Muon spin rotation investigation of the pressure effect on the magnetic penetration depth in YBa2Cu3Ox

Abstract

The pressure dependence of the magnetic penetration depth in polycrystalline
samples of YBa2Cu3Ox with different oxygen concentrations x = 6.45, 6.6, 6.8,
and 6.98 was studied by muon spin rotation (muSR). The pressure dependence of
the superfluid density (p_s) as a function of the superconducting transition
temperature Tc is found to deviate from the usual Uemura line. The ratio
(dTc/dP)/(dp_s/dP) is factor of 2 smaller than that of the Uemura relation. In
underdoped samples, the zero temperature superconducting gap and the BCS ratio
both increase with increasing external hydrostatic pressure, implying an
increase of the coupling strength with pressure. The relation between the
pressure effect and the oxygen isotope effect on the magnetic penetration depth
is also discussed. In order to analyze reliably the muSR spectra of samples
with strong magnetic moments in a pressure cell, a special model was developed
and applied.

Full text

Page 1

arXiv:1107.1341v1 [cond-mat.supr-con] 7 Jul 2011
PREPRINT (July 8, 2011)
Muon spin rotation investigation of the pressure effect on the magnetic penetration
depth in YBa2Cu3Ox
A. Maisuradze,1,2, ∗A. Shengelaya,3A. Amato,1E. Pomjakushina,4and H. Keller2
1Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
2Physik-Institut der Universit¨ at Z¨ urich, Winterthurerstrasse 190, CH-8057 Z¨ urich, Switzerland
3Department of Physics, Tbilisi State University, Chavchavadze av. 3, GE-0128 Tbilisi, Georgia
4Laboratory for Developments and Methods, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
The pressure dependence of the magnetic penetration depth λ in polycrystalline samples of
YBa2Cu3Ox with different oxygen concentrations x = 6.45, 6.6, 6.8, and 6.98 was studied by muon
spin rotation (µSR). The pressure dependence of the superfluid density ρs ∝ 1/λ2as a function of
the superconducting transition temperature Tc is found to deviate from the usual Uemura line. The
ratio (∂Tc/∂P)/(∂ρs/∂P) is factor of ≃ 2 smaller than that of the Uemura relation. In underdoped
samples, the zero temperature superconducting gap ∆0 and the BCS ratio ∆0/kBTc both increase
with increasing external hydrostatic pressure, implying an increase of the coupling strength with
pressure. The relation between the pressure effect and the oxygen isotope effect on λ is also dis-
cussed. In order to analyze reliably the µSR spectra of samples with strong magnetic moments in
a pressure cell, a special model was developed and applied.
PACS numbers:
I. INTRODUCTION
The compound YBa2Cu3Ox was the first high tem-
perature superconductor1(HTS) with a superconducting
transition temperature Tcabove the boiling point of liq-
uid nitrogen, and is one of the most studied HTSs.2Its
superconducting properties are well characterized, even
though some of them are still being heavily discussed.
Detailed muon spin rotation (µSR) studies of the mag-
netic penetration depth λ and the superfluid density
ρs∝ 1/λ2were performed on poly- and single crystals of
YBa2Cu3Ox at ambient pressure.3–10However, the key
question concerning the pairing mechanism responsible
for high temperature superconductivity is still not re-
solved, and is subject of intense debates. Although it is
widely believed that magnetic fluctuations play a domi-
nant role in the pairing mechanism,11oxygen isotope ef-
fect (OIE) studies indicate that lattice degrees of freedom
are essential for the occurence of superdonductivity.12–20
By means of isotope substitution one can probe the in-
fluence of lattice degrees of freedom on superconductiv-
ity without changing the lattice parameters.21There are
no other easily accessible methods which allow to solely
modify the exchange integral J, in order to investigate
its influence on the superconducting state.22However,
the application of hydrostatic pressure changes the in-
teratomic distances in the lattice which in turn modifies
both the lattice dynamics23and the exchange coupling J
between the Cu spins in cuprates.24,25Therefore, a de-
tailed study of the pressure effect (PE) on the supercon-
ducting properties, e.g., the superfluid density ρs∝ 1/λ2,
the gap magnitude ∆0, and the BCS ratio ∆0/kBTc, may
provide important information for testing microscopic
theories of the high-temperature superconductivity.26,27
Up to now, the PE on the superconducting transi-
tion temperature Tcwas studied by resistivity and Hall
effect experiments.28–31Several phenomenological28,32,33
and microscopic models were proposed based on a
Hubbard34,35or a general BCS approach in order to ex-
plain the PE on Tc.36The role of nonadiabatic effects
is discussed in Ref. 37. These models suggest two ba-
sic sources for the PE on Tc: (i) A charge transfer from
the charge reservoir to the superconducting CuO2plane,
which was confirmed by Hall effect experiments,30,31and
(ii) an increase of Tcdue to a pressure dependent pairing
interaction.
The magnetic penetration depth λ is a fundamen-
tal parameter of a superconductor. It is a measure of
the superfluid density according to the relation 1/λ2∝
ns/m∗, where ns is the superconducting carrier dan-
sity and m∗is the corresponding effective mass.5From
the temperature or field dependence of λ one can de-
termine the symmetry of the superconducting gap, its
magnitude and the BCS ratio. The pressure dependence
of λ was previously studied in fine powdered grains of
YBa2Cu3Ox38and YBa2Cu4O839–41by means of mag-
netization experiments. The µSR technique is powerful
and direct method to determine λ in the bulk of a type-
II superconductor.42,43However, due to several technical
difficulties only a small amount of µSR studies of the
penetration depth under pressure were performed so far.
The main technical problems are: (i) The low fraction of
muons stopping in the sample inside the pressure cell and
(ii) the strong diamagnetism of a superconductor which
substantially influences the µSR response of the pressure
cell.
Here, we report on pressure dependent magnetic pen-
etration depth studies in polycrystalline samples of
YBa2Cu3Ox(x = 6.45, 6.6, 6.8, and 6.98) by means of
µSR. We found that the pressure-dependent superfluid
density ρs ∝ 1/λ2vs Tc does not follow the Uemura
relation.6The ratio αp= (∂Tc/∂P)/(∂ρs/∂P) is a fac-
tor ≃ 2 smaller than that of the Uemura relation, but is
quite close to that found in oxygen isotope effect (OIE)

Page 2

2
studies,16,17suggesting a strong influence of pressure on
the lattice degrees of freedom. Interestingly, a small pres-
sure dependence of the superluid density was also found
in the overdoped sample (x = 6.98). The superconduct-
ing gap ∆0 and the BCS ratio ∆0/kBTc both increase
upon increasing the hydrostatic pressure in the under-
doped samples, hence implying an increase of the cou-
pling strength with pressure. Finally, a method of data
analysis for tranverse-field µSR measurements of mag-
netic/diamagnetic samples loaded in a pressure cell is
presented and applied here. This method leads to a sub-
stantial reduction of systematic errors in the data analy-
sis.
The paper is organized as follows: In Sec. II we give
some experimental details. In Sec. III we describe the
method of µSR data analysis and present the experimen-
tal results, followed by a discussion in Sec. IV. The con-
clusions are given in Sec. V. In the Appendix we describe
the method used in this work in order to analyze µSR
spectra obtained for a magnetic/superconducting sample
loaded in a pressure cell.
II.EXPERIMENTAL DETAILS
High quality polycrystalline YBa2Cu3Oxsamples with
x = 6.98, 6.8, 6.6, and 6.45 were prepared from the
starting oxides and carbonate Y2O3, CuO and BaCO3
as described elsewhere.44Transverse field (TF) µSR ex-
periments were performed at the µE1 and πM3 beam
lines of the Paul Scherrer Institute (Villigen, Switzer-
land). The samples were cooled in TF down to 3K, and
µSR spectra were taken with increasing temperature in
applied fields Bapp = 0.1 and 0.5 T. Typical statistics
for a µSR spectrum were 5 − 6 × 106positron events in
the forward and the backward histograms.42,43A CuBe
piston-cylinder pressure cell was used with Dafne oil as
a pressure transmitting medium. The maximum pres-
sure achieved was 1.4 GPa at 3 K. The pressure was
measured by tracking the superconducting transition of
a very small indium plate used as a manometer (cali-
bration constant for In: ∂Tc/∂P = −0.364 K/GPa). In
order to avoid charge transfer effects due to chain reorder-
ing in pressurized YBa2Cu3Ox, the samples were cooled
down below 100 K for the µSR measurements within less
than 1 hour after application of the pressure. This time
is much shorter than the time constant τ = 27.7 h (at
room temperature) for the pressure activated chain re-
ordering process.45Below 100 K τ is much longer than
the typical measurement time of a sample (< 24 h).45
High energy muons (pµ≃ 100 MeV/c) were implanted
in the sample. Forward and backward positron detectors
with respect to the initial muon polarization were used
for the measurements of the µSR asymmetry time spec-
trum A(t) (see Fig. 8).42Cylindrically pressed samples
were loaded into the cylindrical CuBe pressure cell. The
sample dimensions (diameter 5 mm, height 15 mm) were
chosen to maximize the filling factor of the pressure cell.
012345
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
t (µs)
4.5 K
95 K


A(t)
FIG. 1:
YBa2Cu3O6.98 measured at T = 4.5 K and 95 K in an ap-
plied field Bapp = 0.1 T (empty and full circles, respectively).
The fast relaxation of the µSR signal (empty circles) is due to
the formation of a vortex lattice in the superconducting state.
The solid lines are fits of the data to Eq. (1). For a better
visualization the spectra and the fits are shown in a rotating
reference frame of 0.08 T.
(Color online) µSR asymmetry signal A(t) of
The fraction of the muons stopping in the sample was
approximately 40%.
III. RESULTS AND ANALYSIS DETAILS
For type-II superconductors in the vortex state in an
applied field of Bapp ≪ Bc2 (Bc2 is the upper criti-
cal field) the square root of the second moment of the
muon depolarization rate σ is inversely proportional to
the square of the magnetic penetration depth: σ ∝ 1/λ2
(Refs. 4,46,47) and therefore directly related to the su-
perfluid density: ρs ∝ 1/λ2∝ σ. For a polycrystalline
sample of a highly anisotropic and uniaxial superconduc-
tor the dominant contribution to the muon depolariza-
tion originates from the in-plane magnetic penetration
depth λab = λeff/1.31, where λeff is an effective (aver-
aged) magnetic penetration depth.48,49
As was pointed above a substantial fraction of the
µSR asymmetry signal originates from muons stopping in
the CuBe material surrounding the sample. The sample
in the superconducting state induces an inhomogeneous
field in its vicinity (see Appendix). This leads to an ad-
ditional depolarization of the µSR signal arising from the
muons stopping in the pressure cell. Therefore, the µSR
asymmetry time spectra are characterized by two compo-

Page 3

3
nents and may be described by the following expression:
A(t) =A1· exp
?
−1
2(σ2+ σ2
n)t2
?
cos(γµB1t + φ)+ (1)
A2·
?
P(B′)cos(γµB′t + φ)dB′.
Here, A1and A2are the initial asymmetries of the two
components of the µSR signal (A1: sample, A2: pres-
sure cell), γµ is the gyromagnetic ratio of the muon
(γµ= 2π × 135.5342 MHz/T), and φ is the initial phase
of the muon spin polarization. B1is the field in the cen-
ter of the sample (or approximately the mean field in the
sample). The parameter σ denotes the muon depolariza-
tion in the sample due to the field distribution created by
the vortex lattice, while σn= 0.10(2) µs−1is a temper-
ature, doping, and pressure independent depolarization
rate due to the nuclear moments present in the sample.
The total asymmetry is A1+ A2 = 0.275 at 0.1 T and
0.265 at 0.5 T with A1/(A1+ A2) ≃ 0.4 (≃ 40% of the
muon ensemble are stopping inside the sample). P(B′)
represents the magnetic field distribution probed by the
muons stopping in the pressure cell as described in detail
in the Appendix.
Figure 1 exhibits µSR asymmetry time spectra of
YBa2Cu3O6.98above (T = 95 K) and below (T = 4.5 K)
the superconducting transition temperature Tc= 89.6 K
obtained in an applied field of 0.1 T. For a better visu-
alization the spectra and the fits are shown in a rotating
reference frame of 0.08 T. Above Tc only a weak depo-
larization of the muon spin polarization is visible,5while
below Tcthe strong relaxation of the µSR signal reflects
the formation of the vortex lattice in the superconduct-
ing state.3,5,7,43,46Figures 2a, b, and c show the Fourier
transforms (FT) of the µSR time spectra shown in Fig.
1. In Fig. 2d the FT spectra of YBa2Cu3O6.6 below
and above Tc = 60 K are also shown. The narrow sig-
nal around Bapp= 0.1 T in Fig. 2b originates from the
pressure cell, while the broad signal with a first moment
significantly lower than Bapp arises from the supercon-
ducting sample. It can be seen that the signal of the
pressure cell is also modified below Tc due to the dia-
magnetic response of the superconducting sample. The
solid lines are the FTs of the fits to the data using Eq. (1)
(see also Appendix). The good agreement between the
fits and the data demonstrates that the model used here
describes the data rather well.
The whole temperature dependence of the µSR asym-
metry time spectra was fitted globally with the common
parameters Bapp, A1, A2, and σn. Solely the parameters
B1and σ were considered as temperature dependent free
parameters. As shown in the Appendix the field in the
sample is macroscopically inhomogeneous due to the in-
homogeneity of demagnetization effects. B1 is the field
at the point x = y = z = 0 (i.e., the center of the sam-
ple). In addition, the parameters describing the muon
stopping distribution x0,iand σiwere kept the same for
each temperature scan (see Eqs. (A.3) and (A.4) in the
Appendix).
859095
B (mT)

100 105110
0
10
20
30
40
50
8090100110
0
1
2
3
4
5
6
7
96 98
B (mT)
100102
0
10
20
30
40
50
949698100 102 104
B (mT)
0
10
20
30
40
50
60
70
80
x=6.98


FT amplitude (a.u.)
95 K
4.5 K


FT amplitude (a.u.)
B (mT)
95 K
4.5 K
x=6.98
x=6.98
(b)
(d)
(c)

FT amplitude (a.u.)
95 K
4.5 K
(a)
x=6.6
64 K
4.5 K


FT amplitude (a.u.)
FIG. 2: (Color online) Fourier transform (FT) amplitude as
a function of field for the spectra shown in Fig. 1 [panels (a),
(b), and (c)]. Panel (b) is the expanded [along y−axis] view of
panel (a) to show the signal from the sample. Panel (c) is the
expanded [along x−axis] view of panel (a) to show the signal
of the pressure cell. Panel (d) shows the FT of the sample
with x = 6.6 below and above Tc = 60 K. The solid lines are
the FTs of the fitted curves shown in Fig. 1. The FT spectra
are slightly broadened due to a FT apodization of 4 µs−1.
The temperature dependence of the depolarization
rates σ for x = 6.98, 6.8, 6.6, and 6.45 at Bapp = 0.1
and 0.5 T obtained with Eq. (1) are shown in Figs. 3
and 4, respectively. The black empty points correspond
to the data measured at zero pressure, while the full red
points correspond to the data measured at 1.1 GPa (for
x = 6.45, 6.6, and 6.8) and 1.4 GPa (for x = 6.98).
The values of Tc and σ(0) are in good agreement with
previous results.5,6,8,9It is known that the order param-
eter in YBa2Cu3O6.98 has predominantly the form of
∆ = ∆0(ˆp2
y) [ˆ pi= pi/|? p| denotes component of the
unit momentum vector in the reciprocal space along the
i-th axis].11,50,51This implies a linear temperature de-
pendence of the superfluid density ρs down to very low
temperatures due to quasiparticle excitations at the gap-
less line nodes in the ˆ px= ±|ˆ py| directions on the Fermi
surface.43However, in Fig. 3 we clearly see that σ(T)
tends to saturate at low temperatures for YBa2Cu3O6.98
x−ˆp2

Page 4

4
TABLE I: Summary of the results obtained from the temper-
ature dependence of σ at 0.1 and 0.5 T in YBa2Cu3Ox using
Eq. (2). Note that for the sample with x = 6.45 a precise
analysis of σ was not possible due to the occurrence of spin-
glass magnetism below T ≃15 K. Hence, the errors of these
values of σ(0) are rather large.
xP
(GPa)
Bapp
(T)
Tc
(K)
σ(0)
(µs−1) kBTc
∆0
Γu
(K)
6.98
6.98
6.98
6.98
6.8
6.8
6.8
6.8
6.6
6.6
6.6
6.6
6.45
6.45
6.45
6.45
0
1.4
0
1.4
0
1.1
0
1.1
0
1.1
0
1.1
0
1.1
0
1.1
0.1
0.1
0.5
0.5
0.1
0.1
0.5
0.5
0.1
0.1
0.5
0.5
0.1
0.1
0.5
0.5
89.6(4) 4.76(7) 3.87(12) 15(5)
89.5(4) 4.97(7) 3.60(7)
90.0(2) 4.56(7) 2.95(10) 15(5)
89.9(1) 4.72(7) 2.82(7)
77.1(3) 2.07(5) 3.02(12) 0
83.2(5) 2.33(5) 3.48(15) 0
76.4(3) 1.91(5) 2.59(9)
82.3(5) 2.19(5) 2.80(8)
58.9(6) 1.79(5) 3.02(12) 0
62.6(5) 1.95(5) 3.27(12) 0
57.3(6) 1.58(5) 2.92(12) 0
62.3(6) 1.77(5) 2.89(11) 0
45.4(3) 1.17(7) 3.0(5)
49.5(5) 1.22(7) 3.0(5)
45.1(2) 1.00(7) 2.5(2)
48.7(2) 1.14(7) 2.5(2)
15(5)
15(5)
0
0
0
0
0
0
for both applied magnetic fields. Such a behavior was of-
ten observed in µSR studies of polycrystalline samples3,7
and was explained as originating from a strong scatter-
ing of electrons on impurities.52–56
strongly influence the temperature dependence of ρs, but
it has a minor effect on the superconducting transition
temperature Tc. In previous theoretical works it was sug-
gested that such a behavior indicates scattering in the
unitary limit.54,55Thus, the temperature dependence of
the superfluid density ρswas analyzed with the“dirty d-
wave model” of the BCS theory in the unitary limit of
carrier scattering as described in Ref. 52:
This scattering can
ρs∝
1
λ2
ab
=4πe2Nf(vab
f)2
c2
?2π
0
dφ
2π
∞
?
n=0
|∆(φ)|2
n+ |∆(φ)|2)3/2.
(˜ ǫ2
(2)
Here, λab is the in-plane magnetic penetration depth,
∆(φ) = ∆0cos(2φ) · g(t) (t = T/Tc) is the 2D-gap-
function, and ˜ ǫn = Z(ǫn)ǫn are impurity renormalized
Matsubara frequencies: ǫn= (2n+1)πT. ∆0is the max-
imum of the gap function on the Fermi surface and g(t)
represents the temperature dependence of the gap with
g(0) = 1. The parameters Nf and vf are the density of
states at the Fermi level and the Fermi velocity, respec-
tively. The constant e and c represent the electron charge
and the speed of light. The coefficients Z(ǫn) are:52
Z(ǫn) = 1 + Γu
Dn(ǫn)Z(ǫn)
cot2(δ0) + [Dn(ǫn)ǫnZ(ǫn)]2, (3)
0
1
2
3
4
5
0
1
2
3
0 10 20 3040 50 60 708090
-4
-3
-2
-1
0
Bapp = 0.1 T


(a)
σ (µs
-1)

σ (µs
-1)
(b)


∆B (mT)
T (K)
FIG. 3: (Color online) (a) Temperature dependence of σ of
YBa2Cu3Ox measured at Bapp = 0.1 T at zero and applied
hydrostatic pressures for x = 6.45 (♦: P = 0 GPa; ?: P = 1.1
GPa), x = 6.6 (▽: P = 0 GPa; ?: P = 1.1 GPa), x = 6.8
(△: P = 0 GPa; ?: P = 1.1 GPa), and x = 6.98 (◦: P =
0 GPa; •: P = 1.4 GPa).The data were analyzed with
Eq. (1). The solid curves are fits to the data with Eq. (2).
(b) Diamagnetic shift of the field ∆B = B1 − Bapp in the
corresponding samples. B1 is the mean field in the centrer of
the sample (see text and Appendix).
with
Dn(ǫn) =
?
1
?Z(ǫn)2ǫ2
n+ |∆(pf)|2
?
pf
,(4)
and δ0= π/2 in the unitary limit. The angular brackets
?...?pfdenote averaging over the Fermi surface. In order
to find Z(ǫn) and g(t), Eq. (3) is solved together with
the following equation:52
1
2πT
?
ln
?T
Tc
?
+ ψ
?1
2+
Γu
2πT
?
− ψ
?1
2+
Γu
2πTc
??
=
(5)
∞
?
Here, ψ(x) is the digamma function. Note that the im-
purity scattering influences mainly ǫnwhile the temper-
n=0
??
|e(pf)|2
n+ |∆(pf)|2)3/2
(Z(ǫn)2ǫ2
?
pf
−
1
ǫn+ Γu
?
.

Page 5

5
020406080
0
1
2
3
4
5
0
1
2
3

σ (µs

T (K)
Bapp = 0.5 T
-1)
σ (µs
-1)
FIG. 4:
YBa2Cu3Ox measured at Bapp = 0.5 T. The meaning of the
symbols and the solid lines are the same as in Fig. 3(a).
(Color online) Temperature dependence of σ of
ature dependence of the gap g(t) changes only slightly
for a reasonable scattering rate Γu. In the clean limit
(i.e., Γu= 0 and Z(ǫn) = 1, ∀n) the normalized func-
tion g(t) is very close to the analytical approximations
derived from BCS theory.57
Fits of Eq. (2) to σ(T) ∝ 1/λab(T)2measured at var-
ious hydrostatic pressures are presented in Figs. 3 and
4. The corresponding values for ∆0, Tc, σ0, and Γuob-
tained from the analysis are summarized in Table I. The
data for zero and applied pressure and the same doping
x were analyzed simultaneously with the common pa-
rameter Γuwhich characterizes the relaxation rate of the
Cooper pairs on impurities. As shown in Table I the data
for the underdoped samples (x = 6.45, 6.6, and 6.8) are
well described by the clean limit d-wave model, while for
the overdoped sample (x = 6.98) Γu= 15(5) K. Here, we
note that all the studied samples originate from the same
batch and have an identical thermal history, except of the
last process of the oxygen reduction. Therefore, we can-
not explain why only the sample with x = 6.98 exhibits
a saturation of σ in the low temperature limit and why
it has such a high scattering rate Γu = 15(5) K. Con-
sequently, we cannot exclude the possibility of a modifi-
cation of the order parameter in overdoped YBa2Cu3Ox
where the pseudogap state gradually vanishes. Such a be-
havior was also observed previously in optimally doped or
overdoped polycrystalline samples of YBa2Cu3Ox.3,5,7,8
However, in single crystal YBa2Cu3Oxclose to optimum
doping a linear temperature dependence of 1/λ2at low
temperatures was also reported.10,43For the sample with
x = 6.45 only the data above 15 K were analyzed, since
below 15 K the occurrence of field induced spin-glass
magnetic order hinders a precise determination of σ.
0.00.51.01.52.02.54.55.0
0
10
20
30
40
50
60
70
80
90
x=6.98
x=6.8
x=6.6


Tc (K)
σ(0) (µs
-1)
0 GPa, 0.1 T
1.1 GPa, 0.1 T
0 GPa, 0.5 T
1.1 GPa, 0.5 T
x=6.45
FIG. 5: (Color online) Tc vs. σ(0) (Uemura plot) at zero and
applied pressure for YBa2Cu3Ox with x = 6.45, 6.6, 6.8, and
6.98. The solid line is the Uemura line while the dashed line
is a guide to the eye. The dotted lines represent the pressure
effect on Tc and σ(0).
0 20 4060 80
0
50
100
150
200
250
300
350
x=6.98
x=6.8
x=6.6
0 GPa, 0.1 T
1.1 GPa, 0.1 T
0 GPa, 0.5 T
1.1 GPa, 0.5 T

∆0/kB (K)

TC (K)
∆0/kBTC = 3.0
FIG. 6:
YBa2Cu3Ox with x = 6.6, 6.8, and 6.98. The solid line corre-
sponds to ∆0/kBTc = 3 (weak-coupling BSC superconductor:
∆0/kBTc = 1.76). Both ∆0 and ∆0/kBTc increase with in-
creasing pressure.
(Color online) Relation between ∆0 and Tc for
IV.DISCUSSION
The main subject of the present study is the pres-
sure effect on the superconducting gap ∆0 and the su-
perfluid density ρs ∝ σ. The Uemura relation6, imply-
ing the linear relation between Tcand ρsfor underdoped
cuprate superconductors, was established soon after the

Page 6

6
1.52.02.5
100
150
200
250
300
σ(0) (µs
-1)
0 GPa, 0.1 T
1.1 GPa, 0.1 T
0 GPa, 0.5 T
1.1 GPa, 0.5 T


∆0/kB (K)
FIG. 7: (Color online) The gap ∆0 as a function of σ(0) for
the underdoped samples of YBa2Cu3Oxwith x = 6.6 and 6.8.
The linear relation between σ(0) and ∆0 is better fulfilled
under hydrostatic pressure than the Uemura relation Tc vs
σ(0) and ∆0/kBTc vs. Tc (see Figs. 5 and 6). The line is a
guide to the eye.
TABLE II: Values of αp = (∂Tc/∂P)/(∂σ/∂P) for the un-
derdoped YBa2Cu3Ox samples investigated in this work (x =
6.45, 6.6, and 6.8).
xαp (K/µs−1)
0.1 T
–
23(11)
23(7)
αp (K/µs−1)
0.5 T
25(18)
26(6)
21(6)
6.45
6.6
6.8
discovery of HTS1and is one of the important crite-
ria which a microscopic theory of HTS should explain.
The Uemura relation for the data summarized in Ta-
ble I is shown in Fig. 5. As indicated by the dotted
lines the slope αp = (∂Tc/∂P)/(∂σ/∂P) is systemati-
cally smaller than that suggested by the Uemura line
with αU = ∂Tc/∂σ ≃ 40 K/µs−1.
for the underdoped samples investiganted in this work
are summarized in Table II.
netism below ∼ 15 K the error of σ(0) for the sam-
ple with x = 6.45 is rather large. The weighted mean
value of αp ≃ 23(4) K/µs−1is a factor of ≃ 2 smaller
than αU ≃ 40 (Kµs−1). Such a substantial deviation
from the Uemura line (with a lower value of αp) was
also observed by pressure experiments in YBa2Cu4O8us-
ing a magnetization technique.39This is in contrast to
pressure effect results obtained for the organic supercon-
ductor κ-(BEDT-TTF)2Cu(NCS)2which follow the Ue-
mura relation.58Interestingly, a slope with a factor two
The values of αp
Note that due to mag-
smaller than that of the Uemura line was also found by
OIE studies of cuprate superconductors.16This suggests
a strong influence of pressure on the lattice dynamics.
It is known that the pressure dependence of the super-
conducting transition temperature is determined by two
mechanisms: (i) The pressure induced charge transfer to
CuO2 planes ∆nh and (ii) the pairing interaction Veff
which depends on pressure.28,32–37,59
For the underdoped samples the former mechanism
dominates (85-90%) the pressure effect on Tc.28,32,36
Therefore, one can separate the pressure effect on σ
also in two components ∆σ = ∆σch+ ∆σV. The first
term ∆σch≃ (1/αU)(∂Tc/∂P)P follows the Uemura line
and is mainly due to the charge transfer to the plane.
The second term ∆σV ≃ (1/αp− 1/αU)(∂Tc/∂P)P de-
scribes the increase of the superfluid density solely due
to a change of the pairing interaction.
of the superfluid density is equivalent to a decrease of
the effective mass of the superconducting carriers, since
∆σV/σ = ∆λ−2
pressure-induced change of the effective carrier mass can
be written as:
This increase
V/λ−2= −∆m∗
V/m∗.39Therefore, the
dln(m∗
V)/dP = −dln(λ−2
≃ (αU/αp− 1)(∂Tc/∂P)/Tc
≃ 3/TcGPa−1.
V)/dP ≡ −(∆σV/σ)/∆P (6)
Here, Tcand σ are taken at zero pressure and the value
of (∂Tc/∂P) ≃ 4 K/GPa was used. This value is prac-
tically doping independent in underdoped YBa2Cu3Ox
for 6.45 ≤ x ≤ 6.8.32The quantity ∆λ−2
change of the superfluid density solely due to a modi-
fication of the pairing interaction Veff by pressure. It
is remarkable to observe the qualitative agreement be-
tween dln(λ−2
V)/dP and that found in OIE studies for
dlnλ/dlnMOat different carrier dopings (dlnMOis the
relative change of oxygen mass).16Indeed, Eq. (6) pre-
dicts that the pressure effect on m∗V strongly increases
with decreasing Tc.
Another interesting result is the quite small pressure
dependence of σ in the overdoped sample with x = 6.98,
which is approximately a factor of ≃ 2 weaker than that
reported from magnetization measurements.38In Fig. 6
the gap magnitudes ∆0for the samples with x = 6.6, 6.8,
and 6.98 are plotted as a function of Tc. For the under-
doped samples (x = 6.6 and 6.8) both ∆0and ∆0/kBTc
increase upon increasing applied pressure. This suggests
an increase of the coupling strength with increasing pres-
sure. This behavior is different from that found for the
OIE on ∆0, where a proportionality between ∆0and Tc
was found, implying a constant ratio of ∆0/kBTc.18In
the overdoped sample (x = 6.98), Eq. (2) suggests a
small reduction of the coupling strength with increasing
pressure. However, as was mentioned above, the absence
of a linear temperature dependence of σ at low temper-
atures for the sample with x = 6.98 might also indicate
that the superconducting order parameter is not of purely
d-wave character.50,51This, on the other hand, may in-
fluence the result for ∆0and its pressure dependence.
V
describes the

Page 7

7
In Fig. 7 for the underdoped samples (x = 6.6 and 6.8)
∆0 is plotted vs. σ(0), showing a linear correlation be-
tween the two quantities. Note, that this correlation does
not change with the application of hydrostatic pressure.
This is in contrast to what is observed for the Uemura
relation Tcvs. σ(0) and ∆0/kBTcvs. Tc(see Figs. 5 and
6).
V. CONCLUSIONS
The pressure dependence of the magnetic penetration
depth λ of polycrystalline YBa2Cu3Ox (x = 6.45, 6.6,
6.8, and 6.98) was studied by µSR. The pressure de-
pendence of the superfluid density ρs ∝ σ ∝ 1/λ2as
a function of the superconducting transition Tctemper-
ature does not follow the well-known Uemura relation.6
The ratio αp= (∂Tc/∂P)/(∂σ/∂P) ≃ 23(4) K/µs−1is a
factor of ≃ 2 smaller than that of the Uemura relation
observed for underdoped samples. However, the value of
αpis quite close to that found in OIE studies,16indicat-
ing a strong influence of pressure on the lattice degrees
of freedom. We conclude that the contribution of car-
rier doping to the pressure dependence of λ is similar to
the OIE on λ. A weak pressure dependence of the su-
perfluid density ρs was found in the overdoped sample
(x = 6.98). The superconducting gap ∆0 and the BCS
ratio ∆0/kBTcboth increase with increasing applied hy-
drostatic pressure in the underdoped samples, implying
an increase of the coupling strength with pressure. Al-
though the Uemura relation does not hold and the BCS
ratio is increasing with pressure in underdoped samples,
the relation between ∆0 and the µSR relaxation rate σ
is invariant under pressure. Finally, a model to analyze
TF µSR spectra of magnetic/diamagnetic samples loaded
into a pressure cell was developed and successfully used
in this paper (see Appendix), resulting in a substantial
reduction of the systematic errors in the data analysis.
Acknowledgements
We are grateful to M. Elender for his technical support
during the experiment and D. Andreica for providing the
pressure cells. This work was performed at the Swiss
Muon Source (SµS), Paul Scherrer Institut (PSI, Switzer-
land). We acknowledge support by the Swiss National
Science Foundation, the NCCR Materials with Novel
Electronic Properties (MaNEP), the SCOPES grant No.
IZ73Z0-128242, and the Georgian National Science Foun-
dation grant GNSF/ST08/4-416.
Appendix: Field distribution in a pressure cell
loaded with a sample with a non-zero magnetization
Samples with a strong magnetization placed in a pres-
sure cell with an applied magnetic field induce a mag-
FIG. 8: (Color online) (a) Schematic sketch of the µSR pres-
sure instrument GPD at the Paul Scherrer Institute: Cylin-
drical sample (blue); pressure cell (yellow); muon stopping
distribution (red ellipse), and forward and backward positron
detectors (black). (b) Illustration of the surface current on
a slice of a homogeneously magnetized cylindrical sample.
The magnetic field induced by this slice is equivalent to the
magnetic field of the surface currents. (c) Cross section of
the cylindrical sample and the surface current distribution in
xz−plane. (d) Magnetic field map of the surface currents as
illustrated in panels (b) and (c).
netic field in the space around the sample. Typical ex-
amples of such samples are superconductors (strong dia-
magnets), superparamagnets, and ferro- or ferrimagnets.
Thus, muons stopping in a pressure cell (PC) contain-
ing the sample will undergo precession in the vector sum
of the applied field and the field induced by the sample.
This spatially inhomogeneous field leads to an additional
depolarization of the muon spin polarization which de-
pends on the applied field and the induced field together
with the spatial stopping distribution of the muons.
Consider the most simplest case of a sample with the
shape of a round cylinder of hight H and radius R placed
into a cylindrical pressure cell with the same internal ra-
dius R (Fig. 8a). Typical pressure cell radii used for µSR
studies are R = 2.5 - 4 mm. In standard transverse field
(TF) µSR experiments the pressure cell is placed with the
cylinder axis oriented vertically while the magnetic field
is applied perpendicular to the cylinder axis of the pres-
sure cell and the muon beam direction (see Fig. 8). Let
us introduce a cartesian coordinate system with the y-
axis along the sample cylinder axis, and the z-axis along
the direction of the applied field. Thus, the x-axis is
along the initial muon beam direction which is perpen-
dicular to the forward and backward detector planes (see
Fig. 8). The origin of the coordinate system is located in
the center of the sample.
In an applied magnetic field H (along the z-direction)

Page 8

8
the sample has a magnetization M. This magnetization
is the source of an induced field H′(r). Let us assume that
H′is much weaker that the applied field H which is the
case for superconductors in a magnetic field of µ0H ≫
Bc1(Bc1is the first critical field). Thus, one can neglect
the spatial variation of the magnetization due to the ad-
ditional induced field: M = M(H + H′(r)) ≃ M(H).
Typically half (or even more) of all the muons are stop-
ping in the PC outside of the sample volume. The muons
stopping in the macroscopically inhomogeneous field of
the PC contribute to an additional relaxation of the µSR
signal. In order to describe the total µSR time spectrum
(sample and PC) one has to model the field distribu-
tion H′(r). For an applied field H ≫ H′(r) one can ne-
glect the influence of H′
x(r) and H′
spectrum, since only the z-component H′
significantly to the muon depolarization. The induced
magnetic field H′(r) created by a cylindrical sample can
be calculated as follows:60
y(r) on the µSR time
z(r) contributes
H′(r) =
1
4π
?
V
?3(M · (r − r′))(r − r′)
|r − r′|5
−
M
|r − r′|3
?
(A.1)
dr′
Here, the integral is taken over the sample volume V.
For a sample with a constant magnetization the three-
dimensional integral can be replaced by surface integrals.
Let us take one slice of width dz out of the sample cylin-
der and divide it into many small squares dA = dxdy (see
Fig. 8b). The field created by the elementary cell of vol-
ume dV = dxdydz with magnetization M is equivalent
to the field created by the current Iz= Mdz circulating
within this square slice as shown in Fig. 8a. It is obvious
that integration of this field over the whole slice volume
will leave only a current Izflowing over the perimeter of
the slice. The total field of the cylinder is the integral of
the fields created by these slices with constant current Iz
(see Figs. 8b and c).
According to the law of Bio-Savart the field in a point
r created by the elementary currents Idℓ at the surface
of the cylinder (with coordinates rs) is:60
H′(r) =
?
S
I
4π
[dℓs× (r − rs)]
|r − rs|3
. (A.2)
The integration is taken over the surface S of the sam-
ple and dℓsis the elementary length on the surface with
its direction along the current (the subscript s denotes
quantities related to the surfaces of the sample.
The spacial magnetic field distribution around the
ferro/paramagnetic sample calculated with Eq. (A.2) in
x-z plane is shown in Fig. 8d.
pressure cell is the vector sum of this field and the homo-
geneous external field. It is obvious from the figure that
the field along the z-axis is higher(lower) than the exter-
nal field in a ferromagnet(diamagnet). Along the x-axis,
on the other hand, the field is lower(higher) than the ex-
ternal field in a ferromagnet(diamagnet). The maximal
(minimal) induced field in the PC are just on the border
The total field in the
FIG. 9: (Color online) (a) Contour plot of the field distribu-
tion H′
R = 2.5 mm and H = 15 mm (the geometry of the sample
used in the experiment). (b) Contour plot of the muon stop-
ping distribution in the yz−plane. The gray area on the top
of the sample corresponds to the empty pressure cell space
where no muons stop (this space is filled with a low-density
pressure transmission medium).
the sample space. (c) Magnetic field profile of H′
y−axis and (d) magnetic field profile of H′
z(y,z) in the yz−plane for a cylindrical sample with
The dashed line indicates
zalong the
zalong the z−axis.
of the sample/pressure cell along z (x) direction. Note
that demagnetization effects are naturally accounted for
by using Eq. (A.2). Since the sample is not elliptical
this leads to field inhomogenieties within the volume of
the sample (see Fig. 9). As an example Fig. 9 shows the
magnetic field distribution in the yz−plane for a cylin-
drical sample with H = 15 mm and radius R = 2.5
mm, together with fields along z− and y−axes calcu-
lated with Eq. (A.2). Due to demagnetization effects the
magnetic field profiles within the sample has peaks at
the top and bottom edges of the sample where the de-
magnetizing fields are minimal (Fig. 9c). On the other
hand, the field profile within the sample close to the cen-
ter is quite homogeneous, since a cylinder with infinite
hight H is equivalent to an ellipsoid in which the field is

Page 9

9
homogeneous.
In order to calculate the probability field distribution
of a sample in a PC with a substantial first moment a
model for the muon stopping distribution is required.
This distribution may be well approximated by a three-
dimensional Gaussian:61
Ps(x1,x2,x3) =
A
(2π)3/2
3?
i=1
1
σiexp
?
−(xi− x0,i)2
2σ2
i
?
(A.3)
,
where the subscripts i = 1,2,3 correspond to x, y, or
z, respectively. The quantities x0,i determine the mean
value of the muon stopping distribution, σi are corre-
sponding standard deviations, and A is the normalization
factor. The quantities x0,1, x0,2, and x0,3can be deter-
mined quite accurately before starting the experiment
by tuning the momentum of the muon beam and vertical
positioning of the sample. For a sample with nearly the
same density as the pressure cell x0,1≃ x0,2≃ x0,3≃ 0.
Simulations of the stopping distribution with the SRIM
software61yield σ1 = 0.875 mm for copper (the basic
component of the CuBe pressure cell) and the minimal
ratio of σ3/σ1= 3.36. A maximal ratio of σ3/σ1≃ 4 is
estimated for the muon beam collimated by a 4×10 mm
collimator (this uncertainty is related with the degree of
muon beam focusing). The parameter σ2 is in fact the
standard deviation of the function representing the con-
volution of a Gaussian with σ = σ3 over the collimator
profile function along the y-axis. These parameters de-
fine the fraction of muons stopping in the PC and the
sample for a given sample geometry. For a known Ps(r)
one can calculate the magnetic field probability distribu-
tion P(B) in the pressure cell by solving the integral:
P(B) =
?
x2+z2>R2
Ps(r)δ(B − µ0[H + H′
z(r)])dr. (A.4)
Here, δ(x) is the delta function. The integration is taken
over the volume of the pressure cell. Note that this is not
simply the probability field distribution in the pressure
cell, but it is weighted with the muon stopping probabil-
ity distribution Ps(x,y,z). Fits of P(B) to the experi-
mental µSR data are shown in Fig 2. The function P(B)
describes the experimentally measured µSR signal rather
well.
∗Electronic address: alexander.maisuradze@psi.ch
1J.G. Bednorz and K.A. M¨ uller, Z. Phys. B 64, 189 (1986).
2C.W. Chu, P.H. Hor, R.L. Meng, L. Gao, Z.J. Huang, and
Y.Q. Wang, Phys. Rev. Lett. 58, 405 (1987).
3D. R. Harshman, G. Aeppli, E.J. Ansaldo, B. Batlogg,
J.H. Brewer, J. F. Carolan, R.J. Cava, M. Celio, A.C.D.
Chaklader, W.N. Hardy, S.R. Kreitzman, G.M. Luke, D.R.
Noakes, and M. Senba, Phys. Rev. B 36, 2386 (1987).
4J.H. Brewer, et al. Phys. Rev. Lett. 60, 1073 (1988).
5Y.J. Uemura, et al. Phys. Rev. B 38, 909 (1988).
6Y.J. Uemura, et al. Phys. Rev. Lett. 62, 2317 (1989).
7B. P¨ umpin, H. Keller, W. K¨ undig, W. Odermatt, I.M.
Savi´ c, J.W. Schneider, H. Simmler, P. Zimmermann, E.
Kaldis, S. Rusiecki, Y. Maeno, and C. Rossel, Phys. Rev.
B 42, 8019 (1990).
8P. Zimmermann, H. Keller, S.L. Lee, I.M. Savi´ c, M. War-
den, D. Zech, R. Cubitt, E.M. Forgan, E. Kaldis, J.
Karpinski, and C. Kr¨ uger, Phys. Rev. B 52, 541 (1995).
9J.L. Tallon, C. Bernhard, U. Binninger, A. Hofer, G.V.M.
Williams, E.J. Ansaldo, J.I. Budnick, and Ch. Nieder-
mayer, Phys. Rev. Lett. 74, 1008 (1995).
10T.M. Riseman, J.H. Brewer, K.H. Chow, W.N. Hardy,
R.F. Kiefl, S.R. Kreitzman, R. Liang, W.A. MacFarlane,
P. Mendels, G.D. Morris, J. Rammer, J.W. Schneider, C.
Niedermayer, and S. L. Lee, Phys. Rev. B 52, 10569 (1995).
11Patrick A. Lee, Naoto Nagaosa, and Xiao-Gang Wen, Rev.
Mod. Phys. 78, 17 (2006).
12J.P. Franck, in Physical Properties of High Temperature
Superconductors IV, ed. D.M. Ginsberg (World Scientific,
Singapore) p. 189.
13G.M. Zhao, K. Conder, H. Keller, and K.A. M¨ uller, J.
Phys.: Condens. Matter 10, 9055 (1998).
14D. Rubio Temprano, J. Mesot, S. Janssen, K. Conder, A.
Furrer, H. Mutka, and K. A. M¨ uller, Phys. Rev. Lett. 84,
1990, (2000).
15G.M. Zhao, H. Keller, and K. Conder, J. Phys.: Condens.
Matter 13, R569 (2001).
16R. Khasanov, A. Shengelaya, K. Conder, E. Morenzoni,
I.M. Savi´ c, and H. Keller, J. Phys.: Condens. Matter 15,
L17 (2003); J. Phys. Condens. Matter 16, S4439 (2004).
17H. Keller, in, Superconductivity in Complex systems, eds.
K. A. M¨ uller and A. Bussmann-Holder, Structure and
Bonding 114, (Springer-Verlag, Berlin, Heidelberg, New
York 2005) pp. 114-143.
18R. Khasanov, S. Str¨ assle, K. Conder, E. Pomjakushina, A.
Bussmann-Holder, and H. Keller, Phys. Rev. B 77, 104530
(2008).
19R. Khasanov, A. Shengelaya, D. Di Castro, E. Morenzoni,
A. Maisuradze, I. M. Savi´ c, K. Conder, E. Pomjakushina,
A. Bussmann-Holder, and H. Keller, Phys. Rev. Lett. 101,
077001 (2008).
20H. Keller, A. Bussmann-Holder, and K. A. M¨ uller, Mate-
rials Today 11, 38 (2008).
21M. Mali, J. Roos, H. Keller, J. Karpinski, and K. Conder,
Phys. Rev. B 65, 184518 (2002).
22Andr´ e Eckardt and Maciej Lewenstein, Phys. Rev. A 82,
011606(R) (2010).
23M. Calamiotou, A. Gantis, E. Siranidi, D. Lampakis, J.
Karpinski, and E. Liarokapis, Phys. Rev. B 80, 214517
(2009).
24W.A. Harrison, Electronic Structure and the Properties of
Solids (Freeman, San Francisco, 1980).
25R. Ofer, A. Keren, O. Chmaissem, and A. Amato, Phys.
Rev. B 78, 140508(R) (2008).
26J.S. Schilling and S. Klotz, in Physical Properties of High
Temperature Superconductors, edited by D. M. Ginsberg
(World Scientific, Singapore, 1992), Vol. III.
27H. Takahashi and N. Mori, in Studies of High Temperature

Page 10

10
Superconductors, edited by A.V. Narlikar (Nova Science,
New York, 1996), Vol. 16.
28C.C. Almasan, S.H. Han, B.W. Lee, L.M. Paulius, M.B.
Maple, B.W. Veal, J.W. Downey, A.P. Paulikas, Z. Fisk,
and J.E. Schirber, Phys. Rev. Lett. 69, 680 (1992).
29S. Rusiecki, B. Bucher, E. Kaldis, E. Jilek, J. Karpinski, C.
Rossel, B. P¨ umpin, H. Keller, W. K¨ undig, T. Krekels, and
G. Van Tendeloo, J. Less-Common Met. 164, 31 (1990).
30I.D. Parker and R.H. Friend, J. Phys. C 21, L345 (1988).
31C. Murayama, Y. Iye, T. Enomoto, N. Mori, Y. Yamada,
T. Matsumoto, Y. Kubo, Y. Shimakawa, and T. Manako,
Physica C 183, 277 (1991).
32R.P. Gupta and M. Gupta, Phys. Rev. B 51, 11760 (1995).
33J.J. Neumeier and H.A. Zimmermann, Phys. Rev. B 47,
8385 (1993).
34G.G.N. Angilella, R. Pucci, and F. Siringo, Phys. Rev. B
54, 15471 (1996).
35E.V. L. de Mello and C. Acha, Phys. Rev. B 56, 466 (1997);
Physica (Amsterdam) 265B, 142 (1999).
36X.J. Chen, H.Q. Lin, and C.D. Gong, Phys. Rev. Lett. 85,
2180 (2000).
37S. Sarkar, Phys. Rev. B 57, 11661 (1998).
38D. Di Castro, R. Khasanov, A. Shengelaya, K. Conder,
D.-J. Jang, M.-S. Park, S.-I. Lee, and H. Keller, J. Phys.:
Condens. Matter 21, 275701 (2009).
39R. Khasanov, J. Karpinski, and H. Keller, J. Phys.: Con-
dens. Matter 17, 2453 (2005).
40R. Khasanov, T. Schneider, R. Br¨ utsch, D. Gavillet, J.
Karpinski, and H. Keller, Phys. Rev. B 70, 144515 (2004).
41R. Khasanov, T. Schneider, and H. Keller, Phys. Rev. B
72, 014524 (2005).
42S.J. Blundell, Contemporary Physics 40, 175 (1999).
43J.E. Sonier, J.H. Brewer, and R.F. Kiefl, Rev. Mod. Phys.
72, 796 (2000); J.E. Sonier, Rep. Prog. Phys. 70, 1717
(2007).
44A. Maisuradze, A. Shengelaya, B. I. Kochelaev, E. Pom-
jakushina, K. Conder, H. Keller, and K. A. M¨ uller, Phys.
Rev. B 79, 054519 (2009).
45I.M. Fita, R. Puzniak, W. Paszkowicz, A. Wisniewski,
N.A. Doroshenko, and V.P. Dyakonov, Phys. Rev. B 66,
014519 (2002).
46E.H. Brandt, Phys. Rev. B 37, 2349 (1988).
47E.H. Brandt, Phys. Rev. B 68, 054506 (2003).
48W. Barford and J.M.F. Gunn, Physica C 156, 515 (1988).
49I.V. Fesenko, V.N. Gorbunov, and V.P. Smilga, Physica C
176, 551 (1991).
50R. Khasanov, S. Str¨ assle, D. Di Castro, T. Masui, S.
Miyasaka, S. Tajima, A. Bussmann-Holder, and H. Keller,
Phys. Rev. Lett. 99, 237601 (2007).
51R. Khasanov, A. Shengelaya, A. Maisuradze, F. La Mat-
tina, A. Bussmann-Holder, H. Keller, and K. A. M¨ uller,
Phys. Rev. Lett. 98, 057007 (2007).
52D. Xu, S.K. Yip, and J.A. Sauls, Phys. Rev. B 51, 16233
(1995).
53C.H. Choi and P. Muzikar, Phys. Rev. B 39, 11296 (1989).
54P.J. Hirschfeld and N. Goldenfeld, Phys. Rev. B 48, 4219
(1993).
55H. Kim, G. Preosti, and P. Muzikar, Phys. Rev. B 49, 3544
(1994).
56K. Ohishi, R.H. Heffner, G.D. Morris, E.D. Bauer, M.J.
Graf, J.-X. Zhu, L.A. Morales, J.L. Sarrao, M.J. Fluss,
D.E. MacLaughlin, L. Shu, W. Higemoto, and T.U. Ito,
Phys. Rev. B 76, 064504 (2007).
57M. Tinkham, Introduction to Superconductivity, Krieger
Publishing Company, Malabar, Florida (1975).
58M.I. Larkin, A. Kinkhabwala, Y.J. Uemura, Y. Sushko,
and G. Saito, Phys. Rev. B 64, 144514 (2001).
59A. Sahiner, E.D. Crozier, D.T. Jiang, and R. Ingalls, Phys.
Rev. B 59, 3902 (1999).
60R. P. Feynman, R. B. Leighton, M. Sands, The Feynman
Lectures on Physics, Addison Wesley; 2 ed. (2005).
61www.srim.org