High-Temperature-Superconductor Thin Films at Microwave Frequencies

391 Pages·1999·6.246 MB·English

by  Matthias Hein (auth.)

Preface
About the Content
The discovery of the oxide high-temperature superconductors (HTS) by Bed-
norz and Miiller 1 in 1986 added one further Nobel prize to the continuing
history of superconductivity, which was initiated by the fundamental work of
Kammerling Onnes in 8/(91 and 1911 ,2 .3 The high transition temperature
and the related short coherence length, the quasi-two-dimensional electronic
structure and the related anisotropy as well as unconventional features in the
electronic density of states belong to the specific properties of HTS. These
have had multiple impact on interdisciplinary sciences, and are still attracting
increasing attention from scientists and economists. Chemists, physicists and
engineers merged their intellects and interests in order to develop a compre-
hensive understanding and a consequent exploitation of the wide-spread po-
tential that these novel compounds promise. Since their discovery, enormous
technological improvements in the preparation and physical characterization
of high-quality thin HTS films and Josephson junctions have been achieved.
Extended treatments of the phenomenology, possible application and func-
tional demonstration of HTS films and devices at high frequencies accom-
panied this development 4-11. Theoretical and experimental physicists are
still puzzling over spectroscopic data and measurement techniques in order to
resolve the nature of superconductivity in HTS in detail .21 While this de-
bate has markedly enriched the academic view of superconductivity, potential
customers have become aware of promising market opportunities, and efforts
are currently being undertaken to commercialize HTS in communication and
remote sensing systems 13-15.
This book summarizes those physical and technological aspects of HTS
films at microwave frequencies which are considered basic for the present
state of the art. The parallel discussion of the various specific theoretical and
empirical issues aims at emphasizing recent progress in this field. However,
the analysis is supplemented by a review of the corresponding frameworks, in
order to make this exciting field accessible not only to specialists, but also to
advanced students and interested scientists from other disciplines. Regarding
the potential for microwave device applications of HTS, the discussion is
focused on YBa2Cu3Oz-x and T12Ba2CaCu2Oy.
VIII Preface
In line with the general and the specific features of superconducting films,
the microwave properties of the oxide superconductors are compared to those
of the "conventional high-temperature superconductor" Nb3Sn. The com-
pounds with cubic A15 lattice type displayed the highest transition temper-
atures known before the discovery of the superconducting oxides 16. Due
to the much simpler structure and the well-verified theoretical description,
Nb3Sn presents a valuable reference material for HTS. Such a comparison
becomes even more exciting when considering the strong pair coupling and
the short coherence length, which are common to both materials.
About the Strategy Behind this Book
The microwave responses of superconducting films at low excitation levels
(linear regime, Chaps. 1 and )2 and at high levels (nonlinear regime, Chaps.
3 and )4 are treated separately. Both topics are analyzed in terms of well-
approved models and theoretical or numerical approaches as well as in terms
of recent experimental results. The theoretical treatment (Chaps. 1 and )3
forms a bridge from famous summaries of the fundamental aspects of super-
conductivity (like Tinkham's book 71 or the book edited by Parks 18) to
the most specific properties at high frequencies. The experimental data dis-
cussed (Chaps. 2 and )4 refer to the worldwide state of the art, as reported
by many international groups. Of special importance is the development of
a consistent and comprehensive understanding of the microwave response of
the two materials considered, which was not available in the literature before.
Chapter 1 starts with the electrodynamic and the microscopic descrip-
tions of the surface impedance Zs of superconductors, including the impact
of finite film thickness. Some emphasis is put on the phenomenological treat-
ment of Zs in the framework of the two-fluid model. While this approach
developed from the initial understanding of microwave superconductivity by
F. London and H. London ,91 recent extensions account for granularity and
layered structures. In Chap. ,2 the various techniques and procedures to de-
duce the surface impedance of superconducting films from measurement are
summarized from a systematic point of view. The related physical aspects
bear an exciting symmetry between the real and the imaginary parts of the
surface impedance, which is explained and illustrated in terms of comparative
analyses and analogies.
The nonlinear microwave response of superconductors (Chaps. 3 and )4
deserves special attention since it enables the identification of various intrinsic
and extrinsic effects. Furthermore, it currently limits many promising appli-
cations of HTS devices in communication systems. Chapter 3 starts with
the theoretical description of the known mechanisms which introduce mi-
crowave field and power dependences to the surface impedance. Adiabatic,
i.e. quasi-stationary, types of nonlinear response are distinguished from dy-
namic effects, which are related to the intrinsic time scales of the supercon-
Preface IX
ductors. The sources and consequences of microwave heating which dominate
the nonlinear surface impedance of nonideal films over an extended tempera-
ture range are also analyzed. Experimental results on the DC and microwave
magnetic field dependent surface impedance at field levels around the lower
critical field are analyzed in detail in Chap. 4. The knowledge gained from
the DC measurements serves as a valuable basis for identifying magnetic field
limitations at high frequencies and distinguishing them from thermal effects.
The book continues with a brief description of the preparation, handling
and mounting techniques appropriate for both superconductors (Chap. 5).
While the A15 compound has been known for about 30 years, large-grained
phase-pure Nb3Sn films on dielectric substrates became available only re-
cently. Chapter 5 also sketches the preparation of HTS films, and remarks on
the fabrication of engineered grain-boundary Josephson junctions. The lat-
ter part is kept brief, since excellent treatments on the details are available
elsewhere [20]. Chapter 5 finally contains a summary of the present status of
refrigeration technology, which acknowledges its relevance to the integration
of functional superconducting microwave devices.
The last chapter is devoted to a general discussion of possible applications
of high-temperature superconductors in passive microwave devices. The first
part constitutes a bridge from single resonators to filters, which represent the
most prominent class of near-term applications of HTS. Selected examples of
resonant devices are then discussed schematically, in order to illustrate the
potential of HTS at frequencies between 10 MHz and 01 GHz. Passive mi-
crowave components based on the Josephson effect are the subject of the last
part of Chap. 6. Though presently of less economic relevance than passive
film devices, this field combines basic aspects of both microwave and Joseph-
son technology, and thus bears the potential for yielding a large variety of
novel integrable microwave devices.
About the Support for this Work
Many aspects described in this book reflect the advanced state of research and
development at the Departments of Physics and Electrical Engineering at the
University of Wuppertal. The expertise of my colleagues therefore contributed
to the physical, social and financial basis of this work. I am especially grateful
to Dr. B. Aminov, Dr. B. Aschermann, C. Bauer, A. Baumfalk, Prof. H.
Chaloupka, W. Diete, M. Getta, S. Hensen, F. Hill, M. Jeck, Dr. T. Kaiser, J.
Kallscheuer, Dr. S. Kolesov, Dr. M. Lenkens, Prof. G. Miiller, Dr. S. Orbach-
Werbig, M. Perpeet, Prof. H. Piel, J. Pouryamout, M. Reppel, S. SchmSe, P.
Seidel, and H. Schlick.
Furthermore, I appreciate very much the kind support of, fruitful dis-
cussions with, and partially unpublished information from many colleagues:
Prof. S. M. Anlage (University of Maryland), Dr. S. Beuven (ISI, FZ Jiilich),
Prof. A. Braginski (ISI, FZ Jiilich), Dr. E. H. Brandt (MPI Stuttgart), Dr.
X Preface
A. Cassinese (University of Naples), Dr. T. Dahm (MPI Dresden), Dr. M.
Golosovsky (Hebrew University), Prof. R. Gross (Universit/it KSln), Dr. H.-
U. H/ifner (Leybold), Dr. J. Halbritter (IMF, FZ Karlsruhe), Dr. R. Hei-
dinger (IMF, FZ Karlsruhe), Dr. E. Ilfchev (IPHT Jena), Dr. Chr. Jooss
(MPI Stuttgart), Dr. M. Klauda (Bosch), Dr. N. Klein (IFF, FZ Jiilich),
Dr. V. Z. Kresin (Lawrence Berkeley Laboratory), Prof. M. J. Lancaster
(University of Birmingham), Dr. M. Manzel (IPHT Jena), Dr. H.-G. Meyer
(IPHT Jena), Dr. M. Nisenoff (Naval Research Laboratory), Dr. D. E. Oates
(Lincoln Laboratory), Prof. Ya. G. Ponomarev (Moscow State University),
Dr. A. Porch (University of Birmingham), Prof. A. M. Portis (University of
Berkeley), Prof. K. Scharnberg (Universit/it Hamburg), Dr. E. Sodtke (ISI,
FZ Jiilich), Dr. G. Thummes (Universitiit Giet3en), Prof. R. Vaglio (Univer-
sity of Naples), Prof. I. B. Vendik, Prof. O. G. Vendik (Technical University
St. Petersburg), Dr. B. Willemsen (STI), Dr. R. Withers (Bruker), Dr. R.
WSrdenweber (ISI, FZ Jiilich), Dr. J. Wosik (University Houston), and Dr.
C. Zucearo (ISI, FZ Jiilich).
This review of supporters would be incomplete by considering merely the
scientific aspects. I am very much indebted to my wife Sabine, my daughter
Anneli and my son Michael as well as to my mother, who often encour-
aged me during the intense time of writing and reviewing before publication.
They have been very patient and knowledgeable companions during numer-
ous "Highs" and "Lows".
Solingen, April 1999 Matthias Hein
1. Temperature and Frequency Dependent
Surface Impedance
Only fractions of our world -eb
have linearly, but it is amazing
how far linear modeling has pro-
.dedeec
1.1 Physical Framework
This section introduces the electrodynamics needed to understand the basic
concept of surface impedance of superconductors. Starting with Maxwell's
equations and the kinetics of the charges which carry the microwave cur-
rents, the treatment is kept on a phenomenological level. Three characteris-
tic lengths are identified to distinguish between different limiting regimes of
the electrodynamics of superconductors. The physical concept of the surface
resistance and the surface reactance (or, equivalently, the penetration depth)
are subsequently illustrated in terms of the energy dissipated and stored, re-
spectively, in the superconductor. Finally, the effect of finite fihn thickness
on the concept of the surface impedance is discussed.
1.1.1 Field Equations and Characteristic Lengths
Penetration of Microwave Fields into Superconductors
The interaction between a plane electromagnetic wave at circular frequency
= 27r and a metal of conductivity a is described by Maxwell's equations.
These present expressions for the microscopic electric field vector E and
the magnetic induction B, and the macroscopic displacement D and the
magnetic field H. A detailed treatment of the relations between E and D
and between B and H can be found, e.g., in 1. Taking into account the
absence of magnetic monopoles (i.e., divB = 0), and the absence of free
or polarized charges in metals (i.e., divD = 0), the remaining Maxwell's
equations are
OB
curl E - Ot (1.1a)
and
OD
curlH = J + 0~- " (lAb)
The magnetic induction B can be expressed as the rotation of the magnetic
vector potential A: B = curl A. In the absence of magnetizing effects and in
2 .1 Temperature and Frequency Dependent Surface hnpedance
nonmagnetic materials, B is proportional to the magnetic field H, B = #0"H
with #0 = r74 x 10 -7 Vs/Am being the magnetic permeability of vacuum. In
order to find solutions for E and B, (1.1) needs to be supplemented by a
current field relation like, for instance,
J = a.E. (1.2)
With a real, (1.2) describes Ohm's law for isotropic metals. However, as will
be shown below, it also remains valid for superconductors, the conductivity
of which is complex:
7( : 0" 1 -- i(r 2 = al(1 - itan0) . (1.3)
Let us consider a plane electromagnetic wave with harmonic time depen-
dence, with the electric and the magnetic fields spanning the x-y plane and
propagating into a metal along the z direction: E(z, t) = Eo expi(~t - kz).
Combination of (1.1) and (1.2) then yields the wave equation for the electric
field
02 E O E 02 E
Oz 2 -- #or + ZZo#o Ot 2 , (1.4)
with e and r the permittivity of the metal and of vacuum (50#o = c -2
with c the velocity of light). The first term on the right-hand side of (1.4)
results from the microwave transport current J, while the second term is
due to the displacement current OD/Ot. The ratio of the magnitudes of these
two contributions is weighted by a/wr Since this parameter is large com-
pared to unity for any metal at microwave frequencies (a; of order 1010 Hz),
the displacement currents can usually be neglected. The remainder of (1.4)
represents the equation of motion of a wave decaying in the z direction:
E(z, t) = Eo exp{i(wt - kz)} exp(-z/A), (1.5a)
with the supplementing condition
ik + = iw#0a. (1.5b)
The characteristic length A is the microwave penetration depth. It describes
the surface region of a superconductor within which the external field has
decayed to 1/e of its value at the surface. Equation (1.5b) leads to the general
expression
2A = 2 Vfi- + y 2 - 1 _ 2 1 -sin0 (1.6)
~0020"1 /~ ~0020.1 COS 0 '
where the substitution y = al/0.2 = cot0 was used to indicate the "degree
of normal conductivity" (see Sect. 1.2). For normal conductors, 2.0 = 0 and
thus y -+ cc or 0 = 0. In this limit, (1.6) reproduces the skin depth A =
~( (2/w#0o'1) 1/2, with the wave number k = 1/5 following from (1.5b).
=
In the opposite case 0" 1 = 0 (y = 0 or 0 : 7r/2), which corresponds to a
perfect superconductor at zero temperature, A = (1/w#ocr2) 2/1 and k = 0 are
obtained.
1.1 Physical Framework 3
Complex Conductivity
The concept of complex conductivity was initially introduced by Gorter and
Casimir in the framework of the two-fluid model (TFM) 2, and applied by F.
London and H. London to describe the finite conductivity of superconductors
at microwave frequencies 3. However, as discussed in several books ,4 5, the
general idea can be applied to various physical situations (Sect. 1.2). It shall
suffice here to reconstruct the qualitative features of the original TFM to
develop a consistent picture of the electrodynamics of superconductors.
At finite temperatures below the transition temperature To, a supercon-
ductor contains normal electrons and Cooper pairs. The equation of motion of
normal electrons (index "n", mass m and charge -e) takes into account damp-
ing, which results from scattering at time intervals .-7 The scattering time is
assumed to result solely from collisions between electrons and phonons, as
expected for a free electron gas, and to be independent of frequency. This
leads to the Drude model of conductivity of metals:
dvn ~_~
m-~- +m = -e- E. (1.7)
The range that the electrons can travel between subsequent scattering events
is described by the mean free path g = FV (cid:12)9 ,w with the velocity evaluated
at the Fermi level. In contrast, Cooper pairs behave, due to their quantum-
statistical bosonic nature, like free charges of mass 2m and charge -2e (index
"s'). The corresponding equation of motion is
2m dvs
dt = -2e-E. (1.8)
In an electric field, both kinetic contributions add to the total velocity v =
nV + vs. Furthermore, the flux of charges causes a current J = -n. e. v where
n is the number density of charge carriers. In the London gauge (div A = 0
and E = -OA/Ot), (1.8) thus presents a simplified version of the London
equation, which relates the local supercurrent Js to the vector potential A
at the same position 6:
Js - nse2 A . (1.9)
m
Combining (1.7) and (1.8), considering the harmonic time dependence of
the fields, and converting velocities into currents, leads to the current field
relation of superconductors:
)
f
J = a0 tx, 7 ~- ~ ix~/ (cid:12)9 E. (1.10)
Here, 0ro = nr = (1/w'r)an, and nX = nn/n~ (xs = 2ns/n~) denotes
the fractional number densities of normal (paired) electrons, with n~ and an
the number density of electrons at the Fermi level and the conductivity in
the normal state, respectively. Comparing (1.10) with (1.2) illustrates the
4 .1 Temperature and Frequency Dependent Surface hnpedance
physical meaning of the complex conductivity a = 01 - ia2. In addition,
the penetration depth A of a perfect superconductor at T = 0 K equals the
London penetration depth IL = m/#oe2n~.
In the framework of the two-fluid model, Gorter and Casimir assumed
complete pairing, nX + sX = 1, with the temperature dependence of nX de-
scribed by (T/Tc) .4 We will see below that the conditions for complete pairing
can be violated, although the merit of the formalism developed in this section
remains unaffected.
citsiretcarahC Lengths
The current field relations in (1.2), (1.9) and (1.10) imply local conditions,
i.e., the charges carry current at a position r which follows the electric field
at the same position r. Such a local relationship (called "normal skin effect")
is fulfilled in normal metals if the mean free path g is small compared to
the skin depth 6 which describes the spatial variation of E(r) (see (1.5a)).
Since 6 decreases with increasing frequency and g increases with decreasing
temperature, there are ranges of parameters where the application of local
electrodynamics is no longer justified. Rather, the current originates from the
average electric field sensed by the electrons along their mean free path 6:
J(r) f K(r r')E(r')dSr ' . (1.11)
: -
The kernel K(r - r') hence has a range of about .~ This limit, which is
called the "anomalous skin effect", was theoretically analyzed by Reuter and
Sondheimer 7. Further details of this case and corresponding results for the
surface impedance can be found, e.g., in ,8 9.
Similar to the normalconducting case, the local current field relation fails
if the electrodynamic "coherence" range of Cooper pairs, (p, exceeds the
superconducting penetration depth 1. This was first realized, and argued
in analogy to the anomalous skin effect, by Pippard 8. He proposed to
substitute London's equation (1.9) by the nonlocal relation relation
en3 2 f RR. A(r')
J(r)
_ 0~mc74 ~ hG exp(-R/~p)d3r ' , (1.12)
where R = r- r' and R = IR. The "intrinsic coherence length" 0~ is a
constant of the pure metal, determined by FV and the thermodynamic time
constant h/kBTc characterizing the superconducting state (kB = 1.38 x 10 -23
J/K is the Boltzmann constant) ,6 10:
7" Fvh (1.13)
0~ -- "17 2 kBTc
where 7 ~ 1.781 is Euler's constant (see Sect. 1.3.1). The value of ~p varies
between 0~ for clean superconductors (f >> )0~ and f for dirty superconductors
(~ << ~0):
1.1 Physical Framework 5
PC 1 - 1 1 )41.1(
In summarizing the above results, the appropriate electrodynamic description
of superconductors depends on the three basic lengths ,A ~ and .0~ The pen-
etration depth A itself scales with/~ in the local limit, but still for sufficiently
large mean free paths, like /11, 12:
Lil+ oe. )51.1(
Consequently, four limits can be defined by the dimensionless ratios A/~0 and
~/~0 as indicated in Table .1 Nonlocal effects need to be considered especially
in superconductors of the first kind. Mean free path effects become increas-
ingly important in alloyed superconductors. In the case of superconducting
films, the film thickness dF places additional constraints on the locality con-
ditions. We assume here, for simplicity, that dF is sufficiently larger than ~/
and .0~ It shall further be noted that the electrodynamic ("Pippard') coher-
ence length is, in general, to be distinguished from the "Ginzbur~Landau"
coherence length LG~ which describes the spatial variation of the supercon-
ducting order parameter in the presence of magnetic fields. As given in 10,
the microscopic theory yields expressions for LG~ and the Ginzburg-Landau
parameter a = AGL/~GL (see Chap. 3) as functions of 0~ and ,~/ with the
limiting cases reproduced in Table 1.1.
Table 1.1. The four limiting cases describing the electrodynamics of superconduc-
tors, and corresponding equations relating electrodynamic and magnetic parame-
ters 10, ,11 31
Electrodynamics Ginzbur~Landau
Nonlocal, 0G/A ~ 1 Local, oG/A >~ 1 (coefficients evaluated
Eq. (1.12) Eqs. (1.9), (1.10) at T ~ Tc).
Clean limit, PG -~ 0G PG ~- 0G LGG = 0G937.0
0G/g >> 1 ~" ~ )0G~')(258.0 3/1 A = LA a = 0~/LA759.0
Dirty limit, PG = ~ PG -~ ~ LGG --~ 0.852(~0~) 1/2
oG/g << 1 A ~ )g/0G(LA 2/1 A = )g/OG(LA52.1 2U a = 0.720AL/t
1.1.2 Surface Resistance and Surface Reactance
Concept of the Surface Impedance
The concept of the surface impedance facilitates the description of the elec-
trodynamics at the interface between vacuum and the surface of a (su-
per)conductor. The superconductor is assumed here to extend from z -- 0
6 .1 Temperature and Frequency Dependent Surface Impedance
to infinity. The effect of finite thickness dF <_ A of a superconducting sheet is
discussed in Sect. 1.1.3. In accordance with the usual concept of the complex-
valued AC impedance, the "bulk" surface impedance Zs is defined by the
ratio of the electric field at the surface to the total current flowing across a
unit line in the surface:
Es ~E (1.16)
Zs- f~ g(z)dz - Hs
The integration can be performed by using (lAb), leaving Zs as the ratio of
the tangential electric field to the tangential magnetic field at the surface of
the superconductor. Alternatively, by application of (1.2) and (1.5) to (1.16),
Zs can be expressed in the local limit in terms of the complex conductivity
a ,9 :41
Zs = = ~R + iXs. (1.17)
The real part of Zs is called the surface resistance ,~R the imaginary part
the surface reactance Xs. The expression for Zs(a) for arbitrary values of the
conductivity ratio y is found to be
/-~p0 ~
Zs = V -~al t~- + i~+) (1.18a)
with
~:-- 1+y2 Y (V/-f + 2y 4-1 ) soc---- 1(0 (cid:127) (1.18b)
The limiting case for a normal conductor is
Rs = Xs --- Rn ---- (w#0/2Ol) 1/2 , (1.18c)
while for a superconductor in the limit y << 1
Rs = 1/2#2w2oA 3 and .is = (t-dp0/O'2) 1/2 = A0pdt (cid:12)9 (1.18d)
According to 15, the two-fluid model can be modified in order to derive
semi-analytical formulations of ~Z for arbitrary values of g/~0 in relation
to the microscopic understanding of o. Further details of this approach are
postponed until Sect. 1.3.
The implication of nonlocal behavior to the surface impedance of super-
conductors was analyzed by Mattis and Bardeen 61 and discussed in more
detail in 17,18. As before, Zs can be obtained from the frequency dependent
complex conductivity, which now depends additionally on the wave vector of
the charge trajectories. For diffuse scattering of electrons at the surface of
the metal, Zs is given /7( by )1
#owO(, k) dk (1.19a)
~Z = ilrw#0 In 1 + 2k
This equation reduces in the extreme anomalous limit to the relation