Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides
We report the number of deviations from conventional behavior in superconducting properties of ultrathin (1–3 unit-cell (UC)) YBa₀₂Cu₃O₇₋x films sandwiched between semiconducting Pr₀,₆Y₀,₄Ba₂Cu₃O₇₋x layers and for single crystals of cluster superconductor: dodecaboride ZrB₁₂. We have found a qua...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Цитувати: | Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides / V.A. Gasparov // Физика низких температур. — 2006. — Т. 32, № 8-9. — С. 1105–1114. — Бібліогр.: 33 назв. — англ. |
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Gasparov, V.A. 2017-06-11T18:25:14Z 2017-06-11T18:25:14Z 2006 Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides / V.A. Gasparov // Физика низких температур. — 2006. — Т. 32, № 8-9. — С. 1105–1114. — Бібліогр.: 33 назв. — англ. 0132-6414 PACS: 74.25.Nf, 74.72.Bk, 74.70.Ad, 72.15.Gd https://nasplib.isofts.kiev.ua/handle/123456789/120350 We report the number of deviations from conventional behavior in superconducting properties of ultrathin (1–3 unit-cell (UC)) YBa₀₂Cu₃O₇₋x films sandwiched between semiconducting Pr₀,₆Y₀,₄Ba₂Cu₃O₇₋x layers and for single crystals of cluster superconductor: dodecaboride ZrB₁₂. We have found a quadratic temperature dependence of the kinetic inductance, Lk⁻¹(Т) , at low temperatures independent of frequency, with a break in slope at TBKT dc , the maximum of real part of conductance, ωσ₁(T), and a large shift of the break temperature and the maximum position to higher temperatures with increasing frequency ω. We obtain from these data the universal ratio T /Lk⁻¹ (Tdc ) = 25, 25, and 17 nHK for 1-, 2- and 3-UC films, respectively in close agreement with theoretical prediction for vortex—antivortex unbinding transition. Superfluid density of ZrB₁₂ displays unconventional temperature dependence with pronounced shoulder at T/Tc equal to 0.65. Contrary to conventional theories we found a linear temperature dependence of Hс₂(Т) from Tc down to 0.35 K. We suggest that both λ(T) and Hс₂(Т) dependencies can be explained by a two band BCS model with different superconducting gap and Tc. We are grateful to V.F. Gantmakher, A. Hebard, M. Chan, R. Huguenin, C. Lobb, P. Martinoli, D. van der Marel, C. Rogers, D.J. Scalapino, J.-M. Triscone, T. Venkatesan, X.X. Xi for stimulating discussions. This work was partially supported by Russian Ministry of Industry, Science and Technology (MSh-2169.2003.2) and Russian Academy of Sciences Program: New Materials and Structures. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 100-летию со дня рождения Б.Г. Лазарева Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides Article published earlier |
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| title |
Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides |
| spellingShingle |
Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides Gasparov, V.A. К 100-летию со дня рождения Б.Г. Лазарева |
| title_short |
Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides |
| title_full |
Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides |
| title_fullStr |
Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides |
| title_full_unstemmed |
Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides |
| title_sort |
recent observation of anomalous superconducting behavior of ultrathin ybco films and single crystals of cluster borides |
| author |
Gasparov, V.A. |
| author_facet |
Gasparov, V.A. |
| topic |
К 100-летию со дня рождения Б.Г. Лазарева |
| topic_facet |
К 100-летию со дня рождения Б.Г. Лазарева |
| publishDate |
2006 |
| language |
English |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
We report the number of deviations from conventional behavior in superconducting properties
of ultrathin (1–3 unit-cell (UC)) YBa₀₂Cu₃O₇₋x films sandwiched between semiconducting
Pr₀,₆Y₀,₄Ba₂Cu₃O₇₋x layers and for single crystals of cluster superconductor: dodecaboride ZrB₁₂.
We have found a quadratic temperature dependence of the kinetic inductance, Lk⁻¹(Т)
, at low temperatures
independent of frequency, with a break in slope at TBKT
dc , the maximum of real part of
conductance, ωσ₁(T), and a large shift of the break temperature and the maximum position to
higher temperatures with increasing frequency ω. We obtain from these data the universal ratio
T /Lk⁻¹
(Tdc ) = 25, 25, and 17 nHK for 1-, 2- and 3-UC films, respectively in close agreement
with theoretical prediction for vortex—antivortex unbinding transition. Superfluid density of
ZrB₁₂ displays unconventional temperature dependence with pronounced shoulder at T/Tc equal
to 0.65. Contrary to conventional theories we found a linear temperature dependence of Hс₂(Т)
from Tc down to 0.35 K. We suggest that both λ(T) and Hс₂(Т) dependencies can be explained by a
two band BCS model with different superconducting gap and Tc.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120350 |
| citation_txt |
Recent observation of anomalous superconducting behavior of ultrathin YBCO films and single crystals of cluster borides / V.A. Gasparov // Физика низких температур. — 2006. — Т. 32, № 8-9. — С. 1105–1114. — Бібліогр.: 33 назв. — англ. |
| work_keys_str_mv |
AT gasparovva recentobservationofanomaloussuperconductingbehaviorofultrathinybcofilmsandsinglecrystalsofclusterborides |
| first_indexed |
2025-11-25T21:07:27Z |
| last_indexed |
2025-11-25T21:07:27Z |
| _version_ |
1850550735709667328 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9, p. 1105–1114
Recent observation of anomalous superconducting
behavior of ultrathin YBCO films and single crystals
of cluster borides
V.A. Gasparov
Institute of Solid State Physics RAS, Chernogolovka 142432, Moscow Region, Russia
E-mail: vgasparo@issp.ac.ru
Received March 1, 2006, revised April 17, 2006
We report the number of deviations from conventional behavior in superconducting properties
of ultrathin (1–3 unit-cell (UC)) YBa2Cu3O7–x films sandwiched between semiconducting
Pr0.6Y0.4Ba2Cu3O7–x layers and for single crystals of cluster superconductor: dodecaboride ZrB12.
We have found a quadratic temperature dependence of the kinetic inductance, L Tk
�1( ), at low tem-
peratures independent of frequency, with a break in slope at TBKT
dc , the maximum of real part of
conductance, ��1(T), and a large shift of the break temperature and the maximum position to
higher temperatures with increasing frequency �. We obtain from these data the universal ratio
TBKT
dc /Lk
�1 (TBKT
dc ) = 25, 25, and 17 nHK for 1-, 2- and 3-UC films, respectively in close agreement
with theoretical prediction for vortex—antivortex unbinding transition. Superfluid density of
ZrB12 displays unconventional temperature dependence with pronounced shoulder at T/Tc equal
to 0.65. Contrary to conventional theories we found a linear temperature dependence of H Tc2( )
from Tc down to 0.35 K. We suggest that both �(T) and H Tc2( ) dependencies can be explained by a
two band BCS model with different superconducting gap and Tc.
PACS: 74.25.Nf, 74.72.Bk, 74.70.Ad, 72.15.Gd
Keywords: YBCO films, dodecaboride ZrB12, inductance, penetration depth, two band supercon-
ductions.
1. Introduction
The unusual superconducting behavior of high-Tc
superconductors has stimulated efforts to test whether
two-dimensional (2D) behavior is an essential aspect
of these materials. The question of whether an isolated
unit-cell thick layer could exhibit superconductivity
and how his properties relate with bulk materials re-
main controversial. From other side, it has been sug-
gested from Berezinskii—Kosterlitz—Thouless (BKT)
theory that there exist bound pairs of thermally ex-
cited vortices and antivortices (with opposite circula-
tion) in 2D layers below the TBKT and dissociated on
free vortices and antivortices above [1,2]. Although
many observations of the BKT transition in YBCO,
BiSrCaCuO, and TlBaCaCuO compounds have been
reported (see [3] and references therein), detailed
comparison of the experimental data with the theory
by Davis et al. [4] showed disagreements possibly due
to inhomogeneity and vortex pinning. Rogers et al.
reported that the usual BKT transition, i.e., all ther-
mally activated vortices form vortex—antivortex
pairs at temperatures below TBKT, was not observed
in ultrathin Bi2Sr2Cu2O8 films from a low-frequency
noise measurement due to vortex pinning [5]. Repaci
et al. [6] showed from the study of dc I–V curves that
free vortices exist at low temperatures even in
one-unit-cell-thick YBCO films, indicating the ab-
sence of the dc BKT transition. The binding energy
between a vortex—antivortex pair U(r) � 1/r at
r > �eff (here �eff = 2�2/d is the effective penetration
depth and d is the film thickness), diverges at high
distances r. Thus they had pointed out that a precon-
dition [7] for the BKT transition to occur in a super-
conductor, i.e., the sample size Ls< �eff, is not satis-
fied even in YBCO films as thin as one unit cell.
According to the BKT theory extended to finite
frequencies [8–10], higher frequency currents sense
© V.A. Gasparov, 2006
vortex—antivortex pairs of smaller separations. At
high frequency, the electromagnetic response of a 2D
superconductor is dominated by those bound pairs
that have r � l�, where l� = (14D/�)1/2 is the
vortex diffusion length and D is the vortex diffusion
constant. Using the Bardeen—Stephen formula for
free vortices [11], we estimate that l� < 1 �m at
� > 10 MHz, which is much less than �eff = 40 �m for
the 1-UC YBCO film [3]. This implies that it is
possible to detect the response of vortex—antivortex
pairs with short separation lengths at high frequencies
in the samples even though the usual BKT transition
is not present as shown in dc and low-frequency
measurements.
From other side, the recent discovery of super-
conductivity in magnesium diboride [12] has initiated
a substantial interest in potential «high–temperature»
superconducting transition in other borides (see refe-
rences in Refs. 13–16). Yet, only nonstoichiometric
boride compounds (MoB2.5, NbB2.5, Mo2B, W2B,
BeB2.75) demonstrate such transition. A potential clue
to this contradiction may lay in the crystal structure
of boron compounds, in particular in their cluster
structure. Although it is widely accepted that the
layered structure is crucial for high-Tc superconducti-
vity, one can argue that clusters of light atoms are
important for high Tc as well. In particular, there are
a number of rather high-Tc superconductors among
three-dimensional cluster compounds. Those are alkali
metal doped C60 compounds (fullerides) Me3C60
(Me = K, Na, Rb, Cs) with the highest Tc up to 33 K
for RbCs2C60 [17,18]. It is also known that boron
atoms form clusters. These are octahedral B6 clusters
in MeB6, icosahedral B12 clusters in �-rhombohedral
boron, and cubo-octahedral B12 clusters in MeB12.
The quest for superconductivity in these cluster
compounds has a long history. Several superconduct-
ing cubic hexaborides, MeB6, and dodecaborides,
MeB12, have been discovered by Matthias et al. back
in late 1960’s [19] (Me = Sc, Y, Zr, La, Lu, Th).
Many other cluster borides (Me = Ce, Pr, Nd, Eu, Gd,
Tb, Dy, Ho, Er, Tm) were found to be ferromagnetic
or antiferromagnetic [19,20]. It was suggested that
the superconductivity in YB6 and ZrB12 (Tc of 7.1 and
6.03 K, respectively) is exactly due to the effect of a
cluster of light boron atoms. Clearly a systematic
study of ZrB12 is needed to address the question of
superconductivity in this compound.
This has been the motivation for current systematic
study of the temperature dependencies of the fre-
quency and temperature dependences of the complex
sheet conductance, �(�,T), of 1-UC to 3-UC thick
YBCO films sandwiched between semiconducting
Pr0.6Y0.4Ba2Cu3O7–x layers in a frequency range bet-
ween 1 MHz to 30 GHz, as well as �(T) and upper
critical magnetic field, H Tc2( ), in single crystals of
ZrB12. We found a large increase of the transition
temperature as a function of frequency for those films
from 4 MHz to 30 GHz. We observe unusual super-
conducting properties of ZrB12 and argue that these
results can be reconciled by two-band superconducti-
vity with different Tc. Currently these data were
published in Refs. 3, 16, here we present most im-
portant results of these studies.
2. Experimental setup
Ultrathin YBCO layers sandwiched between 100 �
buffer and 150 � cover layers of Pr0.6Y0.4Ba2Cu3O7–x
were grown epitaxially on atomically flat and well-
lattice-matched (100) LaAlO3 substrates using a
multitarget pulsed-laser deposition (PLD) system [3].
For few UC thick films and the substrate used, the
thicknesses are below the critical value for forming
islands and the growth is in the range of layer by layer
Stranski–Krastanov growth. The sample thickness is
rather uniform due to the nature of the growth mode
as characterized by using cross sectional transmission
electron microscopy (TEM), atomic force microscope
(AFM) and in situ RHEED. The three films thickness
we examined were nominally 1-, 2-, and 3-UC thick
and had the c axis normal to the film surface. The
samples were made at different oxygen composition
and therefore we will call them as S1 and S2 ones. The
contacts were made at the edges of the 1�1 cm 3UC
film for van der Pauw four-point resistance mea-
surements.
Under ambient conditions, dodecaboride ZrB12
crystallizes in the fcc structure of the UB12 type
[15,16]. In this structure, the Zr atoms are located at
interstitial openings among the close-packed B12
clusters. Our ZrB12 single crystals were grown using a
floating-zone method [14]. The obtained single crystal
ingots had a typical diameter of about 5 to 6 mm and a
length of 40 mm. To assure good quality of our
samples we performed metallographic and x-ray
investigations of as grown ingots. We discovered that
most parts of the ZrB12 ingot contained a needle like
phase of nonsuperconducting ZrB2. We believe that
ZrB2 needles are due to preparation of ZrB12 single
crystals from a mixture of a certain amount of ZrB2
and an excess of boron [14]. We believe that
unconventional properties observed from other studies
may be due to these sample problems [16]. Therefore,
special care has been taken to cut the samples from
ZrB2 phase free parts.
We used the spark erosion method to cut the single
crystal ingots into rectangular <100> oriented bars of
about 0.5�0.5�8 mm. The samples were lapped with
1106 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9
V.A. Gasparov
diamond paste and etched in hot nitrogen acid to
remove any damage induced by lapping deteriorated
surface layers. A well-defined geometry of the samples
provided for the precise �(T) and superconducting
properties measurements. Temperature was mea-
sured with platinum (PT-103) and carbon glass
(CGR-1-500) sensors. Magnetic measurements of
�(T,H) and �(T,H) were carried out using a super-
conducting coil in applied fields of up to 6 T down to
1.3 K. Additional dc and ac �(H) measurements were
performed in the National High Magnetic Field La-
boratory in Tallahassee, Florida (NHMFL) at tem-
peratures down to 0.35 K. The dc magnetic field was
applied in the direction of the current flow.
For this study, two highly crystalline, supercon-
ducting films of MgB2 were grown on an r-plane
sapphire substrate in a two-step process [22]. De-
position of boron precursor films via electron-beam
evaporation was followed by ex situ postannealing at
890 �C in the presence of bulk MgB2 and Mg vapor.
We investigated films of 500 and 700 nm thickness
with corresponding Tc0’s of 38 K and 39.2 K, res-
pectively. The details of the preparation technique are
described elsewhere [22].
The �(�,T) at RF in thin 1–3 UC YBCO and
MgB2 films was investigated employing a single coil
mutual inductance technique. This technique, origi-
nally proposed in Ref. 23 and lately improved in Ref.
24, has the advantages of the well known two-coil ge-
ometry, and was extensively used for the study of the
�(T) dependence for YBCO and MgB2 films
[3,15,16]. In this radio frequency technique, the
change of inductance L of a one-layer pancake coil
located in the proximity of the film and connected in
parallel with a capacitor C is measured. The LC cir-
cuit is driven by the impedance meter (VM-508
TESLA) operating at 2–50 MHz, with a high fre-
quency stability of 10 Hz. The film is placed at small
distance (� 0.1 mm) below the coil. Both sample and
coil are in a vacuum, but the coil holder is thermally
connected with helium bath, while the sample holder
is isolated and may be heated. During the experiment
the coil was kept at 2.5 K, whereas the sample temper-
ature was varied from 4.2 up to 100 K. Such design al-
lows us to eliminate possible effects in temperature
changes in L and C on the measurements.
The complex mutual inductance M between the
coil and the film can be obtained through
Re ( )
( )
M T L
f
f T
�
�
�
�
�
�
�
�0
0
2
2
1 , (1)
Im ( )
( ) ( ) ( )
( )
M T
f T C Z T Z T
f T
f
�
�
�
�
�
�
�
�
�
1
2
1 1
2
0
2
0
2�
. (2)
Here L, Z(T), f(T), L0, Z0 and f0 are the inductance,
impedance and the resonant frequency of the circuit
with and without the sample, respectively. In the low
frequency regime, where the coil wire diameter is
much thinner than the skin depth at the working
frequency, the expression of the variation of the
M(T) (relative to the case where no sample is in the
coil, M0), as a function of the �(T) may written
as [24]:
M T
M q
ql
d
l
dq( )
( )
� �
�
�
�
�
�
���0
0 1 2 coth
, (3)
where M(q) plays the role of mutual inductance at a
given wave number q in the film plane and depends
on the sample-coil distance h, d is the sample thick-
ness, and l is a complex length defined as l =
= [1/(i��0�1+�2)]1/2, (more details can be found in
Refs. 16, 24). A change in real, Re M(T), and imagi-
nary, Im M(T), parts of M(T) were detected as a
change of resonant frequency f(T) of the oscillating
signal and impedance Z(T) of the LC circuit, and
converted into Lk(T) and �1(T) by both, using
Eqs. (1), (2) with inversion mathematical procedure
and Eq. (3).
The high frequency (100 MHz–1 GHz) measure-
ments were performed using the cavity formed with a
similar spiral coil with no capacity in parallel. The
coil form the radio frequency resonator coupled to a
two coupling loops and is driven by the radio-fre-
quency signal generator/receiver. In this case the
quality factor of the resonator Q and the resonance
frequency were measured and converted to Re M(T)
by Eq. (1) and to Im M(T) by:
Im ( )
( ) ( ) ( )
M T L
f
f T Q T
f
f T Q
�
�
�
�
�
�
�0
0 0
0
1 1
, (4)
where Q(T) and Q0(T) are the quality factors with
and without the samples, respectively.
The MW losses were measured using a resonant
cavity technique with the gold-plated copper cylind-
rical cavity operated in the TE011 mode at 29.9 GHz.
The samples were mounted as a part of the bottom of
the cavity through a thin gold-plated Cu-diaphragm
with a small central hole so that the sample itself
occupies only the holes part of the endplate through a
transparent Teflon film. The resonator was operating
in a transmission configuration. During the experi-
mental run, we measured the amplitude of the trans-
mitted signal at resonance as a function of tem-
Recent observation of anomalous superconducting behavior of ultrathin YBCO films
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 1107
perature. The resonator was coupled weakly to the
input and output waveguides so the amplitude of the
transmitted signal at resonance is proportional to the
square of unloaded quality Q. We assume the MW
electric field E in the film to be uniform along normal
direction and equal to the E without the film. The
�1(�,T) is thus proportional to Q–1(T) = Q–1(T) —
– Q0
–1(T), where Q(T) and Q0(T) are the quality
factors with and without the samples, respectively.
We used similar radio frequency LC technique
[15,16,25] to measure �(T) of ZrB12 samples. This
technique employs a rectangular solenoid coil into
which the sample is placed rather then use of spiral
coil. The connection between parameters of the circuit
and �(T) is described by following equation:
� � �( ) ( )
( ) ( )
( ) ( )
T
f T f
f T fc
�
�
�
� �
� �
0
0
0
2 2
2 2
. (5)
Here � = 0.5(c2�/2��)1/2 is the imaginary part of a
skin depth above Tc, which was determined from the
�(T) measurements close to Tc, f(T) is the resonance
frequency of the circuit at arbitrary T and f(Tc) and
f(0) are the resonance frequency of the circuit at the
superconducting transition and at zero temperature,
respectively.
3. Magnetic penetration depth in YBCO films
Figure 1 displays the Re M(T) and Im M(T) curves
for a 2UC S2 film at three different frequencies from 3
MHz to 500 MHz as measured by different techniques:
LC circuit and spiral coil resonator. The HF data are
normalized to RF Re M(0) data because of different
gap h values used. The most noticeable feature of these
data is rather high shift of the onset point Tc0 of
Re M(T) transition with frequency, not observed in
such measurements on thick films. Notice also, that
the inductive response, Re M(T), starts at lower tem-
peratures than Im M(T), characterized by a peak close
to transition, and this shift is raised with frequency.
We have carried out the mutual inductance mea-
surements on Pr0.6Y0.4Ba2Cu3O7–x films and observed
no any features in the temperature dependences of the
mutual inductance M(T).
The Re M(T) and Im M(T) data are converted to
L Tk
�1( ) and Re �(T) using Eq. (3) and the mathemati-
cal inversion procedure [3] based on the same ap-
proach as in the two-coil mutual inductance method.
Figure 2 shows the L Tk
�1( ) curves in very low perpen-
dicular magnetic fields, and zero field �Re �(T) for
the 1-UC and 2-UC films (S1). We found that L Tk
�1( )
fit well over a wide temperature range by a parabolic de-
pendence: L Tk
�1( ) = Lk
�1(0)[1 – (T/Tc0)2], shown as
thin solid lines in Fig. 2. We emphasize that this qua-
dratics equation fit the data below characteristic tem-
perature which we define as TBKT
dc , and which is below
the positions of the peaks in �Re �(T), which we de-
fine as TBKT
� . The mean field transition temperature,
Tc0, determined by extrapolation of L Tk
�1( ) to 0, is
larger than the onset of transitions of L Tk
�1( ), while is
close to the onset point of �Re �(T) curves. Also, the
Lk
�1 0( ) fitted data are the same for H = 0 and 5 mT
while have different Tc0.
In Fig. 3, we plot �Re �(T) at 8 MHz and
Q–1(T)/Q0
–1(4.2 K) determined from MW data
(30 GHz) for 3-UC sample (S1). The dc resistive tran-
sition of the same sample is also shown in the figure.
According to the Coulomb gas scaling model, the re-
1108 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9
V.A. Gasparov
L
C
� , Z
Fig. 1. The Re M(T) and Im M(T) curves for a 2-UC
film (S2) at different frequencies, MHz: 3 (triangles),
26 (squares) and 500 (circles) calculated from raw f(T),
Z(T) and Q(T) data. The solid lines describe a guide for
the eye. Inset shows experimental set-up.
.
..
.
Fig. 2. Temperature dependence of L Tk
�1( ) for 1- and 2-UC
films (S1) at 8 MHz and different magnetic fields, mT: 0
(circles), 2 (squares), 3 (triangles) and 4 (crosses). The
solid lines curves shows �Re �(T) at zero field. The thin
solid lines are quadratic fits to Lk
�1 (T) below TBKT
dc and
for magnetic field data. Also shown is the theoretical BKT
function (dashed line).
sistance ratio R/Rn is proportional to the number
of free vortices and should follow a universal function
of an effective temperature scaling variable X =
T Tc( 0 – TBKT
dc )/TBKT
dc (Tc0 – T) [8], which can be
approximated by �/�n = C0X exp [–C1(X–1)–1/2]
(here �n is the normal state resistivity, C0 = 1.7 and
C1 = 4.9 are constants). We plot T/X(T) as a func-
tion of T obtained by fitting �/�n data for the 3-UC
film with this equation in the inset of Fig. 3. The best
fit was found with Tc0 = 78.5 K (the point where T/X
= 0) and TBKT
dc = 56 K. Here TBKT
dc represents a nomi-
nal dc BKT transition temperature which is lower
then TBKT
� determined from RF and MW data.
There are three major features observed in our RF
and MW measurements: (i) the large frequency de-
pendence of Tc(�), (ii) a foot jump in temperature
dependence of the L Tk
�1( ), which is destroyed in weak
magnetic fields, and (iii) a maximum in �Re �(T)
with the onset of transition at higher temperatures
than that of the L Tk
�1( ).
Indeed, as we can see from Fig. 1, the TBKT
� value,
determined as the maximum position of losses, in-
creases on 4 K (from 36.6 to 40.7 K) as the frequency
raise from 3 to 500 MHz for 2-UC YBCO S2 film. The
TBKT
� value shift at 30 GHz is much larger: 74 K as
compared with 61.5 K at 8 MHz (see Fig. 3). Even
larger shift of TBKT
� was observed in 1-UC film and no
shift was detected in a 2000 � thick YBCO film.
We determined the TBKT
� as the peak position of
�1(T) point. We also used the TBKT
dc as the point
where Lk
�1 (T) deviates from the square fit (see
Fig. 2), as was used by Hebard et al. [30]. The ratio
TBKT
dc /Lk
�1(TBKT
dc ) thus equal to 25, 25, and 17 nHK
for 1-, 2- and 3-UC S1 films, respectively, is about
constant, which however is larger than the theoreti-
cal estimation TBKT
dc �2(TBKT
dc )/d = �0
2/32�2kB =
= 0.98 cm·K (or TBKT
dc /Lk
�1(TBKT
dc ) = 12.3 nHK) [31]
ignoring small dynamic theory corrections of Ambe-
gaokar et al. [9] to the renormalized coupling cons-
tant KR.
In order to see whether this assumption is correct,
we plot theoretical BKT function L Tk
�1( ) as dashed
straight line on Fig. 2 derived from the universal
relationship:
K
d c
e k T
R
B BKT
� � �2
2 2
216
2� (6)
predicted by theory [4]. Notice however, that this
theoretical dependence is valid for dc case, while the
frequency dependence of TBKT
� is obvious from the
picture discussed above. This is why the critical
temperatures determined from the intercept of dashed
theoretical line with experimental L Tk
�1( ) is lower
then the peak position of �1(T). To see whether this
description is correct, we plot the dependence of the
penetration depth �–2(T) derived from L Tk
�1( ), versus
scaling variable — normalized temperature T/TBKT
�
on Fig. 4. It is obvious, that all data for three studied
S1 films at 8 MHz fall on the same curve, which
proof of our definition of TBKT
� as the peak position
of �1(T).
The Abrikosov vortex lattice parameter av is the
scale limiting the formation of vortex–antivortex
pairs in magnetic field. We can estimate the field Hext
which destroys the vortex pair unbinding from the
following relation: l�� av = (�0/Hext)
1/2. This
estimation gives Hext �1.5 mT at 10 MHz for 1-UC
and 2-UC films in good agreement with the expe-
rimental results (see Fig. 2). It is generally believed
that the inhomogeneity can destroy the interaction of
Recent observation of anomalous superconducting behavior of ultrathin YBCO films
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 1109
�
�
�
�
m
cm
;
.
..
.
.
.
R
e
n
H
–
1
�
Q
/Q
–
1
–
1
0
T
dc
T
rf
T
mw
BKT BKT BKT
Fig. 3. Temperature dependence of dc �(T) (squares),
�Re �(T) (circles) at 8 MHz and �Q–1 (T)/Q0
�1(4.2 K)
at 30 GHz (solid line) for a 3-UC sample (S1). Inset
shows the universal plot: T/X vs T. Arrows indicate the
TBKT values determined from the dc resistivity, loss func-
tion at 8 MHz and MW measurements (30 GHz).
Pe
ne
tra
tio
n
epd
th
,
–
–
2
2
�
m
Fig. 4. The penetration depth �
–2(T) derived from
Lk(T) = �0�
2/d vs normalized temperature T/TBKT
� for:
1-UC (crosses), 2-UC (squares) and 3-UC (circles) films.
vortex—antivortex pairs. However, inhomogeneity
can never help to create the BKT transition.
It is easy to see from above, we report several re-
sults which are in qualitative agreement to the predic-
tion of the dynamic theory of vortex—antivortex pairs
with short separation lengths. These effects must
be from intrinsic effect and can not be the result
of inhomogeneity or due to the skin effect in the
samples [3].
4. Magnetic penetration depth in ZrB12
In the BCS theory the London penetration depth is
identical with �(T) for specular and diffuse surface
scattering and for negligible nonlocal effects. For a
BCS-type superconductor with the conventional
s-wave pairing form, the �(T) has an exponentially
vanishing temperature dependence below Tc/2
(where �T) is almost constant):
� �
�
( ) ( )
( )
exp
( )
T
k T k TB B
� �
�
��
�
�
��
�
�
�
�
�
�0 1
0
2
0
(7)
for clean limit: l > �, and
� �( ) ( )
( )
T
k TB
�
��
�
�
��
0
1
0
2
tanh
(8)
for dirty limit: l < � [13]. Here (0) is the energy
gap and �(0) is the penetration depth at zero tem-
perature. Close to Tc �(T) dependence has a BCS
form [16]:
�
�
( )
( )
T
T
Tc
�
�
��
�
�
��
0
2 1
. (9)
Important problems for �(T) measurements are (i)
determination of basic superconducting parameter
�(0) and (ii) temperature dependence law, to see
whether s-wave or d-wave pairing forms exist. Both
these problems can be addressed from low-T �(T)
dependence according to Eqs. (7) and (8). We used
Eq. (5) to extrapolate the resonance frequency f(T) of
our LC circuit down to zero temperature.
To address the problem with �(0) we plot
�(T) – �(0) data versus BCS reduced temperature:
1 2 1/ T/Tc( )� in the region close to Tc (see the inset
to Fig. 5). The advantage of this procedure is the
insensitivity of such analysis to the choice of f(0) on
this temperature scale. The value of Tc = 5.992 K used
in this data analysis is obtained by getting best linear
fit of the �(T) – �(0) vs 1 2 1/ T/Tc( )� plot. Re-
markably there is only a few millidegrees difference
between Tc obtained from the fit and actual Tc0. We
use the slope of �(T) – �(0) vs 1 2 1/ T/Tc( )� and
Eq. (9) to obtain the value of �(0) = 143 nm.
In order to investigate the temperature dependence
of �(T) in the whole temperature region, in Fig. 6 we
plot the superfluid density �2(0)/�2(T) versus the re-
duced T/Tc for ZrB12 sample using this �(0) deter-
mined from one gap fit close to Tc. One can easily no-
tice an unconventional behavior of superfluid density
with pronounced shoulder at T/Tc equal to 0.65. This
feature can be explained by a model of two independ-
ent BCS superconducting bands with different plasma
frequencies, gaps and Tc’s [16]. We label these two
bands as p- and d-bands in accordance to electron
structure of ZrB12 [26]. It is clear from partial BCS
1110 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9
V.A. Gasparov
Fig. 5. Temperature variations of �(T) for ZrB12 sample.
The inset shows �(T) – �(0) versus BCS reduced
temperature.
�2
�2
Fig. 6. Superfluid density, [�(0)/�(T)]2, of the ZrB12
single crystal sample and MgB2 thin film (crosses). The
predicted behavior of [�(0)/�(T)]2 within the two band
model as described in the text is shown by the solid,
dashed (p-band) and dotted (d-band) lines. The solid line
represent BCS fit of MgB2 data using 2�(0)/kBTc as fit
parameter.
curves, that single gap BCS model is unable to fit a
full [�(0)/�(T)]2 unconventional dependence.
For a dirty limit [27] we can write for two band
superconductor in:
1 2
0 0
2
2 2� �( )
( ) ( )
( ) ( )
( ) (
T
T
k T
T
kp
p
B
p p
d
d
�
tanh tanh
B
d d
T
)
( ) ( )
.
�2 0 0
(10)
Here i is the superconducting energy gap and �i(0)
is residual penetration depth in the p- or d-band.
Using this two gap �(T) BCS-like dependence
and interpolation formula ( ) ( ) tanh [ .T
�0 188
� ( – ) ]T /Tc
/1 1 2 we fit the experimental data with
six fitting parameters: �i, i and Tci . From this fit we
obtain Tc
p= 6.0 K, Tc
d= 4.35 K, p(0) = 0.73 meV,
d(0) = 1.21 meV, �p(0) = 170 nm and �d(0) =
= 260 nm, for p- and d-bands, respectively. Dashed
and dotted lines in Fig. 6 show the contributions of
each p- and d-bands, respectively. Clearly low-tem-
perature dependence of �2(0)/�2(T) is dominated by
the d-band with the smallest Tc, whereas the high-
temperature behavior results from the p-band with
the larger Tc. The reduced energy gap for p-band,
2 p(0)/kBTc
p = 2.81, is rather small relative to the
BCS value 3.52, while d-band value, 2 d(0)/kBTc
d =
= 6.44, is twice as big. Thus we suggest that ZrB12
may have two superconducting bands with different
Tc and order parameters. Notice that this unusual
conclusion may be right for two bands in the limit of
zero interband coupling in agreement with resistivity
data [13]. Also, the (0) of ZrB12 may not be
constant over the Fermi surface.
We based our conclusion on the two-gap model for
dirty-limit superconductors, suggested by Gurevich
[27]. In this model, we can write
1 4
2
2 2
2�
�
( )
( )
T
e
c
N D N Dp p p d d d
�
�
�
1
0
1
02 2� �p d( ) ( )
, (11)
where Ni, i and Di are the density of states, the
energy gap and the diffusivity in p- and d-bands,
respectively. It follows from electron band structure
calculations that the dominant contribution to the
density of states N(EF) is made by the Zr4d and B2p
states, with Nd = 7.3·1021 st/eV·cm3 and Np =
= 8.7·1021 st/eV·cm3, respectively [26]. The B2p
bonding states are responsible for the formation of
B12 intra cluster covalent bonds. In turn, Zr4d bands
are due to Zr sub lattice. We use this two band
approach (Eq. (10)) to obtain p-band diffusivity of
Dp = 57 cm2/s and d-band diffusivity of Dd =
= 17 cm2/s. Note that there is almost a six times
difference between the p- and d-band diffusivity. We
use this result for our discussion of H Tc2( ) data in the
following paragraph.
The important goal of this chapter is comparison
of ZrB12 and MgB2 data. In Fig. 6 we show the tem-
perature variation of a superfluid density,
��(0)/��(T), versus reduced temperature T/Tc for
the best MgB2 film as determined from the one-coil
technique (Eq. (1)) and inversion procedure from
Eq. (3) with �(0) = 114 nm. The solid line represents
BCS single gap calculations by the aid of a single term
of Eq. (10) and using finite energy gap ( (0) =
= 1.93 meV) as the fit parameter. According to Fig. 5,
there is a very good agreement between experimental
data and the single gap BCS curve over the full tem-
perature range. The reduced energy gap 2 (0)/kBTc
is evaluated to be 1.14. It is actually within the range
of values for 3D � bands obtained by PCS on MgB2
single crystals [16]. Both (0) and �(0) are consistent
with microwave measurements on similar c-axis ori-
ented thin films (3.2 meV and 107 nm, respectively)
[28]. Notice that we studied here the penetration
depth in the ab-plane due to the samples being c-axis
oriented thin films. This feature predicts that our
�ab(T) is determined by the small energy gap for the �
band.
5. Upper critical magnetic field
We now turn to the electronic transport data
acquired in magnetic field. Figure 7 displays the
Recent observation of anomalous superconducting behavior of ultrathin YBCO films
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9 1111
�(
H
H
) –
�
(0
),
�
m
�
Fig. 7. Magnetic field variation of �(H) of a single
crystal ZrB12 sample at different temperatures, K: 5.66,
5.53, 5.02, 4.06, 3.45, 2.84, 2.15 and 1.43, from the left
to the right. The solid lines are the guides for the eye. The
arrow describes how Hc2 has been deduced.
penetration depth magnetic field transitions at various
temperatures down to 1.43 K with the fields oriented
along the sample bar. Figure 7 clearly demonstrates a
well defined onset of �(H) transition. We used this
onset to estimate Hc2. Similar approach was used to
�(T) data [16]. Figure 8 presents the Hc2 dependence
defined at the onsets of the finite �(H) and �(H).
Remarkable feature of this plot is an linear increase of
Hc2 with decreasing temperature down to 0.35 K.
To see whether one gap BCS model may work for
ZrB12, we extrapolate H Tc2( ) to zero temperature by
use of the derivative of dH Tc2( )/dT close to Tc and
the assumption that the zero temperature Hc2 0( ) =
= 0.69T dHc c2/dT�Tc. The resulting Hc2 0( ) = 114 mT
is substantially lower than the low temperature onset
data below 3 K. Linear extrapolation of H Tc2( ) to
T = 0 gives Hc2 0( ) = 162 mT. We used this value to
obtain the coherence length �(0), by employing the re-
lations Hc2 0( ) = �0/2��2(0). The latter yields �(0) =
= 45 nm, which is substantially larger than a few ang-
stroms coherence length of high-Tc superconductors.
In contrast to Ref. 21 our estimations agree well with
the Ginzburg–Landau parameter �
� �! Using our
values of �p and �d data we obtain �p = 3.8 and �d =
= 5.8. Both values of �d are larger then 1 2/ that im-
plies that ZrB12 is type II superconductor at all T.
Using the GL expression forH T /c1 0
24( ) ln
� � �� we
obtain: Hc2/Hc1 = 2�2/ln �. From the value Hc2 0( )
obtained above and the Hc1 data from [16], we find �
= 6.3 and �(0) = 280 nm which is in good agreement
with the value �p(0) = 260 nm obtained from two gap
BCS fit for d-band.
In contrast to the conventional BCS theory, H Tc2( )
dependence is linear over an extended temperature
range with no evidence of saturation down to 0.35 K.
Similar linear H Tc2( ) dependence has been observed
in MgB2 (Refs. 29 and 30) and BaNbOx (Ref. 31)
compounds. One can describe this behavior of upper
critical field using the two-gap approach. According
to Gurevich [27], the zero-temperature value of the
Hc2 is significantly enhanced in the two gap dirty
limit superconductor model:
H
k T
D D
g
c
B c
2
0
1 2
0
112 2
( )
.
exp
�
�
�
�
�
�
�
, (12)
as compared to the one-gap dirty limit approximation
Hc2 0( ) = �0kBTc/1.12�D. Here g is a rather compli-
cated function of the matrix of the BCS supercon-
ducting coupling constants.
In the limit of D2 << D1 we can simply approxi-
mate g � �ln (D2/D1)�. The large ratio of D2/D1
leads to the enhancement of Hc2 0( ) and results in the
upward curvature of the H Tc2( ) close to T = 0 [27].
According to our �(T) data (see above), we found
very different diffusivities for p- and d-bands:
Dp/Dd � 3. Thus we can speculate that the limiting
value of Hc2 0( ) is dominated by d-band with lower
diffusivity Dd = 17 cm2/s, while the derivative
dHc2/dT close to Tc is due to larger diffusivity band
(Dp = 56 cm2/s). Indeed, simple estimation of Dp =
= 4�0kB/�2
�dHc2/dT = 39 cm2/s from derivative
dHc2/dT = 0.027 T/K close to Tc gives almost the
same diffusivity relative to one estimated from �(T)
for the p-band. Thus we believe that the two gap theo-
retical model qualitatively explains the unconven-
tional linear H Tc2( ) dependence, which supports our
conclusion about the two gap nature of superconduc-
tivity in ZrB12.
The possibility of the multigap nature of the super-
conducting state was predicted for a multiband super-
conductor with large difference of the e–p interaction
at different Fermi-surface sheets (see Ref. 32 and
references therein). To date MgB2 has been the only
compound with the behavior consistent with the idea
of two distinct gaps with the same Tc. We believe that
our data can add ZrB12 as another unconventional
example of multi gap and multi-Tc superconductor.
Although observed two gap behavior of
�2(0)/�2(T) in ZrB12 is similar to that in high-Tc su-
perconductors, observation of two different Tc in these
bands is unconventional. This also relates to the linear
H Tc2( ) dependence in the wide temperature range up
to Tc. Striking two gap BCS behavior observed calls
certainly for a new study of low-T energy gap and
Hc2(T) of ZrB12 for understand the nature of super-
conductivity in this cluster compounds.
1112 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 8/9
V.A. Gasparov
H
c2
,
m
T
Fig. 8. Temperature variations of H Tc2( ) of ZrB12.
Symbols: the onset H Tc2( ) data determined from �(H)
(circles) and �(H) (squares). Dotted line is the BCS data
determined from the HW formula [33].
6. Conclusion
In summary, we have compared our experimental
results on ultrathin YBCO films with the extended
dynamic theory for BKT transition and found that the
vortex—antivortex pairs with short separation
lengths are present. The unbinding of the vortex pairs
were observable at high frequencies even though a
true BKT transition is absent in the samples. Our
results also indicate that part of the transition
broadening in ultrathin YBCO films can be related to
the dissociation of vortex–antivortex pairs.
While the temperature dependence of �(T) in thin
c-axis oriented thin film MgB2 samples is well des-
cribed by an isotropic s-type order parameter, we find
unconventional behavior of ZrB12 superfluid density
with pronounced shoulder at T/Tc equal to 0.65. The
H Tc2( ) dependences have been deduced from the
�(H) and �(H) data. Both techniques reveal an un-
conventional linear temperature dependence of
H Tc2( ). We conclude therefore that ZrB12 presents a
evidence of the unconventional two-gap superconduc-
tivity with different Tc in the different bands.
We are grateful to V.F. Gantmakher, A. Hebard,
M. Chan, R. Huguenin, C. Lobb, P. Martinoli,
D. van der Marel, C. Rogers, D.J. Scalapino, J.-M.
Triscone, T. Venkatesan, X.X. Xi for stimulating
discussions. This work was partially supported by
Russian Ministry of Industry, Science and Technology
(MSh-2169.2003.2) and Russian Academy of Sciences
Program: New Materials and Structures.
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