The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples
A theory has been developed to explain the anomalous behavior of the magnetic susceptibility of a normal metal-superconductor (NS) structure in weak magnetic fields at millikelvin temperatures. The effect was discovered experimentally (A.C. Mota et al., Phys. Rev. Lett. 65, 1514 (1990)). In cylin...
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irk-123456789-1202112017-06-12T03:05:09Z The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples Gogadze, G.A. Специальный выпуск superconductivity: XX years after the discovery A theory has been developed to explain the anomalous behavior of the magnetic susceptibility of a normal metal-superconductor (NS) structure in weak magnetic fields at millikelvin temperatures. The effect was discovered experimentally (A.C. Mota et al., Phys. Rev. Lett. 65, 1514 (1990)). In cylindrical superconducting samples covered with a thin normal pure metal layer, the susceptibility exhibited a reentrant effect: it started to increase unexpectedly when the temperature lowered below 100 mK. The effect was observed in mesoscopic NS structures when the N and S metals were in good electric contact. The theory proposed is essentially based on the properties of the Andreev levels in the normal metal. When the magnetic field (or temperature) changes, each of the Andreev levels coincides from time to time with the chemical potential of the metal. As a result, the state of the NS structure experiences strong degeneracy, and the quasiparticle density of states exhibits resonance spikes. This generates a large paramagnetic contribution to the susceptibility, which adds up to the diamagnetic contribution thus leading to the reentrant effect. The explanation proposed was obtained within the model of free electrons. The theory provides a good description for experimental results. 2006 Article The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples / G.A. Gogadze // Физика низких температур. — 2006. — Т. 32, № 6. — С. 716–728. — Бібліогр.: 29 назв. — англ. 0132-6414 PACS: 74.50.+r, 74.45.+c http://dspace.nbuv.gov.ua/handle/123456789/120211 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Специальный выпуск superconductivity: XX years after the discovery Специальный выпуск superconductivity: XX years after the discovery |
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Специальный выпуск superconductivity: XX years after the discovery Специальный выпуск superconductivity: XX years after the discovery Gogadze, G.A. The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples Физика низких температур |
| description |
A theory has been developed to explain the anomalous behavior of the magnetic susceptibility
of a normal metal-superconductor (NS) structure in weak magnetic fields at millikelvin temperatures.
The effect was discovered experimentally (A.C. Mota et al., Phys. Rev. Lett. 65, 1514
(1990)). In cylindrical superconducting samples covered with a thin normal pure metal layer, the
susceptibility exhibited a reentrant effect: it started to increase unexpectedly when the temperature
lowered below 100 mK. The effect was observed in mesoscopic NS structures when the N and S
metals were in good electric contact. The theory proposed is essentially based on the properties of
the Andreev levels in the normal metal. When the magnetic field (or temperature) changes, each
of the Andreev levels coincides from time to time with the chemical potential of the metal. As a result,
the state of the NS structure experiences strong degeneracy, and the quasiparticle density of
states exhibits resonance spikes. This generates a large paramagnetic contribution to the susceptibility,
which adds up to the diamagnetic contribution thus leading to the reentrant effect. The explanation
proposed was obtained within the model of free electrons. The theory provides a good
description for experimental results. |
| format |
Article |
| author |
Gogadze, G.A. |
| author_facet |
Gogadze, G.A. |
| author_sort |
Gogadze, G.A. |
| title |
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples |
| title_short |
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples |
| title_full |
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples |
| title_fullStr |
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples |
| title_full_unstemmed |
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples |
| title_sort |
theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| topic_facet |
Специальный выпуск superconductivity: XX years after the discovery |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120211 |
| citation_txt |
The theory of the reentrant effect in susceptibility
of cylindrical mesoscopic samples / G.A. Gogadze // Физика низких температур. — 2006. — Т. 32, № 6. — С. 716–728. — Бібліогр.: 29 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT gogadzega thetheoryofthereentranteffectinsusceptibilityofcylindricalmesoscopicsamples AT gogadzega theoryofthereentranteffectinsusceptibilityofcylindricalmesoscopicsamples |
| first_indexed |
2025-07-08T17:28:07Z |
| last_indexed |
2025-07-08T17:28:07Z |
| _version_ |
1837100686934802432 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6, p. 716–728
The theory of the reentrant effect in susceptibility
of cylindrical mesoscopic samples
G.A. Gogadze
B. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: gogadze@ilt.kharkov.ua
Received February 9, 2006
A theory has been developed to explain the anomalous behavior of the magnetic susceptibility
of a normal metal-superconductor (NS) structure in weak magnetic fields at millikelvin tempera-
tures. The effect was discovered experimentally (A.C. Mota et al., Phys. Rev. Lett. 65, 1514
(1990)). In cylindrical superconducting samples covered with a thin normal pure metal layer, the
susceptibility exhibited a reentrant effect: it started to increase unexpectedly when the tempera-
ture lowered below 100 mK. The effect was observed in mesoscopic NS structures when the N and S
metals were in good electric contact. The theory proposed is essentially based on the properties of
the Andreev levels in the normal metal. When the magnetic field (or temperature) changes, each
of the Andreev levels coincides from time to time with the chemical potential of the metal. As a re-
sult, the state of the NS structure experiences strong degeneracy, and the quasiparticle density of
states exhibits resonance spikes. This generates a large paramagnetic contribution to the suscepti-
bility, which adds up to the diamagnetic contribution thus leading to the reentrant effect. The ex-
planation proposed was obtained within the model of free electrons. The theory provides a good
description for experimental results.
PACS: 74.50.+r, 74.45.+c
Keywords: superconductor-normal metal (proximity) sandwiches, mesoscopic systems, Aharonov–Bohm
effect, Andreev levels, paramagnetic contribution to the susceptibility, reentrant effect.
1. Introduction
Mesoscopic systems [1–3] can exhibit surprising
properties at comparatively low temperatures. For
pure normal metals there is a length scale �N �
� �V /k TF B (VF is the Fermi velocity, T is the tem-
perature, kB is the Boltzmann constant) which has the
meaning of a coherence length in a system with a dis-
turbed long-range order. When this length is compara-
ble with the characteristic dimensions of the system,
the interference effects can come into play. Theo-
retically this was first demonstrated by Kulik [4] for a
thin-wall normal pure-metal cylinder in the vector po-
tential field. It appears that the magnetic moment of
such a system is an oscillating function of the mag-
netic flux through the cross-section of the cylinder,
the oscillation period being equal to the flux quantum
of the normal metal hc/e. The effect is generated by
quantization of the electron motion and due to the
sensitivity of the states of the system to the vector po-
tential field (Aharonov—Bohm effect [5]). Bogachek
and this author showed the existence of oscillating
component with the period hc/e in the magnetic mo-
ment of a singly connected normal cylinder in a weak
magnetic field. Oscillations with this period are pro-
duced by the magnetic surface levels of the cylindrical
sample in a weak magnetic field [6]. The effect of flux
quantization in a normal singly connected cylindrical
conductor was first detected experimentally in 1976
by Brandt et al. when they were investigating the lon-
gitudinal magnetoresistance in pure Bi single crystals
[7,8]. This was actually the first observation of the in-
terference effect of flux quantization in nonsuper-
conducting condensed matter.
Recent advanced technologies of preparation of
pure samples have enabled investigation of the coher-
© G.A. Gogadze, 2006
ent properties of mesoscopic structures taking proper
account of the proximity effect [9]. The samples were
superconducting Nb wires with a radius R of tens of
�m coated with a thin layer d of high-purity Cu or Ag.
The metals were in good contact and the electron
mean free path exceeded the typical scale �N . The
magnetic susceptibilities of copper and silver were
measured. The breakdown field Hb, the supercooled
field Hsc and the superheated field Hsh were esti-
mated as functions of temperature and normal metal
thickness. While continuing their experiments on
these samples, Mota and co-workers [10] detected a
surprising behavior of the magnetic susceptibility of a
cylindrical NS structure (N and S are for the normal
metal and the superconductor, respectively) at very
low temperatures (T � 100 mK) in the external mag-
netic field parallel to the NS boundary.
Most intriguingly, a decrease in the sample temper-
ature below a certain point Tr (at a fixed field) pro-
duced a reentrant effect: the decreasing magnetic sus-
ceptibility of the structure unexpectedly started
growing. A similar behavior was observed with the
isothermal reentrant effect in a field decreasing to a
certain value Hr below which the susceptibility
started to grow sharply. It is emphasized in Ref. 11
that the detected magnetic response of the NS struc-
ture is similar to the properties of the persistent cur-
rents in mesoscopic normal rings. It is assumed [9–12]
that the reentrant effect reflects the behavior of the
total susceptibility � of the NS structure: the paramag-
netic contribution is superimposed on the Meissner ef-
fect-related diamagnetic contribution and nearly com-
pensates it. Anomalous behavior of the susceptibility
has also been observed in AgTa, CuNb and AuNb
structures [11,13].
The reentrant effect revealed by Mota et al. is of
great interest in physics of the quantum proximity ef-
fect in NS sandwiches of ring geometry. We believe
that the effect is not restricted to only NS structures
with the ordinary electron-phonon interaction in su-
perconductors. A modification of the reentrant effect
can well be expected if in place of Nb and Ta high
Tc-superconductors with another type of pairing are
used.
The possibility of the paramagnetic contribution to
the susceptibility of the NS structure needs further
clarification. The NS structure in question is essen-
tially a combination of two subsystems capable of
electron exchange, which corresponds to the establish-
ment of equilibrium in a large canonical ensemble
(with fixed chemical potential). Assume that these
systems are initially isolated with a thick dielectric
layer. It is known that the superconductor response to
the applied magnetic field generates superfluid screen-
ing current near the cylinder surface (Meissner ef-
fect). How does the normal mesoscopic layer respond
to the weak magnetic field? Kulik [4] shows (see
above) that in a weak magnetic field the magnetic mo-
ment of a thin-wall normal cylinder oscillates with the
flux. The magnetic moment oscillations are equivalent
to the existence of persistent current. Since the ener-
gies of the individual states and hence, the total en-
ergy are dependent on the flux, the average current is
nonzero. The current state corresponds to the mini-
mum free energy, therefore the inclusion of weak dissi-
pation would not lead to the decay of the current
state. When the N and S metals are isolated, the quan-
tum states of the quasiparticles in the N metal are
formed at the expense of specular reflection of the
electrons from the dielectric boundaries. The ampli-
tude of the magnetic moment oscillations in the N
layer is small, which is determined by the smallness of
the parameter 1/k RF in the problem and by the para-
magnetic character of the persistent current [4,6]
(when the magnetic field tends to zero, the magnetic
susceptibility is positive). Thus, in the absence of the
proximity effect, the total susceptibility of the NS
structure is only governed by the diamagnetic contri-
bution of the S layer (the paramagnetic contribution
is very small).
When the proximity effect is present in the NS
structure, we assume that the probability of the elec-
tron transit from the superconductor to the N metal is
close to unity. This significantly affects the properties
of the NS structure. The diamagnetic response of the
superconductor persists but new properties appear,
that are brought about by the proximity effect. Now
two kinds of electron reflection are observed in the
normal film — a specular reflection from one bound-
ary and the Andreev reflection from other. Along with
the trajectories closed around the cylinder circle, new
trajectories appear in a weak field, which «screen» the
normal metal. The new trajectories of «particles» and
«holes» confine the quantization area of the triangle
whose base is a part of the NS boundary between the
points of at which the quasiparticle collides with this
boundary. This area is maximum for the trajectories
touching the superconductor. It is shown below that
at certain values of the flux through the triangle area,
the electron density of states experiences flux-depend-
ent resonance spikes. Thus, in the presence of the
proximity effect, the periodic flux-induced oscilla-
tions of the thermodynamic values typical of the nor-
mal layer in the NS structure give way to periodic
resonance spikes with a period equal to a supercon-
ducting flux quantum hc/ e2 [16]. The response of the
normal mesoscopic layer to a weak magnetic field
(H � 10 Oe) is paramagnetic and the susceptibility
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 717
amplitude is large. The picture, however, changes
when the quantized magnetic flux through the trian-
gle area increases and its value divided by hc/ e2 starts
to exceed the highest Andreev «subband» number. A
phase transition occurs in a certain field Hr . As a re-
sult, the N layer is now screened only by the trajecto-
ries of those quasiparticles that do not collide with the
superconducting boundary. Their amplitudes are rat-
her small (see above) against the large diamagnetic re-
sponse. We can thus conclude that the resonance con-
tribution to the paramagnetic susceptibility of the NS
structure can only appear in comparatively weak mag-
netic fields. At this condition the reentrant effect may
be generated. The conclusion correlates well with the
experimental observations [9–14].
The origin of paramagnetic currents in NS structure
was discussed in several theoretical publications.
Bruder and Imry [17] analyze the paramagnetic con-
tribution to susceptibility made by quasiclassical
(«glancing») trajectories of quasiparticles that do not
collide with the superconducting boundary. The au-
thors [17] point to a large paramagnetic effect within
their physical model. However, their ratio between
the paramagnetic and diamagnetic contributions is
rather low and cannot account for experimental re-
sults [9–14].
Fauchere, Belzig, and Blatter [18] explain the large
paramagnetic effect assuming a pure repulsive elec-
tron–electron interaction in noble metals. The proxim-
ity effect in the N metal induces an order parameter
whose phase is shifted by � from the order parameter �
of the superconductor. This generates the paramagnetic
instability of the Andreev states, and the density of
states of the NS structure exhibits a single peak near
the zero energy. The theory in [18] essentially rests on
the assumption of the repulsive electron interaction in
the N metal. Is the reentrant effect a result of specific
properties of noble metals? or Does it display the be-
havior of any normal metal experiencing the proximity
effect from the neighboring superconductor? Only ex-
periment can provide answers to these questions. We
just note that the theories of [17,18] do not account for
the temperature and field dependencies of the paramag-
netic susceptibility and the nonlinear behavior � of the
NS structure. The current theories cannot explain the
origin of the anomalously large paramagnetic reentrant
susceptibility in the region of very low temperatures
and weak magnetic fields.
It is worth mentioning the assumption made by
Maki and Haas [19] that below the transition temper-
ature (� 10 mK) some noble metals (Cu, Ag, Au) can
exhibit p-ware superconducting ordering, which may
be responsible for the reentrant effect. This theory
does not explain the high paramagnetic reentrant ef-
fect either.
In this paper a theory of the reentrant effect is pro-
posed which is essentially based on the properties of
the quantized levels of the NS structure. Levels with
energies no more than � (2� is the gap of the super-
conductor) appear inside the normal metal bounded
by the dielectric (vacuum) on one side and contacting
the superconductor on the other side. The number of
levels n0 in the well is finite. Because of the
Aharonov–Bohn effect [5], the spectrum of the NS
structure is a function of the magnetic flux in a weak
field. The specific feature of the quantum levels of the
structure is that in a varying field H (or temperature
T) each level in the well periodically comes into coin-
cidence with the chemical potential � of the metal. As
a result, the state of the system suffers strong degener-
acy and the density of states of the NS sample experi-
ences resonance spikes.
It is shown that the phenomenon of resonance ap-
pears in a certain interval of weak magnetic fields at
temperatures no higher than a hundred of millikelvins.
Resonance is realizable only in pure mesoscopic N lay-
ers under the condition of the Aharonov–Bohm effect.
The resonance produces a large paramagnetic contribu-
tion � p to the susceptibility of the NS structure. When
� p is added to the diamagnetic contribution �d pro-
duced by the Meissner effect, the total susceptibility
displays the features of the reentrant effect [20].
2. Spectrum of quasiparticles of the
NS structure
Consider a superconducting cylinder with the ra-
dius R which is covered with a thin layer d of a pure
normal metal. The structure is placed in a weak mag-
netic field H( , , )0 0 H oriented along the symmetry axis
of the structure. It is assumed that the field is weak to
an extent that the effect of twisting of quasiparticle
trajectories becomes negligible. It actually reduces to
the Aharonov–Bohm effect [5], i.e. allows for the in-
crement in the phase of the wave function of the
quasiparticle moving along its trajectory in the vector
potential field.
We proceed from a simplified model of NS struc-
ture in which the order parameter magnitude changes
stepwise at the NS boundary. It is also assumed that
the magnetic field does not penetrate into the super-
conductor. The coherent properties observed in the
pure normal metal can be attributed to its large «co-
herence» length �N at very low temperatures.
One can easily distinguish two classes of trajectories
inside the normal metal. One of them includes the tra-
jectories which collide in succession with the dielectric
and NS boundaries. The quasiparticles moving along
718 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
G.A. Gogadze
these trajectories have energies � � and are localized
inside the potential well bounded by a high dielectric
barrier ( � 1 eV) on one side and by the superconduct-
ing gap � on the other side. On its collisions, the
quasiparticle is reflected specularly from the dielectric
and experiences the Andreev scattering at the NS
boundary [15]. We introduce an angle
at which the
quasiparticle hits the dielectric boundary. The angle is
counted off the positive direction of the normal to the
boundary (Fig. 1). In this case the first class contains
the trajectories with
varying within 0 �
� c (
c
is the angle at which the trajectory touches the NS
boundary). The other class includes the trajectories
whose spectra are formed by collisions with the dielec-
tric only, i.e. the trajectories with
� c.
The two groups of trajectories produce signifi-
cantly different spectra of quasiparticles. The distinc-
tions are particularly obvious in the presence of the
magnetic field. The trajectories with
�
c form a
spectrum of Andreev levels which contains a supple-
ment in the form of an integral of the vector potential
field. The spectrum characterizes the magnetic flux
through the area of the triangle between the qua-
siparticle trajectory and the part of the NS boundary.
It is also determines the magnitude of the screening
current produced by «particles» and «holes» in the N
layer. These states are responsible for the reentrant ef-
fect. The trajectories with
� c do not collide with
the NS boundary. The states induced by these trajecto-
ries are practically similar to the «whispering gallery»
type of states appearing in the cross-section of a solid
normal cylinder in a weak magnetic field [6,21]. The
size of the caustic of these trajectories is of the order of
the cylinder radius, i.e. they correspond to high mag-
netic quantum numbers m. The spectrum thus formed
carries no information about the parameters of the su-
perconductor and it is impossible to meet the reso-
nance condition in this case. These states make a para-
magnetic contribution to the thermodynamics of the
NS structure but their amplitude is small ( � 1/k RF ).
It is therefore discarded from further consideration.
Our interest will be concentrated on the trajectories
with
� c.
The spectrum of quasiparticles of the NS structure
can be obtained easily using the multidimensional
quasiclassical method generalized for the case of the
Andreev scattering in the system [16,22]. After colli-
sion with the NS boundary the «particle» transforms
into a «hole». The «hole» travels practically along the
path of the «particle» but in the reverse direction. In
the strict sense, however, the path of the «hole» is
somewhat longer because under the condition of
Andreev elastic scattering the momentum of the «par-
ticle» exceeds that of the reflected «hole». According
to the law of conservation of the angular momentum,
the angle
at which the «hole» comes up to the di-
electric boundary and hence the distance covered by
the «hole» are larger. Eventually, the trajectory of the
quasiparticle becomes closed due to its displacement
along the perimeter of the N layer. However, as the
quasiparticle energy decreases and approaches the
value of the chemical potential, the difference
�
starts tending to zero. Since our further interest is con-
cerned with low-lying Andreev levels, we assume that
the «hole» trajectory is strictly reversible. The dis-
tance covered by the «particle» («hole») between two
boundaries is L0 2� d/ cos
.
According to the multidimensional quasiclassical
method [16,22], there are two congruences of «particle»
rays — towards the dielectric (I) and in the opposite di-
rection (II). There are also two congruences of «hole»
rays — towards the NS boundary (III) and away from it
(IV). The covering space is constructed of four similar
NS structures whose edges are joined in accordance with
the law of quasiparticle reflection from a dielectric and a
NS boundary. At the dielectric boundary the cong-
ruences I and II are joined. The congruences III and IV
are joined independently. The covering space consists of
the outer («particles») and inner («holes») toroidal sur-
faces. Each surface contains only a part of the single
independent integration contour. The path of the «parti-
cle» is 2d. The «hole» travels the same length where-
upon the trajectory of the quasiparticle closes. The total
length of the closed contour along the covering surface
of the NS structure is 4d.
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 719
a
S
H
N
b
N
S
H
Fig. 1. Two classes of trajectories in the normal metal of
NS structure in the magnetic field: trajectories forming the
Andreev levels (a); trajectories colliding only with the di-
electric boundary (b).
It is possible to choose two independent integration
contours within a tours that do not contract into a
point. One condition of quantization relates the caus-
tic radius to the magnetic quantum number m. We re-
place it with an angle of incidence of the quasiparticle
on the dielectric boundary. The other condition of
quantization introduces the radial quantum number n.
Thus, the complete set of quantum numbers describing
the motion of the quasiparticle includes n,
, q, where
q is the quasimomentum component along the symme-
try axis of the cylinder.
Assume that the condition d R�� is obeyed for the
NS structure. We can then neglect the curvature of the
cylinder boundary and assume that it is flat. The con-
dition of quasiclassical quantization can be written as
p A s p A s0 1
0 0
�
�
�
��
�
�
�� � �
�
�
��
�
�
�� �� �| | | |e
c
d
e
c
d
L L
� � ��
�
�
�
�
�2 1
1
�
�
� n /arccos � (1)
where p0 (p1) are the quasimomentum of the «parti-
cle» («hole»), is the «quasiparticle» energy, A is
the vector potential ( , , )0 0 Hy , | |L0 is the trajectory
length covered by the «particle» («hole»). The unity
in the right-hand side of Eq. (1) appears when two
collisions of the quasiparticle with the dielectric
boundary are taken into account [22]. The term
(arccos �/ /�� accounts for the phase delay of the
wave function under the Andreev scattering of
quasiparticles [16]. The quasimomentum p0 and p1 in
Eq. (1) can be expanded in the parameter �/ retain-
ing the first-order terms and replacing n � 1 by n. As a
result, Eq. (1) furnishes the sought for spectrum of
the NS structure in a weak magnetic field (L is the
quasiparticle trajectory):
�
�
�n q
v q
d
n( , ; )
( ) cos
.�
�
�� � ��
�
�
�
�
�
� L
2
1
arccos
tg
(2)
Here v q p q /mFL ( ) *� �2 2 , pF is the Fermi momen-
tum, q is the quasiparticle momentum component
along the cylinder axis, m* is the effective mass of the
quasiparticle, �0 2� hc/ e is the superconducting flux
quantum. The positive
-values refer to «particles»
( )n � 0 , while the negative ones are for «holes»
(n � 0).
The last term in Eq.(2) has the meaning of «phase»
�
�
� �2
0
0
�
A x dx
d
( ) , (3)
which is dependent on the vector potential field and
varies with the angle
characterizing the trajectory
of the quasiparticle.
The spectrum of Eq. (2) is similar to Kulik’s spec-
trum [23] for the current state of an SNS contact.
However, Eq. (2) includes an angle-dependent mag-
netic flux instead of the phase difference of the con-
tacting superconductors.
The value of the «phase» (flux) controls the dia-
magnetic and paramagnetic currents in the NS struc-
ture. To calculate it, we should know the distribution
of the vector potential field inside the normal metal.
The problem of the Meissner effect in superconduc-
tor-normal metal (proximity) sandwiches was solved
by Zaikin [24]. It was shown that the proximity ef-
fect caused the Meissner effect bringing an
inhomogeneous distribution of the vector potential
field over the N layer of the structure: A x Hx( ) � �
� ( ) ( )4�/c j a x d x/( ).� 2 For convenience we intro-
duce the notation a A x dx
d
� � ( )
0
. This expression can
be obtained from the Maxwell equation rotH �
� ( )4�/c j with the boundary conditions A x( )� �0 0
and � � �x A x d H( ) . The screening (diamagnetic)
current j is a function of a, j a j a/s( ) ( )� � � �0 , where
js is the superfluid current and �( )x is function of
flux. Thus, we can write down the self-consistent
equation for a [25,26]:
a
Hd
c
j a d� �
2
3
2
4
3
�
( ) . (4)
The diamagnetic current jd a( ) was calculated in
terms of the microscopic theory as a sum of currents of
quasiparticles («particles» and «holes») for all
quasiclassical trajectories characterized by the angles
� and � [24,26] (below the system of units kB � ��
� �c 1 is used):
j T AT dd
/
n
( , )
sin cos sin[ cos ]
�
�
�
� �
�
��
�
�
� � � �
�
�
� 0
2
0
2
2
2 tg
2
2
20
2
� �
�sh ch tg
�
� �
�
�
n
n
n
/
d
�
�
�
�
�
�
!
!
�
�
cos ( cos )
, (5)
720 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
G.A. Gogadze
where A ek /F� 2 2 2� , � �n n T� �( )2 1 , 2� is the su-
perconductor gap,
� �n n Fd/v� 2 cos , and � is
given by Eq. (3). The function j d ( )� is noted for in-
teresting features. In small magnetic fields (� �� 1)
j jd
s" � �. Such low fields can lead to the effect of
extrascreening of the external magnetic field (see
[24]). When the field increases (� � 1), the current
starts oscillating and for certain «phases» it turns to
zero at regular intervals «phases» �. With high val-
ues of the inequality (� �� 1), the current amplitude
decreases.
3. Resonance spikes in the density of states of
NS structure in weak magnetic fields
In the region of weak magnetic fields, the density
of states of the quasiparticles that are described by the
spectrum of Eq. (2) exhibits sharp singularities. The
spectrum of Eq. (2) is formed by the trajectories of
the quasiparticles which collide with the dielectric
and superconducting boundaries. It encloses a certain
area penetrated by a magnetic flux. At any instant
when the magnetic flux becomes a multiple of the
superconducting flux quantum, the density of states
experiences resonance spikes.
Let us consider the cross-section of a NS structure.
Assume that the superconducting cylinder radius R
and the normal layer thickness d have a mesoscopic
scale. The density of states # ( ) can be calculated pro-
ceeding from the expression
# $
� �
( ) [ ( , )].
, ,
� ��� dq q
n
n (6)
The summation is taken over all quantum numbers
n, q,
and spin %. Since we are not interested in the
contribution from the states formed by the trajectories
of the quasiparticles with
� c, we can write down
#
#
�
�
( ) ( ; )�
�
� d
c
c
, (7)
where #
( ; ) is the contribution to the density of
states from the pre-assigned trajectory with a fixed
.
Eq. (2) for the low-lying Andreev levels ( �� �) is
taken as a spectrum. After integration with respect to
q and introduction of the notation & �� �/ dm2 *, we
can pass on to the dimensionless energy ª � &/ pF .
For #
( , )ª we have the expression
#
� &
� '
' '
( , )
[| | ]
( ) ( )
ª ª
ª
ª
�
� �
� � �
2
2
2
2
2 2 2
p
d
n
n n
F sec sec
sec2
n
� ,
(8)
where '
�� �1 2/ /�tg , and �( )x is the stepwise
Heaviside function. Eq. (8) suggests two cases de-
pending on the parameter n � '.
1. Non-resonance case. If n � (' 0, the energy de-
pendence under the radical sign in Eq. (8) can be ne-
glected for small energies (ª ) *�. Then, the non-
resonance contribution to the density of states is
#
� &
'
�
( )
ª
( )
0
2
2
0
2 2
3
� � �
�
���
�p
d
d
n
F
n
c
sec . (9)
The series in Eq.(9) is calculated readily by the for-
mula in [27]:
1
1
1
1
1
1( )
( )
( )!k n
d
dn
k
n
n
n�
� �
�
���
�
�
�
��
'
�
'
�'ctg .
After calculation of the integral we obtain
#
&
�( )
ª
0 2 0 2
0
2 2
�
�
�
�
�
!
p
d a
a R
d
F �
�
tg , (10)
where 2R/d c� tg
.
2. Resonance case. Now we go back to Eq. (8). We
find # res as
#
�
�
res sec tg sec
tg
� � � �
� �
ª
ª[| | ]
| | |
2
0
2
2
d
b
b
c
n
n n
�
� � bn tg sec
|
,
ª
2 2 2�
�
(11)
where the notations �n n /� � 1 2, b � 2 0a/� are in-
troduced. Equation (11) shows that at certain values
of the flux (b), the radicand in the denominator turns
to zero.
Prior to calculation of # res , let us discuss the ques-
tion of the contribution of different angles
to the
resonance amplitude. It is reasonable to assume that
because of the factor sec2
in the numenator of Eq.
(11), the angles
� c are the main contributors to
the integral. It is convenient to employ in the integral
a new variable of integration x � tg
. Then the neigh-
borhood of the upper limit x c0 � tg
is the main con-
tributor to the integral. Introducing the notation
~a bxn� �� 0 and the small deviation � � � ��x x0 1,
we can write down the equation for the resonance con-
dition as:
( ) (~ ) ~ ( )ª ª ª
~
b ab x a x2 2 2 2
0
2 2
0
22 1 0� � � � � � �� �
. (12)
The point of our interest is the asymptotics # ( ) at
low ª ) *. Eq. (12) is solved to the accuracy within
first-order terms of | |:ª
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 721
�12 0
21,
~ | |ª
�
a
b b
x+ � . (13)
The expression in front of the radical in the denomi-
nator of Eq. (11) has the second order smallness in | |ª ,
i.e. | ~|a 2 � | | ( )ª
2
0
21 � x , which leads to its cancella-
tion with the similar small parameter in the numenator.
The remaining integral is estimated to be a constant
of about unity. The resonance-induced spike of the
density of states always appear when the Andreev
level coincides with the Fermi energy at a certain flux
in the N layer. In the vicinity of the chemical potential
there is a strong degeneracy of the quasiparticle states
with respect to the quantum number q. As a result, a
macroscopic number of q states contribute to the am-
plitude of the effect. Near the resonance, the ratio of
the resonance and nonresonance amplitudes of the
density of states is
#
#
res
( ) | |ª
0 2
1
1� �� . (14)
It is thus shown that a change in the magnetic flux
leads to resonance spikes in the density of states of the
NS structure. The flux interval between the spikes is
equal to the superconducting flux quantum �0.
4. Calculation of susceptibility of NS contact
To explain the reentrant effect, we need to have an
expression for the susceptibility of the NS structure.
We assume that in a weak magnetic field the total sus-
ceptibility of the NS sample consists of two contribu-
tions. Firstly, the response of the superconductor to the
applied magnetic field generates the Meissner effect.
Note that the diamagnetic response is observed in all
fields up to the critical one. The amplitude of the dia-
magnetic current increases monotonously with lower-
ing temperature. On the other hand, the presence of a
pure normal metal in the NS structure produces a para-
magnetic contribution. In a weak magnetic field the
contribution is due to the Aharonov–Bohm effect and
the quantization of the quasiparticle spectrum of the
mesascopic system. When the penetrability of the bar-
rier between the metals is small, the electrons of the
normal metal are reflected specularly from its bound-
aries. As compared to the diamagnetic contribution
from the superconductor, the paramagnetic contribu-
tion produced by the N layer has a small amplitude and
can therefore be neglected. Thus, the paramagnetic and
diamagnetic contributions cannot compete in the ab-
sence of the proximity effect in the NS structure. How-
ever, if the penetrability of the barrier is close to unity,
the mechanism of the Andreev reflection becomes ac-
tive at the NS boundary The quasiparticle spectrum of
the N layer undergoes a significant transformation and
resonance spikes appear in the amplitude of the density
of states in a certain regions of magnetic fields and tem-
peratures. Simultaneously, the distribution of the vec-
tor potential field in the normal layer becomes
inhomogeneous. As shown below at certain values of
the parameters of the problem, the paramagnetic con-
tribution to the susceptibility of the NS structure can
become equal to the diamagnetic contribution. This is
the reason why the reentrant effect appears in pure
mesoscopic NS structures.
Theoretically, the resulting susceptibility includ-
ing the reentrant effect can be represented as a sum of
the paramagnetic contribution � p of the NS structure
caused by the Andreev scattering and the diamagnetic
susceptibility �d of the system in which there is no
proximity effect between the N and S metals. The tem-
perature-induced behavior of the diamagnetic current
in such a system is well known. As the temperature de-
creases, the diamagnetic current amplitude increases
and becomes saturated at temperatures about several
millikelvins. At high temperatures k T V /dB F�� � ,
the diamagnetic current decreases rapidly following
the law j T k Td/ VB F�
� �1 4exp( )� � . Note that in a
NS structure in which the electrons are reflected
specularly at both boundaries of the normal metal, the
susceptibility is negative (i.e. diamagnetic) in the
whole interval of temperatures 0 � �T Tc. However,
we will not use this approach to estimate the resulting
susceptibility. Below we calculate the screening cur-
rent of the NS structure. It naturally allows for the
paramagnetic contribution at certain values of the
magnetic field and temperature. We focus our atten-
tion on calculation of the paramagnetic contribution
in structures with a pronounced proximity effect. This
is important especially in the context of the recent
statement [28] that no paramagnetic reentrance can
occur in NS proximity cylinders in the absence of elec-
tron-electron interaction in the N layer.
Paramagnetic susceptibility of NS contact
The contribution of the states in Eq. (2) to the
paramagnetic susceptibility of the normal layer in a
NS contact can be calculated proceeding from the ex-
pression for the thermodynamic potential (kB � 1)
, � � � ��T q /T
n q
nln [ exp( ( , ) )]
, ,
1
�
�
, (15)
where the summation is taken over the spin (%) and
all the states related to the trajectories of the
quasiparticles with a � ac. The expression for sus-
ceptibility (per unit volumeV of the normal metal) is
found using the formula
722 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
G.A. Gogadze
� � �
�
�
1 2
2V H
,
.
After performing the summation over the spin and
taking into account two signs of the angle
and of the
quasimomentum component q, we arrive at the initial
expression for paramagnetic susceptibility (� is the
chemical potential of the metal):
�
�
�
-
�
�
�d
Tm
d /T
/T2 12
0
2 2*
exp( )
[exp( ) ]�
- � ��� �d dq p q q
c F
n
F
/
n
p
$
�
cos sin ( ) ( ( , )).2
0
2 2 3 2
0
(16)
In Ref. 20 we lost one of the radicals ( )p qF
/2 2 1 2�
in the similar initial expression for �. As a result, the am-
plitude of the paramagnetic contribution appeared to be
underestimated. This mistake is corrected in this work.
It is convenient to present the spectrum in terms of
& �� �/ m d2 * and '
�
� �
1
2
tg
as�
&
'n Fq n p q( , ) cos ( ) .� � �2 2
Now we introduce the dimensionless energy
ª ( )� � & $ / p /F , $ �� �V / dF ( )2 is the distance
between the Andreev levels in the SN structure. Since
� $ / �� 1, the lower limit of the energy integral can be
replaced with �.. By introducing the variable x � tg
and the notation �n n /� � 1 2, b b H T a/� �( , ) 2 0� ,
a A x dx
d
� � ( )
0
, x R/d0 0 2� �tg
and taking into account the parity of the integrand
we obtain, instead of Eq. (16):
�
/
�
�
�
� �
�
�
� ��C
d
/
x dx
bx
bx
n
x
n
n
n
ª ª
ª
ª
( ) ( )
[4
2
0
2
4
002
00
ch �
� 1
1
2
2 2 2
�
� � �
x
bx xn
]
( ) ( )ª�
. (17)
In Eq. (17) the summation is taken over the quan-
tum numbers of the «particles». Here C d/T� �2
0
2� ,
/ $ � /T, n0 is the number of Andreev levels in the po-
tential well and � is the Heaviside step function. It is
seen in Eq. (17) that for the given «subzone» n the
amplitude of the paramagnetic susceptibility increases
sharply whenever the Andreev level coincides with the
chemical potential of the metal. The resonant spike of
susceptibility occurs when �n bx� tends to zero on a
change in the magnetic field (or temperature). Be-
cause of the finite number of Andreev levels, the exis-
tence region of the isothermal reentrant effect is
within 0 � H � Hmax.
Let us calculate the integral over x in Eq. (17). It
contains a singularity under the radical R x( ) �
� � �Ax Bx C2 where A b� �2 2
ª , B bn� �2� ,C �
� ��n
2 2
ª . The singularity is determined by the roots
of the quadratic equation
x
b
b b
bn
n12 2 2 2 2
2 2 2
,
ª
ª
ª
ª
| |
�
�
+
�
� �
�
� .
On introducing the notation
0 � �n/b, the ex-
pression for the roots can be written with a linear ac-
curacy with respect to ª as
x
b12 0 0
21,
| |
.
ª
�
+ � (18)
The main contribution to the integral over x,
Eq. (17), is made by the vicinity of the point ª ) *. If
we exclude the singular points from the interval of in-
tegration, the indefinite integral over x can be calcu-
lated accurately (see the details in the Appendix). Be-
cause the �-function is present under the integral, the
integration intervals ( , )0 1x and ( , )x x2 0 make a finite
contribution to the integral. On substituting the limits
of integration, the expressions obtained have different
powers of the parameter | |ª
�1. We retain only the most
important terms in order | |ª
�4 that determine ampli-
tude of the effect. The discarded terms have higher or-
ders of ª-smallness. The intervals ( , )0 1x and ( , )x x2 0
make contributions of the same order of ª-magnitude.
The region ( , )x x1 2 does not contribute to the integral
at all.
The estimate for the integral over x is
4
3 1
10
2
0
2 2 4
b( ) ª�
. (19)
On substituting Eq. (19) into Eq. (17), the parame-
ter ª
4 drops out of the energy integral and we can
take it quite easily. Taking into account the energy
limits �( | | )ª�n � appearing in the process of calcula-
tion we can obtain the expression for the paramag-
netic contribution to the susceptibility of the NS
structure, which in dimensional units has the form
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 723
�
�
�
�
p
F
F
Bd
V
b H T
V
dk T
n /
n
�
16
3
4
1 22 2
0
2
�
�
�
( , ) ( )
(
th �
�
�
�
�
!
� 1 2 1
1 2
2
2 2
0
0
/
b H T
n /
n
n
)
( , )
.
�
�
�
�
�
�
�
�
�
�
�
�
�
!
!
�
�
(20)
In Eq. (20) the summation over the quantum num-
ber n is taken within finite limits, where n0 has the
meaning of the maximum number of the Andreev lev-
els inside the potential well of the NS structure. Its or-
der of magnitude in n /0 0 � $ , where $ is the dis-
tance between the Andreev levels, $ �� �V / dF 2 , and
2� is the energy gap. The flux b H T a/( , ) � 2 0�
depends on both the magnetic field and temperature.
In the pre-assigned field its value is dictated by the
screening current of the NS structure j � � j a/s�( )�0
(see Eq. (4)). The obtained expression for � p mani-
fests a more rapid decrease susceptibility at the in-
creasing parameter b H T( , ) than it was evidenced by
Eq. (5) in Ref. 20.
We first discuss the isothermal case of a very low
temperature and clear up the qualitative behavior of
susceptibility in Eq. (20). We shall proceed from the
region of very strong magnetic fields (a/�0 1�� ) in
which the second term in Eq.(4) is negligible. Then
the dimensionless flux b H T( , ) �� 1 and the amplitude
of the paramagnetic contribution in Eq. (20) decreases
as b H T( , ) raised to power 3. In comparatively weak
magnetic fields (a/�0 1� ), the function �( )x is actu-
ally an oscillating function of H and here we can
expect the reentrant effect. Indeed as the field
decreases to a certain value and the parameter
b H T /n( , ) 0 becomes 0 1 (n0 is the number of the
Andreev levels in the potential well), the amplitude of
the paramagnetic susceptibility of the NS structure ac-
cepts for the first time an appreciable contribution
from the highest Andreev «subband» (level). On a
further decrease in this field, the contribution from
the highest «subband» persists, but in a certain lower
field an additional contribution appears from the
neighboring lower-lying «subband» n0 1� . Finally, in
a very weak field all the «subbands» of the NS struc-
ture start to contribute and the paramagnetic suscepti-
bility amplitude reaches its peak. However, at H ) 0
( )a/�0 0) , the paramagnetic contribution turns to
zero, as follows from Eq. (20). The reason is that the
resonance condition for the Andreev levels (Eq. (2)),
cannot be realized in a zero field.
Now we change to the case when the temperature of
the NS structure varies but the field is kept constant.
We assume the field to be weak (H � 1 �2 10 1 Oe).
The second term in Eq. (4) for the flux is very impor-
tant. It is highest at millikelvin temperatures. As a
result, the parameter b H T( , ) has the lowest value. In
this temperature region the hyperbolic tangent is close
to unity and the paramagnetic contribution is depend-
ent only on the parameter b H T( , ). Under this condi-
tion, all the «subbands» of the NS structure contrib-
ute to the amplitude of the effect. As the temperature
rises, the parameter b H T( , ) increases smoothly. Si-
multaneously, the argument of the hyperbolic tangent
decreases. At a certain temperature, when the condi-
tion k T V / dB F� �� 4 is met, the contribution from the
lowest «subband» starts dying down and its ampli-
tude is decreasing linearly with growing T. On a fur-
ther rise of the temperature, the contributions from
the higher «subbands» of the spectrum die down in
succession. Finally, at a very high temperature the
paramagnetic contribution tends to zero.
Let us estimate the amplitude of the paramagnetic
contribution. The parameter b H T( , ) is dependent on
the value of the flux a A x dx
d
� � ( )
0
, which at constant T
can be found by solving the self-consistent equation Eq.
(4). In the region of millikelvin temperatures and mag-
netic fields H � 1 �2 10 1 Oe the paramagnetic
contribution has the largest amplitude. We obtain
b H T( , ) � �10 4 in this region of T and H. The coeffi-
cient before the sum in Eq. (20) can be found by
substituting �Ag erg� 8 75 10 12. 1 � , d � 1 �3 3 10 4. cm
VF
Ag
� 11 39 108. cm s/ for the characteristic parame-
ters of the normal Ag layer. We thus obtain
16 3 2 418 102 2
0
2 3� �d / VF� � � . 1 . The product of this
coefficient and the parameter b H T( , ) yields the order of
magnitude of the paramagnetic contribution amplitude.
It is seen that the largest amplitude of the paramagnetic
contribution exceeds that of the diamagnetic contribu-
tion in the vicinity of T � 0.
Full magnetic susceptibility of NS structure in the
presence of proximity effect
Let us consider a structure in which the electrons
experience the Andreev scattering at the NS boundary.
In the presence of magnetic field, the screening cur-
rent is induced in the normal layer due to the Meissner
effect. We estimate the susceptibility generated by
this current.
The total current J is related to the magnetic mo-
ment M as
M
c
JS�
1
0, (21)
where S R0
2� � is the cylinder cross-section (d R�� ).
Let the average current density be j. The total current
724 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
G.A. Gogadze
is then J Sj� , where S dL� (L is the cylinder gene-
ratrix). The density of the screening current in NS
proximity sandwiches was calculated by Zaikin [24,28].
We reproduce the formula for the current density
(see Eq. (5)), which is valid at arbitrary values of tem-
perature and magnetic field. At T V /dF�� � it is
j
ek T
dF
/
n
( ) sin cos
sin[ cos ]
�
�
� � ��
�
4 22
2
0
2
0
2
�
� � �
� �
�
�
tg
sh tg2 2
0
2
� �
�
�
n
/
d
�� cos [ cos ]�
. (22)
Here
� �n n Fd/ V� 2 ( cos ), � �n n T� �( )2 1 and the
phase � follows from Eq. (3). Near T � 0 the summa-
tion of frequencies in Eq. (22) can be replaced with in-
tegration. For � � 1 the response of the current is [28]
j
ek V
d
dF F
/
( ) sin cos cos sin[ cos ]� �� � �
2
3
0
2
2
0
2
�
� � � � � �
�
tg
�
�
/
d
2
� . (23)
If the field is small enough to meet the condition
� �� 1, Eq. (23) reduces to the result that was ob-
tained for the first time in [24]:
j
ek V
d
F F( )� �� �
2
26�
.
(24)
At «phases» � �� 1, the screening current of Eq. (23)
turns to zero.
The current-phase dependence at T � 0.092 K is
plotted in Fig. 2. The dependence is nonlinear and its
amplitude has a maximum at a certain value of �.
Knowing the current-phase dependence, we can deter-
mine the susceptibility of the NS structure using the
equation � � dM/dH. It is seen in Fig. 2 that the sus-
ceptibility � of the NS structure (the derivative of cur-
rent with respect to field) changes its sign at a certain
low value of the magnetic field Hr . The «paramag-
netic» portion of the curve is due to the proximity ef-
fect at the NS boundary and to the Andreev levels in
the N layer.
Let us estimate � in the linear-response regime near
T � 0 when this dependence is described by Eq. (24).
In such weak fields we obtain � �� 3 2
0� 2H T /N ( ) ,
where the «penetration depth» 2N into the normal
metal is dependent on temperature [26]:
2
�
N
ne
m c
T� �2
2
2
0
4
0( ) ( );
*
�
2 2
�N N
A
A A
FT
T
T/T T T
V
d
� �0 � �� �
�
�
�
�
�
�2 2 0
6
2
2
( ) exp( ) .
�
(25)
The estimate of susceptibility in the millikelvin re-
gion is
�
�
2
0 �
R
c
ek V
d
F F N
4
02 2
0
( )
�
.
For the parameters of the problem
d � 1 �3 3 10 4. cm, R � 1 �8 2 10 4. cm,
kF
Ag
�12 108 1. 1 �cm , VF
Ag cm0 11 39 108. /s,
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 725
a
~ |j( |��
j =
j
0
�0.5 1.5 2.5
b
d�
dj
~
�0.5 1.5 2.5
Fig. 2. Dependence of screening current Eq. (5) on «phase»
� at T = 0.092 K. Current is in arbitrary units (a). The de-
rivative of current with respect to «phase» (magnetic field)
changes its sign in the region of low �-values (b). When the
magnetic field tends to zero the full magnetic susceptibility
of NS structure is positive for d = 3.3�10–4 cm.
2N ( )0 � 2 10 61 � cm
we obtain � � �0 06. , which is close to � �
� � 3 4 1 4/ /( )� .
5. Discussion
In this study we have investigated the behavior of a
superconducting cylinder covered with a thin layer of
a pure normal metal. It is assumed that the normal
metal and superconductor are in good contact. The
system was placed in a magnetic field directed along
the NS boundary. The NS structure has mesoscopic
scale dimensions. It is assumed that the mean free path
of the quasiparticles in the N layer exceeds the charac-
teristic length �N F BV /k T� � , which has the mean-
ing of the coherence length for a system with dis-
turbed long-range order. The goal of this study was to
interpret the experiments in which A.C. Mota et al.
[10–14] observed anomalous behavior of the magnetic
susceptibility of a NS structure. This phenomenon was
called a reentrant effect. Until recently it has not been
explained adequately.
Earlier [20] the author clarified the nature of the
reentrant effect. As was found, the origin of the para-
magnetic contribution is closely connected with the
properties of the quantized Andreev levels that are de-
pendent on the magnetic flux varying with both tem-
perature and magnetic field. Typically, the levels in
the NS structure time from time (at certain values of
the field H or temperatures) coincide with the chemi-
cal potential of the metal. As a result, the state of the
system is highly degenerate and the density of states
of the NS structure experiences resonance spikes. The
response of the normal mesoscopic layer to a weak
magnetic field is paramagnetic.
A theory of the reentrant effect has been developed
in this study. We calculated the paramagnetic contri-
bution separately and analyzed its behavior in a vary-
ing magnetic field and at varying temperature. In the
course of this calculation we corrected the mistake
made in [20] which led to underestimation of the ef-
fect amplitude. The paramagnetic response is deter-
mined only by the trajectories of the quasiparticles
that collide with the NS boundary. It is shown that
the reentrant effect can occur in a certain range of
weak magnetic fields at temperatures no higher than
100 mK. We believe that paramagnetic reentrant ef-
fect is an intrinsic effect of mesoscopic NS proximity
structures in the very low temperature limit.
Assume that the temperature of the NS structure is
about 10 3� K and the magnetic field is increasing. As
soon as the field exceeds a certain value Hr , the iso-
thermal reentrant effect must vanish. In strong fields
the Andreev levels cease to make a resonance contribu-
tion to the paramagnetic susceptibility. Now the para-
magnetic contribution is made by the states formed by
the trajectories of the quasiparticles that collide only
with the dielectric boundary. However, their contri-
bution to the resulting susceptibility of the structure
is small because of the smallness of the quasiclassical
parameter of the problem 1/k RF . Under this condi-
tion the susceptibility exhibits diamagnetic behavior
in all strong fields up to the critical one.
A self-consistent calculation of the screening cur-
rent of the NS structure was performed taking into ac-
count the contribution from the Andreev levels. The
analysis of the derived expression suggests the para-
magnetic contribution to current. For example, Fig. 2
illustrates the dependence of the current upon the
phase (magnetic field). The values of the current j to
the left of the extremum �r account for the contribu-
tion of the Andreev levels. The derivative of this curve
with respect to the field is proportional to the mag-
netic susceptibility of the NS structure. It is positive
(«paramagnetic») in the region of low magnetic fields
and negative («diamagnetic») in high fields.
Similar behavior is observed when the susceptibil-
ity of the NS structure is measured as a function of
temperature in a pre-assigned weak magnetic field: it
is «paramagnetic» in the region T Tr� and «diamag-
netic» at T TR� up to the critical temperature. Tem-
perature dependence of magnetic susceptibility in the
NS structure at fixed magnetic field will be investi-
gated in detail in separate publication.
In the absence of the proximity effect in the NS
structure, when the penetrability of the barrier be-
tween the S and N metals is small, the electrons of the
normal metal are reflected specularly from its bound-
aries. In this case the SN structure is a total of two iso-
lated subsystems (normal metal and superconductor)
placed into a magnetic field. Because of the Meissner
effect, diamagnetic current develops near the super-
conductor surface. In normal metal, because of the
Aharonov–Bohm effect, the quantized spectrum of
quasiparticles is dependent on the magnetic flux
through the cross-section of the cylinder. The flux
generates a paramagnetic contribution to the suscepti-
bility whose quasiclassical parameter of the problem
1/k RF is small. Hence, in the absence of the proximity
effect no competition is possible between the paramag-
netic and diamagnetic contributions in the NS struc-
ture, and the reentrant effect is unobservable in such
NS sample.
To conclude, it should be noted that the explana-
tion proposed in this study for the reentrant effect was
developed within a model which does not assume the
electron-electron interaction in the N layer of the NS
structure. In terms of the free-electron model, a large
726 Fizika Nizkikh Temperatur, 2006, v. 32, No. 6
G.A. Gogadze
paramagnetic contribution to the susceptibility of the
NS structure appears in the region of very low temper-
atures in a weak magnetic field. If we increase the
thickness d of the pre-assigned normal metal, this
would lead to a greater number of the Andreev levels
n0 in the potential well and affect the solutions of the
self-consistent equation for a. As a result, the shape of
the curve of the paramagnetic susceptibility would be
slightly «deformed» though its qualitative behavior
would remain the same.
The author is sincerely grateful to A.N. Omel-
yanchouk for helpful discussions and support, to S.I.
Shevchenko for valuable comments. I am indebted to
I.O. Kulik for reading through the text and useful
comments.
Appendix
Let us calculate the integral taken over x in Eq. (15):
J
x bx x dx
bx bx x
n
n n
x
�
� � �
� � � �
�
2 2
4 2 2 2
0
1
1
0
�[ ]
( ) ( ) ( )
.
ª
ª
�
� �
(A.1)
After introducing the notation
0 � �n/b, we can
see that the function in front of the radical in the de-
nominator has a singularity at the point x �
0. Be-
sides, as was noted in the text, the integrand has sin-
gularities at the points x1, x2.
Integral (A.1) can be written as a sum of four
integrals
J dx dx dx dx
x x
x
x
� � � � �� � ��
�
�
�
� � � �
0
0
0
0 1
1
0
0
2
lim
�
�
�
� �
� �
�
�
� �
3
4
5
65
7
8
5
95
dx
x
x
�
2
0
.
It is obvious that the presence of the �-function
makes the second and the third integrals equal to zero.
We first calculate the integral J1:
J
b
x dx
x Ax Bx C
x
1 4 0
2
0
4 2
0
1
1
�
� � ��
�
�lim
( )�
�
, (A.2)
where A b� �2 2
ª , B bn� �2� , C n� ��
2 2
ª . On sub-
stituting the variable
0 1� �x /t, the indefinite in-
tegral becomes
t t dt
t t A
( )
&
0
2
2
1�
� �
� ,
where
� � �( )ª1 0
2 2, &
� 2 0
2
ª . It can be calcu-
lated by the method of undetermined coefficients:
f t dt
t t
A t A t An n
n
( )
( )
& :2
1
1
2
2
� �
� � � � -� � �
�
- � � �
� �
�
& :
& :
t t A
dt
t t
n
2
1
2
if f t( ) is the polinomial to power n. Although the cal-
culation is tedious, it is actually simple. The coeffi-
cients A1, A2, A3, and A4 are readily found as:
A A
/
1
0
2
2
0
2 2
0 0
2
2
0
2 23 1
1
1
� �
�
�
�
�
ª
ª
ª( )
,
( )
( )
,
A
/n
3
2
4
0
2 2
0
2
0
4
2
0
2 3
2
3 1
3 5 2
3 1
� �
�
�
� � �
�
�
ª ª( ) ( )
,
A
/ /n
4
2
0
2
2
0 0
2 2
0 0
2
0
2 3
2 1
1
2 2
1
�
�
�
�
�
�
� ( )
( )
( )
( )ª
.
It is seen that the coefficients have different orders of
ª
�1-magnitude: A A A1 2 4
2, , ª� � , A3
4� ª
� . Finally,
we have to calculate six integrals
J
b
A t R t A R t
t
t
1 4 0
1 2
1
2
0
1
� � �
�
�
�lim { ( ) ( )
�
�
� � � �A t R t A A /
t
R t
1
3
2 1
2
2
&( ) ( )
( )
� ��
�
�
�
�
� �
� 7
8
9
A A
t
R t
A / A
R t
3 2
3 4
2
2
& &
( )
( )
( )
, (A.3)
where R t t t A( ) � � �
&2 and the designations
t /0 01�
, t x1 0 1
1� � � � �
( ) are introduced.
All the six indefinite integrals in expression (A.3) can
be calculated accurately [29]. After substituting the
limits of integration, integrals 1, 2, 3, 4 and 5 are
bounded above on energy, which is due to the term
R t /n( ) ª� �
2 2
0�
, i.e. �( )ª�n � . Taking into
account the determined coefficients Ai (i � 12 3 4, , , ),
we can obtain the final expression for J1::
J
b
b / b /
1 4
0
2
2
0
2 2
0
2
2
0
2 2
1 1 3
3 1
1 6
1
�
( )
( )
( )
( )ª ª
�
�
�
�
�
3
4
5
65
�
�
�
�
� �
�
�
2
3 1
1
5
3
6
1
2
4
0
2 2
0
2
0
4
2
0
2 3
�nb b /
ª ª( )
( )
( )
�
�
�
�
�
�
�
0
2
0
2
3
0
2 5 2
0 0
2
0
2 7 2
2 1
1
2 2
1
b / /
/ /
( )
( )
( )
( )ª ª
The theory of the reentrant effect in susceptibility of cylindrical mesoscopic samples
Fizika Nizkikh Temperatur, 2006, v. 32, No. 6 727
�
�
�
�
�
�
7
8
5
95
b / /( )
( )
( )
( )ª
0
2
2
0
2 2
0
2
0
2 3
2 1
1
2 2
1
(A.4)
Of all the terms in (A.4), the most significant con-
tribution is made by the third term because there is a
factor ª
4 in the numerator of the integral over the en-
ergy in Eq. (17). The contributions of the other terms
are negligible. We thus obtain the estimate
J
b
1
0
2
0
2 2 4
2
3 1
1
�
( ) ª�
. (A.5)
A similar calculation of the integral
J
b
x dx
x Ax Bx Cx
x
4 4 0
2
0
4 2
1
2
0
�
� � ��
�lim
( )�
�
gives a contribution, which is identical in the order of
magnitude with (A.5). As a result, we obtain the J
estimate present in the text, Eq. (19).
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