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Thermally activated penetration of magnetic flux through a surface barrier in high-Tc superconductors
PfllSiCA
Physica C 191 (1992) 219-223 North-Holland
Thermally activated penetration of magnetic flux through a surface barrier in high-Tc superconductors A.E. Koshelev Kamerlingh Onnes Laboratory, Leiden University, PO Box 9506, 2300 RA Leiden, Netherlands
Received 15 November 1991
Thermally assisted penetration of magnetic flux through the surface barrier governs the behavior of the penetration field and the magnetization in HTSC samples with weak volume pinning. The energy barrier for flux penetration through an ideal surface is estimated as a function of external magnetic field and magnetic induction within the sample for both individual and collective penetration regimes.
1. Introduction It is known for a long time that magnetic flux cannot enter freely type II superconductors because of the surface barrier (SB) arising from competition between the interaction of a vortex line with its mirror image and the interaction with the surface shielding current [ 1 ]. In the case of ideal plane surface the Meissner state remains metastab!e due to the SB until the external field H is smaller than the thermodynamic critical field He. In real samples, however, inhomogeneities at the surface suppress the SB and the observed field of flux penetration Hp is always smaller than He. One can expect that the role of the SB is higher in superconductors with large values of the Ginzburg-Landau parameter x, due to the large difference between Hc and the lower critical field H~. In HTSC the relation Hc/Hc~ ,,, 20 and it looks improbable that SB can be totally suppressed by inhomogeneities. Two expected effects of the SB are the increase of the measured penetration .nclu . . . as compared with the thermodynamic value/-/ca and the emergence of the surface contribution to the hysteresis of the magnetization. In most papers devoted to the magnetization measurements in HTSC the surPermanent address: Institute of Solid State Physics, USSR Academyof Sciences, Chernogolovka,Moscowdistr., 142432, Russia.
face effects have been neglected and results have been interpreted on the basis of the simple Bean model. This approach is justified for samples with strong volume pinning when the surface contribution to the hysteresis of the magnetization is small as compared to the volume one. Surface effects, however, lead to the appearance of unusual features in the magnetization behavior which cannot be explained within the simple Bean model [2,3 ] and totally determine the magnetization behavior in the HTSC samples with weak volume pinning [4]. A very important feature of the HTSC is that the probability of the thermally activated penetration (TAP) of flux through SB is not so small as in ordinary superconductors. It changes the regime of the vortex penetration. In ordinary superconductors one can neglect TAP and the flux penetrates into the sample when SB is totally suppressed by the external field at some points of the surface at least. In the HTSC, if the temperature is not too low, the character of the flux penetration is quite different. If the external field is slightly above Hc~ then the rate of the T A P is small and one cannot register the change of the magnetization. This rate steeply increases with the magnetic field. At some field Hp the rate of the TAP becomes large enough to detect the change of magnetization. This field will be experimentally measured as the penetration field. It has nothing to do with both ft¢ and H,.j, and it can preserve a strong
0921-4534/92/$05.00 © 1992 ElsevierScience Publishers B.V. All rights reserved.
220
A.E, Koshelev / Thermally activated penetration
temperat~Jre dependence down to low temperature. "[his physical picture was first proposed in refs. [2 ] an,~ [3 ]. It gives a natural explanation for the freque,atly observed anomaly of the temperature dependence of the penetration field [ 5 ]. An analogous mechanism forms the irreversible behavior of magnetization in pure samples. On increasing of H the magnetic induction B within the sample is determined by the rate of flux penetration. In contrast, on decreasing of H the value o f B is determined by the rate of flux escaping from the sample. The quantities determining the rate of the TAP are the energy barriers for flux penetrating into (U~,) and escaping from (Uo,t) the sample. Hence, the temperature dependence of Hp and the irreversible behavior of the magnetization are determined by the dependences of the energy barriers upon the fields H and B. In addition the derivatives of the energy barriers with respect to B determine the creep rate of the flux. In real samples the surface barrier is substantially suppressed by inhomogeneities and penetration occurs through "weak places" at the surface. Analytical estimates of the energy barrier for such a surface are a very difficult problem. It looks v e ~ interesting and instructive to solve this problem for the simplest case of an ideal plane surface first. The aim of the present papers is to estimate the field dependence of the barrier U,~ for this case.
2. The energy barrier for thermally activated penetration for an ideal plane surface Consider the penetration of flux through the ideal plane surface parallel to the c-axis of the HTSC. An external magnetic field is also applied parallel to the c-axis. For any given value of B the surface barrier stabilizes the vortex lattice in some interval of the external field, H m , , ( B ) < H < H ~ ( B ) (fig. ! ). The value of ft corresponding to thermodynamic equilibrium. H~h(B), also lies within this interwal. Boundaries of metastability, H~,, (B) and H , ~ (B) were found in ref. [ 6 ]. In the region H* << B << H¢2. H*=Oo/(4n).2), ~he boundaries are estimated as
H ~ ( B ) ~. (H~ - B 2 ) ~/~- H~,,(B) .~ B - 0.46H*. in the region Hth(B) < t t < H ~ , . ( B ) the magnetic in-
/
,(B)
£E '°
,!;;',;/ Hcl
H
Hc
Fig. !. Metastability region in B, H-plane and regions differing by the regimes of flux penetration through the surface barrier (la, b are the regions of single vortex penetration, II is the region of collective penetration ).
duction increases with time and the rate of increasing is determined by the barrier U~.(B, H),
dB/dt...exp( - U,,( B, H) /T) .
( 1)
Flux penetrates through the surface by the creation of a critical nucleus consisting of one or several vortex loops. The barrier U,,(B, H) is determined by the energy of the system in the critical state. Several regimes of flux penetration exist. If the field H is not too close to Hth(B) then the critical nucleus consists of one vortex loop, i.e. vortices penetrate through SB independently. In the vicinity of Hth (B) penetration becomes collective, i.e. includes the motion o f a large number of vortex lines. Consider single vortex penetration first. The critical nucleus is a vortex loop the form of which can be described by a single valued function v(z), z being til~. ~.~,.rk~'l~.tltlatt. CtIUII~ t i l e
II~IU.
l " ~ 7 ~ l U t . . t l l l ~ LIIE7 i J ~ l U | -
mation of the vortex lattice caused by the loop, its energ~ U{v(:)i can be written as follows:
U=~ d z [ ~ l n ( L / , )
(;))
t "dz'Z" l:: q × I !+ ? , + ~ (B(v)-li)j,
(2)
A.E. Koshelev / Thermally activated penetration
F is the effective mass anisotropy. The cutoff length L depends on the shape of the loop and can be estimated as U-..,max[v(z)]. The critical configuration vcr(z) gives the extreme value of the energy, 8U/ 8vl~=~c~=0. For the Meissner state, B ( v ) = H e x p ( - v / 2 ) , the barrier Ui. was found in ref. [7]. For an anisotropic 3D superconductor the critical loop has the form of an ellipse (fig. 2 ) and the result for U~, can be written as follows:
221
,I
[Jin
,i
tl *1 ii
:1 ii
t
2 t
1
i i i I
U,.,(H, O) = Uo(H,:/H) ln:'(Hc/H),
(3)
H>> Hcl
H{, H,.(B)
with
75¢4
II H~ H=o,(B)
H
Fig. 3. Schematic field dependences of the barrier for flux penetration for Meissner ( 1 ) and mixed (2) states.
Up - 64nA2F,/2" The estimate for the characteristic barrier Up for the YBa2Cu307 compound is Ups.300 K. An important physical parameter is the size of the critical loop in the z direction, z~. In the region H>>H¢~, Zcr'-.2H*/(HP/Z). The result (3) is valid for layered compounds until Zc, is much greater than the distance between superconducting layers d. This means the field H must be smaller than the crossover field Hcr, H¢,.~2H*/(dF~/2). In the region H>H~, 2D-vortices in a superconducting layer overcome the
SB independently of other layers. The energy barrier in this region is estimated as [2]:
Ui.(H, O) = U2DIn(He~H), U2D::dfb2/ ( 4n2 ) 2 .
For the highly anisotropic Bi and Tl compounds the crossover field is close to Hcl [2] and the characteristic value of the barrier is estimated as U2D~ 420 K. The results (3), (4) can be generalized for the region B >> H*, H - H , h (B) >> tI*. In this region the size L of the loop is much smaller than the distance between the first vortex row and the surface and parameters of the critical loop are determined by the screening current at the surface, which changes with B a n d H a s [6]
Jso = ~
1
(4)
C
(H2-B2) ,/2
(5)
This means the barrier Ui.(H, B) can be obtained from eqs. (3) and (4) by use of the substitution
j/ SAMPLE SURFACE
POSITION OF THE FIRST VORTEX ROW
Fig. 2. Shape of the critical vortex loop in the single vortex penetration regime far from equilibrium ( 1 ) and near crossover to lhe collective penetration (2).
H _ . ( H 2 _ B 2 ) I/2
Single vortex penetration is possibJe until creation of the additional vortex line near the first vortex row of the ideal vortex lattice is energetically favorable. The value of the field, at which this creation becomes unfavorable, can be estimated as Ht = H~h+ o~,H* with or, ~ i. In the vicinity of H, the critical nucleus consists of the long portion of the new vortex line restricted by the two vortex kinks (fig.
222
A.E. Koshelev / Thermally activated penetration
2). The limiting value of the single-vortex energy barrie,,, Ut = U i , ( H = H ~ - 0 ) , approximately equals ,'¢, the energy of the kinks and can be estimated as Uonc 3~ ~n] U1=1.55 (2~-B-~l/21n / [ H J ( H * B ) .
U,, on the field H can be estimated as b].(n,B)
Ut
H*
at H * L n - 3/z << H -
H t h <¢--H * ,
(6) _ ,
At field H~ the transition to two-vortex penetration takes place. This transition is accompanied by a jump in the barrier:
,
I U , ~,H_--2-H~J Ln -3'2 at
H-H,h << H*Ln 3/z •
(9) Here L n = l n ( a / ~ ) .
Uin(H=Ht-O)-Uin(H=H,-O)~U,
.
(7)
Further approaching of the field H to the thermodynamic value H,h(B) leads to increasing of !he number N of vortex loops in the critical nucleus. Addition of a new vortex loop to the nucleus consisting of N loops takes place at field HN when creation of N new vortex lines near the first row becomes energetically unfavorable. Any such addition is accompanied by a j u m p in the barrier Uin of order of U~. Hence, the field dependence of the barrier Uin has the "staircase" appearance (fig. 3). At N>> 1 the critical nucleus is represented by the portion of the new vortex row restricted by steps. The energy of the nucleus consisting of N loops of length L can be estimated as 0o U(N. L ) .~ - ~ ( H - H t h ) N L
+ U~N+~L+
02 aZN a 4rt FLA 2 '
(8)
a = ( cbo/B) '/2
We estimated the barrier for TAP for an ideal plane surface. In the model considered, the rate of TAP is the same at any point of the surface. In real samples, however, penetration is influenced strongly by sto,-cture inhomogeneities and inhomogeneity of the field at the surface. This leads to the creation of "'weak" places at which the rate of the TAP is much higher than in other points of the surface and penetration of flux occurs through these places. A ~trong influence of the quality of the surface on the flux penetration has been demonstrated directly ref. [4]. It has been observed that the penetration field reduccs substantially after irradiation of the surface. Nevertheless, we expect that inhomogeneities reduce the absolute value of the energy barrier but do not change substantially the features o f its field dcpciidences.
Acknowledgements
Here the first term represents the gain in energy due to the creation of the portion o¢ the new vortex row near the surface, the second and third terms are energies of the steps perpendicular and parallel to the field respectively, c:S =asO2o/(4rt)) 2,
3. Discussion
I would like to thank I.F. Schegolev, V.N. Kopylov and T.G. Togonidze for stimulating discussions. This work was supported by the Netherlands Foundation for the Fundamental Research on Matter ( F O M ) .
with a~ ~ 1 .
The last term in eq. ( 8 ) represents :he ncnlccal elastic cnergy of the lattice deformation caused by the nucleus. This term becomes important when N becomes large enough, N > ln3/2(a/~). The barrier U,, is determined by the extremum of U ( N , L ) with respect to N and L. Staircase dependence arises from the fact that the variable N cannot change continuously. Neglecting the jumps, a rough dependence of
References [ 1] C.P. Bean and J.D. Livingston, Phys. Rev. Lett. 12 (1964) 14. [2] V.N. Kopytov. A.E. Koshelev, I.F. Schegolev and T.G. Togonidze, PhysicaC 170 (1990) 291. [3] M.W. McEIfresh. Y. Yeshurun. A.P. Malozemoff and F. Holtzberg, PhysicaA 168 ( 19901 308. [4] M. Konczykowski, k.l. Burlachkov, Y. Yeshurun and F. Hotzberg, Phys. Rev. B 43 ( 1991 ) 13707.
A.E. Koshelev / Thermally activated per~etration [ 5 ] H. Adrian, W. Assrnus, A. Hohr, J. Koalewski, H. Spile and F. Steglich, Physica C 162-164 (1989) 329; A. Umezawa, G.W. Crabtree, K.G. Vanderwoort, U. Welp, W.K. Kwok and J.Z. Liu, Physica C 162-164 (1989) 733; M. Wacenovsky, H.W. Weber, O.B. Hyun and D.K. Finnemore, Physica C 162-164 ( 198f~) 1629.
223
[6 ] F.F. Ternovskii and L.N, Shekhata, ZhETF 62 ( 1972 ~ 2296 (Sov. Phys. JETP 35 (1972) 1202), [ 7 ] V.P. Galaiko, Zh. Eksp. Teor. Fiz. 50 ( 1966 ) 1322 (Soy. Phys. JETP 23 (1966) 1202); B.V. Petukhov and V.R. Cheehiotkin, Zh. Eksp. Teor. Fiz. 65 (1973) 1653 (Sov. Phys. JETP 38 (1974) 827).
Physica C 191 (1992) 219-223 North-Holland
Thermally activated penetration of magnetic flux through a surface barrier in high-Tc superconductors A.E. Koshelev Kamerlingh Onnes Laboratory, Leiden University, PO Box 9506, 2300 RA Leiden, Netherlands
Received 15 November 1991
Thermally assisted penetration of magnetic flux through the surface barrier governs the behavior of the penetration field and the magnetization in HTSC samples with weak volume pinning. The energy barrier for flux penetration through an ideal surface is estimated as a function of external magnetic field and magnetic induction within the sample for both individual and collective penetration regimes.
1. Introduction It is known for a long time that magnetic flux cannot enter freely type II superconductors because of the surface barrier (SB) arising from competition between the interaction of a vortex line with its mirror image and the interaction with the surface shielding current [ 1 ]. In the case of ideal plane surface the Meissner state remains metastab!e due to the SB until the external field H is smaller than the thermodynamic critical field He. In real samples, however, inhomogeneities at the surface suppress the SB and the observed field of flux penetration Hp is always smaller than He. One can expect that the role of the SB is higher in superconductors with large values of the Ginzburg-Landau parameter x, due to the large difference between Hc and the lower critical field H~. In HTSC the relation Hc/Hc~ ,,, 20 and it looks improbable that SB can be totally suppressed by inhomogeneities. Two expected effects of the SB are the increase of the measured penetration .nclu . . . as compared with the thermodynamic value/-/ca and the emergence of the surface contribution to the hysteresis of the magnetization. In most papers devoted to the magnetization measurements in HTSC the surPermanent address: Institute of Solid State Physics, USSR Academyof Sciences, Chernogolovka,Moscowdistr., 142432, Russia.
face effects have been neglected and results have been interpreted on the basis of the simple Bean model. This approach is justified for samples with strong volume pinning when the surface contribution to the hysteresis of the magnetization is small as compared to the volume one. Surface effects, however, lead to the appearance of unusual features in the magnetization behavior which cannot be explained within the simple Bean model [2,3 ] and totally determine the magnetization behavior in the HTSC samples with weak volume pinning [4]. A very important feature of the HTSC is that the probability of the thermally activated penetration (TAP) of flux through SB is not so small as in ordinary superconductors. It changes the regime of the vortex penetration. In ordinary superconductors one can neglect TAP and the flux penetrates into the sample when SB is totally suppressed by the external field at some points of the surface at least. In the HTSC, if the temperature is not too low, the character of the flux penetration is quite different. If the external field is slightly above Hc~ then the rate of the T A P is small and one cannot register the change of the magnetization. This rate steeply increases with the magnetic field. At some field Hp the rate of the TAP becomes large enough to detect the change of magnetization. This field will be experimentally measured as the penetration field. It has nothing to do with both ft¢ and H,.j, and it can preserve a strong
0921-4534/92/$05.00 © 1992 ElsevierScience Publishers B.V. All rights reserved.
220
A.E, Koshelev / Thermally activated penetration
temperat~Jre dependence down to low temperature. "[his physical picture was first proposed in refs. [2 ] an,~ [3 ]. It gives a natural explanation for the freque,atly observed anomaly of the temperature dependence of the penetration field [ 5 ]. An analogous mechanism forms the irreversible behavior of magnetization in pure samples. On increasing of H the magnetic induction B within the sample is determined by the rate of flux penetration. In contrast, on decreasing of H the value o f B is determined by the rate of flux escaping from the sample. The quantities determining the rate of the TAP are the energy barriers for flux penetrating into (U~,) and escaping from (Uo,t) the sample. Hence, the temperature dependence of Hp and the irreversible behavior of the magnetization are determined by the dependences of the energy barriers upon the fields H and B. In addition the derivatives of the energy barriers with respect to B determine the creep rate of the flux. In real samples the surface barrier is substantially suppressed by inhomogeneities and penetration occurs through "weak places" at the surface. Analytical estimates of the energy barrier for such a surface are a very difficult problem. It looks v e ~ interesting and instructive to solve this problem for the simplest case of an ideal plane surface first. The aim of the present papers is to estimate the field dependence of the barrier U,~ for this case.
2. The energy barrier for thermally activated penetration for an ideal plane surface Consider the penetration of flux through the ideal plane surface parallel to the c-axis of the HTSC. An external magnetic field is also applied parallel to the c-axis. For any given value of B the surface barrier stabilizes the vortex lattice in some interval of the external field, H m , , ( B ) < H < H ~ ( B ) (fig. ! ). The value of ft corresponding to thermodynamic equilibrium. H~h(B), also lies within this interwal. Boundaries of metastability, H~,, (B) and H , ~ (B) were found in ref. [ 6 ]. In the region H* << B << H¢2. H*=Oo/(4n).2), ~he boundaries are estimated as
H ~ ( B ) ~. (H~ - B 2 ) ~/~- H~,,(B) .~ B - 0.46H*. in the region Hth(B) < t t < H ~ , . ( B ) the magnetic in-
/
,(B)
£E '°
,!;;',;/ Hcl
H
Hc
Fig. !. Metastability region in B, H-plane and regions differing by the regimes of flux penetration through the surface barrier (la, b are the regions of single vortex penetration, II is the region of collective penetration ).
duction increases with time and the rate of increasing is determined by the barrier U~.(B, H),
dB/dt...exp( - U,,( B, H) /T) .
( 1)
Flux penetrates through the surface by the creation of a critical nucleus consisting of one or several vortex loops. The barrier U,,(B, H) is determined by the energy of the system in the critical state. Several regimes of flux penetration exist. If the field H is not too close to Hth(B) then the critical nucleus consists of one vortex loop, i.e. vortices penetrate through SB independently. In the vicinity of Hth (B) penetration becomes collective, i.e. includes the motion o f a large number of vortex lines. Consider single vortex penetration first. The critical nucleus is a vortex loop the form of which can be described by a single valued function v(z), z being til~. ~.~,.rk~'l~.tltlatt. CtIUII~ t i l e
II~IU.
l " ~ 7 ~ l U t . . t l l l ~ LIIE7 i J ~ l U | -
mation of the vortex lattice caused by the loop, its energ~ U{v(:)i can be written as follows:
U=~ d z [ ~ l n ( L / , )
(;))
t "dz'Z" l:: q × I !+ ? , + ~ (B(v)-li)j,
(2)
A.E. Koshelev / Thermally activated penetration
F is the effective mass anisotropy. The cutoff length L depends on the shape of the loop and can be estimated as U-..,max[v(z)]. The critical configuration vcr(z) gives the extreme value of the energy, 8U/ 8vl~=~c~=0. For the Meissner state, B ( v ) = H e x p ( - v / 2 ) , the barrier Ui. was found in ref. [7]. For an anisotropic 3D superconductor the critical loop has the form of an ellipse (fig. 2 ) and the result for U~, can be written as follows:
221
,I
[Jin
,i
tl *1 ii
:1 ii
t
2 t
1
i i i I
U,.,(H, O) = Uo(H,:/H) ln:'(Hc/H),
(3)
H>> Hcl
H{, H,.(B)
with
75¢4
II H~ H=o,(B)
H
Fig. 3. Schematic field dependences of the barrier for flux penetration for Meissner ( 1 ) and mixed (2) states.
Up - 64nA2F,/2" The estimate for the characteristic barrier Up for the YBa2Cu307 compound is Ups.300 K. An important physical parameter is the size of the critical loop in the z direction, z~. In the region H>>H¢~, Zcr'-.2H*/(HP/Z). The result (3) is valid for layered compounds until Zc, is much greater than the distance between superconducting layers d. This means the field H must be smaller than the crossover field Hcr, H¢,.~2H*/(dF~/2). In the region H>H~, 2D-vortices in a superconducting layer overcome the
SB independently of other layers. The energy barrier in this region is estimated as [2]:
Ui.(H, O) = U2DIn(He~H), U2D::dfb2/ ( 4n2 ) 2 .
For the highly anisotropic Bi and Tl compounds the crossover field is close to Hcl [2] and the characteristic value of the barrier is estimated as U2D~ 420 K. The results (3), (4) can be generalized for the region B >> H*, H - H , h (B) >> tI*. In this region the size L of the loop is much smaller than the distance between the first vortex row and the surface and parameters of the critical loop are determined by the screening current at the surface, which changes with B a n d H a s [6]
Jso = ~
1
(4)
C
(H2-B2) ,/2
(5)
This means the barrier Ui.(H, B) can be obtained from eqs. (3) and (4) by use of the substitution
j/ SAMPLE SURFACE
POSITION OF THE FIRST VORTEX ROW
Fig. 2. Shape of the critical vortex loop in the single vortex penetration regime far from equilibrium ( 1 ) and near crossover to lhe collective penetration (2).
H _ . ( H 2 _ B 2 ) I/2
Single vortex penetration is possibJe until creation of the additional vortex line near the first vortex row of the ideal vortex lattice is energetically favorable. The value of the field, at which this creation becomes unfavorable, can be estimated as Ht = H~h+ o~,H* with or, ~ i. In the vicinity of H, the critical nucleus consists of the long portion of the new vortex line restricted by the two vortex kinks (fig.
222
A.E. Koshelev / Thermally activated penetration
2). The limiting value of the single-vortex energy barrie,,, Ut = U i , ( H = H ~ - 0 ) , approximately equals ,'¢, the energy of the kinks and can be estimated as Uonc 3~ ~n] U1=1.55 (2~-B-~l/21n / [ H J ( H * B ) .
U,, on the field H can be estimated as b].(n,B)
Ut
H*
at H * L n - 3/z << H -
H t h <¢--H * ,
(6) _ ,
At field H~ the transition to two-vortex penetration takes place. This transition is accompanied by a jump in the barrier:
,
I U , ~,H_--2-H~J Ln -3'2 at
H-H,h << H*Ln 3/z •
(9) Here L n = l n ( a / ~ ) .
Uin(H=Ht-O)-Uin(H=H,-O)~U,
.
(7)
Further approaching of the field H to the thermodynamic value H,h(B) leads to increasing of !he number N of vortex loops in the critical nucleus. Addition of a new vortex loop to the nucleus consisting of N loops takes place at field HN when creation of N new vortex lines near the first row becomes energetically unfavorable. Any such addition is accompanied by a j u m p in the barrier Uin of order of U~. Hence, the field dependence of the barrier Uin has the "staircase" appearance (fig. 3). At N>> 1 the critical nucleus is represented by the portion of the new vortex row restricted by steps. The energy of the nucleus consisting of N loops of length L can be estimated as 0o U(N. L ) .~ - ~ ( H - H t h ) N L
+ U~N+~L+
02 aZN a 4rt FLA 2 '
(8)
a = ( cbo/B) '/2
We estimated the barrier for TAP for an ideal plane surface. In the model considered, the rate of TAP is the same at any point of the surface. In real samples, however, penetration is influenced strongly by sto,-cture inhomogeneities and inhomogeneity of the field at the surface. This leads to the creation of "'weak" places at which the rate of the TAP is much higher than in other points of the surface and penetration of flux occurs through these places. A ~trong influence of the quality of the surface on the flux penetration has been demonstrated directly ref. [4]. It has been observed that the penetration field reduccs substantially after irradiation of the surface. Nevertheless, we expect that inhomogeneities reduce the absolute value of the energy barrier but do not change substantially the features o f its field dcpciidences.
Acknowledgements
Here the first term represents the gain in energy due to the creation of the portion o¢ the new vortex row near the surface, the second and third terms are energies of the steps perpendicular and parallel to the field respectively, c:S =asO2o/(4rt)) 2,
3. Discussion
I would like to thank I.F. Schegolev, V.N. Kopylov and T.G. Togonidze for stimulating discussions. This work was supported by the Netherlands Foundation for the Fundamental Research on Matter ( F O M ) .
with a~ ~ 1 .
The last term in eq. ( 8 ) represents :he ncnlccal elastic cnergy of the lattice deformation caused by the nucleus. This term becomes important when N becomes large enough, N > ln3/2(a/~). The barrier U,, is determined by the extremum of U ( N , L ) with respect to N and L. Staircase dependence arises from the fact that the variable N cannot change continuously. Neglecting the jumps, a rough dependence of
References [ 1] C.P. Bean and J.D. Livingston, Phys. Rev. Lett. 12 (1964) 14. [2] V.N. Kopytov. A.E. Koshelev, I.F. Schegolev and T.G. Togonidze, PhysicaC 170 (1990) 291. [3] M.W. McEIfresh. Y. Yeshurun. A.P. Malozemoff and F. Holtzberg, PhysicaA 168 ( 19901 308. [4] M. Konczykowski, k.l. Burlachkov, Y. Yeshurun and F. Hotzberg, Phys. Rev. B 43 ( 1991 ) 13707.
A.E. Koshelev / Thermally activated per~etration [ 5 ] H. Adrian, W. Assrnus, A. Hohr, J. Koalewski, H. Spile and F. Steglich, Physica C 162-164 (1989) 329; A. Umezawa, G.W. Crabtree, K.G. Vanderwoort, U. Welp, W.K. Kwok and J.Z. Liu, Physica C 162-164 (1989) 733; M. Wacenovsky, H.W. Weber, O.B. Hyun and D.K. Finnemore, Physica C 162-164 ( 198f~) 1629.
223
[6 ] F.F. Ternovskii and L.N, Shekhata, ZhETF 62 ( 1972 ~ 2296 (Sov. Phys. JETP 35 (1972) 1202), [ 7 ] V.P. Galaiko, Zh. Eksp. Teor. Fiz. 50 ( 1966 ) 1322 (Soy. Phys. JETP 23 (1966) 1202); B.V. Petukhov and V.R. Cheehiotkin, Zh. Eksp. Teor. Fiz. 65 (1973) 1653 (Sov. Phys. JETP 38 (1974) 827).