Complex microwave conductivity of Pr1.85Ce0.15CuO4-δ thin films using a cavity perturbation method

Abstract

We report a study of the microwave conductivity of electron-doped Pr _1.85 Ce _0.15 CuO _4-δ superconducting thin films using a cavity perturbation technique. The relative frequency shifts obtained for the samples placed at a maximum electric field location in the cavity are treated using the high conductivity limit presented recently by Peligrad etal [Phys. Rev. B 58, 11652 (1998)].Using two resonance modes, TE ₁₀₂ (16.5 GHz) and TE ₁₀₁ (13 GHz) of the same cavity, only one adjustable parameter Γ is needed to link the frequency shifts of an empty cavity to the ones of a cavity loaded with a perfect conductor. Moreover, by studying different sample configurations, we can relate the substrate effects on the frequency shifts to a scaling factor. These procedures allow us to extract the temperature dependence of the complex penetration depth and the complex microwave conductivity, properties that are highly sensitive on the quality of the films.

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APS/123-QED
Complex microwave conductivity of Pr1.85Ce0.15CuO4−δthin films using a cavity
perturbation method
Guillaume Cˆ ot´ e, Mario Poirier, and Patrick Fournier
Regroupement Qu´ eb´ ecois sur les Mat´ eriaux de Pointe, D´ epartement de Physique,
Universit´ e de Sherbrooke, Sherbrooke, Qu´ ebec,Canada J1K 2R1
(Dated: February 5, 2008)
We report a study of the microwave conductivity of electron-doped Pr1.85Ce0.15CuO4−δ super-
conducting thin films using a cavity perturbation technique. The relative frequency shifts obtained
for the samples placed at a maximum electric field location in the cavity are treated using the high
conductivity limit presented recently by Peligrad et al. Using two resonance modes, TE102 (16.5
GHz) and TE101 (13 GHz) of the same cavity, only one adjustable parameter Γ is needed to link
the frequency shifts of an empty cavity to the ones of a cavity loaded with a perfect conductor.
Moreover, by studying different sample configurations, we can relate the substrate effects on the
frequency shifts to a scaling factor. These procedures allow us to extract the temperature depen-
dence of the complex penetration depth and the complex microwave conductivity of two films with
different quality. Our data confirm that all the physical properties of the superconducting state
are consistent with an order parameter with lines of nodes. Moreover, we demonstrate the high
sensitivity of these properties on the quality of the films.
PACS numbers: 74.25.Nf, 74.72.-h, 74.78.-w, 74.78.Bz
I.INTRODUCTION
In high-Tc cuprate superconductors the study of the
pairing symmetry remains a very active field of research
and microwave measurements using resonant structures
have shown to be particularly useful to detect its sig-
natures. Indeed, when a microwave resonant cavity is
loaded with a conducting sample, we can measure shifts
of the resonance frequency and of the quality factor Q,
which can be related to the real and imaginary parts of
the complex conductivity ˜ σ = σ1− iσ2[1, 2, 3]. Based
on the two-fluid model for a superconductor, condensed
pairs and quasiparticles both contribute to ˜ σ which in
turn determine the temperature dependence of the mag-
netic penetration depth λ(T) and the quasiparticle relax-
ation rate T−1
The d-wave symmetry is now well established in hole-
doped cuprates superconductors [5, 6, 7, 8]. However, the
situation remains controversial in electron-doped ones
R2−xCexCuO4−δ (R = Pr,Nd,Sm) [9, 10, 11]. When
single crystals are used, residual microwave absorption is
generally measured in the superconducting state and this
could impede any precise determination of λ(T). This
residual absorption is likely due to an inhomogeneous
oxygen reduction process over the volume of the crystal,
a reduction which is mandatory to induce the supercon-
ducting state. In this respect, thin films of these ma-
terials appear more adapted to microwave experiments.
Indeed, high-quality R2−xCexCuO4−δthin films can be
grown by pulsed laser ablation on appropriate substrates.
Their small thickness and an in-situ post-annealing en-
sures a homogeneous reduction over the complete volume
of the sample. There is, however, an important difference
between microwave measurements of thick samples (sin-
gle crystals) and thin samples (thin films) which has not
been completely overcome yet experimentally.
1 (T)[4].
The perturbation of a microwave cavity by a small
sample of variable conducting properties is an old prob-
lem that has been treated with different approaches for
thin superconducting films [12, 13, 14]. However, a gen-
eral procedure proposing to treat the cavity loaded with
a perfect conductor sample as the unperturbed state and
to find the shifts when the sample becomes a non-perfect
conductor, appears particularly promising for high con-
ductivity thin films. A general expression has been ob-
tained for two intracavity arrangements and analytical
solutions were found for the slab geometry of a thin film
[14, 15]. The general solution for the complex frequency
shift reduces to the Shchegolev formula [16] in the de-
polarization limit when the microwave electric field fully
penetrates the sample, and to the skin depth regime when
the microwave penetration depth becomes smaller than
the size of the sample and the complex frequency shift
is related to the surface impedance [17]. Thin films hav-
ing different thicknesses and conductivity can thus be
treated anywhere between the depolarization regime and
the regime close to the skin depth one [15]. However, the
high-temperature superconducting thin films are grown
on dielectric substrates having relative permittivity ?r
around 20. Hence, when the sample is introduced in the
microwave cavity, the substrate changes the field config-
uration asymmetrically around the sample and modifies
the frequency shifts. A method has been proposed to
mimic this asymmetric solution by introducing the con-
cept of a fictitious substitute sample. This method has
been tested on thin films of hole-doped cuprates mounted
in a cylindrical cavity resonating in the TE111mode [15].
In this paper we present microwave measurements on
optimally doped Pr1.85Ce0.15CuO4−δ(PCCO) thin films
grown on a LaAlO3 substrate mounted in the electric
field of a rectangular cavity. We use the general solution
for the complex frequency shift in microwave electric field
arXiv:0711.4089v1 [cond-mat.supr-con] 26 Nov 2007

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in the high conductivity limit [14]. However, to deduce
the complex conductivity from the measured frequency
shifts, the fictitious substitute sample method could not
be applied to our measurements to take into account the
impact of the dielectric substrate. We rather propose
a modified version of the method presented by Peligrad
et al. exploiting the measurements of the same sample
at two different resonance frequencies and by analyzing
different substrate configurations. Then, the frequency
shifts are transformed into complex penetration depth
and microwave conductivity data. These results obtained
on two films of different quality are discussed in terms of
a two-fluid model. The characteristics of the supercon-
ducting state are found to be very similar to the ones
obtained in the hole-doped cuprates.
II.EXPERIMENT
The PCCO thin films were grown by pulsed laser depo-
sition on standard LaAlO3substrates and their thickness
was measured with a scanning electronic microscope at
grazing incidence giving a precision of 5 nm. Prior to the
microwave measurements, the films were also character-
ized by four-probe resistivity and AC magnetic suscep-
tibility measurements were performed in a PPMS from
Quantum Design between 4 and 300 K. For the two films
investigated here, we give in Table 1 the values of the
thickness d, the critical temperature Tc and the mini-
mum value of the resistivity ρminappearing between 14
and 15 K when a magnetic field of 9T suppresses the
superconducting state [18]. Tc is determined from the
microwave experiment as indicated in the next section.
The microwave measurements were performed with a
cavity perturbation technique in the transmission mode.
We used a rectangular copper cavity for which the TE102
(16.5 GHz) and TE101(13 GHz) resonance modes could
be excited. A synthesized sweeper and a scalar network
analyzer are used to acquire the resonance curve of each
mode after averaging and smoothing. A lorentzian fit of
the curve then yields the resonance frequency f0and the
half width at half maximum (1/2Q). This is done for
the cavity with and without the sample installed in the
maximum electric field position in order to obtain the rel-
ative frequency shifts ∆f/f and ∆(1/2Q). A micrometer
screw allows for the displacement of the sample in and
out of the cavity at low temperatures. With this set-up
the noise level is usually better than 1 kHz. The sample
(thin film + substrate) has a slab geometry with typical
dimensions 2 x 0.2 x 0.5 mm3, 0.5 mm being the thick-
ness of the substrate. The sample and cavity are inserted
in a variable temperature insert that permits to scan the
temperature between 2 and 300 K with a Cernox sensor
and a Lake Shore temperature controller.
TABLE I: Physical parameters of two PCCO films.
Filmd(10−9m)Tc(K)ρmin(µΩ cm)
A225(5)22.5(1)101 ± 6
B170(5)21.5(1)310 ± 40
III.RESULTS AND DISCUSSION
The relative frequency shifts presented in this section
will be treated with the formula valid in the high con-
ductivity limit for a slab oriented with its length along
the maximum electric field [14]. Because our frequency
shifts are measured relative to an empty cavity when
the formula implies shifts relative to a cavity perturbed
by a perfect conductor (unperturbed state), we first de-
scribe the appropriate correction to transform one into
the other. Then, we explain how the substrate effects
are included in the treatment. Finally, the temperature
dependence of the complex in-plane penetration depth˜λ
= λ1 - iλ2 obtained from the frequency shifts and the
subsequent complex conductivity data ˜ σ = 1/iµ0ω˜λ2for
two PCCO films are presented and discussed.
A.The Unperturbed State
Theoretically, when a slab of a perfect conductor is in-
troduced in the electric field of the cavity, the relative
shift in resonance frequency ∆f/f depends only on a
geometrical factor Γ = α/N, where α and N are respec-
tively the filling and the depolarization factors. Then,
the difference in shifts between the empty cavity and
the unperturbed state should be this constant Γ. Since
there is no dissipation in a perfect conductor, there is
no change induced in the ∆(1/2Q) data. An example of
these shifts for sample A, corrected for this Γ factor, is
presented in Fig. 1. The shift ∆(1/2Q) is almost zero
in the superconducting state but increases very rapidly
when Tcis approached; a monotonous increase is further
observed in the normal state. The shift ∆f/f is nega-
tive in the superconducting state with a dip appearing
just below Tcwhich is defined as the temperature where
the rate of increase is maximum (indicated by an arrow);
in the normal state above Tc, the small shift presents a
flat temperature dependence for the 13 GHz mode while
a monotonous increase is observed at 16.5 GHz. The
temperature and frequency dependence of these relative
shifts is fully consistent with a superconducting sample in
the high conductivity limit [14], except for ∆f/f above
Tc for the 16.5 GHz mode. The absence of a flat tem-
perature dependence can be explained by a temperature
dependent Γ(T) for this particular mode.
Indeed, a constant Γ factor implies identical tempera-
ture dependence of the resonance frequency for an empty
cavity and one loaded with a perfect conductor. The

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FIG. 1: (Color online) Temperature dependence of ∆f/f and
∆(1/2Q) at 13 and 16.5 GHz for sample A. Inset: correction
of the ∆f/f data for the TE102 mode at 16.5GHz.
TE101resonance mode at 13 GHz presents only one elec-
tric field lobe and the sample which is located at its center
perturbs the mode in a symmetrical way; we thus expect
a temperature dependence of the resonance frequency
very similar to an empty cavity and, thus, a constant
Γ. The situation is more complex for the TE102 mode
since the sample is located in only one of the two lobes
yielding an asymmetrical perturbation of the mode. The
resonance frequency as a function of temperature is thus
expected to be modified relative to the empty cavity and
this yields a Γ(T) which explains the non monotonous
temperature behavior observed at 16.5 GHz. We should
recall that the temperature dependence of the resonance
frequency is due to a progressive increase of the cavity
volume above 20 K.
Since the comparison of the complex penetration depth
˜λ(T) at two frequencies will validate our approach, it is
important to correct the ∆f/f data at 16.5 GHz . We
show this correction in the inset of Fig. 1. Since the
perturbation of the mode is mainly observed above 20
K, we use a polynomial fit to mimic these effects which
are subsequently substracted from the original data. If
the thin film alone constituted the sample, the constant
Γ could be calculated with sufficient precision by asso-
ciating its geometry to an elongated ellipsoid. However,
the LaAlO3substrate makes Γ an undetermined param-
eter depending on the particular sample and on the reso-
nance mode of the cavity. For each sample, we thus define
two parameters Γ13and Γ16.5keeping in mind that these
parameters are related since the value of the magnetic
penetration depth λ(0) = λ1(0) must be frequency inde-
pendent. We can thus eliminate one degree of freedom
by adjusting accordingly the ratio of the two parameters,
whose values are chosen to yield the best coincidence of
λ1and λ2in the skin depth regime for T > Tcextracted
from the data at 13 and 16.5 GHz (λ1= λ2= δ/2 with
the skin depth δ = (2/ωσ1µ0)1/2). Correction to the
∆(1/2Q) shifts at 16.5 GHz is not necessary.
FIG. 2:
frequency shifts for two substrate thickness 0.42 mm and 0.22
mm at 16.5 GHz for the FS configuration.
(Color online) Comparison of the complex relative
B.Dielectric substrate contribution
Partly included in the parameter Γ, the substrate mod-
ifies also the amplitude of the complex frequency shift
relative to the unperturbed state. We propose here a
modified procedure adapted to our PCCO films which is
based on the comparison of results obtained for different
substrate configurations: i) reduced thickness of the sub-
strate; ii) symmetrical sample, substrate-film-substrate.
In Fig. 2, we compare the relative frequency shifts at
16.5 GHz obtained for thin film B on a 0.42 mm thick
substrate which has been reduced to 0.22 mm after pol-
ishing. The increase of thickness does not affect the tem-
perature dependence of the shifts; it acts merely as an
amplification factor. Would the situation be similar if
the substrate was present on both sides of the film? A
second substrate with identical dimensions was mechan-
ically added onto the film. The relative frequency shifts
for this configuration substrate-film-substrate (SFS) are
compared to the film-substrate (FS) one in Fig. 3 for the
two resonance modes. The temperature dependence of
the shifts is again maintained but the amplification factor
is increased. As expected, the presence of the substrate
increases the frequency shift and decreases the quality
factor as can be easily observed above Tc.
The data of Fig. 2 and Fig. 3 suggest to relate the sub-
strate effects to a scaling factor in the expression connect-
ing the complex frequency shift to the physical properties
of the film. This scaling factor is found by considering
the electric field configuration of Fig. 4: the substrate
on each side of the film acts as an amplifier of the elec-
tric field Esat the interface, a factor ζ on the left and η
on the right. These factors depend on the thickness and
the dielectric constant of the substrate. If we use these

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FIG. 3:
frequency shifts for the FS and SFS configurations at 13 GHz
and 16.5 GHz.
(Color online) Comparison of the complex relative
new interface conditions to find the electric field profile
for the slab geometry and the induced fields˜D(z) and
˜ H(z) (see appendix), we obtain the following complex
frequency shift
∆˜ ω
ω
= κα
N
?
1 −
?tanh(˜ γd/2)
˜ γω2µ0?0d/2+ 1
?
N
?−1
(1)
where ˜ γ = 1/˜λ is the complex wave vector. This equa-
tion is equivalent to the one obtained for the complex
frequency shift from the perfect conductor to an arbi-
trary state (eq.(18) in [15]) except for the scaling factor
κ which is related to the field amplification factors by:
κ =(ζ + η)
4
2?
1 −1
2
?ζ − η
ζ + η
?2?
(2)
This scaling factor κ is evaluated by considering the ex-
perimental data for the FS configuration (Fig. 2) and
the SFS one (Fig. 3) which suggest to write
?∆˜ ω
ω
?
FS
= κFS
?∆˜ ω
ω
?
F
(3)
?∆˜ ω
ω
?
SFS
= κSFS
?∆˜ ω
ω
?
F
(4)
where we have defined the relative complex frequency
shift of the film (F) alone when κ = 1. If we define β
as the ratio κFS/κSFS and we put ζ = η = 1 + ? valid
for the data of Fig. 3 (same substrate geometry on both
sides of the film), we write the scaling factors as
2?
κFS=(2 + ?)
4
1 −1
2
?
?
2 + ?
?2?
,κSFS= (1 + ?)2(5)
This last result allows us to express the factor ?, which
represents the deviation from the no-substrate configura-
tion, as a function of β = κFS/κSFS
? =β − 2 +?(β − 2)2− (2 − β/4)(1 − β)
Because the undetermined constant Γ, defining ∆f/f,
is an adjustable parameter in the final data treatment,
we use the ∆(1/2Q) shifts of Fig. 3 to determine the
factor β: practically the same value, 1.59 and 1.62, is
respectively obtained at 13 GHz and 16.5 GHz which
yields ? ≈ 0.65. There is, of course, a dependence of the
parameter ? on the thickness of the substrate as suggested
in Fig. 2. The best fit is obtained with a value of ?
reduced by a factor 0.43 when the thickness has been
decreased by a factor 0.53. This last result suggests a
quasi-linear relationship between ? and the thickness of
the substrate at these frequencies.
2 − β/4
(6)
FIG. 4:
of the position z from the center of the film of thickness d.
Es is the electric field outside the substrate and ζ and η the
amplification factors of the substrate on both sides of the
films.
(Color online) Electric field profile as a function

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C.Complex penetration depth : d-wave symmetry
In the preceding sections we have described the tools
necessary to extract the complex penetration depth˜λ
and subsequently the conductivity ˜ σ from our relative
frequency shifts data. By adjusting the constant Γ and
using the factor κ determined from Eq.(5) for the FS
configuration , we search for each frequency the best su-
perposition of λ1 and λ2 in the normal state extracted
from Eq.(1).
FIG. 5:
film A as a function of temperature at 13 and 16.5 GHz.
Continuous and broken lines represent the δ/2 calculated from
DC resistivity data.
(Color online) Complex penetration depth of thin
We show in Fig. 5 the complex penetration depth of
the thin film A between 2 K and 40 K at 13 GHz and
16.5 GHz. This measurement is analogous to a surface
impedance one that could have been performed on a sin-
gle crystal. However, thin films offer the advantage of
obtaining directly the absolute value λ(0). In the nor-
mal state for T > Tc, the superposition of λ1 and λ2
is very good for both frequencies, although better at 13
GHz since no correction of the ∆f/f data is necessary.
As expected the penetration depth is larger at 13 than at
16.5 GHz in the normal state, in agreement with the fre-
quency dependence of the skin depth δ. The monotonous
increase of the real and imaginary parts with tempera-
ture compares very well with the δ/2 plots obtained with
the DC resistivity measured on the same film and also
presented in Fig. 5. The microwave value is just a lit-
tle larger than the DC one and this is explained by un-
certainties on the parameters α, N and κ appearing in
Eq.(1). In the superconducting state, we observe no fre-
quency effect on λ1on a wide temperature range although
we imposed the coincidence at the lowest temperatures
only. The extrapolated value towards zero temperature
is λ(0) = 460 ± 100 nanometers; the large uncertainty
is attributed to the adjustment of the scaling factor κ.
This value is larger than the one found in the literature
[19], but is consistent with a higher normal state resis-
tivity. When the temperature is increased from 2 K, λ1
increases quasi-linearly first up to 6-7 K (∼ 17 nm/K)
and quadratically after (inset of Fig. 5); then, it goes
through a small maximum just below Tc. Such a signa-
ture is also observed in other conventional [20] and uncon-
ventional superconductors [21]. When Tcis approached
from above, the initial reduction of quasiparticle density
leads to a first order reduction in screening from quasi-
particles (δ) but only a second order increase in screening
from the superfluid fraction. This leads to a peak that
moves to lower temperatures on increasing frequency as
observed in Fig. 5. In the superconducting state, the
imaginary part λ2expresses a rapid reduction of screen-
ing by quasiparticles below Tc.
FIG. 6:
film B as a function of temperature at 13 and 16.5 GHz. Con-
tinuous and broken lines represent the δ/2 calculated from DC
resistivity data.
(Color online) Complex penetration depth of thin
The complex penetration depth of the thin film B be-
tween 2 K and 40 K at 13 GHz and 16.5 GHz is presented
in Fig. 6. For this sample, the superposition of λ1and
λ2in the normal state is excellent at the two frequencies.
Here the absolute values are a little smaller than the δ/2
data calculated from the DC resistivity. In the supercon-
ducting state, no frequency effects are observed on a wide
temperature range on the real part λ1with an extrapo-
lated value λ(0) = 1850 ± 150 nanometers. When the
temperature is increased from 2 K, λ1increases quadrat-
ically (inset of Fig. 6, ∼ 6 nm/K) and a small peak is still
observed just below Tc. The imaginary part λ2appears
to decrease more rapidly below Tcfor this film compared
to the previous one, the reduction of quasiparticle screen-
ing being more efficient. This is consistent with the data
in Table 1 showing that film A had a higher quality than
film B. This is confirmed by a much lower penetration
depth λ(0) and a quasi-linear increasing rate at low tem-
peratures compared to a quadratic one when impurity
scattering is more important. Thus, these results appear
to be consistent with a d-wave order parameter with lines
of nodes as already suggested by several other studies.