Frequency stable electrical components for wireless communications

The dependence of the resonant frequency of a superconducting planar resonator on the material properties and thicknesses of the superconductors, the dielectric (substrate) and the buffer layers is discussed in this paper by Farhat Abbas. An example of YBCO thin film on rutile with sapphire buffer layers is computed, with the conclusion that if a turning point can be realized at T = 60K, and the temperature controlled to better than 0.1mK, then frequency standards with stabilities of around 10 -15 should be attainable.

T he use of superconducting films in transmission lines has many advantages for signal processing applications such as low dispersion, low loss, and wide bandwidth. In fact, a study of the behaviour of these transmission lines is of interest not only to measure the penetration depth and low-frequency resistance of superconducting thin films, but also to determine the characteristics of thin-film superconducting devices.

Passive microwave devices such as filters, resonators and delay lines require high-quality HTS thin-films and substrate materials.

Workers attempting to grow HTS films directly onto high-quality substrate materials have encountered some serious problems, due to large mismatches of both lattice constants and thermal expansion coefficients of the HTS films and some substrate materials. Also, the interdiffusion between the HTS films and substrate materials has been found to severely degrade the superconducting properties [1-3].

To reduce such problems, it is necessary to investigate suitable buffer layers. The buffer layers should satisfy the minimum five requirements for cryogenic microwave microelectronics applications: prevention of interdiffusion; better thermal expansion coefficient matching; better lattice matching; confinement of the field into the substrate; provide a resonator with a temperature independent frequency.

In this paper, an analysis has been completed to explore the possibility of realizing a resonator with a temperature independent frequency. The design depends on the material properties and thicknesses of the superconductors, the dielectric (substrate), and the buffer layers. The first and second derivatives of propagation phase velocity with respect to temperature for a sample in vacuo are computed for various combinations of material properties and geometric factors. The resonator with a temperature independent frequency, as shown in Figure 1 also depends on the product of dielectric constant of the substrate and the thicknesses of buffer layers, and vice versa.

Field solution
Figure 1 shows the geometry of the normal or superconducting transmission line with dielectric mirrors which we have considered.

The structure consists of a pair of dielectric mirrors (dielectric, region 1) bridged by a central substrate (dielectric, region 2) and sandwiched by a pair of superconducting thin films. Thicknesses of the thin films, dielectric mirrors and substrate are l, d1 and d2 , respectively. The dielectrics, in region 3, are considered to be very thick so that the fields in these regions can be assumed to decay exponentially away from the interfaces.

The procedure used to obtain a field solution is the same as in [1], and it is summarised further on. Consider the propagation of an electromagnetic wave in the zdirection of the transmission line shown in Figure 1. It is assumed that the dielectric thicknesses (d1 and d2 ) and the penetration depth of the high temperature superconductors are very small compared to the width of the line, which in turn is very small compared to the length of the line.

From Figure 1 and the above assumptions, it is clear that the edge effects can be neglected, and there is no y-dependence of the fields and currents.

The two-fluid model is used for the superconductors, in which the total current is the sum of the supercurrent and the normal current. Classical skin effect and London theory are assumed for the normal current and the supercurrent, respectively.

The formulations made in [1] can be used, i.e., we are considering a TM wave:

Here, is the propagation constant along the z direction (taking e -z ), is the angular frequency (assuming e t ), o and o are the permittivity and the permeability of vacuum respectively, r is the dielectric constant of the dielectrics, and are the penetration depth and the conductivity of the superconductors, respectively.

Equation (1) is a secondorder differential equation which has two independent solutions of the form e kx and e kx , where is taken to be the root of 2 with positive real part. In the positive x-direction of the dielectric, region 3, we take only the solution e-K 3 x , and in the negative x-direction we take only the solution e K 3 x , discarding e K 3 x for positive x-direction, and e-K 3 x for negative x-direction. In the superconductors, the dielectric mirrors (dielectrics, region 1) and in the substrate (dielectric, region 2), both solutions are retained in order to satisfy the boundary conditions. In the normal or superconductors, and in regions 1 and 2 of the dielectrics, we need both solutions in order to satisfy the boundary conditions.

With these solutions in the various media, we have twelve arbitrary constants for the amplitudes of the fields (one each in the dielectrics, region 3, two each in the superconductors, the dielectric mirrors (dielectrics, region 1) and the substrate (dielectric, region 2)). There are twelve boundary conditions that must be satisfied, namely the continuity of the tangential fields Ez and Hy at the six boundaries shown in Figure 1. If we ignore any non-linearity in the system, the characteristics of the line are independent of the amplitude of the wave, and eleven of the constants can be determined in terms of the twelfth by using eleven of the twelve boundary conditions. The twelfth boundary condition gives an equation for the propagation constant , which must be satisfied in order for a solution to exist.

The condition is a transcendental equation for which an exact solution cannot be readily obtained. As discussed in [1] the approximations are K1 d1 < < 1 and K2 d2 < < 1, and physically these approximations mean that higher order modes are ignored. With small d1 and d2 , higher order modes will not be excited. With these assumptions, the transcendental equation yields:

Also, note the factor of two in equation (5), which indicates that there are two dielectric mirrors and two normal or superconductor layers. This is because we have considered a symmetric case, i.e., the same Cu or HTS thin films on each side of the substrate separated by the same dielectric mirrors, as shown in Figure 1. The wave velocity relative to that in a vacuum can be written as follows from equation (5):

According to equation (6), the wave is dispersionless even though there is a component of the electric field in the direction of propagation, i.e., the group velocity and phase velocity are equal and independent of frequency. The dispersion relation of the wave has been discussed in [1]. The attenuation of the wave due to losses in each medium and the wave velocity have been obtained by replacing 1 , 2 and in to their complex forms.

Results
The superconducting transmission line with buffer layers shown in Figure 1. can be described by the penetration depth of the superconductors, the dielectric constants r of the r th dielectrics, and the thicknesses dr and of the r th dielectrics and the superconductors.

The temperature dependence of the penetration depth of a superconductor can be described by any one of several models outlined in [4]. Any of those models can be used in our analysis. However, we will concentrate on the following approximate result:

In equation (7), if the Gorter and Casimir model is assumed, then p = 4. However, recently [4] the spin-polaron theory of highTc superconductivity has been explored, in which the charge carriers in a highTc superconductor are considered as biholes obeying the Bose-Einstein statistics and localized within a unit cell of the crystal lattice.

If the charge carrier system in a highTc superconductor is considered as an ideal Bose-Einstein gas, then p = 1.5. In this paper, we will assume p = 4. Using the Gorter and Casimir model, the variations of the first and second derivatives of Vr (dVr /dT and d 2 Vr r/dT 2 ) with respect to temperature and varying dielectrics thicknesses are shown. Also, the temperature dependence of the r can be approximated [5] as 1 = 9.2+2.5x10 11 T 4 (sapphire) and 2 =113.446+0.043T-0.002T 2 +7.724x 10 -6 T 3 -1.072x10 -8 T 4 (rutile).

To provide a resonator with a temperature independent frequency, or a transmission line with temperature independent propagation constant (and therefore phase shift) along the line, it is necessary to choose a buffer layer/main dielectric configuration which will cause the propagation velocity (or equivalently the transmission line wavelength) to be as independent of temperature variations as possible at the selected operating temperature.

Expanding expression (6) in a Taylor series about the operating temperature T0 leads to:

where the partial derivatives are evaluated at T0 , and where T=(T- T0 ). If a certain temperature stability T can be achieved, then the minimum variation in Vr (T) is attained if as many of the lower order partial derivatives as possible can be made zero, or close to zero. The first order approximation is to produce a turning point in Vr (T) by ensuring d Vr /dT = 0 at T0 . But, by judicious choice of geometry factors for a particular combination of dielectrics and superconductors, it is also possible to make d 2 Vr /dT 2 zero, and even possibly higher order terms.

Figure 2 shows the first and second derivatives of propagation phase velocity as a function of temperature for a variety of buffer layer thicknesses, assuming sapphire as the buffer layer and rutile as the main dielectric material. Turning points in Vr (T) can be produced close to any chosen operating temperature in this way.

In Figure 3 the first and second derivatives of propagation phase velocity with respect to T, at T = 60 has been computed as a function of substrate thickness (d2 ) for a variety of buffer layer thicknesses, assuming sapphire as the buffer layer and rutile as the main dielectric material. Turning points in Vr (T) can be produced close to any chosen substrate's thickness in this way.

Conclusions
In this work, we compute the first and second derivatives with respect to temperature of a wave velocity ratio (with respect to free space) for various combinations of material properties. The dependence of resonant frequency on the dielectric constant and thicknesses of the substrate and buffer layers is discussed.

An example of YBCO thin films on rutile with sapphire buffer layers has been computed.

From this example, it may be concluded that if a turning point can be realized at T = 60K, and the temperature controlled to better than 0.1mK, then frequency standards with stabilities of parts in 10 15 should be attainable. We have proposed a new class of planar microwave components which are ultra stable in frequency with temperature.