On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples
Theory of the reentrant effect in susceptibility of mesoscopic cylindrical NS samples is proposed, which is essentially based on the properties of the Andreev levels. The specific feature of the quantum levels of the structure is that in a varying magnetic field (or temperature) each level period...
Збережено в:
| Дата: | 2005 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2005
|
| Назва видання: | Физика низких температур |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/120780 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples / G.A. Gogadze // Физика низких температур. — 2005. — Т. 31, № 1. — С. 120-123. — Бібліогр.: 18 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-120780 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1207802017-06-13T03:04:54Z On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples Gogadze, G.A. Письма pедактоpу Theory of the reentrant effect in susceptibility of mesoscopic cylindrical NS samples is proposed, which is essentially based on the properties of the Andreev levels. The specific feature of the quantum levels of the structure is that in a varying magnetic field (or temperature) each level periodically comes into coincidence with the chemical potential of the metal. As a result, the state of the system becomes strongly degenerate and the amplitude of the paramagnetic contribution to the susceptibility increases sharply. 2005 Article On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples / G.A. Gogadze // Физика низких температур. — 2005. — Т. 31, № 1. — С. 120-123. — Бібліогр.: 18 назв. — англ. 0132-6414 PACS: 74.50.+r, 74.45.+c http://dspace.nbuv.gov.ua/handle/123456789/120780 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Письма pедактоpу Письма pедактоpу |
| spellingShingle |
Письма pедактоpу Письма pедактоpу Gogadze, G.A. On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples Физика низких температур |
| description |
Theory of the reentrant effect in susceptibility of mesoscopic cylindrical NS samples is proposed,
which is essentially based on the properties of the Andreev levels. The specific feature of
the quantum levels of the structure is that in a varying magnetic field (or temperature) each level
periodically comes into coincidence with the chemical potential of the metal. As a result, the state
of the system becomes strongly degenerate and the amplitude of the paramagnetic contribution to
the susceptibility increases sharply. |
| format |
Article |
| author |
Gogadze, G.A. |
| author_facet |
Gogadze, G.A. |
| author_sort |
Gogadze, G.A. |
| title |
On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples |
| title_short |
On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples |
| title_full |
On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples |
| title_fullStr |
On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples |
| title_full_unstemmed |
On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples |
| title_sort |
on the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2005 |
| topic_facet |
Письма pедактоpу |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120780 |
| citation_txt |
On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples / G.A. Gogadze // Физика низких температур. — 2005. — Т. 31, № 1. — С. 120-123. — Бібліогр.: 18 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT gogadzega onthenatureofthereentranteffectinsusceptibilityofmesoscopiccylindricalsamples |
| first_indexed |
2025-07-08T18:34:01Z |
| last_indexed |
2025-07-08T18:34:01Z |
| _version_ |
1837104778270736384 |
| fulltext |
Fizika Nizkikh Temperatur, 2005, v. 31, No. 1, p. 120–123
Letters to the Editor
On the nature of the reentrant effect in susceptibility
of mesoscopic cylindrical 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 October 12, 2004, revised October 22, 2004
Theory of the reentrant effect in susceptibility of mesoscopic cylindrical NS samples is pro-
posed, which is essentially based on the properties of the Andreev levels. The specific feature of
the quantum levels of the structure is that in a varying magnetic field (or temperature) each level
periodically comes into coincidence with the chemical potential of the metal. As a result, the state
of the system becomes strongly degenerate and the amplitude of the paramagnetic contribution to
the susceptibility increases sharply.
PACS: 74.50.+r, 74.45.+c
In 1990 Mota and co-workers [1] detected a surpris-
ing behavior of the magnetic susceptibility of a cylin-
drical NS structure (N and S are for the normal metal
and superconductor, respectively) at very low tempera-
ture (T < 100 mK). The external magnetic field was
applied parallel to the NS boundary. It was most in-
triguing that a decrease in the sample temperature be-
low a certain point Tr (in a fixed field) produced the
reentrant effect: the decreasing magnetic susceptibility
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. The sample were superconducting Nb wires
with a radius R of tens of �m coated with a thin layer d
of very pure Ag. It is emphasized in [2] that the de-
tected magnetic response of the NS structure is similar
to the properties of the persistent currents in meso-
scopic normal rings. It is assumed [1–5] that the
reentrant effect reflects the behavior of the total sus-
ceptibility � of the NS structure: the paramagnetic con-
tribution is superimposed on the Meissner effect-re-
lated diamagnetic contribution and nearly compensates
it. The anomalous behavior of susceptibility was also
observed in AgTa, CuNb and AuNb structures [2,4].
The origin of paramagnetic currents in NS struc-
tures was discussed in a number of theoretical studies.
Bruder and Imry [6] analyzed the paramagnetic
contribution to susceptibility taking into account the
paths of the quasiparticles that do not collide with the
superconducting boundary. The authors note an ap-
preciable paramagnetic effect in the physical situation
under discussion. However, the ratio derived by them
for the paramagnetic and diamagnetic contributions is
rather small, and cannot account for the experimental
results [1–5].
Fauchere, Belzig and Blatter [7] explain the high
paramagnetic effect assuming the pure repulsive elec-
tron-electron interaction in noble metals. The proxim-
ity effect in the normal metal induces the order parame-
ter whose phase is �-shifted against the �s -phase of the
superconductor. This leads to paramagnetic instability
of the Andreev states and the density of states of the NS
structure exhibits a peak near zero energy. The theory
[7] is much based on the assumption of the repulsive
electron interaction in the normal metal. The question
whether the reentrant effect is due to specific proper-
ties of noble metals or is shared by any normal metal
experiencing the proximity effect can be answered only
experimentally. We can just note that the theories in
[6,7] do not describe the temperature and field depen-
dences of the paramagnetic susceptibility of the NS
structure and do not explain the origin of the anoma-
lously large amplitude of the reentrant effect.
It is worth mentioning the assumption made by
Maki and Haas [8] that below the transition tempera-
© G.A. Gogadze, 2005
ture (� 10 mK) some noble metals (Cu, Ag, Au) can
exhibit p-wave superconducting ordering, which may
be responsible for the reentrant effect. This theory
does not explain the high paramagnetic reentrant ef-
fect either.
In this Letter a reentrant effect theory is proposed,
which is essentially based on the properties of the
quantized levels of the NS structure. The levels with
energies no more than �0 (2�0 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 Aha-
ronov—Bohm effect [9], the spectrum of the NS struc-
ture is a function of the magnetic flux in a weak field.
The specific feature of the quantum levels of the struc-
ture is that in a varying field H (or temperature) each
level in the well periodically comes into coincidence
with the chemical potential of the metal. As a result,
the state of the system suffers strong degeneracy and
the density of states of the NS sample experiences re-
sonance spikes. We attribute the reentrant effect to
this resonance.
The resonance spikes of the density of states as a
function of the magnetic field were predicted earlier in
our study on a structure consisting of a normal metal
cylinder coated with a thin superconducting layer [10].
Let us consider a superconducting cylinder of ra-
dius R coated with a thin layer d of a pure normal
metal. The structure is placed in a weak magnetic field
H( , , )0 0 H applied along its symmetry axis. We proceed
from a simplified NS model assuming that the order
parameter modulus of the superconductor changes in a
jump at the NS boundary. We introduce an angle of
incidence of a quasiparticle � onto the dielectric
boundary counted off the cylinder radius. It is evident
that there are two classes of quasiparticle paths inside
the normal metal. One class includes those in which �
varies within 0 � � � � cr (� cr is the angle at
which the path touches the NS boundary). In this case
the quasiparticle collides successively with the dielec-
tric and the NS boundary. The other class with
� �� cr consists of the paths whose spectra are formed
only by the quasiparticle-dielectric collisions. The
spectra of the two classes paths differ considerably.
First we consider the paths with � � � cr . The spec-
trum of the quasiparticles of the NS structure is
readily obtainable by the method of multidimensional
quasiclassics [10,11] generalized for the presence of
the Andreev scattering in the system [12]. We have
� �
� �
�
��
n q
V q
d
n( , )
( ) cos ( )
.
�
�
��
�
�
��
� L
2
1
0 0
arccos
�
�
�
(1)
Here V q p q /mFL ( ) * �2 2 , pF is the Fermi momen-
tum, q is the momentum component along the cylinder
axis, m* is effective mass of the quasiparticle, �0 is the
superconducting flux quantum. The magnetic flux
through the area bounded by a part of the NS boundary
and by the quasiparticle path at an angle � is given by
�( ) ( )� � �2
0
tg A x dx
d
.
The integral of A x( ) can be calculated if we know the
distribution of the vector potential field inside the
normal metal. The problem of the Meissner effect in
superconductor-normal metal (proximity) sandwiches
was solved by Zaikin [13]. The screening current in
the NS structure was calculated in terms of the micro-
scopic theory, and the expression A x Hx( )
�( ) ( ) ( )4 2�/c j a x d x/ can be obtained from the
Maxwell equation rot H j ( )4�/c with the boundary
conditions A x( ) 0 0 and � x A x d H( ) . The
screening current is a function of
a A x dx
d
� ( )
0
, j a j a/s( ) ( ) � � �0 ,
where js is superfluid current and �( )x is oscillating
function of flux (at a/�0 1�� we have j a( )
�j a/s �0). Thus, we can write down the self-consis-
tent equation for a [13–15]:
a H
d
c
j a d
2
3
2
4
3
�
( ) , (2)
j a( ) turns to zero recurrently when a/�0
1 2 1 3 2 2/ /, , , ,� The spectrum of Eq. (1) is similar to
Kulik’s spectrum [16] for the current state of SNS con-
tact. However, Eq. (1) includes an angle-dependent
magnetic flux instead of the phase difference of the
contacting superconductors.
We proceed from the expression for the thermody-
namic potential
� �
��T /TS
S
ln [ exp( )]1 � ,
summation is made with respect to all quantum states
of the spectrum, Eq. (1), and the spin (the Boltzmann
constant kB = 1). We use the approximation of equidis-
tant levels in which the second term in Eq. (1) can be
replaced with 1/2. The contribution to susceptibility
(per unit volume V of the normal metal) can be found
as � � � �( )( )1 2 2/V / H� . Taking into account two
orientations of the spin and two possible signs of � and
q, as well as the finite number of levels n0, we have the
starting expression for susceptibility (L is the cylinder
height, � is the chemical potential of the metal):
On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples
Fizika Nizkikh Temperatur, 2005, v. 31, No. 1 121
�
�
�
� � � � �
�
�
� d
T
d a dqV q qn
pF
ch
cr
2
2
0
2
0
2
�
�
� � �sin ( ) ( ( , ))L��
�
.
n n
n
0
0
(3)
We take an integral over q using the �-function and
introduce the dimensionless energy
ª ( ) � �� �� �/ V / dF� 2 .
Since � ��/ �� 1, the lower limit of the integral of en-
ergy can be replaced with ��. By introducing the
variable x � tg � and the notations a n /n
1 2,
b b H T a/ ( , ) �0, x0 tg cr� � 2R/d and tak-
ing into account the parity of the integrand instead of
Eq. (3) we obtain:
�
�
�
�
��A
d
/n
n
ª ª
ª
| |
[ ], ,
3
2
001 2
0
ch
�
�
� �
� � �
dxx a bx x
a bx a bx x
n
n n
x 2 2
3 2 2
0
1
12
� !| | | |
| | | | ( )
ª
ª
0
� .
(4)
Here A d / R 2 2
0
2� � �( )� , � = ��/T, � is the Heaviside
step function. It is seen in Eq. (4) that for a given
«subzone» n the amplitude of the paramagnetic suscep-
tibility increases sharply whenever the Andreev level
coincides with the chemical potential of the metal. The
resonant spike of susceptibility occurs when a bxn �
tends to zero on a change in the magnetic field (or tem-
perature). Because of the finite number of Andreev
levels, the region of the existence of the isothermal
reentrant effect, is within 0 � H � Hn0
.
The undefined integral over x can be calculated
exactly. The roots of the square trinomial under the
radical to within the first-order terms in ª are
x /b12 0 0
21, | |ª� � �"
( ;�0 a /bn 0 1 0� � �x �
� �x x2 0). The �-function bars the region x x x1 2� �
from consideration. On substituting the limits of inte-
gration, the obtained expressions have different powers
of the parameter | |ª
�1. We retain only the most impor-
tant terms of the order of | |ª
�3. The integral over energy
is taken easily when these terms are substituted in it.
Finally, the susceptibility of the NS structure becomes
�
�
�
�
#
�
�
�
�
�
�
�
�
�
�
$
%
&
'
(
)
d
R
n
V
dT
n
n
F
2
0
2
21
2 4
1
2
1
2
�
th
�
�
�
�
�
�
$
%
&
&
'
(
)
)
�
2
2
3 2
01
0
b H T
/
n
n
( , )
.
, ,�
(5)
The flux b a/ �0 (Eq. (2)) depends on both the
magnetic field and temperature. The screening cur-
rent of the NS structure is j j a/s � �( )�0 , where
j T Td/ Vs F~ exp( )� �1 4� � for T V /dF�� � [13,15].
It is seen from Eq. (5) that b H T( , ) increases with ris-
ing temperature (H is fixed). At the same time the
hyperbolic tangent argument becomes smaller than
unity starting with a certain temperature. As a result,
the amplitude of paramagnetic susceptibility de-
creases rapidly as the temperature rises. For the iso-
thermal reentrant effect appears, the decreasing mag-
netic field causes an increase in the amplitude of
paramagnetic susceptibility. Choosing the character-
istic parameters of the problem d = 3.3·10–4 cm,
R = 8.2·10–4 cm, � ~ 10 12� erg, �0
72 10 * � G/cm2,
we can estimate the coefficient preceding the sum in
Eq. (5) to be � 10 2� . The sum itself has the loga-
rithmic scale � lnn0. The effect is caused by the
paths of the quasiparticles with � � � cr .
The paths of the quasiparticles with � > �cr do not
collide with the NS boundary and their quantum states
are essentially similar to the states of the «whispering
gallery» type that appear in the cross-section of a nor-
mal solid cylinder in a weak magnetic field [17,18].
The caustic of this paths is approximately equal to the
cylinder radius and the spectrum of the states carries no
information about the parameters of the superconduc-
tor. The energy levels cannot be made coincident with
the chemical potential of the metal by varying the mag-
netic field or temperature. As a result, the paramag-
netic contribution into the thermodynamics of the
paths with � > �cr has a much smaller amplitude.
The author is indebted to A.N. Omelyanchouk and
A.A. Slutskin for helpful discussions.
1. P. Visani, A.C. Mota, and A. Pollini, Phys. Rev.
Lett. 65, 1514 (1990).
2. A.C. Mota, P. Visani, A. Pollini, and K. Aupke,
Physica B197, 95 (1994).
3. F.B. Müller-Allinger and A.C. Mota, Phys. Rev. Lett.
84, 3161 (2000).
4. F.B. Müller-Allinger and A.C. Mota, cond-mat/0007331
(2000).
5. R. Frassanito, P. Visani, M. Nideröst, A.C. Mota, P.
Smeibidl, K. Swieca, W. Wendler, and F. Pobell, Proc.
LT 21, Part S4-LT: Properties of Solids I, Prague
(1996), p. 2317.
6. C. Bruder and Y. Imry, Phys. Rev. Lett. 80, 5782
(1998).
7. A.L. Fauchere, W. Belzig, and G. Blatter, Phys. Rev.
Lett. 82, 3336 (1999).
8. K. Maki and S. Haas, cond-mat/0003413 (2000).
9. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485
(1959).
10. G.A. Gogadze, R.I. Shekhter, and M. Jonson, Fiz.
Nizk. Temp. 27, 1237 (2001) [Low Temp. Phys. 27,
913 (2001)].
11. J.B. Keller and S.I. Rubinow, Ann. Phys. (N.Y.) 9,
24 (1960).
122 Fizika Nizkikh Temperatur, 2005, v. 31, No. 1
G.A. Gogadze
12. A.F. Andreev, Zh. Eksp. Teor. Fiz. 46, 1823 (1964)
[Sov. Phys. JETP 19, 1228 (1964)].
13. A.D. Zaikin, Solid State Commun. 41, 533 (1982).
14. W. Belzig, G. Bruder, and G. Schön, Phys. Rev. B53,
5727 (1996).
15. A.L. Fauchere and G. Blatter, Phys. Rev. B56, 14102
(1997).
16. I.O. Kulik, Zh. Eksp. Teor. Fiz. 57, 1745 (1969)
[Sov. Phys. JETP 30, 944 (1969)].
17. E.N. Bogachek and G.A. Gogadze, Zh. Eksp. Teor.
Fiz. 63, 1839 (1972) [Sov. Phys. JETP 36, 973
(1973)].
18. G.A. Gogadze, Fiz. Nizk. Temp. 9, 1051 (1983) [Sov.
J. Low Temp. Phys. 9, 543 (1983)].
On the nature of the reentrant effect in susceptibility of mesoscopic cylindrical samples
Fizika Nizkikh Temperatur, 2005, v. 31, No. 1 123
|