Study of Anisotropy Superconductor using Time-Dependent Ginzburg-Landau Equation

Journal of Natural Sciences Research www.iiste.org
ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online)
Vol.3, No.15, 2013
Study of Anisotropy Superconductor using Time-Dependent
Ginzburg-Landau Equation
Fuad Anwar1,2*, Pekik Nurwantoro1, Arief Hermanto1
1. Department of Physics, University of Gadjah Mada, Sekip Utara, Bulaksumur, Yogyakarta 55281,
Indonesia
2. Department of Physics, University of Sebelas Maret, Jl. Ir. Sutami 36A, Kenthingan, Surakarta 57126,
Indonesia
* E-mail of the corresponding author: fuada70@yahoo.com
Abstract
We have observed an anisotropy superconductor which was immersed in vacuum medium in presence of an
applied magnetic field. The anisotropy properties of superconductor were related with two principal values of the
effective mass of the Cooper pairs, namely mc along the x-axis and mab in the yz-plane. Based on the time-dependent
Ginzburg-Landau and yU methods, the problem was solved and made to be the numerical simulation.
From study using this numerical simulation, we can find that the anisotropy properties can make the critical field
to be lower or higher.
Keywords: anisotropy, superconductor, time-dependent Ginzburg-Landau
1. Introduction
It is well known that the Time Dependent Ginzburg - Landau (TDGL) equations can be used to study the
dynamics of superconductivity phenomenon (Tinkham 1996; Du 2005). These equations have nonlinear nature,
so it will give results more completely to solve them numerically (Du 2005). One of the numerical solutions has
been developed using the gauge invariant variables technique or to be called yU method (Bolech et al. 1995;
Gropp et al. 1996; Winiecki & Adams 2002). In the last decade, the solution of TDGL equations using yU
method has successfully been used to study the dynamics of superconductivity phenomenon in thin films (Barba
et al. 2007; Barba et al. 2008; Barba-Ortega & Aguiar 2009; Barba-Ortega et al. 2010; Barba-Ortega et al. 2012;
Barba-Ortega et al. 2013; Wisodo et al. 2013). However, these studies considered the superconductors having
isotropy properties.
It is also well known that the high Tc superconductors have anisotropy properties (Tinkham, 1996). These
superconductors are viewed as the stack of layers, each layer comprises the ab planes and the c axis is normal to
them. The effective mass of the Cooper pairs is different when measured in the ab planes or along the c axis. The
results in earlier papers show that the anisotropy properties have influence on superconductivity phenomenon
(Hao & Hu 1996; Chapman & Richardson 1998; Achalere & Dey 2008).
In this paper, we study the dynamics of superconductivity phenomenon in an anisotropy superconductor using
the TDGL equations and yU methods. We describe the theoretical formalism and numerical method used for
solving the problem in section 2. The results and their analysis are discussed in section 3. Finally, we conclude in
section 4.
2. Numerical Methods
2.1 The Time-dependent Ginzburg-Landau Equations
The time-dependent Ginzburg-Landau (TDGL) equations are :
m
y y y y y s
99
(1)
(2)
e
i
ö
æ
ö
( ) ( ) ( t ) ( t ) T ( t ) t ( t )
m
Φ t t
e
i
æ
¶
m D t
s
s
s
s
, , ( ) , ( , ) ,
2
, ,
2
2
2 2 2
r r r r r A r r y y b y a y y - + ÷ ÷ø
ç çè
- Ñ = ÷ ÷ø
ç çè
+
¶
h
h
h
h
( ( ) ( ))
Ñ´ Ñ´ -
A r H r
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
ö
÷ø
æ
çè
¶
÷ø
ö
¶
+ - Ñ - æ
e
= Ñ - Ñ -
çè
t
t
t t Φ t
ie
t t t t
s
m i
t t
s
s
ext
,
, , ,
2
, , , ,
2
, ,
1
2
0
0
A r
r r r r r A r r
m
h
h

Vol.3, No.15, 2013
where y is order parameter, A and Hext denote the vector potential and an external magnetic field, es and ms are
the effective charge and the effective mass of the Cooper pairs, a dan b are phenomenological parameters, D is a
phenomenological diffusion constant, F and s are the electrical potential and conductivity (Bolech et al. 1995;
Tinkham 1996; Gropp et al. 1996; Winiecki & Adams 2002).
In this study, we considered an anisotropic superconductor which is immersed in vacuum medium in presence of
an applied magnetic field (The figure 1). Here, an anisotropic superconductor has layered structures, so it has
two principal values of ms, D and s, namely : mc, Dc and sc
ö
æ
¶
2 2
ö
æ
¶
y y
2
2 2
2 2 2 2
ö
æ
¶
y y ay by y
ö
æ
¶
2 2
2 2 2 2
æ
¶
æ
¶
-
¶
¶
ö
¶
-
y x y x
100
along the x-axis and mab, Dab and sab in the yz-plane.
The applied magnetic field is assumed in the z-direction, time-dependent, and spatially uniform, so we have
Ñ´Hext(t)=0, B=Bz(x,y,t)z and A=Ax(x,y,t)x+Ay(x,y,t)y.
Figure. 1 An anisotropy superconductor in presence of an applied perpendicular uniform magnetic field
If we apply the previous assumption to the time-dependent Ginzburg-Landau, we can obtain :
(3)
(4)
æ
¶
m m
e
e
A
¶
A
s
æ
æ
¶
¶
æ
¶
Then, we scale y in y0=(|a|/b)½, t in tc
= xc
ö
A
A
¶
¶
2/Dc, x and y in xc
2
ö
¶
+
¶
+
ö
A
¶
Φ
A
¶
Φ
ö
æ
ö
ö
æ
ö
=(ћ2/2mc|a|)½, Ax and Ay in A0x=μ0Hc2c xc
, F in
F0= xc
A0x/ tc
, sc
and sab in s0c=1/( μ0 kc
2Dc), and we choose F =0, so we can rewrite equations (3) and (4) in the
following form :
superconductor
x
z
y
Hext(t)
y y ay by y
2 2
- + ÷ ÷ø
ç çè
-
¶
+ ÷ø
çè
-
¶
+
- + ÷ ÷ø
ç çè
-
¶
+ ÷ø
çè
-
¶
=
÷ø
çè
+
¶
+ ÷ø
çè
+
¶
y
s
ab
x
s
ab
y
s
c
x
s
c
s
ab ab
s
c c
A
e
i
m y
A
e
i
m x
A
e
i
m y
A
e
i
m x
Φ
e
i
m D t
Φ
e
i
m D t
h
h
h
h
h
h
h
h
h
h
h
h
y
t
y
A y
ie
im y y
x
t
x
A x
ie
im x x
y
y
x
x
x
y
x
y
y
y ab
s
ab
x
x c
s
c
s
ˆ ˆ
2
2
ˆ ˆ
2
2
ˆ
1
ˆ
1
2
2
0 0
÷ ÷
ø
ç ç
è
¶
¶
- ÷ ÷ø
ç çè
-
¶
-
¶
+
÷ø
çè
¶
¶
- ÷ø
çè
-
¶
-
¶
=
÷ ÷
ø
ç ç
è
¶
¶
¶
- ÷ ÷
ø
ç ç
è
¶
¶
¶
y s
y
y
y
y
y s
y
y
y
y
h
h
h
h

Vol.3, No.15, 2013
æ +
+ ÷ø
2 2
2 2
ö
æ
ö
æ +
=
ö
e e 2
¶
æ
¶
¶
æ +
1 1 1 m D iA
÷ + - ÷ø
¶
c m 2m
m = =
l
k = =
c x
Hc = Hc =
a k
2c c ab ab k
l
x
2 = = = ab
= =
m
2
2
2
m Hc
ab
D D
s
ab
s
ò x x + = -
( ( ) ) x y
x
i j U i A y d i N j N i
, ò h h + = -
exp , with = 1,2,…, +1 and = 1,2,…, 1
- +
y y y
U t t t U t t
( ) ( ) 2 ( ) ( ) ( )
+ - -
, 1, , 1, 1,
2
x
y y y
æ
æ
- +
U t t t U t t
t
, , 1 , , 1 , 1
ö
101
(5)
(6)
y e
¶
A
¶
A
In equations (5) and (6), we define :
(7a)
(7b)
(7c)
(7d)
(7e)
(7f)
(7g)
e
ö
¶
-
A
¶
-
¶
A
A
¶
A
1
ö
æ
¶
ö
¶
æ
¶
æ
¶
ö
æ
¶
b
b
ab
c
m
2
2
ab
c
ab
ab
c
ab
where x is the coherence length, l is the penetration depth, and k is the Ginzburg-Landau parameter.
The equations (5) and (6) can be solved by the yU method (Bolech et al. 1995; Gropp et al. 1996; Winiecki &
Adams 2002). In this method, the sample is divided into Nx x Ny cells, with mesh spacings Dx
and Dy. At the cell,
there are three fundamental unknowns, namely y, Ux and Uy. The yi,j is order parameter at position (xi , yj), with i
= 1,2,…Nx+1 and j = 1,2,…,Ny+1. The Ux and Uy are the complex link variables and are related to A by :
(8a)
(8b)
x
x x j
Using the yU method and the Euler method and taking eD = 1, es = 1, sc
=sab = 1, the equations (5) and (6) can
be derived in the following form :
(9)
y ( y y)
e
y
e
e e
2
2
2
2
2 2
21
ö
ç çè
-
¶
÷ ÷
ø
ç ç
è
ö
çè
-
¶
÷ ÷
ø
ç ç
è
¶
÷ ÷
ø
ç ç
è
y
m
m
x
m
m
m D
y
iA
t x
y
y
x
x
i A y
i y y
x
y
x
y
i A x
i x x
y
t
x
t
y x
y c
m
y x
x c
y
ab
x
c
2 ˆ ˆ
2
1
2 ˆ ˆ
2
ˆ ˆ
2 2
2
2 2
÷ ÷
ø
ç ç
è
¶
¶
¶
+ ÷ ÷ø
ç çè
-
¶
-
¶
+
÷ ÷
ø
ç ç
è
¶
¶
¶
- ÷ø
çè
-
¶
-
¶
=
¶
+
¶
y k
y
y
y
y
e
y k
y
y
y
s s y
a
x
a
x
ab
ab
c
and
2
2
2
2
2 h h
= =
and
2
0
2
2
0
2
m a
l
m a
l
s
ab
s
c
e
e
and
2
2
2
2
ab
ab
c
l
k
x
2
and
2
0
2
0
m b
a k
m b
c
c
c
ab
c
Hc
m
2
2
2
k
l
x
e
c
D = 2 e
c
es = 2
( ( ) ) exp , with = 1,2,…, and = 1,2,…, +1 1
, x y
i
y
y y i
y
i j U i A x d i N j N j
j
y
1 ( ) ( )
æ
1
+
2
ö
( ) ( ) 2 ( ) ( ) ( )
( ) ( )
,
2
2 ,
2
2
, ,
e
t t t
t t t t
i j i j
m
m
i j
y
i j i j i j
y
i j
i j
x
i j i j i j
x
i j
i j i j
y y
e
y y
÷ø ö
çè æ
- ÷ ÷
ø
ç ç
è
+ D
ö
÷ ÷ ÷
ø
ç ç ç
è
D
+ D
÷ ÷ ÷
ø
ç ç ç
è
D
+ D = + D
+ - -

Vol.3, No.15, 2013
+ D = - D
( ) ˆ ( ) ˆ ( ) Im ( ) ( ) ( ) ˆ
U t t x U t x i tU t U t t t x
, , , , , 1,
U t (L t L t )x
D
t
y
( ) ( ) ( ) Im ( ) ( ) ( ) ˆ
U t U t t t y
, , 2 , , , , 1
i j i j L =U U U U » -iDxDyB + +
é
× - Ñ - ψ
¶
B B y x
ext z z ¶
j j y = y
y = x
y +
y = y
y
i i y = y
y i N +
i N exp( ) 1, 2, 1, 1 1, ext;z
U y
»U y
U x
+
U x
-iDxDyH j j
j
j
» - D D + +
y
N j U U U U i x yH
x
i U »U U U -iDxDyH +
» - D D + +
x
i N U U U U i x yH
102
(10)
(11)
where,
(12)
( )
D
t
2.2 The Boundary Conditions
In this study, we considered the superconductor is immersed in vacuum medium in presence of an applied
magnetic field, so we have the boundary condition for y and A, namely :
(13)
(14)
ù
A
A
where n denotes the unit normal to the superconductor–vacuum interface.
If we apply the previous research assumptions and scale y in y0=(|a|/b)½, x and y in xc
=(ћ2/2mc|a|)½, Ax and Ay in
A0x=μ0Hc2c xc
, Bz and Bext;z in μ0Hc2c, Hz and Hext;z in
Hc2c, then we can use the yU method and the Euler method
to rewrite equations (13) and (14) in the following form :
at i = 1 :
(15a)
at i = Nx+1 :
(15b)
at j = 1 :
(15c)
at j = Ny+1 :
(15d)
and
at i = 1 :
(16a)
at i = Nx+1 :
(16b)
at j = 1 :
(16c)
at j = Ny+1 :
(16d)
x
2.3 The Numerical Simulation
We begin the numerical simulation with determining the value of Nx, Ny, Dx, Dy, Dt, kc
and em . We also assume
the initial condition of superconductor as in a perfect Meissner state, so we have Hext;z=0, yi,j = 1, Ux
i,j=1 and
Uy
i,j=1. Then, Hext;z is increased linearly with time and with small intervals of DHext;z. When we have a new value
of Hext;z , we compute the new values of yi,j, Ux
i,j and Uy
i,j using equations (9), (10), (11), (15) and (16).
Using this numerical simulation, we can also make magnetization curves. Magnetization can be calculated from :
(17)
where Mz is scaled in Hc2c and Bz is calculated by equation (12).
i j i j
x
c i j
i j i j
x
i j
x
i j
x
i j
x
i j
( ) ( ) ( ) 1 ˆ
2 , , , 1
2
-
D
-
-
+
k
y y
( )
( )( ( ) ( ) 1)ˆ
2 , , 1,
2
U t L t L t y
x
i t
U t t U t
i j i j
y
c i j
i j i j
y
i j
y
i j
m
y
i j
y
i j
-
D
-
D
+ D = -
-
+
k
y y
e
y
y
x
x
exp( ) , 1, i , j
i , j
1 i , j
z;i, j
0 ˆ = úû
êë
e
i s n A
h
y
x
¶
-
¶
= = ;
( ) ( ) ( ) 1, 1, 2, t U t t j
( ) ( ) ( ) 1, , , t U t t N j
N j N j x x x
( ) ( ) ( ) ,1 ,1 ,2 t U t t i
( ) ( ) ( ) , 1 , , t U t t
y y y i N
y
x
x
exp( ) 1, N , j
N , j
1 N , j
ext;z
x x x x
y
y
x
exp( ) ,1 i
1,1 i
,1 i
,2 ext;z
y
y
x
exp( ) , 1 i 1, N
i , N
i , N
ext;z
y y y y
z z ext z M B H ; = -

Vol.3, No.15, 2013
3. Results and Discussion
We try to run the numerical simulation with choosing five cases, namely : em = 1.0, em = 0.5, em = 0.8, em = 1.3,
and em = 2.0. For each case, the values of another input are the same, namely : Nx = 32, Ny = 32, Dx
103
=0.5, Dy
=0.5,
Dt = 0.010, DHext;z= 0.000001, and kc
= 2.0. When em = 1.0, it is the isotropy superconductor case and the others
are the anisotropy superconductor cases.
In figure 2(a), we show the square modulus order parameter curve as a function of magnetic field - Hext 2 y in
the five cases. We can see in the figure, the |y(x,y)|2 will completely vanish at the higher value of the applied
field Hext;z when the value of em increases. It means, by increasing em the value of the surface nucleation field Hc3
will be higher.
In figure 2(b), we show the magnetization curve as a function of magnetic field M-Hext in the five cases. We can
see, as increasing the applied field Hext;z, the magnetization curve will decrease until the minimum value, then
increase until the zero value. When the value of em increases, we find that the minimum value will be located at
the higher value of the applied field Hext;z if em£1 and at the lower value of the applied field Hext;z if em³1. We
also find that the zero value will be located at the higher value of the applied field Hext;z when the value of em
increases. It means, by increasing em, if em£1 the value of the lower critical field Hc1 will be higher and if em³1
the value of Hc1 will be lower. It also means, by increasing em, the value of Hc3 will be higher.
We show the distribution of the square modulus order parameter |y(x,y)|2 on sample for five cases and for the
several values of the applied field Hext;z in the figure 3-7. From these figures, we can see that in the low applied
field Hext;z, |y(x,y)|2 has a high value. As increasing Hext;z, the magnetic field will penetrate into sample to form
vortex and |y(x,y)|2 will decrease. When the value of em increases, we find that |y(x,y)|2 will vanish in the higher
value of the applied field Hext;z, the vortex is formed in the higher value of the applied field Hext;z if em£1 and in
the lower value of the applied field Hext;z if em³1. Once again, these results mean that by increasing em, the value
of Hc3 will be higher, if em£1 the value of Hc1 will be higher and if em³1 the value of Hc1 will be lower.
0 0.5 1 1.5 2 2.5 3 3.5
0
- 0.1
- 0.2
Hext
M
(b)
Hext
2 y
0 0.5 1 1.5 2 2.5 3 3.5
0.8
0.6
0.4
0.2
0
(a)
Figure 2. Plot of (a) ext -H 2 y and (b) M-Hext in the cases of :
— : em = 0.5 — : em = 0.8 — : em = 1.0 — : em = 1.3 — : em = 2.0

Vol.3, No.15, 2013
Figure 3. Plot of |y(x,y)|2 in the case of em=0.5 and (a)Hext;z=0.10 (b)Hext;z=0.20 (c)Hext;z=0.30 (d)Hext;z=0.40
(e)Hext;z=0.50 (f)Hext;z=0.80
The white and the black colours indicate |y(x,y)|2=1 and |y(x,y)|2=0.
Figure 4. Plot of |y(x,y)|2 in the case em=0.8 and (a)Hext;z=0.10 (b)Hext;z=0.20 (c)Hext;z=0.30 (d)Hext;z=0.40
(e)Hext;z=0.50 (f)Hext;z=0.80 (g) Hext;z=1.60
(e)Hext;z=0.50 (f)Hext;z =0.80 (g) Hext;z=1.60 (h) Hext;z=2.20.
104

Vol.3, No.15, 2013
(e)Hext;z=0.50 (f)Hext;z =0.80 (g)Hext;z=1.60 (h)Hext;z=2.20 (i)Hext;z=2.80.
(e)Hext;z=0.50 (f)Hext;z =0.80 (g)Hext;z=1.60 (h)Hext;z=2.20 (i)Hext;z=2.80 (j)Hext;z=3.50
4. Conclusion
We have made a numerical solution of the Time Dependent Ginzburg - Landau (TDGL) equations for
anisotropic superconductor using yU methods. From this simulation, if we set kc
105
in the fixed value and
increase em, we will obtain the value of Hc3 will be higher and the value of Hc1 will be higher if em£1and
will be lower if em³1.
Acknowledgements
We thank to Direktorat Jenderal Pendidikan Tinggi (Ditjen Dikti)-Kementerian Pendidikan dan
Kebudayaan (Kemdikbud)-Indonesia for the support of our research through BPPS scholarship.
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