Magnetic phase diagram of type-II superconductors: From high T-c to low T-c superconductors

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Magnetic phase diagram of type-II superconductors: From high T

c

to low

T

c

superconductors

D. Li

a

, P.J. Lin

b

, B. Rosenstein

b,*

a

Department of Physics, Peking University, Beijing, China

b_{Electrophysics Department, National Chiao Tung University, EF747, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan}

a r t i c l e

i n f o

Article history:

Available online 12 June 2008 PACS: 74.20.De 74.60.w 74.25.Ha 74.25.Dw Keywords: Vortex matter Glass phase Melting Peak effect

a b s t r a c t

Phase diagram of both high temperature superconductors and low temperature superconductors are studied within the Ginzburg–Landau approach. Due to enhanced thermal ﬂuctuation strength, disorder effects are relatively small in high Tcsuperconductors and consequently can be studied analytically.

The vortex glass transition line is different and well separated from both the melting line and from the so-called second peak lines. On contrary, in low temperature superconductors, the disorder effect is dom-inant, as the thermal ﬂuctuation strength is very small. Peak effect appears due to cross over of the col-lective pinning region to stronger pinning region. The location of the peak effect is obtained. The disorder and thermal ﬂuctuation effects on structure phase transition are also studied.

1. Introduction

Theoretical study of the phase diagram of type-II superconduc-tors remains one of the major challenges in condensed matter physics, not only due to its importance to the application of superconductivity, but also its importance for understanding of phase transition. Vortex systems offer a unique testing ground for experimental veriﬁcation of various theoretical concepts like that of the glass phase, overcooled liquid and melting. Based on the Ginzburg–Landau phenomenological theory, we use nonper-turbative analytic methods to obtain various results which can be (even quantitatively) tested experimentally. The article is orga-nized as follows: in Section2we consider the Ginzburg–Landau theory without disorder in Section3 we include weak disorder effects in the model, while in Section 4 the phase diagram of the strong disordered system is studied. In Section5, the struc-ture transition is investigated.

2. Vortex phase diagram of a clean superconductor The model without disorder is deﬁned by free energy:

Z d3x h 2 2mab

r

2

p

i

U

0 A w 2 þ h 2 2mcjozwj 2

a

Tcð1 tÞjwj2þ b 2jwj 4 ; ð1Þ

whereU0= hc/(2e), t = T/Tc, A = (By, 0, 0). The model provides a good

description of thermal ﬂuctuations as long as 1 t b << 1, where b = H/Hc2. Its thermodynamical properties turn however to be

highly nontrivial, even without disorder and within the lowest Landau level (LLL) approximation, in which only the LLL mode is retained and the free energy simpliﬁes (after rescaling):

gLLL¼ Z d3x 1 2jozwj 2 þ aTjwj2þ 1 2jwj 4 : ð2Þ

The simplified model has just one parameter – the (dimensionless) scaled temperature: aT ðt þ b 1Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2=ð

p

2_b2_t2_GiÞ 3 q

with the Ginzburg number deﬁned as Gi 32ð

p

k2Tc

c

=ðU20nÞÞ 2

,

c

2_{= m}

c/mabthe anisotropy parameter, k the magnetic penetration

* Corresponding author. Tel.: +886 3 571 2121 56119; fax: +886 3 5725230. E-mail address:[email protected](B. Rosenstein).

Physica C 468 (2008) 1245–1248

Contents lists available atScienceDirect

Physica C

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depth and n the coherence length. The (effective) LLL model is appli-cable in a surprisingly wide range of ﬁelds and temperatures deter-mined by the condition that the relevant excitation energy e is much smaller than the gap between Landau levels[1]. For a very weakly disordered system like a pure single crystal sample, or even in some disordered system near phase transition temperature, the disorder effect is small, so the result of this model (not including disorder effects) can be also tested experimentally.

The solid energy can be calculated up to two loops order in per-turbation theory or by Gaussian variational method,[2]while the liquid free energy can be obtained by Borel–Pade method[3]to achieve a precision of better than 1%. Comparing the liquid and so-lid free energy it reveals that the melting transition occurs

am

T ¼ 9:5: ð3Þ

The experimental veriﬁcation of this equation can be found in various experiments in YBCO type and even in low temperature type-II superconductors like Nb3Sn[4]. Gaussian variational

calcu-lation also showed that a spinodal line for the solid, the end point of superheated solid, is given by

aspT ¼ 5: ð4Þ

The spinodal line was recently observed and the theoretical pre-diction of the spinodal line was conﬁrmed in various experiments

[5]. The line as

T¼ 5 therefore separates the vortex liquid region of

the phase diagram into two regions: 9.5 < aT< 5 is a normal

vor-tex liquid region, and 5 < aT< 0 is vortex gas region in which ﬂux

line can move freely like particles in a gas. 3. Weak disorder effect

In high temperature superconductors, disorder are relatively weak compared to the thermal ﬂuctuation, and its effect can be ta-ken into account analytically. The disorder can lower the melting transition line in H–T on the vortex phase diagram as the vortex li-quid state can adjust to the disorder better than the vortex lattice state[1,6–8]. On the other hand, the lower temperature part of the phase diagram becomes the glass phase in which ergodicity is broken.

We begin with a simpler two dimensional case. Quenched dis-order is accounted for by random components of coefﬁcients: m1_{! m}1_{½1 þ w}

1ðxÞ;

a

!

a

½1 þ w2ðxÞ; b ! b½1 þ w3ðxÞ with

variances p1, p2, and p3, respectively. Using the replica trick

G ¼ Tlimn!0Z

n₁

n where A is the disorder average of the physics

quantity A, we arrive at the scalar ﬁeld theory Zn

¼ R Qn a¼1 DwaDwaexp Gðwa;wa;wiÞ T

. wi can be averaged out and the

resulting theory can be analyzed nonperturbatively via Gaussian approximation introduced in [9]. Expanding

w

(x) in the basis of the LLL wavefunctions with quasi-momenta k,

w

a(x) /Rd

k/k(x)

w

a(k), the Gaussian effective free energy can be expressed

via the variational matrix parameters, mab¼ hwaðkÞwbðkÞi,

G /X a log m þ aTm rm2 qðm _ Þ2m2 h i aa q 4 X a;b m4 ab; ð5Þ

where the dimensionless parameters are: the 2D LLL temperature

aT¼ bt

p

ffiffiffiffiffiffiffiffi 2Gi p 1=2 ð1 t bÞ;

with the Ginzburg number Gi ¼ 8pe2_k2_T c

c2_h2_L z

2

, and disorder variances r ¼b2p1þð1tÞ2p2

4p2n2pffiffiffiffiffi_2Gi ;q ¼

4bp3

pn2. The glass state is characterized by the loss of ergodicity and reversibility with respect to dynamic processes. This is expressed, formally, by spontaneous breaking of the replica permutation symmetry (RSB). It was shown by Parisi in the context of the spin glass theory, that the correct solution for the theory of

this type is given by the subclass of the matrices mabwhich has a

hierarchical structure and which can be parameterized by the Parisi function m(x), 0 < x < 1. In particular, the well known Edwards– Anderson (EA) glass order parameter corresponds to m(x = 1). The label x reﬂects the hierarchy level and corresponds to the overlap between different valleys in the potential landscape. We ﬁnd that in the disordered liquid (domain to the right of the irreversibility line inFig. 1) the replica symmetric solution is stable, while in the glassy phase (the left side of the line) a nontrivial Parisi function describes a continuous replica symmetry breaking. The irreversibil-ity line for small q is given by

agT¼ 2 ffiffiffi 2 p ðr1=2 r1=2_{Þ þ} 2 þ 5r 2pffiffiffi2r5=2q þ Oðq 2 Þ; 2D ag T¼ 4 r r1=3 þ 5 rþ 4r4=3 3 q þ Oðq2 Þ; 3D ð6Þ

for the case of 2D and 3D, respectively. The glass lines are compared with transport experimental data for the 2D organic superconduc-tor and with the 3D high Tcsuperconductor YBCO data in[8]. On

this line the magnetization M has a cusp, while finite, and its slope dM/dT experiences a jump. This was also recently confirmed in BSCCO[6]. For the range of parameters shown the magnetic field is very small and we have rj_b!0! ð1tÞ2p2

4p2_n2pffiffiffiffiffi_2Gi;q ! 0. Thus we assume that q is very small. If q increases, the glass line will shift higher (parameters Hc2, Gi were fitted in the region in which the melting

line is near Tcand the disorder effect small in[1]). Away from Tc

where the disorder effects will appear, the disorder parameter r is determined from the melting line. The generic phase diagram can be found inFig. 1.

With weak disorder as in high Tcsuperconductor, the melting

line and glass line are usually below line aT= 9.5 (the melting line

of zero disorder). We will call aT= 9.5 as TXline. TXline divides

H g H x Liquid I H c2 Liquid II Solid Bragg glass vortex glass

ODO (second peak)

O D O (melting) Magnetic field (H) Temperature (T)

T c

Fig. 1. Generic phase diagram of the vortex matter: The order–disorder line (red) separates the crystalline phase from the homogenous phase. The glass transition line (blue) separates the glass from the weakly pinned phases, while the pink line is a crossover between two homogeneous phases, locally pinned liquid I and essentially unpinned liquid II. The left inset shows well deﬁned vortex lines pinned by impurities in Bragg glass region and the right inset shows the distribution of the order parameter in the Abrikosov lattice near the melting line. (For interpretation of the references in colour in this ﬁgure legend, the reader is referred to the web version of this article.)

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liquid to two parts: liquid I above TXline and liquid II below TXline

[10]. Liquid II has local crystal structure as the correlation length is larger than the lattice constant. Though TXline is not a phase

tran-sition line, however it can be taken as a dynamical phase trantran-sition line as liquid II is more viscous, therefore has much less resistance than liquid I.

4. Stronger disorder

In low temperature type-II superconductors, the disorder is strong compared to the thermal ﬂuctuation strength. The glass line will be shifted to high temperature even above line TXline.

There-fore the glass phase will be divided to two regions by TX. Below TX,

phase has local crystalline order as shown in previous sections, therefore the pinning is collective. Above TX, the pinning is more

individual. So TXis a crossover line from single pinning to collective

pinning, and peak effect appears at line TX. This explanation of peak

effect was conﬁrmed by experiments[11]and the peak effect ap-pears exactly at line TX.

As discussed above, aT= 5 is the spinodal line representing a

crossover of the vortex liquid into a ‘‘vortex gas”. Above this line, vortices move like particles in a gas, and the ﬂux line can not be pinned, the glass line shall be below the spinodal line. With very strong disorder as in low temperature type-II supercon-ductor, the glass transition line shall be near to this spinodal line.

5. Structural phase transition in the vortex lattice

In this section, we will discuss the distortion of vortex lattice due to the inﬂuence of underlying anisotropy in a–b plane of mate-rial which breaks the in-plane symmetry. With the capability to improve the quality of single crystal in recently years, observation of the structural phase transition (SPT) of vortex lattice are carried out from nonmagnetic borocabide[12–16], and high Tccuprates

such as LSCO and YBCO[17]and CeCoIn5[18]via small angle

neu-tron scattering, scanning tunneling spectroscopy and decoration

experiments. Those materials have 4-fold symmetric crystal lattice, either tetragonal or cubic lattice. The typical case is to apply an external magnetic field along the crystallographic c axis which pre-serves the symmetry in a–b plane; for a fixed temperature, in low field the hexagonal lattice undergoes reorientation with respect to underlying crystal lattice and in high field the rhombic lattice be-come square lattice till normal phase.

The rhombic to square phase transition line, H2(T), observed on

earlier experiments LuNi2B2C [12–15]has a very small positive

slope of the transition line in the T–H plane till it approaches the Hc2(T) region. In some experiments[13,14], it abruptly turns up

and even acquires a negative slope at high ﬁelds, while in other experiments with a closely related material YNi2B2C[16]it

contin-ues the gradual increase even near H2(T). However, in LaSCO, the

transition line exhibits negative slope in the T–H plane.

Previous theoretical studies of SPT ignored the disorder effects, however, we ﬁnd that disorder effect is in fact important for the structure phase transition in low temperature superconductors. To introduce the anisotropy effect of interested tetragonal mate-rial, we add 4-fold symmetric term based on the reason discussed in reference[19]. The gap anisotropy is well represented by two-component Ginzburg–Landau model which can be simply re-duced to a one component GL with additional high derivative term: H4-fold¼ ~

g

4 D 2 x D 2 y 2 DxDyþ DyDx 2 : ð7Þ

The coefﬁcient ~

g

can be positive (usually in low Tcmaterials) or

negative (usually in high Tcmaterials). By solving the Ginzburg–

Landau equation approximately analytically, we found near Hc2

line, the mean field SPT line is temperature independent. While thermal fluctuations influence become stronger, in perturbation approximation, the slope is increasing, seeFig. 2. While taking into account the disorder influence, the slope becomes negative and de-parts from the mean field STP line with increasing disorder strength. One concludes therefore that materials with strong thermal fluctuations exhibit negative slope of structural phase

0.0 0.2 0.4 0.6 0.8 1.0 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 n_d=0.2 superconducting sate _n d=5 n_d=2 n_d=1 n_d=0.8 n_d=0.05 η=0.5

b=H/H

c2

t=T/T

_c Normal state

mean field

Increasing disorder

Thermal fluctuations, Gi=10-3

square

Rhombus

melting line

Fig. 2. The mean ﬁeld structural phase transition line is located at bSPT¼ 0:02387=j~gj which has square lattice beyond it. The SPT with strong thermal ﬂuctuation, Gi = 103,

has negative slope near the Hc2line. And the red lines present various disorder strength which inﬂuence strongly in low temperature. The solid line is melting line which

correspond to Gi = 103

. (For interpretation of the references in colour in this ﬁgure legend, the reader is referred to the web version of this article.)

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transition line (at least well below the melting line). When thermal ﬂuctuations are small and disorder prevails, one expects a positive slope.

6. Conclusion

Phase diagram of both high temperature superconductors and low temperature superconductors can be effectively studied with-in the Gwith-inzburg–Landau approach for ﬁelds signiﬁcantly larger than Hc1. There are three phase transition lines separating various

phases. The vortex glass transition line (replica symmetry break-ing) separates pinned from unpinned phases. The order–disorder line (translation and rotation symmetry breaking) consisting the melting line and from the second peak segments separates homo-geneous from crystalline phase, while the structural transition line (4-fold symmetry breaking) separated different crystalline phases. Acknowledgements

It is a pleasure to thank E. Zeldov, Y. Yeshurun, B. Shapiro for illuminating discussions. This work was supported by Natural Science Foundation of China (#90403002) and NSC ROC92-2112-M009-024.

References

[1] D. Li, B. Rosenstein, Phys. Rev. Lett. 90 (2003) 167004. [2] D. Li, B. Rosenstein, Phys. Rev. B 70 (2004) 144521.

[3] S. Hikami, A. Fujita, A.I. Larkin, Phys. Rev. B 44 (1991) R10400. [4] R. Lortz et al., Phys. Rev. B 74 (2006) 104502.

[5] Z.L. Xiao, O. Dogru, E.Y. Andrei, P. Shuk, M. Greenblatt, Phys. Rev. Lett. 92 (2004) 227004;

A.D. Thakur, S.S. Banerjee, M.J. Higgins, S. Ramakrishnan, A.K. Grover, Phys. Rev. B72 (2005) 134524.

[6] H. Beidenkopf, N. Avraham, Y. Myasoedov, H. Shtrikman, E. Zeldov, B. Rosenstein, E.H. Brandt, T. Tamegai, Phys. Rev. Lett. 95 (2005) 257004. [7] H. Beidenkopf, T. Verdene, Y. Myasoedov, H. Shtrikman, E. Zeldov, B.

Rosenstein, D. Li, T. Tamegai, Phys. Rev. Lett. 98 (2007) 167004. [8] D. Li, B. Rosenstein, V. Vinokur, J. Supercond. 19 (2006) 369. [9] M. Mezard, G. Parisi, J. Phys. (France) 1 (1991) 809.

[10] D.T. Fuchs, E. Zeldov, T. Tamegai, S. Ooi, M. Rappaport, H. Shtrikman, Phys. Rev. Lett. 80 (1998) 4971.

[11] N. Kokubo, K. Kadowaki, K. Takita, Phys. Rev. Lett. 95 (2005) 177005; N. Kokubo, T. Asada, K. Kadowaki, K. Takita, T.G. Sorop, P.H. Kes, Phys. Rev. B 75 (2007) 184512.

[12] P.L. Gammel, D.J. Bishop, M.R. Eskildsen, K. Mortensen, N.H. Andsersen, I.R. Fisher, K.O. Cheon, P.C. Canﬁeld, V.G. Kogan, Phys. Rev. Lett. 82 (2001) 4082.

[13] M.R. Eskildsen, A.B. Abrahamsen, V.G. Kogan, P.L. Gammel, K. Mortensen, N.H. Andersen, P.C. Canﬁeld, Phys. Rev. Lett. 86 (2001) 5148.

[14] T. Park, Y. Sun, M.B. Salamon, A.T. Malinowski, M.F. Hundley, Eun Mi Choi, Heon Jung Kim, Sung-Ik Lee, P.C. Canﬁeld, V.G. Kogan, J.D. Thompson, Phys. Rev. B71 (2005) 054511.

[15] L.Ya. Vinnikov, T.L. Barkov, P.C. Canﬁeld, S.L. Bud’ko, V.G. Kogan, Phys. Rev. B64 (2001) 024504.

[16] D.McK. Paul, C.V. Tomy, C.M. Aegerter, R. Cubitt, S.H. Lloyd, E.M. Forgan, S.L. Lee, M. Yethiraj, Phys. Rev. Lett. 80 (1998) 1517;

C.D. Dewhurst, S.J. Levett, D.McK. Paul, Phys. Rev. B 72 (2005) 014542. [17] R. Gilardi et al., Phys. Rev. Lett. 88 (2002) 217003;

S.P. Brown, D. Charalambous, E.C. Jones, E.M. Forgan, P.G. Kealey, A. Erb, J. Kohlbrecher, Phys. Rev. Lett. 92 (2004) 067004.

[18] M.R. Eskildsen, C.D. Dewhurst, B.W. Hoogenboom, C. Petrovic, P.C. Canﬁeld, Phys. Rev. Lett. 90 (2003) 187001.

[19] D.P. Li, P.-J. Lin, B. Rosenstein, B.Y. Shapiro, I. Shapiro, Phys. Rev. B 74 (2006) 174518;

B. Rosenstein, B.Y. Shapiro, I. Shapiro, Y. Bruckental, A. Shaulov, Y. Yeshurun, Phys. Rev. B 72 (2005) 144512;

B. Rosenstein, A. Knigavko, Phys. Rev. Lett. 83 (1999) 844.