On the interpretation of muon-spin-rotation experiments in the mixed
  state of type-II superconductors
On the interpretation of muon-spin-rotation
experiments in the mixed state of type-II
superconductors
I. L. Landau a,b H. Keller a
aPhysik-Institut der Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich,
Switzerland
bInstitute for Physical Problems, 117334 Moscow, Russia
Abstract
We argue that claims about magnetic field dependence of the magnetic field pene-
tration depth λ, which were made on the basis of muon-spin-rotation (µSR) studies
of some superconductors, originate from insufficient accuracy of theoretical models
employed for the data analysis. We also reanalyze some of already published exper-
imental data and demonstrate that numerical calculations of Brandt [E.H. Brandt,
Phys. Rev. B 68, 54506 (2003)] may serve as a reliable and powerful tool for the
analysis of µSR data collected in experiments with conventional superconductors.
Furthermore, one can use this approach in order to distinguish between conventional
and unconventional superconductors. It is unfortunate that these calculations have
practically never been employed for the analysis of µSR data.
Key words:
type-II superconductors, mixed state, unconventional superconductivity,
muon-spin-rotation experiments, magnetic field penetration depth
PACS: 74.70.Dl, 74.25.Op, 74.25.Ha,76.75.+i
1 Introduction
Muon-spin-rotation (µSR) experiments in the mixed state of type-II supercon-
ductors provide unique information about superconducting properties of the
investigated sample. An important advantage of this method is that muons
probe the bulk of the sample and therefore, the results are not distorted by
possible imperfections of the sample surface. At the same time, in order to
extract quantitative results from µSR measurements, a detailed model of the
magnetic field distribution in the mixed state is needed. As well as we are
Preprint submitted to Elsevier 2 November 2018
http://arxiv.org/abs/0704.1268v2
aware, only the Ginzburg-Landau (GL) theory [1] of the Abrikosov vortex lat-
tice [2] is developed to such a level [3,4,5]. As was recently demonstrated, if an
adequate model is available, not only the magnetic field penetration depth λ
but also the upper critical field Hc2 can be found from µSR data collected in
different applied magnetic fields [6]. It has to be remembered, however, that
theoretical calculations of Refs. [3,4,5] are related to superconductors with one
and isotropic energy gap only. This is why, this kind of analysis should be used
with extreme caution in the case of unconventional superconductors, in which
the applicability of theoretical models is not obvious.
We also point out a very interesting and promising approach which was de-
veloped in Refs. [7,8,9,10]. In these works a microscopic theory was used for
calculation of the mixed state parameters. An important advantage of this ap-
proach is that the results are not limited to conventional superconductors and
it can be used at temperatures well below Tc both for s- and d-wave pairing.
In recent years, µSR measurements were widely used for studying of different
unconventional superconductors such as high-Tc materials, MgB2 and oth-
ers. Some very interesting results were obtained. It was demonstrated that in
some cases the magnetic field penetration depth λ and the superconducting co-
herence length ξ, evaluated from µSR measurements, depend on the applied
magnetic field (see, e.g., [11,12,13,14,15,16,17,18,19]). This result, however,
contradicts the GL theory, which was used as a basis for the data analysis.
This contradiction is a clear sign that the corresponding models are not ade-
quate for describing the magnetic field distribution in the mixed state of these
compounds and rises the question about physical meanings of λ(H) and ξ(H)
obtained in such a way. As we argue below, magnetic field dependences of λ
and ξ cannot be obtained from µSR experiments if the conventional GL the-
ory or the London model were employed for the analysis of experimental data.
Moreover, because in the mixed state the superconducting order parameter is
not spatially uniform, there is no reasonable way to define either λ or ξ. In
other words, the physical meanings of magnetic field dependences of λ and
ξ, evaluated from µSR data, are quite different from traditional definitions of
these two lengths. This circumstance was recognized in Refs. [20,21,22] where
it was pointed out that λ(H), evaluated in such a way, represents some fit-
parameter rather than the magnetic field penetration depth. We underline
that the same should also be addressed to ξ(H) dependences. In the following
section, in order to avoid confusion, we shall use λ0 and ξ0 to denote values λ
and ξ for H → 0.
2 Conventional superconductors
Superconductors with s-pairing and one energy gap we shall consider as con-
ventional, independent of their pairing mechanism. Because the GL theory is
traditionally used for analyses of µSR data, we limit our consideration to this
theory.
The magnetic field penetration depth λ0 together with the zero-field coher-
ence length ξ0 represent two fundamental lengths of the GL theory. If their
values for some particular temperature T are known, one can calculate the
GL parameter
κ(T ) = λ0(T )/ξ0(T ), (1)
the thermodynamic critical magnetic field
Hc(T ) =
2πλ0(T )ξ0(T )
, (2)
the upper critical field
Hc2(T ) =
2κHc(T ) =
, (3)
the lower critical field
Hc1(T ) =
ln κ(T ) + α(κ)√
2κ(T )
Hc(T ) = [ln κ(T ) + α(κ)]
with α(κ) = 0.49693 + exp[−0.41477− 0.775 lnκ− 0.1303(lnκ)2] [5] Further-
more, in the case of conventional superconductors, any characteristics of the
sample for any value of an applied magnetic field may also be calculated and
expressed via λ0 and ξ0. Very detailed numerical calculations of different pa-
rameters of the mixed state for a very wide range of κ and for magnetic fields
ranging from Hc1 to Hc2 are presented in Ref. [5].
Muons probe the distribution of the magnetic induction in the sample. In high-
κ superconductors and low magnetic inductions B, contributions of vortex
cores can be neglected (London limit) and the distribution of the magnetic
induction around a single vortex line may be written as
B(r) =
K0(r/λ0), (5)
where r is the distance from the vortex center, Φ0 is the magnetic flux quantum
and K0 is the modified Bessel function. Because Eq. (5) is obtained from the
London theory, it gives an unphysical divergence of B at r = 0. In order
to improve Eq. (5), an appropriate cutoff has to be introduced [23,24,25]. It
should be remembered, however, that the results of Refs. [23,24,25] can be
considered as sufficiently accurate in low magnetic fields H ≪ Hc2 only. If
this condition is not satisfied, numerical solution of the GL equations must
be used for a reliable analysis µSR data. The magnetic induction distribution
may be calculated as a linear superposition of inductions created by different
vortices (see, for instance, Ref. [25]).
By measuring muon relaxation rates, one obtains the distribution of the mag-
netic induction P (B) experimentally, which allows to calculate the variance
of the magnetic induction
B2(r)−B2
, (6)
where · · · = (1/V )
· · · d3r means spatial averaging over superconductor of
volume V . If the distribution of the magnetic induction around vortices is
known, σ can also be calculated theoretically. According to [5]
σ = F (κ,B/Bc2)/λ
, (7)
where the parameter F depends on κ and B/Bc2. If the value of F is known,
λ0 may straightforwardly be evaluated. In the case of κ ≫ 1 and b ≪ 1,
F ≈ 0.061Φ0. In other situations, reliable results can be obtained from Ref.
[5]. Eq. (7) may also be written as σ = (2πHc2/Φ0)F (κ,B/Bc2)/κ
2. This
representation may be convenient if evaluation of κ is preferable.
While the zero-field value of λ enters the theory, the actual magnetic field
penetration depth is field dependent. According to the original Ginzburg and
Landau publication [1], if the magnetic field is parallel to the sample surface,
λ(H) = λ0 [1 + f(κ)H/Hc] . (8)
The function f(κ) grows monotonically with κ in such a way that for κ ≪ 1
f(κ) ∼ κ/4
2 and f(∞) = 0.125 [1]. Taking into account Eq. (1), we see
that even the magnetic field dependence of λ may be expressed via λ0 and
ξ0. The λ(H) dependence arises due to suppression of the order parameter
|ψ| by the applied magnetic field. In bulk type-II superconductors, Eq. (8)
is applicable in the Meissner state only, i.e., in magnetic fields H < Hc1. If
H ≥ Hc1, the magnetic field penetrates into the bulk of the sample forming a
lattice of Abrikosov vortices.
0 0.5 1.0
10-5 10-4 10-3 10-2 10-1 100
B/Bc2
B/Bc2
κ = 200
κ = 20
κ = 5
Ref. 3
κ = 2
Fig. 1. F multipied by (1− 0.069/κ2) as a function of B/Bc2 according to [5]. The
dashed line shows the F (B/Bc2) according to interpolation formula proposed in [3].
The horizontal line corresponds to F = 0.061Φ0. The inset shows the same curves
on linear scales.
If spatial variations of the order parameter can be neglected, the magnetic
induction decays exponentially on the flat surface of the sample. In the case
of cylindrical geometry (around vortices), the same decay is described by the
Bessel function (see Eq. (5)). Considering µSR experiments in the mixed sate
of type-II superconductors, we can use Eq. (5) if the total volume of vortex
cores is negligibly small compared to the volume of the sample, i.e., κ ≫ 1
and B ≪ Bc2. If one or both of these conditions is not satisfied, spatial
variations of the order parameter have to be taken into account. Numerical
calculations of Brandt are shown in Fig. 1. If κ ≫ 1, the function F (B/Bc2)
is practically independent of κ. As may be seen in Fig. 1, F remains magnetic
field dependent even for very small values of B/Bc2. The maximum on the
F (B/Bc2) dependence at B/Bc2 ≈ 0.17/κ1.2 is the obvious consequence of the
fact that σ vanishes at B/Bc2 → 0. At very low magnetic inductions F is
proportional to
(B/Bc2) [5].
It must be clearly understood that, although the applied magnetic field influ-
ences properties of the sample both in the Meissner and the mixed states, the
physics of this influence is completely different. In the Meissner state, super-
currents are induced in the surface layer of the sample. The density of these
currents is proportional to H and they depress the order parameter, which
leads to an increase of λ (see Eq. (8)). Because the reduction of the order
parameter |ψ| is small (|∆ψ| ≪ |ψ|), one may still introduce the magnetic
field penetration depth in its traditional way.
In the mixed state, the situation is completely different. Because there are
no currents, which are proportional to H , the absolute value of the applied
magnetic field is irrelevant. Only the distance between vortices given by the
magnetic induction B is important. At magnetic inductions B . 0.1Bc2, over-
lapping of vortex cores may be neglected and vortex properties are indepen-
dent of the applied magnetic field [5]. Only the vortex density is changed. At
higher magnetic inductions, vortex cores overlap and not only the vortex den-
sity, but also properties of individual vortices are magnetic field dependent.
Because the local value of the magnetic field penetration depth is inversely pro-
portional to the modulus of the order parameter |ψ|, there is no much sense
to introduce any unique value of λ corresponding to each particular magnetic
field. The correct approach is to calculate some measurable quantities theo-
retically and compare them with experimental results. In this way, however,
only the zero-field value of λ can be evaluated. If the value of λ resulting from
the analysis of experimental data depends on the applied magnetic field, it
means that the theory, which was employed for the analysis, does not describe
the actual experimental situation and the approach to the analysis should be
reconsidered.
If the magnetic field dependence of F is not taken into account or accounted
for incorrectly, the analysis of muon depolarization rates would result in some
effective λeff , which is magnetic field dependent. The knowlege of λeff(H),
however, does not represent any particular interest. This is why it is important
to use reliable models of the mixed sate in order to evaluate λ0.
As is well known, the GL theory is formally applicable at temperatures close
to Tc only. This is why quantitative applicability of theoretical calculations to
the analysis of experimental data at temperatures well below Tc is not obvious.
However, as it was recently demonstrated, the magnetic field dependence of
σ at T → 0 can be very well fitted by calculations of Brandt with two fit-
parameters λ0 and Hc2 [6]. Moreover, the value ofHc2, evaluated in such a way,
coincides with the result of magnetization measurements. We consider this as a
proof that the theoretical σ(H) dependence calculated in framework of the GL
theory can indeed be used for quantitative analysis of isothermal experimental
data even at temperatures T ≪ Tc. Below we reconsider several experimental
µSR studies and demonstrate that their results may perfectly be described by
the conventional GL theory although the magnetic field dependence of λ was
claimed in some of the original publications.
3 Unconventional superconductors
It must be remembered that calculations of Brandt [5], which we have dis-
cussed above, are valid for conventional superconductors only. There are no
reasons to believe that the vortex core structure should be the same in two-
gap superconductors or in superconductors with nodes in the order parameter.
Furthermore, one may assume that the influence of the vortex core region on
the distribution of the magnetic induction should be even stronger than in
the case of conventional superconductors. This is why, if calculations of [5] or
any other calculations based on the conventional GL theory are used for the
analysis of µSR data collected in different magnetic fields, it would produce
an unphysical λ(H) dependence. Although this result does not mean any spe-
cial behavior of the magnetic field penetration depth, it should be considered
as interesting. Indeed, if the conventional GL theory cannot describe the re-
sults of µSR experiments and all other possibilities for this disagreement are
excluded, 1 one may conclude that this superconductor is unconventional.
Superconductors with d-pairing as well as two-gap superconductors are more
complex than conventional ones. For instance, two lengths ξ0 and λ0 are insuf-
ficient for their characterization and some additional information is needed. At
present, there is no experimentally proven theory of the mixed state in uncon-
ventional superconductors. In this sense, magnetic field dependences of muon
relaxation rates cannot be interpreted quantitatively without some additional
assumptions. At the same time, one can try to obtain F (B/Bc2) experimen-
tally in order to compare results for different superconducting materials. Un-
fortunately, concerning high-Tc superconductors, the Hc2(T ) curves are not
yet reliably established.
Interesting theoretical approaches for interpretation of the µSR experiments in
the case of d-pairing was developed in [7,8,9,10,26,27,28,29]. In [26,27,28,29]
was convincingly argued that because of the nodes of the order parameter,
the electrodynamics of the mixed state becomes nonlocal. This nonlocality
effectively increases the vortex core radius and changes the distribution of the
magnetic induction around vortices (see Fig. 6 of Ref. [28]). If this effect is
not taken into account, the magnetic field penetration depth evaluated from
µSR experiments will be overestimated and magnetic field dependent. The
distortion of the results is very clearly demonstrated in Fig. 4 of Ref. [28].
In order to correct the results, the function λeff(B)/λ0 was introduced [28].
Using this function, which is an analog of F (B/Bc2), one can evaluate the
magnetic field penetration depth λ0.
At the same time, in high κ superconductors and at low magnetic inductions,
the total volume of vortex cores is small and contribution of vortex cores
cannot considerably change the muon signal. In this case, one may use F =
0.061Φo for evaluation of λ0 also in unconventional superconductors. Because
λ ∼ 1/
σ, the resulting error is not expected to be big. This means that, if an
experimental σ(H) dependence is available, extrapolation of σ(H) (or 1/
1 For instance, the traditional analysis cannot be used in the case of polycrystalline
samples of anisotropic superconductors.
to H = 0 gives more reliable values of λ0.
4 Analysis of experimental results
In this section, in order to simplify notation, we shall use λ and ξ without
indexes, having in mind the magnetic field penetration depth and the super-
conducting coherence length as they are introduced in the GL theory.
4.1 RbOs2O6, Cd2Re2O7, PrOs4Sb12.
Experimental σ(H) data for a polycrystalline sample of RbOs2O6 are shown
in Fig. 2(a). Because RbOs2O6 is an isotropic superconductor, using of such
samples is justified. This sample was investigated in [30] and experimental
data were analyzed by employing of an interpolation formula proposed in
[3]. Because this formula deviates significantly from more accurate numerical
calculations (see Fig. 1), we reanalyze these data using calculations of Ref.
[5]. As may be seen in Fig. 2(a), experimental data-points for H > 2 kOe
can very well be fitted by the theoretical σ(B/Bc2) curve. This fit results in
Hc2 = (67 ± 10) kOe and λ = (220 ± 5) nm (the value of λ = 260 nm was
obtained in the original publiction). Because the value of Hc2 is obtained by
the extrapolation of the σ(H) curve to σ = 0, the corresponding error margins
are large. It is important to emphasize that Hc2, evaluated in such a way, is in
agreement with the Hc2(T ) curve presented in [30]. This agreement together
with sufficiently high quality of fitting strongly supports our analysis.
We have chosen a high field part of the experimental σ(H) curve for the
analysis because in higher magnetic fields F (B/Bc2) is independent of κ (see
Fig. 1). In principle, analyzing the low field part of the curve, the value of
κ may straightforwardly be evaluated. This, however, is not always feasible.
As was already mentioned, the correct parameter is not H but the magnetic
induction B. The value of B determines intervortex distances and all other
characteristics of the mixed state. In magnetic fields H ≫ Hc1, the difference
(H − B) ≪ H and one can use H instead of B. In low fields, however, the
equilibrium value of B is considerably smaller thanH and the actual difference
(H − B) depends on pinning and on the demagnetizing factor of the sample.
Furthermore, in low fields, the magnetic induction is nonuniform throughout
the sample if its shape is not ellipsoidal. In polycrystalline samples, the sit-
uation complicates even further. Indeed, in such samples, some vortices may
go along intergrain boundaries, which can significantly influence the magnetic
induction distribution. This is the reason that we do not speculate on the
low-field behavior of the σ(H) curve.
0.1 1 10
T = 1.6 K
λ = (220     5) nm+—
Hc2 = (67    10) kOe+—
RbOs2O6
0 1 2 3 4 5 6
H (kOe)
T = 0.1 K
λ = (830     40) nm+—
Hc2 = (5.75    1) kOe+—
Cd2Re2O7
λ = (318     4) nm+—
Hc2 = (21    4) kOe+—
T = 0.1 K
PrOs4Sb12
Fig. 2. σ(H) data for three different superconducting compounds. The solid lines
represent the theoretical σ(B/Bc2) curve fitted to data-points. Only the data marked
by closed symbols were used for fitting. The resulting values of λ and Hc2 are
indicated in the figure. (a) RbOs2O6 sample studied in [30]. The vertical dashed
line indicates the value of Hc1. (b) Cd2Re2O7 sample studied in [31]. The dashed
line represents a linear approximation to a high field part of the theoretical σ(H)
curve. (c) PrOs4Sb12 studied in [32].
Similar results for a Cd2Re2O7 sample studied in [31] are shown in Fig. 2(b).
For the reasons explained above, we disregard the lowest field data-point.
Again in this case, data can be very well fitted with the GL theory, providing
Hc2 = (5.75±1) kOe in agreement with the original data (see [31]). The value
of λ = (830 ± 40) nm is also close to the result λ = 750 nm of Ref. [31].
We also note that approximation of experimental σ(H) data-ponts by a linear
dependence, as it was done in [31] and some other publications, is unjustified.
As may be seen in Figs. 1 and 2(b), the theoretical σ(B) curves are not at all
linear.
Fig. 2(c) presents σ(H) data for a heavy-fermion superconductor PrOs4Sb12
[32]. We do not discuss here different features of this rather unusual supercon-
ductor but limit ourselves to one simple question whether the σ(H) depen-
dence for this compound can be described by the conventional GL theory.
As may be seen in Fig. 2(c) (see also Fig. 4 of Ref. [32]), the values of σ(1kOe)
and σ(2kOe) practically coincide. It was assumed in Ref. [32] that a change
of vortex lattice symmetry or some other important changes of the vortex
structure, which occur in magnetic fields aboveH = 1 kOe, may be responsible
for such a behavior. This explanation seems to be plausible and we, as a
precaution, do not use the highest field data-point in the analysis.
The solid line in Fig. 2(c) represents the best fit of the theoretical σ(H) curve
to the data collected in magnetic fields 0.2 kOe≤ H ≤ 1 kOe. Quite amazingly,
the resulting value of Hc2 = 21 kOe practically coincides with Hc2 = 22.2 kOe
obtained in Ref. [33] from resistivity measurements. The value of λ = (318±4)
nm is also close to the result λ = 290 nm of the original publication.
Table 1
RbOs2O6 (1.6K) Cd2Re2O7 (0.1K) PrOs4Sb12 (0.1K)
Hc2 (kOe) 67 ± 10 5.75 ± 1 21.4 ± 4
κ 34± 1 35± 3 25.5 ± 2.5
λ (nm) 220 ± 5 830 ± 40 318 ± 4
ξ (nm) 7± 0.4 24± 3.5 12.5 ± 1.5
The main characteristics of the superconducting compounds, resulting from
our analysis of the µSR data published in Refs. [30,31,32], are listed in Ta-
ble 1. We emphasize that all parameters were evaluated by fitting of σ(H)
data-points with the theoretical σ(B/Bc2) dependence calculated in [5]. In
all cases, the values of Hc2 practically coincide with results of independent
measurements.
4.2 CeRu2[18].
Calculation of σ(T,H) considered above is not the only way of analysis of µSR
experiments in the mixed state of type-II superconductors. A different method
was employed in Refs. [11,12,13,14,15,16,17,21,22]. In this approach, the dis-
tribution of local fields (the Fourier transform of the muon precession signal)
P (B) was directly analyzed by comparing with corresponding theoretical cal-
culations. In real experiments, however, the P (B) line is usually different from
theoretical predictions. This difference is expected. Indeed, the calculations are
made for a perfect sample and for a perfect vortex lattice. All imperfections,
which cannot be avoided in experiments, influence the P (B) curves. This is
why, in order to approximate experimental data with theoretical calculations,
some gaussian smearing factor is introduced. In such a way, satisfactory agree-
ment between the theory and experiments can be achieved. This is justified
if it is a priori known that the spatial distribution of the magnetic induction
around vortex lines is in agreement with the theory, which is used for the cal-
culations. If it is not the case, introducing of additional Gaussian relaxation
may mask the disagreement and provide misleading results.
For some type-II superconductors, the P (B) curves for different values of the
applied magnetic field are available in the literature. This allows to calculate
σ(H) and to employ the same kind of the analysis as was used above. Below we
present the results of such analysis for single crystals of CeRu2 and vanadium.
Fig. 3(a) shows σ as a function of H for a CeRu2 sample experimentally
investigated in Ref. [18]. The difference to the results displayed in Fig. 2 is that
broadening of the P (B) line resulting from other sources of field inhomogeneity
was not accounted for. In this case σ, evaluated from µSR experiments may
be written as
σ2sc + σ
bg, (9)
where σsc and σbg are the mixed state and background contributions, respec-
tively. As may be seen in [18], σbg is not small and cannot be evaluated from
the data presented in the publication with sufficient accuracy. This is the
reason that we introduce σbg as an additional adjustable parameter. Because
experimental data are insufficient for evaluation of λ, Hc2 and σbg together,
we take the value of Hc2 from the original publication.
2 We also note that
H = 40 kOe is the only data point corresponding to the peak-effect region
2 There is some confusion in Ref. [18]. While Fig. 2 provides Hc2 = 50 kOe, the
value of Hc2 evaluated from Fig. 1 is closer to 45 kOe. Taking into account that
the resulting λ is not very sensitive to some variation of the assumed Hc2 value, we
have chosen Hc2 = 47.5 kOe for the analysis.
0 10 20 30 40 50
H (kOe)
1 10 100
CeRu2
λ = (167   3) nm+—
T = 2 K
Fig. 3. σ (upper panel) and σsc =
σ2 − σ2
(lower panel) versus H for a CeRu2
sample studied in [18]. The solid lines represent the theoretical curves calculated
as explained in the text. The chosen value of Hc2 = 47.5 kOe is indicated in the
figures by vertical lines. Only data-points shown by closed symbols were used for
evaluation of σbg and λ.
(see Fig. 1 in [18]). Because the origin of this effect is not yet established, we
exclude the corresponding data point from the analysis.
As may be seen in Fig. 3(a), all data-points for H ≤ 30 kOe can be fairly well
fitted by the theory, providing λ = (167 ± 3) nm and σbg = (8.4 ± 1) G. The
magnetic field dependence of σsc =
σ2 − σ2bg is shown in Fig. 3(b).
Using the σsc(H) plot presented in Fig. 3(b), we can calculate λ for each of
the data-points. Such calculations were made for two different values of Hc2
and they are presented in Fig. 4. As was expected, the absolute value of λ
is practically independent of the chosen value of Hc2. One can also see that,
contrary to claims of Ref. [18], there is no any noticeable dependence of λ on
0 10 20 30
H (kOe)
Hc2 = 45 kOe
Hc2 = 50 kOe
CeRu2
T = 2 K
Fig. 4. λ as a function of H calculated assuming Hc2 = 45 kOe and Hc2 = 50 kOe.
At the same time, the value of σ(40kOe) deviates quite significantly from
the theoretical curve (see Fig. 3). 3 If this deviation is not an experimental
error, it means that the distribution of the magnetic induction in the case of
the peak-effect is rather different in comparison with the conventional mixed
state. However, one should be extremely careful with such conclusions. In the
case of the peak-effect, the value of σ is rather sensitive even to insignificant
variations of H (see Fig. 13(d) in [18]). In this situation, ∼ 10−5H change
of the external field may explain the difference between σ(40kOe) and the
theoretical curve.
4.3 Vanadium single crystal [22]
We discuss experimental data of Ref. [22] in some detail in order to demon-
strate general problems of interpretation of µSR experiments in the case of
low-κ superconductors. We also discuss some typical errors that can be found
in the literature.
Vanadium is one of the very few pure metals, which displays type-II super-
conductivity at all temperatures. Superconducting characteristics of vanadium
have been rather well investigated (see, for instance, [34,35,36,37]). Although
vanadium has a cubic (bcc) structure, Hc2 depends on the orientation of the
applied magnetic field [35]. According to [35], the value of Hc2 along [111]
direction is approximately 10% higher than that for [001].
3 σ(40kOe) is larger than the corresponding theoretical value. The higher σ means
smaller λ. This conclusion is just opposite to that made in the original publication.
T (K)
Ref. 34, H || [110]
Ref. 35, H || [110]
Ref. 35, H || [111]
0 1 2 3 4 5
Ref. 36, H || [491]
Ref. 37, H || [491]
Vanadium
Fig. 5. Hc2(T ) for three different orientations of vanadium single crystals
The dependences Hc2(T ) for three different orientations are shown in Fig. 5.
As may be seen, results of different studies are in excellent agreement. The
value of Tc may be evaluated as Tc = (5.40± 0.05) K [34,35,36,37]. While Hc2
is orientation dependent, its normalized temperature dependence is practically
universal [35]. This is illustrated in Fig. 6 where Hc2(T )/(TcdHc2/dT )T=Tc is
plotted versus T/Tc. The results of different works, presented in such a way,
nicely collapse onto a single curve. We note that the temperature dependence
of Hc2 is somewhat different from predictions of Helfand-Werthamer (HW)
theory [38].
Vanadium is a low-κ material with κ(0K) = 1.5 for a pure sample investigated
in [34]. This circumstance adds some peculiarities to the mixed state and its
description. First, the condition H ≪ Hc2 is not satisfied even in magnetic
fields down to Hc1, i.e., the London approach, in which vortices are considered
as independent, is inapplicable in the entire range of magnetic fields. In this
situation, the actual magnetic induction distribution in the sample strongly
depends on spatial variations of the order parameter, and the accuracy of the
0.0 0.2 0.4 0.6 0.8 1.0
t = T/Tc
HW theory
Ref. 36, H || [491]
Hc2(0K) = 2.83 kOe
Ref. 37, H || [491]
Hc2(0K) = 2..78 kOe
Ref. 34, H || [110]
Hc2(0K) = 3 kOe
Ref. 35, H || [110]
Hc2(0K) = 3 kOe
Ref. 35, H || [111]
Hc2(0K) = 3.14 kOe
Vanadium
Fig. 6. The normalized upper critical field of vanadium single crystals as a function
of t = T/Tc. The solid line represents the HW theory [38].
corresponding calculations plays a crucial role. Second, if κ ∼ 1, the condition
λ ≫ ξP (ξP = 0.74ξ(0K) is the Pippard coherence length) is not satisfied
at low temperatures and electrodynamics become nonlocal, i.e., quantitative
applicability of the GL theory at T ≪ Tc is questionable. Furthermore, the
results of Refs. [34,35,36] clearly demonstrate that superconducting properties
of vanadium at T ≪ Tc cannot be described by the GL theory and a more
complex approach is necessary. At the same time, as we argue below, experi-
mental σ(H) curves are close to theoretical predictions of Brandt [5] and can
be used for evaluation of the magnetic field penetration depth.
The values of σ were calculated using the P (B) curves presented in Ref. [22].
Clearly visible peaks arising from muons stopped outside the sample were
approximated by Gaussians and subtracted from the data. The resulting values
of σ are plotted in Fig. 7(a). 4 Our analysis gives λ = (49 ± 1.5) nm and
Hc2 = (3.8 ± 0.15) kOe. The estimation of λ is in very good agreement,
with λ = 50 nm, which may be calculated from Hc2(T ) and κ(T ) curves
experimentally measured in [34].
The value of Hc2 evaluated above is just 10% below of Hc2 = 4.2 kOe provided
in the original publication [22]. We note that there are no estimations of
experimental uncertainty for Hc2 in [22]. One can assume that Hc2(0.02K) was
obtained by extrapolation of higher temperature data and the corresponding
error margins are considerable. We also note that Hc2(0.02K) = 4.2 kOe is
well above earlier results (see Fig. 5). Partly this difference may be explained
4 Because the demagnetizing factor of the sample is close to 1, one can safely assume
H = B for all considered magnetic fields.
2 3 4
Vanadium single crystal, H  [111]||
T = 0.02 K
Hc2 = (3.8   0.15) kOe+—
λ = (49   1.5) µm+—
H (kOe)
Fig. 7. (a) σ versus H for a vanadium single crystal studied in [22]. The solid line
represent a fit with the Brandt theory [5]. The resulting values of λ and Hc2 are
indicated in the figure.
by the fact that the sample that we are discussing here was substantially less
pure than those of Refs. [34,35,36,37]. However, such a significant increase of
Hc2 seems to be unlikely. Furthermore, as we show below, the value of Hc2,
evaluated by the analysis of the temperature dependence of muon relaxation
rates, agrees better with the estimate of Hc2 presented in Fig. 7 than with the
value given in [22].
As was already mentioned, experimental results presented in [34] allow for
evaluation both Hc2(T ) and λ(T ) dependences. Using these data, we can also
obtain the expected value of σ for any magnetic field and temperature. Such
results for H = 1.6 kOe are plotted in Fig. 8 for comparison with the µSR
data of Ref. [22]. As may be seen in Fig. 8, the two σ(T ) curves are similar. In
order to emphasize this similarity, we approximate both data sets by the same
functional dependence (see Fig. 8). Considering the results presented in Fig. 8,
one can conclude that the sample investigated in [22] has indeed a somewhat
higher Hc2. The value of Tc(H) may straightforwardly be evaluated from σ(T )
data as the value of T , at which σ vanishes. Such estimate gives Hc2 = 1.6 kOe
at T = 3.38 K. 5 Using this value of Hc2(3.38K) and the normalized Hc2(T )
curve presented in Fig. 6, we can evaluate Hc2(0) = (3.4 ± 0.25) kOe, which
is in reasonable agreement with the estimate made from the analysis of σ(H)
data (see Fig. 7). 6
5 This is well below the value of Tc(1.6kOe) = 3.65 K provided in [22].
6 We use Tc = 5.4 K, as it follows from earlier measurements (the same value is
provided in reference data of Goodfellow Ltd.), assuming that Tc = 5.2 K given in
[22], is a misprint. If, however, we except Tc = 5.2 K, Hc2(0) = (3.65 ± 0.2) kOe,
0 1 2 3
λ according to Ref. 34
σ according to Ref. 34
σ according to Ref. 22
T (K)
2  (µ
Vanadium
Fig. 8. Temperature dependence of σ(1.6kOe) for a vanadium single crystal studied
in [22]. The temperature dependences of σ and λ−2 (right y-axis), evaluated from
experimental data of [34], are shown for comparison. The solid and the dashed lines
are the guides to the eye. The dotted line (f ′(T )) is obtained by scaling of the solid
line (f(T )), i.e., f ′(T ) = 1.29f(1.09T ).
Our results presented in this section are rather different from the conclusions
of Ref. [22]. First and foremost, as it is clearly demonstrated in Fig. 7, the
magnetic field dependence of σ is very close to the result of the GL theory.
Our value of λ for T = 0, which is in excellent agreement with measurements
of Ref. [34], is about 1.5 times smaller than the result of [22] for H = 1.6 kOe.
The temperature dependence of 1/λ2 calculated according to Ref. [34] is also
plotted in Fig. 8. As may be seen, while σ vanishes at T = 3.12 K, the value
of λ(3.12K) remains almost the same as at T = 0. In other words, σ vanishes
at Tc(H) not because of the divergence of λ but because the coefficient F in
Eq. (7) vanishes at this temperature.
It seems important to emphasize that Fig. 10 of Ref. [22] is based on an obvious
misunderstanding. 7 There exists no theory that predicts divergence of λ at
Tc(H). The reference on theoretical calculations of Mühlschlegel [39], given
in [22], is misleading. Indeed, the thermodynamical consideration of Ref. [39]
is based on the fact that the difference between free energies of the normal
and the superconducting states per unit of volume can be written as H2c /8π.
The same difference can also be written as ncp∆ where ncp is be the density
of Cooper pairs and ∆ is the equilibrium (zero-field) superconducting energy
gap. Using this, one obtains ncp(T ) and λ(T ). Nothing in this consideration
can be used to justify Fig. 10 of Ref. [22]. We also note that the temperature
variation ofHc2 should be taken into account if the temperature dependence of
λ is evaluated from measurements in fixed magnetic fields. It can be neglected
the value practically coinciding with the result of Fig. 7.
7 A similar plot one can also find in [30]
only if the condition H ≪ Hc2 is satisfied at all temperatures.
In fact, good agreement with the theory, demonstrated in Figs. 7 and 8, is
rather surprising. As was already mentioned, it is well established that vana-
dium does not obey the GL theory at T ≪ Tc [34,35,36].The most probable
is that the distribution of the magnetic induction in the sample (P (B)) is
different from theoretical predictions, while σ, as a more integral character-
istic of this distribution, remains practically the same. This assumption can
also explain the difference between our results and those of Ref. [22]. Indeed,
the analysis of P (B) functions, carried out in [22], resulted in an unphysical
magnetic field dependence of λ, which clearly demonstrates the inapplicability
of the GL theory to this analysis.
It is important to underline that, although the distribution of the magnetic
induction in the sample cannot be described by the GL theory in low-κ type-II
superconductors at T ≪ Tc, the σ(H) curves can still be used for evaluation of
the magnetic field penetration depth, as it is proven by a very close agreement
between our value of λ(0K) = 49± 1.5 nm and 50 nm calculated from results
of Ref. [34]. We also note that at temperatures closer to Tc the GL theory
should be applicable and both analyses should result in the same λ values.
4.4 YNi2B2C [14]
In order to demonstrate that in some cases the GL theory cannot describe
µSR data, we consider a study of a borocarbide superconductor YNi2B2C [14].
Rare-earth nickel borocarbide superconductors attracted a lot of attention dur-
ing the past decade. Already the very first studies of YNi2B2C demonstrated a
pronounced positive curvature of the Hc2(T ) curve, indicating unconventional
superconductivity [40,41]. Similar conclusions were made from specific heat
data [40,41]. Although YNi2B2C has been extensively studied, the nature of
this unconventionality is still under discussion. While Refs. [42,43,44,45,46,47]
provide evidences of point nodes in the superconducting gap function, other
works point out on multiband superconductivity [48,49,50,51]. The distinction
between these two possibilities is sometimes difficult to make. For instance, as
was recently demonstrated, specific heat data may be fitted equally well by
nodal and two-gap models [52].
Experimental results of Ref. [14] are plotted in Fig. 9 as σ versus H . The
value of Hc2(3K) = 70 kOe for this particular crystal is given in [14]. As may
clearly be seen in Fig. 9, σ(H) data-points cannot be fitted with the theory if
the entire range of magnetic fields is considered. Because there are sufficient
experimental evidences that YNi2B2C is an unconventional superconductor
[42,43,44,45,46,47,48,49,50,51,52], disagreement between the GL theory and
10 100
H (kOe)
λ = 123 nm
YNi2B2C
T = 3 K
λ = 90 nm
Fig. 9. σ as a function of H for a YNi2B2C sample studied in Ref. [14]. The solid
lines are the theoretical σ(H) dependencies calculated for Hc2 = 70 kOe and for
two different values of λ.
experimental data is expected. We also note that in YNi2B2C a transition
from a triangular to a square vortex lattice was observed [53,54]. However,
because this transition occurs in lower magnetic fields, it cannot have any
influence on µSR data presented in Fig. 9. 8
While the totality of data cannot be fitted with the theory, both high-field and
low-fild results may amazingly well be approximated with two different theo-
retical curves, corresponding to two different λ values (Fig. 9). Unfortunately,
insufficient number of data-points does not allow to make unambiguous con-
clusions on this matter, however, if this behavior will be confirmed by a more
detailed study, it may be considered as a rather interesting result, indicating
two gap superconductivity.
In low magnetic fields H ≪ Hc2, most of muons stopped outside vortex cores,
i.e., the magnetic induction distribution in vortex core regions is not very
important for µSR data. In this case, the difference between conventional and
two gap superconductors should not be significant and the resulting σ(H)
curves can be close in these two cases.
In higher magnetic fields, as it was established in studies of MgB2, supercon-
ductivity in one of two bands is completely suppressed and the superconduc-
tor behaves itself as a one gape superconductor but with a smaller number of
Cooper pairs [55,56,57]. This can explain the fact that the two data points for
8 As was demonstrated in [10], although the magnetic induction distributions for
square and triangular lattices are quite different, σ(H) remains practically the same
in both cases.
H ≥ 30 kOe follow a standard theoretical curve with a higher value of λ (see
Fig. 9).
The quantity 1/λ2 is proportional to the density of Cooper pairs. If two gap
superconductivity is assumed, the values of the magnetic field penetration
lengths, evaluated from low- and high-field data, allows evaluation of relative
weights of two superconducting bands. Such estimate gives 54% and 46% for
stronger and weaker gaps, respectively. These values are noticeably different
from the result 71% and 29% obtained in Ref. [52]. At present, however, it is
too early to discuss such differences. Two data points in the high-field range
part of the curve (see Fig. 9) are clearly insufficient in order to make any
definite conclusion about superconductivity in YNi2B2C.
5 Conclusion.
In this work, we applied numerical calculations of Brandt [5] for the analysis
of µSR experiments carried out in the mixed state of several superconducting
compounds. It turned out that this approach may serve as a very powerful tool
for the interpretation of µSR experiments. If the magnetic field dependences
of muon depolarization rates are available, not only λ but also Hc2 can reliably
be evaluated. We show that in the most of considered cases the magnetic field
dependences of σ may very well be described by a single and temperature
independent λ.
In contrast to approximate analytical models, Ref. [5] provides precise numer-
ical solutions of 2-dimensional GL equations for different values of κ (0.85 ≤
κ ≤ 200) and for magnetic fields ranging from Hc1 to Hc2. Using these so-
lutions, different characteristics of the mixed state, including the σ(B/Bc2)
dependences for various κ values, were calculated. As well as we are aware,
these calculations provide the best available description of the magnetic induc-
tion distribution in the mixed state of conventional type-II superconductors.
We also note that numerical calculations of σ(H) are available since 1997 (see
Fig. 3 in [4]). In spite of this, for some mysterious reasons, these theoretical
calculations have practically never been used for the analysis of µSR data.
We also argued that the magnetic field dependence of λ can never be obtained
from analyses of experimental data collected in the mixed state. Indeed, be-
cause the local value of λ is inversely proportional to the absolute value of the
superconducting order parameter, one cannot introduce any single value of λ
in the mixed state.
Calculations of Brandt [5] represent the conventional GL theory and their
validity for the description of unconventional superconductors is question-
able. In fact, there are no reasons to believe that the conventional GL theory
can quantitatively describe either two-gap superconductors or superconduc-
tors with nodes of the order parameter and one should expect disagreement
between Brandt’s theory and experimental results in the case of unconven-
tional superconductors, as it is demonstrated in Fig. 9.
We demonstrated that in conventional superconductors, the results of µSR
experiments may be used for the evaluation of both λ and Hc2. If applicability
of the conventional GL theory is questionable, the knowledge of Hc2 is of
primary importance. Disagreement between the values of Hc2 resulting from
µSR data and independent measurements may be considered as convincing
evidence that this particular superconductor is unconventional.
We are grateful to R. Khasanov for numerous and fruitful discussions.
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	Introduction
	Conventional superconductors
	Unconventional superconductors
	Analysis of experimental results
	RbOs2O6, Cd2Re2O7, PrOs4Sb12.
	CeRu2kado.
	Vanadium single crystal son06
	YNi2B2C ohi
	Conclusion.
	References
