Theory of the Vortex Matter Transformations in High-T
cSuperconductor YBCO
Dingping Li1,2and Baruch Rosenstein2
1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China 2Electrophysics Department and National Center for Theoretical Sciences, National Chiao Tung University,
Hsinchu 30050, Taiwan, Republic of China
(Received 6 June 2002; published 24 April 2003)
Flux line lattice in type II superconductors undergoes a transition into a ‘‘disordered’’ phase such as vortex liquid or vortex glass, due to thermal fluctuations and random quenched disorder. We quanti-tatively describe the competition between the thermal fluctuations and the disorder using the Ginzburg-Landau approach. The following T-H phase diagram of YBCO emerges. There are just two distinct thermodynamical phases, the homogeneous and the crystalline one, separated by a single first order transition line. The line, however, makes a wiggle near the experimentally claimed critical point at 12 T. The ‘‘critical point’’ is reinterpreted as a (noncritical) Kauzmann point in which the latent heat vanishes and the line is parallel to the T axis. The magnetization, the entropy, and the specific heat discontinuities at melting compare well with experiments.
DOI: 10.1103/PhysRevLett.90.167004 PACS numbers: 74.20.De, 74.25.Dw, 74.25.Ha
The H-T phase diagram of a high Tc superconductor is very complex due to the competition between thermal fluctuations (TF) and disorder. At low fields, vortex solid melts into a liquid [1] due to TF. The discontinuity in magnetization [2 – 4] and in the specific heat [5,6] unam-biguously demonstrate that the transition is a first order. Evidence is growing that the solid-glass transition in YBaCuO (YBCO) [7] is also a first order [4,8]. It was suggested [4] that the two first order phase transition lines HT and HmT join at a multicritical point or that the melting line continues as a second order transition be-tween the putative liquid II and a liquid [6]. However, other experiments are interpreted using a concept of ‘‘unified first order transition line’’ [9]: Only character of the transition evolves from the thermal fluctuation dominated to the disorder dominated one without multi-critical points along the line. This single line was dem-onstrated in anisotropic materials BiSrCaCuO (BSCCO) [10], LaSrCuO [11], and NdCeCuO [8], and was claimed recently in YBCO [12].
Theoretically idealized models such as the frus-trated XY model [13,14] or a collection of interacting
pointlike objects subject to both the pinning poten-tial and the thermal bath Langevin force were simu-lated numerically [15,16]. Alternatively, the melting line was located using the phenomenological Lindemann criterion on the solid side either in the framework of the ‘‘cage’’ model [9] or a more so-phisticated approach based on the elasticity theory [17,18]. Although both approaches are very useful in more ‘‘fluctuating’’ superconductors such as BSCCO, a problem arises with their application to YBCO close to Tc: Vortices are far from being pointlike and even their cores significantly overlap. As a conse-quence, the behavior of a dense vortex matter is dif-ferent from that of a system of pointlike vortices and of the XY model.
In this Letter, we quantitatively study the effects of the competition of thermal fluctuations and disorder in the framework of the anisotropic Ginzburg-Landau (GL) model appropriate precisely in the region of interest for YBCO. It was used successfully to describe the ’’clean’’ case [19,20].
The model without disorder is defined by free en-ergy [1]: Z d3x h 2 2mab r 2 i 0 A 2 h 2 2mc j@z j2 Tc1 tj j2 2j j 4; (1)
where 0 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. An ‘‘appli-cability line’’ 1 t b 0:2 is depicted in Fig. 1. The model, however, is highly nontrivial even without disor-der and within the lowest Landau level (LLL) approxi-mation [19] in which only the LLL mode is retained and the free energy simplifies (after rescaling):
gLLL Z d3x 1 2j@z j 2 a Tj j2 1 2j j 4 : (2)
The simplified model has just one parameter — the (dimensionless) scaled temperature: a T t b 1
2=Gi 2b2t2
3
p
with the Ginzburg number Gi 32 2T
c =20 2, 2 mc=mab the anisotropy pa-rameter, the magnetic penetration depth, and the coherence length. The (effective) LLL model is applicable in a surprisingly wide range of fields and temperatures determined by the condition that the relevant excitation energy " is much smaller than the gap between Landau P H Y S I C A L R E V I E W L E T T E R S week ending
25 APRIL 2003
VOLUME90, NUMBER16
levels "c 2 heB=cmab. Within the mean field approxi-mation in the liquid "="c x is a solution of the ‘‘gap equation’’[21]): 2bx 1 t b 2 tpGib p4x 2P1 n0 4x 4n 2 p p4x 4n 2 1= n x p
. The LLL dominance line in Fig. 1 represents a conservative condition "="c 1=20. On the right of the line, the HLL (higher Landau level) TF modes should be considered, while for b < 1 t=13, HLL in solid be-comes important [20]. Apart from the fields smaller than HLLL 3 kG, experimentally observed melting and the solid-glass lines are well within the range of applicability of the LLL approximation.
The disorder is described within the GL approach via a random field Wx with correlator WxWy W#x y [1], while TF are described by partition function Z R
; expf F
R
xWxj xj2 =Tg. Assuming small W, the free energy T lnZ is calculated perturbatively to the second order:
g gclZ x Wx hj xj2i 1 2T Z y Wy hj xj2j yj2i hj xj2ihj yj2i ; (3) where h i and gcldenote the thermal average and effective free energy of the clean system. Averaging the random
field, one obtains in the scaled units g gcl ntgdis, where gdis1
2 R
x hj xj4i hj xj2i2 , and nt / W=T. Recently, a quantitative theory [20] of TF in ’’clean’’ vortex liquids and solids based on the GL model of Eq. (2) was developed and was successfully applied to the fully oxidized YBa2Cu3O7for which the effects of disorder are negligible in whole currently accessible fields range [1,4]. The two loop result for solid in the clean case gsol a2
T=2A 2:85jaTj1=2 2:4=aT is sufficiently precise (on the 0:1% level) all the way to melting. We calculated the one loop disorder correction: gsol 2:14jaTj1=2. An explicit expression for gliqaT, obtained using the Borel-Pade resummation of the renormalized high temperature series, is rather bulky and can be found in [20]. The disorder correction in liquid can be obtained by differ-entiating the ’’clean’’ partition function with respect to parameters: gliq gliq 2aTg0liq=3 g
0
liq2=2. These
results enable us to find the location of the transition line and calculate discontinuities of various physical quanti-ties across the transition line.
It was noted [20] that in a clean system a homoge-neous state exists as a metastable overcooled liquid state all the way down to zero temperature. This is of impor-tance since interaction with disorder can convert the metastable state into a stable one. Indeed, generally a homogeneous state gains more than a crystalline state from pinning, since it can easier adjust itself to the to-pography of the pinning centers. Since at large jaTj, gliq/ a2T is larger than gsol/ jaTj1=2, the transition line shifts to lower fields. The equation for the melting line is daT gliq gsol=gliq gsol nt. The universal function daT, plotted in Fig. 2, turns out to be nonmonotonic. Since nt is a monotonic function of t, one obtains the transition lines for various n in Fig. 1 by ‘‘sweeping’’ the Fig. 2. A peculiar feature of daT is that it has a local minimum at aT 17:2 and a local maxi-mum at aT 12:1 (before crossing zero at aT 9:5.
-40
-30
-20
-10
-0.04
0
0.04
0.08
0.12
aT=-9.5 aT=-17.2 aT=-12.1 aT d (a T )FIG. 2. Nonmonotonic dependence of the disorder function
daT on the LLL scaled temperature aT.
60 70 80 90 0 4 8 12 16 H( T ) T (K) n =0 n=0.08 n=0.12 n =0.3 m e a n fie ld c r o ss o v e r H c 2 (T ) L L L d om in an c e lin e H*(T) MCP Eent(CP) Emag HLLL H m (T ) G L a p p lic a b il ity lin e
FIG. 1. Theoretical first order transition lines for various degrees of disorder separated between a homogeneous and a crystalline phase of vortex matter. The best fit n 0:12 line is compared with experimental melting line HmT [6] and the
solid-vortex glass transition line HT [4]. Equal entropy and
magnetization points are denoted by Eent and Emag. The range
of applicability is restricted by the two lines. Very small fields below HLLLare beyond this range.
P H Y S I C A L R E V I E W L E T T E R S week ending
25 APRIL 2003
VOLUME90, NUMBER16
Therefore between these two points there are three solu-tions to the melting line equation. As a result, starting from the zero field at Tc, the transition field HT reaches a maximum at Eent(point CP in Fig. 1) beyond which the curve sharply turns down (this feature was called ‘‘in-verse melting’’ in [10]) and at Emag backwards. Then it reaches a minimum and continues as the solid (Bragg glass)-vortex glass line roughly parallel to the T axis.
The temperature dependence of the disorder strength nt, as of any parameter in the GL approach, should be derived from a microscopic theory assuming random chemical potential or fitted to experiment. We conjecture (and find to be consistent with the experiments) that the general dependence near Tc is nt n1 t2=t[or W / 1 t2]. The expression approaches the one used at lower temperatures [1] with n 0
T=4 2 2Gi p
3. The best fit for the low field part of the experimental melting line HmT of the optimally doped YBCO (data taken from [6], Tc 92:6, 8:3) gives Gi 2:0 104, H
c2 190 T, ( = 50 (consistent with other experiments [4,22]). This part is essentially independent of disorder. The upper part of the melting curve is very sensitive to disorder: Both the length of the ‘‘finger’’ and its slope depend on n. The best fit is n 0:12. This value is of the same order of magnitude as the one obtained phenom-enologically using Eq. (3.82) in [1]. We speculate that the low temperature part of the ‘‘unified’’ line corresponds to the solid-vortex glass transition HT observed in nu-merous experiments [4,6 –8,12] (squares in Fig. 1 [4]). A complicated shape of the ‘‘wiggling’’ line has been re-cently observed [12].
Magnetization and specific heat of both solid and liquid can be calculated from the above expressions for free energy. Magnetization of liquid along the melting line HmT is larger than that of solid. The magnetization jump is compared in Fig. 3(a) with the SQUID experi-ments [5] in the range 80 – 90 K (triangles) and of the torque experiments (stars [3] and circles [4]). One ob-serves that the results of the torque experiments compare surprisingly well above 83 K while the SQUID data a bit lower than theoretical or the torque one. But those of [4] vanish abruptly below 83 K unlike the theory and are inconsistent with the specific heat experiments discussed below [6,22]. We predict that at lower temperatures (somewhat beyond the range investigated experimentally thus far) magnetization reaches its maximum and changes sign at the point Emag (at which magnetization of liquid and solid are equal).
Entropy jump calculated using the Clausius-Clapeyron relation is compared with an experimental one deduced from the spike of the specific heat [6] [triangles in Fig. 3(b)] and an indirect measurement from the magne-tization jumps [4] (circles). At high temperatures, the theoretical values are a bit lower than the experimental and both seem to approach a constant at Tc. The theoreti-cal entropy jump and the experimental one of [6] vanish at Eent (Fig. 1) near 75 K. Experimentally, such a point
(called Kauzmann points [23]) was established in BSCCO as a point at which the inverse melting appears [10]. Below this temperature, entropy of the liquid be-comes smaller than that of the solid. Note that the equal magnetization point Emag is located at a slightly lower field than the equal entropy point Eent. The Kauzmann point observed at a lower temperature in YBCO in [8] is different from Eent since it is a minimum rather than a maximum of magnetic field. It is also located slightly outside the region of applicability of our solution. The point Eentis observed in [12] in which the universal line is continuous.
In addition to the spike, the specific heat jump has also been observed along the melting line HmT [5,6,22]. Theoretically, the jump does not vanish either at Eent or Emag, but is rather flat in a wide temperature range. Our results are larger than experimental jumps of [6] by a factor of 1.4 to 2 [Fig. 3(c)]. In many experiments there appears a segment of the second order phase transition continuing the first order melting line beyond a certain point. In [6] it was shown that at that point the specific heat profile shows ’’rounding.’’ We calculated the specific heat profile above the universal first order transition line. It exhibits a ’’rounding’’ feature similar to that displayed in [6] with no sign of the criticality. The height of the peak is roughly of the size of the specific heat jump. We therefore propose not to interpret this feature as an evidence for a second order transition above the first order line. -0.01 0 0.01 0.02 0.03 0.04 75 80 85 90 ref. 5 ref. 3 ref. 4 75 80 85 90 -0.4 0 0.4 0.8 1.2 ref.6 ref. 4 72 76 80 84 88 92 0 2 4 6 ref. 6 ∆ M( G ) ∆ S/k B ∆ C/T (m J mo l-1 K -2 ) T (K) a b c E mag Eent T (K) T (K)
FIG. 3. Discontinuities along the first order transition line in YBCO. Jumps of magnetization, entropy, and specific heat are compared with experiments in (a), (b), and (c), respectively.
P H Y S I C A L R E V I E W L E T T E R S week ending
25 APRIL 2003
VOLUME90, NUMBER16
As seen from fitting the optimal doping YBCO, the disorder parameter n in the cases of interest is very small. However, to address a more delicate question of appear-ance of the glass transition between liquid and glass claimed in some experiments [4,6] and simulations [14,15], we used a nonperturbative method, the replica trick, combined with the Gaussian variational approxima-tion [24]. Breaking of the replica symmetry signals ap-pearance of the glass transition in static phenomena. Here we summarize the results, leaving details for a longer presentation. The most general hierarchical homogeneous (liquid) ansatz [24] was considered and found that there is no replica symmetry breaking (RSB) solution.
Now we compare our results with other theories start-ing with models based on the Lindemann criterion [9,17,18]. Near Tc (where effects of disorder are small), the location of the first order transition line HT is qualitatively consistent with that found from the Lindemann criterion [1]. In the intermediate region around Eentour results for the melting line are completely different from all the phenomenological models [9,17,18]. In particular, we do not have a critical point assumed in some of them (the two papers in [17] differ on the continuation of the line beyond the critical point in which two first order lines join together). At temperatures below 70 K, the elasticity theory approach based on the London approximation [17,18] (valid beyond the range of applicability of the GL approach) is expected to smoothly interpolate to the GL approach. The advantage of the present approach is that, in addition to location of the transition line, it enables the determination of disconti-nuities and the calculation of physical quantities away from the transition line.
The comparison with numerical simulations can be made only qualitatively since the disordered GL model has not been simulated. In the XY simulations of [13], a single transition line is parallel to the T axis below certain temperature, while ours is not. In [14] there is, in addition, a slush-liquid transition, while Langevin simu-lation [15] finds the liquid-glass transition. Their tran-sition criterion, however, is dynamical. Absence of RSB in liquid does not generally imply that this state, espe-cially at low temperatures, does not exhibit ‘‘glassy prop-erties’’ in dynamics. In fact overcooled liquids generally are ‘‘glassy.’’ Therefore we propose to consider the glassy properties of the vortex state above the HT line in the context of the ‘‘disordered overcooled’’ liquids. Recent simulations [16] demonstrate that glassy features in dy-namics do not necessarily correspond to the conventional vortex glass scenario. Next we comment on the ’’vortex loops’’ scenario for YBCO. The region in which the loops are relevant according to [25] is below GiHc2, too small to explain the experimental transitions (our fit for Gi 2 104is much smaller than that assumed in [25]). The LLL is inapplicable in this region (see Fig. 1). In strongly anisotropic materials, a model of the Lawrence-Doniah
type is more appropriate [1,26]. We performed a 2D GL calculation and found that in that case there is no ‘‘wiggle’’ of the transition line and speculate that it dis-appears at a certain value of the anisotropy parameter.
We are grateful to E. H. Brandt, X. Hu, and Y. Yeshu-run for discussions, and T. Nishizaki, F. Bouquet, G. Mikitik, E. Zeldov, and Z. Tesanovic for correspon-dence. This work was supported by NSC of Taiwan Grant NSC No. 91-2112-M-009-503, and the Ministry of Science and Technology of China (G1999064602) are acknowledged.
[1] G. Blatter et al., Rev. Mod. Phys. 66, 1125 (1994). [2] E. Zeldov et al., Nature (London) 375, 373 (1995);
R. Liang, D. A. Bonn, and W. N. Hardy, Phys. Rev. Lett. 76, 835 (1996).
[3] M. Willemin et al., Phys. Rev. Lett. 81, 4236 (1998). [4] T. Nishizaki et al., Physica (Amsterdam) 362C, 121
(2001); K. Shibata et al. (unpublished).
[5] A. Schilling et al., Nature (London) 382, 791 (1996); M. Roulin et al., Science 273, 1210 (1996).
[6] F. Bouquet et al., Nature (London) 411, 448 (2001). [7] S. Kokkaliaris et al., Phys. Rev. B 61, 3655 (2000); M. B.
Gaifullin et al., Phys. Rev. Lett. 84, 2945 (2000). [8] Y. Radzyner et al., Phys. Rev. B 65, 100513 (2002). [9] D. Ertas and D. R. Nelson, Physica (Amsterdam) 272C,
79 (1996).
[10] N. Avraham et al., Nature (London) 411, 451 (2001). [11] Y. Radzyner et al., Phys. Rev. B 65, 100503 (2002). [12] D. Pal et al., Phys. Rev. B 63, 132505 (2001); D. Pal et al.,
ibid. 65, 096502 (2002).
[13] P. Olsson and S. Teitel, Phys. Rev. Lett. 87, 137001 (2001). [14] Y. Nonomura and X. Hu, Phys. Rev. Lett. 86, 5140 (2001). [15] A. van Otterlo et al., Phys. Rev. Lett. 81, 1497 (1998). [16] C. Reichhardt et al., Phys. Rev. Lett. 84, 1994 (2000). [17] J. Kierfeld and V. Vinokur, Phys. Rev. B 61, 14928 (2000);
G. P. Mikitik and E. H. Brandt, ibid. 64, 184514 (2001); G. I. Menon, ibid. 65, 104527 (2002).
[18] T. Giamarchi and P. Le Doussal, Phys. Rev. B 52, 1242 (1995); S. E. Korshunov, ibid. 48, 3969 (1993).
[19] E. Brezin et al., Phys. Rev. B 31, 7124 (1985); Z. Tesanovic and A.V. Andreev, ibid. 49, 4064 (1994); G. J. Ruggeri and D. J. Thouless, J. Phys. F 6, 2063 (1976). [20] D. Li and B. Rosenstein, Phys. Rev. Lett. 86, 3618 (2001); Phys. Rev. B 65, 220504 (2002); 65, 024514 (2002); 60, 10460 (1999);60, 9704 (1999).
[21] I. D. Lawrie, Phys. Rev. B 50, 9456 (1994).
[22] K. Deligiannis et al., Physica (Amsterdam) 341C, 1329 (2000).
[23] P. G. Debenedetti, Metastable Liquids. Concepts and
Principles (Princeton University Press, Princeton, 1996).
[24] M. Mezard and G. Parisi, J. Phys. I 1, 809 (1991). [25] A. Subdø and A. K. Nguyen, Phys. Rev. B 60, 15307
(1999); Z. Tesanovic, ibid. 59, 6449 (1999).
[26] J. Hu and A. H. MacDonald, Phys. Rev. B 56, 2788 (1997); A. E. Koshelev, ibid. 56, 11201 (1997); 50, 506 (1994). P H Y S I C A L R E V I E W L E T T E R S week ending 25 APRIL 2003 VOLUME90, NUMBER16 167004-4 167004-4
