Theory of Nernst Effect in Layered Superconductors

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View the table of contents for this issue, or go to the journal homepage for more 2009 J. Phys.: Conf. Ser. 153 012030

Theory of Nernst Effect in Layered Superconductors

Department of Electrophysics, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China

E-mail: [email protected]

Abstract. We calculate, using the time-dependent Ginzburg-Landau (TDGL) equation with thermal noise, the transverse thermoelectric conductivity αxy, describing the Nernst effect, in type-II superconductor in the vortex-liquid regime. The method is an elaboration of the Hartree-Fock. An often made in analytical calculations additional assumption that only the lowest Landau level significantly contributes to αxy in the high field limit is lifted by including all the Landau levels. The resulting values in two dimensions (2D) are significantly lower than the numerical simulation data of the same model, but are in reasonably good quantitative agreement with experimental data on La2SrCuO4 above the irreversibility line (below the irreversibility

line at which αxy diverges and theory should be modified by including pinning effects).

1. Introduction

The electric field is induced in a metal under magnetic field by the temperature gradient ∇T perpendicular to the magnetic field H, phenomenon known as Nernst effect [1] (direction of the electric field being perpendicular to both ∇T and H). Recently the Nernst effect in high Tc superconductors attracted attention both theoretically [1, 2, 3, 4, 5, 6] and experimentally

[7, 8, 9, 10, 11, 12, 13, 14]. In these materials effect of thermal fluctuations is very strong leading to depinning of Abrikosov vortices created by the magnetic field in type II superconductor below second critical field H_c2(T ). 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 microscopic mechanism of superconductivity in cuprates.

In low critical temperature superconductors no sign of superconducting fluctuation was reported as the temperature was raised above Tc2(H) [15]. In sharp contrast, the appearance

of a fluctuation tail above the critical temperature in the Nernst signal was observed in several different high-temperature superconductors [8, 9, 10, 11, 14]. The related Ettignhausen effect was detected as well [7]. At the same time thermal fluctuations in high Tc materials lead to

many other remarkable phenomena, most notably vortex lattice melting and thermal depinning well studied both experimentally and theoretically over the last two decades, so that the theory of the Nernst effect should be consistent with the theory of these phenomena.

Theory of the electronic and heat transport (including the Nernst effect) starting from the phenomenological TDGL equation strongly fluctuating superconductors was developed long ago [1, 2]. More recently within the same framework Ussishkin et al. [3] calculated perturbatively the low-field the Nernst effect for T > Tc due to contribution of Gaussian fluctuations and

diagram is summarized in Fig. 1. There are four phases separated by two transition lines [16]: Bragg glass Glass Liquid Tc Hc2

Figure 1. The thermodynamic phase diagram

the first order melting line (sometimes called the order - disorder line at lower temperatures, dashed line in Fig. 1) and the irreversibility (or glass) continuous transition. The melting line separates crystalline phases from a homogeneous phases, while the glass line (dotted line in Fig. 1) separates pinned phases from the unpinned ones. The mean field Hc2(T ) line (solid

line in Fig. 1) in strongly fluctuating superconductors becomes a crossover. Both pinning and crystalline order lead to a strong reduction of the Nernst signal and will not be considered here. Therefore we will concentrate on the vortex liquid phase (dashed area in Fig. 1) and discuss the melting line and disorder only as limits of applicability of the theory. The quantitative theory of the vortex liquid have been developed recently and it was established that the Hartree-Fock approach for the thermodynamic is close to the convergent 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 done in Ref. [2] to obtain explicit expressions for the transverse thermoelectric conductivity αxy in 2D. Typically

only the lowest Landau level contribution was investigated. We extend it to higher Landau levels necessary for exploring the experimentally accessible parameter region and find 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 η (closely related to the Ginzburg number Gi). The value of the parameter is consistent with the melting line calculated in [18].

2. The Ginzburg-Landau Model in 2D 2.1. 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 Z d2x ¯h 2 2m∗|Dψ|2+ a|ψ|2+ b0 2|ψ| 4_{, (1)}

where A = (−By, 0) describes a constant and practically homogeneous magnetic field (we generally neglect small fluctuations of the magnetic field due to magnetization of order 1/κ2 _{<< 1 in the region of interest) in Landau gauge and covariant derivative is defined}

by D ≡ ∇ − i(2π/Φ0)A, with Φ0 = hc/e∗, e∗ = −2e > 0. For simplicity we assume

a(T ) = α(T − TΛ_{), although the temperature dependence can be easily modified to better}

describe the experimental coherence length. The “mean field” critical temperature TΛ _depends

on the ultraviolet (UV) cutoff Λ specified later. It is higher than measured critical temperature due to thermal fluctuations on the mesoscopic scale. The thickness of a layer is s. We apply this model to describe experiments on overdoped LaSCO [12].

Since we are interested in transport phenomena, it is necessary to introduce some kind of dynamics for the order parameter. The simplest is a gauge-invariant version of the “type A” relaxational dynamics, τ µ ∂ ∂t + i e∗ ¯hφ ¶ ψ = −δF δψ∗ + ζ, (2)

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

τ µ ∂ ∂t + i e∗ ¯hφ ¶ ψ = ¯h 2 2m∗D2ψ − aψ − b0|ψ|2ψ + ζ, (3)

where φ (x) is the scalar potential describing electric field. To incorporate the thermal fluctuations via Langeven method, the noise term ζ (x, t), having Gaussian correlations

shζ∗(x, t)ζ(x0, t0)i = 2T τ δ(x − x0)δ(t − t0), (4)

introduced. Here δ(x − x0_{) is the two-dimensional δ function of the in-plane coordinates.}

2.2. The heat and the electric transport

We start from the definition of the transport coefficients. Generally the electric and heat transport current densities, j(e) and j(h), in metal are related to the applied (sufficiently weak) electric field and the temperature gradient by

j_tr(e)i = σijEj− αij∇jT, (5)

j_tr(h)i=αeijEj− κij∇jT, (6) where, σ, α, α, and κ are the electrical, the thermoelectric, the electrothermal, and the thermale

conductivity components of the conductivity tensor (i, j = x, y). The Onsager relations implies

e

α = T α. The Nernst coefficient (ν_N), under the condition je

tr = 0 , is expressed in terms of the

above coefficients as ν_N = Ey (−∇T )xB = 1 B α_xyσ_xx− α_xxσ_xy σ2 xx+ σ2xy . (7)

currents by the total currents D jhE= − ¯h 2 2m∗ ¿µ ∂ ∂t − i e∗ ¯hφ ¶ ψ∗ µ ∇ − i2π Φ₀A ¶ ψ À + c.c. (9)

These assumptions were extensively discussed in a textbook [1] and [5].

3. The transverse thermoelectric conductivity in the vortex liquid phase

At low temperatures vortex matter organizes itself into a (usually, but not always) hexagonal vortex lattice. When disorder can be effectively neglected (either in very clean materials or when thermal depinning occurs), one can consider transport of the vortex lattice as a whole. Expressions for the electric and the thermal conductivities near Hc2(T ) neglecting thermal

fluctuations were obtained in [2], and according to results the Nernst effect is generally very small compared to one in the vortex liquid. This can be qualitatively 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 Tc superconductors thermal fluctuations are

strong enough (especially for high anisotropy and high magnetic fields) to destroy the expectation value of the condensate hψi = 0. We always assume that thermal fluctuations melted away and in addition temperature is high enough to thermally depin the vortex liquid (avoiding the “vortex glass”). As a consequence impurities in the vortex liquid are neutralized.

Due to thermal fluctuations the expectation value of the order parameter in vortex liquid is zero hψ(x, t)i = 0. Therefore contribution to the expectation values of physical quantities like the electric and the heat current come exclusively from the correlations. The most important is the quadratic one

C(x, t; x0, t0) =­ψ(x, t)ψ∗(x0, t0)®, (10) called the correlation function of the order parameter.

In particular the superfluid density is

h|ψ(x, t)|2i = C(x, t; x, t). (11)

A simple approximation which captures the most interesting fluctuations effects in the Hartree approximation, in which the cubic term in the GL equation Eq. (3) b0_|ψ|2_{ψ is replaced by a}

linear one b0­_|ψ|2®_ψ τ ∂ ∂tψ(x, t) = Ã ¯h2 2m∗D 2_−ea ! ψ(x, t) + ζ(x, t), (12)

leading the “renormalized” value of the coefficient:

e

a = a + b0h|ψ|2i. (13)

The formal solution of this equation is

ψ(x, t) = Z dx0 Z dt0G0(x, t; x0, t0)ζ(x0, t0), (14) 4

where G0 is the equilibrium Green function which is most easily accomplished by expanding G0

in terms of the Landau eigenfunctions. The evaluation of h|ψ(x, t)|2i gives

²_b =e²_b− b0T 2πs m∗_ω B ¯h (αTΛ₎2 Nf X n=0 1 e ²_b+ 2nb, (15)

where the reduced temperature is defined as ² = a/αTΛ_{, ²}

b = ² + b, (with similar expression

for e² and e²b) with b = B/Hc2(0) being the scaled magnetic field, Hc2(0) = Φ0/2πξ2 the

zero-temperature critical field and ξ = (¯h2/2m∗_αT

c)1/2 the zero-temperature coherence length. The

UV cutoff was introduced. It effectively limits the number of Landau levels to Nf = Λ_b − 1. The

“bubble” sum, which diverges logarithmically, can be performed:

b π Nf X n=0 1 2nb +e²b = 1 2πlog Λ + u 0_{, (16)}

where the function u0 _{is related by}

u0(e²_b, b) = 1 2π[f

0

s(e²b/2b) − log(2b)], (17)

to the polygamma function f0 s: f_s0(x) = ∞ X n=1 " 1 n + x − Z _n+1/2 n−1/2 1 (y + x)dy # + · 1 x − log (x + 1/2) ¸ . (18)

Thus the critical temperature Tc is significantly renormalized:

²r_b = ²b+ b 0_T 4πs m∗_ω b ¯h (αTΛ₎2 log Λ =e²b− ηξ2_{T e}∗_H c2(0) 2¯hcTc u 0_(e² b, b), (19)

where η is a dimensionless fluctuation parameter

η = b

0_T c

ξ2_(αTc)2_s, (20)

introduced in [5]. The relation between η and more often used two dimensional Ginzburg number [1, 17], Gi2D ≡ 1₂ ³ 8e2κ2ξ2Tc/πc2¯h2s ´₂ , is η = 4p2Gi2Dπ2. (21)

Let us assume that the weak electric field E is along the y axis, generated by the scalar potential φ = −E_yy. The heat current in the vortex liquid phase is given by [2]

D JhE= − ¯h 2 2m∗ · D (x) µ ∂ ∂t0 − i e∗ ¯hφ ¡ x0¢ ¶ + D∗¡x0¢ µ ∂ ∂t + i e∗ ¯hφ (x) ¶¸ C(x, t; x0, t0)|x=x0_;t=t0, (22) where C(x, t; x0, t0) = 2τ T s Z x1,t1 G(x, t; x1, t1)G∗(x0, t0; x1, t1), (23)

component of the heat current to first the electric field. In the chosen gauge, the electrothermal conductivityαexy = j

h x

Ey (averaged over x) takes a form

e α_xy = e∗T b ¯hπs Nf X n=0 · n + 1/2 2nb +e²b − n + 1 2(n + 1/2)b +e²b ¸ = e∗T (b −e²b) 2¯hbs h u0(e²_b, b) − u0(e²_b+ b, b)i, (25)

where function u0_{was defined in Eq. (17). Using the Onsager relation one obtains the transverse}

thermoelectric conductivity αxy =αexy/T . The vortex liquid energy gap e²b as a function of ²b,

and substituting the results into Eq. (25). Equations (19) and (25) agree with the calculation of Ullah and Dorsey [2].

4. Comparison with experiment and MC simulation

The experiment results of Y. Wang et al. [12] obtained from the Nernst effect and resistivity measurements on an overdoped LaSCO sample with x = 0.2 and Tc = 28K . The comparison is

presented in Fig. 2 (low temperatures in (a) and close to T_c in (b)). The parameters used in the calculation are (see definitions above) Hc2(0) = 45T (thus ξ = 27Ao) and layer spacing s = 16Ao.

The fluctuation parameter is η = 0.25 and provides a reasonable quantitative agreement between theory and experiment. Below irreversibility line where the theory should be modified including both pining and crystalline phase in Fig. 2(a). The deviation develops roughly at the location of the irreversibility line. However, our results are in good quantitative agreement with experiment data for temperature close to Tc in Fig. 2(b), where the numerical simulation gives a nearly

constant α_xy, while the experiment shows more variation.

10 15 20 25 30 0 5 10 15 20 6K 8K 10K 12K x y ( V / ( K m ) ) H(T) (a) 0 5 10 15 20 25 30 0 2 4 6 8 10 16K 20K 24K 28K x y ( V / ( K m ) ) H(T) (b)

Figure 2. Points are α_xy for different temperatures of LaSCO Ref. [12], with x=0.2 (overdoped, Tc = 28K). The dashed line is the simulation value of αxy Ref. [5]. The solid line is the

theoretical value of α_xy, using H_c2(0) = 45T, s = 16Ao_{, η = 0.25.}

5. Conclusion

We obtained, using TDGL equation with thermal noise, explicit expressions for the transverse thermoelectric conductivity αxy in 2D including all Landau levels in type-II superconductor

in the vortex-liquid regime. The method is the Hartree-Fock. We also obtained the relation between the strength of the thermal fluctuation η and the Ginzburg number Gi. We compared the results to the 2D simulation and the experiment results. Our results in 2D are significantly lower than the simulation and experiment data below the irreversibility line at which theory should be modified by including both pinning and crystalline effects, but are in reasonably good quantitative agreement with experimental data on La2SrCuO4 for temperature close to Tc. In

the future work, we will compare with other materials.

Acknowledgments

We are grateful to Y. Wang for proving experiment data, S. Mukerjee for proving simulation data, and Professor A.T. Dorsey for correspondence. This work was supported by NSC of Taiwan.

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