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Complex conductivity of highíT c
single crystals in the microwave band:
Some intermediate results and unsolved problems
M. R. Trunin
Institute of Solid State Physics RAS, 142432 Chernogolovka, Moscow district, Russia
(March 14, 2001)
The analysis of investigations of the microwave surface impedance Zs (T ) and conductivity tensor
Ó
oe(T ) of HTS single crystals as functions of temperature, alongside the procedure of their derivation
from parameters measured directly in experiments, has revealed a number of facts that contradict
previously developed electrodynamic models of the conductivity in these materials. The paper is
dedicated to plausible causes of these discrepancies, which are assumed to be related to specific
features of HTS structures, and to particular questions that should be answered by the research
undertaken in the immediate future, among which the major problem is the nature of residual losses
in HTS.
I. INTRODUCTION
Microwave measurements of the temperature depení
dence of the complex conductivity Ó
oe(T ) = Ó
oe 0 (T ) \Gamma iÓoe 00 (T )
of highíT c superconductors (HTS) have advanced coní
siderably our understanding of the pairing mechanism in
these materials. The real part Ó
oe 0 (T ) is susceptible to the
scattering rate of quasiparticles, as well as their density
of states. The imaginary part Ó oe 00 (T ) is related to the
magnetic field penetration depth Ö(T ). Because of the
high anisotropy of HTS crystals, the conductivity Ó oe is a
tensor characterized by its components oe ab and oe c .
The first part of this paper considers derivation of
oe ab (T ) and oe c (T ) from the measured temperature deí
pendences of the Qífactor and the frequency shift ffif of
the cavity for different sample alignments with respect
to the ac magnetic field H! . In the local electrodynamí
ics, which can be applied to the discussed experiments,
Ó
oe(T ) = i!ï 0 =Z 2
s (T ), where Z s (T ) = R s (T ) + iX s (T ) is
the surface impedance of the sample. At microwave freí
quencies the imaginary part of the surface impedance, the
surface reactance X s , reflects mainly the response of the
superconducting carries and is due to the nondissipative
energy stored in the surface layer of the superconductor.
The real part, the surface resistance R s , is proportional
to the microwave losses and is due to the normal carriers.
In the second part of the paper we will analyze the
common and distinctive features of the temperature deí
pendences of Z ab
s (T ) and oe ab (T ) in the abíplane of various
HTS single crystals. Our attention in this analysis will
be focused on the magnitude of the residual losses, whose
nature in HTS is, perhaps, the central topic of investigaí
tions and discussions. In addition, we will discuss a pheí
nomenological model describing the microwave response
of HTS crystals. Finally, the problem of the anisotropy
in the conductivity of HTS materials will be broached.
II. MEASURED QUANTITIES AND SAMPLES
The most convenient technique for measurements of
the surface impedance of small HTS samples in the XíW
microwave frequency bands is the soícalled `hotífinger'
method. The underlying idea of the method is that a
crystal is set on a sapphire rod at the center of a superí
conducting cylindrical cavity resonating at the frequency
f in the H 011 mode, i.e., at the antinode of a quasií
homogeneous microwave magnetic field (Fig. 1). By varyí
ing the rod temperature, measuring the Q s ífactor and
frequency shift \Deltaf s of the cavity with the sample inside,
and comparing them with the parameters of the empty
cavity, Q e and \Deltaf e , one can determine the sample surface
resistance R s and reactance X s as functions of temperaí
ture using the relations 1
R s (T ) = \Gamma s \Delta(1=Q) = \Gamma s
\Theta
Q \Gamma1
s (T ) \Gamma Q \Gamma1
e (T )
\Lambda
; (1)
X s (T ) = \Gamma2\Gamma s
ffif
f
= \Gamma 2\Gamma s
f
[\Deltaf s (T ) \Gamma \Deltaf e (T ) \Gamma f 0 ] ; (2)
where \Gamma s = !ï 0
R
V H 2
! dV=[
R
S H 2
s dS] is the sample geoí
metrical factor, ! = 2‹f , ï 0 = 4‹ \Delta 10 \Gamma7 H/m, V is the
volume of the cavity, H! is the magnetic field generated
in the cavity, S is the total sample surface area, and H s
is the tangential component of the microwave magnetic
field on the sample surface. In Eq. (1) \Delta(1=Q) is the
difference between the values 1/Q of the cavity with the
sample inside and the empty cavity. In Eq. (2) ffif is the
frequency shift relative to that which would be measured
for a sample with perfect screening and no penetration of
the microwave fields. In the experiment we measure the
difference (\Deltaf s \Gamma \Deltaf e ) = \Deltaf between the resonant freí
quency shifts of the loaded and empty cavity as functions
of temperature, which is equal to \Deltaf (T ) = ffif (T ) + f 0 ,
where f 0 is a constant. The constant f 0 includes both the
perfectíconductor shift and the uncontrolled contribution
caused by opening and closing the cavity.
The efficiency of this method was tested in experiments
with Nb samples 2 . The exponential dependence of the
surface impedance in the lowítemperature range and obí
servation of a coherent peak in the real part of the mií
crowave conductivity of Nb support the assertion that the
BCS theory 3 adequately describes electromagnetic propí
erties of conventional superconductors. Measurements of
1

the London penetration depth ÖL , superconducting gap
\Delta(0), and coherence length ¦ 0 in Nb were in agreement
with literature data.
sample
heater
coupling
lines
sapphire
rod
H
thermometer
Nb
cavity
FIG. 1. Diagram of the microwave cavity used in the
`hotífinger' technique.
In this paper, we will discuss measurements of the
impedance and conductivity of copperíoxide HTS siní
gle crystals, which usually have the shapes of plates
with lateral dimensions a ¦ b ¦ 1 mm and thickí
nesses c ¦ 0:1 mm: YBa 2 Cu 3 O 6:95 (YBCO,T c ‹ 93 K),
Bi 2 Sr 2 CaCu 2 O8+ffi (BSCCO #1, T c ‹ 83 K; BSCCO #2,
T c ‹ 92 K), Tl 2 Ba 2 CaCu 2 O 8\Gammaffi (TBCCO,T c ‹ 112 K),
Tl 2 Ba 2 CuO6+ffi (TBCCO,T c ‹ 90 K). With the excepí
tion of the slightly overdoped sample BSCCO #1, the
doping levels of all other samples were optimal so that
their values of T c were maximal.
Figures 2 and 3 show as examples curves of the surí
face resistance R s (T ) and reactance X s (T ) as functions
of temperature measured in the abíplane of the crystals
BSCCO #1 and BSCCO #2. Equations (1) and (2) iní
dicate that two parameters, \Gamma s and f 0 , are required to
derive absolute values of R s and X s (T ) from measureí
ments of Q(T ) and \Deltaf (T ). The geometrical factor \Gamma s
depends on the shape and dimensions of the sample, and
its position in the microwave cavity with respect to the
microwave field H! distribution. The parameter \Gamma s can
be determined both empirically and theoretically, and
its value is of order of tens of kiloíohms at a frequency
¦ 10 GHz. The constant f 0 can be derived from the
measurements of the microwave response of HTS single
crystals in the normal state, and the determination of
this parameter will be discussed in the following section.
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5 V
##PW#
7##.#
5
UHV
%6&&2###
#
5
V
##W#
7##.#
Dl
DE
##QP#
FIG. 2. Surface resistance Rs(T ) in the abíplane of a
BSCCO #1 single crystal at 9:4 GHz. The inset shows linear
plots of \DeltaÖ ab (T ) and Rs(T ) at low temperatures. The value
of the residual surface resistance Rres ‹ 120
ï\Omega is indicated.
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%6&&2###
5
6
; 6
5
6
####;
6
####W#
7##.#
l##QP#
7##.#
5
6
##PW#
7##.#
FIG. 3. Surface resistance Rs(T ) and reactance Xs(T ) in
the abíplane of a BSCCO #2 single crystal at 9:4 GHz. The
insets show linear plots of Ö(T ) and Rs(T ) at low temperaí
tures.
Let us assess the accuracy of the surface resistance
and field penetration depth determined by the hotífinger
technique. The uncertainty ffiR s in measurements of
2

R s (T ), ffiR s = \Gamma s ffi(Q \Gamma1 \Gamma Q \Gamma1
0 ) = \Gamma s ffiQ=Q 2 , is deterí
mined by the uncertainty ffiQ=Q, which was within 1%
at Q ¦ 10 7 in our experiments. If \Gamma s = 10 k\Omega\Gamma then we
have ffiR s ‹ 10 ï\Omega\Gamma As the temperature drops several
degrees below T c , the increment \DeltaX s (T ) = !ï 0 \DeltaÖ(T ),
and, given X s (T ) expressed in absolute units, one can
determine the magnetic field penetration depth Ö(T ) =
X s (T )=!ï 0 . The uncertainty in the relative value of
the penetration depth is \DeltaÖ(T ), ffi(\DeltaÖ) = (2\Gamma s =!ï 0 ) \Delta
(ffi(\Delta!)=!), which equals several ×
angstrèom. The error in
Ö(T ) can be up to 30% of the actual value Ö(0) and is
largely determined by the measurement accuracy of the
constant f 0 .
Note also that the temperature dependence of the reí
actance can be notably affected by the thermal expansion
of a sample. Since the resonant frequency is determined
by the volume where the microwave field is confined,
the thermal expansion of the sample is equivalent to a
smaller penetration depth of the microwave field, and
this leads to an additional frequency shift \Deltaf th in the
brackets on the right of Eq. (2). It was shown in the earí
lier publication 1 that the contribution of \Deltaf th (T ) to the
total frequency shift of the cavity with a sample inside
is negligible in the range of low temperatures, however,
at T ? 0:9 T c it may be quite considerable, especially
for highly anisotropic BSCCO single crystals in the case
when their abíplanes are oriented perpendicular to the
microwave magnetic field H! : H! kc. In particular, the
curve of X s (T ) in Fig. 3 was plotted with due account of
the thermal expansion of the sample BSCCO #2. Otherí
wise, i.e., when the thermal shift \Deltaf th in Eq. (2) is negí
ligible, the curve of reactance, which follows that plotted
in Fig. 3 up to T ‹ T c , has a smaller slope at T ? T c ,
and the discrepancy is
25m\Omega at T = 150 K.
III. CONDUCTIVITY OF ANISOTROPIC
SUPERCONDUCTORS
At T ? 4 K the relation between the electric field and
current density is local in both the superconducting and
normal states of HTS: j = Ó
oeE, where the conductivity
Ó
oe is a tensor, which is characterized by two components
in the crystals with a tetragonal structure: oe ab in the
abíplanes of CuO 2 and oe c in the direction perpendicular
to the cuprate planes. In the normal state, an ac field
penetrates in the direction of the cíaxis through the skin
depth ffi ab =
p
2=!ï 0 oe ab , and in the CuO 2 planes through
ffi c =
p
2=!ï 0 oe c . In the superconducting state all the paí
rameters ffi ab , ffi c , oe ab = oe 0
ab \Gamma ioe 00
ab , and oe c = oe 0
c \Gamma ioe 00
c are
complex. In the temperature range T ! T c , if oe 0 œ oe 00 ,
the field penetration depths are given by the formulas
Ö ab =
p
1=!ï 0 oe 00
ab , Ö c =
p
1=!ï 0 oe 00
c . In the close neighí
borhood of T c , if oe 0 ? ¦ oe 00 , the decay of the magnetic
field in a superconductor is characterized by the funcí
tions Re (ffi ab ) and Re (ffi c ), which turn to ffi ab and ffi c , reí
spectively, at T Ö T c .
In the hotífinger technique, the components of tensor Ó oe
can be derived from the measurements of the microwave
response for different sample alignments with respect to
the ac magnetic field H! : in the transverse (T) orientaí
tion, H! k c (Fig. 4a), when the screening current flows
in the abíplane of the crystal, and in the longitudinal (L)
orientation, H! ? c (Fig. 4b), with currents running in
the directions of both CuO 2 planes and the cíaxis.
a
c
b
J
ab
J
J
c
ab
H w
H w
H
c w
c
a
b
c
( )
( )
T
L w
a
b
FIG. 4. (a) transverse (T) orientation of a sample with reí
spect to microwave magnetic field, H!kc; arrows on the surí
faces show directions of microwave currents; (b) longitudinal
(L) orientation, H! ? c.
Let us start with the case of the Tíorientation (Fig. 4a).
At a frequency of ¦ 10 GHz and T Ö T c , the field H!
penetrates into the sample to the skin depth ffi ab ¦ 5 \Delta
10 \Gamma3 mm and at T ! 0:9T c to the depth Ö ab ¦ 10 \Gamma4 mm.
Since Ö ab œ c and ffi ab œ c, the impedance Z ab
s of the
crystal in the Tíorientation can be treated as a coeffií
cient in the Leontovich boundary condition 4 at all temí
peratures, and the surface impedance can be related to
the conductivity oe ab by a local equation:
Z ab
s = R s + iX s =
` i!ï 0
oe ab
' 1=2
: (3)
At microwave frequencies the conductivity of HTS
is real if T Ö T c , therefore the constant f 0 in the
Tíorientation in Eq. (2) is obtained, in accordance to
Eq. (3), by equating the imaginary part of the impedance
to the real part in the normal state, i.e., by fitting
the temperature dependence R s (T ) to \DeltaX s (T ) in the
range T Ö T c . In this manner X s (T ) was determined
over the entire range of the studied temperatures, and
Ö ab (T ) = X s (T )=!ï 0 for T ! T c in Fig. 3.
In the superconducting state the conductivity oe ab is
complex and, as follows from Eq. (3), the real part of the
surface impedance R s (T ) is not equal to the imaginary
part X s (T ):
3

Rs (T ) =
r
!ï0 (' 1=2 \Gamma 1)
2oe 00 '
; Xs (T ) =
r
!ï0 (' 1=2 + 1)
2oe 00 '
; (4)
where ' = 1 + (oe 0 =oe 00 ) 2 . It is obvious that R s (T ) !
X s (T ) for T ! T c .
For the temperatures T ! T c , if oe 0 œ oe 00 , Eq. (4) can
be reduced to
Rs ' (!ï0 ) 1=2 oe 0
2(oe 00 ) 3=2
=
1
2
! 2 ï 2
0 oe 0
Ö 3 ; Xs ' (!ï0 =oe 00
) 1=2 = !ï0 Ö: (5)
After calculating R s and X s by Eqs. (1) and (2), we
derive from Eq. (4) the real, oe 0 , and imaginary, oe 00 , parts
of the conductivity:
oe 0 = 2!ï 0 R s X s
(R 2
s +X 2
s ) 2 ; oe 00 = !ï 0 (X 2
s \Gamma R 2
s )
(R 2
s +X 2
s ) 2 : (6)
An important point is that R s (T ) and X s (T ) should be
measured in absolute units in order to determine the coní
ductivity components from Eq. 6. Figure 5 plots oe 0
ab (T )
and oe 00
ab (T ) calculated with the help of Eq. (6) and charí
acterizing the samples BSCCO #1 and BSCCO #2.
Now let us consider the case of the Líorientation of the
crystal with respect to the field H! in the cavity, namely
H! ? c (Fig. 4b). In the superconducting state, curí
rents running in the abíplanes decay over a depth Ö ab ,
whereas those flowing in the direction of the cíaxis have
a penetration depth Ö c . At T ! 0:9 T c these two lengths
are smaller than characteristic sample dimensions, which
allows us to introduce the effective impedance Z ab+c
s in
the Líorientation, which is defined as an average of Z ab
s
and Z c
s with the statistical weights proportional to the
respective areas:
Z ab+c
s ' abZ ab
s + acZ c
s
ab + ac
= bZ ab
s + cZ c
s
b + c
; (7)
where the superscripts of Z s denote the direction of the
screening current. In Eq. (7) we neglect the anisotropy
in the abíplanes and the contribution from the bcífaces
(Fig. 4b), which is a factor ¦ c=a smaller than that of
the abísurfaces. Given measurements of Z ab
s (T ) in the
Tíorientation and of Z ab+c
s (T ) in the Líorientation, one
can obtain the losses R c
s (T ) and the change in \DeltaÖ c (T ) =
\DeltaX c
s (T )=!ï 0 from Eq. (7):
\DeltaÖ c = 1
c
[(b + c) \DeltaÖ ab+c \Gamma b \DeltaÖ ab ] : (8)
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l
DE ### ####QP
s
DE#
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####
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#W
##
P
## #
7##.# #
#
%6&&2###
s
DE#
####
#
#W
##
P
##
#
7##.#
#
l
DE ### ####QP
%6&&2###
s
DE#
#
####
#
#W
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P
##
#
7##.#
#E#
#D#
s
DE#
####
#
#W
##
P
##
#
7##.#
FIG. 5. Conductivities oe 0
ab (T ) (open squares) of two BSCCO single crystals #1 (Fig. 5a) and #2 (Fig. 5b) at 9.4 GHz,
extracted from the surface impedance measurements using Eq. (6). The insets show oe 00
ab (T ) (open circles). Solid lines plot the
calculations based on Eqs. (15), (17)--(19) of the modified twoífluid model with parameter ß = 2 for sample BSCCO #1 and
ß = 3 for BSCCO #2 using the experimental data Tc = 83 K, ffiT c = 2:5 K, !œ(Tc) = 7 \Delta 10 \Gamma3 , fi = 0:3, ff = 1, Rres = 120
ï\Omega for BSCCO #1 and Tc = 92 K, ffiT c = 4:5 K, !œ(Tc) = 9 \Delta 10 \Gamma3 , fi = 2, ff = 2, Rres = 500
ï\Omega for BSCCO #2.
This technique for determination of \DeltaÖ c (T ) was used
in microwave experiments 5--11 at low temperatures, T !
T c . This method, however, does not allow us to derive
Ö c (T ) from measurements of the Qífactor and shifts of
the resonant frequency in the Líorientation, moreover, it
cannot be applied to the range of higher temperatures.
The point is that the size effect plays an important role
in the Líorientation at T ? 0:9 T c . Indeed, whereas the
ratio ffi c =ffi ab ¦ 10 in YBCO single crystals, and in the
frequency band about 10 GHz ffi c ¦ 0:05 mm is compaí
rable to the sample thickness c but less than a and b, in
BSCCO, for example, this ratio can be up to 300, so that
ffi c ¦ a ¦ b. Owing to the size effect, the functional deí
pendence of the effective parameter R ab+c
s;eff (T ) measured
in the normal state deviates from \DeltaX ab+c
s;eff (T ), as a reí
sult, one cannot determine the constant f 0 in Eq. (2) as
it was done previously. In order to analyze our measureí
ments in both the superconducting and normal states
4

of BSCCO crystals in the Líorientation, we used the
procedure 12 based on the utilization of previously known
formulae for the field distribution in an anisotropic long
strip (a AE b; c) in the Líorientation. These formulae neí
glect the effect of bcífaces of the crystal if H! is parallel
to the aíedge of the crystal (Fig. 4b), but take into ací
count the size effect. If we, following Ref. 13 , introduce a
complex susceptibility ï = ï 0 \Gamma iï 00 defined in terms of
the power P = i!ïï 0 H 2
! =2 absorbed in the unit volume
of the sample, we can replace Eqs. (1) and (2) with the
formula
\Delta (1=Q) \Gamma 2i ffif =f = iflïv=V; (9)
which relates the measurements of \Delta (1=Q) and \Deltaf =
ffif +f 0 to the susceptibility ï(T ). In Eq. (9) v is the volí
ume of the sample and fl = V H 2
0 =[
R
V H 2
! dV ] is a constant
characterizing the cavity 1 . At an arbitrary temperature,
the function ï(T ) in the Líorientation is controlled by the
components oe ab (T ) and oe c (T ) of the conductivity tensor
through the penetration depths ffi ab (T ) and ffi c (T ) 12 :
ï = 8
‹ 2
X
n
1
n 2
n tan(ffn )
ff n
+ tan(fi n )
fi n
o
;
ff 2
n = \Gamma b 2
ffi 2
c
i i
2
+
‹ 2
4
ffi 2
ab
c 2
n 2
j
; fi 2
n = \Gamma c 2
ffi 2
ab
i i
2
+
‹ 2
4
ffi 2
c
b 2
n 2
j
; (10)
where the sum is performed over odd integers n ? 0.
In the superconducting state at T ! 0:9 T c we find that
Ö ab œ c and Ö c œ a, and we derive from Eq. (10) a
simple expression for the real part of ï:
ï 0 = 2Ö c
b
+ 2Ö ab
c
: (11)
One can easily check out that, in the range of low temí
peratures, the change in \DeltaÖ c (T ) deriving from Eq. (11)
is identical to that given by Eq. (8).
The advantage of the suggested approach, which takes
account of the size effect, is the possibility of estimatí
ing the constant f 0 in Eq. (9), hence the real oe 0
c (T ) and
imaginary oe 00
c (T ) components of the conductivity tensor
measured along the cíaxis can be determined. The proceí
dure of determination of the constant f 0 for BSCCO #1
and BSCCO #2 single crystals is illustrated by Figs. 6a
and 6b respectively. Using the measurements of oe ab (T )
at T ? T c in the Tíorientation (Fig. 5), alongside the
data on \Delta(1=Q) in the Líorientation (open squares in
Fig. 6), and substituting the imaginary part of ï given
by Eq. (10) with real ffi c in expression (9), we obtain the
curves of ae c (T ) = 1=oe c (T ) at T ? T c shown in the rightí
hand insets to Figs. 6a and 6b for crystals #1 and #2.
Further, using the known functions oe c (T ) and oe ab (T ) at
T ? T c and substituting the real part ï determined by
Eq. (10) in expression (9), we calculate (\Gamma2ffif =f) verí
sus temperature in the normal state, and these calculaí
tions are plotted by the solid lines in Fig. 6. These lines
are approximately parallel to the experimental curves of
\Gamma2\Deltaf =f in the Líorientation (open circles in Fig. 6). The
difference \Gamma2(ffif \Gamma \Deltaf )=f yields the additive constant f 0 .
Given f 0 and \Deltaf (T ) measured in the range T ! T c , we
also obtain ffif (T ), hence ï(T ), in the superconducting
state in the Líorientation. As a result, with due account
of oe ab (T ) in the Tíorientation, we obtain the curves oe 0
c (T )
and oe 00
c (T ) of BSCCO #1 and BSCCO #2 crystals shown
in Figs. 7a and 7b respectively. In sample #1 the value
of Ö c (0) proved to be approximately equal to 50 ïm and
in sample #2 150 ïm. These results are in reasonably
good agreement with our measurements of Ö c (0) in these
crystals that were obtained using other techniques 14;15 .
100 120 140
80 84 88
10 í7
10 í6
100 120 140
4
5
6
100 120 140
100 120 140
0
4
8
86 88 90 92
BSCCO #1
#E#
#D#
2f 0
/f í2df/f
í2Df/f
D(1/Q)
0
8§10 í6
4§10 í6
D(1/Q),
í2Df/f
T (K)
D(1/Q)
í2Df/f
r F
##WˆFP#
T (K)
BSCCO #2
## ##
## ##
0
4§10 í5
2§10 í5
í2df/f
D(1/Q)
í2Df/f
2f 0
/f
D(1/Q),
í2Df/f
T (K)
r F
##WˆFP#
T (K)
D(1/Q)
í2Df/f
FIG. 6. Temperature dependences of \Delta(1=Q) (open squares) and \Gamma2\Deltaf =f (open circles) at H! k b of sample #1 (Fig. 6a)
and sample #2 (Fig. 6b) at T ? Tc . Solid lines show the functions \Gamma2ffif (T )=f deriving from Eqs. (9) and (10). Leftíhand
insets: \Delta(1=Q) and \Gamma2\Deltaf =f as functions of temperature in the neighborhood of Tc . Rightíhand insets: ae c (T ) (triangles).
5

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s
F#
#
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7##.#
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s
F#
##W
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7##.#
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#
s
F#
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#E#
#D#
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s
F#
##W
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## #
7##.#
FIG. 7. Conductivities oe 00
c (T ) (open circles) of BSCCO single crystals #1 (Fig. 7a) and #2 (Fig. 7b) at 9.4 GHz, obtained
by comparing the measurements of \Delta(1=Q) and \Deltaf = ffif + f0 to numerical calculations by Eqs. (9) and (10). The insets show
the behavior of oe 0
c (T ) (open squares).
IV. ANALYSIS OF EXPERIMENTAL RESULTS
IN TRANSVERSE ORIENTATION
A. Surface impedance
In the temperature range T Ö T c , R s (T ) = X s (T ) in
the abíplane of BSCCO (Fig. 3) 9;12;16 , YBCO 7;16--18 and
TBCCO 19 single crystals, and this relation is equivalent
to the condition of the normal skin effect. At temperaí
tures higher than T c in all HTS single crystals, the temí
perature dependence of R s (T ) is adequately described by
the expression 2R 2
s (T )=!ï 0 = ae ab (T ) = ae ab (0) + bT . In
particular, for the crystal BSCCO #2 (Fig. 3) ae ab (0) ‹
13
ï\Omega \Deltacm and b ‹ 0:3
ï\Omega \Deltacm/K.
The issue of the temperature dependence of the
impedance of TBCO 20;21 and HgBaCuO crystals 22 raises
controversy. The problem is that, even if curves of R s (T )
are matched to curves of \DeltaX s (T ) in the range T Ö T c :
R s (T ) = \DeltaX s (T ), the change in the reactance \DeltaX s (T )
in the superconducting region is larger than the change
\DeltaR s (T ), so that we obtain negative values of X s (0).
The problem is also complicated by the circumstance
that the coefficients of thermal expansion for TBCO and
HgBaCuO cannot be found in literature. If we assume
that the thermal expansion coefficient of TBCO in the
range T ? T c equals that of BSCCO 23 or TBCCO 24
and take into account the frequency shift \Deltaf th (T ) in
Eq. (2), the curves of R s (T ) and X s (T ) for TBCO are
parallel to one another in the normal state. But the
attempt to bring these curves to coincidence so as to
satisfy the condition of normal skin effect leads to the
value X s (0) = !ï 0 Ö ab (0) ! 0. Thus, the problem can
be solved by either discovering the cause of the negaí
tive contribution dX s ! 0 in the range T ! T c , which
should be subtracted from the measurements of \DeltaX s (T )
to obtain true values of the reactance, X s (T ) ? 0, which
equals R s (T ) at T ? T c , or explaining the positive difí
ference X s (T ) \Gamma R s (T ) in the normal state of TBCO at
which a reasonable value of X s (0) is obtained. A tení
tative explanation of this discrepancy could be based on
two features of the TBCO structure and growth condií
tions which distinguish this material from BSCCO. It is
known that soícalled cleavage planes can intersect with
the TBCO surfaces, whereas the surfaces of the BSCCO
crystals are smooth. If the intersection lines of these
planes form juts (grooves) on the crystal surfaces in the
form of long lines, and the dimensions of these irregularií
ties (their heights, widths, and periods of their patterns)
are larger than the magnetic field penetration depth in
the direction perpendicular to the surface, the magnetic
field screening by these irregularities brings about a negí
ative contribution dX s ! 0 25 to the measured reactance
X s (T ). The field penetration depth increases with the
temperature, and at a certain temperature T \Lambda ! T c it
becomes comparable to the dimensions of these irregularí
ities. Therefore, the contribution dX s to the reactance
can be neglected at T ? T \Lambda . Another likely cause of
the measurements of X s (T ) to be higher than R s (T ) in
the normal state of TBCO is the size effect in the Tí
orientation. Whereas the elementary cell of the BSCCO
crystal lattice contains two conducting CuO 2 planes, the
elementary cell in TBCO, which has approximately the
same dimensions, includes only one. If we assume that
dissipative highífrequency currents are induced only in
these planes, the screening length c \Lambda in the TBCO crysí
tal should be much smaller than its actual thickness c:
c \Lambda œ c. If c \Lambda ! 3ffi ab , the measured effective parameter
X eff
s (T ) is larger than the effective R eff
s (T ) at T Ö T c
owing to the size effect.
For the frequencies ! ¦ 10 GHz, the common features of
all HTS single crystals are the linear temperature depení
dence of the surface resistance (\DeltaR s (T ) / T ) and of the
surface reactance (\DeltaX s (T ) / \DeltaÖ ab (T ) / T ) at temperí
atures T œ T c (see Refs. 1;26--28 and references therein).
The slopes of \DeltaÖ ab (T ) curves at T œ T c are different.
6

In particular, in YBCO crystals fabricated by different
techniques, the slopes of \DeltaÖ ab (T ) differ by almost one
order of magnitude 18;29;30 .
An important parameter of HTS crystals characterizí
ing their quality is the residual surface resistance R res =
R s (T ! 0), which is the loaníterm taken from the terí
minology of conventional superconductors. There is,
however, a marked difference between these two cases:
whereas R res in conventional superconductors is clearly
defined as the level of the plateau on the R s (T ) curve in
the region T ! T c =4, no plateau has been detected on the
characteristic curves of the R s (T ) in HTS, and R res is set
to the value R s (T = 0), which is obtained by extrapoí
lating to the zero temperature the linear section of the
R s (T ) curve in the region T œ T c (Fig. 2). It was found
in experiments with conventional superconductors 25 that
the parameter R res / ! 2 and is determined by various
defects in the surface layers of samples, therefore, it is
generally accepted that the sample quality is the higher,
the lower R res . The residual surface resistance of HTS
materials is also proportional to the microwave frequency
squared, but its magnitude, even in samples of the highí
est quality, is a factor of several tens higher than R res of
conventional superconductors. If we recall that, in spite
of the intense search for new techniques of HTS crystals
manufacture during the latest five to seven year, no esí
sential progress has been made, and the characteristic
R res has not been reduced notably (moreover, as will be
shown in the following sections, the temperature depení
dence oe 0 (T ) in samples of similar chemical compositions
is radically different at different R res ), it becomes quite
clear that the explanation of the nature of residual losses
has remained a topical problem in the HTS physics.
# ## ## ## ## ###
##
##
##
##
##
##
# ## ##
#
#
#
#####*+]
#####*+]
#####*+]
5 V
##W#
7##.#
5 V
##PW#
7##.#
FIG. 8. Experimental data of BSCCO single crystal 37 at
various frequencies: 14.4 GHz (circles), 24.6 GHz (triangles)
and 34.7 GHz (squares). The solid curves plot the calculated
functions [Rs (T ) + Rres ] with Rres set to 0.29, 0.85 and 1.7
m\Omega\Gamma respectively. The inset shows the temperature depení
dences of the surface resistance at low temperatures.
Let us consider now the differences between the Z ab
s (T )
characteristics of BSCCO, TBCCO and TBCO single
crystals on the one side, whose lattices are tetragonal,
and YBCO crystals with the orthorhombic lattice on the
other. Whereas the linear region of \DeltaR s (T ) / T in the
tetragonal materials, obtained at a frequency ¦ 10 GHz,
can extend to almost T c =2 (Figs. 2, 3, and 8), in YBCO
the linear region terminates near or below T ! T c =3, and
at higher temperatures R s (T ) of YBCO has a broad peak
(Fig. 9). The peak shifts to higher temperatures and dií
minishes in size as the frequency is increased. In higherí
quality YBCO crystals the peak amplitude increases and
R s (T ) reaches its maximum at a lower temperature 31 .
In addition, the Ö ab (T ) 18;29 and R s (T ) 29 curves of some
YBCO single crystals have unusual features in the interí
mediate temperature range T ¦ T c =2.
# ## ## ## ##
##
##
##
##
##
##
######*+]
######*+]
######*+]
######*+]
######*+]
5
V
##W#
7##.#
FIG. 9. Comparison between calculated (lines) and meaí
sured surface resistance Rs(T ) in the abíplane of YBCO siní
gle crystal at 1.14, 2.25, 13.4, 22.7 and 75.3 GHz (symbols)
from Ref. 31 . The solid lines are calculations by Eqs. (4),
(15), and (16) with parameter ß = 9 and experimental values
œ(Tc) = 10 \Gamma13 s, fi = 0:005, and ns(T )=n = oe 00 (T )=oe 00 (0). We
assumed Rres =
0:3m\Omega for 75:3 GHz and zero for all other
frequencies.
B. Complex conductivity
The components oe 0 (T ) and oe 00 (T ) are not measured dií
rectly, but can be derived from measurements of R s (T )
and X s (T ) using Eq. (6). At temperatures not very close
to T c , R s (T ) œ X s (T ) in HTS crystals of high quality,
and Eq. (6) reduces to
oe 0 (T ) = 2!ï 0 R s (T )
X 3
s (T ) ; oe 00 (T ) = !ï 0
X 2
s (T ) : (12)
7

It then follows from Eq. (12) that oe 1 =oe 2 = 2R s =X s œ
1 at T ! T c . The increments \Deltaoe 0 (T ) and \Deltaoe 00 (T ) deí
pend on the increments \DeltaR s (T ) and \DeltaX s (T ) relative to
the respective quantities:
\Deltaoe 0 /
` \DeltaR s
R s
\Gamma 3 \DeltaX s
X s
'
; \Deltaoe 00 / \Gamma \DeltaX s
X s
: (13)
Hence, the curves of oe 00 (T ) are determined by the funcí
tion X s (T ) = !ï 0 Ö(T ) alone and reflect the basic properí
ties of the field penetration depth plotted against temperí
ature, namely, its rectilinear shape at low temperatures
in all highíquality HTS crystals and the features detected
in YBCO in the intermediate temperature range. The
behavior of the real part oe 0 (T ) of the conductivity, as
follows from Eq. (13), is determined by the competition
between the relative increments \DeltaR s =R s and \DeltaX s =X s .
In conventional superconductors, the quantity R s (T ) is
so small at T ! T c that the increment \Deltaoe 0 (T ) ? 0,
i.e., the real part of the conductivity increases with the
temperature (at least in the temperature interval T !
0:8 T c , before the maximum of the BCS coherence peak
is reached). For HTS single crystals \DeltaX s (T ) AE \DeltaR s (T )
at T ! T c . Although R s (T ) ! X s (T ), the parameter
\DeltaR s =R s in Eq. (13) can be smaller than 3\DeltaX s =X s , and
the conductivity oe 0
ab (T ) can drop with the temperature:
\Deltaoe 0
ab (T ) ! 0. This situation occurs in YBCO crystals a
fortiori since \DeltaR s =R s ! 0 on the rightíhand slope of the
R s (T ) peak. Therefore, starting with T = 0, the function
oe 0 (T ) rises, then it achieves its peak value at a certain
T = Tmax , and drops as the temperature increases furí
ther, i.e., the curve of oe 0 (T ) has a peak at T ! T c . It
follows from Eq. (13) that this shape of the curve can
take place if the value of the residual surface resistance
R res is sufficiently small:
R res !
X s (0)
3
\DeltaR s (T )
\DeltaX s (T )
fi fi fi fi T!0
: (14)
The peak of oe 0
ab (T ) shifts to lower temperatures (Tmax
drops) as R res increases, and when R res achieves the
value on the right of Eq. (14), the peak disappears
(Tmax = 0). If R res of the crystal is such that inequalí
ity (14) cannot be satisfied any longer, its conductivity
oe 0
ab (T ) is a monotonically decreasing function of the temí
perature in the range T ! T c . Figures 5a and 5b show
the two possible shapes of the oe 0
ab (T ) curves measured
at a frequency of 9.4 GHz, namely, the presence of the
peak in the sample BSCCO #1 (Fig. 5a; R res ‹ 120 ï\Omega\Gamma
and its absence in BSCCO #2 (Fig. 5b; R res ‹ 500
ï\Omega\Gamma2
The higher the crystal quality, the more clearly the peak
in the conductivity at T ! T c can be seen. The curve
of oe 0
ab (T ) in Fig. 10 corresponds to the characteristic
R s (T ) of the YBCO crystal measured at a frequency of
1.14 GHz (Fig. 9) with R res ¦ 1 ï\Omega\Gamma Near T = 0, oe 0 (T )
increases linearly with T and reaches its peak value at
Tmax , which is always higher than the normal state coní
ductivity oe(T c ). In YBCO single crystals, the temperaí
ture Tmax is close to the temperature at which the peak
of R s (T ) occurs. The peak of oe 0
ab (T ) shifts to higher temí
peratures, and its amplitude diminishes as the microwave
frequency increases. At 0:7 ! T ß T c the real part of the
conductivity does not show a coherence peak, as was preí
dicted by BCS. Usually oe 0 (T ) of HTS single crystals has
a narrow peak near T c (Fig. 10), whose width equals the
width of the transition from the normal to superconductí
ing state on the curve of R s (T ) at microwave frequencies.
# ## ## ## ## ###
#
#
#
#
s
DE#
####
#
#W
##
P
##
#
7##.#
FIG. 10. Real part of the conductivity oe 0
ab (T ) of YBCO
single crystal at 1.14 GHz 31 . The solid line is the calculated
T --dependence of oe 0
ab (T ) using Eqs. (15), (17), and (18) with
ß = 9, œ (Tc) = 10 \Gamma13 s, fi = 0:005 and ffiT c = 0:4 K.
C. Modified twoífluid model
There is a simple description of all the observed feaí
tures of Z ab
s (T ) and oe ab (T ) which was suggested in
Refs. 32;33 and later developed in Refs. 1;27;28;34;35 . The
underlying idea was to generalize the twoífluid model
by Gorter and Casimir 36 to the case of HTS materials,
whose common feature is the high values of T c . In metals,
processes of inelastic scattering of quasiparticles are esí
sential at such high temperatures, therefore the first uní
questionable modification of the twoífluid model should
be the introduction of a temperature dependence of the
'normalífluid' quasiparticle relaxation time œ(T ). Así
suming that scattering processes in this liquid are similar
to those in normal metals, we used the BlochíGrèuneisen
formula (electron--phonon interaction) for the function
œ(T ) in the normal and superconducting states of HTS
and retained the temperatureíindependent impurity reí
laxation time œ(0), which is present in the standard twoí
fluid model:
1
œ =
1
œ(0)
h
1 +
t 5 J5 (ß=t)=J5 (ß)
fi
i
; J5 (ß=t) =
ß=t
Z
0
z 5 e z dz
(e z \Gamma 1) 2
; (15)
where t j T=T c , ß = \Theta=T c (\Theta is the Debye temperaí
ture) and fi is a numerical parameter, which equals, as
8

it follows from Eq. (15), fi = œ(T c )=[œ(0) \Gamma œ(T c )]. Folí
lowing the formal analogy with metals, one can say that
the parameter fi is the characteristic of the `HTS purity':
fi ‹ œ(T c )=œ(0) œ 1 if œ(0) AE œ(T c ). The parameter
of HTS corresponding to \Theta can be estimated as several
hundreds of degrees. At T ! \Theta=10 (ß ? 10t) the secí
ond summand in the brackets on the right of Eq. (15)
is proportional to T 5 ; in the region T ? \Theta=5 (ß ! 5t)
it is proportional to T . Thus, if fi ! 1 the reciprocal
relaxation time (the electron relaxation rate) is constant
and equal to 1=œ(0) over the interval 0 ! T ! T c =3, and
at higher temperatures it increases gradually, starting as
the power function / T 5 in the region T ! T c =2 and
switching to / T around T c , and at T ? T c we have a
linear dependence \Deltaae ab (T ) / 1=œ(T ) / T .
Even though the chosen the form of the function œ(T )
for HTS materials with complex electronic spectra was
overísimplified, it turned out that all common and even
specific features of the R s (T ) and oe 0
ab (T ) curves for both
T ! T c and T ? T c are adequately described by the
modified twoífluid model with only one fitting parameí
ter ß in (15). Indeed, the components of the complex
conductivity are
oe 0
=
nne 2 œ
m
1
1 + (!œ) 2
; oe 00
=
nse 2
m!
h
1 + nn
ns
(!œ) 2
1 + (!œ) 2
i
(16)
and in the case of (!œ) 2 œ 1, which is typical of HTS
crystals at all temperatures in the frequency band about
10 GHz and below, they transform to a very simple form:
oe 0 = nn e 2 œ
m
; oe 00 = n s e 2
m!
: (17)
In (16) and (17) n n;s (T ) are the densities of the frací
tions of normal and superconducting carriers (both have
the same charge e and effective mass m). At temperaí
tures T ß T c the total carrier concentration n = n s + nn
is equal to the concentration of quasiparticles in the norí
mal state. From measurements of R s (T c ) and X s (0), and
slopes dR s =dT and dX s =dT at T œ T c , one can easily
derive, using formulae (4) and (17), the parameters
!œ(T c ) = X 2
s (0)
2R 2
s (T c ) = oe 1 (T c )
oe 2 (0) ; !œ(0) = dR s
dX s
fi fi fi fi T!0
and determine the parameter fi in Eq. (15). Now,
if we use the dependence n s (T )=n = oe 00 (T )=oe 00 (0) =
Ö 2 (0)=Ö 2 (T ) determined in the same experiment and thus
derive the function nn (T )=n = 1 \Gamma n s (T )=n, we can fit,
using Eqs. (15)--(17) and evaluating the parameter ß of
the sample, formula (4) to the measurements of R s (T )
given above and then derive the real part of the conducí
tivity oe 0
ab (T ) of the HTS crystals in the Tíorientation
using formula (6). The solid lines in Figs. 5, 8--10 show
how the experimental data compare with the calculations
by the modified twoífluid model.
Here we have to consider two important points that
have not been discussed as yet, but were implicitly used
in our calculations. Firstly, near T c we took into ací
count the inhomogeneous broadening ffiT c of the superí
conducting transition following the approach 2;27;28 that
describes a peak of the effective conductivity oe 0 (T ) at a
temperature Tm = T c \Gamma ffiT c , which is close to the critical
temperature. The relative amplitude of this peak, which
equals
oe 0 (Tm ) \Gamma oe(T c )
oe(T c ) ‹
ae
fl if fl ? 1
fl 2 if fl ! 0:1 (18)
(fl = ffiT c =[T c !œ(T c )]), is the lower, the more narrow the
superconducting transition is (the smaller ffiT c ). Usually
microwave experiments yield fl ? 1. In particular, the
data published by Hosseini et al. 31 lead to fl ' 7 at
1.14 GHz (Fig. 10), therefore the peak amplitude should
be inversely proportional to the frequency. This asserí
tion is supported by the comparison between the curves
of oe 0
ab (T ) (the lines in Fig. 11) from Ref. 37 obtained at
two different frequencies on the same BSCCO crystal as
was used in Fig. 8.
# ## ## ## ## ###
#
#
#
#
#
#
#
#
######*+]
######*+]
s
DE#
####
#
#W
##
P
##
#
7##.#
FIG. 11. Measurements of oe 0 (T ) of BSCCO single crysí
tal at 14.4 and 34.7 GHz 37 (symbols) and calculations
of oe 0 (T ) (lines) using Eqs. (15), (17), and (18) with
ß = 4, œ(Tc) = 0:8 \Delta 10 \Gamma13 s, fi = 0:1, ffiT c = 2 K,
Rres(14:4
GHz)=0.29m\Omega\Gamma Rres(34:7
GHz)=1.7m\Omega\Gamma Secondly, in comparing the calculations based on the
modified twoífluid model to the measurements of the surí
face resistance, we added to the function R s (T ) deterí
mined by the general formula (4) the value R res , which
was determined in the same experiment and is indepení
dent of the temperature. For this reason, the curves
of oe 0
ab (T ) in Figs. 5 and 11, which were calculated by
Eq. (6), do not tend to zero as T ! 0, even though
the carrier density nn = 0 at T = 0, according to the
twoífluid model, and, as it follows from Eq. (16) or (17),
the conductivity should tend to oe(0) = 0. The paí
rameter R res was not included in formulae (4) and (6)
9

when we compared the data of Fig. 9 (except the upí
per curve) and of Fig. 10 with calculations because in
that case the ratio R res =R s (T c ) was very small, less than
10 \Gamma3 . In most HTS crystals which were investigated,
however, R res =R s (T c ) ? 10 \Gamma3 , and at T œ T c the efí
fect of the residual surface resistance is quite notable.
Another reason for including R res is the increase in the
ratio R res =R s (T c ) / ! 3=2 with the frequency !, which
proved to be essential when we compared the calculated
and experimental curves in Fig. 11 and the upper curve
in Fig. 9.
As was noted above, the cause of the residual losses in
HTS materials has remained unclear. Some researchers
(e.g., Hein et al. 38 ) attributed these losses to the presence
of a fraction n 0 of carriers that remain unpaired at T = 0.
The magnitude of R res was estimated by formula (5)
with the nonzero conductivity oe 0 (0) = n 0 e 2 œ(0)=m from
Eq. (17). One can easily prove, however, that this apí
proach requires that the calculations of R res should satí
isfy inequality (14). If the latter condition is not met,
which may occur in HTS crystals, as was shown previí
ously (see Fig. 5b), the number n 0 should be larger than
the entire concentration n of the carriers.
In developing the traditional approach to the problem,
which attributed the residual surface resistance to various
imperfections of the surface, the researchers took account
of the losses due to weak links 39--41 , twinning planes 41;42 ,
clusters with normal conductivity 43 , etc. Numerical estií
mates, however, indicate that the contributions of these
mechanisms to the residual surface resistance are much
smaller than the measured R res in HTS materials. Moreí
over, a very remarkable fact is that the residual surface
resistance measured at a frequency of 10 GHz was apí
proximately the same, R res ¦ 100 ï\Omega\Gamma in all highíquality
copperíoxide HTS crystals of different chemical compoí
sitions and fabricated by different methods, irrespective
of whether they contained twinning planes or not, and
whether measurements were performed on freshly cleaved
or asígrown surfaces. This fact, apparently, indicates
that the residual losses are inherent to all highíquality
HTS crystals, and their origin is in the features of their
structure, namely, in its conspicuously layered nature.
In other words, a fraction of current flowing in the surí
face layer of an HTS crystal may run through regions
of the layer that are in the normal state and have a fií
nite resistivity ae n . In the phenomenological model under
discussion, this contribution to the resistance can be iní
cluded in the form of a circuit element ae n connected in
parallel to the twoífluid circuit characterized by Eq. (17),
i.e., a resistor ae = 1=oe 0 is shunted by a kinetic inducí
tor l = 1=!oe 00 (the parallel connection of ae and l is in
conformity with the formula relating the current to the
field in the twoífluid model). Obviously, the complex
impedance of the circuit consists of the imaginary part
iX s = i!ï 0 Ö at T ! T c and the sum of two real compoí
nents: R s from Eq. (5) and R 0 = ! 2 ï 2
0 Ö 3 =2ae n . At T = 0,
when R s (0) = 0, the latter summand can play the role
of the residual resistance R res , which is proportional to
! 2 , according to the experimental data. At a frequency
of 10 GHz, and at R res ‹ 100
ï\Omega and Ö(0) ‹ 0:2 ïm,
which are typical parameters of HTS crystals, we obí
tain ae n (0) ‹ 25
ï\Omega \Deltacm, which is the quantity typical of
normal metals. Our procedure of comparison with the
experimental data on R s (T ) also requires that R 0 be iní
dependent of the temperature T œ T c . This is feasible if
ae n (T ) / Ö 3 (T ), i.e., ae n (T ) should be a linear function of
temperature for T œ T c : ae n (t) = ae n (0)(1+1:5fft), where
ff is the slope of the curve of oe 00 (t) at t œ 1 for a specific
HTS sample:
oe 00 (t)=oe 00 (0) = Ö 2 (0)=Ö 2 (t) ' (1 \Gamma fft): (19)
One can easily check out that the coefficients ae n (0)
and 1:5ffae n (0)=T c for the sample BSCCO #2 are apí
proximately equal to ae ab (0) and b in the resistivity
ae ab (T ) = ae ab (0) + bT of this sample in the normal state,
i.e., ae n (T ) ‹ 2R 2 (T )=!ï 0 , where R(T ) is the extrapolaí
tion of the line R s (T ) in the region T ? T c of the normal
state in Fig. 3 to the region of the superconducting state,
T ! T c , down to the point T = 0.
It would be natural to complete this presentation of
the modified twoífluid model with formulae describing
the experimental data on n s (T )=n = oe 00 (T )=oe 00 (0), which
were used in the calculations of the real parts of the
impedance and conductivity in the Tíorientation of HTS
crystals. Several versions of such empirical formulas are
given elsewhere 1;27;28;33;34 . Note only that all these forí
mulas are identical with Eq. (19) at T œ T c because all
curves of oe 00
ab (T ) characterizing highíquality HTS crystals
have rectilinear shapes in the region of low temperatures.
To sum up, one can describe characteristic features
of Z ab
s (T ) and oe ab (T ) curves in highíquality HTS crysí
tals in terms of the modified twoífluid model based on
Eqs. (15)--(19). As follows from these equations, at low
temperatures, t œ 1, all curves have linear sections,
oe 0 / fft=fi, since nn=n ‹ fft and œ ‹ œ(0) ‹ œ(T c )=fi;
\Deltaoe 00 / \Gammafft; R s / fft=fi, in accordance with Eq. (5);
\DeltaX s / \DeltaÖ / fft=2. As the temperature increases, the
curve of oe 0 (t) passes through a maximum at t ! 0:5 if
the residual surface resistance is so small that inequalí
ity (14) is satisfied. This peak is due to the superposií
tion of two competing effects, namely, the decrease in the
number of normal carriers as the temperature decreases,
for t ! 1, and the increase in the relaxation time, which
saturates at t ¦ fi 1=5 . If inequality (14) fails, the function
oe 0 (t) chandes monotonically with the temperature. The
model also accounts for the peculiarities of the surface
impedance and complex conductivity at higher temperaí
tures in HTS single crystals manufactured using different
techniques. The underlying ideas and results of the modí
ified twoífluid model have been analyzed from the viewí
point of modern microscopic theories of the microwave
response of HTS materials in recent publications 27;28 .
10

V. CONDUCTIVITY ALONG CíAXIS
In comparison with the microwave response of the
cuprate layers of HTS materials, the data concerning
their microwave properties in the direction perpendicular
to these layers are scarce. Moreover, the available experí
imental data are controversial, therefore, any discussion
of general regularities in the temperature dependence of
the conductivity components in the direction perpendicí
ular to the CuO 2 layers of HTS crystals is premature.
In particular, there is no consensus in literature about
\DeltaÖ c (T ) at low temperatures. Even in reports on lowí
temperature properties of highíquality YBCO crystals,
which are the most studied objects, one can find both
linear, \DeltaÖ c (T ) / T 6;11 , and quadratic dependences 44
in the range T ! T c =3. In BSCCO materials, the
shape of \DeltaÖ c (T ) depends on the level of oxygen dopí
ing: in samples with maximal T c ' 90 K \DeltaÖ c (T ) / T
at low temperatures 9;10;12 ; at higher oxygen contents
(overdoped samples) the linear function \DeltaÖ c (T ) 12 transí
forms to a quadratic one 10 . The common feature of all
microwave experiments is that the change in the ratio
\DeltaÖ c (T )=Ö c (0) is smaller than in \DeltaÖ ab (T )=Ö ab (0) because
in all HTS Ö c (0) AE Ö ab (0).
0 20 40 60 80 100
0
1
2
3
4
5
6
T (K)
s
`(T)
/
s
`(T c
)
c-axis
a-axis
a-axis
b-axis
b-axis
FIG. 12. Real components of the conductivity tensor
Ó
oe 0 (T )=oe(Tc) measured along the crystallographic axes of the
YBCO crystal at a frequency of 22 GHz 44 at T ! Tc (points).
The solid lines show calculations by Eqs. (15) and (16) from
Ref. 35 .
Almost all measurements of losses indicate that
R c
s (T ) AE R ab
s (T ) in the superconducting state of HTS.
The only exception is the experimental results concerní
ing YBCO crystals with the optimal doping level, which
were obtained by Hosseini et al. 44 and demonstrated the
opposite relation between these parameters in the region
10 ! T ! 65 K. Since measurements of the anisotropy
in the cuprate planes of YBCO were also given in that
publication, we attempted to apply the modified twoí
fluid model to the entire set of those experimental data 44 .
The calculations of the real components of the conducí
tivity tensor are compared with the experimental data
in Fig. 12 35 . The absence of the peak on the curve of
oe 0
c (T ) is due to the very weak temperature dependence
of the relaxation time of normal quasiparticles in the dií
rection of the cíaxis in the region T ! T c : œ c (T ) ‹ const,
fi c AE 1 in Eq. (15). In addition, since the inductive
losses (due to large Ö c ) are notably higher than the ací
tive ones (R c
s and oe 0
c are small), it is plausible that the
microwave círesponse is largely due to the tunneling of
Cooper pairs between the CuO 2 planes. On the other
hand, a peak of oe 0
c (T ) at T ! T c was observed in other
YBCO crystals with the optimal doping level and the reí
sistance R c
s (T ) AE R ab
s (T ) 6;7 , and this peak was similar
to that in oe 0
ab (T ).
The conductivity oe 0
c (T ) of BSCCO crystals either
drops with the decreasing temperature T ! T c 45;46 or
rises, as is shown in Fig. 7 for both samples (even though
oe 0
ab (T ) behaves differently in these two samples, as one
can see in Fig. 5). From the formal viewpoint, the cause
of this growth is quite clear: the residual losses in the
direction of the cíaxis of the BSCCO crystals #1 and
#2 are fairly high, so inequality (14) fails. The quesí
tion of what is the nature of these losses has remained
unanswered by this time.
VI. CONCLUSION
This paper is an attempt to review the main results
and unsolved problems of research into the microwave
response of HTS crystals. Measurements of changes in
the surface impedance and complex conductivity in the
transverse orientation (H! k c) of samples with different
chemical compositions doped to the optimal level could
be systematically described in terms of the modified twoí
fluid model. But even in this specific case, some problems
have remained unsolved (in particular, the temperature
dependence of the TBCO impedance). One of the topical
questions is the cause of residual losses in HTS materií
als, which determine the behavior of oe 0 (T ) in the region
T ! T c . The experimental technique of measuring all
the components of the conductivity tensor is described
in detail, and we hope that this technique will be useful
in comprehensive studies of anisotropic characteristics of
HTS crystals.
VII. ACKNOWLEDGMENTS
I am grateful to my coíauthors Herman J. Fink,
Yu. A. Nefyodov, and D. V. Shovkun for their help
and contribution to this work. Helpful discussions with
11

A. AglioloíGallitto, N. Bontemps, A. Buzdin, V. F. Gantí
makher, A.A. Golubov, J. Halbritter, M. Hein, M. Li
Vigni, E. G. Maksimov, K. Scharnberg, and S. Sridí
har are also acknowledged. This work was supported
by RFBR grant 00í02í17053, DFGíRFBR grant 00í02í
04021, a grant Coll. Int. Li Vigni, CNRSíRAS 4985, and
CLG NATO.
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12