Weakly Coupled Grain Model for High Tc Superconducting
Thin Films Taking Account of Anisotropic Complex
Conductivities

K.Yoshida1, H.Takeyoshi1, H.Morita1, H.Kanaya1, T.Uchiyama2, H.Shimakage2 andZ.Wang2
1

Department of Electronic Device Engineering, Graduate School of Information Science andElectrical Engineering, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581,
JAPAN2KARC Communications Research Laboratory, 588-2 Iwaoka, Iwaoka-cho, Nishi-ku, Kobe 651-2401,
JAPANAbstract: An analytical solution of the London equation for the weakly coupled grain model ofhigh-Tc superconducting thin films has been obtained in the case of finite thickness by taking fullaccount of anisotropic conductivities. Using the solution, we provide general expressions for thetransmission-line parameters of high-Tc superconducting transmission lines. Dependences of theresistance on the grain size, coupling strength and film thickness have been numerically evaluatedand discussed.

1. Introduction
Intensive studies have been made of applications of high-Tc superconducting transmission lines,
which have low loss and low dispersion characteristics, to microwave devices for mobile andsatellite communications. In these applications the precise evaluation of the transmission-line
parameters of superconducting thin films such as the resistance and the kinetic inductance, which
result from the complex conductivity, is essential. It has been pointed out by Hylton et al (1988)(1989a)(1989b) that these basic parameters are greatly influenced by the weaklinks(Josephsonjunctions) inevitably existing in high-Tc thin films, i.e., the so-called "weakly coupled grain (WCG)
model". As a result of this model, it is supposed that the weaklink gives rise to the excess surfaceresistance (Attanasio et al 1991)(Miller et al 1988)(Nguyen et al 1993)(Oates et al 1993) and tends toenhance the kinetic inductance of polycrystalline films compared with that of the single crystal(Porch et al 1993)(Yoshida et al (1996)(1998)). Detailed studies of this model, however, have notbeen made, especially in terms of anisotropic properties of high Tc superconducting films.
In our previous papers (Yoshida et al (1996)(1998)) we have carried out experimental studies on
the WCG model neglecting the anisotropy of high Tc superconducting films. In this paper we make,for the first time, a general theoretical formulation of the effects of anisotropy on the WCG model. InSec.2 an analytical solution of the London equation for the weakly coupled grain model of high Tc
superconducting thin films has been obtained in the case of finite film thickness by taking fullaccount of anisotropic conductivities. Using the solution, we obtained the expressions for thetransmission-line parameters of high Tc superconducting transmission lines in Sec.3. Dependences ofthe resistance on the grain size, coupling strength and film thickness have been numericallyevaluated and discussed.

2. Expression for the Magnetic Field Distribution in the Anisotropic Superconducting
Thin Film with Grain Boundaries

In Fig.1 we show the schematic figure of the weakly coupled grain (WCG) model proposed byHylton et al (1988) for a c-axis oriented high Tc superconducting film, where the c-axis of thesuperconducting film is perpendicular to the substrate. The average grain size is assumed to be a, andthe film thickness is d.
In Fig.2 we show the cross section of the superconducting film representing the WCG model,
where the external current K [A/m] per unit length in the y direction is assumed to be flowing in the zdirection.
grain boundary

Hy = K2aλz

Jzagraingrain boundaryz

K [A /m]grain

Jxsuperconducting
film

dyd

Jxx

substrate

Hy = – K
2Fig. 1 Schematic of weakly coupled grain(WCG) model.

2 λxλzaFig.2 The cross section of the superconducting
thin film

The London equation for the anisotropic superconductor is given as (Hylton et al 1989)(VanDuzer et al 1981)
222 ∂ Hy2 ∂ Hy+ λz= Hyλx22∂z(1)∂xwith the boundary condition at the grain boundary :∂2H y∂H∂ x2λxy−+ λ (0)2 ∂z = 0λ (0)J =Φ02π µ 02λ x J cJatz=± a2(2)withwhere λx and λｚ are the magnetic penetration depth in the x and z direction, respectively, Jc is thecritical current density for the Josephson junction and λJ(0) corresponds to the penetration depth for
an isolated single Josephson junction. According to the symmetry and periodicity of the presentproblem, other boundary conditions for a single grain located in the region –a/2≦z≦a/2 can be
expressed as
H y(x =0) = – K2(3)H y(x =d) = K2(4)The solution of Eq.(1) satisfying boundary conditions Eqs.(2)-(4) is obtained as :sinh 1 (x – d )2λz=(K /2)dsinh ()2λ z

H y(x,z)∞1
Σ 8nπn = 1 d 2 [1 + λ 2( 2nπ ) 2]z d–sin( 2nπ x) cosh (d2[( 2nπ ) cosh ( a2λ xd2
1 + λ 2z ( 2nπ ) ) +d2nπ 2
1 + λ2z( d )z)λx2nπ 21 + λ2z( d )sinh ( a2λ x
λ 2J(5)2nπ 21 + λ2z ( d ) )]It is shown that in the limit of d = + ∞, Eq.(5) coincides with the solution given by Hylton et al(1989a) valid only for the case of a semi-infinite superconductor. Using this solution, the current
distribution can be obtained from the Maxwell equation J = ∇× H .

3. The Equivalent Circuit for the High Tc Superconducting Transmission Line with
Grain Boundaries
For the case of an applied alternating current as Ieｊωｔ, we obtain the expression for the total
current density J as
J = J N + J S = ( [σ1] – j [σ2] ) E(6)with
σ 1x 0 0[σ 1] = 0 σ 1y 0 =0 0 σ 1z

σ 2x 0 0[σ 2] = 0 σ 2y 0 =0 0 σ 2z

1ρc000 ρ1 0ab0 0 ρ1ab1
ωµ 0λ 2c0001
ωµ 0λ 2ab0001
ωµ 0λ 2abwhere JN is the normal-conducting current density, JS is the superconducting current density, E is theelectric field, μ0 is the vacuum permeability, λx = λc is the magnetic penetration depth along the caxis and λy = λz = λab is the magnetic penetration depth in the a-b plane, [σ1] is the normal
conductivity tensor, σ1x = 1 / ρc , σ1y =σ1z = 1 / ρab, ρc is the resistivity in the c direction andρ ab is the resistivity in the a-b plane, [σ 2] is the superconductivity tensor, ω is the angularfrequency.
In Fig.3 we show the equivalent circuit for the transmission line with a unit length made ofanisotropic high Tc superconducting films including grain boundaries. In this figure Lm represents theconventional magnetic inductance per unit length, which is almost independent of the conductormaterial, and LkG and LkJ denote the kinetic inductance of the superconductor per unit length for thegrain and the grain boundary, respectively. RG and RJ represent the resistance per unit length resulted
from the grain and the grain boundary, respectively.
The expressions for the kinetic inductance and the resistance of the grain and the junction can beobtained by calculating the kinetic energy of the superconducting electrons and the balance of thepower consumption, respectively :1L I 2 =2 K1R I 2 =2

Re [ 1 µ 0[λ 2] J S ⋅ J S *] dv
2Re [ 1 (J⋅E *)]dv
2(7)

(8)

where I is the total current as shown in Fig.3 and the volume integral extends over unit length in the zdirection. The obtained results are :ωL kx

RG = 1
DQ x 1+Q –x 2+

ωL kz
Q z 1+Q –z 2

ωL kJ

RJ = 1DQ J 1+Q –J 2

L kx
L kz
+
L kG = 1
D 1 + Q– 2 1 + Q– 2xz(0)L kJ

L kJ = 1
D1 + Q –J 2(9)(10)(11)(12)withL kx =adx 1a 2a dz µ 0λ 2x f x2(x ,z )0–2L kz =adx 1a 2a dz µ 0λ 2x f z2(x ,z )0–2ddσQ x = σ 2x1σQ z = σ 2z1z
QJ =RNI RN= cf Φ0ωL
Φ0L (0)J = 2πIc

L (0)kJ =λJ =(0)d02dx µ 0 λ 2J J z (x , a )2Φ02πµ 0a J c

where Qx and Qz represent the quality factor for the current component Jx and Jz in the grain, QJ
represents the quality factor of the junction, RN is the junction resistance, LJ(0) is the kinetic
inductance for a single junction, Ic is the junction critical current, Lkx and Lkz are the kinetic
inductance per unit area (sheet inductance) associated with the current component Jx and Jz,
respectively, and λJ corresponds to the Josephson penetration depth in the small grain limit (Hyltonet al 1989a). The quantities fx and fz represent the current density for the case of a unit appliedcurrent, i.e, K=1 : They are defined byJx(x,y,z) = fx (x,z) K(y)(13)

Jz(x,y,z) = fz (x,z) K(y)(14)

Equations (9), (10) indicate that the resistances are proportional to the respective kineticinductances.
The quantity D in Eqs.(9)-(12) is defined byI

D=∞
– ∞

2(15)

K 2(y)dy

which corresponds to the geometrical factor representing the characteristic length for a particulartransmission line geometry. The characteristic length D defined by Eq.(15) depends on the geometryof the transmission line. If we assume that the film thickness is sufficiently thin and that thecharacteristic impedance is 50Ω on MgO substrate with the permitivity εr= 9.4, D is determined by
the geometrical configuration of the transmission line almost independent of the internal structure of
the conductor, and we can obtain numerically following the procedure given in Sheen et al (1991) :D = 0.4 W
D = 0.5 W

for coplanar waveguidefor micro-stripline

Eresistance per unit length [ /m]where W is the width of the signal electrode.
In order to evaluate the values for the resistance for various parameters, we first introduce thefollowing normalized parameters ; α= a / ( 2λx ), β= λz2 /λJ(0)2, γ= d / ( 2λz ) , where α is thenormalized grain size, β=λz2 /λJ(0)2 = (4πμ0 /Φ0)λxλz2 Jc represents the strength of the couplingand γ is the normalized thickness.
In Figs.4,5 and 6 we show the dependence of the resistances per unit length on α, β and γ,respectively. In the calculation we used the typical experimental values for YBa2Cu3Ox
superconductors (Friedmann et al 1990)(Hylton et al 1989) : λab = 0.15 [μm], λc = 0.5 [μm], ρc =2.5 × 10-5 [Ωm], ρab= 4.0×10-7 [Ωm], which leads to Qx =1.3×104 and Qz =2.3×103 at ω=2π×109 [Hz]. Junction parameters are : QJ = 4.8 × 102 for it IcRN=1 [mV] and f =109 [Hz]. It must be
mentioned that Eqs.(9) and (10) lead to the resistances proportional to ω2 in the case of Qx , Qz , QJ>> 1. The resistances shown in Figs.4,5 and 6 are calculated at ω=2π× 109 [Hz] in the case of10
D=100 [μm].
RLm LkG LkJRGRJCGRGJ(IcRN=1mV)
RJ(IcRN=10mV)

10.1
0.01
0.001

0.0001
10 ‑50.01

0.1

1normalized grain size

10.10.01
0.001

0.0001E

RG
RJ(IcRN=1mV)
RJ(IcRN=10mV)

10100

Fig.4 The dependence of the resistance perunit length on α for β=γ=1.
tance per unit length [ /m]10Etance per unit length [ /m]Fig.3 The equivalent circuit for the transmission
line with a unit lengthM1010.10.01

RG
RJ(IcRN=1mV)
RJ(IcRN=10mV)

Fig.5 The dependence of the resistance perunit length on β for α=γ=1.

Fig.6 The dependence of the resistance perunit length on γ for α=β=1.

4. Conclusions
In the present paper we obtained general expressions for the the resistance and the kinetic
inductance of the superconducting thin films with grain boundaries by taking full account of
anisotropic conductivities, which include the normalized grain size α, coupling strength β andnormalized thickness γ as free parameters. Detailed discussions with specific samples are to be ournext work.

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