- Email: [email protected]
Labusch parameter of flux line lattice with planar pinning centers
Physica C 418 (2005) 87–92 www.elsevier.com/locate/physc
Labusch parameter of flux line lattice with planar pinning centers R. Laiho a, M. Safonchik a
a,b,*
, K.B. Traito
a
Wihuri Physical Laboratory, Department of Physics, University of Turku, FIN-20014 Turku, Finland b Ioffe Physico-Technical Institute, St. Petersburg 194021, Russia Received 22 October 2004; accepted 14 November 2004
Abstract Field dependence of the Labusch parameter avL ðBÞ is calculated for vortex arrays assuming planar pinning defects parallel to the vortices, taking into account the discreteness of the vortex lattice, i.e. beyond the framework of the elasticity theory. Considering both compressional and shear vortex oscillations the distribution of the vortex displacements are obtained. For the compressional case a region of instability is found near the vortex pinning layers. This region is important for determination of avL ðBÞ. In contrast, the distribution of the vortex displacements for the shear oscillations is smooth and independent of the field. This results in saturation of avL ðBÞ at high fields. Considering also nonlinear effects the dependence of avL on the vortex displacement is obtained up to displacements where depinning of the vortices occurs. 2004 Published by Elsevier B.V. PACS: 74.25.Qt; 74.25.Nf Keywords: Flux lattice; Pinning, AC response
1. Introduction Response of type-II superconductors in the mixed state to external electromagnetic or ultrasonic waves can give valuable information about their flux pinning properties and flux dynamics * Corresponding author. Address: Wihuri Physical Laboratory, Department of Physics, University of Turku, FIN-20014 Turku, Finland. Tel.: +358 23 335938; fax: +358 22 319836. E-mail address: [email protected] (M. Safonchik).
0921-4534/$ - see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physc.2004.11.013
[1,2]. Therefore, a lot of effort is devoted to analyze this problem [3,4]. For specimens with random point defects as the main flux pinning source, the observed linear response is usually described by a single-particle model [5–8] considering small oscillations of the vortices in a quadratic effective potential with curvature aL, known as the Labusch parameter [9]. The phenomenological Labusch constant is a statistical average over some single-peaked distribution of restoring forces. This single-particle
88
R. Laiho et al. / Physica C 418 (2005) 87–92
model agrees reasonably with many experiments performed in a broad range of frequencies, magnetic fields and temperatures, considering the field and temperature dependent Labusch parameter as a fitting parameter [3,10–12]. The single-particle model with a current-dependent Labusch parameter can also be used for description of the ac response of a superconductor in a critical state [13]. However, in the vortex lattice with a low density of pins the single-particle model does not take into account the oscillations of unpinned interstitial vortices. The dynamics of these vortices is determined by the long-range interaction with the pinned vortices. Thus, the overall linear response of the vortex system is determined by two characteristic parameters: the pinning Labusch parameter apL and the ‘‘vortex-cage’’ Labusch parameter avL . This situation is realized for superconducting films with periodic nanoengineered pinning arrays [14,15] or for untwinned YBCO single crystals near the melting temperature [16]. For the point pinning centers avL is determined by the shear modulus c66 [15,16]. Recently, we have investigated electrodynamics of a vortex array with the vortices perpendicular to a periodic planar pinning potential considered to be fixed at their crossing points with the pinning layers (the limit apL ! 1) [17]. In this case the vortex dynamics is determined by the tilt modulus c44 [18] with the joint action of the nonlocal and local parts resulting in two-mode electrodynamics [19, 20]. Thin superconducting screening layers are formed near the pinning layers and the vortex dynamics can be considered as oscillations of weakly pinned interstitial vortices. The resulting effective avL is found to be proportional to the local part of c44 and the thickness of the screening layer. In the case of the vortices parallel to the pinning layers the compressional dynamics is determined by the wave dependent modulus c11(k) which for a dense vortex lattice is [21,22] c11 ðkÞ ¼
B2 1 B/0 ; 4p 1 þ k2 k 2 ð8pkÞ2
ð1Þ
where B is the magnetic induction, k is the London penetration depth and /0 is the magnetic flux quantum. In contrast to c44(k) the local part of
c11(k) is negative, resulting in an unphysical oscillating mode with the wavelength less than the intervortex distance (for tilt waves the short mode decays within a few intervortex distances [19,20]). The negative local part of c11(k) signals about short-wave instability, which can be realized near the pinning layers, i.e. the continuous elasticity theory is not applicable to this case. In this paper we calculate avL numerically for a vortex array with the vortices parallel to planar pinning defects taking into account the discreteness of the vortex lattice, i.e. the calculations are carried out beyond the framework of the elasticity theory. Both compressional and shear vortex oscillations are considered and the distribution of the vortex displacements has been obtained. In the first case a region of the vortex instability with width depending on the magnetic field is found near the pinning layers. This region screens the space between the pinning layers where the distribution of the displacement is quite uniform. This results in avL ðBÞ depending mainly on the instability region. In contrast, the distribution of the vortex displacements for the shear vortex oscillations is smooth and field independent, resulting in absence of the field dependence of avL in a dense vortex lattice. The value of avL is in good agreement with the elasticity theory and is determined by c66 similar to the results obtained for point defects [15,16]. The good agreement is explained by the fact that c66 is positive and has only a local part. The nonlinear effects are also considered obtaining the dependence of avL on the vortex displacement up to displacements where the depinning of the vortices occurs.
2. Model and results We consider the three-dimensional triangular flux line lattice in an isotropic superconductor with vortex chains pinned at parallel planar defects separated by a distance l. The x-axis is perpendicular and the y-axis is parallel to the defect planes. Unpinned interstitial vortices between the defects are assumed to be at positions determined by the external Lorentz force f and long-range repulsive interaction with the pinned vortices at the planar
R. Laiho et al. / Physica C 418 (2005) 87–92
defects. The interaction between two planes containing parallel flux lines at positions (xm, ym) = (0, nb) and (x, y + nb), respectively, is V ðx; yÞ ¼
89
(a)
1 /20 X 2 1=2 K 0 ð½x2 þ ðy þ nbÞ =kÞ; 2 2 8p k n¼1
ð2Þ (b)
where K0(x) is a modified Bessel function. This infinite sum can be evaluated by means of the Poisson summation formula. The result is ! 1 /20 1 jxj=k X esm jxj 2pmy e V ðx; yÞ ¼ cos þ ; b 4pkb 2 sm k m¼1 ð3Þ 2
2 1/2
where sm = [k + (2pm/b) ] . For triangular lattices the position of the neighboring plane is (x0 = 31/2b/2, y0 = b/2). When an external force is applied the vortex chains are distorted from the equilibrium positions to a distance ui. The distributions of the vortex displacements can be found from the minimum of the total energy X X Etot ¼ V i;j ðxi xj ; y i y j Þ ui f ; ð4Þ i;j
i
where Vi,j is the interaction energy between ith and jth chains as given by Eq. (3). A comprehensive modified-Newton algorithm is used for finding this minimum as a function of several variables—vortex chain coordinates. The Labusch parameter avL determines the change of the interaction energy between the vortices (per one vortex) ! X 1 av 2 Etot ð u; uÞ Etot ð0Þ þ ui f ¼ L ð5Þ N 2 i P where u ¼ i ui =N is the average displacement of the chains and N 1 is the number of the chains between the pinning planes. In the case of small vortex displacements avL found from Eq. (3) corresponds to the usual definition of avL as avL ¼ f = u. We consider two cases: (i) compressional vortex displacements with the force directed along x and (ii) shear vortex displacements with the force directed along y. Fig. 1 shows the distribution of the vortex displacements obtained from the minimization of Eq. (4) at different values of the external field for the cases (i) (panel a) and (ii) (panel b).
Fig. 1. Distribution of the vortex displacements ui at different values of the external field (normalized to the characteristic field B* = /0/4pk2) and l/k = 5 for the cases of the compressional oscillations (panel a) and the shear oscillations (panel b).
In this figure the characteristic field B* = /0/4pk2. In the first case a region of large displacements near the pinning planes at 0 and 1 is clearly visible. The instability region is quite narrow as can be expected from the short-wave instability predicted by the elastic theory. Also a clear decrease in the field dependence of the amplitude of the displacement with increasing magnetic field is visible. In contrast, the distribution of ui for the shear displacement is smooth and field independent. According to Eq. (5) avL is completely determined by the distribution of the vortex displacements. Therefore we can calculate avL . Fig. 2 demonstrates the magnetic field dependences of avL for the shear mode at l/k = 10 in units of (/0/4pk2)2. Clearly avL grows steeply in low fields and saturates in high fields. The initial growth is connected with the small value of the shear modulus for a rare flux line lattice [22]. The saturation at high fields can be obtained from the equation of the elastic theory for the dense lattice [1,2], c66
d2 u fB þ ¼ 0; dx2 /0
ð6Þ
where c66 ¼
B/0 : 64p2 k2
ð7Þ
90
R. Laiho et al. / Physica C 418 (2005) 87–92
Fig. 2. Magnetic field dependences of avL for the shear mode at l/k = 10 in units of (/0/4pk2)2. The inset shows the saturation value of avL for different values of l/k.
Solving Eq. (6) with the boundary condition of the vortices fixed at the pinning planes (±l/2) we get u¼
fB ðl2 =4 x2 Þ: 2c66 /0
ð8Þ
Using this u(x) and Eq. (5) one obtains 2 avL ¼ 3ðk=lÞ . The inset to Fig. 2 shows the dependence of the saturation value avL on the distance between the pinning planes. It coincides within line thickness with that obtained in the elastic theory. The inset to Fig. 3 shows the field dependence of avL for the compressional mode at l/k = 5. This dependence has near to square-root behavior. It
differs from that expected from the local elasticity theory predicting a linear field dependence. This difference is explained by breaking of the elasticity theory near the pinning planes where the narrow region of large displacements is observed (see Fig. 1). In contrast, the derivative of the displacement in the space between the pinning planes from 0 to 1 is small. One can estimate the value of avL taking into account only the contribution of the instability region. In this region the current density can be obtained from the London equation j Buc/k24p, where u is the amplitude of the displacement. Then the average force to the vortices between the pinning planes is F 2Bu/0w/k24pl, where w is the width of the instability area. Using the relation F ¼ avL u and the fact that w is of order of the intervortex distance one can obtain avL ¼ 2kB1=2 =l in the units of (/0/4pk2)2. Fig. 3 shows the dependences avL ðBÞ at different l/k in the logarithmic scale (closed circles). It can be concluded that the its asymptotic behavior is described well by power function of B (dotted lines) and the value of the exponent is near to our estimation 1/2. The difference is explained by the approximative value of w and the contribution of the smooth part of the distribution of the vortex displacements. Having the value of avL for the compressional (aX) and the shear (aY) modes one can obtain avL for the arbitrary angle u between f and the x-axis. Because aX and aY are quite different the response is strongly anisotropic and the vortex displacement has two components: parallel to the force, uk, and perpendicular to the force, u?. In this case we can determine the corresponding Labusch parameters ak ¼
f aX aY ¼ 2 uk aX sin u þ aY cos2 u
ð9Þ
f aX aY : ¼ u? ðaX aY Þ sin 2u
ð10Þ
and a? ¼
Fig. 3. Magnetic field dependences avL for the compressional mode at different l/k in the logarithmic scale (closed circles) and fitted with a power function of B (dotted lines). The inset shows the field dependence of avL at l/k = 5.
Fig. 4 shows the angular dependences ak(u) and a?(u) at l/k = 5 and B = 100. The inset demonstrates the field dependences of ak(u) at the different angulars. At large displacements the nonlinear effects become important and the dependence of avL on the
R. Laiho et al. / Physica C 418 (2005) 87–92
91
3. Discussion
Fig. 4. Angular dependences of ak and a? at l/k = 5 and B = 100. The inset demonstrates the field dependences of ak and a? at different.
displacement is essential. Fig. 5(a) shows avL ð uÞ for the compressional mode at different values of B and L/k. Fig. 5 (b) gives the results for the shear mode. At a large displacement depinning occurs. The end point of the curves means the threshold displacement before the depinning. As can be seen from this figure at the depinning the displacement u 0:1–0:4b and the maximum relative reduction of avL is approximately 0.15 avL ð0Þ.
(a)
(b)
Fig. 5. Displacement dependences avL for the compressional mode (a) and for the shear mode (b) for different values of B and L/k.
Finally, we discuss shortly the connection of the considered theory with experiments. First, we should note that in the considered model the pinned vortices are prohibited to move both perpendicular to the planar defects and parallel to them. This is not realized for twin planes which are channels for easy movement. A double layer device consisting of a amorphous Nb3Ge bottom layer covered with a thick NbN layer in parallel channels [23] can be used for testing of the model. The strong pinning in the NbN layer (with the critical current density jc in NbN P 104jc in NbGe) corresponds the considered situation of the weakly pinned interstitial vortices distributed between fixed vortices in one-dimensional stripes. In the definition of avL we assume that the external force changes at distances much larger than l. In this case the long-wave dynamics can be described by an effective-medium theory with parameters averaged over l [17]. Concerning the ac response, the wave number of the electromagnetic wave having the frequency x is for the dense flux line lattice [7,8] 1 1 k2 ¼ 2 ; ð11Þ 1 þ x =ðx k B p ixÞ where xB = B/0/4pk2g, xp ¼ avL =g, and g is the vortex friction coefficient. For the compressional mode the wave vector is directed perpendicular to the pinning planes but for the shear mode it is parallel to them. Eq. (11) gives the value of the 1=2 penetration depth kC ¼ kðB=avL ðBÞÞ . As can be seen from Fig. 2 the condition kC l is satisfied very well for the shear mode. Then one can obtain the surface impedance of the superconductor as x ð12Þ Z¼ : ck The obtained dependence avL ðBÞ for the shear mode results in a monotonous increase of the field dependence of the surface resistance qS = 4pReZ/c with the strength depending on the shear modulus. Hence, experimental investigation of the ac response in this geometry can give information about the shear modulus. In contrast, kC l for the compressional mode in a wide field range (see Fig. 3) and the condition of
92
R. Laiho et al. / Physica C 418 (2005) 87–92
the applicability of the effective-medium theory is realized only at extremely high fields. In this regime Eqs. (11) and (12) predict unusual behavior of the surface resistance qS B1/4, i.e. ac losses decreasing as a function of the increasing field. Because at low field qS / B the ac losses should have a maximum in their field dependence. To obtain avL for the compressional mode in the whole field range the results of investigations of the ultrasound propagation in the mixed state can be used [4,24]. The interaction between the vortices and the pinning centers increases the elasticity and dissipation of sound with the values depending on avL [24]. For the sound frequency of 10 MHz and the velocity ct = 2 · 105 cm/s we obtain the wave length 1.2 mm, which is much longer than l and the effective-medium theory can be applied at any fields. 4. Conclusions The field dependence of the Labusch parameter avL ðBÞ is calculated for the compressional and shear vortex oscillations in a planar pinning potential taking into account the discreteness of the vortex lattice and the nonlinear effects beyond the framework of the elasticity theory. In the compressional case a region of instability of the vortex displacement is found near the pinning layers. This region is important for determination of avL ðBÞ which can attain a maximum in the field dependence of the surface resistance. In contrast, the distribution of the vortex displacements for shear vortex oscillations is smooth and field independent and results in saturation of avL ðBÞ at high fields. Acknowledgment This work was supported by the Wihuri Foundation, Finland.
References [1] G. Blatter, M.V. FeigelÕman, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [2] E.H. Brandt, Rep. Prog. Phys. 58 (1995) 1465. [3] M. Golosovsky, M. Tsindlekht, D. Davidov, Supercond. Sci. Technol. 9 (1996) 1. [4] J. Pankert, G. Marbach, A. Comberg, P. Lemmens, Fro¨ning, S. Ewert, Phys. Rev. Lett. 65 (1990) 3052. [5] J.I. Gittleman, B. Rosenblum, Phys. Rev. Lett. 16 (1966) 734. [6] A.M. Campbell, J. Phys. C 2 (1969) 1492. [7] M.W. Coffey, J.R. Clem, Phys. Rev. Lett. 67 (1991) 386. [8] E.H. Brandt, Phys. Rev. Lett. 67 (1991) 2219. [9] A.M. Campbell, J.E. Evetts, Adv. Phys. 21 (1972) 199. [10] Y. Tsuchiya, K. Iwaya, K. Kinoshita, T. Hanaguri, H. Kitano, A. Maeda, K. Shibata, T. Nishizaki, N. Kobayashi, Phys. Rev. B 63 (2001) 184517. [11] V.F. Correa, G. Nieva, F. de la Cruz, Phys. Rev. Lett. 87 (2001) 057003. [12] T. Banerjee, V.C. Bagwe, J. John, S.P. Pai, R. Pinto, D. Kanjilal, Phys. Rev. B 69 (2004) 104533. [13] R. Prozorov, R.W. Gianetta, N. Kameda, T. Tamegai, J.A. Schlueter, P. Fournier, Phys. Rev. B 67 (2003) 184501. [14] A.V. Silhanec, S. Raedts, M. Lange, V.V. Moshchalkov, Phys. Rev. B 67 (2003) 064502. [15] C.C. de Souza Silva, J.A. Aguiar, V.V. Moshchalkov, Phys. Rev. B 68 (2003) 134512. [16] P. Matl, N.P. Ong, R. Gagnon, L. Taillefer, Phys. Rev. B 65 (2002) 214514. [17] R. Laiho, A.G. Lyublinsky, E. La¨hderanta, K.B. Traito, Phys. Rev. B 63 (2001) 214508. [18] E.H. Brandt, J. Low Temp. Phys. 26 (1977) 709. [19] E.B. Sonin, A.K. Tagantsev, K.B. Traito, Phys. Rev. B 46 (1992) 5830. [20] B. Plac¸ais, P. Mathieu, Y. Simon, E.B. Sonin, K.B. Traito, Phys. Rev. B 54 (1996) 13083. [21] E.B. Sonin, K.B. Traito, Phys. Rev. B 50 (1994) 13547. [22] M.-C. Miguel, M. Kardar, Phys. Rev. B 62 (2000) 5942. [23] M.H. Theunissen, E. Van der Drift, P.H. Kes, Phys. Rev. Lett. 77 (1996) 159. [24] D. Domı´nguez, L. Bulaevskii, B. Ivlev, M. Maley, A.R. Bishop, Phys. Rev. Lett. 74 (1995) 2579.
Labusch parameter of flux line lattice with planar pinning centers R. Laiho a, M. Safonchik a
a,b,*
, K.B. Traito
a
Wihuri Physical Laboratory, Department of Physics, University of Turku, FIN-20014 Turku, Finland b Ioffe Physico-Technical Institute, St. Petersburg 194021, Russia Received 22 October 2004; accepted 14 November 2004
Abstract Field dependence of the Labusch parameter avL ðBÞ is calculated for vortex arrays assuming planar pinning defects parallel to the vortices, taking into account the discreteness of the vortex lattice, i.e. beyond the framework of the elasticity theory. Considering both compressional and shear vortex oscillations the distribution of the vortex displacements are obtained. For the compressional case a region of instability is found near the vortex pinning layers. This region is important for determination of avL ðBÞ. In contrast, the distribution of the vortex displacements for the shear oscillations is smooth and independent of the field. This results in saturation of avL ðBÞ at high fields. Considering also nonlinear effects the dependence of avL on the vortex displacement is obtained up to displacements where depinning of the vortices occurs. 2004 Published by Elsevier B.V. PACS: 74.25.Qt; 74.25.Nf Keywords: Flux lattice; Pinning, AC response
1. Introduction Response of type-II superconductors in the mixed state to external electromagnetic or ultrasonic waves can give valuable information about their flux pinning properties and flux dynamics * Corresponding author. Address: Wihuri Physical Laboratory, Department of Physics, University of Turku, FIN-20014 Turku, Finland. Tel.: +358 23 335938; fax: +358 22 319836. E-mail address: [email protected] (M. Safonchik).
0921-4534/$ - see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physc.2004.11.013
[1,2]. Therefore, a lot of effort is devoted to analyze this problem [3,4]. For specimens with random point defects as the main flux pinning source, the observed linear response is usually described by a single-particle model [5–8] considering small oscillations of the vortices in a quadratic effective potential with curvature aL, known as the Labusch parameter [9]. The phenomenological Labusch constant is a statistical average over some single-peaked distribution of restoring forces. This single-particle
88
R. Laiho et al. / Physica C 418 (2005) 87–92
model agrees reasonably with many experiments performed in a broad range of frequencies, magnetic fields and temperatures, considering the field and temperature dependent Labusch parameter as a fitting parameter [3,10–12]. The single-particle model with a current-dependent Labusch parameter can also be used for description of the ac response of a superconductor in a critical state [13]. However, in the vortex lattice with a low density of pins the single-particle model does not take into account the oscillations of unpinned interstitial vortices. The dynamics of these vortices is determined by the long-range interaction with the pinned vortices. Thus, the overall linear response of the vortex system is determined by two characteristic parameters: the pinning Labusch parameter apL and the ‘‘vortex-cage’’ Labusch parameter avL . This situation is realized for superconducting films with periodic nanoengineered pinning arrays [14,15] or for untwinned YBCO single crystals near the melting temperature [16]. For the point pinning centers avL is determined by the shear modulus c66 [15,16]. Recently, we have investigated electrodynamics of a vortex array with the vortices perpendicular to a periodic planar pinning potential considered to be fixed at their crossing points with the pinning layers (the limit apL ! 1) [17]. In this case the vortex dynamics is determined by the tilt modulus c44 [18] with the joint action of the nonlocal and local parts resulting in two-mode electrodynamics [19, 20]. Thin superconducting screening layers are formed near the pinning layers and the vortex dynamics can be considered as oscillations of weakly pinned interstitial vortices. The resulting effective avL is found to be proportional to the local part of c44 and the thickness of the screening layer. In the case of the vortices parallel to the pinning layers the compressional dynamics is determined by the wave dependent modulus c11(k) which for a dense vortex lattice is [21,22] c11 ðkÞ ¼
B2 1 B/0 ; 4p 1 þ k2 k 2 ð8pkÞ2
ð1Þ
where B is the magnetic induction, k is the London penetration depth and /0 is the magnetic flux quantum. In contrast to c44(k) the local part of
c11(k) is negative, resulting in an unphysical oscillating mode with the wavelength less than the intervortex distance (for tilt waves the short mode decays within a few intervortex distances [19,20]). The negative local part of c11(k) signals about short-wave instability, which can be realized near the pinning layers, i.e. the continuous elasticity theory is not applicable to this case. In this paper we calculate avL numerically for a vortex array with the vortices parallel to planar pinning defects taking into account the discreteness of the vortex lattice, i.e. the calculations are carried out beyond the framework of the elasticity theory. Both compressional and shear vortex oscillations are considered and the distribution of the vortex displacements has been obtained. In the first case a region of the vortex instability with width depending on the magnetic field is found near the pinning layers. This region screens the space between the pinning layers where the distribution of the displacement is quite uniform. This results in avL ðBÞ depending mainly on the instability region. In contrast, the distribution of the vortex displacements for the shear vortex oscillations is smooth and field independent, resulting in absence of the field dependence of avL in a dense vortex lattice. The value of avL is in good agreement with the elasticity theory and is determined by c66 similar to the results obtained for point defects [15,16]. The good agreement is explained by the fact that c66 is positive and has only a local part. The nonlinear effects are also considered obtaining the dependence of avL on the vortex displacement up to displacements where the depinning of the vortices occurs.
2. Model and results We consider the three-dimensional triangular flux line lattice in an isotropic superconductor with vortex chains pinned at parallel planar defects separated by a distance l. The x-axis is perpendicular and the y-axis is parallel to the defect planes. Unpinned interstitial vortices between the defects are assumed to be at positions determined by the external Lorentz force f and long-range repulsive interaction with the pinned vortices at the planar
R. Laiho et al. / Physica C 418 (2005) 87–92
defects. The interaction between two planes containing parallel flux lines at positions (xm, ym) = (0, nb) and (x, y + nb), respectively, is V ðx; yÞ ¼
89
(a)
1 /20 X 2 1=2 K 0 ð½x2 þ ðy þ nbÞ =kÞ; 2 2 8p k n¼1
ð2Þ (b)
where K0(x) is a modified Bessel function. This infinite sum can be evaluated by means of the Poisson summation formula. The result is ! 1 /20 1 jxj=k X esm jxj 2pmy e V ðx; yÞ ¼ cos þ ; b 4pkb 2 sm k m¼1 ð3Þ 2
2 1/2
where sm = [k + (2pm/b) ] . For triangular lattices the position of the neighboring plane is (x0 = 31/2b/2, y0 = b/2). When an external force is applied the vortex chains are distorted from the equilibrium positions to a distance ui. The distributions of the vortex displacements can be found from the minimum of the total energy X X Etot ¼ V i;j ðxi xj ; y i y j Þ ui f ; ð4Þ i;j
i
where Vi,j is the interaction energy between ith and jth chains as given by Eq. (3). A comprehensive modified-Newton algorithm is used for finding this minimum as a function of several variables—vortex chain coordinates. The Labusch parameter avL determines the change of the interaction energy between the vortices (per one vortex) ! X 1 av 2 Etot ð u; uÞ Etot ð0Þ þ ui f ¼ L ð5Þ N 2 i P where u ¼ i ui =N is the average displacement of the chains and N 1 is the number of the chains between the pinning planes. In the case of small vortex displacements avL found from Eq. (3) corresponds to the usual definition of avL as avL ¼ f = u. We consider two cases: (i) compressional vortex displacements with the force directed along x and (ii) shear vortex displacements with the force directed along y. Fig. 1 shows the distribution of the vortex displacements obtained from the minimization of Eq. (4) at different values of the external field for the cases (i) (panel a) and (ii) (panel b).
Fig. 1. Distribution of the vortex displacements ui at different values of the external field (normalized to the characteristic field B* = /0/4pk2) and l/k = 5 for the cases of the compressional oscillations (panel a) and the shear oscillations (panel b).
In this figure the characteristic field B* = /0/4pk2. In the first case a region of large displacements near the pinning planes at 0 and 1 is clearly visible. The instability region is quite narrow as can be expected from the short-wave instability predicted by the elastic theory. Also a clear decrease in the field dependence of the amplitude of the displacement with increasing magnetic field is visible. In contrast, the distribution of ui for the shear displacement is smooth and field independent. According to Eq. (5) avL is completely determined by the distribution of the vortex displacements. Therefore we can calculate avL . Fig. 2 demonstrates the magnetic field dependences of avL for the shear mode at l/k = 10 in units of (/0/4pk2)2. Clearly avL grows steeply in low fields and saturates in high fields. The initial growth is connected with the small value of the shear modulus for a rare flux line lattice [22]. The saturation at high fields can be obtained from the equation of the elastic theory for the dense lattice [1,2], c66
d2 u fB þ ¼ 0; dx2 /0
ð6Þ
where c66 ¼
B/0 : 64p2 k2
ð7Þ
90
R. Laiho et al. / Physica C 418 (2005) 87–92
Fig. 2. Magnetic field dependences of avL for the shear mode at l/k = 10 in units of (/0/4pk2)2. The inset shows the saturation value of avL for different values of l/k.
Solving Eq. (6) with the boundary condition of the vortices fixed at the pinning planes (±l/2) we get u¼
fB ðl2 =4 x2 Þ: 2c66 /0
ð8Þ
Using this u(x) and Eq. (5) one obtains 2 avL ¼ 3ðk=lÞ . The inset to Fig. 2 shows the dependence of the saturation value avL on the distance between the pinning planes. It coincides within line thickness with that obtained in the elastic theory. The inset to Fig. 3 shows the field dependence of avL for the compressional mode at l/k = 5. This dependence has near to square-root behavior. It
differs from that expected from the local elasticity theory predicting a linear field dependence. This difference is explained by breaking of the elasticity theory near the pinning planes where the narrow region of large displacements is observed (see Fig. 1). In contrast, the derivative of the displacement in the space between the pinning planes from 0 to 1 is small. One can estimate the value of avL taking into account only the contribution of the instability region. In this region the current density can be obtained from the London equation j Buc/k24p, where u is the amplitude of the displacement. Then the average force to the vortices between the pinning planes is F 2Bu/0w/k24pl, where w is the width of the instability area. Using the relation F ¼ avL u and the fact that w is of order of the intervortex distance one can obtain avL ¼ 2kB1=2 =l in the units of (/0/4pk2)2. Fig. 3 shows the dependences avL ðBÞ at different l/k in the logarithmic scale (closed circles). It can be concluded that the its asymptotic behavior is described well by power function of B (dotted lines) and the value of the exponent is near to our estimation 1/2. The difference is explained by the approximative value of w and the contribution of the smooth part of the distribution of the vortex displacements. Having the value of avL for the compressional (aX) and the shear (aY) modes one can obtain avL for the arbitrary angle u between f and the x-axis. Because aX and aY are quite different the response is strongly anisotropic and the vortex displacement has two components: parallel to the force, uk, and perpendicular to the force, u?. In this case we can determine the corresponding Labusch parameters ak ¼
f aX aY ¼ 2 uk aX sin u þ aY cos2 u
ð9Þ
f aX aY : ¼ u? ðaX aY Þ sin 2u
ð10Þ
and a? ¼
Fig. 3. Magnetic field dependences avL for the compressional mode at different l/k in the logarithmic scale (closed circles) and fitted with a power function of B (dotted lines). The inset shows the field dependence of avL at l/k = 5.
Fig. 4 shows the angular dependences ak(u) and a?(u) at l/k = 5 and B = 100. The inset demonstrates the field dependences of ak(u) at the different angulars. At large displacements the nonlinear effects become important and the dependence of avL on the
R. Laiho et al. / Physica C 418 (2005) 87–92
91
3. Discussion
Fig. 4. Angular dependences of ak and a? at l/k = 5 and B = 100. The inset demonstrates the field dependences of ak and a? at different.
displacement is essential. Fig. 5(a) shows avL ð uÞ for the compressional mode at different values of B and L/k. Fig. 5 (b) gives the results for the shear mode. At a large displacement depinning occurs. The end point of the curves means the threshold displacement before the depinning. As can be seen from this figure at the depinning the displacement u 0:1–0:4b and the maximum relative reduction of avL is approximately 0.15 avL ð0Þ.
(a)
(b)
Fig. 5. Displacement dependences avL for the compressional mode (a) and for the shear mode (b) for different values of B and L/k.
Finally, we discuss shortly the connection of the considered theory with experiments. First, we should note that in the considered model the pinned vortices are prohibited to move both perpendicular to the planar defects and parallel to them. This is not realized for twin planes which are channels for easy movement. A double layer device consisting of a amorphous Nb3Ge bottom layer covered with a thick NbN layer in parallel channels [23] can be used for testing of the model. The strong pinning in the NbN layer (with the critical current density jc in NbN P 104jc in NbGe) corresponds the considered situation of the weakly pinned interstitial vortices distributed between fixed vortices in one-dimensional stripes. In the definition of avL we assume that the external force changes at distances much larger than l. In this case the long-wave dynamics can be described by an effective-medium theory with parameters averaged over l [17]. Concerning the ac response, the wave number of the electromagnetic wave having the frequency x is for the dense flux line lattice [7,8] 1 1 k2 ¼ 2 ; ð11Þ 1 þ x =ðx k B p ixÞ where xB = B/0/4pk2g, xp ¼ avL =g, and g is the vortex friction coefficient. For the compressional mode the wave vector is directed perpendicular to the pinning planes but for the shear mode it is parallel to them. Eq. (11) gives the value of the 1=2 penetration depth kC ¼ kðB=avL ðBÞÞ . As can be seen from Fig. 2 the condition kC l is satisfied very well for the shear mode. Then one can obtain the surface impedance of the superconductor as x ð12Þ Z¼ : ck The obtained dependence avL ðBÞ for the shear mode results in a monotonous increase of the field dependence of the surface resistance qS = 4pReZ/c with the strength depending on the shear modulus. Hence, experimental investigation of the ac response in this geometry can give information about the shear modulus. In contrast, kC l for the compressional mode in a wide field range (see Fig. 3) and the condition of
92
R. Laiho et al. / Physica C 418 (2005) 87–92
the applicability of the effective-medium theory is realized only at extremely high fields. In this regime Eqs. (11) and (12) predict unusual behavior of the surface resistance qS B1/4, i.e. ac losses decreasing as a function of the increasing field. Because at low field qS / B the ac losses should have a maximum in their field dependence. To obtain avL for the compressional mode in the whole field range the results of investigations of the ultrasound propagation in the mixed state can be used [4,24]. The interaction between the vortices and the pinning centers increases the elasticity and dissipation of sound with the values depending on avL [24]. For the sound frequency of 10 MHz and the velocity ct = 2 · 105 cm/s we obtain the wave length 1.2 mm, which is much longer than l and the effective-medium theory can be applied at any fields. 4. Conclusions The field dependence of the Labusch parameter avL ðBÞ is calculated for the compressional and shear vortex oscillations in a planar pinning potential taking into account the discreteness of the vortex lattice and the nonlinear effects beyond the framework of the elasticity theory. In the compressional case a region of instability of the vortex displacement is found near the pinning layers. This region is important for determination of avL ðBÞ which can attain a maximum in the field dependence of the surface resistance. In contrast, the distribution of the vortex displacements for shear vortex oscillations is smooth and field independent and results in saturation of avL ðBÞ at high fields. Acknowledgment This work was supported by the Wihuri Foundation, Finland.
References [1] G. Blatter, M.V. FeigelÕman, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [2] E.H. Brandt, Rep. Prog. Phys. 58 (1995) 1465. [3] M. Golosovsky, M. Tsindlekht, D. Davidov, Supercond. Sci. Technol. 9 (1996) 1. [4] J. Pankert, G. Marbach, A. Comberg, P. Lemmens, Fro¨ning, S. Ewert, Phys. Rev. Lett. 65 (1990) 3052. [5] J.I. Gittleman, B. Rosenblum, Phys. Rev. Lett. 16 (1966) 734. [6] A.M. Campbell, J. Phys. C 2 (1969) 1492. [7] M.W. Coffey, J.R. Clem, Phys. Rev. Lett. 67 (1991) 386. [8] E.H. Brandt, Phys. Rev. Lett. 67 (1991) 2219. [9] A.M. Campbell, J.E. Evetts, Adv. Phys. 21 (1972) 199. [10] Y. Tsuchiya, K. Iwaya, K. Kinoshita, T. Hanaguri, H. Kitano, A. Maeda, K. Shibata, T. Nishizaki, N. Kobayashi, Phys. Rev. B 63 (2001) 184517. [11] V.F. Correa, G. Nieva, F. de la Cruz, Phys. Rev. Lett. 87 (2001) 057003. [12] T. Banerjee, V.C. Bagwe, J. John, S.P. Pai, R. Pinto, D. Kanjilal, Phys. Rev. B 69 (2004) 104533. [13] R. Prozorov, R.W. Gianetta, N. Kameda, T. Tamegai, J.A. Schlueter, P. Fournier, Phys. Rev. B 67 (2003) 184501. [14] A.V. Silhanec, S. Raedts, M. Lange, V.V. Moshchalkov, Phys. Rev. B 67 (2003) 064502. [15] C.C. de Souza Silva, J.A. Aguiar, V.V. Moshchalkov, Phys. Rev. B 68 (2003) 134512. [16] P. Matl, N.P. Ong, R. Gagnon, L. Taillefer, Phys. Rev. B 65 (2002) 214514. [17] R. Laiho, A.G. Lyublinsky, E. La¨hderanta, K.B. Traito, Phys. Rev. B 63 (2001) 214508. [18] E.H. Brandt, J. Low Temp. Phys. 26 (1977) 709. [19] E.B. Sonin, A.K. Tagantsev, K.B. Traito, Phys. Rev. B 46 (1992) 5830. [20] B. Plac¸ais, P. Mathieu, Y. Simon, E.B. Sonin, K.B. Traito, Phys. Rev. B 54 (1996) 13083. [21] E.B. Sonin, K.B. Traito, Phys. Rev. B 50 (1994) 13547. [22] M.-C. Miguel, M. Kardar, Phys. Rev. B 62 (2000) 5942. [23] M.H. Theunissen, E. Van der Drift, P.H. Kes, Phys. Rev. Lett. 77 (1996) 159. [24] D. Domı´nguez, L. Bulaevskii, B. Ivlev, M. Maley, A.R. Bishop, Phys. Rev. Lett. 74 (1995) 2579.