Scaled Temperature

FIG. 11. 共Color online兲 The melting point and the spinodal point of the crystal. The free energies of the crystalline and the liquid states are equal at melt, while metastable crystal be-comes unstable at spinodal point.

Melting lines of optimally doped untwined 共Welp et al., 1991,1996;Schilling et al., 1996,1997;Willemin et al., 1998; Bouquet et al., 2001兲 YBa2Cu₃O_7−␦ and DyBa₂Cu₃O₇ 共Roulin et al., 1996, 1996; Revaz et al., 1998兲 are also fitted extremely well 共Li and Rosenstein, 2003兲. More recently both NbSe2 and thick films of Nb₃Ge were fitted byKokubo et al.共2007兲in which dis-order is significant but the pristine melting line is be-lieved to be clearly seen in dynamics via peak effect. In Table II parameters inferred from these fits are given where the data for YBCO_7−␦, DyBCO_6.7, and YBCO₇ are taken fromSchilling et al.共1996兲,Roulin, Junod, and Walter共1996兲,Roulin, Junod, et al.共1996兲, andNishizaki et al.共2000兲, respectively. Parameters such as Gi charac-terizing the strength of thermal fluctuations differ a bit from the often mentioned共Blatter et al., 1994兲. Similar fits were made in two dimensions for an organic super-conductor共Fruchter et al., 1997兲. Unlike the Lindemann criterion, the quantitative calculation allows determina-tion of various discontinuities across the melting line 共since we have energies of both phases兲 to which we turn next.

2. Discontinuities at melting a. Magnetization jump

The scaled magnetization, which is defined by m共aT兲

= −df共aT兲/daT, can be calculated in both phases and the difference⌬m=ms− m_lat the melting point a_T^m= −9.5 is

⌬M/Ms=⌬m/ms= 0.018. 共250兲

This was compared by Li and Rosenstein 共2003兲 with experimental results on fully oxidized YBa₂Cu₃O₇ 共Nishizaki et al., 2000兲 and optimally doped untwined YBa₂Cu₃O_7−␦ 共Welp et al., 1991, 1996兲. These samples probably have the lowest degree of disorder not in-cluded in calculations.

b. Specific heat jump

In addition to the delta-function-like spike at melting following from the magnetization jump discussed above experiment shows also a specific heat jump 共Schilling et al., 1996,1997; Bouquet et al., 2001; Lortz et al., 2006兲.

The theory allows us to quantitatively estimate it. The specific heat jump is

⌬c = 0.0075

冉

^{2 − 2b + tt}

冊

^{2− 0.20Gi1/3}共b − 1 − t兲

冉

^tb2

冊

^2/3.

共251兲 It was compared by Li and Rosenstein 共2003兲 with the experimental values of Willemin et al. 共1998兲. See also the comparison with specific heat in NbSn₃ofLortz et al.

共2006兲.

In addition the value of the specific heat jump in the 2D GL model is in good agreement with MC simulations 共Hu and MacDonald, 1993;Kato and Nagaosa, 1993;Hu et al., 1994兲, while the 3D MC result is still unavailable.

3. Gaussian approximation in the crystalline phase and the spinodal line

a. Gaussian variational approach with shift of the field

The Gaussian variational approach in the phase ex-hibiting spontaneously broken symmetry is quite a straightforward, albeit more cumbersome, extension of the method to include the “shift” v共r兲. In our case of one complex field one should consider the most general qua-dratic form,

K =

冕

r,r⬘关␺^*共r兲 − v^*共r兲兴G⁻¹共r,r⬘兲关␺共r⬘兲 − v共r⬘兲兴

+关␺共r兲 − v共r兲兴H^*共r,r⬘^兲关␺共r⬘^{兲 − v共r}⬘兲兴 + c.c. 共252兲 To obtain the shift v and “width of the Gaussian”

which is a matrix containing G and H, one minimizes the Gaussian effective free energy 共Cornwall et al., 1974兲, which is an upper bound on the energy. Assuming hex-agonal symmetry, the shift should be proportional to the zero quasimomentum function, v共r兲=v␸共r兲, with a con-stant v taken real, thanks to the global U共1兲 gauge sym-metry. On LLL, as in perturbation theory, we use the phonon variables O_kand A_kdefined in quasimomentum basis Eqs.共139兲 and 共143兲 instead of␺共r兲,

␺共r兲 = v␸共r兲 + 1

冑

2共2␲兲^3/2

冕

k

e^ikzzc_k␸k共r兲共Ok+ iA_k兲.

共253兲 The phase defined after Eq.共148兲 is quite important for simplification of the problem and was introduced for fu-ture convenience. The most general quadratic form in these variables is

TABLE II. Parameters of high T_csuperconductors deduced from the melting line.

Material Tc Hc2 Gi ␬ ␥a

YBCO_7−␦ 93.07 167.53 1.9⫻10⁻⁴ 48.5 7.76

YBCO7 88.16 175.9 7.0⫻10⁻⁵ 50 4

DyBCO_6.7 90.14 163 3.2⫻10⁻⁵ 33.77 5.3

K =

冕

k

O_kG_OO⁻¹共k兲O−k+ A_kG_AA⁻¹共k兲A−k

+ O_kG_OA⁻¹共k兲A−k+ A_kG_OA⁻¹共k兲O−k, 共254兲 with matrix of functions to be determined together with the constant v by the variational principle. The Gaussian free energy is

f_gauss

vol = a_Tv²+␤_⌬

2 v⁴+ 2^5/2␲ 2共2␲兲³

冕

k

log det共G⁻¹兲

+ 1

2共2␲兲³

冕

k

再 ^冉

^{k2z2+ aT}

^冊

^{关GOO共k兲 + GAA共k兲兴}

+ v²关共2␤k+兩␥k兩兲GOO共k兲

+共2␤k−兩␥k兩兲GAA共k兲兴

冎

^+2␤1⌬

冋

^{2共21␲兲3}

^冕

^k兩␥k兩

⫻关GOO共k兲 − GAA共k兲兴

册

^2+␤2⌬

冋

^{2共21␲兲3}

^冕

^k兩␥k兩

⫻关GOA共k兲兴

册

^{2+4共21␲兲6}

^冕

^{k,l␤k−l关GOO共k兲}

+ G_AA共k兲兴关GOO共l兲 + GAA共l兲兴, 共255兲 leading to the following minimization equations:

v²+ a_T

␤A

= − 1

2共2␲兲³␤_⌬

冕

k

共2␤k+兩␥k兩兲GOO共k兲

+共2␤k−兩␥k兩兲GAA共k兲,

共256兲 2^5/2␲关G共k兲⁻¹兴OO

= k_z²/2 + a_T+ v²共2␤k+兩␥k兩兲 + 1

2共2␲兲³

冕

l

冉

^2␤k−l

+兩␥k兩兩␥l兩

␤_⌬

冊

^{GOO共l兲 +}

冉

^{2␤k−l−兩␥k}␤^兩兩_⌬^␥l兩

冊

^GAA共l兲

and

2^5/2␲关G共k兲⁻¹兴AA

=k_z²

2 + a_T+ v²共2␤k−兩␥k兩兲 + 1 2共2␲兲³

⫻

冕

l

冉

^{2␤k−l+兩␥k}␤^兩兩_⌬^␥l兩

冊

^GAA共l兲

+

冉

^{2␤k−l−兩␥k}␤^兩兩_⌬^␥l兩

冊

^{GOO共l兲25/2␲关G共k兲−1兴OA}

= − 2^5/2␲^GOA共k兲 G_OO共k兲GAA共k兲 − GOA共k兲²

= 4兩␥k兩

␤_⌬ 1 2共2␲兲³

冕

l

兩␥l兩GOA共l兲. 共257兲

These equations look quite intractable, however, they can be simplified.

b. How to eliminate the off-diagonal terms

The crucial observation is that after we have inserted the phase c_k=

冑

␥k/兩␥k兩 in Eq. 共255兲 using our experience with perturbation theory, G_AOappears explicitly only on the right hand side of the last equation. It also implicitly appears on the left hand side due to a need to invert the matrix G. Obviously G_OA共k兲=0 is a solution and in this case the matrix diagonalizes. However, the general solu-tion can be shown to differ from this simple one just by a global gauge transformation. Subtracting the OO equation from the AA equation above 关Eq. 共256兲兴 and using the OA equation, we observe that matrix G⁻¹has a form

G⁻¹⬅

冉

^GGOO−1AO−1^{共k兲 G}共k兲 G^AO−1AA−1^共k兲共k兲

冊

= 1

2^5/2␲

冉

^{kz2/2 +}␮AOk^{␮Ok2 ␮AOk2} 2 k_z²/2 +␮Ak

2

冊

^{, 共258兲}

with

␮Ok

2 = E_k+⌬1兩␥k兩, ␮Ak

2 = E共k兲 − ⌬1兩␥k兩,

共259兲

␮AOk2 =⌬2兩␥k兩,

where ⌬1,⌬2 are constants. Substituting this into the Gaussian energy one finds that it depends on⌬1,⌬2via the combination ⌬=

冑

⌬₁²+⌬₂² only. Therefore without loss of generality we can set⌬2= 0, thereby returning to the G_OA= 0 case.

Using this observation, the gap equations significantly simplify. The function E_kand the constant⌬ satisfy

E_k= a_T+ 2v²␤k+ 2

冓

^␤k−l

^冉

^{␮1Ol+␮1Al}

^冊 冔

^{l, 共260兲}

␤_⌬⌬ = − aT− 2

冓

^␤l

^冉

^{␮1Ol+␮1Al}

^冊 冔

^{l 共261兲}

and the shift equation v²+ a_T

␤A

= −

冓

^{2␤k␮+Ok兩␥k兩+2␤k␮−Ak兩␥k兩}

冔

^{k. 共262兲}

The Gaussian energy共after integration over kz兲 becomes f

vol= v²a_T+␤A

2 v⁴+ f₁+ f₂+ f₃, f₁=具␮Ok+␮Ak典k,

共263兲 f₂= a_T具共␮Ok−1 +␮Ak−1兲 + v²关共2␤k+兩␥k兩兲␮Ok−1

+共2␤k−兩␥k兩兲␮Ak−1兴典k,

f₃=具␤k−l共␮Ok−1 +␮Ak−1兲共␮Ol−1+␮Al−1兲典k,l

+ 1

2␤_⌬^关具兩␥k兩共␮_Ol⁻¹⁻␮_Al⁻¹兲典k兴².

The problem becomes quite manageable numerically af-ter one spots an unexpected small parameaf-ter.

c. The mode expansion Using formula共A18兲

␤k=

兺

n=0

⬁

␹ⁿ␤n共k兲,

共264兲

␤n共k兲 ⬅

兺

兩X兩²=na_⌬²

exp关ik · X兴,

derived in Appendix A and the hexagonal symmetry of the spectrum, one deduces that E_k can be expanded in

“modes,”

E_k=

兺

^{En␤n共k兲. 共265兲}

The integer n determines the distance of a points on reciprocal lattice from the origin, and ␹⬅exp关−a_⌬²/ 2兴

= exp关−2␲^/

冑

3兴=0.0265. One estimates that En⯝␹^naT; therefore the coefficients decrease exponentially with n.

Note that for some integers, for example, n = 2 , 5 , 6, ␤n

= 0. Retaining only first s modes will be called the s mode approximation. We minimized numerically the Gaussian energy by varying v ,⌬, and first few modes of E_k.

The sample results of free energy density for various a_T with three modes are given in TableIII. In practice two modes are also quite enough. We see that in the interesting region of not very low temperatures the en-ergy converges extremely fast. In practice two modes are quite enough.

d. Spinodal point

One can show that above

a_T^spinodal= − 5.5 共266兲

there is no solution for the gap equations. The corre-sponding value in two dimensions is a_T^spinodal= −7 and is consistent with the relaxation time measured in Monte Carlo simulations 共Kato and Nagaosa, 1993兲. The spin-odal point was observed in NbSb₂ 共Xiao et al., 2004;

Thakur et al., 2005; Adesso et al., 2006兲 at the position consistent with the theoretical estimate.

e. Corrections to the Gaussian approximation

The lowest order correction to the Gaussian approxi-mation共sometimes called the post-Gaussian correction兲 was calculated by Li and Rosenstein 共2002a, 2002b, 2002c兲 to determine the precision of the Gaussian ap-proximation. This is necessary in order to fit experi-ments and compare with low temperature perturbation theory and other nonperturbative methods.

A general idea behind calculating systematic correc-tions to the Gaussian approximation was described for

liquid in Sec. III.C and modifications are quite analo-gous to those done for the Gaussian approximation. Re-sults for the specific heat were compared by Li and Rosenstein 共2002c兲. Generally the post-Gaussian result is valid until a_T= −7 and rules out earlier approxima-tions, as the one ofTešanovic´ et al.共1992兲andTešanovic´

and Andreev共1994兲共dotted line兲.

IV. QUENCHED DISORDER AND THE VORTEX GLASS In any superconductor there are impurities present ei-ther naturally or systematically produced using the pro-ton or electron irradiation. The inhomogeneities on both the microscopic and the mesoscopic scales greatly affect thermodynamic and especially dynamic properties of type II superconductors in a magnetic field. The field penetrates the sample in a form of Abrikosov vortices, which can be pinned by disorder. In addition, in high T_c superconductors thermal fluctuations also greatly influ-ence the vortex matter, for example, in some cases ther-mal fluctuations will effectively reduce the effects of dis-order. As a result the T-H phase diagram of the high T_c superconductors is very complex due to the competition between thermal fluctuations and disorder, and it is still far from being reliably determined even in the best stud-ied superconductor, the optimally doped YBCO super-conductor.

It is the purpose of this section to describe the glass transition and static and thermodynamic properties of both the disordered reversible and the irreversible glassy phases. The disorder is represented by the random com-ponent of the coefficients of the GL free energy 关Eq.

共20兲兴 and the main technique used is the replica formal-ism. The most general so-called hierarchical homoge-neous 共liquid兲 ansatz 共Mezard and Parisi, 1991兲 and its stability are considered to obtain the glass transition line and to determine the nature of the transition for various values of the disorder strength of the GL coefficients. In most cases the glassy phase exhibits the phenomenon of replica symmetry breaking when ergodicity is lost due to trapping of the system in multiple metastable states. In this case physical quantities do not possess a unique value but rather have a distribution. We start with the case of negligible thermal fluctuations.

A. Quenched disorder as a perturbation of the vortex lattice 1. The free energy density in the presence of pinning potential

a. GL model with␦^Tcdisorder

We start with space variations of the coefficient of兩⌿兩² 关Eq. 共20兲兴 distributed as white noise 关Eq. 共21兲兴. It can be regarded as a local variation of T_c. As mentioned in Sec.

I other types of disorder are present and might be im-portant, however, as will be shown later are more com-plicated.

Since a pointlike disorder breaks the translational symmetry in all directions including that of the magnetic TABLE III. Mode expansion.

aT −30 −20 −10 −5.5

f −372.2690 −159.5392 −33.9873 −6.5103

field z, one has to consider configurations dependent on all three coordinates and take into account anisotropy.

We restrict to the case m

a

*= m

b

*⬅m^*, F关W兴 =

冕

r

ប²

2m^*兩D⌿兩²+ ប² 2m_c^*兩⳵z⌿兩² +␣共T − Tc兲关1 + W共r兲兴兩⌿兩²+␤

2兩⌿兩⁴, 共267兲 where W共r兲 is the ␦^Tc random disorder 共real兲 field, which we assume to be white noise with variance that can be written in the following form:

W共r兲W共r⬘兲 = n␰²␰c␦³共r − r⬘兲. 共268兲 The dimensionless parameter n is proportional to the density of pinning centers and a single pin’s strength, while␰c⬅␰共m^*/ m_c^*兲^1/2is the coherence length in the field direction. The units used here are the same as before with the addition of ␰c as the unit of length in the z direction. As in previous sections, we confined ourselves mainly to the region in parameter space described well by the lowest Landau level approximation 共LLL兲 de-fined next.

b. The disordered LLL GL free energy in the quasimomentum basis

In the units and the field normalization described in Sec.II.Athe LLL energy becomes

F关W兴 =

冕

r

冋

^{12兩⳵z⌿兩2− aH兩⌿兩2+1 − t2 W共r兲兩⌿兩2}

+1

2兩⌿兩⁴

册

^{, 共269兲}

where a_H=¹₂共1−b−t兲 and

W共r兲W共r⬘^{兲 = n}␦³共r − r⬘^{兲 共270兲} in the new length unit. The order parameter field on LLL can be expanded in the quasimomentum basis de-fined in Sec.III.Aas

⌿共r兲 = 1 共2␲兲^3/2

冕

k

␸k共r兲⌿k, 共271兲

where k⬅共k,kz兲, functions are defined in Eqs. 共134兲 and 共137兲, and the integration measure was defined in Sec.

III.Ato be the Brillouin zone in the x-y plane and the full range of momenta in the z direction. We consider the hexagonal lattice, although modifications required to consider a different lattice symmetry are minor. Using the quasimomentum LLL functions of Eq.共134兲, the dis-order term becomes

F_dis=1 − t 2

冕

r

W共r兲兩⌿共r兲兩²=

冕

k,l

w_k,l⌿_k^*⌿l 共272兲

with

w_k,l= 1 − t 2共2␲兲³

冕

r

W共r兲␸_k^*共r兲␸l共r兲. 共273兲

The remaining terms can be written as F_clean=

冕

k

共kz2/2 − a_H兲⌿_k^*⌿k

+ 1

2共2␲兲³

冕

k,k⬘^,l,l⬘关k,k⬘兩l,l⬘兴⌿_k^*⌿_k^*_⬘⌿l⌿l⬘

⫻

兺

Q

␦共k + k⬘^{− l − l}⬘^{− Q}兲, 共274兲

with关k,l兩k⬘^l⬘^{兴=共1/vol兲兰}r␸_k^*共r兲␸l共r兲␸_k^*_⬘共r兲␸l⬘共r兲 and where Q =共Qជ,0兲 and Qជ is the reciprocal lattice vectors as k , l , k⬘^{, l}⬘satisfy the momentum conservation up to a re-ciprocal lattice vector. 关k,l兩k⬘^l⬘兴 will be equal to zero if k + k⬘^{− l − l}⬘⫽Q.

2. Perturbative expansion in disorder strength a. Expansion around the Abrikosov solution

The GL equations derived from the free energy in the quasimomentum basis are

共k_z²/2 − a_H兲⌿k+␣

冕

l

w_k,l⌿l

+

冕

k⬘^l,l⬘

兺

Q

␦共k + k⬘^{− l − l}⬘^{− Q}兲

⫻关k,k⬘兩l,l⬘兴 共2␲兲³ ⌿_k

⬘

* ⌿l⌿l⬘= 0. 共275兲 The parameter␣= 1 inserted here will help with count-ing orders. The expansion in orders of the disorder strength␣ ^reads

⌿ = ⌿^共0兲+␣⌿^共1兲+␣²⌿^共2兲+ ¯ . 共276兲 The clean case Abrikosov solution of Sec. IIis defined as the quasimomentum zero. Therefore

⌿^共0兲=共2␲兲^3/2

冑

a_H/␤_⌬␦k. 共277兲 The delta function appears due to its long-range trans-lational order. Now Eq. 共275兲 can be solved order by order in␣. Since contributions linear in disorder poten-tial will average to zero, in order to get the leading con-tribution of disorder one should calculate the free en-ergy to the second order in ␣. Multiplying the exact equation 关Eq. 共275兲兴 by ⌿_k^* and integrating over k, one can express the order 4 in⌿ term via simpler quadratic ones,

F =1 2

冕

k

共kz2/2 − a_H兲兩⌿k兩²+␣ 2

冕

k,l

⌿_k^*w_k,l⌿l. 共278兲

Substituting the expansion 关Eq. 共276兲兴 and using delta functions of⌿^共0兲of Eq.共277兲 one gets the following ␣² terms:

F^共2兲= −a_H^3/2共2␲兲^3/2

2␤_⌬^{1/2 关⌿0共2兲*+⌿0共2兲兴 +} 1 2

冕

k

共k_z²/2 − a_H兲

⫻兩⌿_k^共1兲兩²+a_H^1/2共2␲兲^3/2 2␤_⌬^1/2

冕

k

关w0,k⌿_k^共1兲+⌿_k^共1兲*w_k,0兴.

共279兲 Therefore the second order correction to ⌿ is needed only for zero quasimomentum.

b. First order elastic response of the vortex lattice To order␣one obtains the following equation:

共k_z²/2 − a_H兲⌿_k^共1兲+ w_k,0共2␲兲^3/2

冑

^a_␤^H_⌬^{+ 2a}_␤^H_⌬^{␤k⌿k共1兲}

在文檔中 Ginzburg-Landau theory of type II superconductors in magnetic field (頁 33-38)