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Contribution of plastic interactions to the pinning energies in a deformed flux-line lattice in high-Tc superconductors
14 September 1998 PHYSICS
LETTERS
A
Physics Letters A 246 (1998) 353-358
Contribution of plastic interactions to the pinning energies in a deformed flux-line lattice in high-T, superconductors Ali E. Khalil Physics Department, University of Bahrain, P.O. Box 32038, Bahrain
Received 21 April 1998; accepted for publication 22 June 1998 Communicatedby J. Flouquet
Abstract The existence of strong pinning centers in the form of amorphous defects in a deformed flux-line lattice (FLL) considerably modifies the free energy of the vortices in the mixed state. In particular, the contribution of the plastic interactions to the pinning energies as the dislocation fields modify the elastic manifold of the lattice is examined. The question whether the logarithmic dependence of U(J) on the current density is determined by the pinning potential with a long range tail or by disorder is investigated. In this paper we provide an answer to the above question that an important component to the distribution of pinning potentials in the two-dimensional planes arises from disorder in the form of plastic interaction between dislocation fields and random pinning sites in the superconductor. Furthermore, the effect of the plastic deformations as the magnetic field increases is to change the pinning mechanism, therefore, modifying the shape of the pinning potential. @ 1998 Published by Elsevier Science B.V.
1. Introduction Dynamical properties of the flux-line lattice (FLL) in high-T, superconductors are very fundamental to understand its different physical modes and the nature of phase transition in these materials. It was pointed out that the Abrikosov flux-line lattice should have a melting transition at a temperature Tm considerably smaller than the superconducting transition temperature T, as a result of the significant thermal fluctuations [ 11. The study of the pinning forces and flow properties of a deformed FLL could in principle provide some clues to the nature of the melting process and its general behavior that is described by the collective pinning theory in type II superconductors. However, quantitative explanation of many equilibrium and dynamical properties of the system requires detailed 0375-9601/98/$ - see front PI1 SO375-9601(98)00493-9
knowledge of the elastic manifold of the material. Several attempts have been tried to approach this problem where the nonlocal moduli were calculated from Ginzburg-Landau theory [ 21. In addition, the elastic properties of the two-dimensional lattice due to weak random pinning were also investigated [ 31. Furthermore, numerical simulations on the two-dimensional lattice revealed that the nature and size of the pinning forces are determined by the plastic deformations of the FLL which, in general, cannot be explained by the collective pinning theory [ 41. For a superconductor in the mixed state, the cores of the vortices do not overlap and fluctuations in the amplitude ]+_I of the order parameter outside the vortex cores are unimportant. Recent investigations showed that disorder-induced dislocations will cause spatial fluctuations on length scales larger than the super-
matter @ 1998 Published by Elsevier Science B.V. AU rights reserved.
354
A.E. Khdil/Physics
Letters A 246 (1998) 353-358
conducting coherence length [ 51. Although collective pinning theory defines a “correlated volume” as a volume where translational order within the crystal is maintained in the presence of random potential, it cannot account for the effects of plastic deformations on the flux-line lattice (FLL). In addition, the pinning forces result from the incomplete averaging of the random potential over the finite correlated volume. The existence of the correlated volume is due to the interplay between the elasticity of the FLL and the strength of the random potential which causes the distortions [ 61. The appearance of dislocation fields due to local strains in the lattice will affect the dissipation of a moving FLL and therefore, the pinning energies. The generations of these dislocations will further add to the already nonlinear behavior of the Z-V curves [ 71 and the existence of reproducible jumps in the flux flow resistance Rf = dV/dI measurements as was observed earlier experimentally [ 81. Theoretical investigations on the renormalization of the elastic properties of a two-dimensional lattice due to weak random pinning were carried out and the results showed softening in the shear modulus [ 91. Furthermore, variational replica calculations for the effects of random pinning indicated increased hardening in the renormalized shear and bulk moduli [ lo]. In all these approaches, the shape of the pinning potential and the pinning strength distribution influence the activation energy U. A crossover from individually pinned vortices to a regime of elastic vortex interaction is expected, particularly, in BSCCO compound at fields of the order of 1 T [ 111. Alternatively, at small magnetic fields, the intervortex forces are small and the motion of vortices in the field of pinning centers will be affected by dislocation dynamics while, for large magnetic fields, the effects of the elastic forces on the motion of the vortex lattice are important. These effects always influence the magnetization measurements as the temperature or field are increased. Vortex positional fluctuations cause rapid drop of the creep barriers and the irreversible part of the magnetization is suppressed. In the London limit, corresponding to fields I3 much below the upper critical field Bcz (T) , vortex positional fluctuations modify the logarithmic field dependence of the reversible magnetization. The existence of very strong pinning centers in the form of amorphous defects with radii comparable to the Ginzburg-Landau coherence length 6 considerably
modifies the free energy of the vortices in the mixed state. In addition, there is also a large effect on superconducting fluctuations most pronounced in the critical regime. In this case, the localization of vortices on a manifold with random defects leads to the decrease in the mixed state free energy. Therefore, vortex pinning affects the thermodynamic properties of superconductors by lowering the Gibbs free energy of the vortex state. The open question which still remains is whether the logarithmic U(J) law is determined by the pinning potential with a long range tail or by disorder [ 121. In this paper we provide an answer to the above question that an important component of the pinning strength distribution is determined by disorder in the form of plastic interaction between dislocation fields and random pinning sites in the superconductor. Furthermore, the effect of the plastic deformations as the magnetic field increases is to change the pinning mechanism therefore, modifying the shape of the pinning potential.
2. Theoretical considerations The characteristics of the flux-line phases in HTSC near the superconducting transition are governed by dynamical effects in the form of thermal fluctuations [ 131. In addition to these dynamical effects, the existence of static defects on the vortex lattice appears as random potential barriers. The random potential generates local strains which in most cases exceed the elastic limit of the lattice causing dislocations which modify the pinning energies. Thus the generation of dislocations will further contribute to the nonlinear behavior of the Z-V curves that are already observed experimentally. Dislocations with finite displacements are associated with surfaces of misfit in crystals. Translation dislocations which have Burger’s vector equal to or greater than the lattice constant will result in stresses and strains too large to be considered in the elastic limit. The motion of dislocations is influenced by their interaction with defects and for dislocation velocities much smaller than the velocity of sound it is also affected by viscous drag caused by phonons and conduction electrons. At low temperatures, the latter (electronic drag) will predominate and for normal metals it was found that B = rb/ud, where r is the effective
A.E. Khalil/Physics
$_+-I
4-l
Letters A 246 (1998) 353-3.58
the nature of the dis~butio~ of the entire dislocation field, in this case,
$+-I
U=
+i
44 ; I
f
T F.-h
$
4 4-1
tto+ +
Fig. 1. The interaction of a two-dimensional array of edge dislocations with randomly distributed point defects, showing the direction of climb of each dislocation.
stress acting on the dislocation, b is Burgers vector and Ud is the dislocation velocity between defects [ 141. On the other hand, the electronic drag in the superconducting state should be controlled by the same mechanisms that lead to phonon attenuation in superconductors. In this case, the ratio of the drag coefficients in the su~rconducting state to that in the normal state was calculated to be B,/B, = 2f( d(T) ), where f( A( T) ) is the Fermi function of the superconducting energy gap [ 151. In this picture the interaction of dislocations with point defects at low temperature is primarily due to size effects and the distribution of disorder in the material will change to reflect the nature of that interaction (see Fig. 1) The size of the pinning centers will be of the same order as the dilatation centers due to damping and the effective barrier of a defect center and its range will change in such a way to impede the dislocation motion. It was shown earlier that the elastic interaction of defects with the dislocation results in the existence of clouds of point defects extending to a distance 1; = S1/(“-~)~, where 8 = Vo/kT the ratio of the pinning energy to the thermal energy, y = &AB/kT and A3 = B, - B, is the difference between the drag coe%cient of the dislocation at the normal state and the superconducting state [5]. These effects are caused by the interaction between the pinning centers and the dislocation field in the 2D system which is given by
.
Ej*r = 6U= f-ftOVro#))L.
355
(1)
;(Lois the increase in the lattice volume due to a single point defect in the crystal and 700 is the stress tensor at the defect location. The term appearing in (1) will change the pinning energies and in order to calculate the total effect on the lattice, we must take into account
(-&Vrae)[email protected](r)
dr.
(2)
Calculation of the gradient of the stress tensor appearing in the above equation requires using dislocation group dynamics where it is assumed that the velocity of each dislocation is proportional to the total stress field u. The total stress field is the sum of any applied stress field S(r) and the inverse first power stress field of all other dislocations. In this case, the change in the pinning energies due to plastic deformations is given by [51 (3) where A is a constant equal to ~~/2~( 1 - y) for edge dislocations and ,&/27r for screw dislocations, b is the magnitude of the Burger’s vector of the dislocation, ,u is the elastic shear modulus and y Poisson’s ratio. For mixed dislocations (A) will be intermediate between values for screw and edge. Since the PLL is always defective for macroscopic systems, its static and dynamic properties in the presence of pinning and plastic deformations are obtained by minimizing the total distortion energy of the lattice. The distortion energy (plastic energy) of the FLL was minimized with respect to a dislocation density with the result of an equ~lib~um dislocation density which is identified as $i(r) [ 161, t,&(r) =
I’
d2r’d2#‘G$(r-r’)g&-‘-r”)F(r”). (4)
F(r) is the pinning force co~~sponding to a change in the pinning potential A( V,(r) ) in the absence of the dislocation field. This situation corresponds to flux lines pinned at the bottom of a potential well of depth VO.The pinning of flux lines with defects even without external current leads to a distortion of the flux-line lattice. The existence of external currents in the superconducting matrix results in Lorentz-force-induced flux motion which modifies the effective well depth to become VP f AV, on both sides of the flux line, respectively, where AVp = V(r) J/Jo, J is,the current density and JO is the current density without thermal
356
A.E. Khalil/Physics
Letters A 246 (1998) 353-358
activation [ 171. Therefore, the pinning force F(r) appearing in (4) depends on the current and is simply the change in the potential over distances of the order of the superconducting coherence length where F(r)
M Xv(r)/51J/JO,
V(r)=&
;
and
forO
,
0
=&[ln(f)
tl],
forr>l.
(5)
The form of the potential in (5) was determined from measmemen~ of the dependence of activation energy on the current density [ 181. The Green’s functions G$ appearing in (4) are the Fourier’s transform which relates the random force F(r) to the dislocation density and is given M ( 1/2E,)~&j~Ejnk,k,[l/(k2+k,2) by G;(k) 1/k2]}, where G$(r - r’) =
s
d3k (2~)3
exp[ik(r
- r’)]G$k).
(6)
The constant kc = (K/2E,) ‘1’ is a cutoff related to the dislocation core energy and K B 4C~. The dependence of U in Eq. (3) on the magnetic field is provided through the incorporation of the elastic properties of the lattice characterized by different length scales. The elastic moduli of the lattice have strong spatial dispersion and depend on the wave vectors k > l/X, where A is the ~ne~ation depth [ 191. If k < l/a, where a is the lattice constant, then C66 = &$3/C %-A)* and the constant appearing in the Green’s function ko = ( I/877A) (2&B/z&) +. The correlation function of the eIastic displacement has the general form gij( r) m ( 1/27r) [ 6, x tan-” (y/X> + (C&/C11)Eij ln(r/a) -I- Ejkrjrk/r'l. To simplify the calculations and obtain meaningful results that can be compared to experimental measurements several assumptions are introduced. Since for an FLL, the bulk modulus Ctr >> Cs the shear modulus and 3, = (C&/Cft)~ij < 1, therefore the correlation function gij (r) is approximated by
The effects of plastic deformations on the random pinning energies of FLL due to the existence of disloca-
tions in the superconducting matrix can in principle be calculated from Eq. (3). The motion of dislocations will introduce extra dissipation, thus lowering the total energy of the system. At large current (large driving forces), the effect of plastic deformations in the system is minimal resulting in linear I-V curves. As the current is decreased the effects of dislocations become impo~~t and the I-V characteristics start to deviate from their linear behavior.
3. Discussion and results The distributions of distances between point defects and the change in activation energies occur as a result of the plastic interactions. At small fields, fluctuating regions are on the average limited by the typical distance of the interaction region between pinning centers. ~e~while superconducting coherence over large distances will not exist until 6 < L. At large fields the fluctuations are limited by the existence of inhomogeneous domains formed by density variations and localization of groups of point defects. To qualitatively compare the results of the above simulation, Eq. (3) was integrated numerically with a cutoff set equal to the interaction range L. The temperature dependence of iJ was determined by substituting the value t(t) N coherence !5(0)(1t)- i12, for the superconducting length where t = T/TC is the reduced temperature. For each value of the applied magnetic field B, the Green’s function G& appearing in (4), is calculated and, consequently, the equilibrium dislocation density +i( r) is known. In this case, for each field value the activation energy as a function of temperature and current is determined from (3). In order to properly compare the calculations to the ex~~mental data reported in the literature, we chose for our simulation, defined essentially by Eq. (3), the same set of parameters as used in Ref. [ 121 for the highly oriented Bi&$aCu2Os+~ (BSCCO) sample known as B212. The superconducting coherence length is c(O) N 2.5 nm while the transition temperature is Tc = 86.1 K. The quantity Ve is some “average” activation energy given by Vi = 530 K which better fitted the measurements. The value of the penetration depth in BSCCO compound was found to be A M 0.21 pm [20]. The quantity f20 that is the increase in the lattice volume due to a single point de-
A.E. Khalil/Physics Letters A 246 (1998) 353-358
feet was taken from the experimental data to be & N 35 x lo3 nm3 and the magnitude of the Burger’s vector b of the dislocation was set equal to one. The activation energy was first calculated as a function of the applied field B for values extending to 3 T. The results are shown in Fig. 2. The behavior of the activation energy follows a logarithmic decrease for low field values while it follows an almost linear decrease for high field values with a crossover field Bo around 0.65 T. The calculated values of the activation energies are, in general, lower than the measured values since the existence of dislocation fields tend to decrease the activation energy. This result is due to the positive values of the interaction energy relative to the actual negative values of the pinning energies. The agreement between the measured data and the predictions of the model is very clear which confirms the conclusions of Ref. [ 121 that the field dependence of the activation energy is dominated by the plastic deformations of the FLL. The activation energy was calculated next as a function of the current density where JO was set equal to JO M 6 x lo4 A/cm2. The results are shown in Fig. 3 for high field value. The results of the proposed model are in good agreement with the reported measurements. In addition to the field and current dependencies of the activation energy, the temperature dependence was calculated and the results are shown in Fig. 4 for high field value. The results are quantitatively in agreement with the experimental measurements. The change of the pinning potential with the applied field indicates a change in the pinning mechanism due to the influence of plastic deformations on the disorder in the FLL. In order to explore these effects further and examine the question of the logarithmic dependence of U(J) on the current density, the pinning potential at low field values (exponential regime) was calculated for different pinning potential than that given by (5). The random potential function appearing in (5) was replaced by the Gaussian potential V(r) = Voexp( --r2/,$‘) and Eq. (3) was recalculated and no substantial variations in the pinning energy were observed. Furthermore, the integral in Eq. (3) was evaluated for the functional dependence U,, E -&A s[ l/(r - L)“] L@i(r) dr, for the value where n = 3 instead of n = 2 and the results are shown in Fig. 5. The pinning potential is more sensitive to the denominator in the integrand of (3) which is the
357
2000
1500
E 5
1000
500
0 -0.5
0
0.5
1
1.5
2
2.5
3
W-0
Fig. 2. The dependence of the activation energy U on the magnetic field I3 for the sample B212. The circles are the experimental data taken from Fig. 7 of Ref. [ 121, while the squares are the theoretical predictions of Eq. (3).
-200
0
200
400
600 600 J(AIcm*)
1000
1200
1400
Fig. 3. The dependence of the activation energy U on the current density J for the sample B212 at applied field value B = 1.83 T. The circles are the experimental data ( U( J)/U( 0) ) taken from Fig. 6b of Ref. [ 121, while the squares are the theoretical predictions (U(J)/Kj) of Eq. (3).
result of the stress field of all dislocations rather than the shape of the random force function. These features confirm the previous conclusions that the shape of the pinning energies is mostly determined by the nature of the disorder.
4. Conclusions The contribution of the plastic interactions to the pinning energies as the dislocation fields modify the elastic manifold of the lattice has been established. An answer has been presented to the question of the nature of the logarithmic pinning potential U(J) law and its relevance to the existence of disorder and plastic
358
A.E. Khalil/Physics
A 246 (1998) 353-358
250
200
150
P 5 100
50 ‘-
I-
0
50
I
I
I
I
1
60
70
80
90
100
T(K)
Fig. 4. The dependence of the activation energy U on the temperature for the sample B212 at applied field value B = 1.83 T and current density J = 1250 A/cm*. The circles are the experimental data taken from Fig. 5b of Ref. [12], while the squares are the theoretical predictions of Eq. (3).
neous regions formed by density variations and localization of groups of point defects. In addition, the density of vortices increases relative to the density of the disorder and collective behavior starts to dominate. At small fields, individual vortices are pinned by a distribution of pinning strengths with values which are modified by plastic interactions in the FLL. The calculated values of the activation energies are lower than those measured. This result is due to the fact that the dislocation fields tend to decrease the activation energy due to the positive values of the interaction energy relative to the actual negative values of the pinning energies. An important component of the pinning strength distribution is determined by disorder in the form of plastic interaction between dislocation fields and random pinning sites in the superconductor. Furthermore, the effect of the plastic deformations as the magnetic field increases is to change the pinning mechanism thereby modifying the shape of the pinning potential.
0.8
References 6
0.6
3 z
[ 11 D.R. Nelson, Phys. Rev. Lett. 60 (1988)
0.4
0.2
0 0
0.1
0.2
0.3
0.4
0.5
0.6
j (Amperes)
Fig. 5. The dependence of the activation energy U on the current density J for the sample B212 at applied field value B = 0.29 T. The circles are the experimental data (U(J)/U(O) ) taken from Fig. 6a of Ref. [ 121, the squares are the theoretical predictions (U( J)/b) of Eq. (3) with II = 2, the solid triangles are the theoretical predictions (U(J)/Vf) of Eq. (3) with a Gaussian potential, while the empty triangles are the theoretical predictions (U(J)/b) of Eq. (3) with n=3.
deformation in FLL. In the current approach the static interaction between a defect center in the lattice and a moving dislocation field modifies the range and the strength of the pinning well. At small applied fields, fluctuating regions are limited on average by’the typical distance of the interaction region between pinning centers. The appearance of superconducting coherent domains will only exist for spatial distances where .$ < L.At large fields, the fluctuations are limited by the existence of inhomoge-
1973. [2] E.H. Brandt, Phys. Rev. Lett. 63 (1989) 1106. [3] Y.J.M. Brechet, B. Doucot, H.J. Jensen, A.C. Shi, Phys. Rev. B 42 (1990) 2116. [4] A. Brass, H.J. Jensen, A.J. Berlinsky, Phys. Rev. B 39 (1989) 102. [5] A.E. Khalil, Phys. Rev. B 54 (1996) 12437. [ 61 A.I. Larkin, Yu.N. Ovchinnikov, J. Low Temp. Phys. 34 (1979) 409. [7] P Berghuis, A.L.F. van der Slot, PH. Kes, Phys. Rev. Lett. 65 (1990) 2583. [S] S. Bhattacharya, M.J. Higgins, Phys. Rev. B 52 (1995) 64. [9] Y.J.M. Brechet, B. Doucot, H.J. Jensen, A.C. Shi, Phys. Rev. B 42 (1990) 2116. [lo] J.P. Couchaud, M. Mezard, J. Yedidia, Phys. Rev. B 46 (1992) 14686. [ 1 l] M. Inui, PB. Littlewood, S.N. Coppersmith, Phys. Rev. Lett. 63 (1989) 2421. [ 121 C. Attanasio, C. Coccorese, V.N. Kusbnir, L. Maritato, S.L. Prischepa, M. Salvato, Physica C 255 (1995) 239. [ 131 C.J.. van der Beek, M. Konczykowski, T.W. Li, P.H. Kes, W. Benoit, Phys. Rev. B 54 (1996) R792. [ 141 A.D. Brailsford, Phys. Rev. 186 (1969) 959. [ 151 G.P. Huffman, N.P. Louat, Phys. Rev. Lett. 24 (1970) 1055. [ 161 A.-C. Shi, A.J. Berlinsky, Phys. Rev. Lett. 67 (1991) 1926. [ 171 Y.L. Ma, H.F. Li, X.X. Dai, Phys. Rev. B 47 (1993) 12304. [IS] E. Zeldov, N.M. Amer, G. Korcn, A. Gupta, M.W. McElfresh, R.J. Gambino, Appl. Phys. Lett. 56 (1990) 680. [ 191 E.H. Brandt, J. Low Temp. Phys. 26 (1977) 709. [20] A. Schilling, R. Jin, J.D. Guo, H.R. Ott, Physica B 194-196 (1994) 2185.
LETTERS
A
Physics Letters A 246 (1998) 353-358
Contribution of plastic interactions to the pinning energies in a deformed flux-line lattice in high-T, superconductors Ali E. Khalil Physics Department, University of Bahrain, P.O. Box 32038, Bahrain
Received 21 April 1998; accepted for publication 22 June 1998 Communicatedby J. Flouquet
Abstract The existence of strong pinning centers in the form of amorphous defects in a deformed flux-line lattice (FLL) considerably modifies the free energy of the vortices in the mixed state. In particular, the contribution of the plastic interactions to the pinning energies as the dislocation fields modify the elastic manifold of the lattice is examined. The question whether the logarithmic dependence of U(J) on the current density is determined by the pinning potential with a long range tail or by disorder is investigated. In this paper we provide an answer to the above question that an important component to the distribution of pinning potentials in the two-dimensional planes arises from disorder in the form of plastic interaction between dislocation fields and random pinning sites in the superconductor. Furthermore, the effect of the plastic deformations as the magnetic field increases is to change the pinning mechanism, therefore, modifying the shape of the pinning potential. @ 1998 Published by Elsevier Science B.V.
1. Introduction Dynamical properties of the flux-line lattice (FLL) in high-T, superconductors are very fundamental to understand its different physical modes and the nature of phase transition in these materials. It was pointed out that the Abrikosov flux-line lattice should have a melting transition at a temperature Tm considerably smaller than the superconducting transition temperature T, as a result of the significant thermal fluctuations [ 11. The study of the pinning forces and flow properties of a deformed FLL could in principle provide some clues to the nature of the melting process and its general behavior that is described by the collective pinning theory in type II superconductors. However, quantitative explanation of many equilibrium and dynamical properties of the system requires detailed 0375-9601/98/$ - see front PI1 SO375-9601(98)00493-9
knowledge of the elastic manifold of the material. Several attempts have been tried to approach this problem where the nonlocal moduli were calculated from Ginzburg-Landau theory [ 21. In addition, the elastic properties of the two-dimensional lattice due to weak random pinning were also investigated [ 31. Furthermore, numerical simulations on the two-dimensional lattice revealed that the nature and size of the pinning forces are determined by the plastic deformations of the FLL which, in general, cannot be explained by the collective pinning theory [ 41. For a superconductor in the mixed state, the cores of the vortices do not overlap and fluctuations in the amplitude ]+_I of the order parameter outside the vortex cores are unimportant. Recent investigations showed that disorder-induced dislocations will cause spatial fluctuations on length scales larger than the super-
matter @ 1998 Published by Elsevier Science B.V. AU rights reserved.
354
A.E. Khdil/Physics
Letters A 246 (1998) 353-358
conducting coherence length [ 51. Although collective pinning theory defines a “correlated volume” as a volume where translational order within the crystal is maintained in the presence of random potential, it cannot account for the effects of plastic deformations on the flux-line lattice (FLL). In addition, the pinning forces result from the incomplete averaging of the random potential over the finite correlated volume. The existence of the correlated volume is due to the interplay between the elasticity of the FLL and the strength of the random potential which causes the distortions [ 61. The appearance of dislocation fields due to local strains in the lattice will affect the dissipation of a moving FLL and therefore, the pinning energies. The generations of these dislocations will further add to the already nonlinear behavior of the Z-V curves [ 71 and the existence of reproducible jumps in the flux flow resistance Rf = dV/dI measurements as was observed earlier experimentally [ 81. Theoretical investigations on the renormalization of the elastic properties of a two-dimensional lattice due to weak random pinning were carried out and the results showed softening in the shear modulus [ 91. Furthermore, variational replica calculations for the effects of random pinning indicated increased hardening in the renormalized shear and bulk moduli [ lo]. In all these approaches, the shape of the pinning potential and the pinning strength distribution influence the activation energy U. A crossover from individually pinned vortices to a regime of elastic vortex interaction is expected, particularly, in BSCCO compound at fields of the order of 1 T [ 111. Alternatively, at small magnetic fields, the intervortex forces are small and the motion of vortices in the field of pinning centers will be affected by dislocation dynamics while, for large magnetic fields, the effects of the elastic forces on the motion of the vortex lattice are important. These effects always influence the magnetization measurements as the temperature or field are increased. Vortex positional fluctuations cause rapid drop of the creep barriers and the irreversible part of the magnetization is suppressed. In the London limit, corresponding to fields I3 much below the upper critical field Bcz (T) , vortex positional fluctuations modify the logarithmic field dependence of the reversible magnetization. The existence of very strong pinning centers in the form of amorphous defects with radii comparable to the Ginzburg-Landau coherence length 6 considerably
modifies the free energy of the vortices in the mixed state. In addition, there is also a large effect on superconducting fluctuations most pronounced in the critical regime. In this case, the localization of vortices on a manifold with random defects leads to the decrease in the mixed state free energy. Therefore, vortex pinning affects the thermodynamic properties of superconductors by lowering the Gibbs free energy of the vortex state. The open question which still remains is whether the logarithmic U(J) law is determined by the pinning potential with a long range tail or by disorder [ 121. In this paper we provide an answer to the above question that an important component of the pinning strength distribution is determined by disorder in the form of plastic interaction between dislocation fields and random pinning sites in the superconductor. Furthermore, the effect of the plastic deformations as the magnetic field increases is to change the pinning mechanism therefore, modifying the shape of the pinning potential.
2. Theoretical considerations The characteristics of the flux-line phases in HTSC near the superconducting transition are governed by dynamical effects in the form of thermal fluctuations [ 131. In addition to these dynamical effects, the existence of static defects on the vortex lattice appears as random potential barriers. The random potential generates local strains which in most cases exceed the elastic limit of the lattice causing dislocations which modify the pinning energies. Thus the generation of dislocations will further contribute to the nonlinear behavior of the Z-V curves that are already observed experimentally. Dislocations with finite displacements are associated with surfaces of misfit in crystals. Translation dislocations which have Burger’s vector equal to or greater than the lattice constant will result in stresses and strains too large to be considered in the elastic limit. The motion of dislocations is influenced by their interaction with defects and for dislocation velocities much smaller than the velocity of sound it is also affected by viscous drag caused by phonons and conduction electrons. At low temperatures, the latter (electronic drag) will predominate and for normal metals it was found that B = rb/ud, where r is the effective
A.E. Khalil/Physics
$_+-I
4-l
Letters A 246 (1998) 353-3.58
the nature of the dis~butio~ of the entire dislocation field, in this case,
$+-I
U=
+i
44 ; I
f
T F.-h
$
4 4-1
tto+ +
Fig. 1. The interaction of a two-dimensional array of edge dislocations with randomly distributed point defects, showing the direction of climb of each dislocation.
stress acting on the dislocation, b is Burgers vector and Ud is the dislocation velocity between defects [ 141. On the other hand, the electronic drag in the superconducting state should be controlled by the same mechanisms that lead to phonon attenuation in superconductors. In this case, the ratio of the drag coefficients in the su~rconducting state to that in the normal state was calculated to be B,/B, = 2f( d(T) ), where f( A( T) ) is the Fermi function of the superconducting energy gap [ 151. In this picture the interaction of dislocations with point defects at low temperature is primarily due to size effects and the distribution of disorder in the material will change to reflect the nature of that interaction (see Fig. 1) The size of the pinning centers will be of the same order as the dilatation centers due to damping and the effective barrier of a defect center and its range will change in such a way to impede the dislocation motion. It was shown earlier that the elastic interaction of defects with the dislocation results in the existence of clouds of point defects extending to a distance 1; = S1/(“-~)~, where 8 = Vo/kT the ratio of the pinning energy to the thermal energy, y = &AB/kT and A3 = B, - B, is the difference between the drag coe%cient of the dislocation at the normal state and the superconducting state [5]. These effects are caused by the interaction between the pinning centers and the dislocation field in the 2D system which is given by
.
Ej*r = 6U= f-ftOVro#))L.
355
(1)
;(Lois the increase in the lattice volume due to a single point defect in the crystal and 700 is the stress tensor at the defect location. The term appearing in (1) will change the pinning energies and in order to calculate the total effect on the lattice, we must take into account
(-&Vrae)[email protected](r)
dr.
(2)
Calculation of the gradient of the stress tensor appearing in the above equation requires using dislocation group dynamics where it is assumed that the velocity of each dislocation is proportional to the total stress field u. The total stress field is the sum of any applied stress field S(r) and the inverse first power stress field of all other dislocations. In this case, the change in the pinning energies due to plastic deformations is given by [51 (3) where A is a constant equal to ~~/2~( 1 - y) for edge dislocations and ,&/27r for screw dislocations, b is the magnitude of the Burger’s vector of the dislocation, ,u is the elastic shear modulus and y Poisson’s ratio. For mixed dislocations (A) will be intermediate between values for screw and edge. Since the PLL is always defective for macroscopic systems, its static and dynamic properties in the presence of pinning and plastic deformations are obtained by minimizing the total distortion energy of the lattice. The distortion energy (plastic energy) of the FLL was minimized with respect to a dislocation density with the result of an equ~lib~um dislocation density which is identified as $i(r) [ 161, t,&(r) =
I’
d2r’d2#‘G$(r-r’)g&-‘-r”)F(r”). (4)
F(r) is the pinning force co~~sponding to a change in the pinning potential A( V,(r) ) in the absence of the dislocation field. This situation corresponds to flux lines pinned at the bottom of a potential well of depth VO.The pinning of flux lines with defects even without external current leads to a distortion of the flux-line lattice. The existence of external currents in the superconducting matrix results in Lorentz-force-induced flux motion which modifies the effective well depth to become VP f AV, on both sides of the flux line, respectively, where AVp = V(r) J/Jo, J is,the current density and JO is the current density without thermal
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Letters A 246 (1998) 353-358
activation [ 171. Therefore, the pinning force F(r) appearing in (4) depends on the current and is simply the change in the potential over distances of the order of the superconducting coherence length where F(r)
M Xv(r)/51J/JO,
V(r)=&
;
and
forO
,
0
=&[ln(f)
tl],
forr>l.
(5)
The form of the potential in (5) was determined from measmemen~ of the dependence of activation energy on the current density [ 181. The Green’s functions G$ appearing in (4) are the Fourier’s transform which relates the random force F(r) to the dislocation density and is given M ( 1/2E,)~&j~Ejnk,k,[l/(k2+k,2) by G;(k) 1/k2]}, where G$(r - r’) =
s
d3k (2~)3
exp[ik(r
- r’)]G$k).
(6)
The constant kc = (K/2E,) ‘1’ is a cutoff related to the dislocation core energy and K B 4C~. The dependence of U in Eq. (3) on the magnetic field is provided through the incorporation of the elastic properties of the lattice characterized by different length scales. The elastic moduli of the lattice have strong spatial dispersion and depend on the wave vectors k > l/X, where A is the ~ne~ation depth [ 191. If k < l/a, where a is the lattice constant, then C66 = &$3/C %-A)* and the constant appearing in the Green’s function ko = ( I/877A) (2&B/z&) +. The correlation function of the eIastic displacement has the general form gij( r) m ( 1/27r) [ 6, x tan-” (y/X> + (C&/C11)Eij ln(r/a) -I- Ejkrjrk/r'l. To simplify the calculations and obtain meaningful results that can be compared to experimental measurements several assumptions are introduced. Since for an FLL, the bulk modulus Ctr >> Cs the shear modulus and 3, = (C&/Cft)~ij < 1, therefore the correlation function gij (r) is approximated by
The effects of plastic deformations on the random pinning energies of FLL due to the existence of disloca-
tions in the superconducting matrix can in principle be calculated from Eq. (3). The motion of dislocations will introduce extra dissipation, thus lowering the total energy of the system. At large current (large driving forces), the effect of plastic deformations in the system is minimal resulting in linear I-V curves. As the current is decreased the effects of dislocations become impo~~t and the I-V characteristics start to deviate from their linear behavior.
3. Discussion and results The distributions of distances between point defects and the change in activation energies occur as a result of the plastic interactions. At small fields, fluctuating regions are on the average limited by the typical distance of the interaction region between pinning centers. ~e~while superconducting coherence over large distances will not exist until 6 < L. At large fields the fluctuations are limited by the existence of inhomogeneous domains formed by density variations and localization of groups of point defects. To qualitatively compare the results of the above simulation, Eq. (3) was integrated numerically with a cutoff set equal to the interaction range L. The temperature dependence of iJ was determined by substituting the value t(t) N coherence !5(0)(1t)- i12, for the superconducting length where t = T/TC is the reduced temperature. For each value of the applied magnetic field B, the Green’s function G& appearing in (4), is calculated and, consequently, the equilibrium dislocation density +i( r) is known. In this case, for each field value the activation energy as a function of temperature and current is determined from (3). In order to properly compare the calculations to the ex~~mental data reported in the literature, we chose for our simulation, defined essentially by Eq. (3), the same set of parameters as used in Ref. [ 121 for the highly oriented Bi&$aCu2Os+~ (BSCCO) sample known as B212. The superconducting coherence length is c(O) N 2.5 nm while the transition temperature is Tc = 86.1 K. The quantity Ve is some “average” activation energy given by Vi = 530 K which better fitted the measurements. The value of the penetration depth in BSCCO compound was found to be A M 0.21 pm [20]. The quantity f20 that is the increase in the lattice volume due to a single point de-
A.E. Khalil/Physics Letters A 246 (1998) 353-358
feet was taken from the experimental data to be & N 35 x lo3 nm3 and the magnitude of the Burger’s vector b of the dislocation was set equal to one. The activation energy was first calculated as a function of the applied field B for values extending to 3 T. The results are shown in Fig. 2. The behavior of the activation energy follows a logarithmic decrease for low field values while it follows an almost linear decrease for high field values with a crossover field Bo around 0.65 T. The calculated values of the activation energies are, in general, lower than the measured values since the existence of dislocation fields tend to decrease the activation energy. This result is due to the positive values of the interaction energy relative to the actual negative values of the pinning energies. The agreement between the measured data and the predictions of the model is very clear which confirms the conclusions of Ref. [ 121 that the field dependence of the activation energy is dominated by the plastic deformations of the FLL. The activation energy was calculated next as a function of the current density where JO was set equal to JO M 6 x lo4 A/cm2. The results are shown in Fig. 3 for high field value. The results of the proposed model are in good agreement with the reported measurements. In addition to the field and current dependencies of the activation energy, the temperature dependence was calculated and the results are shown in Fig. 4 for high field value. The results are quantitatively in agreement with the experimental measurements. The change of the pinning potential with the applied field indicates a change in the pinning mechanism due to the influence of plastic deformations on the disorder in the FLL. In order to explore these effects further and examine the question of the logarithmic dependence of U(J) on the current density, the pinning potential at low field values (exponential regime) was calculated for different pinning potential than that given by (5). The random potential function appearing in (5) was replaced by the Gaussian potential V(r) = Voexp( --r2/,$‘) and Eq. (3) was recalculated and no substantial variations in the pinning energy were observed. Furthermore, the integral in Eq. (3) was evaluated for the functional dependence U,, E -&A s[ l/(r - L)“] L@i(r) dr, for the value where n = 3 instead of n = 2 and the results are shown in Fig. 5. The pinning potential is more sensitive to the denominator in the integrand of (3) which is the
357
2000
1500
E 5
1000
500
0 -0.5
0
0.5
1
1.5
2
2.5
3
W-0
Fig. 2. The dependence of the activation energy U on the magnetic field I3 for the sample B212. The circles are the experimental data taken from Fig. 7 of Ref. [ 121, while the squares are the theoretical predictions of Eq. (3).
-200
0
200
400
600 600 J(AIcm*)
1000
1200
1400
Fig. 3. The dependence of the activation energy U on the current density J for the sample B212 at applied field value B = 1.83 T. The circles are the experimental data ( U( J)/U( 0) ) taken from Fig. 6b of Ref. [ 121, while the squares are the theoretical predictions (U(J)/Kj) of Eq. (3).
result of the stress field of all dislocations rather than the shape of the random force function. These features confirm the previous conclusions that the shape of the pinning energies is mostly determined by the nature of the disorder.
4. Conclusions The contribution of the plastic interactions to the pinning energies as the dislocation fields modify the elastic manifold of the lattice has been established. An answer has been presented to the question of the nature of the logarithmic pinning potential U(J) law and its relevance to the existence of disorder and plastic
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A.E. Khalil/Physics
A 246 (1998) 353-358
250
200
150
P 5 100
50 ‘-
I-
0
50
I
I
I
I
1
60
70
80
90
100
T(K)
Fig. 4. The dependence of the activation energy U on the temperature for the sample B212 at applied field value B = 1.83 T and current density J = 1250 A/cm*. The circles are the experimental data taken from Fig. 5b of Ref. [12], while the squares are the theoretical predictions of Eq. (3).
neous regions formed by density variations and localization of groups of point defects. In addition, the density of vortices increases relative to the density of the disorder and collective behavior starts to dominate. At small fields, individual vortices are pinned by a distribution of pinning strengths with values which are modified by plastic interactions in the FLL. The calculated values of the activation energies are lower than those measured. This result is due to the fact that the dislocation fields tend to decrease the activation energy due to the positive values of the interaction energy relative to the actual negative values of the pinning energies. An important component of the pinning strength distribution is determined by disorder in the form of plastic interaction between dislocation fields and random pinning sites in the superconductor. Furthermore, the effect of the plastic deformations as the magnetic field increases is to change the pinning mechanism thereby modifying the shape of the pinning potential.
0.8
References 6
0.6
3 z
[ 11 D.R. Nelson, Phys. Rev. Lett. 60 (1988)
0.4
0.2
0 0
0.1
0.2
0.3
0.4
0.5
0.6
j (Amperes)
Fig. 5. The dependence of the activation energy U on the current density J for the sample B212 at applied field value B = 0.29 T. The circles are the experimental data (U(J)/U(O) ) taken from Fig. 6a of Ref. [ 121, the squares are the theoretical predictions (U( J)/b) of Eq. (3) with II = 2, the solid triangles are the theoretical predictions (U(J)/Vf) of Eq. (3) with a Gaussian potential, while the empty triangles are the theoretical predictions (U(J)/b) of Eq. (3) with n=3.
deformation in FLL. In the current approach the static interaction between a defect center in the lattice and a moving dislocation field modifies the range and the strength of the pinning well. At small applied fields, fluctuating regions are limited on average by’the typical distance of the interaction region between pinning centers. The appearance of superconducting coherent domains will only exist for spatial distances where .$ < L.At large fields, the fluctuations are limited by the existence of inhomoge-
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