2.4.1 Free energy
To describe fluctuation of order parameter in thin films or layered superconductors one can start with the GL free energy:
F = s0 Z
dr ~2
2m∗|DΨ|2+ a|Ψ|2+ b0
2|Ψ|4, (2.5)
where A = (−By, 0) describes a constant and practically homogeneous magnetic field (we generally neglect small fluctuations of the magnetic field due to magnetization which are of order 1/κ2 << 1 in the region of interest) in Landau gauge and the covariant derivative is defined by D ≡ ∇+i(2π/Φ0)A, with Φ0 = hc/e∗ being the flux quantum, e∗ = 2|e|. For simplicity we assume linear dependence a(T ) = αTcmf(tmf − 1), tmf = T /Tcmf, although the temperature dependence can be easily modified to better describe the experimental coherence length. The “mean field” critical temperature Tcmf depends on the ultraviolet (UV) cutoff, Λ, specified later. It is higher than measured critical temperature Tc due to strong thermal fluctuations on the mesoscopic scale. The order parameter effective
“thickness” of a layer, s0, is assumed to be small enough, so that order parameter does not vary considerably inside the layer (namely does not exceed the coherence length ξz(T ) along the field direction) and layers are nearly independent. We apply this model to describe experiments not just in BiSCCO and other highly anisotropic materials, but also in overdoped LaSCO [21] and strongly underdoped YBCO [21]. For more isotropic optimally doped or fully doped YBCO [21] an anisotropic 3D GL model (neglecting the layered structure) would be more appropriate. For materials between the two extremes, a more complicated model like the Lawrence-Doniach one should be used.
2.4.2 Relaxation dynamics and thermal fluctuations
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 [8, 34]. In the presence of thermal fluctuations, which on the mesoscopic scale are represented by a complex white noise [8, 35], it reads:
~2γ0
2m∗DtΨ = − δF
δΨ∗ + ζ, (2.6)
called in the present context TDGL equation. Explicitly the TDGL equation for the superconducting order parameter is
~2γ0
2m∗DtΨ = ~2
2m∗D2Ψ − aΨ − b0|Ψ|2Ψ + ζ, (2.7)
where Dt ≡ ∂/∂t − i(e∗/~)Φ is the covariant time derivative with φ (r) = −Ey being the scalar potential describing electric field. To incorporate the thermal fluctuations via Langevin method, the noise term ζ (r, t), having Gaussian correlations
hζ∗(r, t)ζ(r0, t0)i = ~2γ0
m∗s0T δ(r − r0)δ(t − t0), (2.8)
is introduced. Here δ(r − r0) is the two dimensional δ function of the in-plane coordinates, and the inverse diffusion constant γ0/2, controlling the time scale of dynamical processes via dissipation, is real, although a small imaginary (Hall) part is also generally present [36].
Throughout most of the thesis we will use the coherence length, ξ = (~2/2m∗αTcmf)1/2, the zero-temperature correlation length as a unit of length, and Hc2(0) = Φ0/2πξ2 being the zero-temperature critical field (extrapolated by the linear formula from Tc, actual Hc2(T ) at T = 0 is lower) as a unit of magnetic field. After the order parameter field is
rescaled as Ψ2 → (2αTcmf/b0)ψ2. The dimensionless Boltzmann factor in these units is:
F
T = 1
η2Dmftmf
Z dr
·1
2|Dψ|2− (ah+ b
2) |ψ|2+1 2|ψ|4
¸
, (2.9)
where the covariant derivatives in dimensionless units in Landau gauge are Dx = ∂x∂ − iby, Dy = ∂y∂ with b = B/Hc2(0), and the constant is defined as ah = (1 − tmf − b)/2. The dimensionless fluctuations’ strength coefficient is
η2Dmf = q
2Gimf2Dπ, (2.10)
where the Ginzburg number is defined by
Gimf2D = 1 2
µ8e2κ2ξ2Tcmf c2~2s0
¶2
. (2.11)
In analogy to the coherence length and the penetration depth, one can define a characteristic time scale. In the superconducting phase a typical “relaxation” time is tGL = γ0ξ2/2. It is convenient to use the following unit of the electric field and the dimensionless field: EGL = Hc2ξ/ctGL, E = Ey/EGL. The TDGL Eq. (2.7) written in dimensionless units reads
∂tψ − 1
2D2ψ − (ah+ b
2)ψ + |ψ|2ψ − iEyψ = ζ, (2.12)
In terms of dimensionless quantities the Gaussian correlations read:
hζ∗(r, t)ζ(r0, t0)i = 2η2Dmftmfδ(r − r0)δ(t − t0), (2.13)
where the thermal noise was rescaled as ζ = ζ(2αTcmf)3/2/b01/2.
2.4.3 The heat and the electric total and transport currents
The total heat current density is written by [8, 11, 37]:
Jh = − ~2
while the total electric current is
Je= ie∗~
In terms of dimensionless quantities the currents read:
Jh = JGLh jh, jh = − electric current density, respectively. Consistently the conductivity will be given in units of σGL= JGL/EGL= c2γ0/(4πκ2). This unit is close to the normal state conductivity σn in dirty limit superconductors [38]. In general there is a factor k of order one relating the two: σn= kσGL.
An important aspect of the calculation of the electrothermal conductivity, discussed in detail [39], is the need to account for bulk magnetization currents. In the presence of a magnetic field, the system has magnetization current in equilibrium. The total heat current defined by Eq. (2.14) is thus a sum of transport and magnetization parts,
Jh = Jhtr+ Jhmag. (2.18)
The magnetization current is current that circulates in the sample and does not contribute to the net current which measured in a transport experiment. On the other hand, it does contribute to the total microscopic current, and it is thus necessary to subtract it from the total current to obtain the transport current response. In the presence of an applied electric field, it was shown in Ref. [39] that the magnetization current is given by
Jhmag = cM × E, (2.19)
where M is the equilibrium magnetization.
Generally, to define the transport coefficients, the electric and heat transport current densities are related to the applied (sufficiently weak) electric field and the temperature gradient by
Jtr(e)i = σijEj− αij∇jT, (2.20)
Jtr(h)i= eαijEj − κij∇jT, (2.21)
where σ, α, eα, and κ are the electrical, the thermoelectric, the electrothermal, and the thermal conductivity components of the conductivity tensor (i, j = x, y). The Onsager relation implies eα = T α. The Nernst coefficient νN, under the condition J(e)tr = 0 is expressed in terms of the above coefficients as
νN = Ey
(−∇T )xB = 1 B
αxyσxx− αxxσxy
σxx2 + σxy2 . (2.22)
If the system shows no significant Hall effect (only such systems will be considered), then σxy = 0 and the expression simplifies:
νN = αxy
Bσxx. (2.23)
The Nernst signal is defined
eN = Ey
(−∇T )x = BνN. (2.24)
For comparison with experiment, the fluctuation contribution, σxx and eN, should be added to the normal sate contribution, σn and enN. However, the normal state the Nernst signal enN is very small in these materials [12, 23] and will be largely ignored in what follows.
It then follows that the electrothermal conductivity is given by
e
αxy ≡ J(tr)xh Ey
= Jxh Ey
+ cMz. (2.25)
The both terms contribute as will be shown in the following Sections.