Electrical conductivity beyond a linear response in layered superconductors under a magnetic field

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Electrical conductivity beyond a linear response in layered superconductors

under a magnetic field

Bui Duc Tinh,1_{Dingping Li,}2 _{and Baruch Rosenstein}1,3

1_{Department of Electrophysics, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China} 2_{Department of Physics, Peking University, Beijing 100871, China}

3_{Applied Physics Department, Ariel University Center of Samaria, Ariel 40700, Israel} 共Received 29 December 2009; revised manuscript received 12 May 2010; published 29 June 2010兲 The time-dependent Ginzburg-Landau approach is used to investigate nonlinear response of a strongly type-II superconductor. The dissipation takes a form of the flux flow which is quantitatively studied beyond linear response. Thermal fluctuations, represented by the Langevin white noise, are assumed to be strong enough to melt the Abrikosov vortex lattice created by the magnetic field into a moving vortex liquid and marginalize the effects of the vortex pinning by inhomogeneities. The layered structure of the superconductor is accounted for by means of the Lawrence-Doniach model. The nonlinear interaction term in dynamics is treated within self-consistent Gaussian approximation and we go beyond the often used lowest Landau level approximation to treat arbitrary magnetic fields. The I-V curve is calculated for arbitrary temperature and the results are compared to experimental data on high-T_csuperconductor YBa₂Cu₃O_7−␦.

DOI:10.1103/PhysRevB.81.224521 PACS number共s兲: 74.40.Gh, 74.25.fc, 74.20.De

I. INTRODUCTION

Electric response of a high-temperature superconductor 共HTSC兲 under magnetic field has been a subject of extensive experimental and theoretical investigation for years. Mag-netic field in these layered strongly type-II superconductors create magnetic vortices, which, if not pinned by inhomoge-neities, move and let the electric field to penetrate the mixed state. The dynamic properties of fluxons appearing in the bulk of a sample are strongly affected by the combined effect of thermal fluctuations, anisotropy 共dimensionality兲 and the flux pinning.1_{Thermal fluctuations in these materials are far} from negligible and, in particular, are responsible for exis-tence of the first-order vortex lattice melting transition sepa-rating two thermodynamically distinct phases, the vortex solid and the vortex liquid. Magnetic field and reduced di-mensionality due to pronounced layered structure共especially in materials such as Bi2Sr2CaCuO8+␦兲 further enhance the effect of thermal fluctuations on the mesoscopic scale. On the other hand the role of pinning in high-Tc materials is

reduced significantly compared to the low-temperature one, leading to smaller critical currents. At elevated temperatures the thermal depinning1_{further diminishes effects of disorder.} Linear response to electric field in the mixed state of these superconductors has been thoroughly explored experimen-tally and theoretically over the last three decades. These ex-periments were performed at very small voltages in order to avoid effects of nonlinearity. Deviation from linearity, however, are interesting in their own right. These effects have also been studied in low-Tc superconductors

experimentally2,3 _{and theoretically}4,5 _{and recently} experi-ments were extended to HTSC compounds.6,7

Since thermal fluctuations in the low-Tcmaterials are

neg-ligible compared to the intervortex interactions, the moving vortex matter is expected to preserve a regular lattice struc-ture共for weak enough disorder兲. On the other hand, as men-tioned above, the vortex lattice melts in HTSC over large portions of their phase diagram so the moving vortex matter

in the region of vortex liquid can be better described as an irregular flowing vortex liquid. In particular the nonlinear effects will also be strongly influenced by the thermal fluc-tuations.

A simpler case of a zero or very small magnetic field in the case of strong thermal fluctuations was, in fact, compre-hensively studied theoretically8_{albeit in linear response only.} In any superconductor there exists a critical region around the critical temperature兩T−Tc兩ⰆGi·Tc, in which the

fluctua-tions are strong 共the Ginzburg number characterizing the strength of thermal fluctuations is just Gi⬃10−10– 10−7 for low Tc, while Gi⬃10−5– 10−1 for HTSC materials兲. Outside

the critical region and for small electric fields, the fluctuation conductivity was calculated by Aslamazov and Larkin9 _by considering 共noninteracting兲 Gaussian fluctuations within Bardeen-Cooper-Schrieffer 共BCS兲 and within a more phe-nomenological Ginzburg-Landau 共GL兲 approach. In the framework of the GL approach 共restricted to the lowest Landau-level approximation兲, Ullah and Dorsey10 _computed the Ettingshausen coefficient by using the Hartree approxi-mation. This approach was extended to other transport phe-nomena such as the Hall conductivity10 _{and the Nernst} effect.11

The fluctuation conductivity within linear response can be applied to describe sufficiently weak electric fields, which do not perturb the fluctuations’ spectrum.12_{Physically at electric} field, which is able to accelerate the paired electrons on a distance on the order of the coherence length␰, so that they change their energy by a value corresponding to the Cooper pair binding energy, the linear response is already inapplicable.8 _{The resulting additional field-dependent} de-pairing leads to deviation of the current-voltage characteris-tics from the Ohm’s law. The non-Ohmic fluctuation conduc-tivity was calculated for a layered superconductor in an arbitrary electric field considering the fluctuations as nonin-teracting Gaussian ones.13,14 _{The fluctuations’ suppression} effect of high electric fields in HTSC was investigated ex-perimentally for the in-plane paraconductivity in zero

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netic field,15–17 _{and a good agreement with the theoretical} models13,14 _{was found. Theoretically the nonlinear} fluctua-tion conductivity in HTSC has been treated by Puica and Lang.18 _{Below we compare their approach and results to} ours.

In this paper the nonlinear electric response of the moving vortex liquid in a layered superconductor under magnetic field perpendicular to the layers is studied using the time-dependent GL 共TDGL兲 approach. The layered structure is modeled via the Lawrence-Doniach discretization in the magnetic field direction. In the moving vortex liquid the long-range crystalline order is lost due to thermal fluctua-tions and the vortex matter becomes homogeneous on a scale above the average intervortex distances. Although sometimes motion tends to suppress the fluctuations, they are still a dominant factor in flux dynamics. The TDGL approach is an ideal tool to study a combined effect of the dissipative 共over-damped兲 flux motion and thermal fluctuations conveniently modeled by the Langevin white noise. The interaction term in dynamics is treated in self-consistent Gaussian approxima-tion which is similar in structure to the Hartree approximation.8,10,18,19

First the model of Ref. 18 is physically different from ours. The authors in Ref.18 believe that the two quantities, layer distance and thickness in the Lawrence-Doniach for HTSC are equal共apparently not the case in HTSC兲, while we consider them as two independent parameters. Another dif-ference is we use so-called self-consistent Gaussian approxi-mation to treat the model while Ref. 18 used the Hartree approximation.

A main contribution of our paper is an explicit form of the Green’s function共GF兲 incorporating all Landau levels. This allows to obtain explicit formulas without need to cutoff higher Landau levels. In Ref. 18, a nontrivial matrix inver-sion共of infinite matrices兲 or cutting off the number of Lan-dau levels is required. Note that the exact analytical expres-sion of Green’s function of the linearized TDGL equation in dc field can be even generalized also to ac field. The method is very general, and it allow us to study transport phenomena beyond linear response of type-II superconductor such as the Nernst effect and Hall effect. The renormalization of the models is also different from Ref.18. One of the main result of our work is that the conductivity formula is independent of ultraviolet共UV兲 cutoff 共unlike in Ref.18兲 as it should be as the standard 兩⌿兩4 theory is renormalizable. Furthermore self-consistent Gaussian approximation used in this paper is consistent to leading order with perturbation theory, see Ref. 20in which it is shown that this procedure preserved a cor-rect the UV renormalization共is renormalization group invari-ant兲. Without electric field the issue was comprehensively discussed in a textbook of Kleinert.20 _{One can use Hartree} procedure only when UV issues are unimportant. We can also show, if there is no electric field, the result obtained using TDGL model and self-consistent Gaussian approxima-tion will lead the same thermodynamic equaapproxima-tion using self-consistent Gaussian approximation.

The paper is organized as follows. The model is defined in Sec.II. The vortex liquid within the self-consistent Gaussian approximation is described in Sec.III. The I-V curve and the comparison with experiment are described in Sec.IVwhile Sec. Vcontains conclusions.

II. THERMAL FLUCTUATIONS IN THE TIME-DEPENDENT GL LAWRENCE-DONIACH MODEL

To describe fluctuation of order parameter in layered su-perconductors, one can start with the Lawrence-Doniach ex-pression of the GL free energy of the two-dimensional共2D兲 layers with a Josephson coupling between them

FGL= s

⬘

兺

n

冕

d2r

再

ប 2 2mⴱ兩D⌿n兩 2₊ ប 2 2mcd

⬘

2 兩⌿n−⌿n+1兩2 + a兩⌿n兩2+ b

⬘

2兩⌿n兩 4

冎

_, _共1兲

where s

⬘

is the order parameter effective “thickness” and d

⬘

distance between layers labeled by n. The Lawrence-Doniach model approximates paired electrons density of states by homogeneous infinitely thin planes separated by distance d

⬘

. While discussing thermal fluctuations, we have to introduce a finite thickness, otherwise the fluctuations will not allow the condensate to exist共Mermin-Wagner theorem兲. The thickness is of course smaller than the distance between the layers共otherwise we would not have layers兲. The order parameter is assumed to be nonzero within s

⬘

. Effective Coo-per pair mass in the ab plane is mⴱ共disregarding for simplic-ity the anisotropy between the crystallographic a and b axes兲 while along the c axis it is much larger mc. For simplicity we

assume a =␣Tc

mf_{共t−1兲, t⬅T/T} c mf

, although this temperature dependence can be easily modified to better describe the ex-perimental coherence length. The “mean-field” critical tem-perature Tc

mf

depends on UV cutoff,␶c, of the “mesoscopic”

or “phenomenological” GL description, specified later. This temperature is higher than measured critical temperature Tc

due to strong thermal fluctuations on the mesoscopic scale. The covariant derivatives are defined by D⬅ⵜ

+i共2␲/⌽0兲A, where the vector potential describes constant and homogeneous magnetic field A =共−By,0兲 and ⌽0 = hc/eⴱis the flux quantum with eⴱ= 2兩e兩. The two scales, the coherence length␰2₌_ប2_/共2mⴱ_␣_T

c兲, and the penetration depth

␭2_{= c}2_mⴱ_b

_⬘

_/共4_␲_eⴱ2_␣_T

c兲 define the GL ratio ␬⬅␭/␰, which

is very large for HTSC. In this case of strongly type-II su-perconductors the magnetization is by a factor ␬2 _smaller than the external field for magnetic field larger than the first critical field H_c1共T兲, so that we take B⬇H. The electric cur-rent, J = Jn+ Js, includes both the Ohmic normal part

Jn=␴nE 共2兲

and the supercurrent

Js=

ieⴱប

2mⴱ共⌿n ⴱ_D_⌿

n−⌿nD⌿nⴱ兲. 共3兲

Since we are interested in a transport phenomenon, it is nec-essary to introduce a dynamics of the order parameter. The simplest one is a gauge-invariant version of the “type A” relaxational dynamics.21 In the presence of thermal fluctua-tions, which on the mesoscopic scale are represented by a complex white noise,22 _{it reads}

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ប2_␥

_⬘

2mⴱD␶⌿n= − 1 s

⬘

␦FGL ␦⌿nⴱ +␨n, 共4兲

where D_␶⬅⳵/⳵␶− i共eⴱ/ប兲⌽ is the covariant time derivative with ⌽=−Ey being the scalar electric potential describing the driving force in a purely dissipative dynamics. The elec-tric field is therefore directed along the y axis and conse-quently the vortices are moving in the x direction. For mag-netic fields that are not too low, we assume that the electric field is also homogeneous.22 _{The inverse diffusion constant}

␥

⬘

/2, controlling the time scale of dynamical processes via dissipation, is real, although a small imaginary共Hall兲 part is also generally present.23 _{The variance of the thermal noise,} determining the temperature T is taken to be the Gaussian white noise

具␨nⴱ共r,␶兲␨m共r

⬘

,␶

⬘

兲典 =

_, _共6兲

where the covariant derivatives in dimensionless units in Landau gauge are Dx=_⳵x⳵ − iby, Dy=_⳵y⳵ with b = B/Hc2and the

order parameter field was rescaled: ⌿2=共2␣Tc

mf_/b

_⬘

_兲_␺2_{. The} dimensionless fluctuations’ strength coefficient is

␻=

冑2Gi

␲, 共7兲

where the Ginzburg number is defined by

Gi =1

2共8e 2_␬2_␰_T

c

mf_␥_/c2_ប2_兲2_. _共8兲 Note that here we use the standard definition of the Ginzburg number different from that in Ref.24. The relation between parameters of the two models, the Lawrence-Doniach and the three-dimensional anisotropic GL model, is d

⬘

= d␰c

= d␰/␥, s

⬘

= s␰c= s␰/␥, where␥2⬅mc/mⴱis an anisotropy

pa-rameter. In analogy to the coherence length and the penetra-tion depth, one can define a characteristic time scale. In the superconducting phase a typical “relaxation” time is ␶GL =␥

⬘

␰2_{/2. It is convenient to use the following unit of the} electric field and the dimensionless field: EGL= Hc2␰/c␶GL,

E=E/EGL. The TDGL Eq.共4兲 written in dimensionless units reads Hˆ␺n+ 1 2d2共2␺n−␺n+1−␺n−1兲 − 1 − t 2 ␺n+兩␺n兩 2_␺ n=␨n, Hˆ = D_␶−1 2D 2_, _共9兲

while the Gaussian white-noise correlation takes a form

thermal noise was rescaled as␨n=␨n共2␣Tc

mf_兲_3/2_/b

_⬘

_1/2

. The di-mensionless current density is Js= JGLjs, where

js=

i

2共␺n ⴱ_D_␺

n−␺nD␺nⴱ兲. 共11兲

with JGL= cHc2/共2␲␰␬2兲 being the unit of the current density.

Consistently the conductivity will be given in units of ␴GL = JGL/EGL= c2␥

⬘

/共4␲␬2兲. This unit is close to the normal-state conductivity␴nin dirty limit superconductors.25In

gen-eral there is a factor k of order one relating the two: ␴n

= k␴GL.

III. VORTEX LIQUID WITHIN THE SELF-CONSISTENT GAUSSIAN APPROXIMATION

A. Gap equation

Thermal fluctuations in vortex liquid frustrate the phase of the order parameter, so that具␺n共r,␶兲典=0. Therefore the

con-tributions to the expectation values of physical quantities like the electric current come exclusively from the correlations, the most important being the quadratic one 具␺n共r,␶兲␺nⴱ共r

⬘

,␶

⬘

兲典. In particular, 具兩␺n共r,␶兲兩2典 is the

super-fluid density. A simple approximation which captures the most interesting fluctuations effects in the self-consistent Gaussian approximation, in which the cubic term in the TDGL Eq.共9兲, 兩␺n兩2␺n, is replaced by a linear one 2具兩␺n兩2典␺n

冉

Hˆ −b

2

冊

␺n+ 1

2d2共2␺n−␺n+1−␺n−1兲 + ␧␺n=␨n, 共12兲

leading the “renormalized” value of the coefficient of the linear term

␧ = − ah+ 2具兩␺n兩2典, 共13兲

where the constant is defined as ah=共1−t−b兲/2. The average

具兩␺n兩2典 is expressed via the parameter ␧ below and will be

determined self-consistently together with ␧. It differs slightly from a well-known Hartree procedure in which the coefficient of the linearized term is generally different 共see Refs. 20and22and Appendix C for details兲.

Due to the discrete translation invariance in the field di-rection z, it is convenient to work with the Fourier transform with respect to the layer index

␺n共r,␶兲 =

冕

0 2␲/d_dk z 2␲e −inkzd␺ k_z共r,␶兲, ␺k_z共r,␶兲 = d

兺

n einkzd␺ n共r,␶兲, 共14兲

and similar transformation for¯. In terms of Fourier compo-␨ nents the TDGL Eq. 共12兲 becomes

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再

Hˆ −b

2+ 1

d2关1 − cos共kzd兲兴 + ␧

冎

␺k_z共r,␶兲 =␨k_z共r,␶兲.

册

e −共2␧−b+v2_兲␶ e−2␶/d2I0共2␶/d2兲. 共20兲 Here I0共x兲=共1/2␲兲兰₀2␲ex cos␪_d_␪ _{is the modified Bessel}

func-tion. The first pair of multipliers in Eq.共20兲 is independent of the interplane distance d and exponentially decreases for ␶ ⬎共2␧−b+v2_兲−1_{while the last pair of multipliers depends on} the layered structure. The expression in Eq.共19兲 is divergent at small␶, so an UV cutoff␶cis necessary for regularization.

Substituting the expectation value into the “gap equation,” Eq. 共13兲, the later takes a form

␧ = − ah+ ␻tb ␲s

冕

_␶=␶ c ⬁ _f_共␧,_␶_兲 sinh共b␶兲. 共21兲 B. Renormalization

In order to absorb the divergence into a renormalized value ah

r

of the coefficient ah, it is convenient to make an

integration by parts in the last term for small␶c

b

冕

␶=␶c ⬁ _f_共_␶_兲 sinh共b␶兲 ⯝ −

冕

0 ⬁ ln关sinh共b␶兲兴 d d␶

冋

f共␧,␶兲 cosh共b␶兲

册

− ln共b␶c兲. 共22兲 Physically the renormalization corresponds to reduction in the critical temperature by the thermal fluctuations from Tc

mf

to Tc. The thermal fluctuations occur on the mesoscopic

scale. The critical temperature Tc is defined at ␧=0, and ␷

= 0, and at low magnetic field less than Hc1= Hc2

2␬2ln共␬兲共for a typical high-Tcsuperconductor, ␬⯝50, Hc1= 7.8⫻10−4Hc2兲,

the superconductor is at Meissner phase, b = 0, leading to

Tc= Tc mf

再

1 +2␻ ␲s关ln共␶c/d 2_{兲 +}_␥ E兴

冎

, 共23兲

where ␥E= 0.577 is Euler constant, and Eq. 共21兲 can be

re-written as ␧ = − ah r − ␻t ␲s

冕

₀ ⬁ ln关sinh共b␶兲兴d d␶

冋

f共␧,␶兲 cosh共b␶兲

册

+ ␻t ␲s兵␥E− ln共bd 2_兲其, _共24兲 where ah r =1−b−T/Tc

2 , t = T/Tc, and ␻=

冑

2Gi␲, where Gi

=1₂共8e2␬2␰Tc␥/c2ប2兲2 共Tc mf

is now replaced by Tc兲. The

for-mula is cutoff independent. In terms of energy UV cutoff⌳, introduced, for example, in Ref.11, the cutoff “time”␶ccan

be expressed as

␶c= 1/共2e␥E⌳兲. 共25兲

This is obtained by comparing a thermodynamic result for a physical quantity like superfluid density with the dynamic result 共see Appendix B兲. The temporary UV cutoff used is completely equivalent to the standard energy or momentum cutoff Lambda used in thermodynamics 共in which the time dependence does not appear兲. Physically one might think about momentum cutoff as more basic and this would be universal and independent of particular time-dependent real-ization of thermal fluctuations 共TDGL with white noise in our case兲. Roughly 共in physical units兲 ⌳⯝␧F=ប2kF

2_/共2mⴱ_兲. In the next section we will discuss the estimate of Tc

mf

using this value due to the following reason. For high-Tcmaterials

ordinary BCS is invalid and coherence length is of order of lattice spacing 共the cutoff becomes microscopic兲 and there-fore the energy cutoff is of order ␧F. Except the formula to

calculate Tc mf

, all other formulas in this paper is independent of energy cutoff.

IV. I-V CURVE A. Current density

The supercurrent density, defined by Eq. 共11兲, can be ex-pressed via the Green’s functions as

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兲 + c.c. 共26兲 Performing the integrals, one obtains

jy s = ␻t 4␲s␯

冕

_␶=0 ⬁ _f_共␧,_␶_兲 cosh

冉

b␶ 2

冊

2, 共27兲

where the function f was defined in Eq.共20兲. Consequently the contribution to the conductivity is␴¯s_{= j}

y

s_{/E. The}

conduc-tivity expression关Eq. 共27兲兴 is not divergent when expressed as a function of renormalized Tc共the real transition

tempera-ture兲, so it is independent of the cutoff. This is considered in detail in Sec. III Band is indeed different from the Ref.18. In physical units the current density reads

Jy=␴nE

冤

1 + ␻t 4␲s 1 k

1 2+ ␧ b

冊

册

冎

, 共29兲

where␺is the polygamma function.

B. Comparison with experiment

In this section we use physical units while the dimension-less quantities are denoted with bars. The experiment results of Puica et al.,7_{obtained from the resistivity and Hall effect} measurements on an optimally doped YBa2Cu3O7−␦共YBCO兲 films of thickness 50 nm and Tc= 86.8 K. The distance

be-tween the bilayers used the calculation is d

⬘

= 11.68 Å in Ref. 26. The number of bilayers is 50, large enough to be described by the Lawrence-Doniach model without taking care of boundary conditions. In order to compare the fluctua-tion conductivity with experimental data in HTSC, one can-not use the expression of relaxation time ␥

⬘

in BCS which may be suitable for low-Tc superconductor. Instead of this,

we use the factor k as fitting parameter. The comparison is presented in Fig.1. The resistivity

␳= 1 ␴s+␴n , 共30兲 ␴s= ␴n k␴¯s 共31兲

curves were fitted to Eq.共30兲 with the normal-state conduc-tivity measured in Ref.7to be␴n= 1.9⫻104 共⍀ cm兲−1. The

parameters we obtain from the fit are: Hc2共0兲

= TcdHc2共T兲/dT兩T_c= 190 T 共corresponding to ␰= 13.2 Å兲,

the Ginzburg-Landau parameter␬= 45.6, the order parameter effective thickness s

⬘

= 8.5 Å, and the factor k =␴n/␴GL = 0.55, where we take ␥= 7.8 for optimally doped YBCO in Ref. 27. Using those parameters, we obtain Gi = 1.12⫻10−3 共corresponding to ␻= 0.148兲. The order parameter effective thickness s

⬘

can be taken to be equal to the layer distance 共see in Ref. 28兲 of the superconducting CuO plane plus the coherence length 2␰c= 2␰_␥ due to the proximity effect:

3.18 Å + 213.2_7.8 Å = 6.9 Å, roughly in agreement in magni-tude with the fitting value of s

⬘

.

We will now estimate Tc mf

for this sample. For the under-doped YBCO, the radius of the Fermi surface of YBCO was measured in Ref.29, kF= 0.7 Å−1, while the effective mass is

mⴱ= 1.9me. We will assume that the Fermi energy for

under-doped YBCO of Ref. 29 is ␧F=ប2kF2/共2mⴱ兲 and is roughly

the same for the optimal YBCO studied in this paper. The cutoff time in physical units is then, according to Eq. 共25兲,

␶c= 1.39⫻10−17 s. Equation共23兲 gives then Tc mf

= 101.15 K. Using the parameters specified above we plot several theo-retical I-V curves. As expected the I-V curve shown in Figs. 2 and 3 has two linear portions, the flux flow part for E ⰆEGLand the normal Ohmic part for EⰇEGL. In the cross-over region, E⬃EGL, a I-V curve becomes nonlinear due to destruction of superconductivity 共the normal area inside the vortex cores increases to fill all the space兲. In Fig.2the I-V curves are shown for different the magnetic fields, at a fixed temperature T = 0.75Tc. At given electric field, as the

mag-netic field increases, the supercurrent decreases. When the magnetic field reaches H_c2, the I-V curve becomes linear. In Fig.3the I-V curves are shown for different temperatures, at a fixed magnetic field H = 0.5Hc2. At given electric field, as

the temperature increases, the supercurrent decreases. When the temperature reaches Tc, the I-V curve becomes linear.

With decreasing temperature the crossover becomes steeper.

V. DISCUSSION AND CONCLUSION

We quantitatively studied the transport in a layered type-II superconductor in magnetic field in the presence of strong FIG. 1. Points are resistivity for different electric fields of an optimally doped YBCO in Ref. 7. The solid line is the theoretical value of resistivity calculated from Eq.共30兲 with fitting parameters 共see text兲. The dashed line is the theoretical value of resistivity in linear response with the same parameters.

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thermal fluctuations on the mesoscopic scale beyond the lear response. While in the normal state the dissipation in-volves unpaired electrons, in the mixed phase it takes a form of the flux flow. Time-dependent Ginzburg-Landau equations with thermal noise describing the thermal fluctuations are used to describe the vortex-liquid regime and arbitrary flux flow velocities. We avoid assuming the lowest Landau-level approximation, so that the approach is valid for arbitrary values the magnetic field not too close to Hc1共T兲.

Our main objective is to study layered high-Tc materials

for which the Ginzburg number characterizing the strength of thermal fluctuations is exceptionally high, in the moving vor-tex matter the crystalline order is lost and it becomes homo-geneous on a scale above the average intervortex distances. This ceases to be the case at very low temperature at which two additional factors make the calculation invalid. One is the validity of the GL approach 关strictly speaking not far from Tc共H兲兴 and another is effect of quenched disorder. The

later becomes insignificant at elevated temperature due to a very effective thermal depinning. Although sometimes mo-tion tends to suppress fluctuamo-tions, they are still a dominant factor in flux dynamics. The nonlinear term in dynamics is treated using the renormalized self-consistent Gaussian

ap-proximation. The renormalization of the critical temperature is calculated and is strong in layered high-Tcmaterials. The

results were compared to the experimental data on HTSC. Our resistivity results are in good qualitative and even quan-titative agreement with experimental data on YBa₂Cu₃O₇₋_␦ in strong electric fields.

Let us compare the present approach with a widely used Londons’ approximation. Since we have not neglected higher Landau levels, as very often is done in similar studies,1,10_our results should be applicable even for relatively small fields in which the London approximation is valid and used. There is no contradiction since the two approximations have a very large overlap of applicability regions for strongly type-II su-perconductors. The GL approach for the constant magnetic induction works for HⰇH_c1共T兲 while the Londons’ approach works for HⰆHc2共T兲. Similar methods can be applied to

other electric-transport phenomena like the Hall conductivity and thermal-transport phenomena like the Nernst effect. The results, at least in 2D, can be, in principle, compared to nu-merical simulations of Langevin dynamics. Efforts in this direction are under way.

ACKNOWLEDGMENTS

兲 is the Heaviside step function, C and ␤are coefficients.

Substituting the Ansatz Eq. 共A1兲 into Eq. 共18兲, one ob-tains following conditions condition:

␧ −b 2+ ␯2 2 + 1 d2关1 − cos共kzd兲兴 + 1 ␤+ ⳵␶C C = 0, 共A3兲

FIG. 2. The current-voltage curves calculated from Eq. 共28兲 by using the parameters 共see text兲 for different magnetic fields

b = B/H_c2: 0.04 共1兲, 0.1 共2兲, 0.4 共3兲, and 1.0 共4兲 at temperature

t = 0.75.

FIG. 3. The current-voltage curves calculated from Eq.共28兲 by using the parameters共see text兲 for different temperatures t=T/Tc: 0.2共1兲, 0.3 共2兲, 0.4 共3兲, and 1.0 共4兲 at magnetic flied b=0.5.

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⳵␶␤ ␤2 − 1 ␤2+ b2 4 = 0. 共A4兲

The Eq. 共A4兲 determines ␤, subject to an initial condition

␤共0兲=0,

␤=2

btanh共b␶

⬙

/2兲共A5兲

while Eq.共A3兲 determines C

C = b 4␲exp

再

−

冉

␧ − b 2 + v2 2 + 1 d2关1 − cos共kzd兲兴

冊

␶

⬙

冎

⫻ sinh−1

冉

b␶

⬙

2

冊

. 共A6兲

The normalization is dictated by the delta function term in definition of the Green’s function Eq.共18兲.

APPENDIX B: COMPARISON WITH THERMODYNAMICS

From TDGL, we obtain in the case␷= 0 具兩␺n共r,␶兲兩2典 = ␻tb 2␲s

冕

_␶=␶ c ⬁ exp

再

−

冉

2␧ − b + 2 d2

冊

␶

冎

I0

冉

2␶ d2

冊

sinh共b␶兲 . 共B1兲 The superfluid density at b = 0 and␧=0 can be obtained by taking b and␧ to zero limit in the above equation

具兩␺n共r,␶兲兩2典 = ␻t 2␲s

冕

_␶=␶ c/d2 ⬁ _exp_{兵− 2}_␶_其I0共2_␶_兲 ␶ . 共B2兲

Performing the integration by parts, one obtains 具兩␺n共r,␶兲兩2典 ⯝ −

␻t

2␲s兵ln共␶c/d

2_{兲 +}_␥

E其 + O共␶c兲. 共B3兲

In the case without external electric field 共or v=0兲, the equation obtained from TDGL shall approach the thermody-namics result. In thermodythermody-namics method, we shall evaluate the partition function Z =兰D␺nD␺nⴱe−FGL/T, where FGL/T is defined in Eq. 共6兲. The superfluid density in the thermody-namic approach at the phase transition point

具兩␺n共r,␶兲兩2典 = ␻td 共2␲兲3_s

冕

0 k_max dk

冕

0 2␲/d dkz 1 k2 2 + 1 − cos共kzd兲 d2 ⯝ ␻t 2␲s兵ln ⌳ + ln共2d 2_{兲其 + O共⌳}−1_兲, _共B4兲 where⌳=k_max2 /2.

The relation between the cut-off time ␶c and energy UV

cutoff ⌳ is obtained by comparing Eq. 共B3兲 with Eq. 共B4兲

␶c⯝

1

2⌳e␥E. 共B5兲

We also remark that in thermodynamic approach, if we use the self-consistent Gaussian approximation, we will get the exact same equation derived in Eq.共24兲 without electric field derived from TDGL after using Eq.共B5兲.

APPENDIX C: COMPARISON WITH THE HARTREE APPROACH

Here we explain the difference using an example of ther-modynamics. The dynamics is not different since it always can be cast in the Martin-Siggia-Rose form共see Ref. 22兲.

By using the Hartree approximation, one substitute兩␺兩4by 2具兩␺兩2_典兩_␺_兩2 _{in the GL free energy Eq.} _{共6兲 leading the} renor-malized value of the coefficient of the linear term in the TDGL Eq.共12兲

␧ = − ah+具兩␺n兩2典. 共C1兲

In the framework of the variational Gaussian approxima-tion, the GL free-energy Eq.共6兲 is divided into an optimized quadratic part K, and a “small” part V. Then K is chosen in such a way that the energy of a Gaussian state is minimal.1_In liquid phase with an arbitrary homogeneous U共1兲 symmetric state, just one variational parameter ␧ is sufficient. Thus

K = s ␻t

兺

n

冕

d2r

冋

␺nⴱ

冉

− 1 2D 2₋b 2+␧

冊

␺n

册

共C2兲 and the small perturbation becomes

V = s ␻t

兺

n

冕

d2_r

冋

_{共− a} h−␧兲兩␺n兩2+ 1 2兩␺n兩 4

册

_. _共C3兲 The eigenvalue of Nth Landau level is

−1 2D 2_␸ n=

冉

N + 1 2

冊

b␸n. 共C4兲

The Gaussian energy which will be minimized therefore is

fgauss⬅ − log

冋

冕

D␺nD␺¯nexp共− K兲

册

+具V典K, 共C5兲

where

具V典K=

兺

n

关共− ah−␧兲具兩␺n兩2典 + 具兩␺n兩2典具兩␺n兩2典兴. 共C6兲

Minimizing the Gaussian energy with respect to␧, we get the gap equation

␧ = − ah+ 2具兩␺n兩2典. 共C7兲

While the Hartree method is generally simpler, the Gauss-ian method applied in its consistent form conserves Ward identities共electric current兲 and its effective energy is positive definite. In addition it has the correct “large number of com-ponents” limit, unlike Hartree method.

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