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Dynamics of the flux-line lattice in high-Tc oxides
PHYSICA
Physiea C 185-189 (1991) 270-275 North-Holland
D y n a m i c s of the F l u x - L i n e Lattice in High-To Oxides Ernst H e l m u t B r a n d t *
AT~IT Bell Laboratories, Murray Hill, NJ 0797,S, USA An applied a.c. current or magnetic field penetrates into a type-II superconductor by compressional and tilt waves of the flux-line lattice. Accounting for elastic pinning, viscous drag, and linear creep of the flux lines and their images, I derive the complex a.c. penetration depth and linear s.c. resistivity and susceptibility of films and cylinders. Discussed are further nonlinear hysteretic di3sipation, thermally activated depinnivg, thermal fluctuation, flux diffusion, vibrating superconductors, and irreversibility or depinning lines.
1. W h e r e Do Currents Flow in Type-II Superconductors? A type-ll superconductor allows an applied field B, to penetrate partially in form of magnetic flux lines if B a < Bo < Bcz. The lower and upper criticalfields BcI(T) and B~2(T) vanish at the superconducting transition temperature T¢. The penetrating flux lines in the absence of pinning form a regular triangular lattice of spacing a and induction B = @oV~/2a 2 <_ B, ~_ B~2 where ~bo = h/2e -- 2.10 -is T m 2 is the quantum of flux. A sometimes overlooked fact is that the flux-linelattice (FLL) interacts with any applied d.c. or a.c. field and with induced or applied currents only at the specimen surface, in a layer of thickness ~ A, the magnetic penetration depth. One has A(T) oc (Tc-T)-t/2 near To. Any applied field generates a surface shielding current I, which exerts a pressureIsB on flux lines oriented parallei to the surface, and a tangentialforce Io~Poon the flux-line ends sticking out from the surface (magnetic monopoles). A small a.c. magnetic field or current (induced or applied by contacts) generates compressional waves [1] in the FLL when the constant field Bo (which generates this FLL) is parallel to the surface, and it generates bending waves when Bo is perpendicular to the surface, Fig. 1. If the FLL is weakly pinned with an elastic restoring force t~L per unit volume (C~Lis the Labusch parameter [2]), then such compressional or tilt waves penetrate with exponentially decreasing amplitude to a depth Ac = At1 = (CII/OtL)1/2 or A:,. = (C~.~./OL)I/2, respectively, where clt is the compressional modulus and c44 the tilt modulus of the FLL [3]. When the Ginzburg-Landau (GL) parameter ~ and B are not too small, say t¢ > 2 and B~ > B > 2B~1, then the vortex fields overlap and one has c44 = BBo/Ito B2/#o. in this case Art and A44 both equal the Campbell penetration depth Ac = (B2/poOr)l/2. Therefore, the response of the FLL to small d.c. and a.c. fields is
independent of the orientation of B. with respect to the specimen surface, provided pinning and viscous drag are isotropic (see below). For strongly anisotropic superconductors the planar results presented in this paper still apply if A, or,, and the viscosity rI [4] are interpreted as their anisotropic values for currents, wave vector k, or vortex displacements u in the given direction. Another sometimes overlooked fact is that currents in type-II superconductors can have exactly three origins: They are (a) either sudace currents restricted to a depth ~ A, or they are (b,c) caused by vortices within a radius ~ A from the vortex core; averaged over a few FLL cells, the vortex currents vanish in a regular FLL, but a finite "transport current" results if (b) the FLL has a density gradient or if (c) the FLs are curved. The two vortex contributions are easily visualized by writing the current density J as (for B ~, BQ) /JoJ = V x B = V B x !~ + B V x 1~ where 13 = B / B , B = [B I. In bulk samples typically the gradient term dominates, J ~. pfflVB, but in films the current is carried predominantly by the curvature of the flux lines [5]. In superconductors there is no current far from surface or vortices. 2. Linear a.c. Penetration D e p t h By distorting the FLL, an applied d.c. or a.c. field or current can penetrate much deeper into the superconductor than if the FLL were rigidly pinned. With ideally pinned flux lines a superconductor behaves as if it were in the Meissner stzte: A.-.y small field change or current decreases exponentially over the penetration depth A. For weak pinning, however, the perturbation field Bl can penetrate to the larger Campbell depth Ac >> A. A general treatment of the penetration problem has to account for the image vortices, which are introduced to satisfy the boundary condition that J flows parallel to the surface. It has further to account for the nonlocalities of the elastic response of the FLL and of the
"On leave from: Max-Planck-lnstitut f'dr Metallforschung,lnstitut I'fir Physik, tleisenbergstr. 1, D-7000 Stuttgart 80, FRG
E~.~ev'.,erScience PoblJsher¢ a v
E.H. Brondt / Flux-line lattice in high-T¢ oxides applied force, which originate from the long range A of the interaction of the vortices with each other and with the surface currents, and possibly with a nonlocal collective pinning force, smeared over a correlation volume V,. The penetration of a perturbation field Bt(=) into a superconducting half space was obtained recently by using an exponential ansatz [6] or Fourier transforms [7] for Bt(=) and for the vortex displacement field u(=). For local pinning the solution is simply
Bt(x > O) = B, e x p ( - x / A , c )
(1)
2 -_ A2 + e t t / o t £ = A2 + ~ . A.c
(2)
For nonlocal pinning, i.e. when the Labusch parameter aL(k) is dispersive, one gets [7]
Bt(x >_ O) = B,
o. ( - i k / t ) exp(ikx) dk oo ~:2+ [X2 + B21maL(k)l-t"
(3)
Eqs. (1-3) apply to arbitrary orientation of Bo ~ B = ( B = , 0 , B , ) w.r.t, the surface since A, ~. A~ for B > 2 B a . Here and below I assume B, IIz. If nollV, only tilt waves are generated and the penetration depth is reduced to A-~e = A~+ B~/POaL. In general, B~ = [0, e,~ exp(-x/~,~), e,, exp(--x/Aab)].
3. C o n s : - d e r i n g
Flux
Flow
If pinning is absent one has ~ = 0 and thus A~, = Ac = oo. The flux lines then can move freely, feeling only a viscous drag force density - t / v where v = 0 u / ~ is the vortex velocity, r / = B~/pFr the flux-flow viscosity, and PFF "~ p , ( T ) B / B ~ ( T ) the flux-flow resistivity. The superconductor then behaves like a normal conductor with anisotropic linear resistivity p = pFt: for currents £ B, and p = 0 for currents [[ B. Consequently, an a.c. field of frequency o~]2rr will penetrate to a skin depth = (Wpo/2pFF) 1/~. One may say that the flux lines diffuse with diffusivity D = PFF/lio, thus ~ = (w/2D) ~/2. More precisely, using the induction laws 13 = X7 x E and E = B x v = B x J x B / ~ / = (B2/~)J± = PFFJ.L one finds [8] that the internal field B(r, t) follows the equation of motion
271
are now complex, but only the real parts of u and Bt have physical meaning. One has then still (I) with
~L = x2 + B21~o(aL + i,~).
(6)
For # = 0 (6) reproduces (l, 2) and for ol, = 0 gives
B,(=, t) = B. exp(-=lO cos(=IS - ~t) where S is the skin depth from above.
4. C o n s i d e r i n g
Flux
Creep
In high-T~ superconductors (HTSC) at sufficiently large temperatures T thermally activated depinning is observed in form of flux creep, which causes the persistent currents or the irreversible (pinning-caused) magnetization to decay. At large J ;3, Jc (Jc = critical current density) this decay is caused by the usual flux flow and is described by (5), yielding J o t e x p ( - t / r ) with r = d2/r2D for specimens of thickness d at times t ~I~ r [8,10-12]. At current densities J ~ Jc, the currentvoltage curve is highly nonlinear, V cx exp(J/Ji), and the decay is typically logarithmic, J 0¢ const. - In(t/to). I consider hcre the linear (ohmic) respond, which occurs at d >", Jc in the region of usual flux flow, and again at J << JckT/U in the region of thermally assisted flux flow (TAFF) [10]. The TAFF resistivity is strongly T dependent, PTAFF ~" PFF exp(- U / k T ) where U(B, T) is an activation energy. In the TAFF reg;me, eqs. (4-7) apply with p r r replaced by PrAtt, i.e., p, D = p/#o, and l / r / = p / B 2 are reduced by a factor e x p ( U / k T ) . Thermally activated depinning, or any other linear relaxation mechanism, can be incorporated into the complex penetration depth (6) by noting that after a sudden step-like displacement of the vortex lattice the ela,~tic pinning force decreases exponentially in time, with a relaxation time r >> ~'o, e.g. r = r 0 e x p ( U / k T ) , where ro = ~/C~L is the flux-flow relaxation time of pinned flux lines. In general, the elastic pinning force density P(t) caused by a unifo.,rm FLL displacement u(t) is
Pc,(O -- - a L
Z ,i(t - t') e x p ( - t ' / O
(4)
(Fig. 2a). In time Fourier space this means
For B ( r , t ) ~ const. = Bo, (4) may be linearized to give
P~,(~,) -- --~L a(~)/(] -- i/~,~').
f~ = g~tV x p r r B x I~1 x V x B . 13 = DV2B + PFFV x JU + O( IB - B0[ ~ )
(5)
where D = prr(Bo)/#o. Thus, in geometries where the current component JII parallel to B vanishes, flux flow is equivalent to a linear diffusion of the fluz lines, [I = DS72B as shown previously for Bllz [9,10]. Flux flow is easily incorporated into the a.c. penetration problem by replacing aL with oL + iwr~ in (2) or (3). The elastic force density - ~ u is nov,"supplemented by the drag force -~/v = -~lOu/Ot = -~liwu where , ~ , u(x, t) = u(x) exp(ito/), and Bn(x, t) = B~(x) exp(iwt)
(7)
dr'
(S)
(9)
Therefore, flux flow and creep are considered by replacing aL with oL/(1 - i/wr) + iwri. This gives our final I, resulL
~or
t i*l e "
comptex ~
I
a.c.
p
At
ene[r,ttioh uepL.
A2~, or, when r = ~'oexp(~-U/kT) >> 7"o = r//OL, 1 - i/wr The surface impedance is Z, = iWOo~,~.
E.H. lkandt / Flux.line lattice in high.Tc oxides
272
5. L i n e a r a . c . S u s c e p t i b i l i t y Extending the above considerations to superconducting films ([x[ _< d/2) [10-12] and cylinders (r <_ R) [13] in parallel a.c. field one obtains instead of (1) B~(lz[ _< d/2) = Bo cosh(z/~o~)/coeh(d/2,~,c)
(12)
B,(,- _< R) = B. to(,-/~.,)/Xo(R/,~.o).
(13)
with ,~o~ from (11). lo(u) is a modified Bessel function. Averaging B! over the specimen one gets the complex a.c. susceptibility
B is along z) yield the a.c. resistivity P~c(') = E / J = • 2 twpo~,c, which is independent of z, Fig. 2b,
p.Jw) = iwpo)O + PrArF(1 + i w , ) / ( l + itoro),
(19)
P'I'AFF ~" PFFTo/T ~ PFFeXD(--U/~BT ) ~, PFF. Thus,
p,Jw) ~ p,c(0~) ~
praPr itopo(A a + ~ )
for ~o~ 1/1" (20) for 1/1"~I::~<:: l/To (21)
PoJ') ~ pvv for 1/1"o
d
(14)
e with u = ~ .
(I~)
( 2 D / w ) !1~ as in normal conducting metals. For interrne. d i a t e frequencies, poc is imaginary and oc w; the surface
(I~ dlo/du).The decomposition of the very general results (12) to (15) into real and imaginary parts is straightforward. In the inductive case 1/~" <:~ to 1/1"o the argument is red; for the film one then has p(w) ~ 1/u -- 2(A ~ + ~)~l~/d for d ~ and #(to) 1 - u2/3 = 1 - d~/12(A ~ + A~) for d ~ A. In the ohmic (diffusive) cases at low and large frequencies, the known skin effect behavior results, with a complex argument u = (1 + i)(tod~/8D)~l 2 where the flux diffusivities are
currents are then nearly loss free due to strong elastic pinning; the shielding in this case is like in the Meissner state but with a larger penetration depth (~2 + ~)V~. There is always a weak linear damping from (19) even in the shielding state (21) with Re{p,~} = P r A t t +pFFto%'O2 ~lm{po~}. But in a.c. experiments a typically much larger nonlinear (amplitude dependent) attenuation is observed, in particular in the very sensitire measurements on vibrating superconductors [14-20]. These hysteretic losses ezceed the linear viscous drag losses, in particular at low frequencies.
p(w) = (Bt(x))
#(u.,)
tanh(u)
(B_~r)) 2/~(u) -. =.Io(.)
with u =
=
~,
D = ~r~vr/~o =X~/¢ D -- PFF/IAO A~/ro
for t o ~ l / r
(16)
= for 1/~'0 ~ w. (17) One then has explicitly for the slab [10,12]
#(to) = (sinh v + sin v) - i(sinh v - sin v)
(cosh ~ + ~os ~) with v = (toan/2D)t/~ = d/6, Fig. 3a.
(18)
6. L i n e a r R e s i s t i v i t y From (1) the Maxwell equations for the current density J(x) = p~XOB/Ox = B~(x)/poA~ and the electric field E(x) = ito)~o~Bt(x) (J and E are along p when
zl
~Bac, B
• I,~fflffffflff,qfffffflfHf
IB=,-
7. N o n l i n e a r
Resistivities
In a.c experiments and in vibrating superconductors [14-20], in the inductive regime (21) the dissipation is often larger than given by the linear response (19). Rather small a.c. supercurrents lead to depinning processes, even at zero temperature. Due to the elasticity of the FLL and randomness of the pins, there are always a few flux lines which are on the verge of jumping over a flat barrier to a new metastable position. During such jumps elastic energy is dissipated. This non-
Pel
B .-
Pac= p'+ ip"
PFF
compression
tilt ~.44 = ~.O
x '~
FIG. 1. Compressional waves (left) and tilt waves (right) of the FLL (indicated by arrows) penetrate to a depth Ac >> ,~ if pinning is weak. )~ is the penetration depth for Meissner shielding currents.
I
I.
PTAFF
I
FIG. 2. a) Relaxation of the pinning force (8) by TAFF (IJnci~i creep, visco-elastic behavior), b) Complex a.c. resistivity (19) which accounts for elastic pinning, viscous drag, and linear flux creep.
E.H. Bmndt
/
Fluxqine ~attice in ~i~.T¢ oxides
linear dissipation is given by the area of a tiny hysteresis loop in the force-displacement curve P(u) times the frequency u = o~/2x [14]. In contrast, the vis. eous (linear) dissipation is r/va, which means constant attenuation P defined by u(t) o~ cos(~ot)exp(-~t), o r u(w) (x (w~+P) -~ (resonance curve). Hysteretic damping at smallamplitudes u has I" oc u", n > 1, but at large u it is a constant frictional force, thus 1" oc l / u . A different nonlinearity at T -- 0 is that of the d.c. current-voltage curve E ( J ) for J > Jc. One-dimensional (ID) viscous motion of a FL in a periodic potential yields E = P~r(J~ ja)tls for J ~ Je, else g = 0 [21]. Other models propose E = Prr(Je - J). Both models fit some experiments. A more general fit with E = PrF(J~ - J'~)ff'~, 7 > 0, seems appropriate but cannot fit the observe inflection point in E ( J ) which may be caused by an inhomogeneous Jc(r). -
8. T h e r m a l l y A c t i v a t e d Depinning A third nonlinearity of E ( J ) is caused by thermally activated depinning at T > 0. Anderson's [22] model assumes activated jumps of pinned flux-line bundels with and against the Lorentz force, with jump rates v+ and v_, jump volume V, and jump width I. The energy gain (barrier reduction) during one forward jump is W = J B V I . The microscopic parameters V and l are model dependent, but the resulting electric field may by written in terms of the phenomenological parameters [8] d~(B) (critical current density at T = 0, J¢ = U/BVI, iV = dU/d¢), p . ( B , T ) (resistivity at d = d¢), and U ( B , T ) (activation energy for flux jumps),
E ( J ) = pJ = v B = (v+
-
v_)lB
= 2p~J~ e x p ( - U / k ~ T ) sinh(JU/J¢ksT).
(23)
At small current densities J ~ J~ = J¢ksT/U one can linearize (23) and gets the ohmic regime of thermally
~7~
assisted flux flow (TAFF) [I0] as recognized first by Dew-Hughes [23]. At larger currents, E(J) becomes nonlinear (flux creep regime) and goes o~er into the usual flux flow at J ~ Jc. One ha, three resistivity
regimes, Fig. 4, p ~ prrexp(-U/ksT) PT~rF for J ~, Jt (24) p~ exp(Jl&) for J ~ J~ (2~) P~ prF(I - j~Ija)If~ ~ pvr for J ) Je. (26) At lower T and very small currents a diff6~-~t nonlinear E(J) is predicted for the ~mrtex glass ~ state, =
p ~ exp[ - ( J 2 / J ) ° ].
This follows from scaling theories [24-26] or from ther. really activated motion of vortex kinks [27.29]. Both pictures yield an activation energy U(J) c~ I / J °, o ~ 1. which diverges as J - . 0 since larger and larger FLL volumes V have to jump if the FLL is elastic. Experiments confirm the linear TAFF regime [30] and the vortex-glass predictions [31-33] in YBCO films {311,single crystals [30,32], and ceramics [33]. But as yet I personally do not see convincing evidence that there is a finite "glass temperature" Tgt,,, [24,31-33] at which U(T, B) diverges and the pinned FLL ~freezes". The prediction of freezing as d ~ 0 appears to apply to any 3D elastic FLL. A vortex glass s:ate does not occur in 2D F'LL [34-36] nor in flux-line liquids [27,37] where there is no elastic correlation.
9. T h e r m a l Fluctuation A further open question is whether a 3D FLL has a sharp melting transition and is thermally "entangled" [38,39]. The thermal fluctuations (u:) of the FLL are large [37-43], in particular when its correct nonlocal elasticity [3,11,41-48] (dispersive tilt modulus [46,47]) and anisotropy [41,45-47] are considered. One has
(u2)~t'gTk4~rBOocsr,) !u=p.'+ iB" '
~
a
res
b
F~ t~
TAFF
'
m~ ([3) Tc
"T
FIG. 3. a) Complex s.c. susceptibility of HTSC slabs (14, 18). b) Resonance frequency and damping peak of vibrating HTSC. Below the depinning temperature the damping depends on the vibrational amplitude (dashed lines).
(27)
X~B-~'2_--~)
x~-~b'(2~-q)
(c~ = shear modulus of the FLL, ~c = GL parameter). For B << B~2 with c~ = B 4 ' o / ( 1 6 ~ o ~ b ) inserted. (28) yields (u 2) ~ ksTpo)~sA~ksz/(~oB) = kBTpo (4~r/B~,3o)l/2~s~. The factors in (28) are: The local result [38]; the nonlocal correction factor (Bs2/2B~2) ~/2 (Blntc/4B~t) I/2 >> 1; the anisotropy ratio ,X~/X~b 5 for YBCO and ~ 60 for BSCCO. For example, at B = 1T. T = 77K..X~b = 200am yields for YBCO (u2) '/2 = 3.2nm = 0.065a (FL spacing a = 49am). Apart from softening of the FLL (which should actually increase pinning [48-50]) these fluctuations have the effect of smearing the pinning potential, i.e., they increase the pin range rp ,m (~2 + (u2)2)1/~ [37,51]: this decreases pinning. The extension of the static pinning
F,.H. Brandt / Flux-line lattice in high-Tc oxides
274
summation at T = 0 [48-50] to dynamic creep at T > 0 is given in [25,3T,51]. For fluctuations in the 2D vortexpoirt lattice of layered superconductors, and for its dislocation-mediated melting and creep, see [34-36]. 2D melting of this point lattice probably does occur. Both pinning-caused distortions and thermal fluctuations of the FLL can rednce the spatial variation of the magnetic field B(r) in anisotropie or layered HTSC [52].
10. Vibrating Superconductors Tilting a superconductor periodically in a constant magnetic field B, m B by an angle ~b is equivalent to applying an a.c. magnetic field witl~ amplitude B, sin ~b perpendicular to Bo. Superconducting vibrating reeds [14-17l, and platelets glued on a high-quality silicon oscillator [18] or suspended on a wire [19,20], can therefore be explained by the same theory above which explains a.c. experiments. Vibrating superconductors exhibit a sharp peak in their attenuation (Fig. 3b) at a temperature T~(B) which was interpreted as a FLL melting temperature in [19] but very likely indicates the onset of depinning. In conventional superconductors, this depinning-peak occurs near the upper critical field Bcz(T), or temperature To(B), where pinning has to go to zero. In HTSC, this depinning is thermally activated and reflects the maximum in the imaginary part of the a.c. susceptibility, e.g. in eqs. (15) and (18) and Fig. 3a. As stated above, in many cases the dissipation is of the hysteretic type and amplitude dependent; but even then the peak should typically occur close to the linear damping peak, and a change of the vibration amplitude should ahnost not shift its position but may change its
E
height. The attenuation peak occurs when the diffusion length of flux coincides with a typical length or thickness of the specimen. The peak is so sharp because the flux diffusivity D(B, T) changes rapidly with T (and often also with B). The resonance frequency v of a vibrating superconductor typically increases when a field B. is applied, because B, ~pulls" at the flux lines and these couple to the material by pinning, thus providing an additional restoring force. Above the damping peak the frequency enhancement/in vanishes because of complete depinning. Thus, typically/in cc B,~. However, for very short flux lines, liv o¢ aL cc B z/2 as shown in [1'I]. A different restoring force is the anisotropy of the magnetization, which in very weak-pinning specimens can even he negative and thus reduces u [20].
11. Depinning Lines The damping peak defines an "irreversibility line" or "depinning line" which separates in the B, T-plane the regions of rigid pinning and of complete thermal depinning (i.e. the usual flux flow), Fig. 5. Depinning lines can also be obtained from other measurements [11], e.g.: Resistive transition p(B, T); vanishing irreversibility in the magnetization curve; maximum a.c. dissipation; diverging a.c. penetration depth [53]; peak in the ultrasonic attenuation [54]; conduction noise in HTSC films [55]. All these depinning lines define a line D(B,T) = const, where the constant depends on the length and time scales of the experiment. Reduction of the specimen size (grains embedded in epoxy) shifts the depinning line to lower T and B [1'1]. This work was supported by the German Bundesministerium f~r Forschung and Technologie.
log B [T]
j
10 -
g,iss7 z,,,,//
~
F,ux,,o,,,,,
Depinning lines
~ u x _
flow
Flux creep "
Flux creep Jc
~j
FIG. 4. Current-voltage curves of HTSC at various temperatures and fields, schematic. The insert visualizes the jumping flux-line bundel.
vortex-glass? 0.11
0.7
I 0.8
~ i
0.9
'~ "
r'l
..
"]"
1.0 To
FIG. 5, Depinning lines of HTSC in the B, T plane mark the TAFF region and separate the regions of strong pinning and of pin,free flux-flow behavior.
E.H. Brandt / Fluxdine lattice in high-Tc oxides
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275
[27] S. Chakravarty, B.I. Ivlev, and Yu.N. Ovchlnnikov, Phys. Rev. Left. 64 (1990) 3187; Phys. Rev. B42 (1990) 2143. [28] B.I. Ivlev, Yu.N. Ovchinnikov, V.L. Pokrovsk~ Europhys. Lett. 13 (1990) 187; B.I. lvlev and N.B. Kopnin, Phys. Rev. Lett. 64 (1990) 1828; J. Low Temp. Phys. 80 (1990) 161; M. Tachiki and S. Takahaa~, Phydca B 100, (1991) 121. [29] G. Blatter, B. Ivlev, and J. Rhyner, Phys. Rev. Lett. 68 (1991) 2392. D. Feinberg and C. ViUard, Phys. Rev. Lett. 85 (1990) 910. [30] T.T.M. Palstra, B. Battlogg, R.B. van Dover, L.F. Schneemeyer, and J.V. Waszczak, Phys. Rev. B 41 (1990) 6621; Appl. Phys. Left. 54, (1989) 763. [31] R.H. Koch, V. Foglietti, W.J. Gallagher, G. Koren, A. Gupta, and M.P.A. Fisher, Phys. Rev. Lett. 83 (1989) 1511. [32] P.L. Gammel, L.F. Schneemeyer, and D.J. Bishop, Phys. Rev. Lett. 68 (1991) 953. [33] T.K. Worthington, E. Olsson, C.S. Nichols, T.M. Shaw, and D.R. Clarke, Phys. Rev. B43 (1991) 10538. [34] M.V. Feigel'man, V.B. Geshkenbein, A.I. Larkin, Physica C 167 (1990) 177. [35] V.M. Vinokur, P.H. Kes, and A.E. Koshelev, Physica C168 (1990) 29. [36] C.J. van der Beek and P.H. Kes, Phys. Rev. B43 (1991) 12843. [37] V.M. Vinokur, M.V. Feigel'man, V.B. G~shkenbein, and A.I. Larkin, Phys. Rev. Lett. 65 (1990) 259. [38] D.R. Nelson and H.S. Seung, Phy.¢. Ret,. B 39 (1990) 9174. [39] M.c'. Marchetti and D.R. Nelson, Phys. Rev. B 42 (1990) 9938. and Physica C 174 (1991) 40. [40] M.A. Moore, Phys. Re,, _R gO (!989) 917L [41] A. Houghton, R. A. Pelcovits, and A. Sudbo, Phys. Rev. B 40 (1989) 6T63. [42] E.H. Brandt, Phys. Rev. Lett. 63 (1989) 1106. [43] E.H. Brandt, Physica C 162-164 (1989) 1167. [44] E.H. Brandt and U. Essmann, phys. star. sol. (b) 144 (1987) 13 (review on the FLL). [45] A. Sudb0 and E.H. Brandt, Phys. Rev. B 43 (1991) 10482. [46] A. Sudb~ and E.H. Brandt, Phys. Rev. Left. 66 (1991) 1781. [47] E.H. Brandt and A. Sudb~ Physica C(submitted). [48] A.I. Larkin and Yu.N. Ovchinnikov, J. Low Temp. Phys. 43 (1979) 109; [49] R. Kerchner, J. Low Temp. Phys. 46 (1981) 205; 50 f 1 O O 1 ' $ "~Q "Y
[50] E.H. Brandt, J. Low Temp. Phys. 64 (1986) 375. [51] M.V. Feigel'man and V.M. Vinokur, Phys. Rev. B 41 (1990) 8986. [52] E.g. Brandt, Phys. Rev. Left. 66 (1991) 3213; Phys. Rev. B 37 (1988) 2349. [53] A. Hebard, P. Gammel, C. Rice, and A. Levi, Phys. Rev. B 40 (1989) 5243. [54] J. Pankert, Physica C 168 (1990) 335. [55] A. Maeda et al. Physica B 165& 166 (1990) 1363.
Physiea C 185-189 (1991) 270-275 North-Holland
D y n a m i c s of the F l u x - L i n e Lattice in High-To Oxides Ernst H e l m u t B r a n d t *
AT~IT Bell Laboratories, Murray Hill, NJ 0797,S, USA An applied a.c. current or magnetic field penetrates into a type-II superconductor by compressional and tilt waves of the flux-line lattice. Accounting for elastic pinning, viscous drag, and linear creep of the flux lines and their images, I derive the complex a.c. penetration depth and linear s.c. resistivity and susceptibility of films and cylinders. Discussed are further nonlinear hysteretic di3sipation, thermally activated depinnivg, thermal fluctuation, flux diffusion, vibrating superconductors, and irreversibility or depinning lines.
1. W h e r e Do Currents Flow in Type-II Superconductors? A type-ll superconductor allows an applied field B, to penetrate partially in form of magnetic flux lines if B a < Bo < Bcz. The lower and upper criticalfields BcI(T) and B~2(T) vanish at the superconducting transition temperature T¢. The penetrating flux lines in the absence of pinning form a regular triangular lattice of spacing a and induction B = @oV~/2a 2 <_ B, ~_ B~2 where ~bo = h/2e -- 2.10 -is T m 2 is the quantum of flux. A sometimes overlooked fact is that the flux-linelattice (FLL) interacts with any applied d.c. or a.c. field and with induced or applied currents only at the specimen surface, in a layer of thickness ~ A, the magnetic penetration depth. One has A(T) oc (Tc-T)-t/2 near To. Any applied field generates a surface shielding current I, which exerts a pressureIsB on flux lines oriented parallei to the surface, and a tangentialforce Io~Poon the flux-line ends sticking out from the surface (magnetic monopoles). A small a.c. magnetic field or current (induced or applied by contacts) generates compressional waves [1] in the FLL when the constant field Bo (which generates this FLL) is parallel to the surface, and it generates bending waves when Bo is perpendicular to the surface, Fig. 1. If the FLL is weakly pinned with an elastic restoring force t~L per unit volume (C~Lis the Labusch parameter [2]), then such compressional or tilt waves penetrate with exponentially decreasing amplitude to a depth Ac = At1 = (CII/OtL)1/2 or A:,. = (C~.~./OL)I/2, respectively, where clt is the compressional modulus and c44 the tilt modulus of the FLL [3]. When the Ginzburg-Landau (GL) parameter ~ and B are not too small, say t¢ > 2 and B~ > B > 2B~1, then the vortex fields overlap and one has c44 = BBo/Ito B2/#o. in this case Art and A44 both equal the Campbell penetration depth Ac = (B2/poOr)l/2. Therefore, the response of the FLL to small d.c. and a.c. fields is
independent of the orientation of B. with respect to the specimen surface, provided pinning and viscous drag are isotropic (see below). For strongly anisotropic superconductors the planar results presented in this paper still apply if A, or,, and the viscosity rI [4] are interpreted as their anisotropic values for currents, wave vector k, or vortex displacements u in the given direction. Another sometimes overlooked fact is that currents in type-II superconductors can have exactly three origins: They are (a) either sudace currents restricted to a depth ~ A, or they are (b,c) caused by vortices within a radius ~ A from the vortex core; averaged over a few FLL cells, the vortex currents vanish in a regular FLL, but a finite "transport current" results if (b) the FLL has a density gradient or if (c) the FLs are curved. The two vortex contributions are easily visualized by writing the current density J as (for B ~, BQ) /JoJ = V x B = V B x !~ + B V x 1~ where 13 = B / B , B = [B I. In bulk samples typically the gradient term dominates, J ~. pfflVB, but in films the current is carried predominantly by the curvature of the flux lines [5]. In superconductors there is no current far from surface or vortices. 2. Linear a.c. Penetration D e p t h By distorting the FLL, an applied d.c. or a.c. field or current can penetrate much deeper into the superconductor than if the FLL were rigidly pinned. With ideally pinned flux lines a superconductor behaves as if it were in the Meissner stzte: A.-.y small field change or current decreases exponentially over the penetration depth A. For weak pinning, however, the perturbation field Bl can penetrate to the larger Campbell depth Ac >> A. A general treatment of the penetration problem has to account for the image vortices, which are introduced to satisfy the boundary condition that J flows parallel to the surface. It has further to account for the nonlocalities of the elastic response of the FLL and of the
"On leave from: Max-Planck-lnstitut f'dr Metallforschung,lnstitut I'fir Physik, tleisenbergstr. 1, D-7000 Stuttgart 80, FRG
E~.~ev'.,erScience PoblJsher¢ a v
E.H. Brondt / Flux-line lattice in high-T¢ oxides applied force, which originate from the long range A of the interaction of the vortices with each other and with the surface currents, and possibly with a nonlocal collective pinning force, smeared over a correlation volume V,. The penetration of a perturbation field Bt(=) into a superconducting half space was obtained recently by using an exponential ansatz [6] or Fourier transforms [7] for Bt(=) and for the vortex displacement field u(=). For local pinning the solution is simply
Bt(x > O) = B, e x p ( - x / A , c )
(1)
2 -_ A2 + e t t / o t £ = A2 + ~ . A.c
(2)
For nonlocal pinning, i.e. when the Labusch parameter aL(k) is dispersive, one gets [7]
Bt(x >_ O) = B,
o. ( - i k / t ) exp(ikx) dk oo ~:2+ [X2 + B21maL(k)l-t"
(3)
Eqs. (1-3) apply to arbitrary orientation of Bo ~ B = ( B = , 0 , B , ) w.r.t, the surface since A, ~. A~ for B > 2 B a . Here and below I assume B, IIz. If nollV, only tilt waves are generated and the penetration depth is reduced to A-~e = A~+ B~/POaL. In general, B~ = [0, e,~ exp(-x/~,~), e,, exp(--x/Aab)].
3. C o n s : - d e r i n g
Flux
Flow
If pinning is absent one has ~ = 0 and thus A~, = Ac = oo. The flux lines then can move freely, feeling only a viscous drag force density - t / v where v = 0 u / ~ is the vortex velocity, r / = B~/pFr the flux-flow viscosity, and PFF "~ p , ( T ) B / B ~ ( T ) the flux-flow resistivity. The superconductor then behaves like a normal conductor with anisotropic linear resistivity p = pFt: for currents £ B, and p = 0 for currents [[ B. Consequently, an a.c. field of frequency o~]2rr will penetrate to a skin depth = (Wpo/2pFF) 1/~. One may say that the flux lines diffuse with diffusivity D = PFF/lio, thus ~ = (w/2D) ~/2. More precisely, using the induction laws 13 = X7 x E and E = B x v = B x J x B / ~ / = (B2/~)J± = PFFJ.L one finds [8] that the internal field B(r, t) follows the equation of motion
271
are now complex, but only the real parts of u and Bt have physical meaning. One has then still (I) with
~L = x2 + B21~o(aL + i,~).
(6)
For # = 0 (6) reproduces (l, 2) and for ol, = 0 gives
B,(=, t) = B. exp(-=lO cos(=IS - ~t) where S is the skin depth from above.
4. C o n s i d e r i n g
Flux
Creep
In high-T~ superconductors (HTSC) at sufficiently large temperatures T thermally activated depinning is observed in form of flux creep, which causes the persistent currents or the irreversible (pinning-caused) magnetization to decay. At large J ;3, Jc (Jc = critical current density) this decay is caused by the usual flux flow and is described by (5), yielding J o t e x p ( - t / r ) with r = d2/r2D for specimens of thickness d at times t ~I~ r [8,10-12]. At current densities J ~ Jc, the currentvoltage curve is highly nonlinear, V cx exp(J/Ji), and the decay is typically logarithmic, J 0¢ const. - In(t/to). I consider hcre the linear (ohmic) respond, which occurs at d >", Jc in the region of usual flux flow, and again at J << JckT/U in the region of thermally assisted flux flow (TAFF) [10]. The TAFF resistivity is strongly T dependent, PTAFF ~" PFF exp(- U / k T ) where U(B, T) is an activation energy. In the TAFF reg;me, eqs. (4-7) apply with p r r replaced by PrAtt, i.e., p, D = p/#o, and l / r / = p / B 2 are reduced by a factor e x p ( U / k T ) . Thermally activated depinning, or any other linear relaxation mechanism, can be incorporated into the complex penetration depth (6) by noting that after a sudden step-like displacement of the vortex lattice the ela,~tic pinning force decreases exponentially in time, with a relaxation time r >> ~'o, e.g. r = r 0 e x p ( U / k T ) , where ro = ~/C~L is the flux-flow relaxation time of pinned flux lines. In general, the elastic pinning force density P(t) caused by a unifo.,rm FLL displacement u(t) is
Pc,(O -- - a L
Z ,i(t - t') e x p ( - t ' / O
(4)
(Fig. 2a). In time Fourier space this means
For B ( r , t ) ~ const. = Bo, (4) may be linearized to give
P~,(~,) -- --~L a(~)/(] -- i/~,~').
f~ = g~tV x p r r B x I~1 x V x B . 13 = DV2B + PFFV x JU + O( IB - B0[ ~ )
(5)
where D = prr(Bo)/#o. Thus, in geometries where the current component JII parallel to B vanishes, flux flow is equivalent to a linear diffusion of the fluz lines, [I = DS72B as shown previously for Bllz [9,10]. Flux flow is easily incorporated into the a.c. penetration problem by replacing aL with oL + iwr~ in (2) or (3). The elastic force density - ~ u is nov,"supplemented by the drag force -~/v = -~lOu/Ot = -~liwu where , ~ , u(x, t) = u(x) exp(ito/), and Bn(x, t) = B~(x) exp(iwt)
(7)
dr'
(S)
(9)
Therefore, flux flow and creep are considered by replacing aL with oL/(1 - i/wr) + iwri. This gives our final I, resulL
~or
t i*l e "
comptex ~
I
a.c.
p
At
ene[r,ttioh uepL.
A2~, or, when r = ~'oexp(~-U/kT) >> 7"o = r//OL, 1 - i/wr The surface impedance is Z, = iWOo~,~.
E.H. lkandt / Flux.line lattice in high.Tc oxides
272
5. L i n e a r a . c . S u s c e p t i b i l i t y Extending the above considerations to superconducting films ([x[ _< d/2) [10-12] and cylinders (r <_ R) [13] in parallel a.c. field one obtains instead of (1) B~(lz[ _< d/2) = Bo cosh(z/~o~)/coeh(d/2,~,c)
(12)
B,(,- _< R) = B. to(,-/~.,)/Xo(R/,~.o).
(13)
with ,~o~ from (11). lo(u) is a modified Bessel function. Averaging B! over the specimen one gets the complex a.c. susceptibility
B is along z) yield the a.c. resistivity P~c(') = E / J = • 2 twpo~,c, which is independent of z, Fig. 2b,
p.Jw) = iwpo)O + PrArF(1 + i w , ) / ( l + itoro),
(19)
P'I'AFF ~" PFFTo/T ~ PFFeXD(--U/~BT ) ~, PFF. Thus,
p,Jw) ~ p,c(0~) ~
praPr itopo(A a + ~ )
for ~o~ 1/1" (20) for 1/1"~I::~<:: l/To (21)
PoJ') ~ pvv for 1/1"o
d
(14)
e with u = ~ .
(I~)
( 2 D / w ) !1~ as in normal conducting metals. For interrne. d i a t e frequencies, poc is imaginary and oc w; the surface
(I~ dlo/du).The decomposition of the very general results (12) to (15) into real and imaginary parts is straightforward. In the inductive case 1/~" <:~ to 1/1"o the argument is red; for the film one then has p(w) ~ 1/u -- 2(A ~ + ~)~l~/d for d ~ and #(to) 1 - u2/3 = 1 - d~/12(A ~ + A~) for d ~ A. In the ohmic (diffusive) cases at low and large frequencies, the known skin effect behavior results, with a complex argument u = (1 + i)(tod~/8D)~l 2 where the flux diffusivities are
currents are then nearly loss free due to strong elastic pinning; the shielding in this case is like in the Meissner state but with a larger penetration depth (~2 + ~)V~. There is always a weak linear damping from (19) even in the shielding state (21) with Re{p,~} = P r A t t +pFFto%'O2 ~lm{po~}. But in a.c. experiments a typically much larger nonlinear (amplitude dependent) attenuation is observed, in particular in the very sensitire measurements on vibrating superconductors [14-20]. These hysteretic losses ezceed the linear viscous drag losses, in particular at low frequencies.
p(w) = (Bt(x))
#(u.,)
tanh(u)
(B_~r)) 2/~(u) -. =.Io(.)
with u =
=
~,
D = ~r~vr/~o =X~/¢ D -- PFF/IAO A~/ro
for t o ~ l / r
(16)
= for 1/~'0 ~ w. (17) One then has explicitly for the slab [10,12]
#(to) = (sinh v + sin v) - i(sinh v - sin v)
(cosh ~ + ~os ~) with v = (toan/2D)t/~ = d/6, Fig. 3a.
(18)
6. L i n e a r R e s i s t i v i t y From (1) the Maxwell equations for the current density J(x) = p~XOB/Ox = B~(x)/poA~ and the electric field E(x) = ito)~o~Bt(x) (J and E are along p when
zl
~Bac, B
• I,~fflffffflff,qfffffflfHf
IB=,-
7. N o n l i n e a r
Resistivities
In a.c experiments and in vibrating superconductors [14-20], in the inductive regime (21) the dissipation is often larger than given by the linear response (19). Rather small a.c. supercurrents lead to depinning processes, even at zero temperature. Due to the elasticity of the FLL and randomness of the pins, there are always a few flux lines which are on the verge of jumping over a flat barrier to a new metastable position. During such jumps elastic energy is dissipated. This non-
Pel
B .-
Pac= p'+ ip"
PFF
compression
tilt ~.44 = ~.O
x '~
FIG. 1. Compressional waves (left) and tilt waves (right) of the FLL (indicated by arrows) penetrate to a depth Ac >> ,~ if pinning is weak. )~ is the penetration depth for Meissner shielding currents.
I
I.
PTAFF
I
FIG. 2. a) Relaxation of the pinning force (8) by TAFF (IJnci~i creep, visco-elastic behavior), b) Complex a.c. resistivity (19) which accounts for elastic pinning, viscous drag, and linear flux creep.
E.H. Bmndt
/
Fluxqine ~attice in ~i~.T¢ oxides
linear dissipation is given by the area of a tiny hysteresis loop in the force-displacement curve P(u) times the frequency u = o~/2x [14]. In contrast, the vis. eous (linear) dissipation is r/va, which means constant attenuation P defined by u(t) o~ cos(~ot)exp(-~t), o r u(w) (x (w~+P) -~ (resonance curve). Hysteretic damping at smallamplitudes u has I" oc u", n > 1, but at large u it is a constant frictional force, thus 1" oc l / u . A different nonlinearity at T -- 0 is that of the d.c. current-voltage curve E ( J ) for J > Jc. One-dimensional (ID) viscous motion of a FL in a periodic potential yields E = P~r(J~ ja)tls for J ~ Je, else g = 0 [21]. Other models propose E = Prr(Je - J). Both models fit some experiments. A more general fit with E = PrF(J~ - J'~)ff'~, 7 > 0, seems appropriate but cannot fit the observe inflection point in E ( J ) which may be caused by an inhomogeneous Jc(r). -
8. T h e r m a l l y A c t i v a t e d Depinning A third nonlinearity of E ( J ) is caused by thermally activated depinning at T > 0. Anderson's [22] model assumes activated jumps of pinned flux-line bundels with and against the Lorentz force, with jump rates v+ and v_, jump volume V, and jump width I. The energy gain (barrier reduction) during one forward jump is W = J B V I . The microscopic parameters V and l are model dependent, but the resulting electric field may by written in terms of the phenomenological parameters [8] d~(B) (critical current density at T = 0, J¢ = U/BVI, iV = dU/d¢), p . ( B , T ) (resistivity at d = d¢), and U ( B , T ) (activation energy for flux jumps),
E ( J ) = pJ = v B = (v+
-
v_)lB
= 2p~J~ e x p ( - U / k ~ T ) sinh(JU/J¢ksT).
(23)
At small current densities J ~ J~ = J¢ksT/U one can linearize (23) and gets the ohmic regime of thermally
~7~
assisted flux flow (TAFF) [I0] as recognized first by Dew-Hughes [23]. At larger currents, E(J) becomes nonlinear (flux creep regime) and goes o~er into the usual flux flow at J ~ Jc. One ha, three resistivity
regimes, Fig. 4, p ~ prrexp(-U/ksT) PT~rF for J ~, Jt (24) p~ exp(Jl&) for J ~ J~ (2~) P~ prF(I - j~Ija)If~ ~ pvr for J ) Je. (26) At lower T and very small currents a diff6~-~t nonlinear E(J) is predicted for the ~mrtex glass ~ state, =
p ~ exp[ - ( J 2 / J ) ° ].
This follows from scaling theories [24-26] or from ther. really activated motion of vortex kinks [27.29]. Both pictures yield an activation energy U(J) c~ I / J °, o ~ 1. which diverges as J - . 0 since larger and larger FLL volumes V have to jump if the FLL is elastic. Experiments confirm the linear TAFF regime [30] and the vortex-glass predictions [31-33] in YBCO films {311,single crystals [30,32], and ceramics [33]. But as yet I personally do not see convincing evidence that there is a finite "glass temperature" Tgt,,, [24,31-33] at which U(T, B) diverges and the pinned FLL ~freezes". The prediction of freezing as d ~ 0 appears to apply to any 3D elastic FLL. A vortex glass s:ate does not occur in 2D F'LL [34-36] nor in flux-line liquids [27,37] where there is no elastic correlation.
9. T h e r m a l Fluctuation A further open question is whether a 3D FLL has a sharp melting transition and is thermally "entangled" [38,39]. The thermal fluctuations (u:) of the FLL are large [37-43], in particular when its correct nonlocal elasticity [3,11,41-48] (dispersive tilt modulus [46,47]) and anisotropy [41,45-47] are considered. One has
(u2)~t'gTk4~rBOocsr,) !u=p.'+ iB" '
~
a
res
b
F~ t~
TAFF
'
m~ ([3) Tc
"T
FIG. 3. a) Complex s.c. susceptibility of HTSC slabs (14, 18). b) Resonance frequency and damping peak of vibrating HTSC. Below the depinning temperature the damping depends on the vibrational amplitude (dashed lines).
(27)
X~B-~'2_--~)
x~-~b'(2~-q)
(c~ = shear modulus of the FLL, ~c = GL parameter). For B << B~2 with c~ = B 4 ' o / ( 1 6 ~ o ~ b ) inserted. (28) yields (u 2) ~ ksTpo)~sA~ksz/(~oB) = kBTpo (4~r/B~,3o)l/2~s~. The factors in (28) are: The local result [38]; the nonlocal correction factor (Bs2/2B~2) ~/2 (Blntc/4B~t) I/2 >> 1; the anisotropy ratio ,X~/X~b 5 for YBCO and ~ 60 for BSCCO. For example, at B = 1T. T = 77K..X~b = 200am yields for YBCO (u2) '/2 = 3.2nm = 0.065a (FL spacing a = 49am). Apart from softening of the FLL (which should actually increase pinning [48-50]) these fluctuations have the effect of smearing the pinning potential, i.e., they increase the pin range rp ,m (~2 + (u2)2)1/~ [37,51]: this decreases pinning. The extension of the static pinning
F,.H. Brandt / Flux-line lattice in high-Tc oxides
274
summation at T = 0 [48-50] to dynamic creep at T > 0 is given in [25,3T,51]. For fluctuations in the 2D vortexpoirt lattice of layered superconductors, and for its dislocation-mediated melting and creep, see [34-36]. 2D melting of this point lattice probably does occur. Both pinning-caused distortions and thermal fluctuations of the FLL can rednce the spatial variation of the magnetic field B(r) in anisotropie or layered HTSC [52].
10. Vibrating Superconductors Tilting a superconductor periodically in a constant magnetic field B, m B by an angle ~b is equivalent to applying an a.c. magnetic field witl~ amplitude B, sin ~b perpendicular to Bo. Superconducting vibrating reeds [14-17l, and platelets glued on a high-quality silicon oscillator [18] or suspended on a wire [19,20], can therefore be explained by the same theory above which explains a.c. experiments. Vibrating superconductors exhibit a sharp peak in their attenuation (Fig. 3b) at a temperature T~(B) which was interpreted as a FLL melting temperature in [19] but very likely indicates the onset of depinning. In conventional superconductors, this depinning-peak occurs near the upper critical field Bcz(T), or temperature To(B), where pinning has to go to zero. In HTSC, this depinning is thermally activated and reflects the maximum in the imaginary part of the a.c. susceptibility, e.g. in eqs. (15) and (18) and Fig. 3a. As stated above, in many cases the dissipation is of the hysteretic type and amplitude dependent; but even then the peak should typically occur close to the linear damping peak, and a change of the vibration amplitude should ahnost not shift its position but may change its
E
height. The attenuation peak occurs when the diffusion length of flux coincides with a typical length or thickness of the specimen. The peak is so sharp because the flux diffusivity D(B, T) changes rapidly with T (and often also with B). The resonance frequency v of a vibrating superconductor typically increases when a field B. is applied, because B, ~pulls" at the flux lines and these couple to the material by pinning, thus providing an additional restoring force. Above the damping peak the frequency enhancement/in vanishes because of complete depinning. Thus, typically/in cc B,~. However, for very short flux lines, liv o¢ aL cc B z/2 as shown in [1'I]. A different restoring force is the anisotropy of the magnetization, which in very weak-pinning specimens can even he negative and thus reduces u [20].
11. Depinning Lines The damping peak defines an "irreversibility line" or "depinning line" which separates in the B, T-plane the regions of rigid pinning and of complete thermal depinning (i.e. the usual flux flow), Fig. 5. Depinning lines can also be obtained from other measurements [11], e.g.: Resistive transition p(B, T); vanishing irreversibility in the magnetization curve; maximum a.c. dissipation; diverging a.c. penetration depth [53]; peak in the ultrasonic attenuation [54]; conduction noise in HTSC films [55]. All these depinning lines define a line D(B,T) = const, where the constant depends on the length and time scales of the experiment. Reduction of the specimen size (grains embedded in epoxy) shifts the depinning line to lower T and B [1'1]. This work was supported by the German Bundesministerium f~r Forschung and Technologie.
log B [T]
j
10 -
g,iss7 z,,,,//
~
F,ux,,o,,,,,
Depinning lines
~ u x _
flow
Flux creep "
Flux creep Jc
~j
FIG. 4. Current-voltage curves of HTSC at various temperatures and fields, schematic. The insert visualizes the jumping flux-line bundel.
vortex-glass? 0.11
0.7
I 0.8
~ i
0.9
'~ "
r'l
..
"]"
1.0 To
FIG. 5, Depinning lines of HTSC in the B, T plane mark the TAFF region and separate the regions of strong pinning and of pin,free flux-flow behavior.
E.H. Brandt / Fluxdine lattice in high-Tc oxides
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