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Thick superconductors in a perpendicular magnetic field
PHYSICA ELSEVIER
Physica C 282-287 (1997) 343-346
Thick Superconductors in a Perpendicular Magnetic Field Ernst Helmut Brandt ~ aMax-Planck-Institut fiir MetMlforschung, D-70506 Stuttgart, Germany The magnetic moment, ac susceptibility, current density J(r, t) and magnetic field B(r, t) in superconductors may be calculated for various geometries and material laws by time-integration of a one- or two-dimensional
spatial integral ore/" J(r, t). This is shown for disks or cylinders of arbitrary thickness in axial magnetic field.
1. I N T R O D U C T I O N The determination of the electromagnetic properties of type-II superconducting materials from experiments requires reliable calculations of current and flux distributions in the superconductor and of the resulting magnetic moment or ac susceptibility. In most cases a continuum description is sufficient, obtained by averaging the microscopic currents and magnetic and electric fields over the Abrikosov vortices. Such continuum calculations may be classified according to which material laws are used and which geometry is considered.
2. M A T E R I A L
LAWS
2.1. Reversible magnetization
curve The static linear and nonlinear magnetic response of a reversible material is characterized'by a law B = B ( H ) where B is the average induction (flux density) in a volume element which sees an applied field H. For isotropic response one may write B -- B ( H ) H / H = p ( H ) H with p(H) = B ( H ) / H and B(H) = aF/aB, where F(B) is the average free energy density. For reversible type-II superconductors one has B(H) = 0 at H < H~I (lower critical field) and B(H) = poll at H > H~2 (upper critical field). For superconductors with large Ginzburg-Landau parameter t¢ one has Hcl << He2 ~, (2tc2/lntc)H~l and B ~ poll at H >> He1. Below we are mainly interested in the irreversible and time (t) dependent responses. For this one requires an additional material law, e.g., 0921-4534/97/$17.00 © Elsevier Science B.V. All rights reserved PII S0921-4534(97)00261-X
the current-voltage curve. In fact, in irreversible or dynamic problems in many cases (if everywhere in the superconductor B > 2p0H¢l) a good approximation is B = poll, which is equivalent to assuming He1 = 0. T h e magnetic moment m of the superconductor is then given by m ( t ) = !2 fr x J ( r , t ) d a r . From m ( t ) one may derive an ac susceptibility Xo¢(w, H0) if Ha(t) = Ho sinwt. 2.2. B e a n m o d e l A very useful and often realistic material law is the static Bean model [1] which assumes that the vortices do not move as long as J = IJI is less than a critical current density Jc. If J > Jc, the vortices rearrange such that J < Jc everywhere. In the simple Bean model Jc is assumed to be constant, but in general J~(B) depends on B. Of course, all material laws will depend also on the temperature T, but this does not affect isothermal calculations. 2.3. Nonlinear resistivity
A more general dynamic description uses the current-voltage characteristics E = E ( J ) where E = B x v is the local electric field, which is caused by the motion of vortices with velocity v (if annihilation of the vortices is disregarded). In the isotropic case and disregarding the Hall effect ones has E = E ( J ) J / J . A useful and realistic model is the power law E(J) = Ee(d/Jc) n with in general Je = Je(B) and n = n(B). Typically one has n >> 1. In the limit n -~ co this power law yields the Bean model ( E = 0 for J < d c and E = oo for or > Je), while n = 1 means an Ohmic conductor with constant and real re-
344
E.I-L Brandt/Physica C 282-287 (1997) 343-346
sistivity p = E I J = E¢/Jc. For E c( j n a scaling law applies which states that the susceptibility Xae(w, 11o) in general depends only on a combination of frequency and amplitude, e.g., on Ho/w 1/° +w/H~ with a = n - 1. This scaling law nicely shows the transition from Xac(W) (Ohm, ~ = 0) to Xac(Ho) (Bean, ~ -+ co). 2.4. L i n e a r c o m p l e x r e s i s t i v i t y A special case of E(J) is E0 = pac(W)Jo where Pac = P~+ ip" is a linear complex dispersive resistivity and Ha(t) : Hosinwt, E = Re{Eoei'°t}, and J --- Re{Joe i'°t} are assumed, with E0(r) and J0(r) denoting complex amplitudes and Re the real part. So far we are not able to combine the nonlinear real p(J) = E ( J ) / J with the linear complex pae(w) into one general theory, but both resistivities may be used in the calculations below. The Meissner state with complete expulsion of magnetic flux is then realized, in spite of our assumption H~I = 0, either at low Ha if E(J) cx jn with n >> 1, or, if E / J = pa¢(w), at short times [for suddenly switched on Ha(t)] or high frequencies [for periodic Ha(t)].
the sheet current J , = fbbJ(r,_ t)dy which describe 1D nonlocal diffusion of J~. Analytic solutions were obtained [3, 4] for the static Bean model extended to these 1D transverse geometries. The presence of a transport current in addition to the circulating screening current in a Bean strip was considered in [5]. These thin-limit results qualitatively differ from the longitudinal geometry. In particular, the sheet current does not vanish in the flux-free central zone and the perpendicular field Bz (r) exhibits a logarithmic infinity at the specimen edge and a vertical slope at the penetrating flux front, while the longitudinal Bean model predicts constant slope of Bz (r). The extension of the limit d --+ 0 to finite thickness given below, can explain these features The dynamic linear [6] and nonlinear [2] response, flux penetration [7] and flux creep [8] were computed by time integration of a 1D spatial integral for strips and disks. The linear susceptibility Xae(W) was calculated for general complex resistivity [9] pae(w). These linear and nonlinear results recently were generalized to thin rings [2] and to the linear and nonlinear screening of a thin supercoflductor disk between two coils [10].
3. G E O M E T R I E S Various specimen shapes and orientations of the applied field Ha are used in experiments. The general three-dimensional (3D) problem in many cases may be approximated by a two- or even onedimensional problem. If not specified else, we assume here that no current is fed in by contacts. 3.1. P a r a l l e l field, 1D a n d 2D Infinitely long superconductors in parallel field Ha exhibit no demagnetizing effects. For infinite slabs or circular cylinders the problem is 1D, and for general cross-section (e.g. rectangular bars) it is 2D. The linear or nonlinear response is now obtained by solving a 1D or 2D linear or nonlinear diffusion equation. The resulting expressions are often used to evaluate experiments even when this longitudinal geometry does not apply. 3.2. P e r p e n d i c u l a r field, 1D Thin infinite strips and circular disks of width 2a and thickness d = 2b << 2a in perpendicular HallY are described by 1D integral equations for
3.3. P e r p e n d i c u l a r field, 2D Thin flat superconductors of arbitrary shape in transverse (or arbitrarily oriented) Ha require the time integration of a 2D integral equation for the sheet current which describes nonlocal diffusion. Such first-principle computations were performed for thin squares and rectangles [11]. From the measured flux density By(z, z) at the surface of a superconductor the sheet current may be obtained by inversion methods [12]. 3.4. Bars and disks o f arbitrary thickness Infinite bars and circular disks of finite thickness in a perpendicular (or axial) field yield a 2D integral equation which still m a y be timeintegrated on a Personal Computer. This method [13] applies to arbitrary cross-section (e.g. to spheres and elliptic bars) but here we consider only rectangular cross-section 2a x 2b which includes, e.g., thin disks (b << a) and long cylinders (b >> a). For brevity we consider here only cylindrical symmetry. The similar results for strips,
E.H. Brandt/Physica C 282-287 (1997) 343-346
345
bars, and slabs are given in [13]. 4. T H I C K D I S K S I N A X I A L F I E L D The vector potential A ( r , t ) = ~A(r,y,t) caused by the current density J ( r , t) = ~SJ(r, y, t) circulating in a disk of radius a and thickness d = 2b and by the applied field Ba(t) = #oH~(I) (along y) is [13] A(r)=-#0
f0°dr 'f0dy'Qcyl(r,r')J(r') b
°.
r -TBo (1)
with r = (r, y) and r' = (r', y'). The kernel Qcyl(r, r') = f(r, r', y - y') + f(r, r', y + y'),
f(r,
7) =
f0
+
+
cos d: + 2,r'cos
(2)
was obtained by integrating the Green function of the 3D Laplace equation, 1/(47rlra - r ' a D with r3 = (=,y, z), over the angle ~o = arctan(z/x). If desired, f(r, r I, 11) may be expressed in terms of elliptic integrals, but we found it more convenient to evaluate the ~o integral numerically. With B = V x A the induction law B = - V x E takes the form .4 = - E ( J ) since E ( r , t ) = ~E(r, y, t). To fix ideas one may use E(J) = Ee(J/J¢) n. The equation of motion for J(r, y, t) is then E[g(r,t)] = p
/d
7,1 . 2r'Qcyl(r,r')J(r',t) + ~ B , . (3)
This implicit equation may be inverted, J
(41
with E = E [ J ( r ' , t ) ] = EcJ(r', y',t)n/J:. The 1 inverse kernel Q~-yl(r, r )I may be computed by tabulating Qcyl(r, r') (2) on a discrete grid ri, rj and inverting this matrix. Figures 1 and 2 show the magnetic field lines during flux penetration into thick superconducting disks with aspect ratios b/a = 1 and b/a = 0.25, calculated by time integration of Eq. (4) with an exponent n : 51 (almost Bean limit). The depicted applied field values are Ha/Hp = 0.1, 0.2, 0.4, and 0.8, where
Hp =Jobln[
-a
-}-
(1 -}- a 2
1/2]
(5)
Figure 1. Field lines of the magnetic field penetrating into a thick Bean disk with b/a = 1 at applied field~ Ha/Hp = 0.1, 0.2, 0.4, 0.8. The bold line marks the flux front, which separates the central core where B = J = 0 from the outer region with circulating current J = Jc. is the field of full penetration [15] for thick disks and cylinders. These field lines look very similar to those of thick strips [13] with same b/a. Note that the lense-shaped field- and current-free central zone becomes isolated for H~ close to Hp. Figure 3 shows the profiles of the perpendicular induction at the surface, By(r, [y[ = b), and at the middle plane, Bu(r , y = 0), of a Bean disk with b/a = 0.25 in increasing applied field. For this thick disk the profiles at the surface still look similar to the profiles of a long Bean cylinder in parallel field (straight parallel lines), but the profiles on the central plane look more like in a thin disk in perpendicular field [3] (with steep front and with a cusp at the edge). The resulting magnetization curves re(Ha) fall between the limits for long cylinders [1,2] and thin disks [3,2]. When normalized to unity initial slope m~(0) = 1 and unity saturation value m(H~ >
E.H. Brandt/Physica C 282-287 (1997) 343-346
346
0.6
y(,.,b), By(,',
0) i::::.........
O..S 0.4
0.3 0.2 O.t 0
•
B
Figure 2. As Fig. 1 but for
b/a = 0.25.
Hp) = 1 the magnetic moments re(Ha) for all aspect ratios b/a differ only little. REFERENCES
1. C.P. Bean, Rev. Mod. Phys. 36, 31 (1964). 2. E.H. Brandt, Phys. Rev. B (submitted). 3. P.N. Mikheenko and Yu.E. Kuzovlev, Physica C 204, 229 (1993). 4. E.H. Brandt, M. Indenbom, and A. Forkl, Europhys. Lett. 22, 735 (1993). 5. E. Zeldov et al., Phys. Key. B 49, 9802 (1994); E.H. Brandt and M. Indenbom, Phys. Rev. B 45, 12893 (1993). 6. E.H. Brandt, Phys. Rev. Lett. 71, 2821 (1993); Phys. Rev. B 49, 9024 (1994); 50,
4034 (1994).
-I
-0.5
0
r/a
Figure 3. The profiles of the perpendicular magnetic field By at the surface y = =t:b (solid lines) and in the central plane y = 0 (dashed lines) of a thick disk with b/a -- 0.25 (like Fig. 2) during flux penetration. The depicted applied field values are (from bottom to top) Ha/Hp - 0.1, 0.2, 0.4, 0.6, 0.8, 0.9, 1.05. B is in units ~oJea. 7. Th. Schuster, Phys. Rev. Lett. 73, 1424 (1994); Phys. Rev. B 50, 16684 (1994). 8. A. Gurevich and E.H. Brandt, Phys. Rev. Lett. 73, 178 (1994); Phys. Rev. B (submitted); E.H. Brandt and A. Gurevich, Phys. Rev. Lett. 76, 1723 (1996); E.H. Brandt, Phys. Rev. Lett. 76, 4030 (1996). 9. J. K5tzler et al., Phys. Rev. B 50, 3384 (1994); E.H. Brandt, Phys. Rev. B 50, 13833
(1994). 10. J. Gilchrist and E.H. Brandt, Phys. Rev. B 54, 3530 (1996). 11. E.H. Brandt, Phys. Rev. Lett. 74, 3025 (1995); Phys. Rev. S 52, 15442 (1995). 12. W. Xing et al., J. Appl. Phys. 76, 4244 (1994); R. Wijngaarden et hi., Phys. Rev. B 54, 6742 (1996); E.H. Brandt, Phys. Rev. B 46, 8628 (1992); A.E. Pashitski, A. Gurevich, et al. Science (Jan. 1997, in print); Ch. Joos et al., Phys. Rev. B (submitted). 13. E.H. Brandt, Phys. Rev. B 54, 4246 (1996). 14. E.H. Brandt, Rep. Prog. Phys. 58, 1465 (1995). 15. A. Forkl, Physica Scripta T49, 148 (1993).
Physica C 282-287 (1997) 343-346
Thick Superconductors in a Perpendicular Magnetic Field Ernst Helmut Brandt ~ aMax-Planck-Institut fiir MetMlforschung, D-70506 Stuttgart, Germany The magnetic moment, ac susceptibility, current density J(r, t) and magnetic field B(r, t) in superconductors may be calculated for various geometries and material laws by time-integration of a one- or two-dimensional
spatial integral ore/" J(r, t). This is shown for disks or cylinders of arbitrary thickness in axial magnetic field.
1. I N T R O D U C T I O N The determination of the electromagnetic properties of type-II superconducting materials from experiments requires reliable calculations of current and flux distributions in the superconductor and of the resulting magnetic moment or ac susceptibility. In most cases a continuum description is sufficient, obtained by averaging the microscopic currents and magnetic and electric fields over the Abrikosov vortices. Such continuum calculations may be classified according to which material laws are used and which geometry is considered.
2. M A T E R I A L
LAWS
2.1. Reversible magnetization
curve The static linear and nonlinear magnetic response of a reversible material is characterized'by a law B = B ( H ) where B is the average induction (flux density) in a volume element which sees an applied field H. For isotropic response one may write B -- B ( H ) H / H = p ( H ) H with p(H) = B ( H ) / H and B(H) = aF/aB, where F(B) is the average free energy density. For reversible type-II superconductors one has B(H) = 0 at H < H~I (lower critical field) and B(H) = poll at H > H~2 (upper critical field). For superconductors with large Ginzburg-Landau parameter t¢ one has Hcl << He2 ~, (2tc2/lntc)H~l and B ~ poll at H >> He1. Below we are mainly interested in the irreversible and time (t) dependent responses. For this one requires an additional material law, e.g., 0921-4534/97/$17.00 © Elsevier Science B.V. All rights reserved PII S0921-4534(97)00261-X
the current-voltage curve. In fact, in irreversible or dynamic problems in many cases (if everywhere in the superconductor B > 2p0H¢l) a good approximation is B = poll, which is equivalent to assuming He1 = 0. T h e magnetic moment m of the superconductor is then given by m ( t ) = !2 fr x J ( r , t ) d a r . From m ( t ) one may derive an ac susceptibility Xo¢(w, H0) if Ha(t) = Ho sinwt. 2.2. B e a n m o d e l A very useful and often realistic material law is the static Bean model [1] which assumes that the vortices do not move as long as J = IJI is less than a critical current density Jc. If J > Jc, the vortices rearrange such that J < Jc everywhere. In the simple Bean model Jc is assumed to be constant, but in general J~(B) depends on B. Of course, all material laws will depend also on the temperature T, but this does not affect isothermal calculations. 2.3. Nonlinear resistivity
A more general dynamic description uses the current-voltage characteristics E = E ( J ) where E = B x v is the local electric field, which is caused by the motion of vortices with velocity v (if annihilation of the vortices is disregarded). In the isotropic case and disregarding the Hall effect ones has E = E ( J ) J / J . A useful and realistic model is the power law E(J) = Ee(d/Jc) n with in general Je = Je(B) and n = n(B). Typically one has n >> 1. In the limit n -~ co this power law yields the Bean model ( E = 0 for J < d c and E = oo for or > Je), while n = 1 means an Ohmic conductor with constant and real re-
344
E.I-L Brandt/Physica C 282-287 (1997) 343-346
sistivity p = E I J = E¢/Jc. For E c( j n a scaling law applies which states that the susceptibility Xae(w, 11o) in general depends only on a combination of frequency and amplitude, e.g., on Ho/w 1/° +w/H~ with a = n - 1. This scaling law nicely shows the transition from Xac(W) (Ohm, ~ = 0) to Xac(Ho) (Bean, ~ -+ co). 2.4. L i n e a r c o m p l e x r e s i s t i v i t y A special case of E(J) is E0 = pac(W)Jo where Pac = P~+ ip" is a linear complex dispersive resistivity and Ha(t) : Hosinwt, E = Re{Eoei'°t}, and J --- Re{Joe i'°t} are assumed, with E0(r) and J0(r) denoting complex amplitudes and Re the real part. So far we are not able to combine the nonlinear real p(J) = E ( J ) / J with the linear complex pae(w) into one general theory, but both resistivities may be used in the calculations below. The Meissner state with complete expulsion of magnetic flux is then realized, in spite of our assumption H~I = 0, either at low Ha if E(J) cx jn with n >> 1, or, if E / J = pa¢(w), at short times [for suddenly switched on Ha(t)] or high frequencies [for periodic Ha(t)].
the sheet current J , = fbbJ(r,_ t)dy which describe 1D nonlocal diffusion of J~. Analytic solutions were obtained [3, 4] for the static Bean model extended to these 1D transverse geometries. The presence of a transport current in addition to the circulating screening current in a Bean strip was considered in [5]. These thin-limit results qualitatively differ from the longitudinal geometry. In particular, the sheet current does not vanish in the flux-free central zone and the perpendicular field Bz (r) exhibits a logarithmic infinity at the specimen edge and a vertical slope at the penetrating flux front, while the longitudinal Bean model predicts constant slope of Bz (r). The extension of the limit d --+ 0 to finite thickness given below, can explain these features The dynamic linear [6] and nonlinear [2] response, flux penetration [7] and flux creep [8] were computed by time integration of a 1D spatial integral for strips and disks. The linear susceptibility Xae(W) was calculated for general complex resistivity [9] pae(w). These linear and nonlinear results recently were generalized to thin rings [2] and to the linear and nonlinear screening of a thin supercoflductor disk between two coils [10].
3. G E O M E T R I E S Various specimen shapes and orientations of the applied field Ha are used in experiments. The general three-dimensional (3D) problem in many cases may be approximated by a two- or even onedimensional problem. If not specified else, we assume here that no current is fed in by contacts. 3.1. P a r a l l e l field, 1D a n d 2D Infinitely long superconductors in parallel field Ha exhibit no demagnetizing effects. For infinite slabs or circular cylinders the problem is 1D, and for general cross-section (e.g. rectangular bars) it is 2D. The linear or nonlinear response is now obtained by solving a 1D or 2D linear or nonlinear diffusion equation. The resulting expressions are often used to evaluate experiments even when this longitudinal geometry does not apply. 3.2. P e r p e n d i c u l a r field, 1D Thin infinite strips and circular disks of width 2a and thickness d = 2b << 2a in perpendicular HallY are described by 1D integral equations for
3.3. P e r p e n d i c u l a r field, 2D Thin flat superconductors of arbitrary shape in transverse (or arbitrarily oriented) Ha require the time integration of a 2D integral equation for the sheet current which describes nonlocal diffusion. Such first-principle computations were performed for thin squares and rectangles [11]. From the measured flux density By(z, z) at the surface of a superconductor the sheet current may be obtained by inversion methods [12]. 3.4. Bars and disks o f arbitrary thickness Infinite bars and circular disks of finite thickness in a perpendicular (or axial) field yield a 2D integral equation which still m a y be timeintegrated on a Personal Computer. This method [13] applies to arbitrary cross-section (e.g. to spheres and elliptic bars) but here we consider only rectangular cross-section 2a x 2b which includes, e.g., thin disks (b << a) and long cylinders (b >> a). For brevity we consider here only cylindrical symmetry. The similar results for strips,
E.H. Brandt/Physica C 282-287 (1997) 343-346
345
bars, and slabs are given in [13]. 4. T H I C K D I S K S I N A X I A L F I E L D The vector potential A ( r , t ) = ~A(r,y,t) caused by the current density J ( r , t) = ~SJ(r, y, t) circulating in a disk of radius a and thickness d = 2b and by the applied field Ba(t) = #oH~(I) (along y) is [13] A(r)=-#0
f0°dr 'f0dy'Qcyl(r,r')J(r') b
°.
r -TBo (1)
with r = (r, y) and r' = (r', y'). The kernel Qcyl(r, r') = f(r, r', y - y') + f(r, r', y + y'),
f(r,
7) =
f0
+
+
cos d: + 2,r'cos
(2)
was obtained by integrating the Green function of the 3D Laplace equation, 1/(47rlra - r ' a D with r3 = (=,y, z), over the angle ~o = arctan(z/x). If desired, f(r, r I, 11) may be expressed in terms of elliptic integrals, but we found it more convenient to evaluate the ~o integral numerically. With B = V x A the induction law B = - V x E takes the form .4 = - E ( J ) since E ( r , t ) = ~E(r, y, t). To fix ideas one may use E(J) = Ee(J/J¢) n. The equation of motion for J(r, y, t) is then E[g(r,t)] = p
/d
7,1 . 2r'Qcyl(r,r')J(r',t) + ~ B , . (3)
This implicit equation may be inverted, J
(41
with E = E [ J ( r ' , t ) ] = EcJ(r', y',t)n/J:. The 1 inverse kernel Q~-yl(r, r )I may be computed by tabulating Qcyl(r, r') (2) on a discrete grid ri, rj and inverting this matrix. Figures 1 and 2 show the magnetic field lines during flux penetration into thick superconducting disks with aspect ratios b/a = 1 and b/a = 0.25, calculated by time integration of Eq. (4) with an exponent n : 51 (almost Bean limit). The depicted applied field values are Ha/Hp = 0.1, 0.2, 0.4, and 0.8, where
Hp =Jobln[
-a
-}-
(1 -}- a 2
1/2]
(5)
Figure 1. Field lines of the magnetic field penetrating into a thick Bean disk with b/a = 1 at applied field~ Ha/Hp = 0.1, 0.2, 0.4, 0.8. The bold line marks the flux front, which separates the central core where B = J = 0 from the outer region with circulating current J = Jc. is the field of full penetration [15] for thick disks and cylinders. These field lines look very similar to those of thick strips [13] with same b/a. Note that the lense-shaped field- and current-free central zone becomes isolated for H~ close to Hp. Figure 3 shows the profiles of the perpendicular induction at the surface, By(r, [y[ = b), and at the middle plane, Bu(r , y = 0), of a Bean disk with b/a = 0.25 in increasing applied field. For this thick disk the profiles at the surface still look similar to the profiles of a long Bean cylinder in parallel field (straight parallel lines), but the profiles on the central plane look more like in a thin disk in perpendicular field [3] (with steep front and with a cusp at the edge). The resulting magnetization curves re(Ha) fall between the limits for long cylinders [1,2] and thin disks [3,2]. When normalized to unity initial slope m~(0) = 1 and unity saturation value m(H~ >
E.H. Brandt/Physica C 282-287 (1997) 343-346
346
0.6
y(,.,b), By(,',
0) i::::.........
O..S 0.4
0.3 0.2 O.t 0
•
B
Figure 2. As Fig. 1 but for
b/a = 0.25.
Hp) = 1 the magnetic moments re(Ha) for all aspect ratios b/a differ only little. REFERENCES
1. C.P. Bean, Rev. Mod. Phys. 36, 31 (1964). 2. E.H. Brandt, Phys. Rev. B (submitted). 3. P.N. Mikheenko and Yu.E. Kuzovlev, Physica C 204, 229 (1993). 4. E.H. Brandt, M. Indenbom, and A. Forkl, Europhys. Lett. 22, 735 (1993). 5. E. Zeldov et al., Phys. Key. B 49, 9802 (1994); E.H. Brandt and M. Indenbom, Phys. Rev. B 45, 12893 (1993). 6. E.H. Brandt, Phys. Rev. Lett. 71, 2821 (1993); Phys. Rev. B 49, 9024 (1994); 50,
4034 (1994).
-I
-0.5
0
r/a
Figure 3. The profiles of the perpendicular magnetic field By at the surface y = =t:b (solid lines) and in the central plane y = 0 (dashed lines) of a thick disk with b/a -- 0.25 (like Fig. 2) during flux penetration. The depicted applied field values are (from bottom to top) Ha/Hp - 0.1, 0.2, 0.4, 0.6, 0.8, 0.9, 1.05. B is in units ~oJea. 7. Th. Schuster, Phys. Rev. Lett. 73, 1424 (1994); Phys. Rev. B 50, 16684 (1994). 8. A. Gurevich and E.H. Brandt, Phys. Rev. Lett. 73, 178 (1994); Phys. Rev. B (submitted); E.H. Brandt and A. Gurevich, Phys. Rev. Lett. 76, 1723 (1996); E.H. Brandt, Phys. Rev. Lett. 76, 4030 (1996). 9. J. K5tzler et al., Phys. Rev. B 50, 3384 (1994); E.H. Brandt, Phys. Rev. B 50, 13833
(1994). 10. J. Gilchrist and E.H. Brandt, Phys. Rev. B 54, 3530 (1996). 11. E.H. Brandt, Phys. Rev. Lett. 74, 3025 (1995); Phys. Rev. S 52, 15442 (1995). 12. W. Xing et al., J. Appl. Phys. 76, 4244 (1994); R. Wijngaarden et hi., Phys. Rev. B 54, 6742 (1996); E.H. Brandt, Phys. Rev. B 46, 8628 (1992); A.E. Pashitski, A. Gurevich, et al. Science (Jan. 1997, in print); Ch. Joos et al., Phys. Rev. B (submitted). 13. E.H. Brandt, Phys. Rev. B 54, 4246 (1996). 14. E.H. Brandt, Rep. Prog. Phys. 58, 1465 (1995). 15. A. Forkl, Physica Scripta T49, 148 (1993).