Superconductivity – Theory and Applications

314
Figs. 11–13. A large circle drawn at the center in each figure represents the FES of air at the
same frequency. All EFSs of air in these three figures have larger radius than those in the
PhC. If we use the conservation rule mentioned before, they all result in the same conclusion
that the refracted angle is larger than the incident angle. It also indicates that the absolute
value of the effectively refracted index is smaller than 1.0. Because each EFS shrinks with an
increasing frequency, the effectively refracted index is negative. Therefore, the negative
refraction takes place here. When the frequency is higher, the shape of the EFS of the fifth
photonic band is closer to a circle. The circular EFS means that the PhC can be considered as
a homogeneous medium at this frequency. The relations between the incident and refracted
angles for these three frequencies are shown in Figs. 15(a)-(c). On the one hand, the lower
curve of each figure shows the negative refraction, where the refracted angle is defined as
negative for convenience. The negative angles are calculated from lines intersecting with the
EFSs in the first Brillouin zone as shown in Figs. 11-13. It can be seen that the relation
between the incident angle and refracted angle is much like that in a homogeneous medium.
On the other hand, the upper curves for a larger incident angle in Fig. 15 (a) and (b) show
the normal refraction with positive refracted angle. They are calculated from lines
intersecting with the EFSs in the right repeated Brillouin zone as shown in Figs. 11-13. Fig. 7. The EFS of the fourth photonic band in the first Brillouin zone. Fig. 8. The EFS of the fifth photonic band in the first Brillouin zone.

Photonic Band Structure and Transmittance of the Superconductor Photonic Crystal

315

transmission of the finite thickness PhC (Sakoda, 1995a, 1995b, 2004). This method is based on Superconductivity – Theory and Applications

316 Fig. 11. The EFS of the fifth photonic band at 0.81 (2πc/a) in the repeated Brillouin zone. The
largest circle at the center represents the FES of air with the same frequency. Fig. 12. The EFS of the fifth photonic band at 0.83 (2πc/a) in the repeated Brillouin zone. The
largest circle at the center represents the FES of air with the same frequency. Fig. 13. The EFS of the fifth photonic band at 0.85 (2πc/a) in the repeated Brillouin zone. The
largest circle at the center represents the FES of air with the same frequency.

Photonic Band Structure and Transmittance of the Superconductor Photonic Crystal

317

Fig. 14. EFSs of the fifth photonic band when frequencies are 0.81, 0.83, and 0.85 (2πc/a). (a) (b) (c)
Fig. 15. Refracted angles vs. incident angles calculated from Figs. 11–13 for T = 5 K at (a)
0.81, (b) 0.83, and (c) 0.85 (2πc/a).


2
, where N
L
is the periodic number. So the total layers of cylinders are 2N
L
. The
configuration of the PhC is shown in Fig. 17. The first Brillouin zone is shown at the up-right
corner. The region at the left-handed side of the PhC is called the incident region, and that at
the right-handed side of the PhC is called the transmitted region. The plane wave in the
incident region is incident on the left interface. After propagating through the PhC, the
transmitted wave is through the right interface and into the transmitted region. Fig. 17. The PhC structure with finite length in the y-direction and infinite length in the x-
direction.
Because the two-fluid model is only suitable for the currents flowing along the cylinder
direction, we only discuss an E-polarized plane wave incident upon the superconductor
PhC here. Two interfaces are along the ΓΜ direction of the PhC. The 2D wave vector of the
incident wave is denoted by
i
k

= (k
i
sinθ, k
i
cosθ) = (k
i,x,
, k
i,y

reflected and transmitted waves of order n, respectively, G
n
= 2nπ/a
1
is the reciprocal lattice
vector corresponding to the periodicity a
1
, and n is an integer. Each component of the Eq.
(39) is called the nth order phase matching condition. It means that the periodicity along the
ΓΜ direction is like a diffraction grating. The wave-vector components of the nth order
Bragg reflected and transmitted waves normal to the interface are



2
2
,
2
2
nn
ix ix
n
ry
n
xi
kk ifkk
k
i k k Otherwise

 







(41)

Here,
tt
kc


 and ε
t
are the transmitted wave vector and dielectric constant of the
transmitted region, respectively. The electric fields in the incident region and the
transmitted region are given by

0
,
( , ) exp( ) exp( )
n
iz i n r
n
Ex
y
Eikr Rikr



n
are the amplitudes of the electric field of the incident wave, the
reflected Bragg wave, and the transmitted Bragg wave, respectively. The electric field inside
the PhC satisfies the following equation derived from Maxwell’s equations:


22 2
.
222
1
(,) (,) 0
(,, )
EPC PC
LE xy E xy
xy
xy c






 








320
where δ
nm
is the Kronecker’s δ. The boundary value function f
E
(x,y) satisfies the boundary
conditions at each interface:





,0 ,0
Eiz
fx Ex and




.
,,
Etz
f
xL E xL (46)
Moreover, we define






 


 
(48)





.
,0 , 0
E
xxL



(49)
The problem of unknown
E
pc
becomes to deal with the internal field. We have to solve Eq.
(48) to obtain
E
pc
field in the PhC. If we expand ψ
E
(x,y) and ε
-1
(x,y,ω) in Fourier series, we

y
xy



 


(51)
Then the electric field in the PhC is expressed as

00
(,) 1 exp( )
n
PC n n n x
n
yy
Exy T ER ikx
LL












2
2
,
,
2
1
n
nm x n n m m n m
nnmm
nm
m
Ak A
L
c










 




 











 

  


1
2
00
2
,
1
2
m
nnn
TR E
m
c




  




(54)
where
ε
a
=ε
s
(ω) and (,,)ujll


is the center of each cylinder, which are

12 ,
(1, , ) ( , )ull lala Rd





(55)

12,
(2, , ) (( 1 2) ,( 1 2) )ull l al a Rd

,
0
1
11
exp
2(,,)
aL
nm n
L
m
dx i G x y dy
aL xy L

















 







(59)
where
(, )
nm n
GGmL



. After calculating the integration, we obtain

0,0
,
1
ab
f
f



 (60)


GR


 




 













(61)



2
1
121.21
2





(62)
where
f is the filling fraction of the superconductor rods in the calculation domain:

2
1
.
2
L
f
NRaL

 (63)
Finally, we want to solve the unknown coefficients,
A
nm
, R
n
, and T
n
. Eq. (53) is not enough to
solve all unknown coefficients because the number of equations is less than the number of

Superconductivity – Theory and Applications

322



1
,00.
1
(1) 1
mn
nm n t y n n
m
mARiLkTE






(65)
Follow the calculation processes and consider the boundary conditions for the
E-polarized
mode, we can determine the unknown coefficients,
A
nm
, R
n
, and T
n
. In practical calculation,
we restrict the Fourier expansion up to
n = ±N and m = M. So there are (2N + 1)M terms in
the Fourier expansion. The total number of the unknown coefficients is (2

RTE
kk








(66)
where
n and n

represent the summation over the Bragg waves with real wave vectors.
Then, we can use the Eq. (52) to define the transmission and reflection:

2
2
,
0
,
cos
n
ty
t
n
ti
n
k

7. The transmission calculated by internal-field expansion method
In previous Section, we have introduced the internal-field expansion method to calculate the
finite thickness PhC. This method used to calculate the transmission of the electromagnetic
wave propagating through the PhC is faster than the FDTD method if the size of the (2
N +
1)(
M + 2) × (2N + 1)(M + 2) matrix is not very large. In the original references (Sakoda,
1995a, 1995b, 2004), the author concludes that this method can be used for the general two-
dimensional PhC. In the following, we use this method to calculate transmissions of the

Photonic Band Structure and Transmittance of the Superconductor Photonic Crystal

323
superconductor PhC. Obviously, the boundary conditions of the magnetic field in Eqs. (64)
and (65) are no more suitable for the superconductor PhC if superconductor rods are
embedded in air. It is the factor that the boundary conditions of the magnetic field in this
method are dealt with at the interface between two homogeneous media but not between
cylinders and a homogeneous medium. In the latter half part of this section, we try to
overcome this problem by adding a virtual edge region. At the beginning, transmissions are
directly calculated without adding a virtual edge region. Then we investigate the effect on
transmissions after adding it.

200 300 400 500 600 700 800
0.65
0.7
0.75
0.8
0.85
0.9
0.95


324
direction (y-direction) perpendicular to the interface which is along the ΓΜ direction (x-
direction). The number of layers along the
x-direction is assumed to be infinite. The number
of layers along the
y-direction is still 30. The lattice constant along the x-direction is a
1
and
that along the
y-direction is a
2
. We choose a
2
= a = 100 μm and a
1
=
3
a
2
. The radius of all
superconductor cylinders is 0.2
a.
First, the width of the edge region
d is considered to be zero. N is fixed at 5 and M is
determined at the situation when the transmission is convergent. The frequency is chosen at
0.54 (
2πc/a). In Fig. 18, it is found out that M=600 is enough for calculation. Then N=5 and
M=600 are used to calculate transmissions from 0.01 to 1.00 (2πc/a). The transmissions of the
internal-field expansion method have some differences with those of the ADE-FDTD

2πc/a). In frequencies from 0.80 to 0.895 (2πc/a), the transmissions of the ADE-
FDTD method show the third zero-transmission region. This region exists between 0.845
and 0.955 (
2πc/a) in Fig. 19, which is 0.02 (2πc/a) larger than that of the ADE-FDTD method.
Next, we try to extend the boundary away from the edge of the cylinder by increasing the
width
d of the edge region. It is an imaginary boundary between air and the superconductor
PhC because the background medium of the superconductor PhC is also air. In fact, such
edge region doesn’t exist. The nonzero edge region implies that the results should have
something to do with the width of it. Several values of
d=0.5a, 1.0a, 1.5a, and 2.0a are
calculated and all of them are shown in Figs. 5-20(a)-(d). After comparing all results, we find
out that the nonzero edge region only affects transmissions below 0.17 (
2πc/a), where the
dielectric function in Eq. (9) is negative. The transmissions above 0.17 (
2πc/a) are almost
unchanged. So it explicitly reveals that this method is not suitable for negative dielectric
function.
To summarize, some transmissions of the internal-field expansion method are close to those
of the ADE-FDTD method, and some frequency regions have relative shifts between two
methods. Roughly speaking, the shift is about 0.06 multiplying the frequency, so it is
obvious that all zero-transmission regions below 1.00 (
2πc/a) broaden in the internal-field
expansion method. In Fig. 21, both results of the internal-field expansion method and the
ADE-FDTD method are shown, in which the frequency scale of the ADE-FDTD method is Photonic Band Structure and Transmittance of the Superconductor Photonic Crystal

325

a) (b)
0 0.1 0. 2 0.3 0. 4 0.5 0. 6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (2πc/a)
Transmition
0 0.1 0. 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (2πc/a)
Transmission



326
multiplied by 1.06. It can be seen that most results of two methods can match each other
much better after 0.17 (
2πc/a). We find that the calculations of the internal-field expansion
method exists some errors, which need to be overcome. It still cannot solve the problem,
even the edge region is added in calculations. In order to match the results of the PBS and
the ADE-FDTD method, the internal-field expansion method needs to be modified in some
way.
8. Conclusion
This study focuses on the transmissions of the two-dimensional superconductor PhC. The
PhC, composed of copper oxide high-temperature superconductor rods in a triangular array
in air, can be tunable utilizing the temperature modulation. We use the plane wave
expansion method introduced in Section 2 to calculate the PBS of it, which is much like a
metallic PhC system described by the Drude’s model if the normal conducting current is
ignored. The frequency of the fundamental mode in the superconductor PhC is far above
zero. It is the reason that the dielectric function is positive when frequency is more than
(,)
s
p
x
y

, the plasma frequency of the superconducting electron. Because both the electric
susceptibility and magnetic permeability have to be either positive or negative, light has the
ability to propagate through the superconductor.
Then we use the ADE-FDTD method introduced in Section 4 to calculate the transmission
when light is normally incident from air into the superconductor PhC. The results of the ADE-
FDTD method are consistent with the PBS and also verify the frequency of the fundamental
mode is more than ( , )

Photonic Band Structure and Transmittance of the Superconductor Photonic Crystal

327
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15
Electrodynamics of High
Pinning Superconductors
Klimenko E.Yu.
National Research Nuclear University “MEPhI”
Russian Federation
1. Introduction

High pinning superconductors (HPSC) hold much promise for power engineering. There are
some annoying consequences of up-to-date unequal state of these materials electrodynamics.

330
properties along the wire length and the foil area. The niobium–titanium conductors are
convenient for pinning anisotropy studying. Sufficiently high anisotropic pinning is not
complicated with critical field anisotropy. No other commercially available material offers
these advantages. These advantages make it possible to study the general laws of
electrodynamics in technical superconductors [Klimenko et al., 1997], which are almost not
masked by specific features of particular samples. Experiments with niobium–titanium
wires are not complicated by brittleness and high sensitivity to straining; those are typical
features of intermetallic compounds and HTSC. On the other hand, niobium–titanium wires
are by no means a simple material. These wires were displaced from the focus of research,
not even having been exhaustively studied. Later, it was found that niobium–titanium alloys
are two-component (as manifested by a difference in the critical fields of the grain body and
boundaries [Klimenko et al., 2001a]) and are characterized by anisotropic pinning in the
cross section of a wire [Klimenko et al., 2001b].
We used a commercial monofilament copper coated Nb-50wt%Ti wire 0.15 mm in diameter.
Superconducting core diameter was 0.12 mm. Several dozens of voltage-current curves were
recorded in magnetic field range from zero up to B
c2
[Klimenko et al., 2005]. The curves were
converted into ohm-ampere ones. The latter were linear in semi-logarithmic scale at several Fig. 1. Dependence of reduced resistivity of Nb-Ti wire on magnetic field at zero current
[11]. The resistivity was obtained by the extrapolation of ohm-ampere curves to zero
current. Some of the used ohm-ampere curves are shown on a panel. Another panel
illustrates the extrapolation procedure.

Electrodynamics of High Pinning Superconductors
331
orders of resistance values. There were two reasons to use just ohm-ampere curves. Firstly, a

|






]} (1)
The law was derived from assumption that a critical surface (



+



+



=1)in its
traditional form should be replaced with a smooth transition layer. The comparison is not
complicated because expression (1) contains only one fitting parameter: critical current
corresponding to effective resistivity equal to a half of normal one at zero temperature and
zero magnetic field. In the strict sense, T
c
and B
c2
are also fitting parameters, but they must
be closely alligned to the critical values measured at small current. Fig.2 shows that this


)
=

(−) (3)
The part of the conductivity tensor odd with respect to magnetic field is antisymmetric with
respect to transposition of indices and determines the physical phenomena such as Hall
Effect etc. This part does not contribute to the heat generation and is small in
superconductors. Therefore, we restrict ourselves only with the symmetric part of this
tensor. For isotropic superconductors in magnetic field the symmetric part may be
presented in the form

Superconductivity – Theory and Applications
332


=



−



+







(
,
|

|
,
|

|)
=0 (5)
where |B|and |j| are modules of a vector. Due to intrinsic textural inhomogeneity of HPSC
the actually observed transition to resistive state is smoothed at the vicinity of critical state.
The tensor approach requires generalising of (1) for the transverse and longitudinal parts of
electrical conductivity in superconductors:


=

[1+exp



] (6)


=1−




0,
is Kronecker delta. It has nothing common with δ in (1, 6 and 9). By
analogy with (6-8) longitudinal conductivity may be written


=

[1+exp



] (9)

Electrodynamics of High Pinning Superconductors
333


=1−



−
|

|


−
|


kinetic transition from superconducting to normal state. In the limit δ → 0 our scheme tends
to the critical state model [Bean, 1962]. As δ varies in the range 0.05–0.005, the values of T
c

and B
c2
(0) appear to be close to the related thermodynamic quantities (generally, these
parameters may be redefined if needed).
There is no trace of thermal activation process in the experimentally approved HPSC
constitutive law. An alternative model is statistic one. [Baixeras&Fournet, 1967]. Fig. 3. These parameters weakly depends on temperature as well as on magnetic field.
It appeared that (1) described well transition of a model multilinked network consisting of
superconducting elements if their critical temperatures had been normally distributed
around certain mean value (Fig.4a) [Klimenko, 1983, 1985]. This model bears relation to
“bulk inhomogeneous” superconductor such as monofilament wire. Another type of wires
was current important in eighties. It was multifilament wire with broken filaments. This
type needed another approach. We proposed regarding it as “longitudinally
inhomogeneous” [Dorofeev et al., 1980].
=





√




assumed that this line separates voltage-current curves with positive and negative
curvatures, the negative curvature considering as an evidence of the true superconducting
condition. Fig.4c hints that irreversible line, perhaps, has no real sense. In fact, every curve
changes its curvature from negative value to positive one at certain resistance level. The
positive curvature itself arises due to stretching the current coordinate by logarithmic scale
at low currents.

Electrodynamics of High Pinning Superconductors
335
3. Electrodynamics equations
The electrodynamics of isotropic HPSC may be described in terms of the general
quasistationary electrodynamics of a continuous media. In fact all the known HPSC are
anisotropic. However, isotropic electrodynamics is ever considered as a necessary step
[Klimenko et al., 2010].
=−


(13)
= (14)
=

 (15)
Here E and B are the electric and magnetic fields, respectively, j is the current density, and
μ
0
is the magnetic constant. At a boundary of superconductor 1 and normal metal 2 the
following components must be continuous

()
=


)

)
Instead of electric and magnetic fields, it is more conveniently to use the scalar and vector
potentials,
= (19)
=−∇−


(20)
In terms of potentials (19) and (20) the equations of electrodynamics are given by
∇

A

=μ

σ

(∇

φ+



) (21)
∇



thermal conductivity, and the heat generation G(T) is determined as
G()= (24)

Superconductivity – Theory and Applications
336
The constitutive law (6,9) permits to enclose the set of equations.
A package of computer codes was developed on the basis of Eq(21-24) for real geometry and
heat exchange condition. It provides a possibility of stability and AC loss computation for
arbitrary cycles of external magnetic field and current. The results will be soon published on
behalf of the whole team.
4. 2D voltage-current curves
The introduced above tensor conductivity is in contradiction with a widespread belief that
current and electric field are collinear in isotropic superconductor: =



. [Carr, 1983].
W.J. Carr had supposed it as an intuitive generalization of Bean model. However the
generalisation has appeared wrong. It is right only for the case of magnetic field
perpendicular to current, as well as Bean model. Indeed moving vortices generate electric
field in a plane normal to magnetic field. This field must be tilted to the current, if later does
not lie in the plane.
=
[

]
=

[−
(

rolling direction.
We cannot yet offer something being equivalent to the theory described in part 3. An
approach [Klimenko et al., 1997] was developed in frames of critical state model. That time
we used Critical Lorentz Force for critical state description. Critical Lorentz force (scalar)
related to unit superconductor volume is a radius of certain closed surface (called “pinning
surface”) in a space of Lorentz forces. Notice, subsequent reasoning always relates to unit
superconducting volume. In the case of isotropic superconductor critical Lorentz force
doesn’t depend on Lorentz force direction and the pinning surface is 3D sphere. In the case
of anisotropic pinning the critical Lorentz force depends not only on Lorentz force direction
but also on magnetic field direction. So the pinning surface must be constructed in 5D space:
three components of magnetic field plus only two components of Lorentz force because the
Lorentz force is always normal to magnetic field.
The following procedure was proposed as a simplest option of the pinning 5D-surface
construction. It is well known that energy of pinned array of vortices is less than energy of
free one. It means that the pinning array forms a potential well, which may be described with
three parameters: height, width and steepness of a potential barrier. The height is an energy
gain of pinned magnetic flux at rest. The width is a distance between nearby positions of the
flux with the same energy gain. It is obvious that the width equals to the least of two mean
values: distance between pinning centers or between vortices. The barrier steepness is a
maximum derivative of the flux energy with respect to coordinate when the flux is displaced
from the rest position. If shapes or distribution of pinning centers are anisotropic, the barrier
parameters are described with tensors corresponding to certain ellipsoids which main
diameters are aligned with the main directions of the pinning centers array.

Superconductivity – Theory and Applications
338
Tensor U corresponds to the barrier height:
U=



(27)
here =

|

|
is a unit vector in magnetic field direction.
Tenzor L will help to calculate critical Lorentz force in direction l, which is 


=max(


),
L=






00
0




0
00



|
is a unit vector in Lorentz force direction. and




=(L)

 (30) Fig. 7. The depth of potential well as well as the half-width of the potential barrier are
described either with the symmetrical tensors of second valence or the ellipsoids
corresponding to the tensors. The pinning surface main 2D cross sections are shown.