
2p
H0
Ž 11 .
™
Usually, the xm are normalized such that for H0 0 or v ` the ideally diamagnetic susceptibility x Ž0, v . results; this normalization is achieved by dividing all xm , Eq. Ž11., by the magnitude of the initial slope < mX Ž 0 .< s lim H a ™ 0 < EmŽ Ha .rEHa <. In the following, only the third harmonic ac susceptibility x 3 s x 3X y i x 3Y is considered.
d y Q Ž r,r . j˙Ž r ,t . X
q xB˙a Ž t . .
j Ž r ,t . s my1 0
i
2.2. Parallel limit
a
E j Ž r ,t . s m 0
Ž 10 .
For sinusoidal HaŽ t . s H0 sinŽ v t ., one may define the nonlinear complex ac susceptibility xm s xmX y xmY , m s 1, 2, 3, . . . ,
xm Ž H0 , v . s
Ž r,rX .
= A Ž r X . q x X Ba .
b
H0 d xH0 d yj Ž x , y . x.
with x "s x " xX , y "s y s "yX . Formally, Eq. Ž4. may be inverted and written in the form j Ž r . s ymy1 0
j˙Ž r,t .d t. The detailed numerical calculation of the above equations can be found in Refs. w20–23x. In the following, we consider the power law electric field EŽ j . s Ec Ž jrjc . n , as it describes the linear response case for creep exponent n s 1, the Bean critical state for n ` and the flux creep region for 1 - n - `. As the current density is derived from Eq. Ž9., the magnetic moment per unit length of a strip with rectangular cross-section in a perpendicular field Ha I yˆ can be obtained as w20–23x,
™
X
H0 d x H0 d y Q Ž r,r . j Ž r .
y xBa ,
109
X
b
X
X
H0 d x H0 d y Q Ž r,r .
= E j Ž rX ,t . y xX B˙a Ž t . 4 .
Ž 9.
In the general case of nonlinear EŽ j . and arbitrary sweep of BaŽ t ., this integral equation can be numerally time integrated by starting with jŽ r,0. s 0 at time t s 0 and then putting jŽ r,t q d t . s jŽ r,t . q
™
For the case of b `, we numerically solve Eq. Ž2.. The discretion and numerical integration of Eq. Ž2. are carried out by using a simple single step method, as has been discussed in detail in our previous paper w29x. As the profiles for B are derived, the current density j can be obtained by means of ŽEBrE x . s m 0 j, and therefore the magnetic moment and ac susceptibility can be further derived from Eq. Ž10. and Ž11., respectively. For all examples calculated in this work, the values for the parameters used in Eq. Ž2. are: u v m s 1mrs,
Bdrm 0 jc s 1r4p .
110
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
3. Results and discussion The calculated third harmonic ac susceptibility x 3 s x 3X y i x 3Y of a superconducting strip with side ratio bra s 0.03 is plotted in Fig. 1 as a function of
Y
X
Fig. 2. The data of Fig.1 replotted as x 3 vs. x 3 .
the ac field amplitude H0rHp for creep exponent n s 3, 5, 11, 51 Žfrom top to bottom.. Hp is the full penetration field of superconducting strip in perpendicular magnetic field in the Bean limit w31x, Hp s j c
X
Y
Fig. 1. Third harmonic ac susceptibility x 3 s x 3 y i x 3 of a strip with aspect ratio br as 0.03 in a perpendicular ac field with frequency v s Ec rŽ m 0 jc a2 ., as a function of the amplitude H0 r Hp Žthe full penetration field Hp s 0.0861, Eq. Ž12... Plotted X Y are x 3 Žsolid lines. and x 3 Ždotted lines. curves on a semilogarithmic plot for creep exponents ns 3, 5, 11, 51.
b
p
b a2 arctan qln 1q 2 b a b
2a
ž
/
.
Ž 12 .
It can be seen from Fig. 1 that at low or at high H0rHp values, both x 3Y Ždotted lines. and x 3X Žsolid lines. are zero, while in the transition region x 3Y and x 3X oscillate between positive and negative values. A large positive peak can be observed in both x X Ž H0rHp . and x Y Ž H0rHp .. As n is increased, both positive peaks increase, the transition region narrows and shifts to large H0rHp values. This is in agreement with the results of the fundamental ac susceptibility w20–23,32x. For n s 3, the positive peak in x 3Y is lower than that in x 3X , but as n is increased, the peak in x 3Y increases and becomes larger than that in x 3X for n s 51. In order to see the difference between different flux creep exponents n more clearly, we replot the data of Fig. 1 as polar plot of x 3Y vs. x 3X in Fig. 2. For each creep exponent n, a closed curve is observed. As n is increased, the area closed by the curve increases, i.e., both the positive peaks in x Y and x X increase, while the small negative peak in x 3X decreases. There is a region for n s 3, where both x 3X and x 3Y are negative, but this region is not
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
111
observed for larger n, it is not clear yet, whether this is the feature of small n or caused by calculating errors. Shown in Fig. 3 are x 3X and x 3Y of a superconducting strip with side ratio bra s 0.3 as a function of the ac field amplitude H0rHp for creep exponent n s 3, 5, 11, 51 Žfrom top to bottom.. Similar features as in Fig. 1 can be observed. However, note that there are no negative peaks in x 3X curve for all n
X
Y
Fig. 4. Third harmonic ac susceptibility x 3 s x 3 y i x 3 of a strip with aspect ratio br as 0.03, 0.3, 1,3 in a perpendicular ac field with frequency v s Ec rŽ m 0 jc a2 ., as a function of the amplitude H0 r Hp Ž Hp s 0.0861, 0.4237, 0.7206, 0.8958, respectively.. X Y Plotted are x 3 Žsolid lines. and x 3 Ždotted lines. curves on a semilogarithmic plot for creep exponents ns 3.
Fig. 3. Similar as Fig.1 but for br as 0.3.
and the positive peak in x 3Y curve is always smaller than that in x 3X curve. The results for other thickness have also been obtained and shown in Fig. 4, where x 3X and x 3Y as a function of the ac field amplitude H0rHp are plotted for side ratios bra s 0.03, 0.3, 1,
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
112
X
X
Y
Fig. 5. Third harmonic ac susceptibility x 3 s x 3 y i x 3 of a infinite long slab in parallel field, derived from the Bean critical state model Župper panel. and from numerically solving Eq. 2 Žlower panel..
3 Žfrom top to bottom. and creep exponent n s 3. As bra is increased, the positive peaks increase, the transition region narrows and shifts to large H0rHp values.
Y
Y
Fig. 7. Third harmonic ac susceptibility x 3 s x 3 y i x 3 of a strip and a disk with aspect ratio br as 0.03 in a perpendicular ac field with frequency v s Ec rŽ m 0 jc a2 ., as a function of the X amplitude H0 r Hp Ž Hp s 0.0861, 0.1260.. Plotted are x 3 Žsolid Y lines. and x 3 Ždotted lines. curves on a semilogarithmic plot for creep exponents ns 3.
™
In order to compare to the parallel limit, we plot in Fig. 5 x 3Y and x 3X vs. H0rHp for Ž bra `. derived from the Bean critical state model Župper panel. and from numerically solving the flux creep equation Žlower panel., Hp is the full penetration field in the Bean limit Hp s jc a ŽEq. Ž12. with
X
Fig. 6. Polar plot of x 3 vs. x 3 for br as 3, ns 3; br as 3, ns 51 and br as`, ns 3. Also plotted is the curve derived from the Bean critical state model.
Y
X
Fig. 8. The data of Fig. 7 replotted as x 3 vs. x 3 .
™
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
bra `.. For the Bean model, x 3X keeps zero for H0 - Hp , and then a positive peak follows, that is, x 3X is nonnegative. When flux creep effects are taken into account, as can be seen from the lower panel and Fig. 1, before the positive peak, a small negative peak can be observed, and the transition begins at much lower H0rHp values. A direct comparison can be obtained by plotting the polar plot of x 3X vs. x 3X , as is shown in Fig. 6, where x 3Y vs. x 3X for bra s 3a, n s 3, bra s 3, n s 51 and bra s `, n s 3 are plotted, also plotted is the curve derived from the Bean critical state model. It can be seen clearly that even for n s 3 the peaks of the parallel limit Ž bra `. are larger than the peaks of bra s 0.03 for n s 51, which indicates that the sample thickness has a much more prominent effect on ac susceptibility than the flux creep exponent n. From ŽFigs. 1, 3, 4 and 6., we can see that as bra is increase, it goes to the parallel limit, and with both bra and n are increased, it goes to the Bean critical state model, so the calculations are self-consistent. ac susceptibility of disks have also been calculated, the results are shown in Fig. 7, where x 3X and x 3Y of a strip Župper panel. and a disk Žlower panel. with aspect ratio bra s 0.03 are plotted as a function of the amplitude H0rHp Ž Hp s 0.0861, 0.1260. for creep exponents n s 3. The difference is obvious, compared to the results of the strip, the transition region of the disk is smaller and shifts to lower value of H0rHp . When the data of Fig. 7 are shown as polar plot of x 3Y vs. x 3X in Fig. 8, the difference between the peaks for both sample shapes can be clearly seen, which is also observed for other side ratios, not shown here for simplicity. This feature should be contrasted to the case of fundamental ac susceptibility, where the peaks in x 1Y ŽT . curves for different sample shapes are observed to be almost the same w33x.
™
4. Conclusions In summary, third harmonic ac susceptibility of superconducting strips and disks with finite thickness Žcross-sections 2 a = 2 b . in perpendicularly applied magnetic field Ha s H0 sinŽ v t . have been calculated. The flux creep effects have been taken into account by using a power-law electric field EŽ j . s
113
Ec Ž jrjc . n. Detailed results are present for different thickness b and creep exponents n, which are compared with those derived from the Bean critical state model and from numerically solving the flux creep equation for infinite slab in parallel field. In the limits of bra 4 1 and n ` the results from the parallel situation and from the Bean critical state model are reproduced, respectively.
™
Acknowledgements We are grateful to Professor E. H. Brandt for kind offer of the program and for useful discussions.
References w1x G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys 66 Ž1994. 1125. w2x S.Y. Ding, G.Q. Wang, X.X. Yao, H.T. Peng, Q.Y. Peng, S.H. Zhou, Phys. Rev. B 51 Ž1995. 9107. w3x M.J. Qin, X.X. Yao, Physica C 272 Ž1996. 142. w4x A.P. Malozemoff, T.K. Worthington, Y. Yeshurun, F. Holtzberg, P.H. Kes, Phys. Rev. B 38 Ž1988. 3203. w5x F. Gomory, ¨ ¨ T. Takacs, ´ Physica C 217 Ž1993. 297. w6x A. Gupta, P. Esquinazi, H.F. Braun, H.W. Neumuller, Europhys. Lett. 10 Ž1989. 663. w7x H.H.P.M. Emmen, V.A.M. Brabers, W.J.M. de Jonge, Physica C 176 Ž1991. 137. w8x Y. Yeshurun, M.W. McElfresh, A.P. Malozemoff, J. Hagerhorst-Trewhella, J. Mannhart, F. Holtzberg, G.V. Chandrashekhar, Phys. Rev. B 42 Ž1990. 6322. w9x S. Senoussi, S. Hadjoudj, R. Maury, A. Fert, Physica C 165 Ž1990. 364. w10x F.M. Sauerzopf, H.P. Wiesinger, H.W. Weber, Cryogenics 30 Ž1990. 650. w11x M. Wacenovsky, H.W. Weber, O.B. Hyun, D.K. Finnemore, K. Mereiter, Physica C 160 Ž1989. 55. w12x M. Daumling, D.C. Larbalestier, Phys. Rev. B 40 Ž1989. ¨ 9350. w13x D.J. Frankel, J. Appl. Phys. 50 Ž1979. 540. w14x H. Theuss, A. Forkl, H. kronmuller, Physica C 190 Ž1992. ¨ 345. w15x L.W. Conner, A.P. Malozemoff, Phys. Rev. B 43 Ž1991. 402. w16x P.N. Mikheenko, Yu.E. kuzovlev, Physica C 204 Ž1993. 229. w17x J. Zhu, J. Mester, J. Lockhart, J. Turneaure, Physica C 212 Ž1993. . w18x J.R. Clem, A. Sanchez, Phys. Rev. B 50 Ž1994. 9355. w19x J. McDonald, J.R. Clem, Phys Rev. B 53 Ž1996. 8643. w20x E.H. Brandt, Phys. Rev. B 54 Ž1996. 4246. w21x E.H. Brandt, Phys. Rev. B 55 Ž1997. 14513. w22x E.H. Brandt, Phys. Rev. B 58 Ž1998. 6506.
114 w23x w24x w25x w26x w27x
M.J. Qin, C.K. Ong r Physica C 334 (2000) 107–114
E.H. Brandt, Phys. Rev. B 58 Ž1998. 6523. C.P. Bean, Rev. Mod. Phys 2 Ž1964. 31. A.M. Campbell, J. Phys. C 2 Ž1969. 1492. A.M. Campbell, J. Phys. C 4 Ž1971. 3186. L. Ji, R.H. Sohn, G.C. Spalding, C.J. Lobb, M. Tinkham, Phys. Rev. B 40 Ž1989. 10936. w28x T. Ishida, R.B. Goldfarb, Phys. Rev. B 41 Ž1990. 8937.
w29x M.J. Qin, X.X. Yao, Phys. Rev. B 54 Ž1996. 7536. w30x C.J. van der Beek, V.B. Geshkenbein, V.M. Vinokur, Phys. Rev. B 48 Ž1993. 3393. w31x A. Forkl, Phys. Scr. T 49 Ž1993. 148. w32x M.J. Qin, C.K. Ong, Physica C 325 Ž1999. 173. w33x Th. Herzog, H.A. Radovan, P. Ziemann, E.H. Brandt, Phys. Rev. B 56 Ž1997. 2871.