Anisotropic peak effect due to structural phase transition in the vortex lattice

(1)

Ž .

Physica C 332 2000 203–206

www.elsevier.nlrlocaterphysc

Anisotropic peak effect due to structural phase transition

in the vortex lattice

Baruch Rosenstein

)

, Anton Knigavko

Electrophysics Department, National Chiao Tung UniÕersity, Hsinchu, Taiwan

Abstract

Ž .

The recently observed new peak effect in YBCO is explained by softening of the vortex lattice VL due to a structural phase transition in the VL. At this transition, square lattice transforms into a distorted hexagonal one. While conventional peak effect is associated with softening of shear modes at melting, in this case the relevant mode is the point. The squash mode is highly anisotropic and we point out some peculiar effects associated with this feature. q 2000 Published by Elsevier Science B.V. All rights reserved.

Keywords: Vortex lattice; Phase transitons; Critical current density; Peak effect

Conventional peak effect, sudden increase of the critical current, has been observed in great variety of both low and high T superconductors. In conven-c

tional superconductors, the peak effect was theoreti-cally explained a long time ago by Larkin and

w x

Ovchinnikov 1 , while in high T superconductorsc

like untwinned YBCO, it is generally believed that the peak is due to softening of the shear mode just before the first order melting transition of the vortex

Ž . w x

lattice VL takes place 2 . However, recently, an-other peak in critical current in YBCO has been discovered on a line almost parallel to the T-axis starting from the melting line at H ; 9 T and contin-uing to lower temperatures. First it appeared only as

w x

a ‘‘fishtail’’ in magnetization hysteresis loops 3–5 ; but recently, a direct measurement of the critical

w x

current 6 clearly established a line presumably

cor-)

Corresponding author. National Center for Theoretical Sci-ences, National Chiao Tung University, Hsinchu 30043, Taiwan.

Ž

E-mail address: [email protected] B.

Rosen-.

stein .

responding to some transition in vortex matter. As a

Ž .

possible explanation, the transition or crossover

Ž .

from the topologically ordered Bragg glass to

vor-Ž . w x

tex glass or pinned liquid was proposed 7–10 . Independently from these findings, both experi-mental and theoretical advances indicated that in YBCO there is a structural phase transition in VL: ‘‘distorted’’ hexagonal lattice stable at lower mag-netic fields transforms into the square lattice oriented at the angle of q s 458 with respect to the

crystallo-w x

graphic 100 axis at higher fields. Experimental evidence for a significantly distorted hexagonal phase

Ž .

comes from scanning tunneling spectroscopy STM

w₁₁x _{and small angle neutron scattering} _Ž_SANS_. w_{12,13 . Theoretical evidence comes from the deriva-}x

Ž .

tion of the d-wave Ginzburg–Landau GL equations

w x

from certain microscopic models 14–16 . The

the-w x

ory was simplified by Franz et al. 17,18 and by one

w x

of us 19,20 . Various properties of the VL solutions were also studied. In borocarbide superconductors, an analogous phase transition was firmly established

w x

by SANS and STM experiments 24 . This formalism

Ž .

(2)

( ) B. Rosenstein, A. KnigaÕko r Physica C 332 2000 203–206 204

w x

had been proven adequate 21–23 . The location of the phase transition line in YBCO, inferred from the

w x

angle measured in SANS 19,20 , roughly coincides with the line of ‘‘additional’’ peaks in critical cur-rent.

In this note we explain the second line of peaks by softening of the ‘‘squash’’ elastic mode of VL on the line of the structural phase transition. All the relevant elastic moduli of VL around the phase transition are calculated. We find that the characteris-tic width of vortex bundles depends on orientation. This leads to a prediction that the peak current for

w x

orientations along the crystallographic axes 100 and

w_{010 is smaller then for orientations along 1,1,0}x w x

'

w x

and 1,1,0 by a factor of 2 .

We start with a description of the structural phase transition in VL and an estimate of its location on the phase diagram of untwinned YBCO. Qualita-tively, anisotropy of the gap functions, in both the

Ž .

d-wave the dominant component and the s-wave

channels, leads to asymmetric four-lobe shape of

w x

vortex cores 14–16 . This, in turn, causes VL to prefer the square arrangement. The asymmetric fea-tures are most prominent closer to H_c2 where the core is more important, but even away from the H_c2 line, the asymmetry of vortices still distorts the usual hexagonal VL as was clearly demonstrated in

experi-w x

ments 11–13 .

We employ a simplified formulation of GL theory for four-fold symmetric superconductors using only

Ž . w x

one d-wave order parameter field c 17–20 . The free energy in addition to usual GL terms contains a term describing anisotropy:

< 2 2 _<2

w

x

F_anis c s ´

_Ž

_D_{D y D}_y _D_x

_.

c _.

_{Ž .}

₁

Ž U .

Here DD ' = y i e rc A ,i i i i s x, y is the

covari-ant derivative and eU is the charge of the Cooper pair. The material parameter ´ quantifies the devia-tion from the exact rotadevia-tional symmetry. The last term is the only derivative term, which is four-fold symmetric and violates rotational symmetry. We assume that magnetic field is in the c-direction and

Ž

is constant far enough from Hc1 this is a good

.

approximation since k 4 1 . At certain value of ´ there is a phase transition from distorted hexagonal to a more symmetric square lattice first noticed in

Ž .

simulations using the two-field d and s

formula-w x

tion 25,26 . The present formulation was shown

w_{19,20 to be essentially equivalent to the two-field}x

one. However, it contains just one parameter ´ char-acterizing the anisotropy and is simple enough to avoid numerical methods in the relevant regions of the phase diagram.

It is important for calculation of elastic moduli to consider the energy of VL of most general form. It is

Ž

characterized by the lattice vectors a and b with an

.

angle q between them and by an angle w specify-ing the orientation of VL with respect to the

crystal-w x

lographic 1,0,0 axis. One solves the linearized GL equation perturbatively in dimensionless anisotropy parameter h ' ´ m_{a b}eUH. Finally, we minimize the

Ž

free energy analytically with respect to w clearly

.

w s pr4 is one of the minima and numerically over r and s to find the lattice structure.

w x

It was found 19–23 that the transition occurs at

h s 0.00238. For every h - h , there are two de-_c _c

generate minima. One is at r s 1r2, s s s and,

Ž .

correspondingly, q s arctan 2 s . The other minima correspond to the lattice rotated by pr2. On the mean field level the phase transition is a second order phase transition with mean field critical expo-nents. For example, we calculated the dependence of the angle q on h close to the transition point and

Ž .1r2

found that q s 3.3 h y h_c . At lower fields and temperatures, one can use the London approximation

w_{17,18 to study the triangular lattice. However, the}x

peak effect is prominent at fields sufficiently close to

H ._c2

The line of the structural phase transition in VL is parallel to the T axis and goes at certain H sstr

Ž U.

h r ´ m e . We estimate this field using input of_c _{a b} w x

q s 53.5 " 0.58 for H s 2T 12,13 that for the

sam-ple of 6 H ( 6T.str

Using thermodynamic arguments, we calculate all the relevant non-dispersive elastic moduli. The only

w x

modulus that has a dispersion 27,30 , the tilt modu-lus c , is not changed significantly compared to the44

usual case without the asymmetry term. In order to obtain all the 2D elastic moduli of the flux line lattice, we first choose a particular form of distortion and then express the excess free energy correspond-ing to this distortion in terms of elastic moduli. We obtain following two useful combinations of the four elastic moduli: the shear c66 and the ‘‘squash’’

(3)

( )

B. Rosenstein, A. KnigaÕko r Physica C 332 2000 203–206 205

The dependence of the shear modulus on anisotro-py is weak. On the other hand, the squash modulus

< <

vanishes on the transition line linearly in h y h . Itc

is noteworthy that above and below the point h s hc

the coefficients are different:

h 2

°

< < 8.7 1 y H y H_c2

_Ž

T

_.

, h - h_c h_c

~

c ssq h ₂

Ž .

2 < < 5.5 1 y H y Hc2

Ž

T

.

, h ) h .c

¢

_h c

This is similar to the behavior of the soft moduli at structural phase transitions in solids.

Because of vanishing of squash elastic modulus of VL we expect some anomalies in physical properties of the superconductor. For example, vanishing of an elastic modulus in the ferroelastic phase transitions in usual crystals manifests itself via softening of the speed of corresponding branch of the sound. Below we argue that in our case, a peak in critical current should appear once one crosses the transition line.

To determine critical current j we use the ‘‘dy-c

w x

namical approach’’ 28–30 . The VL equation of motion: s B2 _Eu _{d F} _{d F} ₁ elast pin s y y y _{j = B}

_{Ž .}

₃ 2 _Et _{d u} _{d u} _c c

is solved perturbatively in the disorder energy F_pins

3 _{Ž .} 3 _² _{Ž . Ž .:} i K r _Ž _. _Ž _. Hd r´ r , Hd r ´ r ´ 0 e s _{2pF rB W K .}₀

Here s is the normal state conductivity, c is the speed of light and K is the reciprocal lattice vector. The critical current is found to be:

7r2 2 4 cW

Ž

2p BrF0

.

2 j u sc

Ž

.

f u

Ž

.

Ž .

4 Bc c c44 66 sq Ž . < Ž . <3 r 2 < Ž

where f u s cos u q pr4 q _{cos u y} .<3r2

pr4 . Therefore, the critical current along the crystalline axes a or b is smaller by factor of

4

2

'

_{2 s 2 compared to the one along 110 or 110 .}

'

w x w x w x

For untwinned YBCO, one estimates 2 W s U2_Bj2_{n , where n is point pinning centers density}

0 p p

and U is the depth of an individual pinning poten-0

w x

tial. As in the melting peak effect 28,29 , the effect of thermal depinning is taken into account by an

Ž .y1 1r2

additional factor 1 q TrTdp where Tdp is the

depinning temperature. The case of ‘‘small bundles’’ where dispersion of c44 is important can be treated

w x

analogously 2,30 .

Due to different slopes of the moduli csq as

Ž Ž ..

function of h y hc see Eq. 2 the peak shape is asymmetric provided the general 1rB trend is elimi-nated: 1

°

, B - Bstr 8.7 B y B

Ž

str

.

~

j B ;c ₁

Ž .

5 , B ) B ._str

¢

5.5 B y B

Ž

str

.

Of course cutoff is understood when the size of the Larkin domain is no longer large compared to the distance between vortices. In this case, the elasticity theory becomes inapplicable.

To determine the applicability region of the elas-ticity theory, we calculate the correlation length which is the most important characteristics of the mixed state in the collective pinning theory. It is

² 2Ž .:

deduced from the displacement correlator u r w_{30 is given by:}x d3_k 2W

_H

1 y cos k P r

_Ž

_.

G

_Ž

k G

_.

_Ž

y_k

_.

i j i j 3 2p

Ž

.

Ž .

where G_{i j} k is the elastic Green’s function. To determine the correlation length in certain direction

Ž

n within the collective pinning theory

ˆ

size and

. ² 2Ž .:

shape of the Larkin domain one writes u Rnn

ˆ ˆ

2 w x

sj _{30 . The correlator in the c-direction does not}

change compared to the case of hexagonal lattice,

²u2ŽR_c.:s_{2WR r p}_c Ž 3r2_{c c}₆₆ ₄₄._{, while in the ab}

plane it depends on the angle f that n makes with

ˆ

w x

the crystallographic direction 100 :

WRf 2 ²u

_Ž

R

_.

:s _{f f .}

_Ž

_.

_{Ž .}

₆ f _{p c c}2 1r2 1r2 c sq 66 44

The results are significantly different compared to the case of peak effect associated with the VL

² 2Ž .:

melting where c vanishes and u R s

66 a b

Ž 2 3r2 1r2.

WR r 2p c_{a b} ₆₆ c₄₄ . We see that 1rc_sq replaces 1rc . In the present situation, Larkin domain is not66

only asymmetric with respect to a, b vs. c direc-tions. Due to particular orientation of the soft mode

(4)

( ) B. Rosenstein, A. KnigaÕko r Physica C 332 2000 203–206 206

destroying the square lattice the correlation length becomes asymmetric within the ab plane as well:

p3r2_{c c j}2 _{c c}1r2_c1r2_j2

66 44 sq 66 44

R sc , R sf .

Ž .

7

2W Wf f

Ž

.

The dynamical approach calculation of j can be_c supplemented by a simpler and more intuitive deriva-tion from the correladeriva-tion volume. The critical current in certain direction f with respect to the crystal is determined by equating the Lorentz force to the pinning force. The pinning energy for the relaxed lattice is linked to the in plane elastic energy due to the displacement of order j in the direction u q pr2

w x

caused by the Lorentz force 30 . The elastic energy

Ž . Ž .2

is U u ; c_c _sq jrR_uqpr_{2 V , where V is the cor-}_c _c

relation volume. Therefore, the critical current ob-tained from the balance of the Lorentz force and the pinning force is:

2 c U u_c

_Ž

_.

j 2 j u sc

Ž

.

; c jsq 0

ž /

f u q pr2

Ž

.

B j V_c R_a 8

Ž .

'

Ž .

where j s cH r 3 6 pl is the depairing current.0 c

This agrees with the dynamical approach result and shows in addition its range of applicability. Obvi-ously too close to the transition, the calculation is invalid.

To summarize the structural phase transition in the VL of YBCO leads to the anisotropic peak effect via vanishing of the ‘‘squash’’ elastic modulus. We calculated the value of the peak in critical current and its shape.

Acknowledgements

Ž .

One of the authors B.R. is indebted to V. Kogan and T.K. Lee for discussions. The work is supported by the grant of NSC of Taiwan a89-2112-M-009-039.

References

w x_{1 A.I. Larkin, Yu.N. Ovchinnikov, J. Low Temp. Phys. 34} Ž1979 409..

w x_{2 A.I. Larkin, M.C. Marchetti, V.M. Vinokur, Phys. Rev. Lett.}

Ž .

75 1995 2992.

w x3 K. Deligiannis, Phys. Rev. Lett. 79 1994 2121.Ž .

w x_{4 H. Kupfer, Th. Wolf, C. Lessing, A.A. Zhukov, X. Lanson,}

R. Meier-Hirmer, W. Schauer, H. Wuhl, Phys. Rev. B 58

Ž1998 2886..

w x_{5 N. Kobayashi, T. Nishizaki, T. Naito, S. Awaji, K.}

Watan-Ž .

abe, Physica B 246–247 1998 433.

w x_{6 G.W. Crabtree, W.K. Kwok, U. Welp, D. Lopez, J.A.} Ž .

Fendrich, in: S. Bose, R. Kossowski Eds. , Proceedings of the NATO Advanced Study Institute, Kluwer Academic Pub-lishing, The Netherlands, 1998, referring to D. Lopez et al., Peak effect deep in the vortex solid of untwinned YBCO, preprint.

w x7 T. Giamarchi, P. Le Doussal, Phys. Rev. B 52 1995 1242.Ž . w x8 T. Giamarchi, P. Le Doussal, Phys. Rev. B 55 1997 6577.Ž . w x_{9 A. van Otterlo, R.T. Scalettar, G.T. Zimanyi, Phys. Rev.}

Ž .

Lett. 81 1998 1497.

w_{10 A. van Otterlo, R.T. Scalettar, G.T. Zimanyi, R. Olsson, A.}x

Petrean, W.K. Kwok, V. Vinokur.

w_{11 I. Maggio-Aprile, Ch. Renner, A. Erb, E. Walker, O. Fischer,}x

Ž .

Phys. Rev. Lett. 75 1995 2754.

w_{12 B. Kermer, W.Y. Shih, R.W. Erwin, J.W. Lynn, F. Dogan,}x

Ž .

I.A. Aksay, Phys. Rev. Lett. 73 1994 3459.

w_{13 B. Keimer, J.W. Lynn, R.W. Erwin, F. Dogan, W.Y. Shih,}x

Ž .

I.A. Aksay, J. Appl. Phys. 76 1994 6778.

w_{14 P.I. Soininen, C. Kallin, A.J. Berlinsky, Phys. Rev. B 50}x Ž1994 13883..

w15 Y. Ren, J.H. Xu, C.S. Ting, Phys. Rev. Lett. 74 1995 3680.x Ž . w16 R. Joynt, Phys. Rev. 41 1990 4271.x Ž .

w_{17 M. Franz, I. Affleck, M.H.S. Amin, Phys. Rev. Lett. 79}x Ž1997 1555..

w18 M. Franz, I. Affleck, M.H.S. Amin, Phys. Rev. B 55 1997x Ž .

R704.

w_{19 D. Chang, C.Y. Mou, B. Rosenstein, C.L. Wu, Phys. Rev.}x

Ž .

Lett. 80 1998 145.

w_{20 D. Chang, C.Y. Mou, B. Rosenstein, C.L. Wu, Phys. Rev. B}x

Ž .

57 1998 7955.

w_{21 Y. De Wilde, M. Iavarone, U. Welp, V. Metlushko, A.E.}x

Koshelev, I. Aranson, G.W. Crabtree, P.C. Canfield, Phys.

Ž .

Rev. Lett. 78 1997 4273.

w_{22 V.G. Kogan, P. Miranovic, Lj. Dobrosavljevi-Grujic, W.E.}x

Ž .

Pickett, D.K. Christen, Phys. Rev. Lett. 79 1997 741.

w_{23 V.G. Kogan, M. Bullock, B. Harmon, P. Miranovich, Lj.}x

Dobrosavljevic-Grujic, P.L. Gammel, D.J. Bishop, Phys. Rev.

Ž .

B 55 1997 R8693.

w24 U. Yaron et al., Nature 282 1996 236.x Ž .

w25 J.H. Xu, Y. Ren, C.S. Ting, Phys. Rev. B 52 1995 7663.x Ž . w_{26 M. Franz, C. Kallin, P.I. Soininen, A.J. Berlinsk, A.L. Fetter,}x

Ž .

Phys. Rev. B 53 1996 5795.

w27 E.H. Brandt, J. Low Temp. Phys. 26 1977 735.x Ž .

w28 A.I. Larkin, Yu.N. Ovchinnikov, Sov. Phys. JETP 38 1974x Ž .

854.

w29 M.V. Feigel’man, V.M. Vinokur, Phys. Rev. B 41 1990x Ž .

8986.

w_{30 G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin,}x

Ž .