J Supercond Nov Magn (2014) 27:359–363DOI 10.1007/s10948-013-2316-z

O R I G I NA L PA P E R

Fluctuation Hall Conductivity Beyond Linear Responsein Layered Superconductors Under a Magnetic Field

Bui Duc Tinh · Le Minh Thu · Le Viet Hoa ·Nguyen Manh Cuong

Received: 5 June 2013 / Accepted: 15 July 2013 / Published online: 14 August 2013© Springer Science+Business Media New York 2013

Abstract We calculate fluctuation Hall conductivity of astrongly type-II superconductor in strong electric fields byusing the time dependent Ginzburg–Landau approach. Ther-mal fluctuations, represented by the Langevin white noise,are assumed to be strong enough to melt the Abrikosov vor-tex lattice created by the magnetic field into a moving vor-tex liquid. The layered structure of the superconductor isaccounted for by means of the Lawrence–Doniach model.The nonlinear interaction term in dynamics is treated withinself-consistent Gaussian approximation and we go beyondthe often used lowest Landau level approximation to treatarbitrary magnetic fields. The results are compared to ex-perimental data on high-Tc superconductor YBa2Cu3O7−δ .

Keywords Time-dependent Ginzburg–Landau · Hallconductivity · Type-II superconductor

1 Introduction

The influence of superconducting fluctuations on off-diago-nal components of the magnetoconductivity tensor (usuallydenoted as the excess Hall effect) in high temperature su-perconductor (HTSC) has received considerable experimen-tal and theoretical attention over the past few years [1–5].Though a general consensus seems to be achieved now re-garding the existence and the temperature dependence of

B.D. Tinh (B) · L.M. Thu · L.V. HoaDepartment of Physics, Hanoi National University of Education,136 Xuanthuy, Caugiay, Hanoi, Vietname-mail: tinhbd@hnue.edu.vn

N.M. CuongInstitute of Physics, Vietnam Academy of Scienceand Technology, 10 Dao Tan, Ba Dinh, Hanoi, Vietnam

the excess Hall effect, theoretical predictions of its sign arestill controversial. Several theoretical approaches have at-tempted to explain the complex features of the Hall resis-tivity temperature dependence, but no consensus has beenachieved. The Hall anomaly might originate from the pin-ning force [6], nonuniform carrier density in the vortexcore [7, 8], or can be calculated in the time dependentGinzburg–Landau (TDGL) model [9, 10].

The non-Ohmic fluctuation conductivity was calculatedfor a layered superconductor in an arbitrary electric fieldconsidering the fluctuations as noninteracting Gaussianones [11–14]. Physically at electric field, which is able toaccelerate the paired electrons on a distance of the orderof the coherence length ξ so that they change their energyby a value corresponding to the Cooper pair binding en-ergy, the linear response is already inapplicable [15]. Theresulting additional field dependent depairing leads to devi-ation of the current–voltage characteristics from the Ohm’slaw. The fluctuations’ suppression effect of high electricfields in HTSC was investigated experimentally for the in-plane paraconductivity in zero magnetic field [16–18], anda good agreement with the theoretical models [13, 14] wasfound. Theoretically, the nonlinear fluctuation conductivityin HTSC has been treated [19] and a comparison with re-cently experimental data was done well in [20].

The nonlinearity of the off-diagonal (Hall) componentsof the fluctuation magnetoconductivity tensor in a high elec-tric field was treated in [21]. However, the authors in [21]believe that the two quantities, layer distance and thicknessin the Lawrence–Doniach for HTSC are equal (apparentlynot the case in HTSC), and a nontrivial matrix inversion (ofinfinite matrices) or cutting off the number of Landau levelsis required. Recently, the fluctuations’ suppression effect ofhigh electric fields in HTSC was investigated experimentallyfor the Hall conducitivity in a magnetic field [5].

360 J Supercond Nov Magn (2014) 27:359–363

In this paper, fluctuation Hall conductivity for a lay-ered superconductor in vortex lipuid phase in strong electricfields is calculated by using the time dependent Ginzburg–Landau (TDGL) approach. The layered structure is mod-eled via the Lawrence–Doniach discretization in the mag-netic field direction. We consider layer distance and thick-ness as two independent parameters. The interaction term inTDGL is treated in self-consistent Gaussian approximationwhile [21] used the Hartree approximation. Self-consistentGaussian approximation used in this paper is consistent toleading order with perturbation theory; see [22, 23] in whichit is shown that this procedure preserved a correct ultravio-let (UV) renormalization (is RG invariant). One can use theHartree procedure only when UV issues are unimportant.We can also show if there is no electric field, the result ob-tained using the TDGL model and self-consistent Gaussianapproximation will lead the same thermodynamic equationusing a self-consistent Gaussian approximation [20]. A maincontribution of our paper is an explicit form of the Greenfunction incorporating all Landau levels. This allows to ob-tain explicit formulas without the need to cutoff higher Lan-dau levels. Note that the exact analytical expression of Greenfunction of the linearized TDGL equation in DC field canbe even generalized also to the AC field. The method is verygeneral, and it allows us to study transport phenomena be-yond the linear response of a type-II superconductor like theNernst effect. One of the main results of our work is thatthe conductivity formula is independent of UV cutoff (un-like in [21]) as it should be as the standard |Ψ |4 theory isrenormalizable.

The paper is organized as follows. The model is definedin Sect. 2. The Green’s function of TDGL in Gaussian ap-proximation is described in Sect. 3. The Hall conductivityand the comparison with experiment are described in Sect. 4,while Sect. 5 contains conclusions.

2 Dissipative Dynamics of Vortices and Electric Fieldsin the Mixed State for Hall Effect

In order to calculate the Hall conductivity, the imaginarypart of the relaxation time in the TDGL equation must be in-troduced to break the particle-hole symmetry and allow fora nonvanishing Hall current [15]. The TDGL equation [20]then takes form:

�2(γ ′ + iγ ′′)

2m∗ DτΨn = − 1

s′δFGL

δΨ ∗n

+ ζn. (1)

Here, Dτ ≡ ∂/∂τ − i(e∗/�)Φ is the covariant time deriva-tive, with Φ = −Ey being the scalar electric potential de-scribing the driving force in a purely dissipative dynamics.FGL is the Lawrence–Doniach (LD) expression of the GL

free energy

FGL = s′ ∑

n

∫d2r

{�

2

2m∗ |DΨn|2 + �2

2mcd ′2 |Ψn − Ψn+1|2

+ a|Ψn|2 + b′

2|Ψn|4

}, (2)

s′ is the order parameter effective “thickness” and d ′ dis-tance between layers labeled by n. Effective Cooper pairmass in the ab plane is m∗ (disregarding for simplicitythe anisotropy between the crystallographic a and b axes),while along the c axis it is much larger-mc. For simplic-ity, we assume a = αT

mfc (t − 1), t ≡ T/T

mfc , although

this temperature dependence can be easily modified to bet-ter describe the experimental coherence length. The “meanfield” critical temperature T

mfc depends on UV cutoff, τc,

of the “mesoscopic” or “phenomenological” GL descrip-tion, specified later. The covariant derivatives are defined byD ≡ ∇ + i(2π/Φ0)A, where the vector potential describesconstant and homogeneous magnetic field A = (−By,0)

and Φ0 = hc/e∗ is the flux quantum with e∗ = 2|e|.The variance of the thermal noise, determining the tem-

perature T is taken to be the Gaussian white noise:

⟨ζ ∗n (r, τ )ζm

(r′, τ ′)⟩ = �

2γ ′

m∗s′ T δ(r − r′)δ

(τ − τ ′)δnm. (3)

The electric current the supercurrent is

Js = ie∗�

2m∗(Ψ ∗

n DΨn − ΨnDΨ ∗n

). (4)

Throughout most of the paper, we use the coherencelength ξ as a unit of length, Hc2 = Φ0/2πξ2 as a unitof the magnetic field, τGL = γ ′ξ2/2 as a unit of time,EGL = Hc2ξ/cτGL as a unit of electric field. After rescalingEq. (1) by x → ξx, y → ξy, s′ → ξs/γ, d ′ → ξd/γ, τ →τGLτ,B → Hc2b,E → EGLE , T → tmf T mf ,Ψ 2

n →(2αT

mfc /b′)ψ2

n (γ 2 = mc/m∗), one obtains equation

(1 + iη)Dτψn − 1

2D2ψn + 1

2d2(2ψn − ψn+1 − ψn−1)

− 1 − tmf

2ψn + |ψn|2ψn = ζn, (5)

while the Gaussian white-noise correlation takes a form

⟨ζn

∗(r, τ )ζm

(r′, τ ′)⟩ = 2ωt

sδ(r − r′)δ

(τ − τ ′)δnm, (6)

where η = γ ′′/γ ′. Usually γ ′′ is small in comparison withγ ′ by a ratio of the order of Tc/EF [15]. The covariant timederivative in dimensionless units is Dτ = ∂

∂τ+ ivby with

v = E/b being the vortex velocity and the thermal noise was

J Supercond Nov Magn (2014) 27:359–363 361

rescaled as ζn → ζn(2αTmfc )3/2/b′1/2. The dimensionless

fluctuations’ strength coefficient is

ω = √2Giπ, (7)

where the Ginzburg number is defined by

Gi = 1

2

(8e2κ2ξT

mfc γ /c2

�2)2

. (8)

The dimensionless current density is Js = JGLjs where

js = i

2

(ψ∗

nDψn − ψnDψ∗n

), (9)

with JGL = cHc2/(2πξκ2) being the unit of the current den-sity. Consistently, the conductivity will be given in unitsof σGL = JGL/EGL = c2γ1/(4πκ2). This unit is close tothe normal state conductivity σn in dirty limit superconduc-tors [24]. In general, there is a factor k of order one relatingthe two: σn = kσGL.

3 The Green’s Function of TDGL in GaussianApproximation

A simple approximation which captures the most interest-ing fluctuations effects in the self-consistent Gaussian ap-proximation, in which the cubic term in the TDGL Eq. (5),|ψn|2ψn, is replaced by a linear one 2〈|ψn|2〉ψn

(1 + iη)Dτψn −(

1

2D2 + b

2

)ψn

+ 1

2d2(2ψn − ψn+1 − ψn−1) + εψn = ζn, (10)

leading the “renormalized” value of the coefficient of thelinear term:

ε = −ah + 2⟨|ψn|2

⟩, (11)

where the constant is defined as ah = (1 − tmf − b)/2. Theaverage 〈|ψn|2〉 is expressed via the parameter ε below andwill be determined self-consistently together with ε.

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

ψn(r,τ ) =∫ 2π/d

0

dkz

2πe−inkzdψkz(r, τ ),

ψkz(r,τ ) = d∑

n

einkzdψn(r, τ ),

(12)

and similar transformation for ζ . In terms of Fourier com-ponents, the TDGL Eq. (10) becomes{(1 + iη)Dτ − 1

2D2 − b

2

+ 1

d2

[1 − cos(kzd)

] + ε

}ψkz(r, τ )

= ζkz(r, τ ). (13)

The noise correlation is

⟨ζ ∗kz

(r, τ )ζk′z

(r′, τ ′)⟩ = 4πωt

d

sδ(r − r′)δ

(τ − τ ′)δ

(kz − k′

z

).

(14)

The relaxational linearized TDGL equation with a Langevinnoise, Eq. (13), is solved using the retarded (G = 0 for τ <

τ ′) Green function (GF) Gkz(r, τ ; r′, τ ′):

ψn(r, τ ) =∫ 2π/d

0

dkz

2πe−inkzd

∫dr′

×∫

dτ ′Gkz

(r, τ ; r′, τ ′)ζkz

(r′, τ ′). (15)

The GF satisfies{(1 + iη)Dτ − 1

2D2 − b

2

+ 1

d2

[1 − cos(kzd)

] + ε

}Gkz

(r, r′, τ − τ ′)

= δ(r − r′)δ

(τ − τ ′). (16)

The Green function is a Gaussian

Gkz

(r, r′, τ ′′) = Ckz

(τ ′′)θ

(τ ′′) exp

[ib

2X

(y + y′)

]

× exp

[−X2 + Y 2

2β− v(1 + iη)X

], (17)

with X = x − x′ − vτ ′′, Y = y − y′, τ ′′ = τ − τ ′. θ(τ ′′) isthe Heaviside step function, C and β are coefficients.

Substituting the Ansatz (17) into Eq. (16), one obtains thefollowing:

β = 2

btanh

(b

2

τ ′′

1 + iη

), (18)

while

Ckz

(τ ′′) = b

4πexp

{−

(ε − b

2+ v2(1 + iη)2

2

+ 1

d2

[1 − cos(kzd)

]) τ ′′

1 + iη

}

× sinh−1(

b

2

τ ′′

1 + iη

). (19)

362 J Supercond Nov Magn (2014) 27:359–363

The thermal average of the superfluid density (density ofCooper pairs) is

⟨∣∣ψn(r, τ )∣∣2⟩ = 2ωt

d

s

∫ 2π/d

0

dkz

2π

∫dr′

×∫

dτ ′∣∣Gkz

(r − r′, τ − τ ′)∣∣2

= ωtb

2πs

∫ ∞

τ=τc

f (ε, τ )

sinh(bτ), (20)

where

f (ε, τ ) = exp(uv2)e−[2ε−b+(1+η2)v2]τ

× e−2τ/(1+η2)d2I0

(2τ/

(1 + η2)d2). (21)

Here, I0(x) = (1/2π)∫ 2π

0 ex cos θ dθ is the modified Bessel

function, and u(τ, b) = 4b[1 − cos( bητ

1+η2 )]/ cosh( bτ

1+η2 ). Thefirst pair of multipliers in Eq. (21) is independent of theinter-plane distance d and exponentially decreases for τ >

(2ε − b + v2)−1, while the last pair of multipliers dependson the layered structure. The expression (20) is divergentat small τ , so an UV cutoff τc is necessary for regular-ization. Note that η is small a small parameter (the orderof 10−3 − 10−2), reflecting also the small Hall angle [15].Therefore, the expression (20) can be expanded in first orderin small η. By substituting it into Eq. (11) and keeping zeroorder, Eq. (11) takes a form after the renormalization of thecritical temperature [20]

ε = −arh − ωt

πs

∫ ∞

0ln

[sinh(bτ)

] d

dτ

[f (ε, τ )

cosh(bτ)

]

+ ωt

πs

{γE − ln

(bd2)}, (22)

where arh = 1−b−T/Tc

2 , t = T/Tc, γE = 0.577 is Euler con-stant, and ω = √

2Giπ where Gi = 12 (8e2κ2ξTcγ /c2

�2)2

(T mfc is now replaced by Tc).

4 The Hall Conductivity

4.1 Current Density

The supercurrent density, defined by Eq. (9), can be ex-pressed via the Green’s functions as

j sx = i

2

⟨ψ∗

n (r, τ )

(∂

∂x− iby

)ψn(r, τ )

⟩+ cc

= iηtd

s

∫ 2π/d

0

dkz

2π

∫

r′,τ ′Gkz

(r, r′, τ

)

×(

∂

∂x− iby

)Gkz

(r, r′, τ

) + c.c. (23)

Performing the integrals, one obtains

jx = ωtd

sπv

∫ ∞

τ=0u

[2η + iu

(1

β− 1

β∗

)]exp

(uv2)f (ε, τ ),

(24)

where the function f was defined in Eq. (21).The expression (24) for Hall current is obtained without

any assumption about the imaginary part of the relaxationtime γ ′′ in the TDGL equation. Under the assumption η = 0,the hall current density Eq. (24) is exactly equal to zero. Thatis why as the next step we expand the expression (24) up tothe first order over small η and obtain.

The fluctuation Hall conductivity in physical units isgiven by

σ sxy = J s

x

E= σGLηωt

8πs

×∫ ∞

τ=0

2 tanh(bτ/2) − bτ [1 − tanh2(bτ/2)]sinh2(bτ/2)

f (τ ),

(25)

In order to compare with experimental data, normal conduc-tivity σn

xy should be added. Thus, the total Hall conductivitywill be consequently

σxy = σ sxy + σn

xy (26)

4.2 Comparison with Experiment

Hall effect measurements on an optimally doped YBCOfilms of thickness 50 nm and Tc = 86.8 K was done in [5] inwhich the resistivity of the same sample was fitted in [20].The fitting parameters were remained the same, namely,Hc2(0) = TcdHc2(T )/dT |Tc = 190 T (corresponding to ξ =13.2 Å), the Ginzburg–Landau parameter κ = 45.6, the or-der parameter effective thickness s′ = 8.5 Å, and the fac-tor k = σn/σGL = 0.55, where we take γ = 7.8 for opti-mally doped YBCO in [25]. The comparison is presented inFig. 1. The Hall conductivity curves were fitted to Eq. (26)with the normal-state conductivity measured in [5] to beσn

xy = 42 (� cm)−1. We found that the best fits were ob-tained with η = −0.0017, and inferred empirically. The ab-solute value of η obtained from our fitting is consistent withits value [26, 27]. The negative value of the hole-particleasymmetry parameter η (this means a negative σ s

xy ) im-plies a positive energy-derivative of the density of states atεF when the carriers are holes in the normal state. As sug-gested by Kopnin and Vinokur [28], one possibility to ex-plain this behavior is that the Fermi surface of a metal in thenormal state has both hole-like and electronic pockets. TheHall anomaly may thus depend on the doping level, as it wasreported by Nagaoka et al. [29]. Very recently, Angilella et

J Supercond Nov Magn (2014) 27:359–363 363

Fig. 1 Points are Hall conductivity for different electric line is thetheoretical value of resistivity calculated from Eq. (26) with fitting pa-rameters (see text)

al. [30] have found that, close to an electronic topologicaltransition of the Fermi surface, in the hole-like doping range,the fluctuation Hall conductivity has indeed an opposite signwith respect to the normal-state one, giving additional strongsupport that the Hall resistivity sign reversal is intrinsic anddepends on the details of the structure of the electronic spec-trum.

5 Discussion and Conclusion

We have calculated the fluctuation Hall conductivity for alayered superconductor in an arbitrary in-plane electric fieldand perpendicular magnetic field in the frame of the TDGLtheory with thermal noise describing the thermal fluctua-tions using the self-consistent Gaussian approximation. Wehave obtained explicit formulas including all Landau levelswithout any assumption about the imaginary part of the re-laxation time γ ′′ in the TDGL equation. It is then easy toget the expression for the fluctuation Hall conductivity un-der assumption that the imaginary part of the relaxation timeis very small.

The results were compared to the experimental data onHTSC. Our the fluctuation Hall conductivity results are ingood qualitative and even quantitative agreement with ex-perimental data on YBCO in strong electric fields.

Acknowledgements We are grateful to I. Puica for providing detailsof experiments and to B. Rosenstein and D. Li for discussions and en-couragement. This work was supported by the National Foundation for

Science and Technology Development (NAFOSTED) of Vietnam un-der Grant No. 103.02-2011.15.

References

1. Hagen, S.J., Lobb, C.J., Greene, R.L., Forrester, M.G., Kang, J.H.:Phys. Rev. B 41, 11630 (1990)

2. Kang, W.N., Kim, D.H., Shim, S.Y., Park, J.H., Hahn, T.S., Choi,S.S., Lee, W.C., Hettinger, J.D., Gray, K.E., Glagola, B.: Phys.Rev. Lett. 76, 2993 (1996)

3. Kokubo, N., Aarts, J., Kes, P.H.: Phys. Rev. B 64, 014507 (2001)4. Puica, I., Lang, W., Gäob, W., Sobolewski, R.: Phys. Rev. B 69,

104513 (2004)5. Puica, I., Lang, W., Siraj, K., Pedarnig, J.D., Bauerle, D.: Phys.

Rev. B 79, 094522 (2009)6. Wang, Z.D., Dong, J.M., Ting, C.S.: Phys. Rev. Lett. 72, 3875

(1994)7. van Otterlo, A., Feigelman, M.V., Geshkenbein, V.B., Blatter, G.:

Phys. Rev. Lett. 75, 3736 (1995)8. Kato, Y.: J. Phys. Soc. Jpn. 68, 3798 (1999)9. Troy, R.J., Dorsey, A.T.: Phys. Rev. B 47, 2715 (1993)

10. Kopnin, N.B., Ivlev, B.I., Kalatski, V.A.: J. Low Temp. Phys. 90,1 (1993)

11. Harault, J.P.: Phys. Rev. 179, 494 (1969)12. Gorkov, L.P.: JETP Lett. 11, 32 (1970)13. Varlamov, A.A., Reggiani, L.: Phys. Rev. B 45, 1060 (1992)14. Mishonov, T., Posazhennikova, A., Indekeu, J.: Phys. Rev. B 65,

064519 (2002)15. Larkin, A., Varlamov, A.: Theory of Fluctuations in Superconduc-

tors. Clarendon Press, Oxford (2005)16. Soret, J.C., Ammor, L., Martinie, B., Lecomte, J., Odier, P.,

Bok, J.: Europhys. Lett. 21, 617 (1993)17. Fruchter, L., Sfar, I., Bouquet, F., Li, Z.Z., Raffy, H.: Phys. Rev.

B 69, 144511 (2004)18. Lang, W., Puica, I., Peruzzi, M., Lemmermann, L., Pedarnig, J.D.,

Bäuerle, D.: Phys. Status Solidi C 2, 1615 (2005)19. Puica, I., Lang, W.: Phys. Rev. B 68, 212503 (2003)20. Tinh, B.D., Li, D., Rosenstein, B.: Phys. Rev. B 81, 224521 (2010)21. Puica, I., Lang, W.: Phys. Rev. B 70, 092507 (2004)22. Kovner, A., Rosenstein, B.: Phys. Rev. D 39, 2332 (1989)23. Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, and

Polymer Physics. World Scientific, Singapore (1995)24. Kopnin, N.: Vortices in Type-II Superconductors: Structure and

Dynamics. Oxford University Press, Oxford (2001)25. Li, D., Rosenstein, B.: Phys. Rev. B 65, 220504(R) (2002)26. Lang, W., Heine, G., Schwab, P., Wang, X.Z.: Phys. Rev. B 49,

4209 (1994)27. Lang, W., Puica, I., Göb, W., Sobolewski, R.: Int. J. Mod. Phys. B

17, 3496 (2003)28. Kopnin, N.B., Vinokur, V.M.: Phys. Rev. Lett. 83, 4864 (1999)29. Nagaoka, T., Matsuda, Y., Obara, H., Sawa, A., Terashima, T.,

Chong, I., Takano, M., Suzuki, M.: Phys. Rev. Lett. 80, 3594(1998)

30. Angilella, G.G.N., Pucci, R., Varlamov, A.A., Onufrieva, F.: Phys.Rev. B 67, 134525 (2003)