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Magnetic response of disk shaped YBa2Cu3O7−δ thin film to a perpendicular magnetic field
PHYSICA0 ELSEVIER
Physica C 276 (1997) 167-172
Magnetic response of disk shaped YBa2Cu307_ ~ thin film to a perpendicular magnetic field M.J. Qin a, X.N. Xu a, A.M. Sun a, X.X. Yao a, j.p. Zuo b, S.Z. Yang b a Department of Physics and National Laboratory of Solid State Microstructures, Nanfing University, Center for Advanced Studies in Science and Technology of Microstructures, Nanfing 210093, China b Department of Electronic Science and Engineering, Nanfing University, Nanfing 210093, China
Received 28 November 1996; revised manuscript received 13 January 1997
Abstract
Magnetic measurements have been performed on a disk shaped YBa2Cu30 7_ 8 thin film in a perpendicular field. The critical state model developed by Mikheenko and Kuzovlev and Zhu et al. is extended to analyze the demagnetizing effects of the film. Using the field-dependent critical current density through the two-current model, the virgin magnetic moment as a function of the applied field and the hysteresis loops at several temperatures can be well fit by the extended model. The current density profiles and the flux density profiles of the film have been obtained and discussed. PACS: 77.75; 74.60J
1. Introduction
Since the discovery of high temperature superconductors, most magnetic data have been taken on thin films and single crystals (usually in the form of platelets resembling thin film rectangles or disks) with the 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 [1-4]. 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 qualita-
tively [5-8]. Because the theoretical model was not available, they used a numerical calculation to treat this problem instead. More recently, Mikheenko and Kuzovlev [9] presented an exact critical state model for thin superconducting disks in a perpendicular field. They took into account a current distribution in the vortex-free region of the disk (Meissner state), in addition to the current density I j l = J c, which is assumed constant and flows in the vortex-penetrated region. Zhu et al. [10] 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 the dc and ac response of superconducting disks and strips under a perpendicular field by taking into account the field-dependent critical current density
0921-4534/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved Pll S 0921-4534(97)00041-5
M.J. Qin et al. / Physica C 276 (1997) 167-172
168
[11,12]. However, this model is applicable only at low field and needs to be extended to describe experimental data at high field. In contrast to the theoretical works reported, there is a lack of systematic experimental data on thin films to relate the magnetic moment to its critical current density in this model. We have attempted to take these points into account in this paper by performing magnetic measurements on a disk shaped YBa2Cu307_ 8 thin film in a perpendicular field.
tion r = a, the current density distribution and zcomponent of the field in the film plane can be approximated [9] as the linear combination of Eq. (1), R
T
aJa l(r) =
d,
R r -Haf r F(7)W(l,
Ha)dl,
a
(2) '0,
2. Theoretical
model
H We start by considering a disk shaped thin film with radius R and thickness d where d << R. We suppose that the disk is in the x - y plane, centered on the z axis, and we use cylindrical coordinates r = (x 2 +y2)1/2, ~b = t a n - l ( y / x ) and z. When applying a weak field normal to the surface of the film (Ha ]1Z), if the effective field at the edge of the film is less than the lower critical field Hel, no vortex penetrates into the disk. Because of the strong demagnetizing effects, the flux lines are severely curved around the sample, resulting in azimuthal screening currents flowing over the entire surface of the disk. It can be shown that the induced current density, averaged over the film thickness and the z-component of the field in the film plane can be written as
I(r) = - H a F ( r / R ) , Hz(r, z ~ O) = Ha[1 + Q ( r / R ) ] ,
(1)
(4
~ ,
r
O,
r>R,
{i
r
)
2
=
__
1
1
r/R
~( r/R) 2 - 1
_sin-l__+
+ 1]W(I, Ha) dl,
a
+ 1]W(/, Ha) dl,
r>R,
(3) where W(l, H a) is a weight function and satisfies the normalization equation
L
Ha) dl = 1.
(4)
In the vortex penetrated region (a < r < R), the screening current density is equal to the critical current density R / r \ I ( r ) = - H a£ F t T ) W (l, Ha)dl = lc(Hz(r )), (5)
which can be used to obtain the weight function W(l, Ha). If It(r) is a function of the field, Eqs. (2)-(5) must be solved numerically. The magnetic moment of the film can be related to its critical current density through
r
=
F
r
HafaR[Q(ff)
a
where
Q[r,
Hz( r, z ---~O) =
r
r>R.
When the local field at the edge of the film exceeds Hc~, the vortex begins to penetrate into the disk. Supposing that vortices penetrate to the loca-
m = ~ f ; I ( r ) r 2 dr.
(6)
We now consider the situation when the applied field is increased from zero to the maximum value of H a and then is decreased to H b with [ H b I < Ha. In this case, the disk is free of flux in the region r < a, the flux density changes in the outer regions (b < r < R) but is unchanged in the inner regions (a < r <
169
M.J. Qin et al. / Physica C 276 (1997) 167-172
b). Accordingly, the current density and the z-component of the field in the film plane take a form like Eqs. (2) and (3), respectively,
I --Hbf : (Fr ) -[ G(I, Hb) dl, l(r) = [--lib
r
frRF(r)G(l, ' 7Hb) dl, a
The film with a radius R = 4 mm and thickness d -- 2000 .~ and T~ = 87 K was c-axis oriented, with the c-axis perpendicular to the disk's plane. Magnetic measurements have been carried out on a home-made vibrating sample magnetometer using a zero field cooled process. The applied field with a sweep rate of 10 G / s was applied normal to the surface of the film ( H a II c).
(7) 07
H Hz(r, z~O)=
r
( r + llG(I,
[Qt7)
_
4. Results and discussions
/-s,,) d,,
a
"r
r>R, (8) where G(l, Hb) is a new weight- function and also satisfies the normalization equation
faRG(l, Hb) dl =
1.
(9)
Shown in Figs. 1 and 2 , respectively, are the virgin magnetic moment as a function of the applied field and the hysteresis loops of the film at several temperatures indicated by different symbols. In order to fit Eqs. (2)-(10) to the experimental data, the field dependence of the critical current density should be employed. It is found that using the Kim type expression [14] I¢(H)= leo/(1 + I H [ / H o) cannot fit the virgin magnetic moment and the hysteresis loops well simultaneously; therefore, we choose the twocurrent model as discussed by Senoussi [15] for the
But now, we require two equations to solve for G(l, Ha), which come from the critical state model and our knowledge of the flux density in the inner regions ( a < r < b).
0.20
--HbfrRF(1)G(l, Hb) dl=Ie(Hz(r)), b < r < R,
(10)
Hbfar[Q(l) + l]G(l, Hb) dl=Hz(r), a
m can be derived from Eqs. (7), (8) and (6), respectively.
3. Experimental
E
0.15
0)
"E 60K
0~
E O.lO 0 E A
, 0~
0.0~
66K 691(
g 0.05
0
All experiments reported here have been performed on a disk shaped YBa2Cu307_ 8 thin film deposited on a single crystal SrTiO 3 substrate using the off-axis dc magnetron sputtering technique [13].
63.6K
~
50
100
150
200
250
Applied Field (Gs) Fig. I. The virgin magnetic moment of the film as a function of the applied field at several temperatures indicated by different symbols. Sofid lines am fits with Eqs. (2)-(6) and Eq. (11).
170
M.J. Qin et al./Physica C 276 (1997) 167-172
YBCO FilmHllc
m
51K
o
56.2K, ,K
"5" E
0,1
0
E olo O E ¢O "~
-0.1 L6K ~K ~K
-0.2
+
72.5K
t q2 K -3000
-2000
-1000
0
1000
2000
3000
Applied Field (Gs) Fig. 2. The hysteresis loops of the film at several temperatures indicated by different symbols. Solid lines a r e fits with Eqs. (2)-(11),
calculation of the hysteresis loop and the virgin magnetic moment
Iho
Ic(H)
1,o
= 1 + I H I/Hho + 1 + I H I / H ] 0 '
(11)
where Ih0 and 110 are independent of the field, Hho and H]o are characteristic fields. The first term on the right hand side would describe the high field behavior whereas the second one would correspond t o the low field domain. Another problem associated with the fitting procedure is that Eqs. (2)-(10) can be used only when a is larger than the maximum value of d, A and 2 A2/d where A is the London penetration depth and 2 A2/d is the two-dimensional screening length, because the cutoff length for Eq. (1) is d if d > A or 2A2/d if d < A. [16,17]. Therefore, Eqs. (2)-(10) are applicable only at low field. At high field, vortices penetrate into the center of the disk, a becomes smaller than d, A or 2A2/d; then Eqs. (2)-(10) break down. This condition can also be clearly seen from the physical meaning of Eq. (2). The linear combination of Eqs. (1) and (2) means that we can always treat the vortex free region ( r < a ) as thin film (a >> d), even when a has reached or is smaller than, the values of d, A or 2AZ/d. Therefore, in order to satisfy this assumption, a should always be larger than d, A or 2 AZ/d.
When fitting Eqs. (2)-(11) to the experimental data, it is found convenient to fix the location of the critical state region (the value of a) and treat the external field as unknown. With this fitting procedure, it is easy to extend the theoretical model of Mikheenko and Kuzovlev, and Zhu et al. to fit the experimental data at high field. When a has reached the film thickness d, we suppose that a further increase of the field will result in vortices penetrating into the center of the film, because the screening currents have been saturated and cannot completely screen the field higher than the value at which a = d. In other words, the region of r < d is regarded as a point. In this case we can fix the field at the center of the film and treat the external field as unknown. If a or the field at the center of the film is known, we start with an initial estimate of Ic(Hz(r))= constant, W(l, Ha) and then Hz(r) can be obtained numerically from Eqs. (5) and (3), respectively. With respect to the calculated value of Hz(r), Ic(Hz(r)) varies according to Eq. (11). Inserting the new expression for Ic(Hz(r)) into Eq. (5) and calculating iteratively until the critical current density Ic(Hz(r)) is self-consistent, then W(l, Ha) is determined by Eq. (5), and then I(r), Hz(r) and m can be derived from Eqs. (2), (3) and (6), respectively. Finally, the
3000
i
I ¥ eo
I
21)00 A
1500
~'N
5t~
1000
500
0
'
0
,
1000
,
2000
,
T7,2~
3000
Applied Field (Gs) Fig. 3. The calculated critical current densities of the film using Eq. (12) as a function of the applied field at several temperatures indicated by different symbols. Solid lines are calculated critical current densities as a function of the applied field using Eq. (11).
171
M.J. Qin et al. / Physica C 276 (1997) 1 6 7 - 1 7 2
applied field is determined by Eq: (4). The same fitting procedure is applied in the decreasing field case. Adjusting Ih0, Ii0, Hh0 and Hl0 as fitting parameters, we obtain the fitting results, shown as solid lines in Figs. 1 and 2 for the virgin magnetic moment and the hysteresis loops, respectively. The similarities are evident. Shown in Fig. 3 is the field dependence of the critical current densities at several temperatures indicated by different symbols, using the B e a n critical state model for a cylinder in an axial field
I c = 3 Am/2"rrR 3,
•
8000
i
I
A <=>
s \
T=51K 6000 11 i-
1 / t /-
~
{
,.-/
4000
/
/
}iL . ....
/
/ 2000
/
(12)
which is extensively used for the calculations of I~ in thin films. The solid lines in Fig. 3 are calculated Io(Hz) curves using Eq. (11). There is considerable agreement between these two results at high fields. At low fields smaller than about 500 Gs, however, disparities between these two results can be clearly seen from Fig. 3. These results can be understood by considering that at high fields, especially when the applied field has penetrated into the center of the film, the critical currents flow in the whole sample, and the flux lines are not so severely curved as at low fields. In this case, the self-field generated by the currents and therefore the demagnetizing effects can be neglected relative to the applied field. Then the Bean critical state model (Eq. (12)) can be used. However, at low applied fields, the self-field generated by the currents may become of the same order of magnitude as the applied field. Then some features different from a cylinder magnetized in a axial field (Eq. (12)) are expected. This can be seen clearly from Figs. 4a and b, in which we plotted the z-component of the flux density profiles and the current density profiles in the film at T = 51 K for several values of the low applied field, respectively. It can be seen from Fig. 4a that the field is strongly enhanced at the edge of the film. Because the apparent divergence of Hz(r) at r = R, the field at the edge of the film is infinitely larger for the zero thickness sample. For a real case of finite thickness d, the field is proportional to In(d/R). As can be seen from Fig. 4b, in the current density profiles there exist cusplike peaks at the values of r - - - a where Hz(r)= 0, which are caused by the field dependence of I c and a strong suppression of I~ at the edge where Hz(r) is large. Also, the cusplike
)
I YBC° Filrn HIIe
0 0,00
i
0.25
0 50
0.75
1.00
1.25
r/R
-500 ~i : "' "
"
" "
-'-'" T=51 K
-100o
~-15oo
[
',. ',
'~ : :
-2000
!
:
~.
"
"
',
:,
:
-25oo1-1~ ! ! IL -3ooo,, ," ~ 000
,,
0.25
. . . .
'.
',
~
~ .... ,
.
\,
.
~
•
..... ,
0,50
,
,
/
'-,--
0.75
4
4,00
r/R Fig. 4. (a) The calculated z-component of the flux density profiles of the film in the film plane (z = 0) at several low applied fields using Eq. (2)-(5). (b) The corresponding calculated current density profiles.
features in Fig. 4b will be rounded off on the length scale o f d i f A < d o r 2 A 2 / d i f d
5. Conclusions In summary, we have extended the approach of Mikheenko and Kuzovlev, and Zhu et al. to calculate the critical state behavior of the current density profiles, the flux density profiles, the virgin magnetic
172
M.J. Qin et al. / Physica C 276 (1997) 167-172
m o m e n t and the hysteresis loops o f a disk shaped s u p e r c o n d u c t i n g thin film. M a g n e t i c m e a s u r e m e n t s have been carried o u t on a d i s k s h a p e d Y B a 2 C u 3 0 7 - 8 thin film. T h e m a g n e t i c m o m e n t as a f u n c t i o n o f the applied field and the hysteresis loops o f the f i l m can be w e l l fitted by the a p p r o a c h described. T h e critical current density profiles and the z - c o m p o n e n t o f the flux density profiles h a v e been o b t a i n e d and discussed.
Acknoledgements T h i s project was supported by the National C e n t e r for R e s e a r c h and D e v e l o p m e n t on S u p e r c o n d u c t i v i t y o f China.
References [1] Y. Yeshurun, M.W. McElfresh, A.P. Malozemoff, J. Hagerhorst-Trewhella, J. Mannhart, F. Holtzberg and G.V. Chandrashekhar, Phys. Rev. B 42 (1990) 6322. [2] S. Senoussi, S. Hadjoudj, R. Maury and A. Fert, Physica C 165 (1990) 364.
[3] F.M. Sauerzopf, H.P. Wiesinger and H.W. Weber, Cryogenics 30 (1990) 650. [4] M. Wacenovsky, H.W. Weber, O.B. Hyun, D.K. Finnemore and K. Mereiter, Physica C 160 (1989) 55. [5] M. D~iumling and D.C. Larbalestier, Phys. Rev. B 40 (1989) 9350. [6] D.J. Frankel, J. Appl. Phys. 50 (1979) 540. [7] H. Theuss, A. Forkl and H. Kronmiiller, Physica C 190 (1992) 345. [8] L.W. Conner and A.P. Malozemoff, Phys. Rev. B 43 (1991) 402. [9] P.N. Mikheenko and Yu.E. Kuzovlev, Physica C 204 (1993) 229. [10] J. Zhu, J. Mester, J. Lockhart and J. Turneaure, Physica C 212 (1993) 216. [11] J.R. Clem and A. Sanchez, Phys. Rev. B 50 (1994) 9355. [12] J. McDonald and J.R. Clem, Phys. Rev. B 53 (1996) 8643. [13] H. Zhang, Z.J. Sun, S.Z. Yang, Z.M. Ji, P.H. WU, S.Y. Zhang, H.C. Zhang and Y. Hang, J. Vac. Sci. Technol. A 11 (1993) 390. [14] Y.B. Kim, C.F. Hempstead and A.R. Stmad, Phys. Rev. Lett. 9 (1962)306. [15] S. Senoussi, J. Phys. III 2 (1992) 1041. [16] J. Pearl, in: Proc. Ninth Int. Conf. on Low Temperature Physics, eds, J.G. Daunt, D.V. Edwards, F.J. Milford and M. Yaqub (Plenum, New York, 1965), Part A, p. 566; J. Pearl, Appl. Phys. Lett. 5 (1964) 65. [17] P.G. deGennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966).
Physica C 276 (1997) 167-172
Magnetic response of disk shaped YBa2Cu307_ ~ thin film to a perpendicular magnetic field M.J. Qin a, X.N. Xu a, A.M. Sun a, X.X. Yao a, j.p. Zuo b, S.Z. Yang b a Department of Physics and National Laboratory of Solid State Microstructures, Nanfing University, Center for Advanced Studies in Science and Technology of Microstructures, Nanfing 210093, China b Department of Electronic Science and Engineering, Nanfing University, Nanfing 210093, China
Received 28 November 1996; revised manuscript received 13 January 1997
Abstract
Magnetic measurements have been performed on a disk shaped YBa2Cu30 7_ 8 thin film in a perpendicular field. The critical state model developed by Mikheenko and Kuzovlev and Zhu et al. is extended to analyze the demagnetizing effects of the film. Using the field-dependent critical current density through the two-current model, the virgin magnetic moment as a function of the applied field and the hysteresis loops at several temperatures can be well fit by the extended model. The current density profiles and the flux density profiles of the film have been obtained and discussed. PACS: 77.75; 74.60J
1. Introduction
Since the discovery of high temperature superconductors, most magnetic data have been taken on thin films and single crystals (usually in the form of platelets resembling thin film rectangles or disks) with the 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 [1-4]. 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 qualita-
tively [5-8]. Because the theoretical model was not available, they used a numerical calculation to treat this problem instead. More recently, Mikheenko and Kuzovlev [9] presented an exact critical state model for thin superconducting disks in a perpendicular field. They took into account a current distribution in the vortex-free region of the disk (Meissner state), in addition to the current density I j l = J c, which is assumed constant and flows in the vortex-penetrated region. Zhu et al. [10] 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 the dc and ac response of superconducting disks and strips under a perpendicular field by taking into account the field-dependent critical current density
0921-4534/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved Pll S 0921-4534(97)00041-5
M.J. Qin et al. / Physica C 276 (1997) 167-172
168
[11,12]. However, this model is applicable only at low field and needs to be extended to describe experimental data at high field. In contrast to the theoretical works reported, there is a lack of systematic experimental data on thin films to relate the magnetic moment to its critical current density in this model. We have attempted to take these points into account in this paper by performing magnetic measurements on a disk shaped YBa2Cu307_ 8 thin film in a perpendicular field.
tion r = a, the current density distribution and zcomponent of the field in the film plane can be approximated [9] as the linear combination of Eq. (1), R
T
aJa l(r) =
d,
R r -Haf r F(7)W(l,
Ha)dl,
a
(2) '0,
2. Theoretical
model
H We start by considering a disk shaped thin film with radius R and thickness d where d << R. We suppose that the disk is in the x - y plane, centered on the z axis, and we use cylindrical coordinates r = (x 2 +y2)1/2, ~b = t a n - l ( y / x ) and z. When applying a weak field normal to the surface of the film (Ha ]1Z), if the effective field at the edge of the film is less than the lower critical field Hel, no vortex penetrates into the disk. Because of the strong demagnetizing effects, the flux lines are severely curved around the sample, resulting in azimuthal screening currents flowing over the entire surface of the disk. It can be shown that the induced current density, averaged over the film thickness and the z-component of the field in the film plane can be written as
I(r) = - H a F ( r / R ) , Hz(r, z ~ O) = Ha[1 + Q ( r / R ) ] ,
(1)
(4
~ ,
r
O,
r>R,
{i
r
)
2
=
__
1
1
r/R
~( r/R) 2 - 1
_sin-l__+
+ 1]W(I, Ha) dl,
a
+ 1]W(/, Ha) dl,
r>R,
(3) where W(l, H a) is a weight function and satisfies the normalization equation
L
Ha) dl = 1.
(4)
In the vortex penetrated region (a < r < R), the screening current density is equal to the critical current density R / r \ I ( r ) = - H a£ F t T ) W (l, Ha)dl = lc(Hz(r )), (5)
which can be used to obtain the weight function W(l, Ha). If It(r) is a function of the field, Eqs. (2)-(5) must be solved numerically. The magnetic moment of the film can be related to its critical current density through
r
=
F
r
HafaR[Q(ff)
a
where
Q[r,
Hz( r, z ---~O) =
r
r>R.
When the local field at the edge of the film exceeds Hc~, the vortex begins to penetrate into the disk. Supposing that vortices penetrate to the loca-
m = ~ f ; I ( r ) r 2 dr.
(6)
We now consider the situation when the applied field is increased from zero to the maximum value of H a and then is decreased to H b with [ H b I < Ha. In this case, the disk is free of flux in the region r < a, the flux density changes in the outer regions (b < r < R) but is unchanged in the inner regions (a < r <
169
M.J. Qin et al. / Physica C 276 (1997) 167-172
b). Accordingly, the current density and the z-component of the field in the film plane take a form like Eqs. (2) and (3), respectively,
I --Hbf : (Fr ) -[ G(I, Hb) dl, l(r) = [--lib
r
frRF(r)G(l, ' 7Hb) dl, a
The film with a radius R = 4 mm and thickness d -- 2000 .~ and T~ = 87 K was c-axis oriented, with the c-axis perpendicular to the disk's plane. Magnetic measurements have been carried out on a home-made vibrating sample magnetometer using a zero field cooled process. The applied field with a sweep rate of 10 G / s was applied normal to the surface of the film ( H a II c).
(7) 07
H Hz(r, z~O)=
r
( r + llG(I,
[Qt7)
_
4. Results and discussions
/-s,,) d,,
a
"r
r>R, (8) where G(l, Hb) is a new weight- function and also satisfies the normalization equation
faRG(l, Hb) dl =
1.
(9)
Shown in Figs. 1 and 2 , respectively, are the virgin magnetic moment as a function of the applied field and the hysteresis loops of the film at several temperatures indicated by different symbols. In order to fit Eqs. (2)-(10) to the experimental data, the field dependence of the critical current density should be employed. It is found that using the Kim type expression [14] I¢(H)= leo/(1 + I H [ / H o) cannot fit the virgin magnetic moment and the hysteresis loops well simultaneously; therefore, we choose the twocurrent model as discussed by Senoussi [15] for the
But now, we require two equations to solve for G(l, Ha), which come from the critical state model and our knowledge of the flux density in the inner regions ( a < r < b).
0.20
--HbfrRF(1)G(l, Hb) dl=Ie(Hz(r)), b < r < R,
(10)
Hbfar[Q(l) + l]G(l, Hb) dl=Hz(r), a
m can be derived from Eqs. (7), (8) and (6), respectively.
3. Experimental
E
0.15
0)
"E 60K
0~
E O.lO 0 E A
, 0~
0.0~
66K 691(
g 0.05
0
All experiments reported here have been performed on a disk shaped YBa2Cu307_ 8 thin film deposited on a single crystal SrTiO 3 substrate using the off-axis dc magnetron sputtering technique [13].
63.6K
~
50
100
150
200
250
Applied Field (Gs) Fig. I. The virgin magnetic moment of the film as a function of the applied field at several temperatures indicated by different symbols. Sofid lines am fits with Eqs. (2)-(6) and Eq. (11).
170
M.J. Qin et al./Physica C 276 (1997) 167-172
YBCO FilmHllc
m
51K
o
56.2K, ,K
"5" E
0,1
0
E olo O E ¢O "~
-0.1 L6K ~K ~K
-0.2
+
72.5K
t q2 K -3000
-2000
-1000
0
1000
2000
3000
Applied Field (Gs) Fig. 2. The hysteresis loops of the film at several temperatures indicated by different symbols. Solid lines a r e fits with Eqs. (2)-(11),
calculation of the hysteresis loop and the virgin magnetic moment
Iho
Ic(H)
1,o
= 1 + I H I/Hho + 1 + I H I / H ] 0 '
(11)
where Ih0 and 110 are independent of the field, Hho and H]o are characteristic fields. The first term on the right hand side would describe the high field behavior whereas the second one would correspond t o the low field domain. Another problem associated with the fitting procedure is that Eqs. (2)-(10) can be used only when a is larger than the maximum value of d, A and 2 A2/d where A is the London penetration depth and 2 A2/d is the two-dimensional screening length, because the cutoff length for Eq. (1) is d if d > A or 2A2/d if d < A. [16,17]. Therefore, Eqs. (2)-(10) are applicable only at low field. At high field, vortices penetrate into the center of the disk, a becomes smaller than d, A or 2A2/d; then Eqs. (2)-(10) break down. This condition can also be clearly seen from the physical meaning of Eq. (2). The linear combination of Eqs. (1) and (2) means that we can always treat the vortex free region ( r < a ) as thin film (a >> d), even when a has reached or is smaller than, the values of d, A or 2AZ/d. Therefore, in order to satisfy this assumption, a should always be larger than d, A or 2 AZ/d.
When fitting Eqs. (2)-(11) to the experimental data, it is found convenient to fix the location of the critical state region (the value of a) and treat the external field as unknown. With this fitting procedure, it is easy to extend the theoretical model of Mikheenko and Kuzovlev, and Zhu et al. to fit the experimental data at high field. When a has reached the film thickness d, we suppose that a further increase of the field will result in vortices penetrating into the center of the film, because the screening currents have been saturated and cannot completely screen the field higher than the value at which a = d. In other words, the region of r < d is regarded as a point. In this case we can fix the field at the center of the film and treat the external field as unknown. If a or the field at the center of the film is known, we start with an initial estimate of Ic(Hz(r))= constant, W(l, Ha) and then Hz(r) can be obtained numerically from Eqs. (5) and (3), respectively. With respect to the calculated value of Hz(r), Ic(Hz(r)) varies according to Eq. (11). Inserting the new expression for Ic(Hz(r)) into Eq. (5) and calculating iteratively until the critical current density Ic(Hz(r)) is self-consistent, then W(l, Ha) is determined by Eq. (5), and then I(r), Hz(r) and m can be derived from Eqs. (2), (3) and (6), respectively. Finally, the
3000
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I
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1500
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5t~
1000
500
0
'
0
,
1000
,
2000
,
T7,2~
3000
Applied Field (Gs) Fig. 3. The calculated critical current densities of the film using Eq. (12) as a function of the applied field at several temperatures indicated by different symbols. Solid lines are calculated critical current densities as a function of the applied field using Eq. (11).
171
M.J. Qin et al. / Physica C 276 (1997) 1 6 7 - 1 7 2
applied field is determined by Eq: (4). The same fitting procedure is applied in the decreasing field case. Adjusting Ih0, Ii0, Hh0 and Hl0 as fitting parameters, we obtain the fitting results, shown as solid lines in Figs. 1 and 2 for the virgin magnetic moment and the hysteresis loops, respectively. The similarities are evident. Shown in Fig. 3 is the field dependence of the critical current densities at several temperatures indicated by different symbols, using the B e a n critical state model for a cylinder in an axial field
I c = 3 Am/2"rrR 3,
•
8000
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I
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s \
T=51K 6000 11 i-
1 / t /-
~
{
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4000
/
/
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/
/ 2000
/
(12)
which is extensively used for the calculations of I~ in thin films. The solid lines in Fig. 3 are calculated Io(Hz) curves using Eq. (11). There is considerable agreement between these two results at high fields. At low fields smaller than about 500 Gs, however, disparities between these two results can be clearly seen from Fig. 3. These results can be understood by considering that at high fields, especially when the applied field has penetrated into the center of the film, the critical currents flow in the whole sample, and the flux lines are not so severely curved as at low fields. In this case, the self-field generated by the currents and therefore the demagnetizing effects can be neglected relative to the applied field. Then the Bean critical state model (Eq. (12)) can be used. However, at low applied fields, the self-field generated by the currents may become of the same order of magnitude as the applied field. Then some features different from a cylinder magnetized in a axial field (Eq. (12)) are expected. This can be seen clearly from Figs. 4a and b, in which we plotted the z-component of the flux density profiles and the current density profiles in the film at T = 51 K for several values of the low applied field, respectively. It can be seen from Fig. 4a that the field is strongly enhanced at the edge of the film. Because the apparent divergence of Hz(r) at r = R, the field at the edge of the film is infinitely larger for the zero thickness sample. For a real case of finite thickness d, the field is proportional to In(d/R). As can be seen from Fig. 4b, in the current density profiles there exist cusplike peaks at the values of r - - - a where Hz(r)= 0, which are caused by the field dependence of I c and a strong suppression of I~ at the edge where Hz(r) is large. Also, the cusplike
)
I YBC° Filrn HIIe
0 0,00
i
0.25
0 50
0.75
1.00
1.25
r/R
-500 ~i : "' "
"
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-'-'" T=51 K
-100o
~-15oo
[
',. ',
'~ : :
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~.
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',
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:
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.
\,
.
~
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0,50
,
,
/
'-,--
0.75
4
4,00
r/R Fig. 4. (a) The calculated z-component of the flux density profiles of the film in the film plane (z = 0) at several low applied fields using Eq. (2)-(5). (b) The corresponding calculated current density profiles.
features in Fig. 4b will be rounded off on the length scale o f d i f A < d o r 2 A 2 / d i f d
5. Conclusions In summary, we have extended the approach of Mikheenko and Kuzovlev, and Zhu et al. to calculate the critical state behavior of the current density profiles, the flux density profiles, the virgin magnetic
172
M.J. Qin et al. / Physica C 276 (1997) 167-172
m o m e n t and the hysteresis loops o f a disk shaped s u p e r c o n d u c t i n g thin film. M a g n e t i c m e a s u r e m e n t s have been carried o u t on a d i s k s h a p e d Y B a 2 C u 3 0 7 - 8 thin film. T h e m a g n e t i c m o m e n t as a f u n c t i o n o f the applied field and the hysteresis loops o f the f i l m can be w e l l fitted by the a p p r o a c h described. T h e critical current density profiles and the z - c o m p o n e n t o f the flux density profiles h a v e been o b t a i n e d and discussed.
Acknoledgements T h i s project was supported by the National C e n t e r for R e s e a r c h and D e v e l o p m e n t on S u p e r c o n d u c t i v i t y o f China.
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[3] F.M. Sauerzopf, H.P. Wiesinger and H.W. Weber, Cryogenics 30 (1990) 650. [4] M. Wacenovsky, H.W. Weber, O.B. Hyun, D.K. Finnemore and K. Mereiter, Physica C 160 (1989) 55. [5] M. D~iumling and D.C. Larbalestier, Phys. Rev. B 40 (1989) 9350. [6] D.J. Frankel, J. Appl. Phys. 50 (1979) 540. [7] H. Theuss, A. Forkl and H. Kronmiiller, Physica C 190 (1992) 345. [8] L.W. Conner and A.P. Malozemoff, Phys. Rev. B 43 (1991) 402. [9] P.N. Mikheenko and Yu.E. Kuzovlev, Physica C 204 (1993) 229. [10] J. Zhu, J. Mester, J. Lockhart and J. Turneaure, Physica C 212 (1993) 216. [11] J.R. Clem and A. Sanchez, Phys. Rev. B 50 (1994) 9355. [12] J. McDonald and J.R. Clem, Phys. Rev. B 53 (1996) 8643. [13] H. Zhang, Z.J. Sun, S.Z. Yang, Z.M. Ji, P.H. WU, S.Y. Zhang, H.C. Zhang and Y. Hang, J. Vac. Sci. Technol. A 11 (1993) 390. [14] Y.B. Kim, C.F. Hempstead and A.R. Stmad, Phys. Rev. Lett. 9 (1962)306. [15] S. Senoussi, J. Phys. III 2 (1992) 1041. [16] J. Pearl, in: Proc. Ninth Int. Conf. on Low Temperature Physics, eds, J.G. Daunt, D.V. Edwards, F.J. Milford and M. Yaqub (Plenum, New York, 1965), Part A, p. 566; J. Pearl, Appl. Phys. Lett. 5 (1964) 65. [17] P.G. deGennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966).