Supplementary information:
Microwave complex conductivity of the YBCO thin
films as a function of static external magnetic field
J. Krupka1,*, J. Judek2, C. Jastrzębski2, T. Ciuk1, J.
Wosik3,4 and M. Zdrojek2
1
Institute of Microelectronics and Optoelectronics, Warsaw University of Technology, Koszykowa 75, 00-662
Warsaw, Poland
2
Faculty of Physics, Warsaw University of Technology,
Koszykowa 75, 00-662 Warsaw, Poland
3
Electrical and Computer Engineering Department, University of Houston, Houston, TX 77204, United States
4
Texas Center for Superconductivity, University of Houston, Houston, TX 77204, United States
plates. If appropriate model for the temperature dependence of σ 2 ( T,ω ) is known then one can determine the
resonance shift value at a reference temperature (in our
case at 4.2K) and σ 2 (T =4 .2 K, ω ). The procedure of adjustment of the value of the reference resonant frequency
shift and the reference σ 2 ( T = 4 .2 K, ω ) value is of great
importance and it is equivalent to finding the London
penetration depth at the temperature of 4.2K. In order to
do that rigorous electromagnetic modelling of the sapphire resonator terminated by YBCO films deposited on
MgO substrates had to be performed in order to find relationship between the fres-fres,ref and the complex conductivity of YBCO films. Then the resonance frequencies
and Q-factors of sapphire resonator terminated by YBCO
films and independently by Cu plates at two different
Terminated
YBCO
by bulk Cu plates
1. Adjustment procedure for setting the initial conductivity value at 4.2K
In sapphire rod resonator technique the complex
conductivity σ( ω ) =σ 1 ( ω ) – i· σ 2 ( ω ) values can be determined from measured values of the resonance frequency shifts and Q-factors due to losses in YBCO films.
The last quantity (Q-factor in YBCO) can be evaluated
from the measured Q-factor values by subtracting the
losses in the lateral surface of the cavity and losses in
sapphire with very good accuracy because these additional losses are much smaller than the losses in YBCO
films. However the resonance frequency shift value
which is defined as the difference between the resonance
frequency of sapphire resonator terminated by perfectly
conducting plates and the resonator terminated by YBCO
films cannot be determined from measurements with satisfactory accuracy. In practice the reference measurements are performed on the sapphire resonator terminated by two flat, polished with optical quality, copper
plates. The major problem is that flatness of YBCO films
and Cu plates is not perfect and repeatability of the resonance frequency after assembling-disassembling cycle is
the order of few MHz (typically 2-3MHz). With such uncertainty of the resonance frequency it is only possible to
measure thin YBCO samples (30nm thickness) when the
resonance frequency shifts are the order of 40MHz. For
thick samples the expected resonance frequency shifts
are the order of few MHz so additional measurements are
required to determine the absolute resonance frequency
shift values.
Common approach in such a case is to perform
variable temperature measurements for the sapphire resonator terminated by YBCO films under test and then
measurements for the same resonator terminated by Cu
T (K)
f (MHz)
Q_YBCO
f (MHz)
Q_Cu
4.2
28156.557
331021.11
28158.305
8600
70
28153.487
151627.96
28156.087
7600
Table I. Measured resonance frequencies and Q-factors of sapphire resonator terminated by YBCO films and Cu plates at two
different temperatures
temperatures have to be measured. Results of such measurements are presented in Table I.
It should be noted that measured resonance frequency-temperature variations depend on two major factors namely on temperature variations of sapphire resonator itself and the temperature variations of σ 2 (or London penetration depth). The temperature variations of
FIG. 1. Imaginary part of complex conductivity versus
resonant frequency shift (difference between resonance
frequency of the cavity with and without sample). Black
squares denote date obtained from rigourous electromagnetic
modelling and the solid red line denotes fit used in
interpolation. Green dots mean the values that match
simultanously two constraints and the solution of e-m
modeling.
sapphire resonator can be found as the difference between the resonance frequencies at 70K and 4.2K measured with copper plates while the total variations as the
difference between the resonance frequencies at 70K and
4.2K measured with YBCO films. It should be noted that
the resonance frequency variations due to the temperature change of the skin depth for copper plates is small
and can be neglected (Q-factors with Cu plates are close
each other at 4.2K and 70K as it is seen in the last column
of Table I). Therefore the resonance frequency shift due
to the temperature change of σ 2 for YBCO from the
measurement results in Table I can be found as:
(𝑓70 − 𝑓4.2 )YBCO = (𝑓70 − 𝑓4.2 )YBCO+sapphire − (𝑓70 −
𝑓4.2 )Cu(sapphire)
(1)
In our previous work1Error! Bookmark not
defined. the temperature studies of the complex conductivity of various YBCO films from the same manufacturer (THEVA) as those that are used in our current work
have been performed. We revealed that temperature dependence of our YBCO films well suits to the following
model: σ2(T)= σ2(0)·[1-(T/Tc)4], despite the fact that the
model rather applies to s-wave superconductors, while
YBCO has been known as a d-wave superconductor.
From this model one can calculate the ratio of conductivities at 4.2K and 70K as:
𝜎2 (𝑇=70K)
𝜎2 (𝑇=70K)
= 0.59
(2)
Having known theoretical dependence of the resonance
frequency shift on the conductivity σ2 as well as the two
constrains: measured resonance frequency shift (1) and
conductivity ratio (2) one can uniquely determine the absolute value of σ2 at 4.2K as it is shown in FIG. 1.
2. Discussion on differences between complex conductivity and surface impedance approaches
In the fundamental electrodynamics (Maxwell
equations in the complex form that are valid for time harmonic electromagnetic fields) the following basic relations are introduced:
a) complex permittivity that includes conductivity
term
𝜀 = 𝜀0 𝜀r = 𝜀0 (𝜀r′ − i𝜀r" − i
𝜎
𝜔𝜀0
)
(3a)
b) complex wave impedance
𝑍𝑤 = √
𝜇
𝜀
(3b)
For nonmagnetic materials (such as dielectric or
YBCO) both the complex permittivity and the complex
wave impedance can be alternatively used to characterize
bulk material. At microwave frequencies electromagnetic fields in conductors and superconductors are atten-
uated and exponentially decay in the direction perpendicular to the surface with can be described by one parameter namely the attenuation constant 𝛼 which is defined as
the real part of the propagation constant 𝛾.
𝛾 = 𝛼 + i𝛽 = √−𝜔 2 𝜀𝜇
(4)
It should be noted that for superconductors the
conductivity
𝜎
is
the
complex
quantity
𝜎 = 𝜎1 − i𝜎2 with the imaginary part 𝜎2 which is larger
than the real part 𝜎1 . Substituting the complex conductivity into equations (3a) and (4) one can notice that electromagnetic fields are attenuated in superconductors mainly
(but not entirely) due to the presence of the imaginary
part of conductivity. In metals or in superconductors in
the normal state (at temperatures above the critical temperature) attenuation is entirely due to the presence of the
real part of conductivity. The length over which the electromagnetic field decays by a factor of e is called the penetration depth λ for superconductors and skin depth for
normal conductors. In both cases It can be evaluated from
the same formulae as λ = 1/α , or as 𝛿 = 1/α .
If the thickness of a sample made from superconducting or conducting material is larger than the penetration depth or the skin depth then the electromagnetic
fields in the internal part of the sample are negligibly
small and the sample can be considered as being made of
thin material having thickness equal to the penetration
depth or the skin depth and its properties can be alternatively described either by the complex wave impedance
(which is often called the surface impedance) or by the
complex permittivity (complex conductivity). The wave
(surface) impedance approach is commonly used in the
electrodynamics of superconductors.
The problem becomes more complicated when
the thickness of the sample becomes smaller than the
penetration depth or the skin depth. In such cases the
complex wave impedance – which is defined on the surface of the sample depends not only on the material properties of the sample (i.e. its complex permittivity or complex conductivity) but also on the properties of layers that
lay behind the conducting or superconducting film. Impedance transformation approach which is valid for layer
structures and for TEM electromagnetic waves in free
space still can be successfully used for analysis of resonators containing such structures (with some limitations
that were discussed in [1]) but one cannot interpret the
wave impedance of the layered structure as the material
property of the conducting (superconducting) film. For
this reason the complex permittivity (complex conductivity) approach is more general than the complex impedance approach and what is more important it allows to be
directly implemented in the modern electromagnetic simulators.
1
J. Krupka, J. Wosik, C. Jastrzębski, M. Zdrojek,
T.Ciuk, and J. Mazierska, IEEE Trans. Appl.
Supercond 23, 1501011, (2013).
* Author to whom correspondence should be addressed. Electronic mail: krupka@imio.pw.edu.pl