- Email: [email protected]
Investigation of the magnetic flux density distribution, current distribution and magnetization curve of YBaCuO thin films
Journal of
ALIL@~
AND COMU~O~D~ Journal of Alloys and Compounds 251 (1997) 146-149
ELSEVIER
Investigation of the magnetic flux density distribution, current distribution and magnetization curve of Y - B a - C u - O thin films A. Forkl, C. Joofi, R. Warthmann, H. Kronmiiller, H.-U. Habermeier* Ma.t-Planck.hlstitut fiir Memliforschung. h~stilut ffir Physik. Heisenbergstmsse l. D-70569 Stuttgart. Germany
Abstract
in high quality Y~Ba,Cu~O7, thin films with circular, triangular and quadratic shape, the magnetic flux density distribution (MFDD) and magnetizationcurve are measured. Comparison of calculated and measured MFDD shows good agreement and yields the critical current densitiesof the films. Thesej~-values agree with the j,-values calculated from the maximummagnetization which is reached at the t
t
external field /~H.. Within the Bean model the field /~,Hp should agree with the penetration field ~ H . necessary for full flux ~netration. The measurements show that I~tt;, is smaller than/~H~, by a factor of about 2, This is understood by taking into account a field dependent critical current density. Furtheron a formula is presented for the magnetic stray-field in the center of a regular polygon with homogeneous thickness which gives also the penetration field/~H.. t
Keywords: Y~Ba~Cu~O thin lilms
1. introduction After zero field ct~ling of a type !i superconductor and application of magnetic fields, I~H,,, the magnetic flux penetrates in several steps into the sample. In the first step. Meissner shielding currents appear in the surfitce of the superconductor shielding the inner of the sample. Second° ly, tbr ~H~ above a geometryodependent critical field, the magnetic flux penetrates p~aly into the superconductor building up a flu~ front which moves towards the sample center as/Aft,, increases. A critical current density j,. flows in these flux filled regions. When I'.,H, is equal to the Soocalled penetration field./A~H;,, the flux front reaches the center of the supercondu,:ior and in a third step (/~H~> ~Hp) the whole sample fills up with magnetic flux and the critical current density decreases until ~ H . is equal to the upper critical field, where j¢ is :J,'ro. All existing currents j(r) produce a self-field. IA~H,~,(r). and a magnetization/A~M given by:
V x H,~.(r) = j(r),
(I) (2)
v
where V is the volume of the sample. In highoT~-supercon° *Corresponding author. ~25~,83881971517,~ © 1997 Elsevier Science S,A, All rights reserved Pll S0925-8388t 96102789.2
ductors, especially in Y-Ba=Cu=O, the contribution of the Meissner currents can be neglected for external magnetic fields much larger than the earth field. Only the contribution of the critical currents has to be taken into account. Models descri!~ing the currem distribution j(r) have to take into account that there exisls a Iowofield-region (/~1t,,< p~}H~,) where the flux tYonl has partly penetrated and a high=field=region (~H,, >/.t, Hp) where no flux free region exists. In the low-field-region, a critical current flows in the flux filled parts of the sample and inner shielding currents flow in the flux free parts. Under the assumption of Bean's critical state model [I], i,e. a field independent critical current density, and neglecting the surface Meissner currents, the inner shielding currents can be calculated for thin samples with different geometries [2,3]. Calculations of the magnetic flux density distribution (MFDD) agree well with measurements performed with the magneto-optical Faraday-effect [4]. Also, the calculated magnetization curves M(H,) show g ~ d agreement with measurements 151, except for the fact that the calculations yield no maximum l~r M(H,), since, within the above considerations, the flux never reaches the sample center. This lack can be avoided by calculating the magnetization and the MFDD of the high-field-region, i.e. by assuming a critical cur~nt density flowing in the whole sample volume, in this case the calculated MFDD agrees also well with investigations by the MOFE. Furtheron the calculated
147
A. Forkl et al. I Journal of AIIo.x:+ and Compottnds 25! (19971 146-149
magnetization should agree with the measured maximum m a g n e t i z a t i o n l/,oMmax and, following former considerations [6], the self-field tzoH~f(r = 01 in the sample center is equal to the penetration field/ZoHp. To prove this we have measured for the first time the MFDD and the magnetization curve of the same Y - B a C u - O films possessing different geometries, i.e. circular. triangular and square shaped samples.
2. Experimental
c-Axis oriented Y iBa2CU3OT_.,.-films with a thickness of 230 to 290 nm were evaporated onto SrTiO3-substrates by laser ablation [71. Then circular, triangular and quadratic samples were patterned by standard lithographic techniques. After evaporation of AI (thickness d^~ =200 nm) and EuSIEuF2 (thickness dE,=250 nm) the samples were cooled down in zero magnetic field from room temperature to Helium temperature. For this a cryostat was used, which is designed for light microscopical investigations. The polarization vector of a linearly polarized light beam passing through the evaporated magneto-optical EuS/EuF 2 layer is rotated by an angle which is proportional to the local magnetic flux density, B,, parallel to the light beam direction. In externally applied magnetic fields, the occurrence of magnetic flux is then seen as an increase of the intensity of the reflected light after passing through an analyzer. These changes are detected by a low light
L$
level video camera which is connected to a digital image processing system for quantitative analysis. The observed flux patterns are represented in Fig. l(a)-(c) for different geometries and a certain magnetic field applied perpendicular to the film plane. Bright areas belong to regions with penetrated magnetic flux. A detailed description of flux penetration and the according current distribution can be found in [81. The penetration field .uoHp can be seen directly in the microscope and is obtained by an accuracy of ___8mT. Furtheron, flux density profiles B~(r) are measured in the fully penetrated state. By comparison with calculated profiles critical current densities J¢.n are calculated. Magnetization curves were measured in a conventional Squid magnetometer at the same samples.
3. C a l c u l a t i o n s
For external fields applied perpendicular to the film plane, the critical current density distribution in the highfield-region can be assumed as shown in Fig. I(d)-(f). The currents flow parallel to the sample edge. in the case of the triangular and square shaped sample, the currents change direction at the angle bisectors (dashed line in r.~g. l(d) and Fig. I(e)). The self-field of such regular polygons with thickness D can be calculated by dividing the sample into N isosceles triangles in which the current flows homogeneously in one direction (N= 2, 3, 4 and ~ belongs to strip, triangle, square and circle). The self-field of such a basic brig is known analytically 181 and thus the sell-field of the whole sample is obtained by a superposition. In Fig. I(g)-(i) the calculated MFDD is represented. The magnetic field/~Jt,~,(r=0) in the center of such a regular polygon with thickness D is equal to the penetration tield ~H;, and is given according to Biot-Savart's law by p,,H,, = - - ~ "N
D I)'sin ~ ~ ,"r . 4R. alan 2R\ ¢~ + ..D. sin -~. In )
m
-
D
rr \,~l
+D
- I - ~ ' cos ~ +
[] (
~" O
rr)
1,
rr
~"
t °
• In
+ 2J/+\/
Fig. I. Experimentally determined magnetic flux density distribution pattern of a triangular, square and circular shaped YIBa,Cu~Ov ,-tilm alter zero-lield cooling and application of an external magnetic field ~H,, of (a) 13.5 roT, (b) 100.3 mT and (c) 150.7 mT. (d)-(f) show the current distribution in the high-field-region and accordingly, (g)-(i) the calculated flux density distribution•
rr
+ t - Yt~ .co, ~ * \/b
~(
n" D rr) D ~r 2. sil)~ + ~-. cos~ - ~-•sin ~.cos ~
(3)
where ~p= I+((DI2R).cos(xt/N))" and R the height of one isosceles triangle. The calculation of the magnetization of such a sample according to Eq. (2) yields ~M -
p~j~R 3
(4)
A. Forkl et ol. I Jourmd {#'Allo)'s and Coml~otmds 251 (1997) 14{5-149
148
Table I Experimentally determined values of the 3 different samples (thickness D, inner radius R, penetration field 14,Hp. maximum magnetization /.~Mm,,
appearing at pt,H'~. and saturation field/~,H,,~) Sample D (nm) ' R (p.m)
/~H;, (mT)
/4~M...... (T)
/~,H~ (mT)
p,H,~ (mT)
Triangle Square Circle
175 165 185
100.6 111.8 165.6
85 75 81
190 170 200
280 290 230
580 1000 IO00
4. Results and discussion Table 1 gives the thickness D and the inner radius R of the investigated triangular, circular and quadratic shaped sample together with the experimental results, i.e. the penetration field port, as obtained# by the MOFE investigations; the external field /~)Hp at which the maximum magnetization/~M,,~, appears. Comparison of the measured flux density profiles B,(r) with calculated profiles yield the critical current density j~.,. Furtheron, from the measured/~Mm,, and/tulip the critical current densities J,.M and J,.tt can be calculated according to Eq. (3) and Eq. (4). All these j,-values are listed in Table 2. We have also measured the remanent magnetization /~M, after field cooling of the sample, application of a maximum external field ~)H.,,..~ and reducing ~)H~ to zero. in Fig. 2 the remanent magnetization /~M, is represented in dependence of/~)H,,,.,.~. The determined jo/ovalues are smaller than the j~.~)values, The reason is that the calculated B,(r)-distribution shows a sharp peak in the center with a half-widtl0 of a few microns, which is near to the lateral resolution of the microscope. Within such a small distance the sharp ino c,'ease of B/(r), if it exists, can not be detected. So the critical currents calculated from/~1~ are underestimated. From the considerations of the Section I it is expected that the penetration field /~Ht, necessary for full flux penetration agrees within a certain accuracy with ~HI, at which ~ M reaches its maximum value/.~M,,,~. Obviously the measured values disagree by a factor of about 2. The reason is that with increasing flux deasity the critical current density decrea~s, especially at the border (see figure 6 in [91), With increasing external field /4~H, the inner currents also reach a critical value. This contribution to /4~M cannot compensate the decrease of the magnetization due to the decrease of the critical currents at the
border. So the maximum of M(H~) appears before the flux front reaches the sample center. The Jc.M" and it.B-values are comparable. They can be regarded as an averaged value. In the case of J:.M, the border currents contribute more to /.toM than the inner currents (see Eq. (2)), whereas in the case of Jc.tt, all currents contribute in the same way to the mean value. Both effects, i.e. change of current distribution and different averaging, lead to comparable Jc.t~- and j:.M-values. The remanent magnetization/~M,, obtained after application of a maximum external field/.toH~.m,~ and reducing H. to zero, shows an increase with increasing poH,,.m,~ until a saturation value/~)M,~, is reached at/4)H.,,,. With increasing /~)H,.,,,~, the flux front penetrates deeper into the sample, and thus the volume part in which critical currents flow increases until the flux front reaches the sample center. So the remanent magnetization should reach the saturation at/~Ho. This is obviously the case for all three samples (see Table I ). From measurements of/4~M, in dependence of/4~H,,.,,,,, the penetration field ~ H p can be measured.
5.
Summary
We have investigated the flux distribution and the thin
magnetization curve of superconducting Y - B a - C u - O
172
-po1~at.............~. l,i
"
i
168
~166
Sample
j,.n(10" Am
Triangle Square Circle
2.0 1.4 2.0
~)
j,.H(10" Am 1.2 0.8 1.6
:)
j, u(lO" 2. I 1.3 2.0
Am
:)
162
'%0
PoHaat
m I
164 'Fable 2 Values of the critical current density calculated from the measured flux profiles B,(r), ~netration field I~tt,, and the maxintum magnetization ~,M~, (for details see text)
tim " m . . o o
170
I
,oo ,so 2oo PoHa,n~
360 [roT]
Fig. 2. Remarmnt magnetization ~,M, of the circuhr structured fihn in dependence of the maximum applied magnetic fieldt~,H~......
A. Forkl et al. I Journal of Alloys and Compmmds 251 (1997) 146-149
films possessing different geometries. Critical currents are calculated from different characteristic features, i.e. the flux density profiles, the penetration fields and the maximum magnetizations. The different jc-values can be understood by taking into account that they represent an averaged mean value and that the local critical current density decreases with increasing local flux density. For the calculation ofjc from the penetration field//~Hp, a formula is given for regular polygons with thickness D.
References Ill C.E Bean, Phys. Rev. Lett.. 8 (1962) 250. [2l E.H. Brandt. M.V. lndenbom and A. Forkl, Europh3s. Lett.. 22 ( 1993} 735.
149
[3l P.N. Mikheenko and Yu.E. Kuzovlev, Physica C. 204 (1993) 229. [41 P. B:'fill, D, Kirchgtissner and P. Leiderer, Physica C, 182 { 1991 ) 339; R. Knorpp, A, Forkl, H.-U. Habermeier and H, Kronmiiller, Physica C, 230 (1994) 128; D. Glatzer, A. Forid, H. Theuss, H.-U. Habermeier and H. Kronmiiller, Physica status solidi (b). 170 (1992) 549. [5] Y. Mawatari, A. Sawa and H. Obara, Ph.vsica C, 258 (1996) 121. [6] A. Forkl, Phy~ica Scripta. T49 (1993) 148. 171 H.-U. Haberrneier, Eur. J. Solid State lnorg. Chem.. 28 (1991) 619, [8] A. Forkl and H. Kronmiiller, Phys. Rev. B, 52 (1996) 16 130. [9] H. Theuss, A. Forkl and H. Kronmfiller, Physica C. 190 (1992) 345.
ALIL@~
AND COMU~O~D~ Journal of Alloys and Compounds 251 (1997) 146-149
ELSEVIER
Investigation of the magnetic flux density distribution, current distribution and magnetization curve of Y - B a - C u - O thin films A. Forkl, C. Joofi, R. Warthmann, H. Kronmiiller, H.-U. Habermeier* Ma.t-Planck.hlstitut fiir Memliforschung. h~stilut ffir Physik. Heisenbergstmsse l. D-70569 Stuttgart. Germany
Abstract
in high quality Y~Ba,Cu~O7, thin films with circular, triangular and quadratic shape, the magnetic flux density distribution (MFDD) and magnetizationcurve are measured. Comparison of calculated and measured MFDD shows good agreement and yields the critical current densitiesof the films. Thesej~-values agree with the j,-values calculated from the maximummagnetization which is reached at the t
t
external field /~H.. Within the Bean model the field /~,Hp should agree with the penetration field ~ H . necessary for full flux ~netration. The measurements show that I~tt;, is smaller than/~H~, by a factor of about 2, This is understood by taking into account a field dependent critical current density. Furtheron a formula is presented for the magnetic stray-field in the center of a regular polygon with homogeneous thickness which gives also the penetration field/~H.. t
Keywords: Y~Ba~Cu~O thin lilms
1. introduction After zero field ct~ling of a type !i superconductor and application of magnetic fields, I~H,,, the magnetic flux penetrates in several steps into the sample. In the first step. Meissner shielding currents appear in the surfitce of the superconductor shielding the inner of the sample. Second° ly, tbr ~H~ above a geometryodependent critical field, the magnetic flux penetrates p~aly into the superconductor building up a flu~ front which moves towards the sample center as/Aft,, increases. A critical current density j,. flows in these flux filled regions. When I'.,H, is equal to the Soocalled penetration field./A~H;,, the flux front reaches the center of the supercondu,:ior and in a third step (/~H~> ~Hp) the whole sample fills up with magnetic flux and the critical current density decreases until ~ H . is equal to the upper critical field, where j¢ is :J,'ro. All existing currents j(r) produce a self-field. IA~H,~,(r). and a magnetization/A~M given by:
V x H,~.(r) = j(r),
(I) (2)
v
where V is the volume of the sample. In highoT~-supercon° *Corresponding author. ~25~,83881971517,~ © 1997 Elsevier Science S,A, All rights reserved Pll S0925-8388t 96102789.2
ductors, especially in Y-Ba=Cu=O, the contribution of the Meissner currents can be neglected for external magnetic fields much larger than the earth field. Only the contribution of the critical currents has to be taken into account. Models descri!~ing the currem distribution j(r) have to take into account that there exisls a Iowofield-region (/~1t,,< p~}H~,) where the flux tYonl has partly penetrated and a high=field=region (~H,, >/.t, Hp) where no flux free region exists. In the low-field-region, a critical current flows in the flux filled parts of the sample and inner shielding currents flow in the flux free parts. Under the assumption of Bean's critical state model [I], i,e. a field independent critical current density, and neglecting the surface Meissner currents, the inner shielding currents can be calculated for thin samples with different geometries [2,3]. Calculations of the magnetic flux density distribution (MFDD) agree well with measurements performed with the magneto-optical Faraday-effect [4]. Also, the calculated magnetization curves M(H,) show g ~ d agreement with measurements 151, except for the fact that the calculations yield no maximum l~r M(H,), since, within the above considerations, the flux never reaches the sample center. This lack can be avoided by calculating the magnetization and the MFDD of the high-field-region, i.e. by assuming a critical cur~nt density flowing in the whole sample volume, in this case the calculated MFDD agrees also well with investigations by the MOFE. Furtheron the calculated
147
A. Forkl et al. I Journal of AIIo.x:+ and Compottnds 25! (19971 146-149
magnetization should agree with the measured maximum m a g n e t i z a t i o n l/,oMmax and, following former considerations [6], the self-field tzoH~f(r = 01 in the sample center is equal to the penetration field/ZoHp. To prove this we have measured for the first time the MFDD and the magnetization curve of the same Y - B a C u - O films possessing different geometries, i.e. circular. triangular and square shaped samples.
2. Experimental
c-Axis oriented Y iBa2CU3OT_.,.-films with a thickness of 230 to 290 nm were evaporated onto SrTiO3-substrates by laser ablation [71. Then circular, triangular and quadratic samples were patterned by standard lithographic techniques. After evaporation of AI (thickness d^~ =200 nm) and EuSIEuF2 (thickness dE,=250 nm) the samples were cooled down in zero magnetic field from room temperature to Helium temperature. For this a cryostat was used, which is designed for light microscopical investigations. The polarization vector of a linearly polarized light beam passing through the evaporated magneto-optical EuS/EuF 2 layer is rotated by an angle which is proportional to the local magnetic flux density, B,, parallel to the light beam direction. In externally applied magnetic fields, the occurrence of magnetic flux is then seen as an increase of the intensity of the reflected light after passing through an analyzer. These changes are detected by a low light
L$
level video camera which is connected to a digital image processing system for quantitative analysis. The observed flux patterns are represented in Fig. l(a)-(c) for different geometries and a certain magnetic field applied perpendicular to the film plane. Bright areas belong to regions with penetrated magnetic flux. A detailed description of flux penetration and the according current distribution can be found in [81. The penetration field .uoHp can be seen directly in the microscope and is obtained by an accuracy of ___8mT. Furtheron, flux density profiles B~(r) are measured in the fully penetrated state. By comparison with calculated profiles critical current densities J¢.n are calculated. Magnetization curves were measured in a conventional Squid magnetometer at the same samples.
3. C a l c u l a t i o n s
For external fields applied perpendicular to the film plane, the critical current density distribution in the highfield-region can be assumed as shown in Fig. I(d)-(f). The currents flow parallel to the sample edge. in the case of the triangular and square shaped sample, the currents change direction at the angle bisectors (dashed line in r.~g. l(d) and Fig. I(e)). The self-field of such regular polygons with thickness D can be calculated by dividing the sample into N isosceles triangles in which the current flows homogeneously in one direction (N= 2, 3, 4 and ~ belongs to strip, triangle, square and circle). The self-field of such a basic brig is known analytically 181 and thus the sell-field of the whole sample is obtained by a superposition. In Fig. I(g)-(i) the calculated MFDD is represented. The magnetic field/~Jt,~,(r=0) in the center of such a regular polygon with thickness D is equal to the penetration tield ~H;, and is given according to Biot-Savart's law by p,,H,, = - - ~ "N
D I)'sin ~ ~ ,"r . 4R. alan 2R\ ¢~ + ..D. sin -~. In )
m
-
D
rr \,~l
+D
- I - ~ ' cos ~ +
[] (
~" O
rr)
1,
rr
~"
t °
• In
+ 2J/+\/
Fig. I. Experimentally determined magnetic flux density distribution pattern of a triangular, square and circular shaped YIBa,Cu~Ov ,-tilm alter zero-lield cooling and application of an external magnetic field ~H,, of (a) 13.5 roT, (b) 100.3 mT and (c) 150.7 mT. (d)-(f) show the current distribution in the high-field-region and accordingly, (g)-(i) the calculated flux density distribution•
rr
+ t - Yt~ .co, ~ * \/b
~(
n" D rr) D ~r 2. sil)~ + ~-. cos~ - ~-•sin ~.cos ~
(3)
where ~p= I+((DI2R).cos(xt/N))" and R the height of one isosceles triangle. The calculation of the magnetization of such a sample according to Eq. (2) yields ~M -
p~j~R 3
(4)
A. Forkl et ol. I Jourmd {#'Allo)'s and Coml~otmds 251 (1997) 14{5-149
148
Table I Experimentally determined values of the 3 different samples (thickness D, inner radius R, penetration field 14,Hp. maximum magnetization /.~Mm,,
appearing at pt,H'~. and saturation field/~,H,,~) Sample D (nm) ' R (p.m)
/~H;, (mT)
/4~M...... (T)
/~,H~ (mT)
p,H,~ (mT)
Triangle Square Circle
175 165 185
100.6 111.8 165.6
85 75 81
190 170 200
280 290 230
580 1000 IO00
4. Results and discussion Table 1 gives the thickness D and the inner radius R of the investigated triangular, circular and quadratic shaped sample together with the experimental results, i.e. the penetration field port, as obtained# by the MOFE investigations; the external field /~)Hp at which the maximum magnetization/~M,,~, appears. Comparison of the measured flux density profiles B,(r) with calculated profiles yield the critical current density j~.,. Furtheron, from the measured/~Mm,, and/tulip the critical current densities J,.M and J,.tt can be calculated according to Eq. (3) and Eq. (4). All these j,-values are listed in Table 2. We have also measured the remanent magnetization /~M, after field cooling of the sample, application of a maximum external field ~)H.,,..~ and reducing ~)H~ to zero. in Fig. 2 the remanent magnetization /~M, is represented in dependence of/~)H,,,.,.~. The determined jo/ovalues are smaller than the j~.~)values, The reason is that the calculated B,(r)-distribution shows a sharp peak in the center with a half-widtl0 of a few microns, which is near to the lateral resolution of the microscope. Within such a small distance the sharp ino c,'ease of B/(r), if it exists, can not be detected. So the critical currents calculated from/~1~ are underestimated. From the considerations of the Section I it is expected that the penetration field /~Ht, necessary for full flux penetration agrees within a certain accuracy with ~HI, at which ~ M reaches its maximum value/.~M,,,~. Obviously the measured values disagree by a factor of about 2. The reason is that with increasing flux deasity the critical current density decrea~s, especially at the border (see figure 6 in [91), With increasing external field /4~H, the inner currents also reach a critical value. This contribution to /4~M cannot compensate the decrease of the magnetization due to the decrease of the critical currents at the
border. So the maximum of M(H~) appears before the flux front reaches the sample center. The Jc.M" and it.B-values are comparable. They can be regarded as an averaged value. In the case of J:.M, the border currents contribute more to /.toM than the inner currents (see Eq. (2)), whereas in the case of Jc.tt, all currents contribute in the same way to the mean value. Both effects, i.e. change of current distribution and different averaging, lead to comparable Jc.t~- and j:.M-values. The remanent magnetization/~M,, obtained after application of a maximum external field/.toH~.m,~ and reducing H. to zero, shows an increase with increasing poH,,.m,~ until a saturation value/~)M,~, is reached at/4)H.,,,. With increasing /~)H,.,,,~, the flux front penetrates deeper into the sample, and thus the volume part in which critical currents flow increases until the flux front reaches the sample center. So the remanent magnetization should reach the saturation at/~Ho. This is obviously the case for all three samples (see Table I ). From measurements of/4~M, in dependence of/4~H,,.,,,,, the penetration field ~ H p can be measured.
5.
Summary
We have investigated the flux distribution and the thin
magnetization curve of superconducting Y - B a - C u - O
172
-po1~at.............~. l,i
"
i
168
~166
Sample
j,.n(10" Am
Triangle Square Circle
2.0 1.4 2.0
~)
j,.H(10" Am 1.2 0.8 1.6
:)
j, u(lO" 2. I 1.3 2.0
Am
:)
162
'%0
PoHaat
m I
164 'Fable 2 Values of the critical current density calculated from the measured flux profiles B,(r), ~netration field I~tt,, and the maxintum magnetization ~,M~, (for details see text)
tim " m . . o o
170
I
,oo ,so 2oo PoHa,n~
360 [roT]
Fig. 2. Remarmnt magnetization ~,M, of the circuhr structured fihn in dependence of the maximum applied magnetic fieldt~,H~......
A. Forkl et al. I Journal of Alloys and Compmmds 251 (1997) 146-149
films possessing different geometries. Critical currents are calculated from different characteristic features, i.e. the flux density profiles, the penetration fields and the maximum magnetizations. The different jc-values can be understood by taking into account that they represent an averaged mean value and that the local critical current density decreases with increasing local flux density. For the calculation ofjc from the penetration field//~Hp, a formula is given for regular polygons with thickness D.
References Ill C.E Bean, Phys. Rev. Lett.. 8 (1962) 250. [2l E.H. Brandt. M.V. lndenbom and A. Forkl, Europh3s. Lett.. 22 ( 1993} 735.
149
[3l P.N. Mikheenko and Yu.E. Kuzovlev, Physica C. 204 (1993) 229. [41 P. B:'fill, D, Kirchgtissner and P. Leiderer, Physica C, 182 { 1991 ) 339; R. Knorpp, A, Forkl, H.-U. Habermeier and H, Kronmiiller, Physica C, 230 (1994) 128; D. Glatzer, A. Forid, H. Theuss, H.-U. Habermeier and H. Kronmiiller, Physica status solidi (b). 170 (1992) 549. [5] Y. Mawatari, A. Sawa and H. Obara, Ph.vsica C, 258 (1996) 121. [6] A. Forkl, Phy~ica Scripta. T49 (1993) 148. 171 H.-U. Haberrneier, Eur. J. Solid State lnorg. Chem.. 28 (1991) 619, [8] A. Forkl and H. Kronmiiller, Phys. Rev. B, 52 (1996) 16 130. [9] H. Theuss, A. Forkl and H. Kronmfiller, Physica C. 190 (1992) 345.