On existence of a paramagnetic contribution to the susceptibility of a mesoscopic cylindrical normal metal-superconductor structure
The density of states N(ε) of a mesoscopic cylindrical structure consisting of a normal pure metal and a superconductor has been calculated using the Gorkov–Green functions. It is shown that magnetic fluxes of certain values cause resonance spikes of N(ε) suggesting a large-amplitude paramagnetic...
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irk-123456789-1168572017-05-17T03:02:59Z On existence of a paramagnetic contribution to the susceptibility of a mesoscopic cylindrical normal metal-superconductor structure Gogadze, G.A. Свеpхпpоводимость, в том числе высокотемпеpатуpная The density of states N(ε) of a mesoscopic cylindrical structure consisting of a normal pure metal and a superconductor has been calculated using the Gorkov–Green functions. It is shown that magnetic fluxes of certain values cause resonance spikes of N(ε) suggesting a large-amplitude paramagnetic contribution which accounts for the reentrant effect detected (P. Visani, A.C. Mota, and A. Pollini, Phys. Rev. Lett. 65, 1514 (1990)). 2008 Article On existence of a paramagnetic contribution to the susceptibility of a mesoscopic cylindrical normal metal-superconductor structure / G.A. Gogadze // Физика низких температур. — 2008. — Т. 34, № 3. — С. 225–230. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 74.45.+c http://dspace.nbuv.gov.ua/handle/123456789/116857 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Gogadze, G.A. On existence of a paramagnetic contribution to the susceptibility of a mesoscopic cylindrical normal metal-superconductor structure Физика низких температур |
| description |
The density of states N(ε) of a mesoscopic cylindrical structure consisting of a normal pure metal and a
superconductor has been calculated using the Gorkov–Green functions. It is shown that magnetic fluxes of
certain values cause resonance spikes of N(ε) suggesting a large-amplitude paramagnetic contribution
which accounts for the reentrant effect detected (P. Visani, A.C. Mota, and A. Pollini, Phys. Rev. Lett. 65,
1514 (1990)). |
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Article |
| author |
Gogadze, G.A. |
| author_facet |
Gogadze, G.A. |
| author_sort |
Gogadze, G.A. |
| title |
On existence of a paramagnetic contribution to the susceptibility of a mesoscopic cylindrical normal metal-superconductor structure |
| title_short |
On existence of a paramagnetic contribution to the susceptibility of a mesoscopic cylindrical normal metal-superconductor structure |
| title_full |
On existence of a paramagnetic contribution to the susceptibility of a mesoscopic cylindrical normal metal-superconductor structure |
| title_fullStr |
On existence of a paramagnetic contribution to the susceptibility of a mesoscopic cylindrical normal metal-superconductor structure |
| title_full_unstemmed |
On existence of a paramagnetic contribution to the susceptibility of a mesoscopic cylindrical normal metal-superconductor structure |
| title_sort |
on existence of a paramagnetic contribution to the susceptibility of a mesoscopic cylindrical normal metal-superconductor structure |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2008 |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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http://dspace.nbuv.gov.ua/handle/123456789/116857 |
| citation_txt |
On existence of a paramagnetic contribution
to the susceptibility of a mesoscopic cylindrical normal
metal-superconductor structure / G.A. Gogadze // Физика низких температур. — 2008. — Т. 34, № 3. — С. 225–230. — Бібліогр.: 20 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT gogadzega onexistenceofaparamagneticcontributiontothesusceptibilityofamesoscopiccylindricalnormalmetalsuperconductorstructure |
| first_indexed |
2025-07-08T11:12:46Z |
| last_indexed |
2025-07-08T11:12:46Z |
| _version_ |
1837077025165148160 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 3, p. 225–230
On existence of a paramagnetic contribution
to the susceptibility of a mesoscopic cylindrical normal
metal-superconductor structure
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 November 5, 2007
The density of states N( )� of a mesoscopic cylindrical structure consisting of a normal pure metal and a
superconductor has been calculated using the Gorkov–Green functions. It is shown that magnetic fluxes of
certain values cause resonance spikes of N( )� suggesting a large-amplitude paramagnetic contribution
which accounts for the reentrant effect detected (P. Visani, A.C. Mota, and A. Pollini, Phys. Rev. Lett. 65,
1514 (1990)).
PACS: 74.45.+c Proximity effects; Andreev effect; SN and SNS junctions.
Keywords: mesoscopic systems, Aharonov–Bohm effect, Andreev levels, reentrant effect.
1. Introduction
The technological advance in preparation of pure sam-
ples has enabled investigations of the coherent properties
of mesoscopic structures taking into account the proxim-
ity effect [1]. The samples were superconducting (S) Nb
wires (cylinders with a radius of tens of microns) coated
with a thin layer of a normal (N) pure metal (Cu or Ag ).
The structure was placed in a weak magnetic field.
A.C. Mota et al. [2,3] who measured the magnetic suscep-
tibility � of such structures could observe its rather sur-
prising reentrant-type behavior: at lowering temperatures
the (diamagnetic) susceptibility (in a constant field)
changed in accordance with theory but it unexpectedly
started growing at T � 100 mK. A similar behavior was ob-
served with the isothermal reentrant effect in a decreasing
magnetic field: the susceptibility started to grow sharply
in a held decreasing below a certain value. The effect was
observed only on mesoscopic NS structures. The authors
[2] interpreted the discovered phenomenon as a new co-
herent quantum effect in pure NS structures. They as-
sumed that a paramagnetic contribution could appear for
some reason in the NS structure in addition to the diamag-
netic current.
The origin of the paramagnetic currents in NS struc-
tures has been discussed in several theoretical publica-
tions. Bruder and Imry [4] analyze the paramagnetic con-
tribution to susceptibility made by quasiclassical («glanc-
ing») trajectories of quasiparticles that do not collide with
the superconducting boundary. The authors [4] point to a
large paramagnetic effect within their physical model.
However, their ratio between the paramagnetic and dia-
magnetic contributions is rather low and cannot account
for the experimental results [2,3].
Fauchere, Belzig, and Blatter [5] explain the large
paramagnetic effect assuming a pure repulsive elec-
tron-electron interaction in noble metals. The proximity
effect in the N metal induces an order parameter whose
phase is shifted by � from the order parameter � of the su-
perconductor. 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 Ref. 5 essentially rests on the assumption of the
repulsive electron interaction in the N metal. Is the
reentrant effect a result of specific properties of the noble
metals, or does it display the behavior of any normal
metal experiencing the proximity effect from the neigh-
boring superconductor? Only experiment can provide an-
swers to these questions. We just note that the theories of
Refs. 4 and 5 do not account for the temperature and field
dependences of the paramagnetic susceptibility and the
nonlinear behavior � of the NS structure. The current the-
ories cannot explain the origin of the anomalously large
© G.A. Gogadze, 2008
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 [6] that below the transition temperature
( �10 mK) some noble metals (Cu, Ag, Au) can exhibit
p-ware superconducting ordering, which may be respon-
sible for the reentrant effect. This theory does not explain
the high paramagnetic reentrant effect either.
Note that the NS structure generally has three contri-
butions to its magnetic susceptibility. First, this is a dia-
magnetic response induced by the electron excitation
specularly reflected from the dielectric boundary and
scattered (Andreev scattering) at the NS boundary. The
Andreev levels form in the film [7] when the normal layer
thickness is small. The peculiar feature of the quantum
proximity effect is the magnetic susceptibility diamagne-
tism (Meissner effect) modified by the Andreev levels.
Another contribution is related to the electron trajecto-
ries that do not collide with the NS boundary («whisper-
ing mode»). They generate persistent current in the nor-
mal layer and make a weak paramagnetic contribution to
the magnetic moment of the system.
Finally, a large-amplitude persistent paramagnetic cur-
rent is induced in the normal layer due to the Aharo-
nov–Bohm effect [8] caused by strong degeneracy of the
system when the Andreev level superimposes on the Fer-
mi level of the metal. In this case resonance spikes are ob-
served in the density of states. The spectrum of quasi-
particles (sec Eq. (4)) includes an angle � at which they
hit the dielectric boundary. For a pre-assigned �-value,
resonance occurs at a certain magnetic flux through the
quantized area enclosed by the trajectory. For other an-
gles � the resonance-inducing fluxes are slightly differ-
ent. The total contribution to the density of the states of a
NS structure is a sum of contributions from all trajecto-
ries. It is found [9,10] that the high paramagnetic re-
sponse can occur in a certain range of weak magnetic
fields and at temperatures no higher than 100 mK.
We obtained a large paramagnetic contribution � p to
the susceptibility of a NS structure within the model of
free electrons. When � p is added to the diamagnetic con-
tribution � d , the resulting total susceptibility features the
reentrant effect. Theoretically [9,10] the effect is expect-
ed in samples in which quasiparticles have large mean
free paths comparable with the cylindrical N layer perim-
eter. This was observed experimentally in Refs. 2,3.
As the magnetic field (or temperature) increases and
eliminates a prerequisite to resonance, the large paramag-
netic contribution disappears. The paramagnetic contri-
bution from the «whispering mode» persists but it is small
due to the smallness of the quasiclassical parameter of the
problem �1/ ( )k RF (�k F is the Fermi momentum, R is
the radius of the cylinder) and cannot effect the total
susceptibility of the system.
When the magnetic field increases considerably (or
the temperature approaches Tc of the superconductor) the
diamagnetic susceptibility has only the «classical» contri-
bution from the motion of the Cooper pains inside the
superconducting layer near the NS boundary. This contri-
bution persists in the whole range of temperatures (and
magnetic fields) where the superconducting state exists.
The Meissner effect in pure NS structure has been in-
vestigated recently by Galaktionov and Zaikin [11] who
used the Gorkov microscopic equations of superconduc-
tivity [12]. They calculated the diamagnetic current of a
NS structure taking into account the proximity effect. The
result obtained is essentially similar to that in Zaikin’s
first publication on the subject [13]. At the same time it
was stated [11] that a paramagnetic contribution to the
susceptibility of a NS structure is absent if there is no
strong electron–electron repulsion in the N layer.
The goal of this study is to show the existence of a
paramagnetic contribution to the susceptibility of a NS
structure using the Green function approach. The density
of states N ( )� of such a structure has been calculated. It is
shown that magnetic flux of certain values induce reso-
nance spikes of N ( )� . The model of free electrons was
applied.
Note that a large paramagnetic contribution is unob-
tainable with the Eilenberger–Green functions [13]. The
Eilenberger equations [14] were derived by integrating
the Gorkov microscopic equations of superconductivity
with respect to the quasipartieles energies. The remark-
able effect of quasiparticle state degeneracy occurs in a
narrow interval of energies approximately equal to the en-
ergy gap of the superconductor (see above). This is the in-
terval in which the density of states of a NS structure ex-
hibits resonance features. The integration of the Gorkov
Green functions over energies reduces the large contribu-
tion from the quantized Andreev levels to its average
value. As a result the paramagnetic contribution to the
thermodynamics of the NS structure disappears. To detect
this contribution one should proceed from the exact Gor-
kov equations for the Green functions of a contact in a
magnetic field.
2. The density of states of a NS structure
A mesoscopic cylindrical NS contact consisting of a
thin pure normal-metal layer (0 � �x d) and a bulk super-
conductor has been considered (see Fig. l). We neglect the
curvature of the cylindrical NS surface, which is permis-
sible if R d�� . The assumption of a flat NS boundary
largely simplifies the consideration, whereas the value of
the screened current is practically similar for cylindrical
and flat geometries. The currents are distinctive in param-
eter d R/ �� 1(see [11]). We proceed from the free-elec-
tron model and assume a stepwise variation of the order
parameter at the NS boundary. The mean free paths of the
226 Fizika Nizkikh Temperatur, 2008, v. 34, No. 3
G.A. Gogadze
quasiparticle exceed the characteristic dimensions of the
structure.
The density of states of a NS contact, is found as
N G iR( ) Im ( )�
�
�
� �
1
11Sp . (1)
The Spur operation is performed over the variables deter-
mining the Green matrix function. G R
11 is the analytical
continuation of the11-th component of the Green function
0. It is assumed that the NS structure is homogeneous
along the interface.
The Fourier transformation of the normal (G) and
anomalous (F � ) Green functions in the coordinates along
the NS boundary yields
G
d
G x x
n n
i
� �
�
( , )
( )
( , , )
||
||
( )
|| || ||r r
k
k
k r r
�
��
� �
2
22
e .
The Gorkov equations become [12]:
i H x
x i H
G x x
F x
n
n c
n
n
�
�
�
�
�
�
�
�
��
�
�
��
�
�
� ( )
* ( ) �
( , , )
( ,
||�
�
k
x
x x
�
�
�
�
�
�
�
�
�
� ��
�
�
�
�
�
, )
( )
||k
�
0
.
(2)
Here � �n n T
�( )2 1 is the Matsubara frequency; �( )x is
the superconducting order parameter. The Hamiltonian �H
is determined as �
*
~
*
( )
||
H
m x m
V x
�
�
�
� � �
1
2 2
2
2
2
k
� , A is
the vector potential field;
~
( )|| || ||k k A
�
e
c
x ; � — is the
Fermi energy andV x( ) is the potential of the metal bound-
ary. �H c is found from �H by reversing the sign of the elec-
tron charge e. It is assumed that the London penetration
depth of the field is small in a bulk superconductor and
the field inside the superconductor of our structure is
zero.
The Green function of a NS structure was calculated
for pure metals in a zero magnetic field by Arnold [15].
The Green function of the composite structure was found
from the Green functions of isolated systems G x xN ( , )�
and G x xS ( , )� and was then used to calculate the tunnel
density of states at the NS boundary. The Green function
of a NS structure in a magnetic field was obtained by
Galaktionov and Zaikin [11]:
G x x
i
n
x
� �
�( , , )||�
�
k
2
v
th , (3)
where �
�
�
� �
2 n
x
d
i
v
tan . Here �
� �
2
0
0
�
A x dx
d
( ) , vx is the
velocity component along the normal to the NS boundary,
� is the angle at which a quasiparticle hits the dielectric
boundary of the normal layer, �
0 2hc e/ is the supercon-
ducting flux quantum.
It is seen that the function G x x
n� ( , , )||� k is constant in
x but dependent on the magnetic flux. Such dependence
disappears in a bulk metal (d
�). If the normal metal
has a finite thickness, its electron states are obtainable
from the poles of the Green function. For low-lying states
the spectrum of the Andreev levels in the N layer is
� �
� � �
�n q
q
d
n( , ; )
( ) cos
�
� � ��
�
�
�
�
�
v
2
1
2
tan
. (4)
Here v( ) / *q p q mF
�2 2 , vF is the Fermi velocity, q is
the momentum component along the symmetry axis of the
cylinder, m* is the effective mass of a quasiparticle, n
0,
� 1, � 2,...
A x( ) is found by solving the Maxwell equation for the
boundary condition A x( )
0 0,
�
�
A
x
Hx d| [13]:
A x Hx
c
j d
x
x( )
� ��
�
�
�
�
�
4
3 2
�
, (5)
where j is the current density in the N layer. We assume
that this current includes both the diamagnetic and para-
magnetic components. Integration of both sides of Eq. (5)
over the normal layer thickness gives a sell-consistent
equation for a A x dx
d
� � ( )
0
:
a
Hd
c
j a d
�
2
3
2
4
3
�
( ) .
The parameter a is dependent both on the magnetic field
and the temperature.
The expression for the density of states depends from
the Green function of the structure. Substituting Eq. (3) in
the Green function and using its analytical continuation,
we can obtain after summation over the spin
N
m
dq d
d
p
p
n x
F
F
c
c
( )
*
( )
Im�
� �
�
�
�
�
�
� �
�
� �
�� �
4
2
2
2 2
tan tan
v
!
"
#
$ .
On existence of a paramagnetic contribution to the susceptibility
Fizika Nizkikh Temperatur, 2008, v. 34, No. 3 227
S N
J
0 d
H
z
x
Fig. 1. Normal-metal layer (thickness d) in proximity to a bulk
superconductor. A magnetic field H parallel to the surface is
applied, driving screening current J along the surface.
The angle � is measured from the positive direction of
the normal to the boundary. The spectrum of the Andreev
levels (Eq. (4)) is formed by the quasiparticle paths in the
N layer whose angles vary within 0 � | |� � � c . � c is
the angle at which the quasiparticle trajectory touches the
NS boundary [10]. Those trajectories are responsible for
a large paramagnetic contribution to the susceptibility
and hence for the reentrant effect. Another group includes
the trajectories with � �� c that collide only with the di-
electric boundary. They induce states practically coincid-
ing with the «whispering gallery» type of states occurring
in the cross-section of a normal solid cylinder in a weak
magnetic field [16]. These trajectories generate paramag-
netic contribution of small amplitudes (see the Introduc-
tion) and are therefore discarded from this consideration.
The main contribution to the density of states comes
from the vicinity of the tangent poles. Expanding the nu-
merator and denominator into a series near the Andreev
levels, we obtain (
0)
N
d
dq p q
d
i
p
n
F
n
F c
( ) Im�
�
�
� �
�
� �
� ��� �
2
3
0
2 2
0
. (6)
We then use the relation
1 1
x i
P
x
i x
�
��� ( ),
where P is the principal quantity and obtain an expression
similar to that in [10]:
N
d
dq p q d q
p
n
F n
F c
( ) [ ( , ; )]�
�
� � � � �
�
� � � ��� �
2
2
0
2 2
0
.
(7)
The density of states can be written as
N d
c
( ) ( ; )� �% � �
�
�
0
,
where % � �( ; ) is the contribution to the density of states
from the pre-assigned trajectory with a fixed �. After in-
tegration with respect to q and introduction of the nota-
tion & �
� / ( *)2dm , we can pass on to the dimensionless
energy º /
� &pF . We obtain the expression for % �(º; )
% �
� &
�' ( �
( (
(º; ) º
[ º ]
( ) ( ) º
� �
� � �
2
2
2
2
2 2 2
p
d
n
n n
F sec sec
sec 2�n
� , (8)
where (
�
�
� �
1
2
tan
and '( )x is the Heaviside step func-
tion. Equation (8) suggests two cases depending on the
parameter n � (.
Nonresonance case
If n � )( 0 the energy dependence under the radical
sign in Eq. (8) can be neglected for small energies. Then,
the nonresonance contribution to the density of states is
%
� &
�
�
(
�
( ) º
( )
0
2
2
0
2
3
2
�
�� �
��
��
p
d
d
n
F
n
c
sec
. (9)
The series in Eq. (9) is calculated by the formula of
Ref. 17:
1
1
1
1
1
1( )
( )
( )!k n
d
dn
k
n
n
n�
�
�
��
��
�
�
��
(
�
(
�(cot ,
which yields
%
�
&
�
�
� �
�
( ) º
[ ]
cos cos [ ]
0 2
0
2 3
2
�
�
��
p
d
dF
c
sin tan
tan
. (10)
After integration over � we have
%
�
�( ) *
º0
2
0 2
0
4
2
2 2
�
�
�
�
!
"
#
$k m
a
a R
d
F tan ,
where 2R d c/ � tan � .
Resonance case
Now we go back to Eq. (8) and find % res as
%
�
res
*
4
3
2k mF *
º
*
� �
� �
� d
a b
a b a b
c
n
n n0
2
2
�
�
�' � �
� �
sec tan sec
tan tan
[ º ]
( ) ( ) 2 2 2�
�
º sec �n
,
(11)
where the notations a nn
�1 2/ , b a
�2 0/ are intro-
duced. Equation (11) shows that at certain values of the
magnetic flux, the radicand in the denominator tends to
zero. Our interest is concentrated on the asymptotics of
%(º ) at low energies º.
Prier to estimation of % res , we shall consider the con-
tributions of different angles � to the resonance ampli-
tude. It is reasonable to assume that because of the factor
sec 2� in the numerator of Eq. (11), the angles � �� c are
the main contributors to the integral. It is convenient to
introduce a new variable of integration x
tan �. Then
the neighborhood of the upper limit x c0
tan � is the
main contributor to the integral. Introducing the notation
~a a bxn
� 0 and the small deviation +
� ��x x0 1, we
can write down the equation for the resonance condi-
tion as:
( º ) (~ º ) ~ º ( )b ab x a x2 2 2 2
0
2 2
0
22 1 0� � � � � �
+ + . (12)
228 Fizika Nizkikh Temperatur, 2008, v. 34, No. 3
G.A. Gogadze
The solution of Eq. (12) to the accuracy within the first
order terms of |º | gives:
+1 2
0
2
0
21 1
,
~
( )
|º |
� �
�
�
�
a
b x b x
. (13)
The expression in front of the radical in the denominator
is of second order smallness in |º |, i.e. | ~|a 2
� |º | ( )2
0
21� x ,
which leads to its cancellation with the similar term in the
numerator.
The remaining integral is estimated to be a constant of
about unity. Resonance-induced spikes of the density of
states appear when the Andreev level coincides with the
Fermi energy at a certain flux in the N layer. In the vicin-
ity 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
contributes significantly to the amplitude of the effect.
Near the resonance, the ratio of the resonance and
nonresonance amplitudes of the density of states is
%
%
res
( ) |º |0 2
1
1� �� . (14)
Thus, we have shown that a change in the magnetic
flux leads to resonance spikes in the density of states of
the NS contact. The flux interval between the spikes is
equal to the superconducting flux quantum �0.
3. Discussion
The flux quantization effect and the paramagnetic con-
tribution to the susceptibility of a thin-wall pure metal
cylinder in the vector potential field were predicted
by Kulik [18]. In pure normal metals there is a length
+N F Bk T
�v / , which has the meaning of a coherence
length of a system with disturbed long-range order. When
the temperature lowers, this length becomes equal to the
characteristic dimensions of the system, which can lead to
interference effects in the system. Kulik [18] shows that
the magnetic moment of a thin-wall cylinder is an oscil-
lating function of the magnetic flux through the cross sec-
tion of the cylinder, its oscillation period being equal to
the flux quantum of the normal metal hc e/ . The effect is
generated by quantization of the electron motion along
the perimeter of the cylinder and is due to the sensitivity
of the states of the system to the vector potential field
(Aharonov–Bohm effect [8]). Bogachek and Gogadze
[16] investigated the coherent quantum effect in singly
connected normal cylindrical conductors in a weak mag-
netic field. The authors proved the existence of an oscil-
lating component with a flux period hc e/ in the magnetic
moment The oscillation amplitude is small due to the
smallness of the quasiclassical parameter of the problem
1/ k RF . The amplitude of the effect decreases exponen-
tially as the radius R increases. As a result, the persistent
current disappears in macroscopic systems. The effect of
flux quantization in pure Bi whiskers was first detected
experimentally by Brandt et al. ([19,20]). That was the
first observation of the interference effect of the flux
quantization in nonsupercondncting condensed matter.
Mota et al. [2,3] investigated a mesoscopic NS struc-
ture in a magnetic field. If the electric contact between the
N and S elements is good, the electrons penetrate easily
from the superconductor to the normal layer and thus sig-
nificantly affect the properties of the NS system. There
are two types of electron collisions in a normal film — a
specular reflection from one boundary and the Andreev
reflection from the another. Along with the quasiparticle
trajectories closed around the circular perimeter of the
cylinder, new trajectories appear in a weak field, which
screen the normal metal. The new trajectories of the «par-
ticles» and «holes» confine the area of a triangle whose
base in a part of the NS boundary between the points at
which the quasiparticle collides with this boundary. This
area is maximum for the trajectories touching the super-
conductor. At certain values of the magnetic flux through
the triangle area, the electron density of states experi-
ences resonance spikes [9,10]. Their existence was
proved in the standard calculation of the density of states
N qn( ) [ ( , , )]� � � � �
� �� . Here we demonstrate this
using the Greens functions of Gorkov.
Thus, the quantum proximity effect transforms the pe-
riodic flux-induced oscillations of the thermodynamic va-
lue with a period hc e/ into periodic resonance spikes with
a period equal to a superconducting flux quantum hc e/ 2 .
The response to a weak magnetic field (H �10 Oe) is
paramagnetic and the susceptibility amplitude becomes
anomalously large. The resonance features disappear
when the magnetic flux increases and its value divided by
hc e/ 2 starts the exceed the highest Andreev «subband»
numbei (see Eq. (4)). Only the quasiparticle trajectories
that do not collide with the superconducting boundary
contribute to the susceptibility, but their amplitudes are
rather small (see above). Under this condition, the experi-
ment only registers a large diamagnetic response
(Meissner effect) We can therefore conclude that, the res-
onance contribution to the paramagnetic susceptibility
can only appear in comparatively weak magnetic fields.
In this case the experiment shows the reentrant effect
[2,3].
The author is grateful to A.N. Omelyanchouk for help-
ful discussions.
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