Stationary Josephson effect in a weak-link between nonunitary triplet superconductors
A stationary Josephson effect in a weak-link between misorientated nonunitary triplet superconductors is investigated theoretically. The non-self-consistent quasiclassical Eilenberger equation for this system has been solved analytically. As an application of this analytical calculation, the curr...
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Rashedi, G. Kolesnichenko, Yu.A. 2017-06-14T12:25:46Z 2017-06-14T12:25:46Z 2005 Stationary Josephson effect in a weak-link between nonunitary triplet superconductors / G. Rashedi, Yu.A. Kolesnichenko // Физика низких температур. — 2005. — Т. 31, № 6. — С. 634-639. — Бібліогр.: 15 назв. — англ. 0132-6414 PACS: 74.20.Rp, 74.50. + r, 74.70.Tx, 85.25.Cp, 85.25.Dq https://nasplib.isofts.kiev.ua/handle/123456789/121475 A stationary Josephson effect in a weak-link between misorientated nonunitary triplet superconductors is investigated theoretically. The non-self-consistent quasiclassical Eilenberger equation for this system has been solved analytically. As an application of this analytical calculation, the current-phase diagrams are plotted for the junction between two nonunitary bipolar f-wave superconducting banks. A spontaneous current parallel to the interface between superconductors has been observed. Also, the effect of misorientation between crystals on the Josephson and spontaneous currents is studied. Such experimental investigations of the current-phase diagrams can be used to test the pairing symmetry in the above-mentioned superconductors. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Свеpхпpоводимость, в том числе высокотемпеpатуpная Stationary Josephson effect in a weak-link between nonunitary triplet superconductors Article published earlier |
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Stationary Josephson effect in a weak-link between nonunitary triplet superconductors |
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Stationary Josephson effect in a weak-link between nonunitary triplet superconductors Rashedi, G. Kolesnichenko, Yu.A. Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Stationary Josephson effect in a weak-link between nonunitary triplet superconductors |
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Stationary Josephson effect in a weak-link between nonunitary triplet superconductors |
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Stationary Josephson effect in a weak-link between nonunitary triplet superconductors |
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stationary josephson effect in a weak-link between nonunitary triplet superconductors |
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Rashedi, G. Kolesnichenko, Yu.A. |
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Rashedi, G. Kolesnichenko, Yu.A. |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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A stationary Josephson effect in a weak-link between misorientated nonunitary triplet superconductors
is investigated theoretically. The non-self-consistent quasiclassical Eilenberger equation
for this system has been solved analytically. As an application of this analytical calculation,
the current-phase diagrams are plotted for the junction between two nonunitary bipolar f-wave
superconducting banks. A spontaneous current parallel to the interface between superconductors
has been observed. Also, the effect of misorientation between crystals on the Josephson and spontaneous
currents is studied. Such experimental investigations of the current-phase diagrams can be
used to test the pairing symmetry in the above-mentioned superconductors.
|
| issn |
0132-6414 |
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https://nasplib.isofts.kiev.ua/handle/123456789/121475 |
| citation_txt |
Stationary Josephson effect in a weak-link between nonunitary triplet superconductors / G. Rashedi, Yu.A. Kolesnichenko // Физика низких температур. — 2005. — Т. 31, № 6. — С. 634-639. — Бібліогр.: 15 назв. — англ. |
| work_keys_str_mv |
AT rashedig stationaryjosephsoneffectinaweaklinkbetweennonunitarytripletsuperconductors AT kolesnichenkoyua stationaryjosephsoneffectinaweaklinkbetweennonunitarytripletsuperconductors |
| first_indexed |
2025-11-24T23:42:05Z |
| last_indexed |
2025-11-24T23:42:05Z |
| _version_ |
1850500727600840704 |
| fulltext |
Fizika Nizkikh Temperatur, 2005, v. 31, No. 6, p. 634–639
Stationary Josephson effect in a weak-link between
nonunitary triplet superconductors
G. Rashedi1,3 and Yu.A. Kolesnichenko2
1Institute for Advanced Studies in Basic Sciences, Zanjan, 45195-1159, Iran
2B.Verkin Institute for Low Temperature Physics and Engineering of National Academy of Sciences of
Ukraine, 47, Lenin Ave., Kharkov 61103, Ukraine
3Department of Physics, Faculty of Science, University of Shahrekord, Shahrecord, P.O.Box 115, Iran
E-mail: rashedy@www.iasbs.ac.ir
Received November 5, 2004
A stationary Josephson effect in a weak-link between misorientated nonunitary triplet super-
conductors is investigated theoretically. The non-self-consistent quasiclassical Eilenberger equa-
tion for this system has been solved analytically. As an application of this analytical calculation,
the current-phase diagrams are plotted for the junction between two nonunitary bipolar f-wave
superconducting banks. A spontaneous current parallel to the interface between superconductors
has been observed. Also, the effect of misorientation between crystals on the Josephson and spon-
taneous currents is studied. Such experimental investigations of the current-phase diagrams can be
used to test the pairing symmetry in the above-mentioned superconductors.
PACS: 74.20.Rp, 74.50. + r, 74.70.Tx, 85.25.Cp, 85.25.Dq
1. Introduction
In recent years, the triplet superconductivity has
become one of the modern subjects for researchers in
the field of superconductivity [1–3]. Particularly, the
nonunitary spin triplet state in which Cooper pairs
may carry a finite averaged intrinsic spin momentum
has attracted much attention in the last decade [4,5].
A triplet state in the momentum space k can be de-
scribed by the order parameter � � ��( ) = ( ( ) )k d ki y� � �
in a 2�2 matrix form in which �� j are 2�2 Pauli matri-
ces ( � ( � , � , � ))� � � �� x y z . The three dimensional complex
vector d(k) (gap vector) describes the triplet pairing
state. In the nonunitary state, the product
� ( ) � ( ) ( ) ( ) ( ( ) ( )) �� �k k d k d k d k d k† � � � � �� �i � is not
a multiple of the unit matrix. Thus in a non-unitary
state the time reversal symmetry is necessarily broken
spontaneously and a spontaneous moment m k( ) �
� � �id k d k( ) ( ) appears at each point k of the momen-
tum space. In this case the macroscopically averaged
moment � m k( ) integrated on the Fermi surface
does not vanish. The value m(k) is related to the net
spin average by Tr[ � ( ) � � ( )]� �k k†� j . It is clear that the
total spin average over the Fermi surface can be non-
zero. As an application, the nonunitary bipolar state
of f-wave pairing symmetry has been considered for
the B-phase of superconductivity in the UPt3 com-
pound which has been created at low temperatures T
and small values of the magnetic field H [5,6].
In the present paper, the ballistic Josephson weak
link via an interface between two superconducting
bulks with different orientations of the crystallo-
graphic axes is investigated. This type of weak link
structure can be used for the demonstration of the
pairing symmetry in the superconducting phase [7].
Consequently, we generalize the formalism of paper
[8] for the weak link between triplet superconducting
bulks with a nonunitary order parameter. In the paper
[8], the Josephson effect in the point contact between
unitary f-wave triplet superconductors has been stud-
ied. Also, the effect of misorientation on the charge
transport has been investigated and a spontaneous cur-
rent tangential to the interface between the f-wave su-
perconductors, has been observed.
In this paper the nonunitary bipolar f-wave model
of the order parameter is considered. It is shown that
the current-phase diagrams are totally different from
the current-phase diagrams of the junction between
the unitary triplet ( axial and planar) f-wave super-
conductors [8]. Roughly speaking, these different
characters can be used to distinguish between
nonunitary bipolar f-wave superconductivity and the
other types of superconductivity. In the weak-link
© G. Rashedi and Yu.A. Kolesnichenko, 2005
structure between the nonunitary f-wave supercon-
ductors, the spontaneous current parallel to the inter-
face has been observed as a fingerprint for unconven-
tional superconductivity and spontaneous time reversal
symmetry breaking. The effect of misorientation on the
spontaneous and Josephson currents is investigated. It
is possible to find the value of the phase difference in
which the Josephson current is zero but the spontane-
ous current, which is produced by the interface and is
tangential to the interface, is present. In some config-
urations and at the zero phase difference, the Josephson
current is not generally zero but has a finite value.
This finite value corresponds to a spontaneous phase
difference which is related to the misorientation be-
tween the gap vectors d.
The arrangement of the rest of this paper is as fol-
lows. In Sec. 2 we describe the configuration that we
have investigated. For a non-self-consistent model of
the order parameter, the quasiclassical Eilenberger
equations [9] are solved and suitable Green functions
have been obtained analytically. In Sec. 3 the formulas
obtained for the Green functions have been used for the
calculation of the current densities at the interface. An
analysis of numerical results will be presented in Sec. 4
together with some conclusions in Sec. 5.
2. Formalism and basic equations
We consider a model of a flat interface y = 0 be-
tween two misorientated nonunitary f-wave supercon-
ducting half-spaces (Fig. 1) as a ballistic Josephson
junction. In the quasiclassical ballistic approach, in
order to calculate the current, we use «transport-like»
quasiclassical Eilenberger equations [9] for the energy
integrated Green functions
�
g F m( � , , )v r
v g i gF m� � � � �
� � � �
[ , ] ,
�3 0� (1)
and the normalization condition
� � �
gg � 1, where
�m T m� �( )2 1 are discrete Matsubara energies
m = 1, 2, ... T is the temperature, vF is the Fermi ve-
locity and
�
� �3 3�
� �I in which � (� j j � 1, 2, 3) are
Pauli matrices. The Matsubara propagator
�
g can be
written in the form:
�
g
g g i
i g g
�
� � � �
� � � �
1 1 1 2 2
2 1 3 4 2 4
g g
g g
� ( �) �
� ( �) � �
� � �
� � � � ��2
�
�
��
�
�
��, (2)
where the matrix structure of the off-diagonal self en-
ergy
�
� in the Nambu space is
�
� �
�
�
�
�
��
�
�
���
0
0
2
2
d
d
� �
� �
� �
� �
i
i
. (3)
The nonunitary states, for which d d� �� 0 are in-
vestigated. Fundamentally, the gap vector (order pa-
rameter) d has to be determined numerically from the
self-consistency equation [1], while in the present pa-
per, we use a non-self-consistent model for the gap
vector which is much more suitable for analytical cal-
culations [10]. Solutions to Eq. (1) must satisfy the
conditions for the Green functions and the gap vector
d in the bulks of the superconductors far from the in-
terface as follow:
�
g
i i
i in
m n n n n
n n
�
�
� � �
��
( �) [ ] � �
� [
1 2
2
� � � � �
� �� �
A d d A
d d A n n] � � ( �) �� � � �
�
�
��
�
�
���
� � �2 21 A
(4)
where
A
d d
d d d d d d
n
n
m n n m n n n n
i
�
�
� � � � � � �
�
� � �
n
2 2 2 2( ) ( )
(5)
and
� n
m n n n n
m n n m n n
�
� � � �
� � � � �
� �
�
2 2 2 2
2 2
[( ) ( ) ]
(
d d d d
d d d d� �� �) ( )2 2d dn n
(6)
d d v( ) ( , � ) exp�� � �
�
�
�
�
�12 2, FT
i
�
�
(7)
where � is the external phase difference between the
order parameters of the bulks and n = 1, 2 label the
left and right half spaces respectively. It is clear that
poles of the Green function in the energy space are in
� n � 0. (8)
Consequently,
Stationary Josephson effect in a weak-link between nonunitary triplet superconductors
Fizika Nizkikh Temperatur, 2005, v. 31, No. 6 635
c2
b2
a2
(i)
d (k)exp(– )1 i^
2
�
(ii)
d (k)exp( )2 i^
2
�
�
�
y
z c2 b2
a2
c1
b1
a1
Fig. 1. Scheme of a flat interface between two supercon-
ducting bulks which are misorientated as much as �.
( ) ( )� � � � � �� �E n n n n
2 2 2 0d d d d (9)
and
E in n n n� � � � �� �d d d d
2
(10)
in which E is the energy value of the poles. The equa-
tion (1) has to be supplemented by the continuity
conditions at the interface between superconductors.
For all quasiparticle trajectories, the Green functions
satisfy the boundary conditions both in the right and
left bulks as well as at the interface. The system of
equations (1) and the self-consistency equation for
the gap vector d [1] can be solved only numerically.
For unconventional superconductors such solution
requires the information about the interaction be-
tween the electrons in the Cooper pairs and the na-
ture of unconventional superconductivity in novel
compounds which in most cases are unknown. Also, it
has been shown that the absolute value of a self-con-
sistent order parameter is suppressed near the inter-
face and at the distances of the order of the coherence
length, while its dependence on the direction in the
momentum space almost remains unaltered [11]. This
suppression of the order parameter changes the ampli-
tude value of the current, but does not influence the
current-phase dependence drastically. For example, it
has been verified in Ref. 12 for the junction between
unconventional d-wave superconductors, in Ref. 11
for the case of unitary «f-wave» superconductors and
in Ref. 13 for pinholes in 3He, that there is good
qualitative agreement between self-consistent and
non-self-consistent results for not very large angles of
misorientation. It has also been observed that the re-
sults of the non-self-consistent model in [14] are simi-
lar to experiment [15]. Consequently, despite the fact
that this solution cannot be applied directly to a
quantitative analysis of a real experiment, only a
qualitative comparison of calculated and experimen-
tal current-phase relations is possible. In our calcula-
tions, a simple model of the constant order parameter
up to the interface is considered and the pair breaking
and scattering on the interface are ignored. We be-
lieve that under these strong assumptions our results
describe the real situation qualitatively. In the frame-
work of such a model, the analytical expressions for
the current can be obtained for a certain form of the
order parameter.
3. Analytical results
The solution of Eq. (1) allows us to calculate the
current densities. The expression for the current is:
j r v v r( ) ( ) ( � , , )� �2 0 1i eTN gF F m
m
�
(11)
where ... stands for averaging over the directions of
an electron momentum on the Fermi surface �v F and
� �N 0 is the electron density of states at the Fermi
level of energy. We assume that the order parameter
is constant in space and in each half-space it equals its
value (7) far from the interface in the left or right
bulks. For such a model, the current-phase depend-
ence of a Josephson junction can be calculated analy-
636 Fizika Nizkikh Temperatur, 2005, v. 31, No. 6
G. Rashedi and Yu.A. Kolesnichenko
0 0.25 0.5 0.75 1.0
� �/2
–0.12
–0.08
–0.04
0
0.04
0.08
0.12
� �= /8
� �= /6
� �= /5
j
/j y
0
Fig. 2. Component of the current normal to the interface
(Josephson current) versus the phase difference � for the
junction between nonunitary bipolar f-wave bulks,
T/Tc � 015. , geometry (i) and different misorientations.
Currents are given in units of j eN vF0 0
2
0 0�
�
( ) ( )� .
0 0.25 0.5 0.75 1.0
� �/2
–0.08
–0.04
0
0.04
0.08 � �= /8
� �= /6
� �= /5
j
/j y
0
Fig. 3. Component of the current normal to the interface
(Josephson current) versus the phase difference � for the
junction between nonunitary bipolar f-wave bulks,
T/Tc � 015. , geometry (ii) and different misorientations .
tically. It enables us to analyze the main features of
current-phase dependence for any model of the
nonunitary order parameter. The Eilenberger equa-
tions (1) for Green functions
�
g, which are supple-
mented by the condition of continuity of solutions
across the interface, y = 0, and the boundary condi-
tions at the bulks, are solved for a non-self-consistent
model of the order parameter analytically. In the bal-
listic case the system of equations for functions gi
and gi can be decomposed into independent blocks of
equations. The set of equations which enables us to
find the Green function g1 is:
v g iF
� ( )k d g d g� � � � ��
1 3 2 ; (12)
vF
� ( )k g d g d g� � � � � ��
�2 3 2 ; (13)
v igF m
�k g g d d g� � � � � � �2 2 12 2
; (14)
v igF m
�k g g d d g� � � � �� �
�3 3 12 2
, (15)
where g g g� � �1 4. The Eqs. (12)–(14) can be
solved by integrating over the ballistic trajectories of
electrons in the right and left half-spaces. The general
solution satisfying the boundary conditions (4) at in-
finity is
g a s tn m
n
n n1 2( ) exp( )� � �
�
� ; (16)
g A C� � � � �( ) exp( )n m
n
n n ns t2 2
�
� ; (17)
g
d d A d d C
2
22
2 2
( )n n n n
n
n n n n
n m
s ti ia
s
n�
� �
�
� �
�
�
� �
�
�
e ;
(18)
g
d d A d d C
3
22
2 2
( )n n n n
n
n n n n
n m
s ti ia
s
n�
� �
�
� �
�
� � � �
�
� �
�
�
e ,
(19)
where t is the time of flight along the trajectory,
sgn sgn( ) ( )t y s� � and � � sgn( ).vy By matching the
solutions (16)–(19) at the interface � �y t� �0 0, , we
find constants an and Cn. Indices n = 1, 2 label the
left and right half-spaces, respectively. The function
g g g1 1
1
1
20 0 0( ) ( ) ( )( ) ( )� � � � which is a diagonal term
of the Green matrix and determines the current den-
sity at the interface, y = 0, is as follows:
g
B
1
2 2 1
2
1 1 2
2
2 1
0( )
[ ( ) ( ) ]
[ ( )
�
� � � � � �
� �
� �
�
�
d d d d
d d
� �
� 1 2
2( )]�
� �
(20)
where B i� � � � �d d A + A1 2 1 2 2 1( )( )( ).�
�
� �
We consider a rotation
�
R only in the right supercon-
ductor (see Fig. 1), i.e., d k d k2 1
1( � ( �)) =
� �
R R� ; �k is the
unit vector in the momentum space. The crystallo-
graphic c-axis in the left half-space is selected parallel
to the partition between the superconductors (along
the z-axis in Fig. 1). To illustrate the results obtained
by computing the formula (20), we plot the cur-
rent-phase diagrams for two different geometries.
These geometries correspond to the different orienta-
tions of the crystals in the right and left sides of the
interface (Fig. 1).
(i) The basal ab-plane in the right side has been ro-
tated around the c-axis by �; � | | �c c1 2.
(ii) The c-axis in the right side has been rotated
around the b-axis by �; � | | �b b1 2.
Further calculations require a certain model of the
gap vector (order parameter) d.
4. Analysis of numerical results
In the present paper, the nonunitary f-wave gap
vector in the B-phase (low temperature T and low
field H) of superconductivity in UPt3 compound has
been considered. This nonunitary bipolar state which
explains the weak spin-orbit coupling in UPt3 is [5]:
d v x y( , ) ( ) ( �( ) � )T T k k k ik kF z x y x y� � �� 0
2 2 2 . (21)
The coordinate axes �, �, �x y z are selected along the crys-
tallographic axes �, �, �a b c in the left side of Fig. 1. The
function � �0 0� ( )T describes the dependence of the
gap vector on the temperature T (our numerical cal-
culations are done at the low value of temperature
T/Tc � 01. ). Using this model of the order parameter
(21) and solution to the Eilenberger equations (20),
Stationary Josephson effect in a weak-link between nonunitary triplet superconductors
Fizika Nizkikh Temperatur, 2005, v. 31, No. 6 637
0 0.25 0.5 0.75 1.0
� �/2
–0.1
–0.05
0
0.05
0.1
0.15
� �= /8
� �= /6
� �= /5
j
/j x
0
Fig. 4. The x-component of the current tangential to the
interface versus the phase difference � for the junction be-
tween nonunitary bipolar f-wave superconducting bulks,
T/Tc � 015. , geometry (i) and the different misorientations.
we have calculated the current density at the inter-
face numerically. These numerical results are listed
below.
1. The nonunitary property of Green’s matrix diago-
nal term consists of two parts. The explicit part
which is in the B mathematical expression in Eq. (20)
and the implicit part in the �12, and d1,2 terms. These
�12, and d1,2 terms are different from their unitary
counterparts. In the mathematical expression for �12,
the nonunitary mathematical terms A12, are pre-
sented. The explicit part will be present only in the
presence of misorientation between gap vectors,
B i� � � � �d d A + A1 2 1 2 2 1( )( )( )�
�
� � , but the
implicit part will be present always. So, in the ab-
sence of misorientation (d1||d2) although the implicit
part of nonunitary exists the explicit part is ab-
sent.This means that in the absence of misorientation,
current-phase diagrams for planar unitary and
nonunitary bipolar systems are the same but the maxi-
mum values are slightly different.
2. A component of current parallel to the interface
jz for geometry (i) is zero similar to the unitary case
[8] while the other parallel component jx has a finite
value (see Fig. 4). This latter case is a difference be-
tween unitary and nonunitary cases. Because in the
junction between unitary f-wave superconducting
bulks all parallel components of the current (jx and
jz ) for geometry (i) are absent [8].
3. In Figs. 2,3, the Josephson current jy is plotted for
a certain nonunitary model of f-wave and different
geometries. Figures 2, 3 are plotted for the geometry
(i) and geometry (ii), respectively. They are
completely unusual and totally different from their
unitary counterparts which have been obtained in [8].
4. In Fig. 2 for geometry (i), it is observed that by in-
creasing the misorientation, some small oscillations
appear in the current-phase diagrams as a result of the
non-unitary property of the order parameter. Also,
the Josephson current at the zero external phase dif-
ference � = 0 is not zero but has a finite value. The
Josephson current will be zero at the some finite values
of the phase difference.
5. In Fig. 3 for geometry (ii), it is observed that by
increasing the misorientation the new zeros in cur-
rent-phase diagrams appear and the maximum value
of the current will be changed non-monotonically. In
spite of the Fig. 2 for geometry (i), the Josephson
currents at the phase differences � � 0, � �� , and
� �� 2 are zero exactly.
6. The current-phase diagram for geometry (i) and
x-component (Fig. 4) is totally unusual. By increas-
ing the misorientation, the maximum value of the cur-
rent increases. The components of current parallel to
the interface for geometry (ii) are plotted in Fig. 5
and Fig 6. All the terms at the phase differences
� � 0, � �� , and � �� 2 are zero. The maximum value
of the current-pase diagrams is not a monotonic func-
tion of the misorientation.
5. Conclusions
Thus, we have theoretically studied the super-
currents in the ballistic Josephson junction in the
638 Fizika Nizkikh Temperatur, 2005, v. 31, No. 6
G. Rashedi and Yu.A. Kolesnichenko
0 0.25 0.5 0.75 1.0
� �/2
–0.03
0
0.03
0.06
� �= /8
� �= /6
� �= /5
j
/j x
0
Fig. 5. Current tangential to the interface versus the
phase difference � for the junction between nonunitary bi-
polar f-wave superconducting bulks, T/Tc = 0.15, geome-
try (ii) and the different misorientations (x-component).
0 0.25 0.5 0.75 1.0
� �/2
–0.02
0
0.02
� �= /8
� �= /6
� �= /5
j
/j z
0
Fig. 6. Current tangential to the interface versus the
phase difference � for the junction between nonunitary bi-
polar f-wave superconducting bulks, T/Tc � 015. , geometry
(ii) and the different misorientations (z-component).
model of an ideal transparent interface between two
misoriented UPt3 crystals with nonunitary bipolar
f-wave superconducting bulks which are subject to a
phase difference �. Our analysis has shown that
misorientation between the gap vectors creates a cur-
rent parallel to the interface and different misorien-
tations between gap vectors influence the spontaneous
parallel and normal Josephson currents. These have
been shown for the currents in the point contact be-
tween two bulks of unitary axial and planar f-wave
superconductor in [8] separately. Also, it is shown
that the misorientation of the superconductors leads
to a spontaneous phase difference that corresponds to
the zero Josephson current and to the minimum of the
weak link energy in the presence of the finite sponta-
neous current. This phase difference depends on the
misorientation angle. The tangential spontaneous cur-
rent is not generally equal to zero in the absence of the
Josephson current. The difference between unitary
planar and nonunitary bipolar states can be used to
distinguish between them. This experiment can be
used to test the pairing symmetry and recognize the
different phases of UPt3.
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