3.1 Theoretical calculation of α from quasi-static analysis and conformal mapping technique
3.1.1 Quasi-TEM mode of CPW
Consider an electromagnetic wave propagating on a transmission line along the z−axis. The field quantities E,~ H, and the current density~ J~ can be written in a general form as (with a harmonic time dependenceejωt omitted):
X(x, y, z) =~ [~xt(x, y) +xz(x, y)ˆz]e−jβz (3.1)
whereβ is the propagation constant and the vectorX~ is decomposed into its transverse component
~xt and longitudinal componentxzz.ˆ
The solutions to Maxwell’s equations show that a CPW made of perfect conductor (perfect CPW) immersed in a homogenous media can support a TEM mode. In this “pure” TEM mode, the longitudinal components of E~ and H~ vanish while the transverse component of current density J~ vanishes:
ez= 0, hz= 0, jt= 0. (3.2)
A superconducting CPW differs from the above case in two aspects. First, a conventional CPW is usually made on a substrate (see Fig. 3.1), so the regions on the top and bottom of the CPW are filled with media of different dielectric constants. This inhomogeneity gives rise to longitudinal components ez, hz and transverse component jt. Second, the superconductor has a finite surface impedance which gives rise to longitudinal componentsez on the surface. The conclusion is that a superconducting CPW cannot support a pure TEM mode.
However, both theory and lab measurements show that the propagation mode in a supercon- ducting CPW is quasi-TEM, where non-TEM field components are much smaller than the TEM components. For instance, jtcontributed by the inhomogeneity from the substrate/air interface is
estimated to be on the order of (see Appendix C) jt
jz ∼ w
λ (3.3)
where wis the transverse dimension of the transmission line andλis the actual wavelength of the wave in propagation. A lithographed CPW line used in MKIDs usually has a transverse dimension of 10–100µm (the distance between the two ground planes) while the wavelength is usually thousands of microns. Thus,jt/jz∼1% andjtis indeed small as compared tojz.
On the other hand, according to the definition of surface impedance, superconductor contributes anez on the metal surface that is estimated by
ez
et ≈Zs
Z0
(3.4) whereZsis the surface impedance andZ0 is the characteristic impedance of the transmission line.
For CPW made of superconducting Al,Zsis on the order of mΩ (e.g., surface reactanceXs≈2 mΩ for T=0 K, f=5 GHz, andλeff = 50 nm) andZ0= 50 Ω. Thus,ez/et∼10−4in our case. Even for a normal metal,Zsis usually much smaller thanZ0 and soez is always much smaller thanet.
It can be shown [51] that for the quasi-TEM mode of CPW, the transverse fields ~et and ~ht
are solutions to two-dimensional static problems, from which the distributed capacitance C and inductanceLcan be derived.
In the electrostatic problem, because~etquickly attenuates to zero over the Thomas- Fermi length (on the order of one ˚A, which is always much smaller than the film thickness that we use) into the superconductor, the electric energy inside the superconductor has a negligible contribution to the capacitance C and the electric field ~et outside the superconductor is almost identical to that for a perfect conductor. Therefore,~et can be solved with the introduction of an electric potential Φ, which satisfies the Laplace’s equation
∇2Φ = 0 (3.5)
outside the superconductor, and has constant values at the surfaces of the superconductors which are now treated as perfect conductors. For a CPW, we assume Φ =V on the center strip and Φ = 0 on the two ground planes. ~et is given by~et=∇Φ. The distributed capacitanceC can be obtained either fromC =Q/V where Qis the total charge on the center strip, or from C= 2we/V2 where weis the total electric energy (per unit length).
In the magnetostatic problem,~ht penetrates into the superconductor by a distance given by the effective penetration depthλeff. In general, the magnetic field can be derived by solving the Maxwell equations together with the Mattis-Bardeen equation (Eq. 2.11). This leads to the following two
equations of the vector potentialAz(x, y):
∇2Az= 0 outside the superconductor
∇2Az=R
K(x−x′, y−y′)Az(x′, y′)dx′dy′ inside the superconductor (3.6) whereK(x−x′, y−y′) is the Mattis-Bardeen kernel appropriate for the two dimensional problems.
To join these two equations, we require~htto be continuous at the superconductor surfaces.
Similar to the electrostatic problem, if the penetration depth is much smaller than the film thickness, λeff ≪t, the magnetic field outside the superconductor is almost identical to that for a perfect conductor. Therefore,~htcan be solved from the first Laplace equation in Eq. 3.6, with the perfect conductor boundary condition—Azis constant at superconductor surfaces or~htis parallel to the surfaces, and with the constraint that a total currentIis flowing in the center strip and returns in the two ground planes for a CPW. The transverse field~ht can be obtained from~ht =∇ ×Az. The (distributed) geometric inductance Lm can be obtained either fromLm=φ/I whereφ is the total magnetic flux per unit length going through the gap between the center strip and the ground planes, or fromLm= 2wm/I2 where wm is the total magnetic energy. To account for the stored energy and dissipation inside the superconductor, we use the surface current derived for the perfect conductor (equal to~hton the surface) and apply the surface impedance Zs of the superconductor to the calculation of the (distributed) kinetic inductanceLkiand (distributed) resistanceR.
However, if the penetration depth is much larger than the film thickness, λeff ≫t, so that the film is fully penetrated by the magnetic field, the perfect-conductor approximation for the surface current and the near magnetic field outside the superconductor is no longer a good approximation, because the perfect-conductor boundary condition—ht parallel to the superconductor surfaces—
fails at the edges of the superconducting film; as the film becomes thinner and thinner, the surface current derived for a perfect conductor will become more and more singular at these edges, while in fact both the magnetic field and current density will become less and less singular due to the longer penetration. In this case, one has to solve faithfully the two equations in Eq. 3.6. However, for λeff ≫t, the relationship between J~ and A~ becomes local (see Section 2.4.1.3) and the second equation in Eq. 3.6 can be replaced by the London equation:
∇2Az= 1
λ2LAz (3.7)
which is easier to solve than the original differential-integral equation.
So far the electrostatic problem and the magnetostatic problem are independent, which is enough in many cases where only the distributed parametersLandC, or the characteristic impedanceZ0= pL/C and the phase velocityvp = 1/√
LC are required. The two static problems can be further linked by applying the relationships between the voltage and current given by the transmission line
f
z1
z2
z3 z4
z5 z6
z7
=∞ w1
w2 w3
w4
w5 w6
w7
α1π
Figure 3.2: Schwarz-Christoffel mapping equations.