Table Of ContentMathematical model of vortex penetration phenomenon
∗
Rongchao Ma
Department of Physics, University of Alberta, Edmonton, AB, Canada T6G 2G7
(Dated: January 4, 2011)
Vortex penetration affects the stability of a superconducting system and limits the possible ap-
plication of the system. However, the mathematical description to this phenomenon is currently
unavailable. HereIpresentedamathematicalmodelinwhichIconsideredtheeffectsofbulkpinning
and internal field repulsive force on vortex hopping. Thereafter, I proposed a series expansion to
1 the activation energy and derived a general formula to describe the time dependence of the vortex
1 penetration process. With these formulas, I can analyze the experimental data and calculate the
0 activation energy of the vortex penetration phenomenon. The results are accurate for the time
2 dependenceof theinternal field measurements in a Bi Sr CaCu O superconductor.
2 2 2 8+x
n
a PACSnumbers: 74.25.Op,74.25.Uv,74.25.Wx
J
2
I. INTRODUCTION is important to construct a mathematical model which
includes the effects of the bulk pinning and internalfield
]
n repulsive force, and can predicts the time scale of the
An appliedmagnetic field can penetrate into a type-II
o
vortex penetration process.
c superconductor in the form of quantized vortices (each
InthisarticleItheoreticallyshowthatthevortexpen-
- carry a single flux quantum Φ ) when the applied mag-
0
r etration process is time dependent because of the sur-
p netic field increases to a value above the lower critical
u field of the superconductor.1–6 The vortices first pene- faceattractiveforce,bulkpinningforceandinternalfield
s trate into the superconductor from the surfaces which repulsive force. First, I discussed the field dependence
. of activation energy of the vortex penetration process.
t are parallel to the applied magnetic field, and then go
a to the center of the superconductor.7,8 Vortex motion Next, I discussed the time dependence of activation en-
m
is usually retarded9 during the penetration process be- ergybasedontheArrenhiusrelation. Finally,Ideriveda
- timedependentequationfortheinternalfieldinavortex
d cause the vortices are subjected to surface attractive
n force7, pinning forces, internal field repulsive force and penetration process.
o damping force10,11. The internal magnetic field (or pen-
c etrated magnetic field) increases gradually and finally
[ reaches a saturated value. De Gennes12 first studied the II. MODEL
2 vortex penetration in terms of free energy. A number
v of works studied the vortex entry conditions13–20, and As mentioned before, the vortices inside a type-II su-
7 other works investigated the vortex penetration into the perconductor are subjected to various forces and vor-
8
bulkofsuperconductorsovertheBean-Livingstonsurface tex motion is generally retarded. The experimental
8
barrier21–27. Theexperimentalstudiesofthevortexpen- observations29 have shown that, at lower temperatures
1
. etration were generally carried out with Hall sensors25 and under small applied fields, the time scale of vortex
9 and magneto-optical imaging technique6,28 which pro- penetration process in a superconducting crystal sample
0
vides direct observations of vortex front. extends to infinity long. Therefore, it is reasonable to
0
It is known that vortex entry conditions and vortex assume that the vortices are at some metastable states
1
: penetration over surface barrier were studied. How- under an intermediate applied field Hc1 < Ha < Hirr,
v
ever, the Bean-Livingston surface barrier reduces expo- where H is lower critical field and H is irreversible
i c1 irr
X nentially with respect to the distance to the surface of field. Inthissense,thevortexpenetrationprocesscanbe
r the superconductor.7 The vortex motion inside a macro regardedas a vortices diffusion process, i.e., a process of
a size superconductor is then mainly determined by the vorticeshoppingbetweenadjacentpinningcenters. Inan
bulk pinning and internal field repulsive force. The the- analogytothatoffluxrelaxation30,wecandescribedthe
oretical description to this phenomenon is currently un- vortex penetration process with the Arrhenius relation:
available. On the other hand, the recent experimental ν = ν e−Ua/kT, where ν is attempting frequency, U is
0 0 a
observations29 have shown that the internal magnetic the activation energy of the vortices, k is the Boltzmann
field ofa high-T superconductoris stronglytime depen- constant and T is temperature.31 This relation implies
c
dent after a magnetic field (below irreversible field) is that the activation energy U plays an important role in
a
applied to the superconductor. The time scale depends the vortices hopping.
on the sample geometry, temperature and applied field In this work I am going to use some similar concepts
magnitude. This time dependence implies that a vortex and mathematical tools used in the studies of flux re-
systemis unstableduring the vortexpenetrationprocess laxation. But it should be emphasized that the physical
and a superconducting device may not work properly at contents of these two processes are completely different.
this stage. For the purpose of application, therefore, it Invortexpenetrationprocessthevorticesarepushedinto
2
thesuperconductorbyexternaldrivingforce;however,in denoted as U (T,λ,ξ) and a (T,λ,ξ).
c l
flux relaxation process the vortices jump out of the su- The total activation energy of a vortex, U (B), is the
a
perconductor because of various reasons32. Let us first summation of U and U (B), that is
BL b
discuss the possible expressionofthe fielddependence of
activation energy U in the vortex penetration process. n
a U (B)=U +U (B)=U + a Bl (3)
a BL b 0 l
l=1
X
A. Field dependence of activation energy
whereU =U +U istheactivationenergyatvanishing
0 BL c
internal field, or the activation energy at time t =0. U
0
According to Bean and Livingston7, a vortex close to isdeterminedbythe sample geometry,externalfieldand
thesurfaceofasuperconductorissubjectedtoattractive pinning ability.
imaging force and external field repulsive force. The ad- From Eq.(1) we see that the activation energy U (B)
a
ditionofthese two forcesresultsin the surfacebarrierto
is a local function, i.e., a function of position x. This is
the vortexpenetration, UBL, whichisdeterminedby the not explicitly shown in Eq.(3) because we are interested
samplegeometryandexternalfieldmagnitudeH . Early
a in the field dependence of U in the current work.
a
studies have shown that7
Φ 2 2x Φ
UBL(x)= 0 K0 − 0Hae−x/λ (1) B. Time dependence of activation energy
4πλ λ 4π
(cid:18) (cid:19) (cid:18) (cid:19)
where Φ0 is flux quantum, λ is penetration depth and The above discussion has shown that in vortex pene-
x is the distance between the vortexand sample surface. tration process the activation energy U is an increasing
a
ThefirsttermontherightsideofEq.(1)iscausedbythe function of the internal field B. It is also known that
attractiveimagingforce,andthesecondtermontheright B increases gradually with time t. Therefore, U is an
a
side is caused by the applied field driving force. Eq.(1) increasing function of t.
showsthatUBL(x)reducesexponentiallywithrespectto The rate of change of the internal field, dB/dt, is pro-
x, that is, x→∞, UBL(x)→0. Thus,the vorticesaway portional to vortex hopping rate. On the other hand,
from the sample surface are almost uninfluenced by the B is an increasing function of t in the vortex penetra-
surface barrier. tion process, i.e., dB/dt > 0 (in relaxation dB/dt < 0).
Ontheotherhand,avortexinsideasuperconductoris According to the Arrhenius relation, we have33
subjected to bulk pinning force and internal field repul-
siveforce,whichpreventsthevorticesmotionandreduces dB =Ce−Ua/kT (4)
the vortices hopping rate. According to the Arrhenius dt
relation, this internal field repulsive force increases the
where C is a positive proportional constant. Eq.(4)
activation energy. Therefore, the activation energy is an
can also be derived from the diffusion equation of flux
increasingfunction ofthe internal field B. Let U be the
b flow (conservation of flux).34 Comparing to that of flux
activationenergyrelatedtothebulkpinningandinternal
relaxation33, Eq.(4) has differences in sign and initial
field repulsive force, we have dU /dB >0.
b
conditions. These cause the differences between the
The exact form of U (B) is usually unknown because
b
mathematical models of vortex penetration and flux re-
ofthecomplexityoftheinteractionbetweenvorticesand
laxation.
pinning centers, as well as the reaction of vortices to ex-
The exact solution of Eq.(4) is currently unavailable,
ternalfielddrivingforces. Butfromapuremathematical
but we can find an approximate solution to it. Consider
consideration,we canexpressU (B) asaTaylorseriesof
b
B (or magnetization M).32 Let us now write out the se- applying a magnetic field to a superconductor at time
t=0. The activation energy of a vortex at this moment
ries expression of U (B) explicitly,
b
is U , whichis alsothe activationenergyatzerointernal
0
n field (See Eq.(3)). As time increases to t, the activation
Ub(B)=Uc+ alBl (2) energy increases to Ua. Rewrite Eq.(4) and integrate it
l=1 on both sides,
X
where U = U (0), a = U′(0), a = U′′(0)/2!, ···,
c b 1 b 2 b
an = Ub(n)(0)/n!. The parameter Uc is the pinning po- UaeUa/kTdU = tCdUadt (5)
tentialinside the bulkofthe superconductor. The coeffi- a dB
ZU0 Z0
cients a (l >2) represent the weight of the contribution
l
from inelastic deformation and interaction between the With logarithmic accuracy33, we obtain the following
vortices,a2representstheweightofthecontributionfrom equation:
elastic deformation of the vortices and a represents the
1
weight of the contribution from Lorentz force.32 These t
U (t)=U +kTln 1+ (6)
coefficients are functions of temperature T and can be a 0 τeU0/kT
(cid:18) (cid:19)
3
where τ = kT/[C(dU /dB)] is a short time scale where
a
parameter35. Eq.(6) shows that U has a strong effect
0
t
on the time dependence of the activation energy Ua(t). w(t)=kTln 1+ (8)
τeU0/kT
(cid:18) (cid:19)
C. Time dependence of internal field
To find outthe time dependence ofinternalfieldB(t),
Our final purpose is to obtain the time dependence of letusnowinvertEq.(7)by expandingB[w(t)] asaseries
internal field B(t). Since the vortex motion in a vor- of w(t),
tex penetrationprocessis retardedby variousforces,the
n n l
internal field increases gradually with increasing time. t
B[w(t)]= b wl(t)= b kTln 1+
This relation can be obtained by canceling out the ac- l l τeU0/kT
l=1 l=1 (cid:20) (cid:18) (cid:19)(cid:21)
tivation energy U from Eq.(3) (field dependence of U ) X X
a a (9)
and Eq.(6) (time dependence of U ), that is,
a
n Similartoearlierstudies32,wehavethecoefficientsbl’s
w(t)= a Bl(t) (7)
l
l=1
X
1
b =
1
a
1
1 a
b = − 2
2 a2 a
1 (cid:18) 1(cid:19)
2
1 a a
b = 2 2 − 3
3 a3 a a (10)
1 " (cid:18) 1(cid:19) (cid:18) 1(cid:19)#
···
1 1 l(l+1)···(l−1+s+t+u+···) a s a t a u
b = (−1)s+t+u+···· · 2 3 4 ···
l al l s!t!u!··· a a a
1 s,t,u··· (cid:18) 1(cid:19) (cid:18) 1(cid:19) (cid:18) 1(cid:19)
X
···
wheres+2t+3u+···=l−1. Onconsideringthesymme- a by doing a commutation to the coefficients b ↔ a ,
l l l
trybetweenEq.(7)andEq.(9),weobtainthecoefficients that is
1 1 l(l+1)···(l−1+s+t+u+···) b s b t b u
a = (−1)s+t+u+···· · 2 3 4 ··· (11)
l bl l s!t!u!··· b b b
1 s,t,u··· (cid:18) 1(cid:19) (cid:18) 1(cid:19) (cid:18) 1(cid:19)
X
Eq.(9) describes the time dependence of internal the internal field B. It includes the contributions from
field B(t). This relation can also be measured the surface barrier, deformation of vortices, interaction
experimentally.29 With these relations we can calculate between vortices and possibly other unknown sources.
the field dependent activation energy U (B) using the Figure 1 shows that Eq.(9) is accurate for the experi-
a
following procedure: first, fitting the experimental data mental data of a Bi Sr CaCu O single crystal.
2 2 2 8+x
with Eq.(9), we obtain the fitting parameters U and b
0 l At lower temperatures and lower applied fields, the
as shownin Figure 1. Next, substituting b into Eq.(11),
l inelastic deformation and interactionbetween vortices is
weobtainthecoefficientsa . Finally,substitutingU and
l 0 notsignificant. Therefore,wecansimplify the activation
a into Eq.(3), we obtain the activation energy U (B).
l a energy U (B) in a manner analogousto that used in the
a
TheactivationenergyU (B)isacombinedresponseto studies of flux relaxation32.
a
4
III. DISCUSSION
0.25
1. Measurement of U and U .— The bulk pinning
c BL
T)0.20 T = 25 K potential Uc canbe obtainedby measuringthe time evo-
3 lution of internal field B(t) at a position x >> λ inside
- 0 Ha= 0.05 T
10.15 the sample where UBL(x)=0. Fitting the experimental
X
( dataB(t)withEq.(9),wehavethevalueofUc atthepo-
B(t)0.10 sition x. In case of random pinning centers, Uc is a con-
stant and can be represented by the statistical average
Theory:
value fξN1/2, where f is the pinning force of individual
0.05 Experiment: pinningcenter,ξiscoherencelengthandN isthenumber
of the pinning centers in a correlationvolume.31,36.
0.00 0 1 2 3 4 The surface barrier UBL can be calculated using
10 10 10 10 10 Eq.(1),butwecanalsomeasureitwiththehelpofEq.(9).
Time (s) First, using the above stated procedure to measure the
bulk pinning potential U . Next, fixing a sensor at a po-
FIG. 1: (Color online) Time dependence of vortex penetra- c
′
sitionx closetothe sample surfaceandmeasuringB(t),
tion. The scattering points are the experimental data of
′
paliBedi2Smra2gCnaetCicu2fiOe8ld+xofsi0n.g0l5e Tcr.ys(tTahl,eadtat2a5 iKs furonmderRaefn. 2a9p)- wreelaotbiotnaiUnBthLe(xv′a)lu=eUof0(Ux0′()x−)Uuc,siwngecEaqn.(fi9n).dFoiuntatllhye,uvasilnuge
′
The solid black line is the theoretical fit. The fitting re- of U (x).
BL
sults are: B(t) = b1w(t)+b2w2(t)+b3w3(t), where w(t) = If a superconductor is anisotropic, the superconduct-
(25k)×ln{1+t/[τeU0/(25k)]}, U0 = (1.27±0.01) ×(25k), ing parameter λ is anisotropic. According to Eq.(1), the
τ(−=0.090.±820±.010).×01,10b−14/=(25(k0).72,5b±=0.001.0)0×. 10−4/(25k), b2 = surface barrier UBL is also anisotropic. Therefore, when
3 estimating the surface barrier we must have a clear idea
about which crystalline axis is perpendicular to the sur-
face.
2. Inflection point of B(t)−t curve.— As shown in
Figure 1, the B(t) − t curve displays a concave shape
at short time and then changes to a convex shape with
First, let us consider putting a = 0 (l > 2) in Eq.(3).
l
increasing time. Let us now discuss the possible reason
This is equal to ignoring the inelastic deformation and
for this phenomenon.
interaction between vortices. Thus, we obtain the non-
interacting elastic vortices which are described by the The fitting results indicate that it is accurate enough
quadratic activation energy bykeepingtermsuptosecondorder. Thus,wecanwrite
out the time dependent internal field as B(t)=b w(t)+
1
Ua(B)=U0+a1B+a2B2 (12) b2w2(t). The second derivative of B(t) is,
d2B b t
=g(t) 1− 1 −ln 1+ (16)
Substituting Eq.(6) into Eq.(12), we have (choose one dt2 (cid:20) 2b2kT (cid:18) τeU0/kT(cid:19)(cid:21)
of the solutions which is an increasing function of time),
whereg(t)=2b [kT/(τeU0/kT+t)]2. Puttingd2B/dt2 =
2
B(t)= a1 1+4a2w(t)−1 (13) 0, we obtain the inflection point time
2a a2
2 (cid:20)r 1 (cid:21) t∗ =τeU0/kT[e1−b1/(2b2kT)−1] (17)
where w(t) is defined by Eq.(8).
If t < t∗, then d2B/dt2 > 0; the curve is concave. If
Further putting a = 0 in Eq.(12) is equal to assum- t > t∗, then d2B/dt2 < 0; the curve is convex. One can
2 ∗
ing thatthe vorticeshaveverylargeelasticmodulus and easily see that t is a decreasing function of tempera-
the elastic deformation can be ignored. We obtain the ture T (note that b1 and b2 has different sign). On the
∗
non-interacting rigidvortices which are described by the otherhand, t is anincreasingfunctionofU0 andU0 is a
linear activation energy decreasing function of the applied field Ha (see Eq.(1)).
∗
Therefore, t is a decreasing function of H . These re-
a
U (B)=U +a B (14) sultscanexplaintheexperimentalobservationsinRef.29
a 0 1
∗
which show that: 1. t becomes shorter and finally goes
to zero with increasing H or increasing T. 2. There
a
Substituting Eq.(6) into Eq.(14), we have is no concave shape in the B(t)− t curves of the thin
films, onlyconvexshapeswereobserved. This is because
1
thethinfilmsampleshavesmallsurfacebarrierU and
B(t)= w(t) (15) BL
a1 consequentlysmallU0, whichresultsinthe unobservable
5
∗
short t . Eq.(18)indicatesthatthesurfaceimagingforcecauses
3. Effects of surface barrier on flux relaxation.— flux relaxationtheory to be a local theory.
Eq.(1) shows that surface barrier affects vortex penetra-
tion. Itisthennaturaltoexpectthatthissurfacebarrier
also affects flux relaxation.
IV. CONCLUSION
In flux relaxation, the external field is usually zero
and the second term in Eq.(1) vanishes. But the first
Thegeneralfielddependentactivationenergyofavor-
term in Eq.(1) still exists. The first term is caused by
tex penetration process was proposed and time depen-
the surface imaging force (attractive), which helps to
dence of internal field was derived. Although the vor-
draw the vortices out of the superconductor and reduces
tex penetration is an inverse process to flux relaxation,
the activation energy of the vortices. Early studies32
there is no symmetry between the mathematical mod-
have shown that in flux relaxation the time dependence
of internal field is B(t) = n b wl(t), where w(t) = els of these two processes because of the surface bar-
U − kTln(1+t/τ) and b arl=e1colnstants. By further rier. Finally, it should be mentioned that I have ignored
c lP the effects of damping force on the vortex penetration
considering the surface imaging force, w(t) should be re-
when constructing the mathematical model. The damp-
placed by
ingforcemayaddsignificantinfluenceonthevortexpen-
2 etration under large applied field where the velocity of
Φ 2x t
w(t)=U − 0 K −kTln 1+ (18) vortices is large.
c 0
4πλ λ τ
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
∗ Electronic address: [email protected] 18 B. J. Baelus, A. Kanda, N. Shimizu, K. Tadano, Y.
1 A. A.Abrikosov, Soviet Phys. JETP 5, 1174 (1957). Ootuka, K. Kadowaki, and F. M. Peeters, Phys. Rev. B
2 S. Bascom, Jr. Deaver, William M. Fairbank, Phys. Rev. 73, 024514 (2006).
Lett. 7, 43 (1961). 19 A. D. Hernandez and A. Lopez, Phys. Rev. B 77, 144506
3 W. H. Kleiner, L. M. Roth and S. H. Autler, Phys. Rev. (2008).
133, A1226 (1964). 20 EranSelaandIanAffleck,Phys.Rev.B79,024503(2009).
4 P. L. Gammel, D. J. Bishop, G. J. Dolan, J. R. Kwo, C. 21 L. Burlachkov, Phys.Rev.B 47, 8056 (1993).
A.Murray,L.F.Schneemeyer,andJ.V.Waszczak,Phys. 22 John R.Clem, J. Appl.Phys.50, 3518 (1979).
Rev.Lett. 59, 2592 (1987). 23 Yasunori Mawatari and John R. Clem, Phys. Rev. Lett.
5 C.A.Murray,P.L.Gammel,andD.J.Bishop,Phys.Rev. 86, 2870 (2001).
Lett. 64, 2312 (1990). 24 Andrey A. Elistratov, Denis Yu. Vodolazov, Igor L. Mak-
6 P. E. Goa, H. Hauglin, M. Baziljevich, E. Il’yashenko, P. simov, and John R. Clem, Phys. Rev. B 66, 220506(R)
L. Gammel and T. H. Johansen, Supercond.Sci. Technol. (2002)
14, 729 (2001). 25 M. Pissas and D. Stamopoulos, Phys. Rev. B 64, 134510
7 C. P. Bean, J. D. Livingston, Phys. Rev. Lett. 12, 14 (2001).
(1964). 26 Y.M.Wang,A.Zettl,S.Ooi,andT.Tamegai,Phys.Rev.
8 R. W. De Blois, W. De Sorbo, Phys. Rev. Lett. 12, 499 B 65, 184506 (2002).
(1964). 27 Serkan Erdin, Phys.Rev.B 69, 214521 (2004).
9 A. S. Joseph, W. J. Toasch, Phys. Rev. Lett. 12, 219 28 S.Kasahara,Y.Tokunaga,N.Kameda,M.Tokunaga,and
(1964). T. Tamegai, Phys. Rev.B 71, 224505 (2005).
10 John Bardeen, M. J. Stephen, Phys. Rev. 140, A1197 29 R.Ma, Ph.D. thesis, Universityof Alberta, 2009.
(1965). 30 P. W. Anderson,Phys. Rev.Lett. 9, 309 (1962).
11 A. Gurevich and G. Ciovati, Phys. Rev. B 77, 104501 31 Michael Tinkham, Introduction to Superconductivity
(2008). (McGraw-Hill, NewYork,1996).
12 P. G. de Gennes, Sol. St. Comm. 3, 127 (1965). 32 Rongchao Ma, J. Appl.Phys. 108, 053907 (2010).
13 L. Kramer, Phys. Rev.170, 475 (1968). 33 V.B.Geshkenbein,andA.I.Larkin,Sov.Phys.JETP68,
14 Ryuzo Kato, Yoshihisa Enomoto, Sadamichi Maekawa, 639 (1989).
Phys. Rev.B 47, 8016 (1993). 34 M. R. Beasley, R. Labusch,and W.W. Webb,Phys. Rev.
15 CarlosBolech,GustavoC.Buscaglia, andA.Lopez,Phys. 181, 682 (1969).
Rev.B 52, R15719 (1995). 35 M. V. Feigel’man and V. B. Geshkenbein, and V. M. Vi-
16 G. R. Berdiyorov, L. R. E. Cabral, and F. M. Peeters, J. nokur,Phys. Rev.B 43, 6263 (1991).
Math. Phys. 46, 095105 (2005). 36 A.I.LarkinandYu.N.Ovchinnikov,J.LowTemp.Phys.
17 B.J.Baelus, K.Kadowaki,andF.M.Peeters, Phys.Rev. 34, 409 (1979).
B 71, 024514 (2005).
