Theory of Nernst effect in high-T-c superconductors

Theory of Nernst effect in high-T

_c

superconductors

Bui Duc Tinh and Baruch Rosenstein

Department of Electrophysics, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China 共Received 14 September 2008; published 26 January 2009兲

We calculated, using the time-dependent Ginzburg-Landau equation with thermal noise, the transverse thermoelectric conductivity␣_xyand the Nernst signal e_N, describing the Nernst effect, in type-II supercon-ductor in the vortex-liquid regime. The Gaussian method used is an elaboration of the Hartree-Fock utilized by Ullah and Dorsey 关Phys. Rev. Lett. 65, 2066 共1990兲兴. An additional assumption often made in analytical calculations that only the lowest Landau level significantly contributes to physical quantities of interest in the high-field limit is lifted by including all the Landau levels. The resulting values in two dimensions are significantly lower than the numerical simulation data of the same model but are in reasonably good quanti-tative agreement with experimental data on La2SrCuO4above the irreversibility line共below the irreversibility line at which ␣xydiverges and theory should be modified by including pinning effects兲. The values of eN

calculated in three dimensions are also in good quantitative agreement with experimental data for temperature close to T_con YBa₂Cu₃O₇. For each of the materials we consider, the melting and the irreversibility lines are also fitted with the same set of parameters using a recent quantitative Ginzburg-Landau theory.

DOI:10.1103/PhysRevB.79.024518 PACS number共s兲: 74.40.⫹k, 74.25.Ha, 74.25.Dw

I. INTRODUCTION

The electric field is induced in a metal under magnetic field by the temperature gradient ⵜT perpendicular to the magnetic field H, which is a phenomenon known as Nernst effect1 _{共direction of the electric field being perpendicular to}

bothⵜT and H兲. Recently the Nernst effect in high-Tc

super-conductors attracted attention both theoretically1–6 _and

experimentally.7–15 _{In these materials, effect of thermal}

fluc-tuations is very strong leading to depinning of Abrikosov vortices created by the magnetic field in type-II supercon-ductor below second critical field Hc2共T兲. In the mixed state

the Nernst effect is large due to vortex motion while in the normal state and in the vortex lattice or glass states, it is typically smaller. The Nernst effect therefore is a probe of thermal fluctuations phenomena in the vortex matter but in principle could shed some light on the underlying micro-scopic mechanism of superconductivity in cuprates.

The appearance of a fluctuation tail above the critical tem-perature in the Nernst signal was observed in strongly type-II superconductors, both with low Tc such as NbSe2 and

Nb0.15Si0.85 films,15 and several different high-temperature

materials.8–11,14 _{The related Ettingshausen effect was}

de-tected as well.7_{In particular, the Nernst effect was observed}

well above T_c2共H兲 and even above Tc in Bi2Sr2CaCu2O8+␦

共Ref.14兲 共BiSCCO兲, strongly underdoped YBa2Cu3Oy共Refs.

11, 12, and14兲 共YBCO兲, and La2−xSrxCuO4 共Refs. 10, 11,

13, and 14兲 共LaSCO兲. These layered materials are highly anisotropic and can be described by a quasi-two-dimensional 共2D兲 model. Due to reduced dimensionality the effect of thermal fluctuations is greatly enhanced. However even in less anisotropic materials such as the hole-doped cuprate Nd2−xCexCuO4 共Ref. 14兲 共NCCO兲, and weakly anisotropic

and overdoped or fully oxidized YBCO_6.99,14_{the effect}

per-sists. Fluctuations in these materials cannot be described by a 2D model and generalization to anisotropic three-dimensional共3D兲 models is required. Measurements of eNin

fields H up to 45 T 共Ref. 14兲 reveal that the vortex-Nernst

signal eN has a characteristic “tilted-hill” profile, which is

qualitatively distinct from that of quasiparticles. The hill pro-file, which is observed above and below Tc, underscores the

continuity between the vortex-liquid state below Tcand the

Nernst region above Tc.

Recently, the study of the Nernst effect in NbSe2reveals a

large quasiparticle contribution with a magnitude comparable and a sign opposite to the vortex signal.15_{A large negative}

Nernst coefficient, persisting at temperatures well above Tc

= 7.2 K, was found in this metal. The quasiparticle contribu-tion to the Nernst signal attains a magnitude comparable to the vortex signal in the superconducting state. More recently, in experiment on amorphous thin films of the conventional superconductor Nb0.15Si0.85,15a Nernst signal was generated

by short-lived Cooper pairs in the normal state. In these amorphous films, the contribution of free electrons to the Nernst signal is negligible. The extremely short mean-free path of electrons in amorphous Nb0.15Si0.85 damps the

normal-state Nernst effect and allows a direct comparison of the data with theory. In the zero-field limit and close to Tc,

the magnitude of the Nernst coefficient was found to be in quantitative agreement with a theoretical prediction by Ussishkin et al.,3 _{invoking the superconducting correlation}

length as its single parameter. At high temperature and finite magnetic field, the data were found to deviate from the the-oretical expression. In electron-doped cuprate 共NCCO兲 the quasiparticle contribution to the Nernst signal is large.14_The

quasiparticle contribution actually dominates the Nernst sig-nal far below Tc. Nevertheless, the vortex signal retains its

characteristic tilted-hill profile which is easily distinguished from the monotonic quasiparticle contribution.

We will concentrate on the vortex-Nernst effect in type-II superconductor of the overdoped LaSCO,12_{and underdoped}

and overdoped YBCO,12 _{where e}

N is intrinsically strongly

nonlinear in H and generally much larger than that in non-magnetic normal metal. The observation of the Nernst effect above Tcalong with other strong fluctuation effects was

in-terpreted as a support for the preformed pair scenario for the mechanism of the transition to the superconducting state. At

the same time thermal fluctuations in high-Tcmaterials lead

to many other remarkable phenomena, most notably vortex lattice melting and thermal depinning that are well studied both experimentally and theoretically over the last two de-cades, so that the theory of the Nernst effect should be con-sistent with the theory of these phenomena. Most impor-tantly, the material parameters determining the fluctuation strengths can be determined from these better studied effects since in many recent experiments at least the melting line was measured on the same samples.

Theory of the electronic and the heat transport共including the Nernst effect兲 based on the phenomenological time-dependent Ginzburg-Landau共TDGL兲 equations with thermal noise describing strongly fluctuating superconductors was developed long time ago.1,2 _{More recently within the same}

framework Ussishkin et al.3 _{calculated perturbatively the}

low-field Nernst effect for T⬎Tc due to contribution of

Gaussian fluctuations and obtained results in agreement with a microscopic Aslamazov-Larkin1calculation. Mukerjee and Huse5 _{numerically simulated the two-dimensional TDGL}

equation with Langevin thermal noise for T⬍Tc and

ob-tained results in reasonable agreement with experimental data on LaSCO共Ref.12兲 at lower temperature but the trans-verse thermoelectric conductivity became independent of magnetic field at higher temperatures in contrast to experi-ment. The simulation of this system, even in 2D, is difficult and it was one of our goals to supplement it with a reliable analytical expression in the region of the vortex liquid, namely, in the region above the melting line 共see Fig. 1兲 at which the vortex matter becomes homogeneous on a scale of several lattice spacings and the crystalline symmetry is lost. In this phase the pinning is ineffective and, unlike in the vortex glass phase, vortices actively promote the Nernst ef-fect. Recent understanding of the vortex matter phase dia-gram is summarized in Fig. 1. There are four phases sepa-rated by two transition lines:16 _{the first-order melting line}

共sometimes called the order-disorder line at lower tempera-tures, dashed line in Fig.1兲 and the irreversibility 共or glass兲 continuous transition. The melting line separates crystalline phases from homogeneous phases while the glass line 共dot-ted line in Fig.1兲 separates pinned phases from the unpinned ones. The mean-field H_c2共T兲 line 共solid line in Fig. 1兲 in strongly fluctuating superconductors becomes a crossover. Both pinning and crystalline order lead to a strong reduction in the Nernst signal and therefore these phases will not be considered here. We concentrate on the vortex-liquid phase

共dashed area in Fig. 1兲, and discuss the melting line and disorder only as limits of applicability of the theory and for determining the material parameters. The quantitative Ginzburg-Landau 共GL兲 theory of the vortex liquid has been developed recently and it was established that the Hartree-Fock approach for the thermodynamics is close to the con-vergent Borel-Pade one in the wide region of the vortex-liquid phase.17

In this paper we revisit the Hartree-Fock calculation in TDGL originally performed in Ref. 2 to obtain explicit ex-pressions for the transverse thermoelectric conductivity ␣xy

and the Nernst signal eNin both 2D and 3D. Typically only

the lowest Landau level 共LLL兲 contribution was

investigated.18 _{We extend it to higher Landau levels}

neces-sary for exploring the experimentally accessible parameter region and find a range of applicability of the results due to approximations made, disorder, and crystallization. In this theory the strength of the thermal fluctuations is described by just one dimensionless adjustable parameter ␩ 共closely re-lated to the Ginzburg number Gi兲. This parameter determines simultaneously the location of the melting line measured on the same samples in recent experiments on Nernst effect. The expression of Ref. 17for the melting line is in good agree-ment with many experiagree-ments in very wide range of materials 共as was established recently in Ref. 19兲 and Monte Carlo 共MC兲 simulation. Then fitting of the transverse thermoelec-tric conductivity and related quantities practically has no free parameters共of course there is a certain freedom in determin-ing mean-field parameters such as H_c2and Tcbut the range is

limited by experimental values兲. The value fitted from the Nernst effect turns out to be consistent with that derived from the melting line calculated in Ref.17. We will present the fitting of the melting line for the overdoped LaSCO,12

and underdoped and overdoped YBCO.12

The paper is organized as follows. The model is defined in Sec. II. The transverse thermoelectric conductivity in the vortex-liquid phase and extension to anisotropic 3D model are described in Sec. III. The comparison with experiment and MC simulation data is described in Sec.IV. We conclude in Sec.V. The Appendix calculates magnetization in the vor-tex liquid within the Gaussian approximation.

II. GINZBURG–LANDAU MODEL IN 2D A. Relaxation dynamics and thermal fluctuations

To describe fluctuation of order parameter in thin films or layered superconductors, one can start with the Ginzburg-Landau free energy:

F = s

冕

d2x ប 2 2mⴱ兩D␺兩 2_{+ a兩␺兩}2₊b

⬘

2兩␺兩 4_{, 共1兲}

where A =共−By,0兲 describes a constant and practically ho-mogeneous magnetic field 共we generally neglect small fluc-tuations of the magnetic field due to magnetization which are of order 1/␬2Ⰶ1 in the region of interest兲 in Landau gauge

and the covariant derivative is defined by D⬅ⵜ

−i共2␲/⌽0兲A, with ⌽0= hc/eⴱ, eⴱ= −2e⬎0. For simplicity

we assume linear dependence a共T兲=␣共T−T⌳_{兲 although the} FIG. 1. The thermodynamic phase diagram.

temperature dependence can be easily modified to better de-scribe the experimental coherence length. The “mean-field” critical temperature T⌳depends on the ultraviolet共UV兲 cut-off⌳ specified later. It is higher than measured critical tem-perature due to strong thermal fluctuations on the mesoscopic scale. The thickness of a layer, s, is assumed to be small enough so that order parameter does not vary con-siderably inside the layer 关namely, does not exceed the co-herence length␰z共T兲 along the field direction兴, and layers are

nearly independent. We apply this model to describe experi-ments not just in BiSCCO and other highly anisotropic ma-terials but also in overdoped LaSCO 共Ref.12兲 and strongly underdoped YBCO.12_{For more isotropic optimally doped or}

fully doped YBCO 共Ref. 12兲, an anisotropic 3D GL model 共neglecting the layered structure兲 would be more appropriate. For materials between the two extremes, a more complicated model such as the Lawrence-Doniach one should be used.

Since we are interested in transport phenomena, it is nec-essary to introduce some kind of dynamics for the order pa-rameter. The simplest is a gauge-invariant version of the “type A” relaxational dynamics,

␶

冉

⳵ ⳵t+ i eⴱ ប␾

冊

␺= − ␦F ␦␺ⴱ+␨, 共2兲

called in the present context TDGL equation. Explicitly the TDGL equation for the superconducting order parameter is

␶

冉

⳵ ⳵t+ i eⴱ ប␾

冊

␺= ប2 2mⴱD 2_{␺− a␺− b}

_⬘

_兩␺兩2_{␺+␨, 共3兲}

where␾共x兲 is the scalar potential describing electric field. To incorporate the thermal fluctuations via Langevin method, the noise term ␨共x,t兲 having Gaussian correlations,

s具␨ⴱ共x,t兲␨共x

⬘

,t

⬘

兲典 = 2T␶␦共x − x

⬘

兲␦共t − t

⬘

兲, 共4兲 is introduced. Here ␦共x−x

⬘

兲 is the two-dimensional␦ func-tion of the in-plane coordinates, and the relaxafunc-tion-time rate

␶in the TDGL equation is given by20

␶= ␲ប

3

16mⴱ␰2T. 共5兲

B. Heat and the electric transport

The total heat current density in GL model reads jh= − ប 2 2mⴱ

冓

冉

⳵ ⳵t− i eⴱ ប␾

冊

⌿ⴱ

冉

ⵜ− i 2␲ ⌽0 A

冊

⌿

冔

+ c.c., 共6兲 while the total electric current is

je= − ie ⴱ_ប 2mⴱ

冓

⌿ ⴱ

冉

_{ⵜ− i}2␲ ⌽0 A

冊

⌿

冔

+ c.c. 共7兲

An important aspect of the calculation of the electrothermal conductivity, discussed in detail,21_{is the need to account for}

the magnetization currents. In the presence of magnetic field, a system has the magnetization current in equilibrium. The total heat current defined in Eq. 共6兲 is thus a sum of the transport and the magnetization parts,

jh= j_trh+ j_magh . 共8兲

In the presence of an applied electric field, it was shown in Ref. 21that the magnetization current is given by

j_magh = cM⫻ E, 共9兲

where M is the equilibrium magnetization.

Generally, to define the transport coefficients, the electric and heat transport current densities, j共e兲and j共h兲, in metal are related to the applied共sufficiently weak兲 electric field and the temperature gradient by

j_tr共e兲i=␴ijEj−␣ijⵜjT, 共10兲 j_tr共h兲i=␣˜ij

Ej−␬ijⵜjT, 共11兲

where ␴, ␣, ␣˜ , and␬ are the electrical, the thermoelectric, the electrothermal, and the thermal-conductivity components of the conductivity tensor 共i, j=x,y兲. The Onsager relation implies ␣˜ = T␣. The Nernst coefficient ␯N, under the

condi-tion j_tr共e兲= 0, is expressed in terms of the above coefficients as

␯N= Ey 共− ⵜT兲xB = 1 B ␣xy␴xx−␣xx␴xy ␴xx 2 +␴xy 2 . 共12兲

If the system shows no significant Hall effect 共only such systems will be considered兲, then␴xy= 0 and the expression

simplifies:

␯N= ␣xy

B␴xx

. 共13兲

The Nernst signal is defined eN=

Ey

共− ⵜT兲x

= B␯N. 共14兲

For comparison with experiment, the fluctuation contribu-tion,␴xxand eN, should be added to the normal-state

contri-bution, ␴n and eN n

. However, in the normal state the Nernst signal eN

n

is very small in these materials3,14 _{and will be}

largely ignored in what follows.

It then follows that the electrothermal conductivity is given by ␣ ˜xy⬅ j_共tr兲xh Ey = jx h Ey + cMz. 共15兲

Both terms contribute as will be shown in the following sec-tions.

III. TRANSVERSE THERMOELECTRIC CONDUCTIVITY IN THE VORTEX-LIQUID PHASE

A. Melting of the vortex solid, vortex glass, and the range of validity of the Gaussian approximation

At low temperatures vortex matter organizes itself into a 共usually, but not always兲 hexagonal vortex lattice. When dis-order can be effectively neglected 共either in very clean ma-terials or when thermal depinning occurs兲, one can consider transport of the vortex lattice as a whole. Expressions for the

electric and the thermal conductivities near H_c2共T兲 that ne-glect thermal fluctuations were obtained in Ref. 2, and ac-cording to results the Nernst effect is generally very small compared to one in the vortex liquid. This can be qualita-tively understood as a result of rigidity of the lattice. Below the melting line the situation in this respect does not change much. Moreover due to unavoidable presence of disorder, the vortex lattice is pinned forming a Bragg glass in most of its domain.16_{However in high-T}

csuperconductors thermal

fluc-tuations are strong enough 共especially for high anisotropy and high magnetic fields兲 to destroy the expectation value of the condensate具␺典=0. We always assume that thermal fluc-tuations melted away and in addition temperature is high enough to thermally depin the vortex liquid 共avoiding the “vortex glass”兲. As a consequence impurities in the vortex liquid are neutralized. To determine the range of validity of the above assumptions, one has to estimate the location of the melting and the irreversibility lines. Within the LLL ap-proximation 共which is valid near melting in wide range of parameters17_{兲, the line separating the crystalline and the}

ho-mogeneous phases is given in 2D by aT

2D_{⬅ − 共2Gi}

2D兲−1/4共␲bt兲−1/2共1 − t − b兲 = − 13.6, 共16兲

where aT is the dimensionless “LLL scaled” temperature

with Gi2D⬅ 1 2

冉

8e2␬2␰2Tc ␲c2ប2s

冊

2 , 共17兲

being a 2D analog of the Ginzburg parameter characterizing the strength of thermal fluctuations on the mesoscopic scale. Scaled magnetic field is b = B/Hc2共0兲 with Hc2共0兲

=⌽₀/2␲␰2_{being the zero-temperature critical field}

共extrapo-lated by the linear formula from Tc, actual Hc2共T兲 at T=0 is

lower兲,␰=共ប2/2mⴱ␣Tc兲1/2being the zero-temperature

corre-lation length, and t = T/Tc. Equation 共16兲 determines the

melting line in Fig.1and in turn the melting line fixes the Gi in all the fits to experimental data below. This expression was obtained from the comparison of the calculated free en-ergies of the vortex lattice共expansion to two loop order兲 and of the vortex liquid within the Borel-Pade approach. The corresponding value and definition for 3D are

aT3D= − 21/3共Gi3D兲−1/3共␲bt兲−2/3共1 − t − b兲 = − 9.5, 共18兲 where Gi_3D⬅1 2

冉

8e2_␬2_␰T c␥ ␲c2_ប2

冊

2 , 共19兲

and␥⬅

冑

mc/mⴱ is an anisotropy parameter.

In the presence of disorder, vortex matter can be pinned. It leads to several phenomena. On the one hand the vortex lattice is destroyed effectively at large fields but on the other hand vortices are pinned and cannot take advantage of ther-mal fluctuations. The irreversibility or the vortex glass line determining the region in which thermal fluctuations over-power the quench by disorder is given in 2D by16,22

aT g_{⬅ 4}2r − 1

冑

2r , 共20兲 where r =Gi2D −1/2 4t 共1 − t兲 2_{n, 共21兲}

and dimensionless parameter n characterizes the disorder strength.17 _{This determines the dotted line in Fig.1.}

B. Vortex liquid within the Gaussian approximation

Due to thermal fluctuations the expectation value of the order parameter in vortex liquid is zero 具␺共x,t兲典=0. There-fore contribution to the expectation values of physical quan-tities such as the electric and the heat currents come exclu-sively from the correlations. The most important is the quadratic one,

C共x,t;x

⬘

,t

⬘

兲 = 具␺共x,t兲␺ⴱ共x

⬘

,t

⬘

兲典, 共22兲 called the correlation function of the order parameter. In par-ticular the superfluid density is

具兩␺共x,t兲兩2_{典 = C共x,t;x,t兲. 共23兲}

A simple approximation which captures the most interesting fluctuation effects in the Gaussian approximation 共see Ref. 23 for details兲, in which the cubic term in the GL equation Eq. 共3兲 b

⬘

兩␺兩2␺is replaced by a linear one 2b

⬘

具兩␺兩2典␺:

␶⳵

⳵t␺共x,t兲 =

冉

ប2

2mⴱD

2_{− a˜}

冊

_{␺共x,t兲 +␨共x,t兲, 共24兲}

leading the “renormalized” value of the coefficient: a

˜ = a + 2b

⬘

具兩␺兩2_{典. 共25兲}

The formal solution of this equation is

␺共x,t兲 =

冕

dx

⬘

冕

dt

⬘

G0共x,t;x

⬘

,t

⬘

兲␨共x

⬘

,t

⬘

兲, 共26兲

where G₀is the equilibrium Green’s function. In the Landau gauge, one has

G0共x,t;x

⬘

,t

⬘

兲 = 1 4␲2

冉

mⴱ␻B ប

冊

1/2

冕

␻,y˜0 G0共y˜,y˜

⬘

,␻,y˜0兲 ⫻e−i共mⴱ␻B/ប兲1/2˜y0共x−x⬘兲_ei␻共t−t⬘兲_, 共27兲

where ˜ =y 共mⴱ␻B/ប兲1/2y with ␻B= eⴱB/mⴱc, and ˜y0=

−共ប/mⴱ_␻

B兲1/2kx, with kx as the x component of the vector

momentum and G0共y˜,y˜

⬘

,␻,y˜0兲 =

冉

mⴱ␻B

ប␲

冊

1/2

exp关− 共y˜ − y˜0兲2/2 − 共y˜

⬘

− y˜₀兲2_/2兴

兺

n

1 2nn!

Hn共y˜ − y˜0兲Hn共y˜

⬘

− y˜0兲

共i␶␻+ En兲

, 共28兲 with the energy eigenvalues,

En=

冉

n +

1

2

冊

ប␻B+ a˜ , 共29兲

共Hnare the Hermite polynomials兲. Averaging over the noise,

Eqs. 共4兲 and 共26兲, the equilibrium correlation function 关Eq. 共22兲兴 is C0共x,t;x

⬘

,t

⬘

兲 = 2␶T s

冕

x₁,t1 G0共x,t;x1,t1兲G0ⴱ共x

⬘

,t

⬘

;x1,t1兲 = − T 2␲2s

冉

mⴱ␻B ប

冊

1/2

冕

␻,y˜0 ImG0共y˜,y˜

⬘

,␻,y˜0兲 ␻ ⫻e−i共mⴱ␻B/ប兲1/2˜y0共x−x⬘兲_ei␻共t−t⬘兲_, 共30兲

which enters the self-consistent equation 共sometimes called gap equation兲 关Eq. 共25兲兴, determining a˜. In equilibrium, 具兩␺共x,t兲兩2_{典 is} 具兩␺共x,t兲兩2_{典 =} T 2␲s mⴱ␻B ប

兺

n 1 En . 共31兲

Thus Eq. 共25兲 becomes

⑀b=˜⑀b− b

⬘

T ␲s mⴱ␻B ប共␣T⌳兲2

兺

n=0 N_f 1 ⑀ ˜b+ 2nb , 共32兲

where the reduced temperature is defined as ⑀= a/␣T⌳, ⑀b

=⑀+ b共with similar expression for˜ and⑀ ˜⑀b兲. The UV cutoff

⌳ was introduced. It effectively limits the number of Landau levels to Nf=⌳b− 1. The “bubble” sum, which diverges

loga-rithmically, can be performed as b ␲

兺

n=0 N_f 1 2nb +˜⑀b = 1 2␲log⌳ + u

⬘

, 共33兲

where the function u

⬘

is related by u

⬘

共˜⑀b,b兲 =

1

2␲关fs

⬘

共˜⑀b/2b兲 − log共2b兲兴, 共34兲 to the polygamma function fs

⬘

:

fs

⬘

共x兲 =

兺

n=1 ⬁

冋

1 n + x−

冕

_n−1/2 n+1/2 ₁ 共y + x兲dy

册

+

冋

1 x− log共x + 1/2兲

册

. 共35兲 Thus the critical temperature Tcis significantly renormalized:

⑀b r =⑀b+ b

⬘

T 2␲s mⴱ␻B ប共␣T⌳兲2log⌳ =˜⑀b−␩u

⬘

共˜⑀b,b兲, 共36兲 where␩is a dimensionless fluctuation parameter

␩= b

⬘

Tc

␰2_共␣T

c兲2s

, 共37兲

introduced in Ref.5. The relation between␩used to describe thermal transport and the more often used two-dimensional Ginzburg number1,17 _Gi

2D, see Eq.共17兲, is

␩= 4

冑

2Gi2D␲2. 共38兲

C. Expectation value of the heat current in linear response to electric field

Let us assume that the weak electric field E is along the y axis, generated by the scalar potential␾= −Eyy. The heat and

the electric currents in the vortex-liquid phase can be written as jh= − ប 2 2mⴱ

冋

D共x兲

冉

⳵ ⳵t

⬘

− i eⴱ ប␾共x

⬘

兲

冊

+ Dⴱ共x

⬘

兲

冉

⳵ ⳵t + ie ⴱ ប␾共x兲

冊

册

C共x,t;x

⬘

,t

⬘

兲兩x=x⬘;t=t⬘, 共39兲 je=

冋

បe ⴱ 2mⴱi共ⵜ− ⵜ

⬘

兲 − eⴱ2 2mⴱcA共x兲

册

C共x,t;x

⬘

,t

⬘

兲兩x=x⬘;t=t⬘, 共40兲 where C共x,t;x

⬘

,t

⬘

兲 =2␶T s

冕

x₁,t1 G共x,t;x1,t1兲Gⴱ共x

⬘

,t

⬘

;x1,t1兲, 共41兲 with G as the Green’s function of the linearized TDGL equa-tion in the presence of the scalar potential. One finds correc-tion to the Green’s funccorrec-tion to linear order in the electric field G共x,t;x

⬘

,t

⬘

兲 = G0共x,t;x

⬘

,t

⬘

兲 − i eⴱ␶ ប

冕

x₁,t1 ␾共x1兲 ⫻G0共x,t;x1,t1兲G0共x1,t1;x

⬘

,t

⬘

兲. 共42兲

The transverse thermoelectric conductivity is obtained by ex-panding the correlation function to linear order in the electric field. The correlation function C in terms of the Green’s function G0 using Eqs.共30兲, 共41兲, and 共42兲 takes the form

C共x,t;x

⬘

,t

⬘

兲 = C0共x,t;x

⬘

,t

⬘

兲 + C1共x,t;x

⬘

,t

⬘

兲, 共43兲 where C₁共x,t;x

⬘

,t

⬘

兲 = ie ⴱ_␶ ប

冕

x₁,t1 ␾共x1兲关G0ⴱ共x

⬘

,t

⬘

;x1,t1兲C0共x,t;x1,t1兲 − G0共x,t;x1,t1兲C0ⴱ共x

⬘

,t

⬘

;x1,t1兲兴 = ie ⴱ_␶TE y 2␲2_បs

冉

ប mⴱ␻B

冊

1/2

冕

d␻ ␻ ⫻

冕

y ˜₀,y˜₁ y ˜₁关G₀ⴱ共y˜

⬘

,y˜₁,␻,y˜₀兲

⫻ImG0共y˜,y˜1,␻,y˜0兲 − G0共y˜,y˜1,␻,y˜0兲

⫻ImG0共y˜

⬘

,y˜1,␻,y˜0兲兴

⫻e−i共mⴱ␻B/ប兲1/2˜y0共x−x⬘兲_ei␻共t−t⬘兲_. 共44兲

In order to determine the transverse thermoelectric conduc-tivity, we need to compute the x component of the heat cur-rent to the first order in the electric field. In the chosen gauge, the heat current along the x direction under condition

j_tre共x兲= 0 also contains two terms. The term coming from C0

vanishes,

j₀共h兲x=បT␻B 2␲2s

冕

_␻,y˜

0

共y˜ − y˜0兲ImG0共y˜,␻,y˜0兲 = 0, 共45兲

because ImG0共y˜,␻, y˜0兲 is an odd function of␻. It is possible

to interpret easily that C₀is the equilibrium correlation func-tion which does not contribute to the current. Considering C1, j₁共h兲x=2e ⴱ2_E y␶3BT ␲2_mⴱ_cs

兺

nm 1 2nn! 1 2mm!

冕

_−⬁ +⬁_d␻ 2␲␻ 2 ⫻ 1 共En2+␶2␻2兲 1 共Em2+␶2␻2兲

冕

−⬁ +⬁ dy˜₀

冕

−⬁ +⬁ dy˜₁˜y₁共y˜₀− y˜兲 ⫻exp关− 共y˜ − y˜0兲2−共y˜1− y˜0兲2兴Hn共y˜ − y˜0兲Hn共y˜1

− y˜₀兲Hm共y˜ − y˜0兲Hm共y˜1− y˜0兲

=e

ⴱ_E

yT共b −˜⑀b兲

2បbs 关u

⬘

共˜⑀b,b兲 − u

⬘

共˜⑀b+ b,b兲兴. 共46兲

In order to calculate the transport coefficient ␣xy using the

Onsager relation关Eq. 共15兲兴, we need the equilibrium magne-tization. This is calculated in the Appendix with the result given in Eq. 共A10兲. Together with Eq. 共46兲 one obtains the transverse thermoelectric conductivity

␣xy⬅ ␣ ˜xy T = eⴱ 2ប␲bs关␲共b −˜⑀b兲u

⬘

共˜⑀b,b兲 −␲共b −˜⑀b兲u

⬘

共˜⑀b + b,b兲 −⳵bu共˜⑀b,b兲 +␩tu

⬘

⳵bu

⬘

兴. 共47兲

Analogous calculation of the electrical conductivity␴yy= jye

Ey 共averaged over x兲 results in

␴yy= eⴱ2 16sប

兺

n 共n + 1兲

冉

1 2nb +˜⑀b + 1 2共n + 1兲b +˜⑀b − 2 2共n + 1/2兲b +˜⑀b

冊

= ␲e ⴱ2 32sបb2关共2b −˜⑀b兲u

⬘

共˜⑀b,b兲 −˜⑀bu

⬘

共˜⑀b+ 2b,b兲 − 2共b −˜⑀b兲u

⬘

共˜⑀b+ b,b兲兴. 共48兲

D. Extension to anisotropic 3D model

For 3D materials with asymmetry along the z axis, the GL model takes the form

F =

冕

d3x ប 2 2mⴱ兩D␺兩 2₊ ប 2 2mc 兩⳵z␺兩2+ a兩␺兩2+ b

⬘

2兩␺兩 4_{. 共49兲}

The TDGL equation for the superconducting order parameter in the Gaussian approximation is now

␶

冉

⳵ ⳵t+ i eⴱ ប␾

冊

␺=

冉

ប2 2mⴱD 2₊ ប 2 2mc ⳵z 2 − a˜

冊

␺+␨. 共50兲 The gap equation can be written as

⑀b r

=˜⑀b−␩3Dtu3D

⬘

, 共51兲

where u_3D

⬘

=⳵u_3D/⳵⑀˜b. The function u3Dcan be written in the

following form: u3D共˜⑀b,b兲 = 1

冑

2␲b 3/2_v

冉

˜⑀b 2b

冊

, 共52兲 with v共x兲 =

兺

n=0 ⬁

冋

冑

n + x −2 3

冉

x + n + 1 2

冊

3/2 +2 3

冉

x + n − 1 2

冊

3/2

册

−2 3

冉

x − 1 2

冊

3/2 . 共53兲

The dimensionless fluctuation parameter␩3Dis

␩3D=

b

⬘

Tc ␰2_共␣T

c兲2␰z

, 共54兲

with ␰z=共ប2/2mc␣Tc兲1/2 as the zero-temperature correlation

length along the field direction.

The relation between␩3Dused to describe thermal

trans-port in this case and three-dimensional Ginzburg number1,17

Gi3D, see Eq.共19兲, is

␩3D= 4

冑

2Gi3D␲2. 共55兲

The transverse thermoelectric conductivity is

␣xy= eⴱ 2␲ប␰zb 关␲u3D共˜⑀b,b兲 −␲u3D共˜⑀b+ b,b兲 +␲共b −˜⑀b兲u3D

⬘

共˜⑀b,b兲 −␲共b −˜⑀b兲u3D

⬘

共˜⑀b+ b,b兲 −⳵bu3D+␩3Dtu3D

⬘

⳵bu3D

⬘

兴, 共56兲

while the electrical conductivity

␴yy= ␲eⴱ2 32␰zបb2 关u3D共˜⑀b,b兲 + u3D共˜⑀b+ b,b兲 − 2u3D共˜⑀b+ b,b兲 + 2共b −˜⑀b兲u3D

⬘

共˜⑀b,b兲 − 2˜⑀bu3D

⬘

共˜⑀b+ b,b兲 − 2共b −˜⑀b兲u3D

⬘

共˜⑀b+ b,b兲兴. 共57兲

IV. COMPARISON WITH EXPERIMENT AND MC SIMULATION

Here we compare the results to 2D simulation results of Mukerjee and Huse5 _{and several recent experiments on}

high-Tccuprates.

A. Two-dimensional thermal fluctuations: LaSCO

The experiment results of Wang et al.12 _{were obtained}

from the Nernst effect and resistivity measurements on an overdoped LaSCO sample with x = 0.2 and Tc= 28 K. The

comparison is presented in Fig. 2 关low temperatures in 共a兲 and close to Tcin共b兲兴. The parameters used in the calculation

layer spacing s = 12 Å. The fluctuation parameter is ␩ = 0.18 and provides a reasonable quantitative agreement be-tween theory and experiment. Below the irreversibility line, where the theory should be modified, both pinning and crys-talline phase are included in Fig. 2共a兲. The deviation devel-ops roughly at the location of the irreversibility line. How-ever, our results are in good quantitative agreement with experimental data for temperature close to Tc in Fig. 2共b兲,

where the numerical simulation gives a nearly constant␣xy,

while the experiment shows more variation.

In Fig.3the melting line of overdoped LaSCO of Ref.12 is fitted using Gi_2D= 1.41⫻10−5_{, corresponding to␩⬵0.21}

which is consistent with the adjusted value of␩when we fit the transverse thermoelectric conductivity. The glass 共irre-versibility兲 line is estimated from Fig.2共a兲, where values of

␣xyare lower than simulation and experiment data.

B. Two-dimensional thermal fluctuations: underdoped YBCO

We also compared the results to the experiment on an underdoped YBCO sample with y = 6.5 and Tc= 50 K in Ref.

12. The parameters used in the calculation are Hc2共0兲

= 72 T共thus␰= 22.51 Å兲, s=9 Å, and the normal-state

con-ductivity ␴n= 7.14⫻105 共⍀m兲−1 in Ref.24. The fluctuation

parameter in this case is fitted to be␩= 0.51. Our values are in good quantitative agreement with experimental data for temperature close to Tcin Fig.4. We find that the theoretical

value of eN has a characteristic “tilted-hill” profile observed

in experiment.11,12,14 _{In Fig. 5 we present the fitting of the}

melting line for underdoped YBCO in Ref. 12 that gives Gi_2D= 1.15⫻10−4_{and␩⬵0.59, which is consistent with the}

adjusted value of␩ when we fit the Nernst signal eN.

C. Three-dimensional thermal fluctuations: overdoped YBCO

We also used the results calculated in three dimensions to compare to the experiment on an overdoped YBCO sample with y = 6.99 and Tc= 93 K in Ref.12. The parameters used

in the calculation are H_c2共0兲=350 T 共thus ␰= 9.70 Å兲, ␰z

= 1.4 Å, and the normal-state conductivity ␴n= 9.45

⫻105 _共⍀m兲−1 _{in Ref.24. The fluctuation parameter in this}

case is fitted to be ␩3D= 1.79. Our values are also in good

quantitative agreement with experimental data for tempera-ture close to Tcin Fig.6. In Fig.7we also present the fitting

FIG. 2. Points are ␣xy for different temperatures of LaSCO in

Ref.12, with x = 0.2共overdoped, T_c= 28 K兲. The dashed line is the simulation value of ␣xyin Ref. 5. The solid line is the theoretical

value of␣_xy, using H_c2共0兲=45 T, s=12 Å, and␩= 0.18.

FIG. 3. Comparison of the experimental melting line for over-doped LaSCO in Ref.12with our fitting.

FIG. 4. Points are eNfor different temperatures of YBCO in Ref.

12, with y = 6.5共underdoped, T_c= 50 K兲. The solid line is the the-oretical value of eN, using Hc2共0兲=72 T, s=9 Å, ␴n= 7.14

of the melting line for overdoped YBCO in Ref.12that gives Gi3D= 0.001 and ␩3D⬵2.09, which is also consistent with

the adjusted value of␩3Dwhen we fit the Nernst signal eN.

V. CONCLUSION

Time-dependent Ginzburg-Landau equations with thermal noise describing strong thermal fluctuations on the mesoscopic scale are used to describe strongly type-II super-conductor in the vortex-liquid regime both in 2D共describing strongly layered high-Tcsuperconductors兲 and 3D 共less

lay-ered superconductors such as optimally doped YBaCuO兲. Using GL theory developed earlier, we estimated the region in the parameter space in which, on one hand, vortex crystal is effectively destroyed by thermal fluctuations and, on the other hand, disorder 共significantly “weakened” by thermal fluctuations兲 is not strong enough to significantly affect the transport. Under these conditions we obtained explicit ex-pressions for the transverse thermoelectric conductivity ␣xy

and the Nernst signal eN including all Landau levels were

obtained using a Gaussian approximation. It is very similar

to the Hartree-Fock approximation utilized in Ref. 2but has a virtue of being a variational principle.

The results are presented using both the strength of the thermal fluctuation ␩, and the more often used Ginzburg number Gi in the 2D and 3D. The applicability region con-sidered coincides with domain on the phase diagram in which the signal is large. We compared the results to the available 2D numerical simulations of the same model and the experiments on high-Tcmaterials. Our results in 2D are

significantly lower than the available numerical simulation in Ref. 5 below the irreversibility line at which theory should be modified by including both pinning and crystalline corre-lation effects. However within the applicability region theory is in good qualitative and even quantitative agreement with experimental data on both La₂SrCuO₄ and underdoped YBa₂Cu₃O_6.5for temperatures close to Tc.

We also compared the values of eN calculated in three

dimensions with experiment data for temperature close to Tc

on YBa2Cu3O7, and this comparison is also in good

quanti-tative agreement. The Ginzburg numbers Gi were taken out from the fitting of melting lines of La2SrCuO4, YBa2Cu3O6.5,

and YBa2Cu3O7 on the same samples. The Ginzburg

num-bers Gi are consistent with the adjusted values of␩when we fit the transverse thermoelectric conductivity and the Nernst signal. The irreversibility line of La₂SrCuO₄ was fitted as well with the same set of parameters.

ACKNOWLEDGMENTS

We are grateful to A. Varlamov for enlightening discus-sions and encouragement, to Y. Wang for proving the experi-ment data, to S. Mukerjee for proving the simulation data, to our colleagues J. Y. Juang and C. W. Luo for the valuable discussions, and to A. T. Dorsey for the correspondence. This work was supported by NSC of R. O. C. Contract No. 952112M009048 and MOE ATU program.

APPENDIX: MAGNETIZATION IN THE VORTEX LIQUID WITHIN THE GAUSSIAN APPROXIMATION

In order to calculate magnetization, it is simpler to use the statistical mechanics rather than the 共equivalent兲

time-FIG. 5. Comparison of the experimental melting line for under-doped YBCO in Ref.12with our fitting.

FIG. 6. Points are eNfor different temperatures of YBCO in Ref.

12, with y = 6.99共overdoped, T_c= 93 K兲. The solid line is the the-oretical value of eN, using Hc2共0兲=350 T, ␰z= 1.4 Å, ␴n= 9.45

⫻105 _共⍀m兲−1_{, and␩} 3D= 1.79.

FIG. 7. Comparison of the experimental melting line for over-doped YBCO in Ref.12with our fitting.

dependent Langevin approach. We use the coherence length,

␰, as a unit of length, and Hc2=⌽0/2␲␰2 as a unit of

mag-netic field. After the order-parameter field is rescaled as⌿2

→共2␣Tc/b

⬘

兲␺2, the Boltzmann factor can be written as

f =F T= 2 ␩t

冕

d 2_{x关兩D␺兩}2_+共⑀ b− b兲兩␺兩2+兩␺兩4兴, 共A1兲

where the dimensionless covariant derivatives are D =ⵜ−iA. In the framework of the variational Gaussian approximation, the free energy关Eq. 共A1兲兴 is divided into an optimized qua-dratic part K, and a “small” part V. Then K is chosen in such a way that the energy of a Gaussian state is minimal.17 _In

liquid phase with an arbitrary homogeneous U共1兲 symmetric state, just one variational parameter˜⑀b is sufficient. Thus

K = 2

␩t

冕

d

2_x关␺ⴱ_{共− D}2_{− b +˜⑀}

b兲␺兴, 共A2兲

and the small perturbation becomes V = 2 ␩t

冕

d 2_x

冋

_共⑀ b−˜⑀b兲兩␺兩2+ 1 2兩␺兩 4

册

_{. 共A3兲}

The Gaussian energy which will be minimized therefore is fgauss⬅ − log

冋

冕

D␺D␺¯ exp共− K兲

册

+具V典K

= b ␲

兺

n=0 ⬁ log共2nb +˜⑀b兲 + 共⑀b−˜⑀b兲 b ␲n=0

兺

⬁ 1 2nb +˜⑀b +␩t 2

冉

b ␲

兺

n=0 ⬁ 1 2nb +˜⑀b

冊

2 , 共A4兲

where n is the Landau-level index. Both terms are ultraviolet divergent, namely, at large n the sums diverge. An UV mo-mentum cutoff was introduced for regularization as within the Langevin approach in Sec. II. To extract the divergent part, one can divide fgauss into an infinite part with⌳ and a

finite part, u:

f_gauss= 1

2␲关⌳共log ⌳ − 1兲 + 共˜⑀b− b兲log ⌳兴 + u共˜⑀b,b兲. 共A5兲 The finite part u can be simplified as

u共˜⑀b,b兲 =

b

␲fs共˜⑀b/2b兲 +

b

␲共1/2 −˜⑀b/2b兲log共2b兲, 共A6兲

where the function fs is defined as

fs共x兲 = log x − 共x + 1/2兲关log共x + 1/2兲 − 1兴 +

兺

n=1 ⬁

冋

log共n + x兲 −

冕

n−1/2 n+1/2

log共y + x兲dy

册

, 共A7兲

which is basically −ln⌫共x兲 plus a constant. The total free energy within Gaussian variational approximation for all Landau levels is therefore,

fgauss共˜⑀b兲 = 1 2␲⌳共log ⌳ − 1兲 − ␩t 2

冉

1 2␲log⌳

冊

2 +共⑀b r_{− b兲} ⫻

冉

1 2␲log⌳

冊

+共⑀b r − b兲u

⬘

+ u +␩t 2共u

⬘

兲 2_{. 共A8兲}

Minimizing the energy, we get the gap equation

⑀b r

=˜⑀b−␩tu

⬘

共˜⑀b,b兲, 共A9兲

consistent with the time-dependent approach关Eq. 共36兲兴. Magnetization 2D can be obtained by taking the first de-rivative of Gibbs energy with respect to magnetic field b:

M_2D= − Hc2

8␲␬2␩t⳵bfgauss= −

eⴱT

2␲បcs共⳵bu −␩tu

⬘

⳵bu

⬘

兲. 共A10兲 Similar calculation in 3D results in

M_3D= − e

ⴱ_T

2␲បc␰z

共⳵bu3D−␩3Dtu3D

⬘

⳵bu3D

⬘

兲. 共A11兲

1_{A. Larkin and A. Varlamov, Theory of Fluctuations in} Supercon-ductors共Clarendon, Oxford, 2005兲.

2_{S. Ullah and A. T. Dorsey, Phys. Rev. Lett. 65, 2066共1990兲; S.} Ullah and A. T. Dorsey, Phys. Rev. B 44, 262共1991兲. 3_{I. Ussishkin, S. L. Sondhi, and D. A. Huse, Phys. Rev. Lett. 89,}

287001共2002兲.

4_{I. Ussishkin, Phys. Rev. B 68, 024517共2003兲.}

5_{S. Mukerjee and D. A. Huse, Phys. Rev. B 70, 014506共2004兲.} 6_{S. Tan and K. Levin, Phys. Rev. B 69, 064510共2004兲.} 7_{T. T. M. Palstra, B. Batlogg, L. F. Schneemeyer, and J. V.}

Waszc-zak, Phys. Rev. Lett. 64, 3090共1990兲.

8_{J. A. Clayhold, A. W. Linnen, Jr., F. Chen, and C. W. Chu, Phys.} Rev. B 50, 4252共1994兲.

9_{C. Hohn, M. Galffy, and A. Freimuth, Phys. Rev. B 50, 15875} 共1994兲.

10_{Z. A. Xu, N. P. Ong, Y. Wang, T. Kakeshita, and S. Uchida,} Nature共London兲 406, 486 共2000兲.

11_{Y. Wang, Z. A. Xu, T. Kakeshita, S. Uchida, S. Ono, Y. Ando,} and N. P. Ong, Phys. Rev. B 64, 224519共2001兲.

12_{Y. Wang, N. P. Ong, Z. A. Xu, T. Kakeshita, S. Uchida, D. A.} Bonn, R. Liang, and W. N. Hardy, Phys. Rev. Lett. 88, 257003 共2002兲.

13_{C. Capan, K. Behnia, Z. Z Li, H. Raffy, and C. Marin, Phys.} Rev. B 67, 100507共R兲 共2003兲.

14_{Y. Wang, L. Li, and N. P. Ong, Phys. Rev. B 73, 024510共2006兲.} 15_{R. Bel, K. Behnia, and H. Berger, Phys. Rev. Lett. 91, 066602} 共2003兲; A. Pourret, H. Aubin, J. Lesueur, C. A. Marrache-Kikuchi, L. Berge, L. Dumoulin, and K. Behnia, Nat. Phys. 2, 683共2006兲; A. Pourret, H. Aubin, J. Lesueur, C. A. Marrache-Kikuchi, L. Berge, L. Dumoulin, and K. Behnia, Phys. Rev. B

76, 214504共2007兲.

16_{H. Beidenkopf, N. Avraham, Y. Myasoedov, H. Shtrikman, E.} Zeldov, B. Rosenstein, E. H. Brandt, and T. Tamegai, Phys. Rev. Lett. 95, 257004 共2005兲; D. P. Li, B. Rosenstein, and V. Vi-nokur, J. Supercond. Novel Magn. 19, 369共2006兲.

17_{D. Li and B. Rosenstein, Phys. Rev. B 60, 9704 共1999兲; 65,} 024513 共2001兲; 65, 220504 共2002兲; 70, 144521 共2004兲; Phys. Rev. Lett. 90, 167004共2003兲.

18_{D. J. Thouless, Phys. Rev. Lett. 34, 946 共1975兲; G. J. Ruggeri} and D. J. Thouless, J. Phys. F: Met. Phys. 6, 2063共1976兲; A. J. Bray, Phys. Rev. B 9, 4752共1974兲.

19_{N. Kokubo, K. Kadowaki, and K. Takita, Phys. Rev. Lett. 95,} 177005共2005兲; N. Kokubo, T. Asada, K. Kadowaki, K. Takita,

T. G. Sorop, and P. H. Kes, Phys. Rev. B 75, 184512共2007兲. 20_{W. E. Masker, S. Marcelja, and R. D. Parks, Phys. Rev. 188, 745}

共1969兲.

21_{N. R. Cooper, B. I. Halperin, and I. M. Ruzin, Phys. Rev. B 55,} 2344共1997兲.

22_{H. Beidenkopf, T. Verdene, Y. Myasoedov, H. Shtrikman, E.} Zeldov, B. Rosenstein, D. Li, and T. Tamegai, Phys. Rev. Lett.

98, 167004共2007兲.

23_{B. Rosenstein and V. Zhuravlev, Phys. Rev. B 76, 014507} 共2007兲.

24_{J. Y. Juang, M. C. Hsieh, C. W. Luo, T. M. Uen, K. H. Wu, and} Y. S. Gou, Physica C 329, 45共2000兲.