Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction
We evaluate the Josephson current through a quasi-1D quantum wire coupled to bulk superconductors. It is shown that the interplay of Rashba spin–orbit interaction and Zeeman splitting results in the appearence of a Josephson current even in the absence of any phase difference between the supercon...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1197222017-06-09T03:03:04Z Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction Krive, I.V. Gorelik, L.Y. Shekhter, R.I. Jonson, M. Свеpхпpоводимость, в том числе высокотемпеpатуpная We evaluate the Josephson current through a quasi-1D quantum wire coupled to bulk superconductors. It is shown that the interplay of Rashba spin–orbit interaction and Zeeman splitting results in the appearence of a Josephson current even in the absence of any phase difference between the superconductors. In a transparent junction (D≅1) at low temperatures this anomalous supercurrent Jan appears abruptly for a Zeeman splitting of the order of the Andreev level spacing as the magnetic field is varied. In a low transparency (D << 1) junction one has Jan √D under special (resonance) conditions. In the absence of Zeeman splitting the anomalous supercurrent disappears. We have investigated the influence of dispersion asymmetry induced by the Rashba interaction in quasi-1D quantum wires on the critical Josephson current and have shown that the breakdown of chiral symmetry enhances the supercurrent. 2004 Article Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction / I.V. Krive, L.Y. Gorelik, R.I. Shekhter, M. Jonson // Физика низких температур. — 2004. — Т. 30, № 5. — С. 535-543. — Бібліогр.: 27 назв. — англ. 0132-6414 PACS: 74.80.Fp, 31.30.Gs, 71.70.Ej, 73.20.Dx http://dspace.nbuv.gov.ua/handle/123456789/119722 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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DSpace DC |
| language |
English |
| topic |
Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| spellingShingle |
Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Krive, I.V. Gorelik, L.Y. Shekhter, R.I. Jonson, M. Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction Физика низких температур |
| description |
We evaluate the Josephson current through a quasi-1D quantum wire coupled to bulk superconductors.
It is shown that the interplay of Rashba spin–orbit interaction and Zeeman splitting results
in the appearence of a Josephson current even in the absence of any phase difference between
the superconductors. In a transparent junction (D≅1) at low temperatures this anomalous
supercurrent Jan appears abruptly for a Zeeman splitting of the order of the Andreev level spacing
as the magnetic field is varied. In a low transparency (D << 1) junction one has Jan √D under
special (resonance) conditions. In the absence of Zeeman splitting the anomalous supercurrent disappears.
We have investigated the influence of dispersion asymmetry induced by the Rashba interaction
in quasi-1D quantum wires on the critical Josephson current and have shown that the breakdown
of chiral symmetry enhances the supercurrent. |
| format |
Article |
| author |
Krive, I.V. Gorelik, L.Y. Shekhter, R.I. Jonson, M. |
| author_facet |
Krive, I.V. Gorelik, L.Y. Shekhter, R.I. Jonson, M. |
| author_sort |
Krive, I.V. |
| title |
Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction |
| title_short |
Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction |
| title_full |
Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction |
| title_fullStr |
Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction |
| title_full_unstemmed |
Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction |
| title_sort |
chiral symmetry breaking and the josephson current in a ballistic superconductor–quantum wire–superconductor junction |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2004 |
| topic_facet |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119722 |
| citation_txt |
Chiral symmetry breaking and the Josephson current in a ballistic superconductor–quantum wire–superconductor junction / I.V. Krive, L.Y. Gorelik, R.I. Shekhter, M. Jonson // Физика низких температур. — 2004. — Т. 30, № 5. — С. 535-543. — Бібліогр.: 27 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT kriveiv chiralsymmetrybreakingandthejosephsoncurrentinaballisticsuperconductorquantumwiresuperconductorjunction AT gorelikly chiralsymmetrybreakingandthejosephsoncurrentinaballisticsuperconductorquantumwiresuperconductorjunction AT shekhterri chiralsymmetrybreakingandthejosephsoncurrentinaballisticsuperconductorquantumwiresuperconductorjunction AT jonsonm chiralsymmetrybreakingandthejosephsoncurrentinaballisticsuperconductorquantumwiresuperconductorjunction |
| first_indexed |
2025-07-08T16:29:14Z |
| last_indexed |
2025-07-08T16:29:14Z |
| _version_ |
1837096926616485888 |
| fulltext |
Fizika Nizkikh Temperatur, 2004, v. 30, No. 5, p. 535–543
Chiral symmetry breaking and the Josephson current in a
ballistic superconductor–quantum wire–superconductor
junction
I.V. Krive1,2, L.Y. Gorelik1, R.I. Shekhter1, and M. Jonson1
1Department of Applied Physics, Chalmers University of Technology and Göteborg University
SE-412 96 Göteborg, Sweden
2B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of
Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine
Email: krive@ilt.kharkov.ua
Received November 7, 2003, revised December 15, 2003
We evaluate the Josephson current through a quasi-1D quantum wire coupled to bulk supercon-
ductors. It is shown that the interplay of Rashba spin–orbit interaction and Zeeman splitting re-
sults in the appearence of a Josephson current even in the absence of any phase difference between
the superconductors. In a transparent junction (D�1) at low temperatures this anomalous
supercurrent Jan appears abruptly for a Zeeman splitting of the order of the Andreev level spacing
as the magnetic field is varied. In a low transparency (D << 1) junction one has J Dan � under
special (resonance) conditions. In the absence of Zeeman splitting the anomalous supercurrent dis-
appears. We have investigated the influence of dispersion asymmetry induced by the Rashba inter-
action in quasi-1D quantum wires on the critical Josephson current and have shown that the break-
down of chiral symmetry enhances the supercurrent.
PACS: 74.80.Fp, 31.30.Gs, 71.70.Ej, 73.20.Dx
1. Introduction
Quantum wires (QWs) have the potential of being
the basic elements in future nanometer-scale elec-
tronic devices. Electron transport in QWs is coherent
and ballistic. It results in a number of spectacular
phenomena such as conductance quantization, persis-
tent current oscillations in ring-shaped wires (quan-
tum rings), etc. At present, QWs are realized experi-
mentally in the form of long and narrow channels
( ~L d F�� � , where �F is the electron Fermi
wavelength) in a two-dimensional electron gas
(2DEG) (see Ref. 1) and as conducting 1D and
quasi-1D molecular systems. The most successful and
promising realization of 1D conductors is single-wall
carbon nanotubes [2]. It has been shown both theore-
tically [3] and experimentally [4] that in carbon
nanotubes the effects of Coulomb interaction are sig-
nificant and that they transform the conduction elec-
tron system into a Luttinger liquid. In quantum wires
formed in 2DEG the electron–electron interaction is
less pronounced (presumably due the screening effects
of the nearby bulk metallic electrodes) and electron
transport in these systems can in many cases be suc-
cessfully described by Fermi liquid theory.
It has long been known [5] that electrons confined
to a plane (e.g., in MOSFET structures or in he-
terostructures) experience a strong spin–orbit interac-
tion originating from interface electric field (Rashba
spin–orbit interaction). Recently it was shown experi-
mentally that the strength of the Rashba coupling can
be controlled by a gate voltage [6]. The Rashba effect
leads to various interesting suggestions in spintronics
and it has been a subject of active theoretical and ex-
perimental studies in recent years (see, e.g., Ref. 7
and references therein).
For quasi-1D electron systems the influence of the
Rashba interaction on thermodynamic and transport
properties of quantum wires were considered in
Refs. 8 and 9. There it was shown that in the presence
of an in-plane confinement potential and spin–orbit
interactions the electron spectrum is qualitatively
© I.V. Krive, L.Y. Gorelik, R.I. Shekhter, and M. Jonson, 2004
modified. Chiral symmetry, which is usually assumed
to be present in QWs, is violated, resulting in the
appearence of a dispersion asymmetry. To be more pre-
cise, the right-moving spin-up* (left-moving spin-down)
and left-moving spin-up ( right-moving spin-down)
electrons have different Fermi velocities [8]. This im-
plies that electrons in quantum wires with Rashba in-
teraction are «chiral particles», and their spin projec-
tions are correlated with the direction of motion.
Being interested in low-energy (E F�� � ) properties
of quantum wires, we can classify these particles as
belonging to two subbands («1» and «2») character-
ized by their Fermi velocities (see Fig. 1). Notice that
this electron spectrum pertains to a weak or moderate
spin–orbit interaction. As was demonstrated in Ref. 10
for a strong Rashba interaction, the projection of elec-
tron spin is strongly correlated with the direction of
motion, and left- and right-moving electrons with the
Fermi energy always have opposite spin projections.
The unusual spectral properties of electrons in
QWs have to show up in situations when spin degrees
of freedom are nontrivially involved in the electron
dynamics. Here we consider the Josephson current in a
long S–QW–S junction for an electron spectrum with
dispersion asymmetry and large Zeeman splitting. Re-
cently the combined effect of Zeeman and spin–orbit
interactions on the Josephson current in a short ballis-
tic junction formed in a 2DEG was theoretically stud-
ied [11]. Notice that interaction induced dispersion
asymmetry in the electron spectrum is a specific pro-
perty of quasi-one-dimensional geometry [8]. For a
pure 2D geometry spin–orbit interaction does not lead
to a chiral symmetry breaking, and the peculiar effects
produced by chiral particles do not manifest them-
selves in the short and wide SNS junction considered
in Ref. 11. In particular, we show that the combined
effect of Rashba and Zeeman interactions results in
the appearence of an anomalous Josephson current
J Jan � �( )� 0 in a long S–QW–S junction. For a
transparent junction the induced by Rashba and
Zeeman interaction supercurrent at low temperatures
is a step-like periodic or quasiperiodic function of
magnetic field (see Fig. 2). The periodicity depends
on the ratio of the Fermi velocities and is controlled
by the strength of the spin–orbit interaction. The am-
plitude of the anomalous current at T = 0 is of the or-
der of the critical Josephson current in a long S–N–S
junction and it appears abruptly at finite values of
Zeeman splitting of the order of the Andreev level
spacing (Fig. 2). For a low-transparency junction
(D << 1), realized by introducing a scattering barrier
(impurity) into the normal region, the maximum am-
plitude of Jan at special (resonance) conditions is of
the order of D. Notice that this unusual dependence
536 Fizika Nizkikh Temperatur, 2004, v. 30, No. 5
I.V. Krive, L.Y. Gorelik, R.I. Shekhter, and M. Jonson
* Under the conditions of Rashba effect the electron spin lies in the plane of the heterostructure transition layer and it
is always perpendicular to the electron momentum. The terms spin-«up» and spin-«down» determine two opposite
spin projections orthogonal to the line of the quantum wire at the 2D interface.
�(p)
p�F
2 2
1 1
Fig. 1. Schematic energy spectrum of 1D electrons with
dispersion asymmetry. The particles with energies close to
the Fermi energy �F have an almost linear dependence on
momentum and are classified by their Fermi volocities
(v F1 — subband 1, v F2 — subband 2).
2 4 6 8 10 12
0.6
0.4
0.2
0.8
–0.2
–0.4
–0.6
b
J
J/
a
n
0
2
1
3
4
� �/z L
Fig. 2. The dependence of the normalized anomalous
Josephson current J J Fan J ev L0 0( � ) on the dimensionless
Zeeman splitting � � �Z L L Fv L( � � ). (a) asymmetry pa-
rameter �a � 03. . The different plots (1–4) correspond to
different temperatures T = (0.1, 0.5, 3, 5)T* where T �
� �L 2
. (b) �a � 07. ; T = (0.1, 0.5, 1, 3)T*.
0.6
0.4
0.2
0.8
–0.2
–0.4
–0.6
2 4 6 8 10 12J
J/
a
n
0
a
2
1
3
4
� �/z L
on transparency, which corresponds to tunneling of a
single electron through a barrier (its entangled part-
ner resonantly passes through the structure) holds
also for the critical Josephson current even in the ab-
sence of dispersion asymmetry. This effect can be
interpreted as a tunnel splitting of the de-
Gennes—Saint-James bound state [12] shifted by
magnetic field to the vicinity of the Fermi level. Anal-
ogous effects of giant critical supercurrents in tunnel
SIS («I» stands for insulator) and SINIS structures
were discussed in Ref. 13 (see also Ref. 14, where res-
onance effects are considered for a persistent current
in a normal metal ring).
When the Zeeman splitting vanishes, the anoma-
lous supercurrent and all the above described reso-
nance effects disappear. Then the Josephson current in
a low-D junction is small (�D). What is the influence
of dispersion asymmetry on the critical current? There
is a general statement [15] that spin–orbit interaction
in systems with the Aharonov—Bohm geometry sup-
presses persistent currents. Although the theorem di-
rectly concerns normal ring-shaped conductors, it also
holds for linear hybrid systems with Andreev mirrors
due to the analogy between persistent currents in a
normal 1D ring and Josephson currents in a long SNS
junction. We show here that the cited statement is not
valid when the spin–orbit interaction is accompanied
by chiral symmetry breaking. Rashba spin–orbit inter-
action in quantum wires always enhances the critical
current.
2. Anomalous Josephson current
The Josephson current, being an equilibrium super-
current between two superconductors, can be calcu-
lated from the general thermodynamic relation
J
e
�
�
��
�
, (1)
where
is the thermodynamic potential of the junc-
tion considered and � is the phase difference between
the two superconductors. We have included a factor
of 2, which usually appears in Eq. (1) in combination
with the electric charge, into the definition of
. This
factor originates from spin degeneracy, and in the
presence of Zeeman splitting
� �� � and
� �� .
In general both the Andreev bound states ( | |,E � �
� is the superconducting order parameter) and the
continuum scattering states (E � | |� ) contribute to the
supercurrent. In two limiting cases — short
( ,L v LF�� ��0 � � is the junction length) and long
(L �� �0) junctions — only bound states are relevant.
This statement is well known for the case of short
junctions (see, e.g., Ref. 16 and references therein).
For a long junction it has been shown that the
Josephson current through a long SNS junction does
not depend on | |� at all [17]. Then, one can formally
put | |� � � and sum over all Andreev bound states
E nn ( , , ,...)� � �0 1 2 with the natural assumption that
the supercurrent vanishes in the limit L � �. This
procedure (analogous to Casimir energy evaluation in
quantum field theory; see, e.g., Ref. 18) reproduces
all known results for a long ballistic SNS junction. In
what follows we will consider only long junctions.
To get analytical results we consider a single bar-
rier junction of length L, where the barrier is located
at the point x l L� � ; the distance l is measured from
the left bank of the junction (see Fig. 3). We start
with the general equation for Andreev bound state en-
ergies expressed in terms of scattering matrices of elec-
trons ( �SE) and holes ( �S E�
) in the normal region [19]
det( � � � � )1 0� ��
� A E A E AS r S r , (2)
where � A i E� �exp[ arccos( )]2 � � –1 in our case
(E �� �), and �rA is a diagonal matrix that only de-
pends on the superconducting phases (see Appendix).
Since the presence of a magnetic field violates T-sym-
metry, the two-channel scattering of spin-1/2 elec-
trons is described by a 4 4� nonsymmetric unitary
matrix �SE . The normal backscattering in our ap-
proach (we neglect spin flips induced by a weak
Rashba interaction) is always accompanied by a
change of «channel index» 1 � 2 (see Fig. 1). This
allows us to parametrize the scattering matrix by 7 in-
dependent real parameters (see Appendix).
After straightforward (although rather lengthy)
transformations, Eq. (2) results in the transcendental
equation for Andreev level energies of the form
cos[( ) ] cos[( ) ]( ) ( )E R EZ L Z L l� � � ��
�
��
�
�
�
2 2 2� �
� � � ��D E Z Lcos[( ) ]( )�
� ��
2
0� . (3)
Here � Z Bg H� � is the Zeeman energy splitting
(g � 2 for bare electrons, and we assume that the mag-
netic field H is locally applied only on the quantum
Chiral symmetry breaking and the Josephson currentin a ballistic superconductor
Fizika Nizkikh Temperatur, 2004, v. 30, No. 5 537
� � ��������
�������
�������
�������
�������
� � ��������
�������
�������
�������
�������
� � ��������
�������
�������
�������
�������
� � ��������
�������
�������
�������
�������
L
�
S S
Fig. 3. A long (L �� �0) ballistic S–N–S junction with a
scattering barrier (shaded region).
wire), � � �1, D is the transparency of the junction
D + R = 1, and
� x
F F
x
v v
( )� � �
�
!!
"
#
$$
�
1 1
1 2
. (4)
In the limit H = 0 and v vF F1 2� (absence of spin–or-
bit interaction) Eq. (3) is reduced to a well-known
spectral equation for Andreev levels in a long SNS
junction with a single barrier [16,20].
For a transparent junction (D = 1) the Andreev
bound states are described by two independent sets of
energy levels
E nn L,
( ) ( )
�
�
� %
1 1 11
2 2
� � �
��
!
"
#
$� , (5)
E mm L,
( ) ( )
�
�
� %
2 2 21
2 2
� � �
��
!
"
#
$� ,
where the integers n m, , , ,... and� � � � �0 1 2 1� are
the standard quantum numbers of the Andreev–Kulik
spectrum in a perfectly transmitting SNS junction
[21]; � L
j
jFv L( ) � � and % j Z L
j� � �( ) is the phase
acquired due to Zeeman interaction. Notice that for a
given band index («1» or «2») the relative sign be-
tween the superconducting phase difference � and the
magnetic phase % j is fixed and is different for chan-
nels «1» and «2». This is a direct consequence of the
chiral properties of the electrons in our model. In the
absence of spectral asymmetry the two energy spectra
Eq. (5) correspond to Zeeman-split Andreev levels.
Knowing explicitly the energy spectrum Eq. (5), it
is straightforward to evaluate the Josephson current.
It takes the form
J T H
eT
( , , )� � �
2
�
� � �
�
�
��
�
�
&( )
sin ( )
sinh( )
sin ( )
( )
1
2
1
1
1
1
2k
k L
k
kT
k� %
� %
� sinh( )( )2 2
kT L�
'
(
)
*)
+
,
)
-)
.
(6)
It is readily seen from Eq. (6) that an anomalous
Josephson current J Jan . �( )� 0 appears only if both
the Zeeman splitting (� Z) and dispersion asymmetry
(v vF F1 2� ) are nonzero. Here we use the term «ano-
malous» just to define the supercurrent at � � 0.
Actually this current is induced by Zeeman splitting
in a 1D electron system with dispersion asymmetry.
The direction (sign) of Jan is totally determined by
the sign of asymmetry parameter (see Eq. (8)). At
high temperatures T L
j� �( ) the anomalous supercur-
rent is exponentially small. In the low-temperature
region (T L
j�� �( )) it is a piecewise constant function
of magnetic field represented by the series
/ 0J H
e
L k
v k v kan
k
F F
k
( )
( )
sin sin�
�
�
�
�
�
&
% %
1 1
1 1 2 2
1
.
(7)
For rational values v v p q p qF F1 2 � �( are the inte-
gers) Jan is a periodic function of magnetic field with
period �
�H q gL B� 2 1�( ) ; otherwise it is a quasi-
periodic function.
It is convenient to introduce the mean Fermi velocity
v v vF F F� �( )1 2 2 and the asymmetry parameter � a ,
� a
F F
F F
v v
v v
�
�
�
1 2
1 2
, (8)
which determines the strength of Rashba spin–orbit
interaction in a 1D quantum wire. The dependence of
the normalized anomalous supercurrent J J Jan 0 0( �
� ev LF ) on the dimensionless Zeeman splitting
% � �� � �Z L L Fv L( )� for � a � 0 3. and for differ-
ent temperatures is shown in Fig. 2,a. In the limit of
strong asymmetry (this range of parameters, however,
seems to be unrealistic[8] ), when only one of the two
channels («1» or «2») contributes to Eq. (6), the de-
pendence of the anomalous current on the magnetic
field becomes analogous to the well-known phase de-
pendence of the Josephson current [21]. The approach
to this simple behavior passes via the stages of stair-
case-like dependences (see Fig. 2,b). Notice that we
plotted the figures assuming that v vF F1 2� . The in-
terchange v vF F1 21 makes the supercurrent Eq. (7)
change sign.
3. Giant critical current in a magnetically
controlled low-transparency junction
Now we consider the limit D << 1 pertaining to
low-transparency SNS junctions. As is well known
(see, e.g., Refs. 16,20), the supercurrent in this limit
for a single barrier junction in the absence of Zeeman
and spin–orbit interactions is described by the simple
formula J Jc( ) sin� �� , where the critical current at
low temperatures T L�� � is of the order of D
( )~J Dev Lc F . Interesting physics for low-D junc-
tions appears when resonant electron tunneling occurs.
This is, for instance, the case for the symmetric dou-
ble-barrier ballistic junction considered in Refs. 13,16.
There it was shown that for resonance conditions (re-
alized for a special set of junction lengths) a giant
critical supercurrent appears, J Dc b� , where Db is
the transparency of a single barrier. Analogous results
were obtained for the persistent current in a ballistic
ring with a double barrier [14]. Notice that for the
symmetric structure considered in Ref. 13 the normal
current (transmission coefficient, which determines the
current through a double-barrier structure with normal
538 Fizika Nizkikh Temperatur, 2004, v. 30, No. 5
I.V. Krive, L.Y. Gorelik, R.I. Shekhter, and M. Jonson
leads) at resonance conditions does not depend on bar-
rier transparency at all. This means that for the hybrid
structure considered in Ref. 13, the superconductivity
actually suppresses electron transport.
We show below that in a magnetically controlled
single-barrier junction there are conditions when su-
perconductivity in the leads strongly enhances elec-
tron transport, and a mesoscopic hybrid structure is
characterized by a giant critical current J Dc � .
We start with the case of a symmetric single-barrier
junction, when the scattering barrier is situated in the
middle of the normal region, i.e., l = L/2 in Eq. (3).
Then � x�
� �0 0( ) , and the second cosine term in the spec-
tral equation Eq. (3) is equal to one:
cos ( )E Z L l��
!
"
#
$
2
3
4
5
6
7 ��
��
�
2
12� . (9)
When the condition (9) is fulfilled, the spectral equa-
tion is reduced to the much simpler expression
cos ( )2 1
2 2
E Z L��
!
"
#
$
2
3
4
5
6
7 ���
��
� ��
!
"
#
$ �
2
3
4
5
6
7
�D E Z Lsin ( )2 1
2 2
�
� ��� . (10)
By using Eq. (10), one can readily evaluate the par-
tial supercurrent j{�8 characterized by 3 quantum num-
bers { ( , , )�8 � 9� n where n � � � � �0 1 2 1, , ,...; ,� 9 :
j
e E e
D E L Z L
:�8
�8
�8
�
�9
� � � �
�
�
�
� �
� ��
� �
� �
{
{
( ) ( )cos
1
2
1
2
�!
"
#
$
2
3
4
5
6
7
��
!
"
#
$
2
3
4
4
5
6
7
7
��
�
�
� �
�8L
L
ZE( )
( )
{sin
1
2 2 2
� 9� � � � �:�8L L Z LD E( ) ( ) ( )cos� � �� ��
!
"
#
$
2
3
4
5
6
7
1
2
1
2
�
, (11)
where E En{ , ,�8 � 9. is a solution of Eq. (10). The
Josephson current at T = 0 is a sum of partial currents
over all occupied states.
The resonance current (of order D) is formed by
noncompensated partial currents carried by the
Andreev levels in the vicinity of the Fermi energy,
i.e., for E{�8 � �0 when D � 0. Such levels exist only
for a discrete set of Zeeman splittings
� Z
k
L
k
k( )
( )
( )
, , , ,...�
�
�
�
2 2 1
0 1 2
�
(12)
At a given � Z
k( ) (controlled, e.g., by an external local
magnetic field) two (� � �1) Andreev levels contrib-
ute to the resonance Josephson current Jr , which can
be represented in the form
J J Dr
a
Z
k
L a
Z
( )
sin
sin
( ) ( )
(
�
�
� � �
�
�
��
!
"
#
$�
0
21
2
1
2
1
2
1
2
�
� k
L a
) ( )� � �� ��
!
"
#
$
;
;
; ;
;
;
,
(13)
where J ev LF0 � , and the asymmetry parameter � a
is defined in Eq. (8).
In the absence (� a � 0) of spin–orbit interaction
Eq. (13) has the typical form of a resonance Josephson
current (see, e.g., Ref. 16) associated with the contri-
bution of a single Andreev level. One can interpret
this result as follows. Let us assume for a moment that
the potential barrier is infinite. Then, a symmetric
SNINS junction («I» stands for the insulator «layer»)
breaks into two identical INS-hybrid structures. In
each of the two systems de Gennes–Saint-James ener-
gy levels [12] with spacing 2
�v LF are formed. For
a finite barrier these levels are split with the charac-
teristic splitting energy � ~ D L L� ��� . The tun-
nel-split levels, being localized already on the whole
length L between the two superconductors, are noth-
ing but the Andreev energy levels, i.e., they depend
on the superconducting phase difference. Although
the partial current of a single level is large (of the or-
der of D, see Ref. 13,16), the current carried by a
pair of split levels is small (~ D) due to a partial can-
cellation. A Zeeman splitting of order � L shifts the
set of Andreev levels so that the Fermi energy lies in
between the split levels. Now only the lower state is
occupied, and this results in an uncompensated, large
Josephson current. In other words only one of the two
electrons of the Cooper pairs that form the super-
current tunnels through the barrier. Its entangled
partner at resonance conditions passes through the hy-
brid SFIFS structure («F» denotes the region with
nonzero Zeeman splitting) without backscattering.
Since the quantized electron–hole spectrum is formed
by Andreev scattering, the resonance structure for a
single barrier junction disappears when the leads are
in the normal state. Hence, electron transport through
a normal region in our case is enhanced by supercon-
ductivity.
Chiral symmetry breaking and the Josephson currentin a ballistic superconductor
Fizika Nizkikh Temperatur, 2004, v. 30, No. 5 539
The effect of chiral symmetry breaking on the
physical picture described above is to additionally
split the degenerate Andreev levels. A dispersion
asymmetry � a � 0 lifts the left–right symmetry of
electron transport through the junction and splits the
doubly degenerate Andreev levels at � � 0. This re-
sults in appearence of giant anomalous Josephson cur-
rent (see Eq. (13)) at � � 0.
We saw that it is indispensable that Eq. (9) holds
in order for resonant transport through the single bar-
rier hybrid structure to occur. This equation can be
satisfied not only for the symmetric junction consid-
ered above. One can easily check that for a fixed value
of Zeeman splitting � Z
k( ) given by Eq. (12) there is a
set of points where a barrier still supports resonant
transport. These points, determined by their coordi-
nates xm
k( ) measured from the middle of a junction, are
(m is an integer)
x
m
k
L m km
k( ) ,� �
�
� � �
2 1
0 1 2. (14)
The temperature dependence of the D-currents is
determined by the energy scale � ~ D L� , and at
temperatures T � �, which are much less then � L, all
resonance effects are washed out.
4. Influence of chiral symmetry breaking on the
critical current
There is a general statement [15] that spin–orbit
interaction in 1D systems with the Aharonov–Bohm
geometry produces additional reduction factors in the
Fourier expansion of thermodynamic or transport
quantities. This statement holds for spin–orbit
Hamiltonians for which the transfer matrix is factor-
ized into spin–orbit and spatial parts. In a pure 1D
case the spin–orbit interaction is represented by the
Hamiltonian H 1D
so
so x zp( )
( ) �� � 9 , which is the product
of spin-dependent and spatial operators, and thus it
satisfies the above-described requirements. However,
as was shown by direct calculation in Ref. 8, spin–or-
bit interaction of electrons in 1D quantum wires
formed in 2DEG by an in-plane confinement potential
cannot be reduced to the Hamiltonian H 1D
so( ). Instead,
a violation of left–right symmetry of 1D electron
transport, characterized by a dispersion asymmetry
parameter � a , appears. We show now that in quantum
wires with broken chiral symmetry the spin–orbit in-
teraction enhances a persistent current.
There is a close analogy between the Josephson cur-
rent in a long SNS junction and the persistent current
in a normal metal ring. For a long (L vF�� � � ) SNS
junction Andreev boundary conditions can be recast
[22] in the form of twisted boundary conditions for
chiral ( right- or left-moving) fermions on a ring with
circumference 2L pierced by the magnetic flux
< <0 1 2 2� � �
, where <0 � hc e is the normal
flux quantum. Due to this mapping the corresponding
formulas for the persistent current in a normal dia-
magnetic (with odd number of spinless fermions) 1D
ring and the formulas for the Josephson current coin-
cide up to numerical factors. Here we consider the in-
fluence of dispersion asymmetry in the electron spec-
trum on the off-resonance supercurrent through a
single barrier SNS junction (notice that resonace ef-
fects disappear in zero magnetic field).
In a low-transparency 1D SNS junction the critical
current in the presence of dispersion asymmetry is of
the form J f Dev Lc a F� ( )� . To evaluate the func-
tion f a( )� analytically we consider two limiting
cases: (i) an asymmetric junction l = 0, and (ii) a sym-
metric junction l = L/2.
The first case models a junction with strong normal
backscattering at one of the two interfaces. In zero
magnetic field the spectral equation, Eq. (3), in the
limit considered is reduced to
cos( ) cos( )( ) ( )E
D
EL L� � ��� �� ��
2
. (15)
The energy spectrum and the partial supercurrents in
the limit D << 1 are
E nn
L
( )
( )
0
2
1
2
� ��
!
"
#
$�
�
,
j
e D
nn
a n
L
a,
( )
( )
( ) sin� �
�
� ��� � ��
!
"
#
$ �2
3
4
5
6
7
�
�
1
2
1
2
1
�
.
(16)
540 Fizika Nizkikh Temperatur, 2004, v. 30, No. 5
I.V. Krive, L.Y. Gorelik, R.I. Shekhter, and M. Jonson
0.60.40.2 0.8 �
1.1
1.2
1.3
1.4
1.5
J
(
)
J
(
0
)
�
/
c
c
a
s
a
a
Fig. 4. The dependence of the normalized critical current
J Jc a c a( ) ( )� � � 0 in a low-transparency (D << 1) S–N–S
junction on the parameter of dispersion asymmetry �a.
The curve labelled by «s» corresponds to the case of sym-
metric junction (l = L/2), while the curve «a» describes
a strongly asymmetric junction (l = 0).
By summing partial currents over quantum numbers
of occupied states (En
( )0 0� ) at T = 0 one gets
J j Ja
n
a
n
c
a
a
( )
,
( )
;
( )( )sin� �
�� ��
�
& �
�
� �
1 1
,
J
ev
L
Dc
a
a
F a
a
( )( )
cos( )
�
�
�
�
�
4
1
2
2
. (17)
The critical current in the absence of spin–orbit inter-
action J Dev Lc
a
F
( )( )0 4� coincides with the known
results (see, e.g., Ref. 16). The normalized current
J Jc
a
c
a( ) ( )( ( )�0 0 is shown in Fig. 4 (curve «a»).
For a symmetric junction the analogous calculation
leads to the expression
J
ev
L
D J
ev
L
Dc
s
a
F a a
a
c
s F( ) ( )( )
( )
sin( )
; ( )�
� �
�
�
�
�
1
0
2
.
(18)
The curve labelled by «s» in Fig. 4 demonstrates the
dependence of the critical current in a symmetric
junction on the spin–orbit parameter � a . We see that
the spin–orbit Rashba interaction in quasi-1D quan-
tum wires always enhances the critical current. The
qualitative explanation of the unusual impact of
chiral symmetry breaking on the critical Josephson
current is as follows. The Josephson current (as any
other thermodynamic persistent current) is a sum of
partial currents of all occupied energy levels. The
partial currents of adjacent energy levels are opposite
in sign, and for chiral invariant systems (i.e., without
dispersion asymmetry) they almost perfectly cancel
each other to produce a net current of the order of a
single-level current. When chiral invariance is broken
the absolute values of partial currents of adjacent en-
ergy levels are different, and the concellation of cur-
rents for distant levels is less perfect. It results in an
enhancement of the critical Josephson current. The
effect is, however, numerically not large.
5. Conclusion
In quantum wires formed in a two-dimensional
electron gas (2DEG) by lateral confinement the
Rashba spin–orbit interaction is not reduced to a pure
1D Hamiltonian H 1D
so
so x zp( )
�� � 9 . As was shown in
Ref. 8 the presence of an in-plane confinement poten-
tial qualitatively modifies the energy spectrum of the
1D electrons so that a dispersion asymmetry appears.
As a result the chiral symmetry is broken in quantum
wires with Rashba coupling. Although the effect was
shown [8] not to be numerically large, the breakdown
of symmetry leads to qualitatively novel predictions.
We have considered here the influence of dispersion
asymmetry and Zeeman splitting on the Josephson
current through a superconductor/quantum wire/su-
perconductor junction. We showed that the violation
of chiral symmetry in a quantum wire results in quali-
tatively new effects in a weak superconductivity. In
particularly, the interplay of Zeeman and Rashba in-
teractions induces a Josephson current through the hy-
brid 1D structure even in the absence of any phase dif-
ference between the superconductors. At low tempera-
tures (T v LF�� � ) the anomalous supercurrent can be
of the order of the critical Josephson current. For a
transparent junction with small or moderate dispersion
asymmetry (characterized by the dimensionless para-
meter � a F F F Fv v v v� � �( ) ( )1 2 1 2 ) it appears, as a
function of the Zeeman splitting � Z, abruptly at
� Z Fv L~ � . In a low-transparency (D << 1) junc-
tion, the anomalous Josephson current under special
(resonance) conditions is of the order of D. In zero
magnetic field the anomalous supercurrent disappears
(as it should) since the spin–orbit interaction itself re-
spects T-symmetry. However, the influence of the
spin–orbit interaction on the critical Josephson cur-
rent through a quasi-1D structure is still anomalous.
Contrary to what holds for chiral invariant systems
with the Aharonov–Bohm geometry, where spin–orbit
effects suppress persistent currents [15], the break-
down of chiral symmetry results in an enhancement of
the supercurrent.
All the phenomena described above are absent in a
2D-junction when the effects of transverse mode
quantization are neglected [11]. We have considered
the limiting case of a single (transverse) channel be-
cause this is the case for which the effects induced by a
dispersion asymmetry in the electron spectrum are
most pronounced. The anomalous supercurrent
Eq. (7) is a sign-alternating function of the transverse
channel index, since for neighboring channels the spin
projections of the chiral states are opposite [8]. Be-
sides, the absolute value of the dispersion asymmetry
parameter � a
j( ) decreases with transverse-channel num-
ber j. So, for a multichannel junction the effects re-
lated to a dispersion asymmetry phenomenon will be
strongly suppressed, and they completely disappear in
the pure 2D case.
We evaluated the Josephson current through a
S–QW–S junction in a model of noninteracting elec-
trons. In QWs the effects of electron–electron interac-
tion can be significant, and here we comment on how
interaction effects could modify the obtained results.
Electronic properties of 1D QWs are usually de-
scribed by a Luttinger liquid ( LL ) model (see, e.g.,
Ref. 23). The supercurrent in a S–LL–S junction with
repulsive electron–electron interaction strongly de-
Chiral symmetry breaking and the Josephson currentin a ballistic superconductor
Fizika Nizkikh Temperatur, 2004, v. 30, No. 5 541
pends on the quality of electric contact of a nanowire
with the bulk superconductors. For adiabatic con-
tacts, when only Andreev scattering takes place at the
interfaces, the Josephson current through a perfect
wire is not renormalized by interaction [22,24]. For
tunnel contacts electron–electron interaction in a wire
renormalizes the barrier transparencies (Kane–Fisher
effect [25]), and for repulsive interaction the critical
current is strongly suppressed [26].
We are interested in spin–orbit effects in a
S–QW–S junction. It is reasonably to expect that
Eqs.(6),(7) derived for a perfect junction (without
normal backscattering) will be valid even for interact-
ing electrons. One could expect also that in a tunnel-
ing regime (D << 1) the interaction effects in
Eqs. (13),(17),(18) can be estimated by replacing the
bare transparency D by the interaction-renormalized
one [22] D D d Lg
K~ ( ) =
� �1 1 (here d << L is the
width of the wire and K= is the LL correlation param-
eter). In quantum wires formed in 2DEG the Coulomb
interaction is not strong [1] and it can be controlled
by the gate electrodes. For weakly interacting elec-
trons (a conceivable case for «semiconducting»
QWs), when the electron picture of charge transport
through a 1D wire is still valid [27], the interaction
effects can actually be accounted for by replacing the
bare transmission coefficient by the renormalized one.
The authors thank E. Bezuglyi, A. Kadigrobov, and
V. Shumeiko for fruitful discussions. This research is
supported by the Royal Swedish Academy of Sciences
(KVA) and by the Swedish Research Council (LYG,
RIS). IVK acknowledges the hospitality of Depart-
ment of Applied Physics, Chalmers University of
Technology and Göteborg University.
6. Appendix
We consider electron transport through a normal
region of length L with a local scatterer placed a dis-
tance l from the left bank, at point x = l. In our
two-channel chiral model (the term «chiral» here
means that left- and right-moving particles with a
given Fermi velocity have opposite spin polarizations;
see Fig. 1) backscattering by a nonmagnetic impurity
corresponds to a backward interchannel scattering (
«1» � «2»). In the presence of a magnetic field and
interchannel scattering the general 4�4 nonsymmetric
S-matrix can be parametrized as follows
�
| | exp ( ) | | exp ( )
| | exp (
S
r i t i
t iE
R R R R�
� �9 = 9 � 9 � 9 �
9
1 3 3 3
3 � 9 � 9 = 9 �L L L Lr i� �
�
!!
"
#
$$
3 1 3) | | exp ( )
.
Here 913, are the Pauli matrices, t t iR L R L, ,| | exp( )� � ;
r r iR L R L, ,| | exp( )� = and | | | |t r2 2 1� � . Unitarity of the
S-matrix (SS I† � ) implies the relations
� � � �R R L L� � � , = = � �R L R L� � � . (19)
Thus, the scattering matrix in our problem is de-
scribed by 7 independent real parameters. For a point
scatterer with energy-independent scattering ampli-
tudes (t r0 0, ) one readily gets that | | | |t t D2
0
2� . (D is
the junction transparency). The phases are
�R L l k� � � �( ) , =R L l k� � �( ) , �R Lk� �
��
� R Lk� �
�� , �L lk� �, = >L lk� � �L Lk� �
�� ,
� L Lk� �
�� ,
(20)
where
k k k� �
��
�
��� � , k k k�
? ? ?� �99 9 91
2 1 2( ),
k E k
E g H
vj jF
B
jF
9 9 �
( )� � �
� � 2
�
(21)
and k v jj jF, ( , )� 1 2 are the Fermi momentum and the
Fermi velocity in the jth channel.
The diagonal matrix �rA in Eq. (2) takes the form
�
exp( ) �
exp( ) �
r
i i I
i i I
A �
�
�
!!
"
#
$$
�
�
2 0
0 2
, (22)
where � is the superconducting phase difference and �I
is the 2�2 unit matrix.
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