Nanda, K K

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In d ia n  J. Phys. 70A (6), 753-758 (1996)I J P  A— an international journalV ortex  overlapping in high tem perature  su perconductorsK K NandaInstitute of Physics, Bhubaneswar-751005, IndiaWe have argued that the overlapping of vortices in the mixed state of high temperature superconductors (HTSCs) is important in the intermediate field range because of large Ginzburg-Landau pa­rameter k . Because of the anisotropic behavior, this effect is more pronounced for the applied field B a parallel to c-axis than that for parallel to afc—plane. In case of conventional superconductors, though the overlapping occurs near the upper critical field (J5c2), the overlapping is stronger compared to HTSCs.Keywords: Mixed state; flux lattice; penetration depth; type-II su­perconductorsPACS num ber(s): 74.60.Ec; 74.72.-hT h e  sm a ll  v a lu e  o f  c o h e ren c e  le n g th  f  a n d  la rg e  value o f  p e n e tra t io n  d e p th  A o f th e  h ig h  te m p e r a tu r e  s u p e rc o n d u c to rs  (H T S C s) d iffe ren tia te s  itse lf  fro m  con­v e n tio n a l ty p e - I I  s u p e rc o n d u c to rs .  T h u s , th e  value o f G in zb u rg -L an d a u  (G L ) p a r a m e te r  k a n d  th e  u p p e r  c r it ic a l  field (B c2 ) is ex tre m e ly  h igh  co m p ared  to  c o n v e n tio n a l ty p e - I I  su p e rc o n d u c to rs .In  th e  m ix e d  s t a t e  th e  s e p a ra t io n  d b e tw een  th e  tw o  lines is g iven  by d =  y/<f>o(B, w h e re  <f>o is th e  flu x  q u a n tu m  a n d  B  is th e  m ag n e tic  flux  den sity . T h e  m a g n e t ic  f lu x  d e n s ity  B  in c re a se s  on  in c re a s in g  th e  ap p lied  m ag n e tic  field a n d  h e n c e , th e  in te r s p a c in g  b e tw e en  v o rtice s  d ec reases. F or low  ap p lied  fields, th e  s e p a ra t io n  b e tw e e n  th e  lines is m u ch  la rg e r  th a n  th e  p e n e tra t io n  d e p th  a n d  th e  lin es  a r e  e s se n tia lly  in d e p e n d e n t.  W h e n  d becom es co m p arab le  w ith  A, th e  e le c tro m a g n e tic  re g io n s  o f  a d ja c e n t  flux  lines s t a r t  to  overlap .A c c o rd in g  to  th e  L o n d o n  th e o ry , th e  d is tr ib u tio n  o f  lo ca l field is g iven  by  [1],(1 )© 1996IACS754 KKNandat. e i t  d ecay s likeh ( r ) = 2 ^ A 2 / n r  f ° r ^ < T < X  a n d_  _ ^ _ ( ^ ) 1/ V r/A f o r r »  A 2 ttA2 t(a)Fig. 1S tru c tu re  o f an  isolated vortex line in a  m ateria l w ith  (a) /c=10 and  (b) k= 2 0 .V o r te x  o v e r la p p in g  in h igh  te m p e ra tu re  su p erc o n d u c to rs 755This implies that h(r) diverges logarithmically as In $ as r -0 . In reality this divergence is cut off at r ~  ( ,  where 11/>2 | starts dropping to zero. Thus, Mr) is regular at the center of the vortex line and is maximum which is given by,For the samples whose A is large, the field value at the center of vortex line is less and the decay of field h(r) with distance is slow. For conventional type-II super­conductors, the k value is small compared to HTSCs and the decay of magnetic field h(r) is faster in the former case as is evident from Fig 1. The field value h(r) goes to zero at a short distance from the center of the vortex lines. Thus, in case of conventional type-II superconductors, the value of A is such that it is comparable to the interspacing between vortices near the upper critical field jBc2 and thus, the vortex overlaps near Bc2. Whereas, the value of A of any HTSCs is so large that it is comparable to d even at low applied fields. Thus, the vortices overlap for low field values and the vortex overlapping mechanism can be applied to understand the mixed state.A single isolated vortex line consists of a normal core of radius £ where the magnetic field strength at the center is h(0) -  ((f>0/2irX2)K 0(^/X) and decays as K 0(t/X )  with distance and at a distance of A it becomes h(A) = (^0/27rA2)Ar0(l)  i.c reduces by a factor [1]- If the applied field is such that flux-line lattice(FLL) with d comparable to A, the vortices start overlapping, the superposition of magnetic field contributions of individual flux line increases the magnetic field strength, even at the centre of the vortices. The decay of the field becomes slower and the ratio h(0)/h(X) will be obtained at a distance greater than A. We will have an effective penetration depth Ae/ /  larger than A. If the applied magnetic field is increased further, the field strength B increases and the interspacing between vortices d = (<f>0/ H)1^ 2 decreases, overlapping of electromagnetic regions of vortices becomes stronger, thereby increasing the value of A further i.e. the penetration depth is field-depdendent and is given by,where / 2( B) is the spatial average of / 2, f being the magnitude of the order parameter, normalised to its value in the zero magnetic field and is given by P  = 1 -  X ,  where X  = B /B c2 [2]. It can be noted from Eq. (4) that as the upper critical field is small in case of conventional superconductors compared to HTSCs, the overlapping in former case is stronger than the later at a particular field i. e. when the interline separation is comparable to A, the vortices will start(3)Xeff = X P P iB ) ) 1'2, (4)756 K K N a n d aoverlapping and will be stronger compared to HTSCs.It has been found from magnetization measurement on a grain-aligned Y(123) crystal that the magnetization behaves as ( T - T c)2 near the transition tempera­ture Tc [3,4] and the change in specific heat by the application of magnetic fields A C  is not linear in Ba [3] which can be understood by the vortex overlapping mechanism [5]. As the HTSCs are anisotropic, the magnetic field dependence of specific heat depends on the orientation of Ba with respect to the crystallographic axes. For any general orientation of fl„, the magnetization is [5],M  = w v M w i ,and the change in specific heat is,< h B a P r 1,3<0) t16x2r cA2,(0) 1 - *(5)(6)with f 2 = ( 1  -  and Am = (A2^ ) 1/3, where c(0) =  (sin20 + 7 2co$2B)l l 2,6 is the angle between Ba and crystallographic c-axis, 7  is the anisotropy of the effective mass, B #  is the upper critical field for Ba parallel to c-axis and Aa& and Ac are the penetration depth parallel and perpendicular to ab-plane respectively. Here, we have considered the GL temperature depnedence of £ and A [1 ]. Though the magnetization is approximately logarithmic in Ba, the results obtained from this mechanism at low fields are in between that obtained from the usual London theory and the variational approach proposed by Hao et al. Expanding the logaritmic term near Bc2 we see that the magnetization behaves as (T -  Tc)2 that has been observed experimentally [3,4].We analyse the specific heat data of Athreya et al. [3] and obtain the B ^ ,  Aaj(0) and 7  as follows. In the mean field region v —\ f l .  The least square fitted value of Bc2(0) and A„i(0) are found to be 296 T and 1936 A from the specific heat data for Ba || c-axis. We take Tc=93 K. The value of A„b(0) is comparable with Aai(0)=1400±500A obtained from magnetization measurement [6] and the value of B c2 is also in excellent aggreement with that obtained from magnetiza­tion measurement [7]. Using this Ao{,(0), UC2(0) and the data for Ba 1  c-axis, the least squares fitted value of 7  is estimated to be 6 .1 which is in reasonable agreement with earlier reported value [8]. The least square procedure obtains the A„b=2334 A and 7 = 4.3  where we have used the usual London theory result [9]. Fig. 2 shows the comparison of the data of Athreya et al [3] and the least-square(x fw /r >0 ovv o m x  C ,e^ „ g Wf,» lem ptm ure , v m m d u c M sThe field dependence of AC of a grain-aligned Y(123) crystal. The data is obtained from Fig 5 of Ref 3. The solid lines are the theoretical curves according to Eq (2) and the dashed lines are the usual London results [9].fitted theoretical curves for a grain-aligned Y(123) crystal The solid lines are the theoretical curves according to Eq. (2) and the dashed lines are the usual London results [9]. It is noted that the vortex overlapping mechanism gives be70A (6) 14758 K K Nandater fit to the experimental data and the overlapping is weak when the applied field is parallel to ah-plane. In this direction, the field would penetrate in the form of Josephson vortices with mutual distance d = <f>o/Bs [10], where s is the interlayer spacing. These form a FLL consisting of isosceles triangles [11]. If the applied field is parallel to o-axis, h(r) decays with distance along 6-axis with decay length Ac and decays with decay length Aa& along c-axis [12], i. e. the field h(r) decays faster along c-axis and slowly along 6-axis as Ac > A„{,. Also, the field strength at the center of the vortex line is low compared to that when the field is applied along c-axis. In the intermediate field range the distance between vortices d = <j>o/Bs is so large that the overlapping of vortices is weak.In conclusion, we have given an argument that the overalapping of vortices in the mixed state of HTSCs is because of large k . Thus, the vortex overlapping mechanism could be used to understand the thermodynamic properties of the HTSCs in the mixed state.R efe rences[1] A L Fetter and P C Hohenberg, in Superconductivity, edited by R D Parks (Marcel, Dekker, New York, 1969), Vol. 2, Chap 14, p 817[2] RP Huebner, Magnetic Flux Structures m Superconductors, Vol. 6 of Springer Se­ries in Solid State Sciences (Springer-Verlag, Berlin, 1979); E H Brandt, J. Super­conductivity 6 201 (1993)[3] K S Athreya et al, Phys. Rev. B 38 11846 (1988)[4] U Welp et a i,  Phys. Rev. Lett. 62 1908 (1989)[5] K K Nanda, Physica C 245 341 (1995)[6] L Krusin-Elbaum, R L Greene, F Hotzberg, A P Malozemoff, and Y Yeshurun, Phys. Rev. Lett. 62 217 (1989)[7] A Bezinge, J L Jorda, A Junod, and J Muller, Sohd State Commun. 64 79 (1987)[8] A Junod, E Bonjour, R Calemczuk, J Y Henry, J Muller, G Triscone, and J C Vallier, Physica C 211 304 (1994)[9] S B Ota, Phys. Rev. B 43 1237 (1990); S B Ota, K K Nanda and S N Behera, Physica B 194-196 1387 (1994)[10] P H Kes, J Aarts V M Vinokur and C J van der Beek, Phys. Rev. Lett. 64 1063 (1990)[11] P H Kes, Physica C 153-155 1121 (1988); M Tachiki and S Takahashi, Solid State Commn. 70 291 (1989)[12] J R Clem et al, J. Low Temp. Phys. 88 213 (1992)

Vortex Overlapping in High Temperature Superconductors

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Vortex Overlapping in High Temperature Superconductors

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