where αis the kinetic inductance fraction. Eq. 2.54 will be derived in the next two chapters. The measuredδfr/frandδ1/Qrcan be fitted to the calculatedδXs/XsandδRs/Xswith a simple linear fitting model according to Eq. 2.54. The energy gap ∆0 and the kinetic inductance fraction αare taken as two fitting parameters. Tc can be measured by experiment. Other parameters are taken from the literature and set fixed. The data offr(T) andQr(T) from an Al resonator is fitted in this way and the result is shown in Fig. 2.4, which shows a good agreement between data and calculation.
2.2.6.3 Frequency dependence ofZs
0 2 4 6 8
30 40 50 60 70 80 90
hf/kT
C
λ
eff(nm)
T=0 T=0.7T
c
T=0.8T
c
T=0.9T
c
Figure 2.5: Frequency dependance of the effective penetration depthλeff of an Al bulk supercon- ductor. The material parameters are from Table 2.1 except ∆0= 1.70kTC.
The frequency dependance of effective penetration depth of Al is calculated and plotted in Fig. 2.5. As a verification of our calculation, we use exactly the same material parameters used by Popel in Fig. 15 of Ref. [40]. Comparing Fig. 2.5 to Fig. 15 of Ref. [40], we find that the frequency dependence calculated by “surimp” is identical to that calculated by Popel.
x x
z=d z=0
Figure 2.6: Configuration of a plane wave inci- dent onto a superconducting thin film
By(0)
z=d z=0 By(0) z=2d
z=-2d z=-d
… …
z=3d z=4d
Figure 2.7: Field configuration used by Srid- har to calculateZs of a thin film assuming specular scattering boundary condition at both interfaces
Again, to apply the Mattis-Bardeen equations one has to assume either specular scattering or diffusive scattering at the surface. For specular scattering boundary at both sides of the film, the problem can be solved by mirroring the field and current repeatedly to fill the entire space (see Fig. 2.7) and applying the equation
−d2Ax(z) dz2 =
Z ∞
−∞
K(η)Ax(z′)dz′, η=z′−z (2.55) which can easily be solved in a similar manner as in the bulk case. The result is derived by Sridhar[47]
to be
Zs=iµ0ω d
+∞
X
n=−∞
1
q2n+K(qn) (2.56)
whereqn=nπ/d. Comparing Eq. 2.56 to Eq. 2.35, we see that the only change is that the integral in the bulk case has been replace by an infinite series in the thin film case.
For the diffusive scattering boundary condition, the equation is
−d2Ax(z) dz2 =
Z d 0
K(η)Ax(z′)dz′, η=z′−z (2.57) and unfortunately has to be solved numerically.
2.3.2 Numerical approach
There are two tasks in the numerical calculation of surface impedance of a thin film: evaluating the kernel functionK(η) and solving the integro-differential equation of Eq. 2.57.
2.3.2.1 Implementing the finite difference method
N
2
1
0 N-1
Figure 2.8: Thin film divided into N slices
The integro-differential equation of Eq. 2.57 can be solved numerically by the finite difference method (FD). To implement FD method, we first divide the film intoNthin slices of equal thickness t=d/N (see Fig. 2.8). Then we follow the standard procedures to convert Eq. 2.57 into a discrete FD equation. On the left-hand side, we employ the simple three-point approximation formula
d2Ax(z)
dz2 ≈(An+1−2An+An−1)/t2. (2.58) On the right-hand side, we apply the simple extended trapezoidal rule to approximate the integral as a sum
Z d 0
K(η)Ax(z′)dz′ ≈t XN n′=0
Knn′A(n′) (2.59)
where
Knn′ =
1
2K(|n−n′|t) if n′= 0 or N
1 t
R(n+1/2)t
(n−1/2)tK(|nt−x′|)dx′ if n′= n K(|n−n′|t) otherwise
. (2.60)
So we get the following FD equations
An+1−2An+An−1=−t3 XN n′=0
Knn′A(n′), n= 1, ..., N−1. (2.61)
2.3.2.2 Boundary condition
Eq. 2.61 providesN−1 linear equations withN+ 1 unknowns (A0 ... AN), so two more equations are needed. The two additional equations come from the boundary conditions at the interfaces at z= 0 andz=d. In the configuration of Fig. 2.6, both sides of the film are connected to free space and the electromagnetic wave is incident from z = 0 into the film. In this case, atz = 0 one can assume either boundary conditions of the first kind
A(z)|z=0= 1⇒A0= 1 (2.62)
or of the second kind
Hy(0) = dA(z) dz
z=0= 1⇒A1−A0=t (2.63)
where a two point formula fordA(z)/dz is used. Physically, the former specifies a vector potential and the latter specifies a magnetic field on thez= 0 surface.
At the interface of z = d, one usually assumes that the transmitted wave sees the free space impedance
Z0= −jωµ0
A(z)
dA(z) dz
z=d⇒(1 +jωµ0
Z0 t)AN −AN−1= 0, (2.64) Because the free space impedanceZ0≈377 Ω is usually much larger than the surface impedance of the film, boundary condition Eq. 2.64 is virtually equivalent to
Hy(d) = dA(z) dz
z=d
= 0⇒AN =AN−1 (2.65)
which physically forces a zero magnetic field on thez=dsurface.
2.3.2.3 Retrieving the results
With proper boundary conditions, the FD problem is ready to solve. The N+ 1 linear equations can easily be solved with standard numerical algorithms. Unfortunately, we can not utilize a sparse algorithm to accelerate the calculation.
For thin films, the transmitted wave is often important and can not be neglected. In these cases, it is often not enough to consider only the surface impedance at the surfacez= 0. We can generalize
the concept of surface impedance and define a pair of impedances for the thin film Z11 = Ex(0)
Hy(0) =−jωµ0 Ax(z)|z=0 dAx(z)/dz|z=0
=−jωµ0t A0
A1−A0
(2.66) Z21 = Ex(d)
Hy(0) =−jωµ0 Ax(z)|z=d
dAx(z)/dz|z=0 =−jωµ0t AN
A1−A0 (2.67)
Instead of returning one impedance, bothZ11andZ21are calculated from the solution and reported by our numerical program.
In the case that the electromagnetic wave is incident from one side of the thin film (d < λeff), with the other side exposed to the free space, we have Hy(d)≈ 0 (see the previous discussion of Eq. 2.65) and Hy(0) = K = Rd
0 Jxdz according to Ampere’s law, where K represents the sheet current (current flowing in the entire film thickness). Therefore the electric fields at the two surfaces are given by
Ex(0) =Z11K, Ex(d) =Z21K. (2.68) In the anti-symmetric excitation case, as in a TEM mode of a superconducting coplanar waveg- uide (discussed in more detail in the next chapter), the electromagnetic wave is incident from both sides of the film with Hy(0) =−Hy(d). One can decompose this problem into two problems, each with a wave incident from one side. It can be shown that in this anti-symmetric excitation case,
Hy(0) = −Hy(d) =K/2
Ex(0) = Ex(d) = (Z11+Z21)K/2. (2.69)
2.3.3 Numerical results
With slight modification to “surimp”, a program “surimpfilm” is developed to calculate the surface impedance of a superconducting thin film. The program takes ω and T as independent variables and the same 5 material parameters. It takes the film thicknessd and the number of subdivisions to the filmN as two additional parameters. Besides, all 3 types of boundary conditions (specifying value ofA, value ofH or the load impedanceZ) can be applied to both the top side and the bottom side of the film. The values of generalized surface impedance Z11 and Z12 are returned from the program.
2.3.3.1 λeff of Al thin film
We use “surimpfilm” to calculate the thickness dependence of surface impedance for Al atT = 0 K andf = 6 GHz. The result is shown in Fig. 2.9. We see thatZ11approaches its bulk value when the thickness is large compared to the bulk penetration depth (roughlyd >3λeff) andZ12 goes to zero, implying no magnetic field penetrates through. As the thickness of the film is reduced, both Z11
10
110
210
310
010
110
210
310
4d (nm) λ
eff(nm)
~1/d
2λ
eff1λ
eff2Figure 2.9: Effective penetration depth of Al thin film vs. thickness. f = 6 GHz, T = 0 K, and other material related parameters are from Table 2.1, except that the mean free pathl is set to the film thicknessd. In the calculation,N = 400 division is used. The two effective penetration depths are defined asλeff1 =jωµZ110 andλeff2 =jωµZ210.
andZ21increase. Ultimately when the film is very thin, Z11 andZ21become equal, which implies that the film is completely penetrated. We also notice that the impedance goes as 1/d2, which will be explained later in this chapter.