Evidence for nonmonotonic magnetic field penetration in a type-I superconductor
V.F. Kozhevnikov1∗, C.V. Giuraniuc1, M.J. Van Bael1, K. Temst2, C. Van Haesendonck1,
T.M. Mishonov3, T. Charlton4, R.M. Dalgliesh4, Yu.N. Khaidukov5, Yu.V. Nikitenko5,
V.L. Aksenov5, V.N. Gladilin1,6, V.M. Fomin6, J.T. Devreese6, and J.O. Indekeu1
Laboratorium voor Vaste-Stoffysica en Magnetisme Katholieke Universiteit Leuven, 3001 Leuven, Belgium
Instituut voor Kern- en Stralingsfysica, Katholieke Universiteit Leuven, 3001 Leuven, Belgium
Department of Theoretical Physics, St Clement of Ohrid University of Sofia, 1164 Sofia, Bulgaria
ISIS Science Division, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom
Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
Theoretische Fysica van de Vaste Stoffen, Universiteit Antwerpen, 2020 Antwerpen, Belgium
(Dated: October 24, 2021)
Polarized neutron reflectometry (PNR) provides evidence that nonlocal electrodynamics governs the
magnetic field penetration in an extreme low-κ superconductor. The sample is an indium film with
a large elastic mean free path (11 µm) deposited on a silicon oxide wafer. It is shown that PNR
can resolve the difference between the reflected neutron spin asymmetries predicted by the local and
nonlocal theories of superconductivity. The experimental data support the nonlocal theory, which
predicts a nonmonotonic decay of the magnetic field.
PACS numbers: 74.20.-z, 74.25.Ha, 78.70.Nx
In this paper we pose and answer experimentally the
following fundamental questions. Are nonlocal electrody-
namics effects measurable in superconductors? Can the
nonmonotonic decay of magnetic field penetration pre-
dicted by the nonlocal theory be observed? To what ex-
tent can Polarized Neutron Reflectometry (PNR) resolve
the difference between local and nonlocal diamagnetic re-
sponses expected for strongly type-I superconductors?
Nonlocality is a key concept of superconductivity the-
ory, but its experimental verification is still not estab-
lished. In the Meissner state, a magnetic field ap-
plied parallel to the surface located at z = 0 causes
the magnetic induction B(z) to penetrate over a depth
λ ≡ B(0)−1
B(z)dz. In the London (local) limit,
appropriate to most type-II superconductors, B(z) ∝
exp(−z/λL), where λL is the London penetration depth.
In 1953, to explain the variation of λ in type-I supercon-
ductor Sn caused by adding In, Pippard proposed that
the current density is related to the average of the vector
potential over a region of size ξ0 (the Pippard coherence
length) [1]. The smaller the Ginzburg-Landau parameter
κ, the more important this nonlocal effect.
Nonlocal theory predicts that B(z) deviates from a
simple exponential decay. B(z) is nonmonotonic and,
moreover, changes sign at a specific depth [1]. In the
pure limit (ξ0 ≪ ℓ, where ℓ is the elastic mean free
path) B(z) is determined by the intrinsic parameters
λL(T = 0) and ξ0, and by the temperature T . The devia-
tion is most significant in “extreme” type-I superconduc-
tors (κ ≪ 1/
2), such as Al (κ ≈ 0.01) and In (0.06).
For these the results of the Pippard theory are identi-
cal to those of the Bardeen-Cooper-Schrieffer theory [2].
On the other hand, the local approximation is safe for
κ > 1.5 [3], i.e., marginal nonlocal effects are predicted
for some type-II superconductors such as Nb (κ ≈ 1).
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FIG. 1: Magnetic-induction profiles B(z) in a semi-infinite In
sample. The dashed (solid) line corresponds to the local (non-
local) relation between current density and vector potential.
B(z) in In, calculated in local and nonlocal approaches,
in the pure limit with ξ0 = 0.38 µm and λL(0) = 0.025
µm [4], for T = 1.8 K, is shown in Fig. 1. Details about
the formalism can be found in Ref. 3. In the nonlocal
approach B(2λL)/B(0) ≈ 1/e. The sign reversal is ex-
pected at z ≈ 5.5λL, and the amplitude of the reversed
field is at most about 3% of the field at the surface.
An observation of sign reversal was reported in Ref.
5. An external AC magnetic field H with amplitude up
to 30 Oe was applied parallel to a hollow cylindrical Sn
film about 2 µm thick, and a strongly attenuated (108
times) signal with reversed phase was detected inside the
cylinder at T = 2.88 K and H ≈ 25 Oe. This phase
difference was interpreted as a change of sign in the pen-
etrating field. However, this is questionable because the
phase difference drops back to zero at a larger (30 Oe)
field, whereas the critical field Hc at 2.9 K is 115 Oe [6].
http://arxiv.org/abs/0704.1872v2
Nowadays B(z) can be measured directly using po-
larized neutron reflectometry (PNR) [7] and low-energy
muon spin rotation (LE-µSR) [8] techniques. We com-
ment briefly on the latter before focusing on the former.
In the LE-µSR technique positive muons polarized per-
pendicularly to the applied field are implanted in a sam-
ple over a distance determined by the muon energy. B(z)
is obtained from measuring the Larmor precession fre-
quency of the muon spins at stopping distance. In other
words, the implanted muons serve as tiny sensors of the
magnetic field inside the sample. However, in practice the
muon precession is progressively damped with depth due
to a broad distribution of stopping distances [9]. From
our prospectus, this is the main difficulty in applying the
LE-µSR technique to fields with a sharp profile.
Recently, the LE-µSR technique was used to measure
B(z) in Pb, Nb, and Ta [9]. Most interesting is the obser-
vation of a non-exponential shape of B(z) for all studied
metals. The nonlinearity of the semi-log plots for B(z)
is marginal, which is exactly what should be expected
theoretically in view of the fairly high κ of the studied
samples. For example, κ of pure Nb (residual resistivity
ratioRRR = 1600) is 1.3 at 3 K and 1.0 at 7 K [10]. How-
ever, in Ref. 9 κ of less pure Nb (RRR=133) is reported
to be 0.7 at 2.96 K and 7.6 K. This and some other in-
consistencies with well established literature data suggest
that the muon probing results may contain some hidden
uncertainties. Therefore, additional experiments would
be worthwhile, in particular on low-κ superconductors.
The PNR technique is based on the change of the neu-
tron index of refraction in a magnetized medium. When
a collimated neutron beam polarized along the magnetic
field is incident on a flat, laterally uniform sample under
a grazing angle, its specular reflectivity R is determined
by the profile of the neutron scattering potential below
the surface. R is measured versus momentum transfer
Q = 4π sin θ/λn, where θ is the angle of incidence and
λn the neutron wavelength. The scattering potential con-
sists of a nuclear and a magnetic part, which results in
different reflectivities R+ and R− for neutrons with spins
polarized parallel (up) and anti-parallel (down) to the
applied field, respectively. Direct information about the
sample magnetization is obtained by combining R+ and
R−; the combination s = (R+ − R−)/(R+ + R−) is the
spin asymmetry. B(z) can be found by fitting s(Q) data
with s(Q) calculations based on theoretical models for
B(z). PNR has been applied for measuring the penetra-
tion depth and for detecting surface superconductivity in
Nb [11, 12], high-Tc cuprates [13, 14, 15] and Pb [16, 17].
The nonlocal effect in B(z) measured with PNR was
discussed in Refs. 11, 12, 16 and 17. Although some devi-
ation from exponential decay was noticed in Refs. 12 and
16, no solid confirmation of the nonlocal theory was ob-
tained. The authors of Ref. 17 correctly pointed out that
experiments with low-κ type-I superconductors are de-
sirable to verify nonlocality, but their overall conclusion
was that PNR is incapable of detecting nonlocality in
any superconductor. In this light it is very interesting to
reassess the problem of nonlocality with state-of-the-art
PNR applied to a low-κ material such as In, since during
the last decade many neutron source facilities have sig-
nificantly progressed in neutron flux, neutron optics and
detector technology.
The design of the sample for the PNR study is based
on the following requirements. The irradiated surface
must be flat and possess minimal possible roughness. The
sample must be thick enough to have the same electro-
magnetic properties as the bulk material. Degradation
of the surface quality with increasing thickness limits the
film thickness. Neutrons reflected back from the sub-
strate should have a negligible effect on the reflectivity
in a region close to the critical edge of total reflection, Qc,
where the reflectivity is most sensitive to the magnetic
properties.
Two approaches can meet these requirements. One
is to deposit a thick film on a flat substrate that reflects
least. This can be achieved if the neutron refraction index
of the substrate is larger than that of the sample. This
approach was taken in the experiments on Nb [11, 12] and
Pb [16, 17]. In fact, this was the only option, in view of
the negligibly small absorption of neutrons in Nb and Pb.
However, In is a strong absorber, which enables one to
rely on substrates with a refractive index smaller than
that of In, provided the thickness of the indium film is
properly optimized. In this approach a second plateau or
“hill”, associated with total reflection from the sample-
substrate interface, is expected in the reflectivity curve
R(Q). This should yield additional information about
the sample structure. Modelling shows that an indium
thickness of 2.5 µm is appropriate. Such a sample was
fabricated in the present work.
High purity indium (99.9999%) was deposited by ther-
mal evaporation on the polished side of a silicon oxide
wafer at room temperature. The substrate size was 2×2
cm2×1 mm. The base pressure and the evaporation rate
were 4×10−8 mbar and 60-70 Å/s, respectively. The
nominal film thickness, as recorded by a quartz monitor,
was 2.5 µm. Several smaller area samples were simulta-
neously fabricated for the film characterization.
The root-mean-square (rms) surface roughness σ
probed with an atomic force microscope (AFM) yielded
2.0, 6.7 and 8.0 nm at the scale of 1, 5 and 10 µm, re-
spectively. A scan range up to 10 µm was not sufficient
to reach saturation of the roughness. Consequently, 8.0
nm is a lower bound on the roughness at the scale of
the neutron coherence length (≈ 100 µm [18]). In our
simulations, effects due to surface roughness are mod-
elled using Névot-Croce factors [7], where the roughness
is characterized by σ; it was allowed to vary to fit the
experimental data.
Another parameter associated with the sample surface
is the thickness of the indium oxide film. When exposed
to air, In, like its neighbors in the Periodic Table, Al and
Ga, instantly forms a protective oxide layer. A surface of
indium in air remains lustrous for years. This suggests
that the oxide layer is very thin, perhaps of the order
of a few monolayers, and should not affect the neutron
reflectivity. This is consistent with the negative result of
Rutherford backscattering measurements performed on
our sample: no oxide film has been detected.
The electromagnetic properties of the sample were
characterized by measurements of the DC magnetiza-
tion M and the electrical resistivity. The shape of the
M(H)-curves is typical for type-I superconductors. The
obtained phase diagram Hc(T ) agrees well with the lit-
erature data [6]. Tc of our sample (3.415 K) matches the
tabulated value of 3.4145 K [19], and RRR = 540. Corre-
spondingly, ℓ ≈ 11µm is much larger than ξ0. Therefore,
our sample is a type-I superconductor in the pure limit.
PNR experiments were performed both on the RE-
MUR reflectometer [20] at the Joint Institute for Nuclear
Research (Dubna) and on the CRISP instrument [21] at
ISIS (Oxford). Both sets of measurements confirm that
splitting of the R+(Q) and R−(Q) curves is achievable for
our sample. The ISIS data, which are the most detailed,
allow a quantitative analysis to which we now turn.
CRISP operates with a spin-polarized polychromatic
pulsed neutron beam. The angle of incidence and the
instrumental resolution ∆Q/Q were set to 0.24 degrees
and 3%, respectively.
The reflectivity in the Meissner state was measured at
T = 1.8 K and at magnetic fields of 77, 140, 166, and 194
Oe (Hc(1.8 K) = 205 Oe). The obtained data sets are
shown in Fig. 2. The R(Q)-dependencies exhibit a hill
caused by the total reflection from the substrate. The
splitting between R+ and R− is clearly visible near Qc;
different magnitudes of the error bars are due to different
times of exposure. The data obtained at 77 and 166 Oe
have the smallest statistical error and will be used for
further discussion.
The data for the reflectivity in the normal state are
shown in Fig. 3. Solid curves are simulations, in which
the sample was represented by a pure In film on a SiO2
substrate. In the simulations the angular beam resolution
was allowed to vary due to the unknown uncertainties of
the instrumental resolution and of the geometrical factor
(as only part of the beam covers the sample).
The simulation curve near Qc is mostly controlled
by the resolution (see also [16]). The next segment,
down to the foothill, is determined by the roughness of
the sample surface. The location of the ascending part
(0.011 < Q(Å−1) < 0.014) is governed by the film thick-
ness. The curve segment following the hill is determined
by the substrate scattering properties. No attempts were
made to achieve a better fit for that segment, because
there the spin asymmetry is indistinguishable from zero.
The best fit (Fig. 3) was obtained for the model sample
with σ = 14 nm and ∆Q/Q = 2.5 %. Fitting the ascend-
ing part enables one to determine the film thickness in
situ. The statistical error of the reflectivity data in this
region being ± 5%, the thickness was found to be 2400
± 30 nm, in agreement with the nominal thickness of 2.5
µm. These parameters were further used for simulating
the spin asymmetry. Attempts to introduce an indium
0.008 0.010 0.012 0.014
spin up
spin down
77 Oe
140 Oe
166 Oe
194 Oe
momentum transfer (A-1)
FIG. 2: Reflectivity of polarized neutrons measured in the
Meissner state. Qc is the momentum transfer for total neu-
tron reflection from the outer surface. The scale is shown for
the data at H = 194 Oe; the other data have been shifted for
clarity.
0.01 0.02 0.03 0.04
0.010 0.012
momentum transfer (Å-1)
FIG. 3: Neutron reflectivity at T = 4.6 K. Curves 1 and
2 are simulations for a film thickness of 2.40 and 2.50 µm,
respectively. The inset shows the data for the full range of
Q-values.
oxide layer on top of the sample yielded no reasonable fit
for any appreciable thickness (> 1nm) of the oxide layer.
This is consistent with our expectation that the indium
oxide layer does not affect the neutron reflectivity.
For our simulations of the reflectivity in the Meissner
state, the magnetic field profiles shown in Fig. 1 were as-
sumed on both sides of the sample. The spin asymmetry
data for fields 77 Oe and 166 Oe, along with simulations
77 Oe
0.008 0.009 0.010 0.011 0.012
 b 166 Oe
momentum transfer (Å-1)
FIG. 4: Spin asymmetry at T = 1.8 K and H = 77 Oe (a)
and 166 Oe (b). The curves are simulations performed within
the local (dashed line) and nonlocal (solid line) approaches.
for the local and nonlocal field distributions, are shown
in Fig. 4.
For field 77 Oe (Fig. 4a), the results of the “nonlocal”
simulation fit the experimental data somewhat better,
but no clear discrimination between the local and nonlo-
cal approaches is possible due to insufficient accuracy of
the data at this field. A significantly clearer distinction
is apparent for field 166 Oe due to the larger amplitude
of s(Q). As can be seen from Fig. 4b, the quality of the
fits, with B(z) calculated in the local and nonlocal ap-
proaches, is different. The nonlocal simulation fits the
experimental data definitely better. It is worth stress-
ing that no adjustable parameters have been used for the
simulations of spin asymmetry.
In conclusion, nonlocal electrodynamics effects are
measurable in extreme type-I superconductors. State-of-
the-art PNR measurements performed on low-κ super-
conductor In, combined with simulation, unambiguously
support the nonlocal theory and at the same time demon-
strate consistency with the literature data for λL(0) and
ξ0. Consequently, evidence has been gathered for the
nonmonotonic decay and sign reversal of the penetrating
magnetic field predicted by the nonlocal electrodynamics
approach.
We thank A. Volodin for AFM, S. Vandezande for
electrical conductivity, and A.P. Kobzev for Ruther-
ford backscattering measurements. This research has
been supported by the KULeuven Research Council
(F/05/049, GOA/2004/02), project G.0237.05 of FWO-
Vlaanderen, IUAP P5/1, the European Commission 6th
Framework Programme through Key Action: Strength-
ening the European Research Area, Research Infrastruc-
tures (Contract HII3-CT-2003-505925), Russian State
contract 2007-3-1.3-07-01, INTAS grant 03-51-6426 and
RFBR project 06-02-16221.
[1] A.B. Pippard, Proc. R. Soc. (London) A 216, 547 (1953).
For an excellent historical account of nonlocality, see A.B.
Pippard, IEEE Trans. Magnetics MAG-23, 371 (1987).
[2] M. Tinkham, Introduction to Superconductivity
(McGraw-Hill, New York, 1996).
[3] J. Halbritter, Z. Physik 243, 201 (1971).
[4] P. Valko, M.R. Gomes, and T.A. Girard, Phys. Rev. B
75, 140504(R), 2007; K.S. Wood and D. Van Vechten,
Nucl. Instr. Meth. A 314, 86 (1992); I.N. Khlyustikov
and A.I. Buzdin, Advances in Physics 36, 271 (1987).
[5] K.E. Drangeid and R. Sommerhalder, Phys. Rev. Lett.
8, 467 (1962).
[6] D.K. Finnemore and D.E. Mapother, Phys. Rev. 140,
A507 (1965).
[7] X-Ray and Neutron Reflectivity: Principles and Appli-
cations, edited by J. Daillant and A. Gibaud (Springer,
Berlin, 1999).
[8] E. Morenzoni, F. Kottmann, D. Maden, B. Matthias, M.
Meyberg, Th. Prokscha, Th. Wutzke, and U. Zimmer-
mann, Phys. Rev. Lett. 72, 2793 (1994).
[9] A. Suter, E. Morenzoni, N. Garifianov, R. Khasanov, E.
Kirk, H. Luetkens, T. Prokscha, and M. Horisberger,
Phys. Rev. B 72, 024506 (2005); A. Suter, E. Moren-
zoni, R. Khasanov, H. Luetkens, T. Prokscha, and N.
Garifianov, Phys. Rev. Lett. 92, 087001 (2004).
[10] D.K. Finnemore, T.F. Stromberg, and C.A. Swenson,
Phys. Rev. 149, 231 (1966).
[11] G.P. Felcher, R.T. Kampwirth, K.E. Gray, and R. Felici,
Phys. Rev. Lett. 52, 1539 (1984).
[12] H. Zhang, J.W. Lynn, C.F. Majkrzak, S.K. Satija, J.H.
Kang, and X.D. Wu, Phys. Rev. B 52, 10395 (1995).
[13] R. Felici, J. Penfold, R.C. Ward, E. Olsi, and C. Mata-
cotta, Nature 329, 523, (1987).
[14] A. Mansour, R.O. Hilleke, G.P. Felcher R.B. Laibowitz,
P. Chaudhari, and S.S.P. Parkin, Physica B 156-157,
867 (1989).
[15] V. Lauter-Pasyuk, H.J. Lauter, V.L. Aksenov, E.I. Ko-
rnilov, A.V. Petrenko, and P. Leiderer, Physica B 248,
166 (1998).
[16] K.E. Gray, G.P. Felcher, R.T. Kampwirth, and R.
Hilleke, Phys. Rev. B 42, 3971 (1990).
[17] M.P. Nutley, A.T. Boothroyd, C.R. Staddon, D.McK.
Paul, and J. Penfold, Phys. Rev. B 49, 15789 (1994).
[18] K. Temst, M.J. Van Bael, and H. Fritzsche, Appl. Phys.
Lett. 79, 991 (2001).
[19] Handbook of Physical Quantities, edited by I.S. Grigoriev
and E.Z. Meilikhov (CRC, New York, 1997).
[20] V.L. Aksenov, K.N. Jernenkov, S.V. Kozhevnikov, H.
Lauter, V. Lauter-Pasyuk, Yu. V. Nikitenko, and
A.V. Petrenko, www.jinr.ru/publish/Preprints/2004/
047(D13-2004-47)e.pdf.
[21] D.G. Bucknall, S. Langridge, and R.M. Dalgliesh,
www.isis.rl.ac.uk/largescale/crisp/documents/manual/
crisp manual.htm.
	References

  Polarized neutron reflectometry (PNR) provides evidence that nonlocal
electrodynamics governs the magnetic field penetration in an extreme low-k
superconductor. The sample is an indium film with a large elastic mean free
path (11 mkm) deposited on a silicon oxide wafer. It is shown that PNR can
resolve the difference between the reflected neutron spin asymmetries predicted
by the local and nonlocal theories of superconductivity. The experimental data
support the nonlocal theory, which predicts a nonmonotonic decay of the
magnetic field.

Introduction to Superconductivity
(McGraw-Hill, New York, 1996).
[3] J. Halbritter, Z. Physik 243, 201 (1971).
[4] P. Valko, M.R. Gomes, and T.A. Girard, Phys. Rev. B
75, 140504(R), 2007; K.S. Wood and D. Van Vechten,
Nucl. Instr. Meth. A 314, 86 (1992); I.N. Khlyustikov
and A.I. Buzdin, Advances in Physics 36, 271 (1987).
[5] K.E. Drangeid and R. Sommerhalder, Phys. Rev. Lett.
8, 467 (1962).
[6] D.K. Finnemore and D.E. Mapother, Phys. Rev. 140,
A507 (1965).
[7] X-Ray and Neutron Reflectivity: Principles and Appli-
cations, edited by J. Daillant and A. Gibaud (Springer,
Berlin, 1999).
[8] E. Morenzoni, F. Kottmann, D. Maden, B. Matthias, M.
Meyberg, Th. Prokscha, Th. Wutzke, and U. Zimmer-
mann, Phys. Rev. Lett. 72, 2793 (1994).
[9] A. Suter, E. Morenzoni, N. Garifianov, R. Khasanov, E.
Kirk, H. Luetkens, T. Prokscha, and M. Horisberger,
Phys. Rev. B 72, 024506 (2005); A. Suter, E. Moren-
zoni, R. Khasanov, H. Luetkens, T. Prokscha, and N.
Garifianov, Phys. Rev. Lett. 92, 087001 (2004).
[10] D.K. Finnemore, T.F. Stromberg, and C.A. Swenson,
Phys. Rev. 149, 231 (1966).
[11] G.P. Felcher, R.T. Kampwirth, K.E. Gray, and R. Felici,
Phys. Rev. Lett. 52, 1539 (1984).
[12] H. Zhang, J.W. Lynn, C.F. Majkrzak, S.K. Satija, J.H.
Kang, and X.D. Wu, Phys. Rev. B 52, 10395 (1995).
[13] R. Felici, J. Penfold, R.C. Ward, E. Olsi, and C. Mata-
cotta, Nature 329, 523, (1987).
[14] A. Mansour, R.O. Hilleke, G.P. Felcher R.B. Laibowitz,
P. Chaudhari, and S.S.P. Parkin, Physica B 156-157,
867 (1989).
[15] V. Lauter-Pasyuk, H.J. Lauter, V.L. Aksenov, E.I. Ko-
rnilov, A.V. Petrenko, and P. Leiderer, Physica B 248,
166 (1998).
[16] K.E. Gray, G.P. Felcher, R.T. Kampwirth, and R.
Hilleke, Phys. Rev. B 42, 3971 (1990).
[17] M.P. Nutley, A.T. Boothroyd, C.R. Staddon, D.McK.
Paul, and J. Penfold, Phys. Rev. B 49, 15789 (1994).
[18] K. Temst, M.J. Van Bael, and H. Fritzsche, Appl. Phys.
Lett. 79, 991 (2001).
[19] Handbook of Physical Quantities, edited by I.S. Grigoriev
and E.Z. Meilikhov (CRC, New York, 1997).
[20] V.L. Aksenov, K.N. Jernenkov, S.V. Kozhevnikov, H.
Lauter, V. Lauter-Pasyuk, Yu. V. Nikitenko, and
A.V. Petrenko, www.jinr.ru/publish/Preprints/2004/
047(D13-2004-47)e.pdf.
[21] D.G. Bucknall, S. Langridge, and R.M. Dalgliesh,
www.isis.rl.ac.uk/largescale/crisp/documents/manual/
crisp manual.htm.
	References
