Melting and decoupling in the vortex system of layered superconductors

Physica C 332 Ž2000. 66–70 www.elsevier.nlrlocaterphysc

Melting and decoupling in the vortex system of layered superconductors Gianni Blatter ) , Matthew Dodgson, Vadim Geshkenbein Theoretische Physik, ETH-Honggerberg, CH-8093 Zurich, Switzerland ¨ ¨

Abstract We derive the melting and defect-unbinding lines in the pancake-vortex system of layered superconductors with zero Josephson coupling. A finite Josephson coupling transforms the defect-unbinding into a topological decoupling transition. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Josephson coupling; Pancake-vortex system; Defect-unbinding lines

1. Introduction The pancake-vortex lattice in a layered superconductor defines a tunable soft matter system with astonishing properties, the most widely discussed phenomena being the vortex lattice melting w1x and layer decoupling w2x transitions. At high magnetic fields Žhere always considered to be parallel to the c-axis. the crystal consists of weakly coupled two-dimensional Ž2D. pancake-vortex lattices, the coupling ensuring the existence of true long-range order, otherwise absent in a pure 2D system. At small magnetic fields, the 3D crystal is made from weakly interacting 1D stacks of pancake vortices. The coupling between the 2D layers at high fields and defining the 1D stacks at low fields involves both electromagnetic forces and supercurrents flowing across the layers, the Josephson coupling. Here, we

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Corresponding author.

ignore the latter and concentrate on the pancakevortex system where the interlayer coupling is entirely due to the weak but long-range electromagnetic coupling. On the one hand, this simplification will allow us to construct a simple mean-field type description of the vortex lattice melting transition, which goes beyond the Lindemann analysis w3x. Also, we will find a new interstitial-vacancy defect-unbinding transition w4x in the 3D pancake-vortex crystal for which we can offer an exact description in terms of a Kosterlitz–Thouless scenario Žwith a finite Josephson coupling, this topological transition triggers the decoupling of the layers.. On the other hand, real materials, such as the Bi- and Tl-based copper-oxides, as well as artificially layered materials, such as the YrPr–Ba–Cu–O system, do come with a very weak Josephson coupling, and an approximate description through the zero Josephson coupled limit appears to make sense. We first analyze the vortex lattice melting, and thereafter, the defect-unbinding transition. We will end with a discussion on how a finite Josephson coupling modifies these transitions.

0921-4534r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 9 . 0 0 6 4 6 - 2

G. Blatter et al.r Physica C 332 (2000) 66–70

Throughout the present work we concentrate on the statistical mechanics aspects of the pure system and ignore effects of disorder. 2. Vortex lattice melting

where u 0 denotes the displacement of the pancakevortex Žin layer n s 0. out of the stack’s center. The elastic constant a s follows from pairwise summation over the pancake-vortex interaction energies,

Fab s

We begin with the transitions at high and low magnetic fields w5x, the field scale being defined by the London penetration depth l, Bl ' F 0rl2 , where F 0 s hcr2e is the flux quantum.

67

Ý n/0

as s

E 2 Vn Ž < u n y u 0 < .

¦

;

Eu 0a Eu 0b

´0 d 2 ² u 0 :th q 2 l2

s a s da b ,

th

,

Ž 2.

and has to be determined self-consistently together with the mean squared displacement

2.1. High fields B 4 Bl The interaction between pancake-vortices is dominated by the in-plane currents producing shear forces, where we can neglect the electromagnetic interaction between the planes. Melting then is due to dislocation-unbinding and is realized at the temperature Tm2 D s ´ 0 dr8'3 p f ´ 0 dr70. Here, ´ 0 s ŽF 0r 4pl. 2 is the basic energy scale and d denotes the layer separation.

² u 20 :th s

2.3. Intermediate fields B ; Bl In order to find the melting line in the intermediate field regime, we concentrate again on the lowfield single-stack evaporation and remind ourselves of the fact that the BKT transition temperature can be obtained from a self-consistency analysis, which looks for an instability of the stack. We view an individual pancake-vortex in the stack Žsay in layer n s 0. as subject to the potential created by all the other pancake-vortices in the stack. Next, the free energy of this pancake-vortex is approximated by a harmonic potential 1 as Fs Ž u 0 . s Fa b u 0a u 0b s u 20 , Ž 1. 2 2

a2

´ 0 dr2T y 1

,

Ž 3.

Vn Ž R . s

™

s

diverging at T BK T , as expected. Here, we have made use of the pancake-vortex interaction potential w7x

2.2. Low fields B < Bl The pancake-vortices arranged in 1D stacks are held together by the electromagnetic interaction. The interaction in the plane is screened Žon the l scale w6x., and we neglect it in the B 0 limit. Concentrating on a single stack, the latter then evaporates w6x at the Berezinskii–Kosterlitz–Thouless ŽBKT. temperature TBK T s ´ 0 dr2, a factor 35 larger than the above high-field melting temperature.

2 l2

2T

F 02 d 2 4p

d 3q

H Ž 2p .

q2 3

2 qH Ž 1 q l2 q 2 .

e iŽ q z n dqq H R . ,

Ž 4. which is weak, of order Ž drl. ´ 0 d, but long-range, extending over lrd layers. The above analysis then introduces two concepts which we want to extend into the finite field regime: Ži. the idea of a substrate potential generalizing the elastic constant a s in Eq. Ž1., and Žii. the self-consistent harmonic approximation for the effective elastic constant Ž2.. Accordingly, we have to redefine the elastic energy to include interplanar shear and compression enerb gies, F s Ž 1r8p 2 . HBZ d 2 Ku 0aK Fa b Ž K; T . u 0yK Ž a 0 s F 0rB the lattice constant.,

(

Fab s

1

½ ¦

0 Ri

q

Ý n/0

E 2 Vn Ž R . ER aER b

i

E 2 V0 Ž R .

¦

Ý Ž 1 y ei K R . a2

ER a ER b

;5

;

R iq d u 0 i

,

R iq d u n i

the first term describing interplanar elastic modes while the second term generalizes Eq. Ž2. and d u n i ' u nŽ R i . y u 0 Ž0.. The elastic matrix Fa bŽ K;T . de-

G. Blatter et al.r Physica C 332 (2000) 66–70

68

pends on ² u 20 :th and has to be determined self-consistently with ² u 20 :th s

d2 K

H Ž 2p .

T 2

2

c66 K q a sra02 T

q

2

c11 K q a sra02

,

Ž 5.

where we use the usual relation between the elastic moduli c66 , c11 , c 44 s a sra20 and the elastic matrix w5x. This task has to be carried out numerically, and a typical result of such an analysis is shown in Fig. 1. In order to improve the accuracy of the analysis, we have gone beyond the simple self-consistent harmonic approximation ŽSCHA. by taking additional diagrams into account within the so-called two-vertex SCHA, see Ref. w8x for details. The above stability analysis of the crystal puts an upper limit on the melting line — the true transition point has to be obtained from a comparison of the free energies of the solid and liquid phases. While the free energy of the solid directly follows from adding up the ground state energy of a 2D crystal subject to a substrate potential and the contributions from harmonic fluctuations, the free energy of the

liquid is obtained from Monte Carlo simulations of a 2D Coulomb liquid w9x, exploiting the fortunate fact that we know the exact free energy at the temperature T0 s ´ 0 d w10x. The result is shown in Fig. 1; the melting line comes to lie below but still close to the instability line. From the change of slope at Tm in the free energy, we can extract the latent heatrjump in entropy Žper pancake-vortex. at the transition: we find that D Spv f 0.4 k B at high fields and a logarithmic increase D Spv ; k B lnŽ a 0rl. at low fields with a 0 ) l.

3. Defect-unbinding It has recently been shown w11,12x that a defect in a 2D crystal with long-range interactions is effectively screened by the elastic response of the medium: in a layered superconductor the pancake-vortex system is incompressible w Vpc Ž R . s 2´ 0 d lnŽ Rrj .x and an interstitial defect displaces the other vortices by an amount u s a 20 Rr2p R 2 , removing exactly one pancake-vortex from the defect’s neighborhood. The elastic energy Eel s y´ 0 d lnŽ Lra0 . Ž L is the system size. then exactly compensates the log-divergent

Fig. 1. Left: Instability line for the pancake-vortex lattice in the B–T plane calculated with the two-vertex-SCHA. The line goes asymptotically to Tm2D at high fields and ends at TBKT at zero field. The left inset shows the two-vertex-SCHA results for the shear modulus c66 , the substrate strength a s , and the pancake fluctuation width ² u 2 : with increasing temperature at the field B s Bl . The right inset is a zoom of the instability line at low fields, and shows the result Ždashed. of the simple SCHA scheme. Right: Comparing the free energies of the vortex solid and liquid phases, the melting line Žsolid. is found below but still close to the instability line Ždashed..

G. Blatter et al.r Physica C 332 (2000) 66–70

69

term in the elastic self-energy of the defect, rendering the defect’s energy of order of the core energy, Ei s hi ´ 0 d with hi a numerical of order unity. The electromagnetic coupling produces a substrate potential acting on the 2D crystal, which spoils the perfect screening; as a result, only the fraction 1rŽ1 q g . with gs

a02 4pl

2

ln

a0

f

Hc 1

d

Ž 6.

B

of the defect ‘charge’ is screened, and the logarithmic self-energy is only partly compensated. Similarly, the logarithmic interaction between defects is partially screened, a vacancy-interstitial defect pair separated by R interacting via the potential Vi sc ,Õ s 2 ´ 0 d

g 1qg

ln

R

.

a0

Ž 7.

An immediate consequence of this screened loginteraction between defects is the appearance of a defect-unbinding transition at a reduced temperature Tdef s g ´ 0 dr2Ž1 q g .. Solving for B, we obtain the defect-unbinding line at Bdef Ž T . s

F0 4pl

ln 2

a0 d

ž

´0 d 2T

/

y1 .

Ž 8.

At high temperatures, we have to take the thermal smoothing of the substrate potential into account, and using Eq. Ž2., we arrive at the expression Bdef Ž T

™T

BKT

.s

F0

´ 0 Ž 0. d

2

T

4pl Ž 0 .

ž

1y

T TBK T

2

/ Ž 9.

valid close to T BK T . Comparison with the melting line Bm ŽT . shows that the latter undercuts the defect-unbinding line, resulting in a phase diagram, as shown in Fig. 2. The finite field defect-unbinding transition described above can be naturally related to the zero-field vortex-unbinding transition, destroying the superfluid stiffness at TBK T . Quite interestingly, it comes to lie eight times lower than the transition found before by Daemen et al. w13x from a self-consistent analysis and by Horovitz and Goldin w14x using an RG analy-

Fig. 2. Phase diagram for a layered superconductor with zero Josephson coupling. The solid line marks the defect-unbinding transition Bdef ŽT . that transforms into the decoupling transition Bdec ŽT . in the case of a finite weak Josephson coupling with ´ - d r l. The dashed line shows the melting line Bm ŽT . calculated within a self-consistent analysis Žparameters used are lŽ0. f ˚ df15 A, ˚ Tc f100 K.. 2000 A,

sis. The factor 8 is not a coincidence but is related to the fact that a free energy of the type F s Ý d2 R

H

n

2´ 2 q

d2

´0 d 2p

Ž =wn q a .

2

Ž 1 y cos fn , nq1 . q Hd 3 r

B2 8p

,

Ž 10 .

describing a layered superconductor in the London limit Žhere, ´ 2 s mrM is the anisotropy parameter describing a finite Josephson coupling., produces two-phase transitions: ignoring the Josephson term Ž ´ s 0. vortices with energy ´ 0 d lnŽ Lrj . unbind at a temperature T BKT s ´ 0 dr2. On the other hand, with the Josephson term present, we obtain, after mapping to a Coulomb gas, a ‘roughening’ transition at an eight times higher temperature. This ‘‘roughening’’ transition is nothing but the vortex-loop transition proposed by Friedel w15x to trigger a zero-field layer decoupling transition. Expressing the displacement field of the pancake vortices through the superconducting phase Žusing the relation w K s Ž2p ira20 .Ž u K = K.rK 2 . one can show w4x that the elastic free energy reduces to an expression similar to Eq. Ž10. but with a reduced superfluid stiffness

70

G. Blatter et al.r Physica C 332 (2000) 66–70

Ž gr1 q g . ´ 0 d — hence, we obtain the same pair of transitions at finite field, a defect-unbinding transition at Tdef s w grŽ1 q g .x ´ 0 dr2 within the individual layers, and an interlayer vortex-loop transition at an eight times higher temperature, which is nothing but the transition found by Daemen et al. w13x and by Horovitz and Goldin w14x.

4. Decoupling The above analysis strictly applies for zero Josephson coupling ´ s 0, but the results remain approximately valid as long as ´ - drl. In fact, the melting line derived above crosses over to a Bm A = Ž1 y TrTc . 3r2 behavior when the Josephson coupling becomes relevant as ´ ) drlŽT . due to the divergence of lŽT . close to Tc w16x. Let us then study the effect of a finite Josephson coupling on the defect-unbinding transition. With a small but finite Josephson coupling, the defects trigger the decoupling of the layers and the defect-unbinding line Bdef ŽT . turns into a decoupling line Bdec ŽT .. For fields above Bdec ŽT ., the system develops a finite c-axis resistivity w17x with rc proportional to the density of free mobile defects n d . Close to the unbinding transition, free defects appear in the solid with a density n d ; ay2 expwy2 br= 0 Ž1 y TrTdef .1r2 x Žwith b a non-universal constant.. At higher temperatures, the core energy Ecore s h´ 0 d determines n d ; ay2 expŽyEcorerT .. Second, let us 0 estimate the shift of the defectrdecoupling transition due to a finite Josephson coupling. Comparing the 2 Josephson energy jdef EJ within the coherence area 2 y1 jdef f n d with temperature, we obtain the upward shifted decoupling transition Žsee also Ref. w7x. Tdec Ž B . f Ecorerln Ž Ecorer´ 0 d . Ž BrBL . .

Ž 11 .

While the result Ž8. is valid at low fields, the steep upward shift described by Eq. Ž11. becomes effective when g f Hc1rB - hrlnŽ dr´l., with h s Ecorer´ 0 d the numerical quantifying the core energy of the defects. The latter has been investigated numerically by Frey et al. w18x and by Olive and Brandt w19x, who found values h f 0.15–0.2. The quantitative condition for the applicability of the result Ž8. thus depends both on the anisotropy parameter ´ and the defect core-energy Ecore . At even higher fields B ) Ž8rh . lnŽ dr´l. Hc1, the decoupling line crosses over into the loop transition found by Daemen et al. w13x and by Horovitz and Goldin w14x.

References w1x D.R. Nelson, Phys. Rev. Lett. 60 Ž1988. 1973. w2x L. Glazman, A. Koshelev, Phys. Rev. B 43 Ž1991. 2835. w3x M.J.W. Dodgson, A.E. Koshelev, V.B. Geshkenbein, G. Blatter, cond-matr99. w4x M.J.W. Dodgson, V.B. Geshkenbein, G. Blatter, condmatr9902244. w5x G. Blatter et al., Rev. Mod. Phys. 66 Ž1994. 1125. w6x J. Clem, Phys. Rev. B 43 Ž1991. 7837. w7x M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, Physica C 167 Ž1990. 177. w8x M.J.W. Dodgson, V.B. Geshkenbein, M.V. Feigel’man, G. Blatter, cond-matr99. w9x J.M. Caillol et al., J. Stat. Phys. 28 Ž1982. 325. w10x A. Alastuey, B. Jancovici, J. Phys. ŽParis. 42 Ž1981. 1. w11x M. Slutzky, R. Mints, E.H. Brandt, Phys. Rev. B 56 Ž1997. 453. w12x E.H. Brandt, Phys. Rev. B 56 Ž1997. 9071. w13x L. Daemen et al., Phys. Rev. Lett. 70 Ž1993. 1167. w14x B. Horovitz, T.R. Goldin, Phys. Rev. Lett. 80 Ž1998. 1734. w15x J. Friedel, J. Phys. France 49 Ž1988. 1561. w16x G. Blatter et al., Phys. Rev. B 54 Ž1996. 72. w17x A.E. Koshelev, Phys. Rev. Lett. 76 Ž1996. 1340. w18x E. Frey, D.R. Nelson, D.S. Fisher, Phys. Rev. B 49 Ž1994. 9723. w19x E. Olive, E.H. Brandt, Phys. Rev. B 57 Ž1998. 13861.