- Email: [email protected]
Studies on high-Tc superconducting microstrip resonator
a __
__ li!B
PHYSICA ti
ELSEVIER
Physica C 292 (1997) 83-88
Studies on high-T, superconducting
microstrip resonator
G.P. Srivastava a,*, Vincent Mathew ‘, Agnikumar G. Vedeshwar a Department
of Electronic
h Department
Science, Uniwrsity
of Physics and Astrophysics,
of Delhi South Campus, NW Delhi Uniwrsity
of Delhi. Delhi
’
110021, lndiu
11UOO7. Indiu
Received 17 December 1996; revised 18 August 1997: accepted 26 August 1997
Abstract We present a theoretical analysis of a high-T, superconducting microstrip resonator using spectral domain method. Fourier transformed impedance Green’s functions are derived and the resonant frequencies are calculated for three different substrates, as a function of the resonator length, in Galerkin’s procedure. Effects of thermal expansion of the substrate and surface impedance of the superconducting film are incorporated in the theory to analyze the temperature dependence of both the unloaded Q and the resonant frequency. The effects of these two mechanisms are analyzed separately and compared. It has been found that the thermal expansion of the substrate is mainly responsible for linear shift in the resonant frequency below 0.8 T,. The effect of surface impedance is found to be dominant only in a temperature range close to the transition temperature for shift in both resonant frequency and Q. 0 1997 Elsevier Science B.V. PACS:
85.X-j;
Kepwords:
84.40.Cb:
Microstrip
84.4O.Y~
resonator;
Superconducting
devices; Spectral domain method; Impedance
1. Introduction The growing interest in microwave studies of high-temperature superconducting (HTSC) materials has two origins. First, microwave studies provide a better understanding of the physics of these materials at high frequencies. Second, these materials find many applications in the microwave and millimeter wave devices and circuits [l]. Again, HTSC passive microwave devices are employed in the experimental investigation of the microwave properties of the superconducting materials [2]. Due to these reasons,
* Corresponding
author. Fax: +9l
0921-4534/97/$17.00 PII SO92 I -4534(97)0
I I 6886427.
0 1997 El.\evier Science B.V. All rights reserved 1697.3
Green‘s functions
analytical and experimental studies on the microwave superconducting devices are gaining much importance. In developing passive superconducting devices, theoretical and experimental analysis is very important for generating sufficient design data or for devising good design models. Full wave analysis methods like the spectral domain method [3] are usually employed in the analysis of microwave and millimeter wave devices and circuits which can produce results with sufficient accuracy. It has been applied successfully to many problems involving HTSC microstrip transmission lines recently [4,5]. In this paper we extend this method for the analysis of HTSC microstrip resonator.
84
G.P. Sricasrava
et al. / Physica C 292 (1997) 83-88
The dependence of the performance of high-T, superconducting (HTSC) circuits on the operating temperature is one of the important factors to be understood for properly designing such circuits. Experimental results [6] show considerable influence of temperature on HTSC circuit performance. This influence of operating temperature on resonant frequency and other parameters of a superconducting microstrip resonator can be attributed mainly to two mechanisms [6]: ( I> The thermal expansion of the substrate causes the resonator patch (HTSC film) to elongate in its length and changes the structural parameters. (Expansions in the other dimensions are neglected.) (2) The kinetic inductance of the superconducting material increases with temperature which will increase the surface impedance of the strip. Therefore, it is necessary to analyze the effects of these two processes on the performance of a HTSC resonator in order to obtain an exact view of this problem. As a preliminary investigation, we consider the simple case of a rectangular YBCO thin film HTSC microstrip resonator as represented schematically in Fig. 1. In the following, we present an outline of the theoretical formulation of the problem. First we calculate the resonant frequency of the resonator as a function of the resonator length. The effect of thermal expansion on the resonant frequency is studied separately for two substrate materials. The effects of the two mechanisms are compared and the combined effect of both these mechanisms is studied for the c__
2a
m--M
I
I
case of the typical LaAlO, substrate. The results of the numerical computation are discussed.
2. Theoretical
formulation
The advantages of using the spectral domain method, i.e. Galerkin’s procedure in Fourier transform domain for the analysis of microstrip like transmission structures are well discussed in the literature [7]. Out of the two mechanisms for temperature dependence, one enters the analysis through the structural parameter (thermal expansion) and the other through the material property (surface impedance) of the HTSC film. We discuss first the spectral domain formulation and then the temperature dependence. 2. I. Spectral domain formulation The spectral domain method is a very standard tool used in the rigorous analysis of field problems connected with microstrip circuit structures [7]. In spectral domain method, we write down the field components at the superconducting strip in the Fourier transform domain. Applying the boundary conditions and incorporating the complex conductivity of the superconductor, we derive the dyadic Green’s functions which will form the impedance matrix in the Fourier transformed domain. Expanding the strip current in a suitable basis functions, the resonant frequency is calculated in Galerkin’s procedure. The geometry of the HTSC resonator under consideration is schematically represented in Fig. 1. With respect to the geometry of the resonator, the following transformation is applied to all quantities for formulating the problem in the Fourier transform domain:
xexp[i( / t Substrate Fig. 1. The geometry of the HTSC microstrip resonator.
ax + Pz)]dxdz
(1)
where (Y and /3 are the transform variables [3]. The electric and magnetic field components in the substrate and in the air media are of hybrid nature and can be expressed as superpositions of the TE and TM electromagnetic fields with respect to the y-di-
G.P. Srioastaua et al./Phyica
rection. In Fourier transform
domain they are written
C 292 (1997) 83-88
fi;;x =[(w,k;
as, a,y,P)
= (k2 - P’)$;
(2.1)
H;;( (Y,Y,P)
= (kZ - P’)&
(2.2)
Exj< WY$)
= -Odi
- iwpi(Gi/aY)
(2.3)
fix;< ci.Y,p)
= -C&G;
+
i;;(
85
- “‘)~o
+pr(ki - ~2)yltanh(yld)]/A 6: = G_x,,
(8c) (84
with,
iwei(a&/ay)
A = [ y,tanh(y,d)
(2.4)
where the subscript i = I,0 designates the substrate or air region. Here +(x,y,z) and $(x,y,z) are two scalar potentials, the transforms of which satisfy,
+ ~,r,,]
x [Ylcow,4
+ wol
(8e)
The resonant frequency for the fundamental mode is calculated from Eq. (7) in Galerkin’s procedure. Expanding the strip currents JI, and i: using a set of basis functions,
(3a) (9a)
J:-= ; d,.L,(a$)
with y;=a2++s,prk;
(da)
y; = a* + p2 - k,2
(4b)
k, = (&r /4”*k,
(5a)
k, = ( co /.Q)”
w
(5b)
Dyadic Green’s functions are derived by applying the complex resistive boundary conditions for the superconducting strip [4], nx(E++Y)=O 12X (H+-H-)
f,
[
K;;;;“‘c, +
,,R, [ KI.3
Kj$‘d,]
= 0
( lOa)
c,, + K,‘,$d,]
= 0
( lob)
where 1= 1,2, . . , N and,
= -(l/Z,)nx
(n XE’)
(6b)
X~~,x((Y,p,ko)j”,,(LY,P)dp (lla) Kj,t,i”)(k,)= -,
where, Z, is the surface impedance of the HTSC film which will be discussed later and,
e.r,;=
+ pryltanh(Ml/A
(*a)
- P*)Y,tanh(M)]/A
,r;loli;,bP) X~~,,,~((Y,p,ko)jl*,(a,p)dp (11~)
Kk%)
=
nc, ~?r,(c4) d,(
[(Er+$-P*)YO +/-+;
/o?-hW
X~.~,~(CY,P,kO)j:-m((Y’P)dp (lib) K,‘.,“‘(k,) =
= o$[~,~
c, and d, satisfy [3]:
The coefficients
(6a)
in spectral domain. Here n is the unit vector normal to the HTSC strip and the superscripts k refer to the field components on the top and bottom of it. The resulting equation is given by,
&
(9b)
lIl=l
(8b)
Resonant frequency and (lob) by setting
wP&)j:-m( a>P)dP (lid) is calculated from Eqs. (lOa> the coefficient matrix equal
zero. The same set of basis functions used in Ref. [3] has been used here. Once the resonant frequency is estimated. the spectral domain method can again be used to find out the attenuation constant (k,) and propagation constant (k,) at the same resonant frequency for an infinitely long microstrip line with the same cross sectional geometry. This procedure is detailed in Ref. [4]. Now the unloaded Q can be estimated from the relation [8], ’
Q,, = 27r{ I - exp( -47rk;,/k,)}
2.2. Tetnperature
( 12)
dependence
The temperature dependence of both resonant frequency and unloaded Q of the microstrip resonator can be computed by the incorporation of the thermal expansion of substrate and temperature dependence of surface impedance of the strip material. The effect of thermal expansion of the substrate along its length can be represented by L(T)
= L(O){ I + CY’T}
(13)
where cy’ is the coefficient of thermal expansion the substrate. The surface impedance is given by [I], iwp Z,=R,+iX,= i c( w,T)
1
of
l/2
(14)
where (r = U, - i (T? is the complex conductivity of the superconductor given by the two-fluid model. The temperature dependence of R, and X, results from that of U, and A,~ (London penetration depth) respectively. The main contribution to R, comes from normal electrons with density n, which has a temperature dependence of t” (t = T/q) in two-fluid enmodel and e P’/‘liT (A is the superconducting ergy gap) in BCS theory. The temperature dependence of (T is governed by the ratio of superconducting electron to normal electron densities as tz,/n, = (1 - t’)/t’ in two-fluid model which is quite close to that predicted by BCS theory. However, the difference between the values of surface resistance, R,. calculated by two-fluid model and BCS theory is not too high. Both are qualitatively similar. Therefore, it is quite customary to employ two-fluid model in
most of the calculations concerning superconducting microwave devices in the absence of a well developed microscopic theory as in the case of high temperature superconductors [9].
3. Results and discussion In the calculations, we have taken YBCO thin film as the resonator patch (strip) with parameters ?=92 K and A,=0.15 pm at T=O K. The dimensions used are a = 5 mm, w = I mm and d = 2 mm. The relative permittivity and the coefficient of thermal expansion for three commonly used substrate materials [ 1, IO] are listed in Table I. The resonant frequencies are calculated as a function of resonator length using the above formulation and are shown in Fig. 2. The variation of resonant frequency with the resonator length is of the same form as that of normal resonators [3]. In order to see the effect of thermal expansion on the resonant frequency, it has been calculated separately for LaAlO, and MgO substrates as a function of operating temperature in the range O-90 K. The results are shown in Fig. 3 as a shift in resonant frequency with temperature from its standard value at T = 0 K. The results are in good agreement with the experimental results of Ref. [6]. Since resonant frequency varies inversely as strip length, this result is the direct consequence of linear thermal expansion of the substrate. Slope of the frequency shift as a function of temperature increases with thermal expansion coefficient of the substrate. Even though the frequency shift seems to be very small in the case of substrates with low (Y’, it cannot be neglected or underestimated for substrates with high cr’. Therefore. this analysis should certainly be helpful for the
Table I Properties
of’ the substrate
materials
used in this work
Substrate material
pr
E,
Thermal expansivity (X 10-h K-‘)
LaAIO, (LAO) YAIO, (YAO) M&O
I.0 I .o I.0
24.0 16.0 9.6
10.0 6.0 8.0
Source: Refs.
[I, IO].
G.P. Sriumwn
et al. / Physica C 292 (I9971 83-8X
: 4-o
0010
-
87
I, 2 ,3 Theory
5t
5
31 3
I
1
___L___
7
5
Resonator
9
Length
Fig. 2. The shift in the resonant resonator.
z
II
2L(mm)
frequency
L 0
I
I
20
40 Temperature
I
I
60
80
\
A_-L.--IL
0020 0
20
40
60
Temperature
with the length of the
choice of substrates as well as resonant frequency in designing. The combined as well as separate effect of thermal expansion of the substrate and the temperature dependence of surface impedance of the strip are studied for LaAIO, substrate. The normalized shift in resonant frequency (A.f/fO where Af=f-.fo) is illustrated in Fig. 4. The contribution to frequency shift due to surface impedance is almost negligible up to 0.77T, above which frequency shift is unacceptably steep. If the resonator is operated below 0.8T,, the whole contribution to the frequency shift is due to substrate thermal expansion which can be minimized with the choice of convenient substrate of
-20
-0
1
100
(K)
Fig. 3. Shift in the resonant frequency with temperature due to the thermal expansion of the substrate for the case of LaAIO, and MgO substrates.
80
100
(K)
Fig, 4. Normalized shift in the resonant frequency (from its value at T = 0) with temperature for the case of LaAIO, substrate. Structural parameters are same as those of Fig. 3. Curve (I ) shows the frequency shift due to thermal expansion of the substrate. Curve (2) shows the shift due to change in surface impedance. Curve (3) shows variation due to combined effect of these two. Dark triangles represent the experimental result [6].
low enough CX’. It is a usual practice to operate superconducting devices well below Tc, because most of the superconductivity parameters vary sharply near T,. The overestimation of the upper limit for operating temperature would be 0.8T,. Therefore as far as resonant frequency is concerned, surface impedance has least contribution in shifting the resonant frequency when resonator is operated below O.XT, as can be realized from curve (3) of Fig. 4. The experimental results for the variation of resonant frequency of Ref. [6] are also shown in the same figure. which shows good agreement between the calculations and experiment. The variation of resonant frequency with temperature as calculated here also agrees qualitatively with the experimental results of Ref. [l 11. However, a better quantitative agreement can be expected if we incorporate the appropriate material parameters of Bi,Sr,CaCuzO, of Ref. [l I] in the . calculation ot Z,. The variation of unloaded Q of the YBCO microstrip resonator with temperature is shown in Fig. 5. The continuous curve shows the variation of calculated Q, normalized to its value at T = 0 K. The variation of Q calculated here agrees well with the reported experimental results of Ref. [6] which are also shown in Fig. 5. The absolute values in the
88
G.P. Srimstma
et al. / Physica C 292 (1997) 83-88
4. Conclusion
o
oIII_____L__. -lJ 0 20
40
Temperature
60
80
100
(K)
Fig. 5. Variation of the Q with temperature for LaAIO, substrate. Continuous curve represents the theoretical result and dark triangles represent the experimental result [6]. (Structural parameters are same as those of Fig. 3).
two cases would be different because the structural parameters are different. It should be noted that the behavior of Q, is similar to the variation of superconducting gap parameter with temperature even though we have used only two-fluid model in our calculation. This is because of the qualitative similarity between the surface impedance calculated by the two-fluid model and BCS theory. Routinely, R, is determined using the measured Q for microstrip resonators or superconducting cavities. In the literature, the measured variations of Q with temperature for a superconducting device fabricated by different methods differ quantitatively which we attribute to the variation of R, with temperature arising from various physical processes like surface roughness, Josephson tunneling at grain boundaries (weak links), granularity, flux pinning, etc. Even though we have presented calculations using a two-fluid model here, the present method can be used to analyze the various situations mentioned above by incorporating the suitable functional form of Z, in the calculation. The contribution of these mechanisms to Q can be calculated either separately or in combination to generate the behavior of Q which can be utilized to understand the measured Q and hence the material. The detailed analysis is in progress.
The present analysis based on spectral domain method shows a good potential for its application in the analysis of superconducting microwave devices like microstrip resonators. A linear shift in the resonant frequency with temperature below 0.8T, is found to be mainly due to the substrate thermal expansion. This analysis can be used to calculate Q of the microstrip resonator due to various physical processes by simply incorporating the relevant functional dependence of Z,.
Acknowledgements One of the authors (VM) is grateful to University Grants Commission (UGC), Govt. of India, for the financial assistance.
References [I I Z.Y. Shen, High-temperature Superconducting Microwave Circuits, Artech House, London, 1994. [2] A.M. Portis, Electrodynamics of High-temperature Superconductors, World Scientific, Singapore, 1993, p. 105. [3] T. Itoh, IEEE Trans. MTT 22 (1974) 946. [4] J.M. Pond, C.M. Crowne, W.L. Canter, IEEE Trans. MTT 37 (19891 181. [5] D. Nghiem, J.T. Williams, D.R. Jackson, IEEE Trans. MIT 39 (1991) 1553. [6] C. Wilker, Z.Y. Shen, P. Pang, P.W. Face, W.L. Holstein, A.L. Mathews, D.B. Laubacher, IEEE Trans. MTT 39 (1991) 1462. [7] D. Mirshekar-Syahkal. Spectral Domain Method for Microwave Integrated Circuits, Wiley, New York, 1990. [8] J. Kessler, R. Dill, P. Russer, IEEE Trans. MIT 39 (1991) 1566. [9] M. Tinkham, Introduction to Superconductivity, McGrawHill, New York, 1996. [IO] E.K. Kollmann, O.G. Vendik, A.G. Zaitsev, B.T. Melekh, Superconduct. Sci. Technol. 7 (1994) 609. [I 11 G. Godel, N. Gold. J. Hasse, J. Bock, J. Halbritter, Superconduct. Sci. Technol. 7 (1994) 745.
__ li!B
PHYSICA ti
ELSEVIER
Physica C 292 (1997) 83-88
Studies on high-T, superconducting
microstrip resonator
G.P. Srivastava a,*, Vincent Mathew ‘, Agnikumar G. Vedeshwar a Department
of Electronic
h Department
Science, Uniwrsity
of Physics and Astrophysics,
of Delhi South Campus, NW Delhi Uniwrsity
of Delhi. Delhi
’
110021, lndiu
11UOO7. Indiu
Received 17 December 1996; revised 18 August 1997: accepted 26 August 1997
Abstract We present a theoretical analysis of a high-T, superconducting microstrip resonator using spectral domain method. Fourier transformed impedance Green’s functions are derived and the resonant frequencies are calculated for three different substrates, as a function of the resonator length, in Galerkin’s procedure. Effects of thermal expansion of the substrate and surface impedance of the superconducting film are incorporated in the theory to analyze the temperature dependence of both the unloaded Q and the resonant frequency. The effects of these two mechanisms are analyzed separately and compared. It has been found that the thermal expansion of the substrate is mainly responsible for linear shift in the resonant frequency below 0.8 T,. The effect of surface impedance is found to be dominant only in a temperature range close to the transition temperature for shift in both resonant frequency and Q. 0 1997 Elsevier Science B.V. PACS:
85.X-j;
Kepwords:
84.40.Cb:
Microstrip
84.4O.Y~
resonator;
Superconducting
devices; Spectral domain method; Impedance
1. Introduction The growing interest in microwave studies of high-temperature superconducting (HTSC) materials has two origins. First, microwave studies provide a better understanding of the physics of these materials at high frequencies. Second, these materials find many applications in the microwave and millimeter wave devices and circuits [l]. Again, HTSC passive microwave devices are employed in the experimental investigation of the microwave properties of the superconducting materials [2]. Due to these reasons,
* Corresponding
author. Fax: +9l
0921-4534/97/$17.00 PII SO92 I -4534(97)0
I I 6886427.
0 1997 El.\evier Science B.V. All rights reserved 1697.3
Green‘s functions
analytical and experimental studies on the microwave superconducting devices are gaining much importance. In developing passive superconducting devices, theoretical and experimental analysis is very important for generating sufficient design data or for devising good design models. Full wave analysis methods like the spectral domain method [3] are usually employed in the analysis of microwave and millimeter wave devices and circuits which can produce results with sufficient accuracy. It has been applied successfully to many problems involving HTSC microstrip transmission lines recently [4,5]. In this paper we extend this method for the analysis of HTSC microstrip resonator.
84
G.P. Sricasrava
et al. / Physica C 292 (1997) 83-88
The dependence of the performance of high-T, superconducting (HTSC) circuits on the operating temperature is one of the important factors to be understood for properly designing such circuits. Experimental results [6] show considerable influence of temperature on HTSC circuit performance. This influence of operating temperature on resonant frequency and other parameters of a superconducting microstrip resonator can be attributed mainly to two mechanisms [6]: ( I> The thermal expansion of the substrate causes the resonator patch (HTSC film) to elongate in its length and changes the structural parameters. (Expansions in the other dimensions are neglected.) (2) The kinetic inductance of the superconducting material increases with temperature which will increase the surface impedance of the strip. Therefore, it is necessary to analyze the effects of these two processes on the performance of a HTSC resonator in order to obtain an exact view of this problem. As a preliminary investigation, we consider the simple case of a rectangular YBCO thin film HTSC microstrip resonator as represented schematically in Fig. 1. In the following, we present an outline of the theoretical formulation of the problem. First we calculate the resonant frequency of the resonator as a function of the resonator length. The effect of thermal expansion on the resonant frequency is studied separately for two substrate materials. The effects of the two mechanisms are compared and the combined effect of both these mechanisms is studied for the c__
2a
m--M
I
I
case of the typical LaAlO, substrate. The results of the numerical computation are discussed.
2. Theoretical
formulation
The advantages of using the spectral domain method, i.e. Galerkin’s procedure in Fourier transform domain for the analysis of microstrip like transmission structures are well discussed in the literature [7]. Out of the two mechanisms for temperature dependence, one enters the analysis through the structural parameter (thermal expansion) and the other through the material property (surface impedance) of the HTSC film. We discuss first the spectral domain formulation and then the temperature dependence. 2. I. Spectral domain formulation The spectral domain method is a very standard tool used in the rigorous analysis of field problems connected with microstrip circuit structures [7]. In spectral domain method, we write down the field components at the superconducting strip in the Fourier transform domain. Applying the boundary conditions and incorporating the complex conductivity of the superconductor, we derive the dyadic Green’s functions which will form the impedance matrix in the Fourier transformed domain. Expanding the strip current in a suitable basis functions, the resonant frequency is calculated in Galerkin’s procedure. The geometry of the HTSC resonator under consideration is schematically represented in Fig. 1. With respect to the geometry of the resonator, the following transformation is applied to all quantities for formulating the problem in the Fourier transform domain:
xexp[i( / t Substrate Fig. 1. The geometry of the HTSC microstrip resonator.
ax + Pz)]dxdz
(1)
where (Y and /3 are the transform variables [3]. The electric and magnetic field components in the substrate and in the air media are of hybrid nature and can be expressed as superpositions of the TE and TM electromagnetic fields with respect to the y-di-
G.P. Srioastaua et al./Phyica
rection. In Fourier transform
domain they are written
C 292 (1997) 83-88
fi;;x =[(w,k;
as, a,y,P)
= (k2 - P’)$;
(2.1)
H;;( (Y,Y,P)
= (kZ - P’)&
(2.2)
Exj< WY$)
= -Odi
- iwpi(Gi/aY)
(2.3)
fix;< ci.Y,p)
= -C&G;
+
i;;(
85
- “‘)~o
+pr(ki - ~2)yltanh(yld)]/A 6: = G_x,,
(8c) (84
with,
iwei(a&/ay)
A = [ y,tanh(y,d)
(2.4)
where the subscript i = I,0 designates the substrate or air region. Here +(x,y,z) and $(x,y,z) are two scalar potentials, the transforms of which satisfy,
+ ~,r,,]
x [Ylcow,4
+ wol
(8e)
The resonant frequency for the fundamental mode is calculated from Eq. (7) in Galerkin’s procedure. Expanding the strip currents JI, and i: using a set of basis functions,
(3a) (9a)
J:-= ; d,.L,(a$)
with y;=a2++s,prk;
(da)
y; = a* + p2 - k,2
(4b)
k, = (&r /4”*k,
(5a)
k, = ( co /.Q)”
w
(5b)
Dyadic Green’s functions are derived by applying the complex resistive boundary conditions for the superconducting strip [4], nx(E++Y)=O 12X (H+-H-)
f,
[
K;;;;“‘c, +
,,R, [ KI.3
Kj$‘d,]
= 0
( lOa)
c,, + K,‘,$d,]
= 0
( lob)
where 1= 1,2, . . , N and,
= -(l/Z,)nx
(n XE’)
(6b)
X~~,x((Y,p,ko)j”,,(LY,P)dp (lla) Kj,t,i”)(k,)= -,
where, Z, is the surface impedance of the HTSC film which will be discussed later and,
e.r,;=
+ pryltanh(Ml/A
(*a)
- P*)Y,tanh(M)]/A
,r;loli;,bP) X~~,,,~((Y,p,ko)jl*,(a,p)dp (11~)
Kk%)
=
nc, ~?r,(c4) d,(
[(Er+$-P*)YO +/-+;
/o?-hW
X~.~,~(CY,P,kO)j:-m((Y’P)dp (lib) K,‘.,“‘(k,) =
= o$[~,~
c, and d, satisfy [3]:
The coefficients
(6a)
in spectral domain. Here n is the unit vector normal to the HTSC strip and the superscripts k refer to the field components on the top and bottom of it. The resulting equation is given by,
&
(9b)
lIl=l
(8b)
Resonant frequency and (lob) by setting
wP&)j:-m( a>P)dP (lid) is calculated from Eqs. (lOa> the coefficient matrix equal
zero. The same set of basis functions used in Ref. [3] has been used here. Once the resonant frequency is estimated. the spectral domain method can again be used to find out the attenuation constant (k,) and propagation constant (k,) at the same resonant frequency for an infinitely long microstrip line with the same cross sectional geometry. This procedure is detailed in Ref. [4]. Now the unloaded Q can be estimated from the relation [8], ’
Q,, = 27r{ I - exp( -47rk;,/k,)}
2.2. Tetnperature
( 12)
dependence
The temperature dependence of both resonant frequency and unloaded Q of the microstrip resonator can be computed by the incorporation of the thermal expansion of substrate and temperature dependence of surface impedance of the strip material. The effect of thermal expansion of the substrate along its length can be represented by L(T)
= L(O){ I + CY’T}
(13)
where cy’ is the coefficient of thermal expansion the substrate. The surface impedance is given by [I], iwp Z,=R,+iX,= i c( w,T)
1
of
l/2
(14)
where (r = U, - i (T? is the complex conductivity of the superconductor given by the two-fluid model. The temperature dependence of R, and X, results from that of U, and A,~ (London penetration depth) respectively. The main contribution to R, comes from normal electrons with density n, which has a temperature dependence of t” (t = T/q) in two-fluid enmodel and e P’/‘liT (A is the superconducting ergy gap) in BCS theory. The temperature dependence of (T is governed by the ratio of superconducting electron to normal electron densities as tz,/n, = (1 - t’)/t’ in two-fluid model which is quite close to that predicted by BCS theory. However, the difference between the values of surface resistance, R,. calculated by two-fluid model and BCS theory is not too high. Both are qualitatively similar. Therefore, it is quite customary to employ two-fluid model in
most of the calculations concerning superconducting microwave devices in the absence of a well developed microscopic theory as in the case of high temperature superconductors [9].
3. Results and discussion In the calculations, we have taken YBCO thin film as the resonator patch (strip) with parameters ?=92 K and A,=0.15 pm at T=O K. The dimensions used are a = 5 mm, w = I mm and d = 2 mm. The relative permittivity and the coefficient of thermal expansion for three commonly used substrate materials [ 1, IO] are listed in Table I. The resonant frequencies are calculated as a function of resonator length using the above formulation and are shown in Fig. 2. The variation of resonant frequency with the resonator length is of the same form as that of normal resonators [3]. In order to see the effect of thermal expansion on the resonant frequency, it has been calculated separately for LaAlO, and MgO substrates as a function of operating temperature in the range O-90 K. The results are shown in Fig. 3 as a shift in resonant frequency with temperature from its standard value at T = 0 K. The results are in good agreement with the experimental results of Ref. [6]. Since resonant frequency varies inversely as strip length, this result is the direct consequence of linear thermal expansion of the substrate. Slope of the frequency shift as a function of temperature increases with thermal expansion coefficient of the substrate. Even though the frequency shift seems to be very small in the case of substrates with low (Y’, it cannot be neglected or underestimated for substrates with high cr’. Therefore. this analysis should certainly be helpful for the
Table I Properties
of’ the substrate
materials
used in this work
Substrate material
pr
E,
Thermal expansivity (X 10-h K-‘)
LaAIO, (LAO) YAIO, (YAO) M&O
I.0 I .o I.0
24.0 16.0 9.6
10.0 6.0 8.0
Source: Refs.
[I, IO].
G.P. Sriumwn
et al. / Physica C 292 (I9971 83-8X
: 4-o
0010
-
87
I, 2 ,3 Theory
5t
5
31 3
I
1
___L___
7
5
Resonator
9
Length
Fig. 2. The shift in the resonant resonator.
z
II
2L(mm)
frequency
L 0
I
I
20
40 Temperature
I
I
60
80
\
A_-L.--IL
0020 0
20
40
60
Temperature
with the length of the
choice of substrates as well as resonant frequency in designing. The combined as well as separate effect of thermal expansion of the substrate and the temperature dependence of surface impedance of the strip are studied for LaAIO, substrate. The normalized shift in resonant frequency (A.f/fO where Af=f-.fo) is illustrated in Fig. 4. The contribution to frequency shift due to surface impedance is almost negligible up to 0.77T, above which frequency shift is unacceptably steep. If the resonator is operated below 0.8T,, the whole contribution to the frequency shift is due to substrate thermal expansion which can be minimized with the choice of convenient substrate of
-20
-0
1
100
(K)
Fig. 3. Shift in the resonant frequency with temperature due to the thermal expansion of the substrate for the case of LaAIO, and MgO substrates.
80
100
(K)
Fig, 4. Normalized shift in the resonant frequency (from its value at T = 0) with temperature for the case of LaAIO, substrate. Structural parameters are same as those of Fig. 3. Curve (I ) shows the frequency shift due to thermal expansion of the substrate. Curve (2) shows the shift due to change in surface impedance. Curve (3) shows variation due to combined effect of these two. Dark triangles represent the experimental result [6].
low enough CX’. It is a usual practice to operate superconducting devices well below Tc, because most of the superconductivity parameters vary sharply near T,. The overestimation of the upper limit for operating temperature would be 0.8T,. Therefore as far as resonant frequency is concerned, surface impedance has least contribution in shifting the resonant frequency when resonator is operated below O.XT, as can be realized from curve (3) of Fig. 4. The experimental results for the variation of resonant frequency of Ref. [6] are also shown in the same figure. which shows good agreement between the calculations and experiment. The variation of resonant frequency with temperature as calculated here also agrees qualitatively with the experimental results of Ref. [l 11. However, a better quantitative agreement can be expected if we incorporate the appropriate material parameters of Bi,Sr,CaCuzO, of Ref. [l I] in the . calculation ot Z,. The variation of unloaded Q of the YBCO microstrip resonator with temperature is shown in Fig. 5. The continuous curve shows the variation of calculated Q, normalized to its value at T = 0 K. The variation of Q calculated here agrees well with the reported experimental results of Ref. [6] which are also shown in Fig. 5. The absolute values in the
88
G.P. Srimstma
et al. / Physica C 292 (1997) 83-88
4. Conclusion
o
oIII_____L__. -lJ 0 20
40
Temperature
60
80
100
(K)
Fig. 5. Variation of the Q with temperature for LaAIO, substrate. Continuous curve represents the theoretical result and dark triangles represent the experimental result [6]. (Structural parameters are same as those of Fig. 3).
two cases would be different because the structural parameters are different. It should be noted that the behavior of Q, is similar to the variation of superconducting gap parameter with temperature even though we have used only two-fluid model in our calculation. This is because of the qualitative similarity between the surface impedance calculated by the two-fluid model and BCS theory. Routinely, R, is determined using the measured Q for microstrip resonators or superconducting cavities. In the literature, the measured variations of Q with temperature for a superconducting device fabricated by different methods differ quantitatively which we attribute to the variation of R, with temperature arising from various physical processes like surface roughness, Josephson tunneling at grain boundaries (weak links), granularity, flux pinning, etc. Even though we have presented calculations using a two-fluid model here, the present method can be used to analyze the various situations mentioned above by incorporating the suitable functional form of Z, in the calculation. The contribution of these mechanisms to Q can be calculated either separately or in combination to generate the behavior of Q which can be utilized to understand the measured Q and hence the material. The detailed analysis is in progress.
The present analysis based on spectral domain method shows a good potential for its application in the analysis of superconducting microwave devices like microstrip resonators. A linear shift in the resonant frequency with temperature below 0.8T, is found to be mainly due to the substrate thermal expansion. This analysis can be used to calculate Q of the microstrip resonator due to various physical processes by simply incorporating the relevant functional dependence of Z,.
Acknowledgements One of the authors (VM) is grateful to University Grants Commission (UGC), Govt. of India, for the financial assistance.
References [I I Z.Y. Shen, High-temperature Superconducting Microwave Circuits, Artech House, London, 1994. [2] A.M. Portis, Electrodynamics of High-temperature Superconductors, World Scientific, Singapore, 1993, p. 105. [3] T. Itoh, IEEE Trans. MTT 22 (1974) 946. [4] J.M. Pond, C.M. Crowne, W.L. Canter, IEEE Trans. MTT 37 (19891 181. [5] D. Nghiem, J.T. Williams, D.R. Jackson, IEEE Trans. MIT 39 (1991) 1553. [6] C. Wilker, Z.Y. Shen, P. Pang, P.W. Face, W.L. Holstein, A.L. Mathews, D.B. Laubacher, IEEE Trans. MTT 39 (1991) 1462. [7] D. Mirshekar-Syahkal. Spectral Domain Method for Microwave Integrated Circuits, Wiley, New York, 1990. [8] J. Kessler, R. Dill, P. Russer, IEEE Trans. MIT 39 (1991) 1566. [9] M. Tinkham, Introduction to Superconductivity, McGrawHill, New York, 1996. [IO] E.K. Kollmann, O.G. Vendik, A.G. Zaitsev, B.T. Melekh, Superconduct. Sci. Technol. 7 (1994) 609. [I 11 G. Godel, N. Gold. J. Hasse, J. Bock, J. Halbritter, Superconduct. Sci. Technol. 7 (1994) 745.