5.4 Fluctuation ac conductivity
5.4.1 Linear response to electric field
The supercurrent density, defined by Eq. (3.9), was already expressed via the GF Eq.
(3.47). In a similar manner one also has
jys= iηtd scalar potential. One finds correction to the GF to linear order in the electric field
Gkz(r, t, r0, t0) = G0kz(r, t, r0, t0) + i
where φ(r1, t1) and E(t1) are the scalar electric potential and electric field in dimensionless units respectively, τ1 = t − t1, and τ2 = t1− t0.
Substituting the full GF (5.20) into expression (5.19), the supercurrent density takes form:
and
Carrying out the integration (5.22) over r0 one obtains
jy0s = 0, (5.24)
while carrying out the integration (5.23) over r1, r0 one can find
jy1s = b
By putting ω = 0 in the expression (5.26) one gets back to dc current as:
jy1s = ηt
which is consistent with the expression (3.49) in linear response case.
By doing the Fourier transform expression (5.26) with respect to frequency, then one
obtains complex conductivity as:
The complex conductivity (5.28) can be given in terms of the real part and the imaginary part
In 2D case, the expression (5.32) and (5.33) become
σ1(ω) = 1
σ2(ω) = 1
which agree with the results of A.T. Dorsey [118], and A. Larkin and A. Varlamov [8].
This results also agree with the results of Schmidt [116], which were derived by using the Kubo formula.
5.4.2 Comparison with experiment
Here we compare the results with the experimental results of Y. Tsuchiya et al. [114], obtained from the the microwave surface impedance measurements at ω/2π=31.7 GHz on an overdoped YBCO slightly overdoped single crystal with Tc=91.2 K. The layer distance used the calculation is d0 = 11.68 ˚A in Ref. [60]. The comparison is presented in Fig. 5.1.
The complex resistivity is
ρs(ω) = 1
The change of resistivity from the zero-field is defined as
δρ1 = ρ1(B) − ρ1(0), (5.38)
where returning to physical units
ρ1 = k σn
σ1
σ12+ σ22, (5.39)
0 2 4 6 8 10 12 14 16 0
1 2 3 4 5 6
H(T)
1
(m)
T=70 K
T=80 K
T=85 K
Figure 5.1: Points are resistivity for different temperatures of an overdoped YBCO in Ref.
[114]. The solid line is the theoretical value of resistivity calculated from Eq. (5.38) with fitting parameters (see text).
The change of resistivity curves were fitted to Eq. (5.38) with the normal-state conduc-tivity measured in Ref. [114] to be σn = 3.3 × 106 (Ω m)−1. The parameters we obtain from the fit are: Hc2(0) = 180 T (corresponding to ξ = 13.5 ˚A), κ = 47.8, s0 = 5.88 ˚A, and k = σn/σGL = 0.74, where we take γ = 7.8 for optimally doped YBCO in Ref. [42].
Using those parameters, we obtain Gi = 1.56 × 10−3 (corresponding to η = 0.176). The order parameter effective thickness s0 can be taken to be equal to the layer thickness (see in Ref. [61]) of the superconducting CuO2 plane plus the coherence length 2ξz = 2ξγ due to the proximity effect: 3.18 ˚A+213.67.8 ˚A= 6.96 ˚A, roughly in agreement in magnitude with the fitting value of s0.
We also compare the change of resistivity Eq. (5.38) with the experimental results of T.
Hanaguri et al. [115], obtained from the the microwave surface impedance measurements at 40.8 GHz on a Bi2212 single crystal with Tc=91 K. The layer distance and the normal-state conductivity used the calculation are d0 = 19.6 ˚A in Ref. [61] and σn = 1.42 × 104 (Ωcm)−1 in Ref. [48], respectively. The comparison is presented in Fig. 5.2. The best fitting parameters are: Hc2(0) = 178 T (corresponding to ξ = 13.6 ˚A), κ = 49.5, s0 = 3.41
0 20 40 60 80 100 120 0.0
0.5 1.0 1.5 2.0 2.5 3.0 3.5
T=70 K
T=50 K
1
(cm)
H (m T)
Figure 5.2: Points are resistivity for different temperatures of an overdoped Bi2212 in Ref.
[115]. The solid line is the theoretical value of resistivity calculated from Eq. (5.38) with fitting parameters (see text).
˚A, k = 0.67, and γ = 34 which give Gi = 0.036. The order parameter effective thickness s0 can be roughly estimated: 3.32 ˚A+213.634 ˚A= 4.11 ˚A. The fitting parameters of ac resistivity are consistent with the fitting ones of dc resistivity in section 3.4.2.
5.5 Summary
We calculated the complex conductivity and resistivity in a layered Type-II superconduc-tor under magnetic field in the presence of strong thermal fluctuations on the mesoscopic scale in linear response. Time dependent Ginzburg-Landau equations with thermal noise describing the thermal fluctuations is used to describe the vortex-liquid regime and arbi-trary flux flow velocities. The nonlinear term in dynamics is treated using the renormalized Gaussian approximation. The renormalization of the critical temperature is calculated.
We obtained explicit expressions for the complex conductivity σs and resistivity ρs in-cluding all Landau levels, so that the approach is valid for arbitrary values the magnetic field not too close to Hc1(T ).
The results were compared to the experimental data on HTSC. Our the resistivity results are in good qualitative and even quantitative agreement with experimental data on YBCO and Bi2212. We found that the change in the resistivity was linear in magnetic field at low temperature in YBCO and Bi2212. This linear behavior was was well described in terms of the Coffey-Clem unified theory [114, 122] of vortex motion with B-independent, and frequency-independent vortex dynamics parameters, and essentially not different from that in conventional superconductor. However, there is a clear derivation from the B-linear behavior due to thermal effect. The thermal fluctuation was included in the present approach, so that our results should be applicable for above and below Tc.
Conclusion and future work
We quantitatively studied electrical transport and thermal transport phenomena in Type-II superconductor in magnetic field in the presence of strong thermal fluctuations on the mesoscopic scale in the linear response and also beyond the linear one. While in the normal state the dissipation involves 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 is used to describe strongly Type-II superconductor in the vortex-liquid regime in 2D, 3D, and layered superconductor. 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(T ).
Our main objective is to study layered high-Tc materials for which the Ginzburg num-ber characterizing the strength of thermal fluctuations is exceptionally high, in the moving vortex matter the crystalline order is lost and it becomes homogeneous on a scale above the average inter-vortex 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 (strictly speaking not far from Tc(H)) and another is effect of quenched disorder. The later becomes insignificant at elevated temperature due to a very effective thermal depinning. Although sometimes motion tends to suppress fluctuations, they are still a dominant factor in flux dynamics. We also 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 (significantly “weakened” by thermal fluctuations) is not strong enough to significantly affect the transport.
The nonlinear term in TDGL equation is treated using the self-consistent Gaussian approximation. In linear response we solve the linearized TDGL by expanding Green’s function to linear order in the electric field. This allows us to obtain explicit expres-sions for the transverse thermoelectric conductivity αxy, the Nernst signal eN in 2D and 3D, and the ac conductivity in layered superconductor including all Landau levels. In nonlinear response we also have an explicit form of the Green function incorporating all Landau levels. This allows to obtain explicit formulas for electrical conductivity and Hall conductivity beyond linear response without need to cutoff higher Landau levels. The results are presented using both the strength of the thermal fluctuation η and more often used the Ginzburg number Gi in 2D and 3D.
We compared the transverse thermoelectric conductivity αxy and the Nernst signal eN to the available 2D numerical simulation of the same model and the experiments on LaSCO and YBCO. Our the resistivity and Hall conductivity results were compared with experimental data on YBCO in strong electric fields as well as on Bi2212 in linear case.
The renormalization of the critical temperature is calculated and is strong in layered high-Tc materials. The change of ac resistivity was also compared to the experiment on YBCO and Bi2212. Our comparisons show a good agreement with several experiment and numerical simulation on HTSC.
Let us compare the present approach with a widely used Londons’ approximation.
Since we haven’t neglected higher Landau levels, as very often is done in similar studies [11, 43], 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 superconductors.
The GL approach for the constant magnetic induction works for H >> Hc1(T ), while the Londons’ approach works for H << Hc2(T ). 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 numerical simulations of Langevin dynamics. Efforts in this direction are under way.
In the future work, we will calculate ac conductivity beyond linear response and also consider pinning and crystalline correlation effects on the transport properties in Type-II superconductors.
Appendix A
Derivation of Green’s function
In this appendix we outline the method for obtaining the Green’s function in strong electric field for the linearized equation of TDGL (see Eq. (3.18))
½ H −b b
2+ 1
d2[1 − cos(kzd)] + ε
¾
Gkz(r, r0, t − t0) = δ(r − r0)δ(t − t0), (A.1)
where bH = Dt− 12D2, the covariant time derivative Dt and the covariant derivatives D in Landau gauge are as follows
Dt= ∂
∂t+ ivby; Dx = ∂
∂x − iby; Dy = ∂
∂y. (A.2)
The Green’s function is a Gaussian
Gkz(r, r0, τ ) = exp
·ib
2X (y + y0)
¸
gkz(X, Y, τ ) , (A.3)
where
gkz (X, Y, τ ) = Ckz(τ )θ (τ ) exp µ
−X2+ Y2
2β − vX
¶
, (A.4)
with X = x − x0− vτ, Y = y − y0, τ = t − t0. θ (τ ) is the Heaviside step function, C and β are coefficients.
Substituting the Ansatz (A.3) into Eq. (A.1), one obtains coefficients
Substituting Eq. (A.9) and Eq. (A.10) into Eq. (A.1), one obtains pre-exponential factor which has only quadratic and constant parts
1
Then one obtains following conditions condition:
ε − b
The Eq. (A.12) determines β, subject to an initial condition β(0) = 0,
β = 2
b tanh (bτ /2) , (A.13)
while Eq. (A.11) determines C:
The normalization is dictated by the delta function term in definition of the Green’s function Eq. (A.1).
It is easy to obtain the Green’s function G0kz(r, r0, τ ) for TDGL Eq. (A.1) in case without electric field by putting v = 0 in Eq. (A.3)
G0kz(r, r0, τ ) = exp
Comparison with the Hartree approach
Here we explain the difference using an example of thermodynamics. The dynamics is not different since it always can be cast in the Martin-Siggia-Rose form (see [35]).
By using the Hartree approximation, one subsitube |ψ|4 by 2
|ψ|2®
|ψ|2 in the GL free energy Eq. (3.6) leading the “renormalized” value of the coefficient of the linear term in the TDGL Eq. (3.10)
ε = −ah +
|ψn|2®
. (B.1)
In the framework of the variational Gaussian approximation, the GL free energy Eq.
(3.6) 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 [43]. In liquid phase with an arbitrary homogeneous U(1) symmetric state, just one variational parameter ε is sufficient. Thus
K = s
ηmftmf X
n
Z dr
· ψ∗n
µ
−1
2D2− b 2+ ε
¶ ψn
¸
, (B.2)
and the small perturbation becomes
The eigenvalue of Nth Landau level is
−1
2D2ϕn = (N + 1
2)bϕn. (B.4)
The Gaussian energy which will be minimized therefore is
ggauss ≡ − log
Minimizing the Gaussian energy with respect to ε
∂ggauss
The derivative of the first term in Eq (B.7) gives
∂
while the second term gives
∂ hV iK
∂ε = −
|ψn|2®
+ (−ah− ε) ∂
∂ε
|ψn|2® + 2
|ψn|2® ∂
∂ε
|ψn|2®
. (B.9)
Substituting Eq. (B.8) and Eq. (B.9) into Eq. (B.7) one obtains gap equation
ε = −ah + 2
|ψn|2®
. (B.10)
While the Hartree method is generally simpler, the Gaussian method applied in it’s con-sistent form conserves Ward identities (electric current) and its effective energy is positive definite. In addition it has the correct “large number of components” limit, unlike Hartree method.
Appendix C
Comparison with thermodynamics
From TDGL, we obtained the superfluid density Eq. (3.36) in the case b = 0, υ = 0 :
|ψn(r, τ )|2®
' −ηmftmf 2πs {ln¡
τcut/d2¢
+ γE} + O (τcut) . (C.1)
In the case without external electric field (or v = 0), the equation obtained from TDGL shall approach the thermodynamics result. In thermodynamics method, we shall evaluate the partition function Z = R
DψnDψn∗e−FGL/T where FGL/T is defined in Eq.
(3.6).
The superfluid density in the thermodynamic approach at the phase transition point
|ψn(r, τ )|2®
= ηmftmfd (2π)3s
Z kmax
0
dk Z 2π/d
0
dkz 1
k2
2 + 1−cos(kd2 zd)
' ηmftmf
2πs {ln Λ + ln¡ 2d2¢
} + O¡ Λ−1¢
, (C.2)
where Λ = kmax2 /2.
The relation between the cutoff “time” τcut and energy UV cutoff Λ is obtained by comparing Eq. (C.1) with Eq. (C.2)
τcut ' 1
2ΛeγE. (C.3)
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. (3.44) without electric field derived from TDGL after using Eq. (C.3).
Appendix D
Derivation of Green’s function of TDGL for Hall effect
In the same way we outline the method for obtaining the Green’s function in strong electric field for the linearized equation of TDGL (see Eq. (4.8))
½
(1 + iϑ)Dt− 1
2D2− b 2 + 1
d2[1 − cos(kzd)] + ε
¾
G0kz(r, r0, t−t0) = δ(r−r0)δ(t−t0), (D.1)
In Landau gauge the covariant time derivative Dt and the covariant derivatives D are as follows
Dt= ∂
∂t+ ivby; Dx = ∂
∂x − iby; Dy = ∂
∂y. (D.2)
The Green’s function is a Gaussian
G0kz(r, r0, τ ) = exp
·ib
2X (y + y0)
¸
gk0z(X, Y, τ ) , (D.3)
where
g0(X, Y, τ ) = C0(kz)θ (τ ) exp
·
−X2 + Y2
2β0 − v(1 + iϑ)X
¸
, (D.4)
with X = x − x0− vτ, Y = y − y0, τ = t − t0. θ (τ ) is the Heaviside step function, C0 and β0 are coefficients.
Substituting the Ansatz (D.3) into Eq. (D.1), one obtains coefficients
DxG0 =
Substituting Eq. (D.9) and Eq. (D.10) into Eq. (D.1), one obtains pre-exponential factor which has only quadratic and constant parts
1
Then one obtains following conditions condition:
ε − b
EFFECT
while Eq. (D.11) determines C0:
C0 = b 4π exp
½
− µ
ε − b
2+ v2(1 + iϑ)2
2 + 1
d2[1 − cos(kzd)]
¶ τ
¾ sinh−1
µbτ 2
¶
. (D.14)
The normalization is dictated by the delta function term in definition of the Green’s function Eq. (D.1).
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