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RAPID COMMUNICATIONS PHYSICAL REVIEW B VOLUME 62, NUMBER 22 1 DECEMBER 2000-II Tilt modulus and angle-dependent flux lattice melting in the lowest Landau level approximation G. Mohler and D. Stroud Department of Physics, Ohio State University, Columbus, Ohio 43210 共Received 23 May 2000; revised manuscript received 7 August 2000兲 For a clean high-T c superconductor, we analyze the Lawrence-Doniach free energy in a tilted magnetic field within the lowest Landau level approximation. The free energy maps onto that of a strictly c-axis field, but with a reduced interlayer coupling. We use this result to calculate the tilt modulus C 44 of a vortex lattice and vortex liquid. The vortex contribution to C 44 can be expressed in terms of the squared c-axis Josephson plasmon frequency 2pl . The transverse component of the field has very little effect on the position of the melting curve. I. INTRODUCTION II. LOWEST LANDAU LEVEL APPROXIMATION WITH TILTED FIELD This paper is concerned with the tilt modulus C 44 of the vortex system in a high-T c or otherwise layered superconductor. C 44 is an elastic constant which measures the free energy cost of applying a small transverse field in addition to a field applied parallel to the c axis. It is a relevant parameter in many physical processes, such as collective pinning,1 the ‘‘peak effect,’’1,2 and the transition to the Bose glass state.3 Previous investigations have shown that C 44 is very strongly dependent on the wave vector k of the transverse field.4,5 In addition, it is finite in the flux liquid as well as the flux solid phase and is strongly affected by different kinds of disorder.1,6,7 A number of authors have analyzed C 44 in various approximations. Several groups4,5,8 have calculated C 44(k) in a flux lattice as a function of the wave vector k of the transverse magnetic field. Other authors have considered C 44 in the flux liquid state,7,9 and in the presence of disorder in both the solid and liquid phases.6,7 In this paper, we calculate C 44 for a layered superconductor at high fields. We describe the superconductor using a Lawrence-Doniach10 free energy functional, and we evaluate fluctuations in the lowest Landau level 共LLL兲 approximation, appropriate for strong c-axis magnetic fields.11 This LLL approach accounts well for the position of the flux lattice melting curve in the magnetic-field/temperature plane,11,12 as well as the value of the magnetization in both solid and liquid phases,13 both in Bi2 Sr2 CaCu2 O8⫹ ␦ and the less anisotropic YBa2 Cu3 O7⫺x . To obtain C 44 , we generalize the LLL approximation to apply to fields with a finite ab component. For a defect-free system in the LLL approximation, the tilted field free energy can be exactly mapped onto the usual free energy for a field B储 c, but with a weaker interlayer coupling which depends on tilt angle. As a byproduct of this transformation, we can also analyze the behavior of the flux lattice melting curve in a magnetic field tilted at an angle to the c axis. We find that this curve is little affected by the presence of a transverse magnetic field component at fields where the LLL approximation is applicable, consistent with available 共but limited兲 experiment. At fixed external magnetic field H⫽H z ẑ⫹H x x̂, the Lawrence-Doniach Gibbs free energy in a form which incorporates the field energy is10 0163-1829/2000/62共22兲/14665共4兲/$15.00 PRB 62 G 关 ,A兴 ⫽d 兺n 冕 再 d 2 r a 共 T 兲 兩 n 共 r兲 兩 2 ⫹t 兩 n 共 r兲 e ⫺i(2 /⌽ 0 )A z d ⫺ n⫺1 共 r兲 兩 2 ⫹ 1 2m ab ⫹V 冏冉 冊 冏 2 q b ⫺iបⵜ⬜ ⫺ A⬜ n 共 r兲 ⫹ 兩 n 共 r兲 兩 4 c 2 冎 兩 B⫺H兩 2 8 ⬅G̃ 关 ,A兴 ⫹V 兩 B⫺H兩 2 . 8 共1兲 Here n is the order parameter in the n th layer, d is the distance between layers, q⫽⫺2 兩 e 兩 , and t⫽ប 2 /(2m c d 2 ) is the interlayer coupling energy. In a cuprate superconductor such as Bi2 Sr2 CaCu2 O8⫹ ␦ , d represents the repeat distance in the c direction. “⬜ is the xy component of the gradient operator, V is the system volume, and the magnetic induction B⫽“⫻A, where the vector potential A⫽(A⬜ ,A z ). We choose the gauge A⫽⫺B z yx̂⫹B x yẑ and expand the n ’s as a linear combination of LLL states, to obtain n (r) 2 2 2 2 (T)/4b 2 ) 1/4兺 k c k,n e ikx⫺(y⫺kl ) /2l , where k⫽2 p/ ⫽( 冑3a H L x is the quantized momentum (p a positive integer兲, l⫽ 冑⌽ 0 /2 B z is the magnetic length, a H (T)⫽ 关 a(T) ⫹2t 兴关 1⫺B z /H c2 (T) 兴 , and sample volume is V⫽L x L y L z . In terms of measurable quantities, a(T)⫹2t⫽ ⫺បqH c2 (T)/(2m ab c), where m ab is the transverse effective mass and H c2 (T) is the upper critical field as a function of temperature T. Substituting this expansion into Eq. 共1兲 and integrating over x and y yields R14 665 ©2000 The American Physical Society RAPID COMMUNICATIONS R14 666 G̃⫽ G. MOHLER AND D. STROUD 2 aH 共 T 兲 l 2d b Nx 兺 k,n 再 ⫺ 兩 c k,n 兩 2 ⫺ * c k,n⫺1 e ⫺i2 B x dl ⫻ 共 c k,n ⫹ 3 1/4 兺 5 5/2 q ,q 1 * c k,n c k⫹q 2 2 k/⌽ 0 corresponding, at low temperatures, to a tilted Abrikosov lattice. t exp„⫺ 共 B x dl/⌽ 0 兲 2 … 兩 a H共 T 兲兩 ⫹c.c. 兲 c* c e 1 ,n k⫹q 2 ,n k⫹q 1 ⫹q 2 ,n 2 2 ⫺(q 1 ⫹q 2 )l 2 /2 PRB 62 冎 III. TILT MODULUS C 44 is defined by , 冉 冊 1 2G C 44⫽ V 2 共2兲 where N x ⫽(L x /l)( 冑3/4 ) . This expresses the Gibbs free energy as a function of the expansion coefficients c k,n for a tilted magnetic field. This expression differs from that for B储 ẑ in only two respects:11 共i兲 there is an extra phase factor in the interlayer coupling; and 共ii兲 the strength of the interlayer coupling is renormalized by an exponentially decaying factor. The phase factor in Eq. 共2兲 can be eliminated by introduc2 ing a new set of coefficients c̃ k,n ⬅c k,n e ⫺i(2 B x dl k/⌽ 0 )n , in 2 * c k,n⫺1 e ⫺i2 B x dl k/⌽ 0 Eq. 共2兲. The product term becomes c k,n * c̃ k,n⫺1 . This transformation does not affect the free ⫽c̃ k,n energy term which contains products of four coefficients. To see this, note that when such a typical product is transformed, it picks up a phase factor 2 e i(2 B x dl n/⌽ 0 )(k⫺(k⫹q 1 )⫺(k⫹q 2 )⫹(k⫹q 1 ⫹q 2 )) , where each (k ⫹q i ) term is an integer modulo N ⌽ , N ⌽ being the number of flux lines in the sample. The terms in the exponential can be summed to yield zero, leaving this term unaltered. Similarly, 兩 c k,n 兩 2 ⫽ 兩 c̃ k,n 兩 2 . In terms of the new variables, the free energy is thus 1/2 G̃⫽ 2 aH 共 T 兲 l 2 dN x b 兺 k,n 再 ⫺ 兩 c̃ k,n 兩 2 ⫺ * c̃ k,n⫺1 ⫹c.c 兲 ⫹ ⫻ 共 c̃ k,n ⫻ 兺 q ,q 1 * c̃ k,n c̃ k⫹q 2 t⬘ 兩 a H共 T 兲兩 5 t ⬘ ⫽te ⫺( B x dl/⌽ 0 ) 2 2 ⫺(q 1 ⫹q 2 )l 2 /2 冎 , 2 ⫻ 冊 兺 k,n 再 兩 B⫺H兩 2 . 8 ⫺ 兩 c k,n 兩 2 ⫺ 1/4 兺k c̃ k,n e ⫺ik(2 B dl /⌽ )n e ikx⫺(y⫺kl ) /2l , x 2 0 3 1/4 兺 2 5/2 q 1 ,q 2 共7兲 t exp„⫺ 共 B x dl/⌽ 0 兲 2 … 兩 a H共 T 兲兩 2 2 2 共5兲 * c̃ k,n c̃ k⫹q c̃ * 2 c̃ 1 ,n k⫹q 2 ,n k⫹q 1 ⫹q 2 ,n 2 2 2 /l e ⫺(q 1 ⫹q 2 ) 冎 . 共8兲 Then, writing 2 / 2 ⫽B z2 2 / B 2x , taking B z ⬃H z , and evaluating the derivatives from Eqs. 共7兲, with the result 共4兲 is the renormalized interlayer coupling. Thus, the tilted B-field free energy is identical in form to the free energy for a field B储 ẑ, but with a weaker interlayer coupling. The ground state solution for the redefined amplitudes c̃ k,n are identical to those for the c k,n ’s in a strictly longitudinal field but weaker B储 ẑ. However, the order parameter n picks up a B x dependency, 4b 2 k,n 兩 B⫺H兩 2 8 Then, writing / →H z / H x for small H x , we find that c the second term in Eq. 共7兲 contributes a term C 44 2 2 ⫽H z /(4 )⬃B z /(4 ) in the LLL regime, where the magnetization is small. This is the compressive contribution to the tilt modulus. To calculate the remaining 共vortex-related兲 contribution to v C 44 , which we denote C 44 , we first write G̃/k B T⫽H/T, 2 2 where T⫽bk B T/a H (T) l d and ⫹ where 冉 ⬅G v ⫹V 冕兿 * dc̃ k,n e ⫺G̃/k B T ⫹V dc̃ k,n * c̃ k,n⫺1 ⫹c.c. 兲 ⫻ 共 c̃ k,n 共3兲 n 共 r兲 ⫽ G⫽⫺k B Tln 5/2 c̃ * c̃ e 1 ,n k⫹q 2 ,n k⫹q 1 ⫹q 2 ,n 冑3a H2 共 T 兲 where is the angle between H and the c axis, and where G, the Gibbs free energy appropriate to an experiment at constant H and T, is given by H⫽N x 3 1/4 共6兲 , ⫽0 v ⫽ C 44 冉冑 冊 冉 冑 冑 冓兺 3 8 1/2 ⫻ Bz td 3 兩 a 共 T 兲 ⫹2t 兩 k,n ⌽ 0 bL y L z 冊 共 1⫺B z /H c2 兲 冔 * c k,n⫺1 ⫹c.c. 兲 T , 共 c k,n 共9兲 where we have dropped the tildes on the c k,n ’s because this expression is evaluated at B x ⫽0. Equation 共9兲 can be expressed in terms of measurable quantities using the identities a(T)⫹2t⫽បqH c2 (T)/2m ab c and b⫽2 2 (qប/m ab c) 2 , where ⫽ ab (B z ,T)/ ab (B z ,T) is the ratio of the ab penetration depth and coherence length. We also write t ⫽ប 2 /(2m ab d 2 ␥ 2 ), where ␥ 2 is the anisotropy parameter. The result is RAPID COMMUNICATIONS PRB 62 TILT MODULUS AND ANGLE-DEPENDENT FLUX . . . v C 44 ⫽ 1 32 ␥ 2 2 ⫻ 冓兺 k,n 关 H c2 共 T 兲 ⫺B z 兴 B z 冔 R14 667 Nx N ⌽N z * c k,n⫺1 ⫹c.c. T . c k,n 共10兲 Finally, the total tilt modulus is C 44⬃ B z2 4 v ⫹C 44 . 共11兲 In a triangular Abrikosov lattice, for example, only N y ⫽N /N x of the N ⌽ coefficients in each layer are nonzero. The nonzero coefficients are c 2pN x ,n ⫽(2/ A ) 1/2, c (2p⫹1)N x ,n ⫽i(2/ A ) 1/2, where p is an integer ranging from 0 to N y and  A ⫽1.159595 . . . is the Abrikosov ratio.14 Substituting these values into Eq. 共10兲, using N y /L y ⫽2N x / (L x 冑3), and adding the compressive term, we obtain C 44⫽ B z2 4 ⫹ 1 8  A 2␥ 2 „H c2 共 T 兲 ⫺B z …B z . 共12兲 Equation 共12兲 can be compared to a previous estimate9,15 LVG 2 C 44 ⫽(B z2 /4 ) 关 1⫹1/(4 ˜ ab (B z ,T) ␥ 2 n s ) 兴 , where n s is the superfluid number density for a two-dimensional Bose system related to the three-dimensional flux line system by a path integral formalism.3 Writing ab (B z ,T)⫽ ab (B z ,T), 2 (B z ,T)⬃2 „H c2 (T)⫺B z …/⌽ 0 , and approximating n s 1/ ab by n 0 ⫽B z /⌽ 0 , the number of bosons 共i.e., flux lines兲 per LVG ⬃B z2 /(4 )⫹ 关 „H c2 (T) unit area,6,7 we obtain C 44 2 2 ⫺B z …B z 兴 /(8 ␥ ) This result is in agreement with our own result, Eq. 共12兲, except for the factor (1/ A )⬃0.86, which arises from the nonuniformity of n s in the Abrikosov lattice phase. Our C v44 is closely connected to the so-called Josephson plasmon frequency pl , as calculated in the same LLL approximation.16 2pl (B z ,T) is the squared plasma frequency of the Josephson junction formed between adjacent ab layers, and is given by16 F 2 2pl 共 B,T 兲 ⫽ 共 M pl 兲 ⫻ 冓兺 k,n FIG. 1. Vortex contribution C v44 to the tilt modulus, plotted versus temperature for Bi2 Sr2 CaCu2 O8⫹ ␦ at B z ⫽2 T, as evaluated using Eq. 共14兲 and 2pl from Ref. 16. C v44 is discontinuous at the flux lattice melting temperature T M (B z ) 共indicated by arrow兲. Inset: enlargement of C v44 near and above T M . v that 2pl , and hence C 44 , have discontinuities at flux lattice melting in clean Bi2 Sr2 CaCu2 O8⫹ ␦ . 17,18 IV. FLUX LATTICE MELTING IN A TILTED MAGNETIC FIELD Equations 共3兲 and 共4兲 have some striking implications for the LLL phase diagram of a clean layered superconductor. When B储 c, this phase diagram depends on only two parameters, namely g⫽a H 冑 l 2 d/(bk B T) and ⫽t/ 兩 a H 兩 . 12 Our results show that, even with a transverse field B x , the phase diagram still depends on only two parameters, except that is now replaced by ⬘ ⫽ exp关⫺(B2x d2)/(2Bz⌽0)兴. As found previously,11–13 this phase diagram contains a single firstorder melting line separating a triangular vortex lattice from a vortex liquid. In the regime where the LLL approximation is adequate, our results show that this first-order line should persist in a tilted magnetic field 共cf. Fig. 2兲. Furthermore, in most high-T c materials, the line is shifted very little by a A 关  A N x / 共 2N ⌽ N z 兲兴 2 冔 * c k,n⫺1 ⫹c.c. 兲 T , 共 c k,n 共13兲 F 2 2 where M is the pl ⫽ 冑关 „H c2 (T)⫺B…cq 兴 / 关 ⑀ 0 ␥ ប  A 兴 mean-field Josephson plasmon frequency and ⑀ 0 is an interlayer dielectric constant. Combining Eqs. 共10兲 and 共13兲 gives v C 44 共 B z ,T 兲 ⑀ 0 2pl 共 B z ,T 兲 ⫽ B zប 2 8⌽ 0 q 2 . 共14兲 v (B z ,T) for B z ⫽2 T in clean Figure 1 shows C 44 Bi2 Sr2 CaCu2 O8⫹ ␦ , using this relation 2pl as calculated in v Ref. 16. Like 2pl , C 44 has a discontinuity at the flux lattice melting transition, and remains finite in the flux liquid state. Experiments on 2pl are consistent with this result, indicating FIG. 2. Phase diagram of a clean high-T c material in the LLL approximation, including a transverse magnetic field component B x . The parameters ⬘ and g are defined in the text. Points and spline fit are given by Hu and MacDonald 共Ref. 12兲. RAPID COMMUNICATIONS R14 668 G. MOHLER AND D. STROUD nonzero B x . 共For Bi2 Sr2 CaCu2 O8⫹ ␦ , t ⬘ ⬃t exp关⫺7.5 ⫻10⫺7 B 2x /B z 兴 .兲 We have found no experimental melting data in tilted magnetic fields at fields where the LLL approximation is applicable. In BiSr2 Ca2 Cu2 O8⫹x , the low-field melting temperature has been reported19 to depend only on the c component of H, which, though obtained in a very different regime, would be consistent with the result of our calculations. It would be of great interest to have further tests of the LLL predictions in the relevant high-field regime. PRB 62 melting temperature will be little affected by the application of an oblique magnetic field in the range where the LLL approximation is valid, except possibly at angles nearly parallel to the layers. This prediction appears consistent with some existing 共but low-field兲 experiments.19 We also obtain an expression for the zero-wave-vector tilt modulus C 44 , in good agreement with previous estimates by other means.9,15 v Finally, the vortex contribution C 44 is proportional to the squared Josephson plasmon frequency 2pl , as calculated in the same LLL approximation, and remains finite in the vortex liquid as well as the vortex solid phase. V. SUMMARY In this paper, we have extended the LLL approximation for high-T c superconductors to treat fields tilted at an angle to the layer perpendicular. The resulting free energy has exactly the same form as the usual case, except that the effective interlayer coupling is reduced. For high-T c materials, this reduction is small; hence, we predict that the flux lattice 1 A.I. Larkin and Yu.N. Ovchinnikov, J. Low Temp. Phys. 34, 409 共1971兲. 2 T. Belincourt, Phys. Rev. 114, 969 共1959兲; W.K. Kwok et al., Phys. Rev. Lett. 73, 2614 共1994兲. 3 D.R. Nelson and V.M. Vinokur, Phys. Rev. Lett. 68, 2398 共1992兲. 4 A. Sudbo” and E.H. Brandt, Phys. Rev. Lett. 66, 1781 共1991兲. 5 T.R. Goldin and B. Horovitz, Phys. Rev. B 58, 9524 共1998兲. 6 U.C. Tauber and D.R. Nelson, Phys. Rep. 289, 157 共1997兲. 7 P. Benetatos and M.C. Marchetti, Phys. Rev. B 59, 6499 共1999兲. 8 A.E. Koshelev and P.H. Kes, Phys. Rev. B 48, 6539 共1993兲. 9 A.I. Larkin and V.M. Vinokur, Phys. Rev. Lett. 75, 4666 共1995兲. 10 W.E. Lawrence and S. Doniach, in Proceedings of the 12th International Conference on Low Temperature Physics, Kyoto, 1970, ACKNOWLEDGMENTS This work was supported by the Midwest Superconductivity Consortium through Purdue University, Grant No. DE-FG 02-90 ER45427, and by NSF Grant No. DMR9731511. edited by E. Kanda 共Keigaku, Tokyo, 1971兲, p. 361. Z. Tešanović and L. Xing, Phys. Rev. Lett. 67, 2729 共1991兲; R. Šašik and D. Stroud, Phys. Rev. Lett. 75, 2582 共1995兲. 12 J. Hu and A.H. MacDonald, Phys. Rev. B 56, 2788 共1997兲. 13 R. Šašik and D. Stroud, Phys. Rev. Lett. 72, 2462 共1994兲. 14 R. Šašik, Ph.D. thesis, The Ohio State University, 1996. 15 V. B. Geshkenbein 共unpublished兲. 16 I.-J. Hwang and D. Stroud, Phys. Rev. B 59, 3896 共1999兲. 17 T. Shibauchi et al., Phys. Rev. Lett. 83, 1010 共1999兲. 18 Y. Matsuda et al., Phys. Rev. Lett. 75, 4512 共1995兲. 19 B. Schmidt, M. Konczykowski, N. Morozov, and E. Zeldov, Phys. Rev. B 55, 8705 共1997兲. 11