COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Simulation on Vortex Effect for Superconducting Devices
S. L. Lei 1, M. I. Lao 2*, I. N. Chan 2
1
2
Department of Mathematics, University of Macau, Macau, China
Department of Electrical and Electronics Engineering, University of Macau, Macau, China
Email: [email protected]
Abstract: The 2D time-dependent Ginzberg-Landau (TDGL) equations in the non-equilibrium state are applied for
the study on the properties of thin-film superconductor under a magnetic field. The numerical study includes of
modeling the dynamics of vortices in superconducting flux flow transistor (SFFT) and the oscillation change of
order parameter in a narrow superconducting ring. The discretion method to the equations by staggered grid is
established during the simulation. The simulation results are analyzed and shown relation to the measured data or the
real circuit.
Key words: vortex, time-dependent Ginzberg-Landau (TDGL), high-temperature superconductors (HTSC),
superconducting flux flow transistor (SFFT), superconducting quantum interference device (SQUID)
INTRODUCTION
The discovery and potential application of high-temperature superconductors (HTSC) enhance recently study on the
dynamic properties of vortex superconductivity [1-4]. One of studied model is the well-known time dependent
Ginzberg-Landau equations (TDGL) of superconductivity [5, 6]. In the Ginzburg-Landau theory the electromagnetic
field can be applied to a superconductor and the related electron motion can be described by solving the pair of
non-linear partial differential equations. Abrikosov pointed out that vortex could exit in the superconductor under
external magnetic field if the parameters of penetration depth and coherent length are large. In other words, the
external magnetic field forming flux lines penetrates into the superconductor. The superconductor with this property is
named type-II superconductor. The quantized flux line regarded as a vortex is encircling by superconducting current in
the nearby superconducting region.
Two main devices are for study subjects in this paper: the Superconducting Flux Flow Transistor (SFFT) proposed by
Martens et al. [7] and the superconducting ring. The study in superconducting ring shows the relation to the
Superconducting Quantum Interference Device (SQUID) [8]. We use Ginzburg-Landau theory to describe the
behavior of vortices for a SFFT device and the oscillation motion of order parameter in the superconducting ring. The
discretized Helmholtz energy function is used in the numerical scheme. The boundary conditions depended on the type
of devices and the values of different applied magnetic field are set for each side at grid edge. The simulation on the
dynamic behavior of vortex in a SFFT is compared with the result from Hiroya Andoh [9] and the oscillation motion of
order parameter is studied further to relate to the dc SQUID’s current [10].
THE SEMI-DISCRETIZED TDGL (SDTDGL) MODEL
In the GL model the electro-magnetic state of a superconductor is determined by two functions (ψ, A) in the studied
region and time. The Helmholtz energy in Ginburg-Landau theory [11] can be written as
2
⎛
1 4
∇
2
2⎞
G (ψ , Α) = ∫ Ω ⎜ − ψ + ψ + ( − i Α)ψ + ∇ × Α − Η ⎟ d Ω
⎜
⎟
2
κ
⎝
⎠
(1)
where ψ is the complex-values order parameter ψ = ψ 1 + iψ 2 whose complex conjugate denoted by a superscript *.
In our work, Α is a two-dimensional real-valued magnetic vector potential with two components expressed as Α =
⎯ 313 ⎯
(Ax, By), Η is a z-direction uniform applied field, κ is the GL parameter and Ω is the region occupied by the
superconducting sample.
Cited from Erhan Coskun and others [11, 12], the relation between G, A and ψ can be written as
∂ψ
δG
=−
in Ω
δψ *
∂t
(2)
∂Α
1 δG
=−
in Ω
2 δΑ
∂t
(3)
It expresses that the gradient flow of energy function with ψ and A is related to the opposite change with time to the
ψ and A, respectively. With the help of using the bond variables as
iΚ
W ( x, y ) = e ∫
iΚ
V ( x, y ) = e ∫
y
x
A (ζ , y )d ζ
(4)
B ( x ,η )dη
(5)
Then, W and V have the expressions as
∂ x (W *ψ ) = (∂ x − κ iA)ψ
(6)
∂ y (V *ψ ) = (∂ y − κ iB )ψ
(7)
And Eq. (1) can be rewritten in two dimensional form as
2
2
⎛ 1
⎞
1 4⎞
1
2
⎛
G (ψ , A, B) = ∫ Ω ⎜ − ψ + ψ ⎟ dxdy + ∫ Ω ⎜ ∂ x (W *ψ ) + ∂ y (V *ψ ) + (∂ x B − ∂ y Α − Η ) 2 ⎟ dxdy
⎜κ
⎟
κ
2
⎝
⎠
⎝
⎠
(8)
And the boundary conditions are expressed as
(∇ × Α) × n = Η × n on Γ
(
∇
κ
(9)
− iΑ)ψ ⋅ n = 0 on Γ
(10)
where Γ denotes the boundary of Ω and n is the outward surface normal direction. Eq. (4) is for the magnetic
field at the boundary. Eq. (5) states that the normal component of super-current vanishes at the boundary.
During the calculation we discretize the variables in energy function (1) onto the staggered grid-points as shown in
Fig. 1.
Figure 1: Staggered Grid for evaluation points of ψ(△), A(□), B(◎) and H(☆)
In staggered grid, the lattice points of ψ , A, B and H are evaluated, respectively, as the following:
ψ ij = ψ (ihx , jhy )
(11)
1
Aij = A((i + )hx , jhy )
2
(12)
1
Bij = B (ihx ,( j + )hy )
2
(13)
⎯ 314 ⎯
Η ij = Η (hx , hy )
(14)
in where hx and hy are the spatial increments in the x-direction and y-direction. By this way, the discrete TDGL
is obtained by minimizing the discretized energy function G. The SDTDGL can be rewritten as
s
⎛ eiAκ hxψs − 2ψ + e − iAκ hxψr eiB↓κ hyψ ↓ − 2ψ + e − iBκ hyψ ↑ ⎞
⎜
⎟ + hx hy N (ψ )
+
⎜
⎟
hx2
hy2
⎝
⎠
G
⎛ B − B + B ↓ − B2 A↑ − 2 A + A↓ ⎞ hy
G
∂A
= − hx ⎜
+
− N ( A,ψ ,ψ )
⎟
⎜
⎟ κ
∂t
hx
hy
⎝
⎠
∂ψ hx hy
= 2
k
∂t
H
G
H
⎛ A − A + A↑ − A3 B − 2 B + B ⎞ hx
∂B
↑
= − hy ⎜
+
⎟⎟ − N ( B,ψ ,ψ )
⎜
∂t
h
h
κ
y
x
⎝
⎠
(15)
(16)
(17)
where
2
N (ψ ) := (1 − ψ )ψ
r
r
r
r
r
N ( A,ψ ,ψ ) := (ψ 1ψ 2 − ψ 2ψ 1 )cos( Aκ hx ) − (ψ 1ψ 1 + ψ 2ψ 2 )sin( Aκ hx )
(19)
N ( B,ψ ,ψ ↑ ) := (ψ 1ψ 2↑ − ψ 2ψ 1↑ ) cos( Bκ hy ) − (ψ 1ψ 1↑ + ψ 2ψ 2↑ )sin( Bκ hy )
(20)
The discreted ideal boundary conditions from (9) and (10) are the following
r
↓
⎛
B−B⎞
↓ ikhy B
↓
ψ =ψ e
, A = A −⎜Η −
⎟ hy on ΩT
hx ⎠
⎝
r
⎛
B−B⎞
↑ − ikhy B
↑
ψ =ψ e
, A = A + ⎜Η −
⎟ hy on Ω B
hx ⎠
⎝
r
r ⎛
⎜
⎝
ψ = ψ e − ikh A , B = B − ⎜ Η +
s
x
s
s ⎛
⎜
⎝
ψ = ψ eikh A , B = B + ⎜ Η +
x
A↓ − A ⎞
⎟ hx on Ω L
hy ⎟⎠
(18)
(21)
(22)
(23)
A↑ − A ⎞
⎟ hx on Ω R
hy ⎟⎠
(24)
where Ω L , Ω R , Ω B and ΩT denoted the left, right, bottom and top edges of Ω , in sequence.
SIMULATION RESULTS
The simulation is written in MatLab program and run in desktop and notebook computers. The simulated SFFT is
outlined in Fig. 2. The current Ic in control line and bias current I in body provide a magnetic field difference across
the two sides of body. The situation is simulated by the different values of magnetic field at the boundaries during
the calculation. The effect of special and time steps are also investigated for higher accuracy of the results.
Figure 2: Layout of SFFT device
⎯ 315 ⎯
The variables changed during simulation include the material parameter κ , the magnetic field, scale and step of time
and space, and the boundary condition under special device and provided circumstance. The boundary condition is
also included the effect of material surface. For the result shown in Fig. 3, the typical data is given as followed: scale
is 30*30, grid point is 121*121, time step is 0.0125, the values of H at boundary of top, bottom, left and right are
1.0, 1.0, 0.4 and 1.0 respectively. The contour lines in the 2D plot describe the vortex in the sample. The density of
vortex shown by the distance between magnetic fluxes near surface and in the body is depended on the material
property and the applied magnetic field strength. The size of individual vortex is also varied by the strength field. If
combined with the calculation of flux motion rate, the relation among Lorentz force, viscous force and pinning force
can be further studied to fit for real samples.
Figure 3: Vortices motion in a magnetic field difference
The motion of flux is from right to left due to the field difference that causes larger current in the left side. The
larger current produces larger Lorentz force to move the flux. For example, a flux located at point (x, y) = (19.38,
5.75) at t = 53.75 moves to point (x, y) = (19.08, 5.76) at t = 55 as shown in Fig. 3. The arbitrary time unit in figures
is for calculation reference only. The calculated speed of flux is greater than that of other flux in the right side. With
this simulation result, it shows a distribution of bias current in the sample which will complicate the calculation for a
real large-size SFFT. To simulate a small size of sample, results shown in Fig. 4 and Fig. 5 are under the same
condition as that in Fig. 3 but with smaller scale. The graphs show a uniform arrangement and motion of fluxes. The
simulation also shows that the motion is changed according to the change of field strength at the boundary.
Therefore, the relation between the applied control current to the measured voltage can be found by the interaction of
uniform moving flux with uniform bias current.
Figure 4: 3D graphics for ψ
Figure 5: Flux motion in a time sequence
⎯ 316 ⎯
To simulate a superconducting ring, the boundary condition for the internal area is added during the calculation. The
value of order parameter ψ at grid points in the center region is set to zero but the value of vector potential at grid
points is followed to the change of the nearby regions because the flux and field can enter the whole region. One
interested result form the simulation of a superconducting ring is the periodic change of order parameter along the
ring as shown in Fig. 6 where selected graphs are shown in a series for reference. But the change pattern within a
period is actually different and related to the use of parameters during the simulation; for example, the strength of
magnetic field and the width and size of the ring sample. However, the oscillation period regarding to the time step
is basically kept in a constant under the suitable parameters. In other words, it is oscillating in a fixed frequency. To
find a related real sample for interpreting this simulation result, a possible device is the SQUID in which a bias
current is flowing. The boundary condition in the simulation is similar to that of SFFT where a current flowing
between top and bottom sides. Therefore, the left and right sides are the paths of current. In Fig. 6, the order
parameter representing the electron density is alternating along the two paths. That indicates the change of current in
the paths. To be a result, the measured voltage across the top side and the bottom side will alternate. As known for a
dc SQUID with an appropriate bias current, the measured voltage oscillates in applied magnetic flux. However, the
relation between the dc SQUID and the simulated superconducting ring is under further study and comparison.
Figure 6: Periodic change of order parameter in a superconducting ring
The graphs #2 and #11 in Fig. 6 show the minimum of order parameter in both paths that implies a minimum current
across the sides of top and bottom. This minimum appears repeatedly in the simulation with really equal time step
period as shown in Table 1 from one of simulation results.
Table 1 List of time steps for ψ minimum appeared in both sides
Time series →
1
2
3
4
5
6
time steps ↓
29
2475
4908
6325
7725
9144
10547 11964 13385 14783
188
2616
5044
6464
7865
9280
10703 12103 13524 14922
332
2757
5181
6600
8021
9420
10840 12241 13660 15062
474
2897
5336
6739
8160
9557
10979 12384 13799 15203
617
3040
5478
6878
8296
9700
11116 12538 13936 15355
1772
3183
5612
7019
8433
9855
11255 12675 14076 15496
1913
3327
5752
7175
8571
9991
11395 12815 14216 15631
2052
3471
5888
7312
8710
10130 11535 12951 14372 15771
2194
4628
6028
7448
8851
10267 11689 13090 14509 15909
2335
4770
6170
7587
9005
10406 11828 13230 14646 16051
⎯ 317 ⎯
7
8
9
10
Irregularity in the oscillation occurs during the very beginning of simulation and in the steps between 617 and 1772,
3471 and 4628. The other data appears a repeated period in the range from 130 to160 time steps; with an average
value of 145+15 time steps. Depended on the setting for simulation, one step for the data in Table 1 is after a 20
circles of calculation.
CONCULTION
Based on the SDTDGL method, a simulation and computation program for analyzing superconducting device is
established. Using the program on devices of SFFT and superconducting ring shows results either be similar to the
published data or be novel for further study by comparison to other theories and well known results. In other way,
the simulation results in fact reflect the program is reliable. Due to the time dependence nature of the equation
applied in the simulation, the program can actually research the non-stable state during the operation of the device. It
will provide more information for a device not only on the steady state.
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