Frustration phenomena in Josephson point contacts between single-band and three-band superconductors
Within the formalizm of Usadel equations the Josephson effect in dirty point contacts between single-band and three-band superconductors is investigated. The general expression for the Josephson current, which is valid for arbitrary temperatures, is obtained. We calculate current-phase relations f...
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| Zitieren: | Frustration phenomena in Josephson point contacts between single-band and three-band superconductors / Y.S. Yerin, A.N. Omelyanchouk // Физика низких температур. — 2014. — Т. 40, № 10. — С. 1206-1213. — Бібліогр.: 33 назв. — англ. |
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Yerin, Y.S Omelyanchouk, A.N. 2017-06-08T04:42:14Z 2017-06-08T04:42:14Z 2014 Frustration phenomena in Josephson point contacts between single-band and three-band superconductors / Y.S. Yerin, A.N. Omelyanchouk // Физика низких температур. — 2014. — Т. 40, № 10. — С. 1206-1213. — Бібліогр.: 33 назв. — англ. 0132-6414 PACS 74.50.+r, 74.78.Na, 74.20.Rp https://nasplib.isofts.kiev.ua/handle/123456789/119676 Within the formalizm of Usadel equations the Josephson effect in dirty point contacts between single-band and three-band superconductors is investigated. The general expression for the Josephson current, which is valid for arbitrary temperatures, is obtained. We calculate current-phase relations for very low temperature and in the vicinity of the critical temperature. For three-band superconductors with broken time-reversal symmetry (BTRS) point contacts undergo frustration phenomena with different current-phase relations, corresponding to φ-contacts. For three-band superconductors without BTRS we have close to sinusoidal current-phase relations and absence of the frustration, excepting the case of very low temperature, where under certain conditions two ground states of the point contact are realized. Our results can be used as the potential probe for the detection of the possible BTRS state in three-band superconducting systems This work was supported by DKNII (Project No. M/231-2013) and by BMBF (UKR-2012-028). en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур III Международный семинар по микроконтактной спектроскопии Frustration phenomena in Josephson point contacts between single-band and three-band superconductors Article published earlier |
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| title |
Frustration phenomena in Josephson point contacts between single-band and three-band superconductors |
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Frustration phenomena in Josephson point contacts between single-band and three-band superconductors Yerin, Y.S Omelyanchouk, A.N. III Международный семинар по микроконтактной спектроскопии |
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Frustration phenomena in Josephson point contacts between single-band and three-band superconductors |
| title_full |
Frustration phenomena in Josephson point contacts between single-band and three-band superconductors |
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Frustration phenomena in Josephson point contacts between single-band and three-band superconductors |
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Frustration phenomena in Josephson point contacts between single-band and three-band superconductors |
| title_sort |
frustration phenomena in josephson point contacts between single-band and three-band superconductors |
| author |
Yerin, Y.S Omelyanchouk, A.N. |
| author_facet |
Yerin, Y.S Omelyanchouk, A.N. |
| topic |
III Международный семинар по микроконтактной спектроскопии |
| topic_facet |
III Международный семинар по микроконтактной спектроскопии |
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2014 |
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English |
| container_title |
Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
Within the formalizm of Usadel equations the Josephson effect in dirty point contacts between single-band
and three-band superconductors is investigated. The general expression for the Josephson current, which is valid
for arbitrary temperatures, is obtained. We calculate current-phase relations for very low temperature and in the
vicinity of the critical temperature. For three-band superconductors with broken time-reversal symmetry (BTRS)
point contacts undergo frustration phenomena with different current-phase relations, corresponding to φ-contacts.
For three-band superconductors without BTRS we have close to sinusoidal current-phase relations and absence
of the frustration, excepting the case of very low temperature, where under certain conditions two ground
states of the point contact are realized. Our results can be used as the potential probe for the detection of the possible
BTRS state in three-band superconducting systems
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119676 |
| citation_txt |
Frustration phenomena in Josephson point contacts between single-band and three-band superconductors / Y.S. Yerin, A.N. Omelyanchouk // Физика низких температур. — 2014. — Т. 40, № 10. — С. 1206-1213. — Бібліогр.: 33 назв. — англ. |
| work_keys_str_mv |
AT yerinys frustrationphenomenainjosephsonpointcontactsbetweensinglebandandthreebandsuperconductors AT omelyanchoukan frustrationphenomenainjosephsonpointcontactsbetweensinglebandandthreebandsuperconductors |
| first_indexed |
2025-11-24T16:06:41Z |
| last_indexed |
2025-11-24T16:06:41Z |
| _version_ |
1850850658385657856 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 10, pp. 1206–1213
Frustration phenomena in Josephson point contacts
between single-band and three-band superconductors
Y.S. Yerin and A.N. Omelyanchouk
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv 61103, Ukraine
E-mail: yerin@ilt.kharkov.ua
Received July 1, 2014, published online August 21, 2014
Within the formalizm of Usadel equations the Josephson effect in dirty point contacts between single-band
and three-band superconductors is investigated. The general expression for the Josephson current, which is valid
for arbitrary temperatures, is obtained. We calculate current-phase relations for very low temperature and in the
vicinity of the critical temperature. For three-band superconductors with broken time-reversal symmetry (BTRS)
point contacts undergo frustration phenomena with different current-phase relations, corresponding to φ-con-
tacts. For three-band superconductors without BTRS we have close to sinusoidal current-phase relations and ab-
sence of the frustration, excepting the case of very low temperature, where under certain conditions two ground
states of the point contact are realized. Our results can be used as the potential probe for the detection of the pos-
sible BTRS state in three-band superconducting systems.
PACS: 74.50.+r Tunneling phenomena; Josephson effects;
74.78.Na Mesoscopic and nanoscale systems;
74.20.Rp Pairing symmetries.
Keywords: Josephson effect, three-band superconductivity, φ-contact, frustration, broken time-reversal symmetry.
1. Introduction
The symmetry of the order parameter of recently discov-
ered iron-based superconductors still remains a controversial
question and an unresolved challenge. The initial hypothesis
that the order parameter has two components with opposite
signs (s±-wave symmetry) is casted doubt on by numerous
data obtained during the angle-resolved photoemission spec-
troscopy [1] and in experiments on the temperature depend-
ence of the specific heat capacity [2–4]. These results indi-
rectly indicate the presence in these compounds super-
conducting chiral state like spin-triplet p-wave in strontium
ruthenate [5] or recently proposed d + id wave superconduc-
tivity in graphene [6].
At the same time it is well known that chiral supercon-
ductivity can lead to an interesting phenomenon in such
compounds, namely the broken time-reversal symmetry
(BTRS): phases of the multicomponent order parameter
undergo frustration, leading to the emergence of several
“equal in rights” ground states of the superconductor.
BTRS in superconducting oxypnictides and chal-
cogenides is discussed earlier in the frame of the s + id [7]
and s± + is++ [8] symmetry of the order parameter. These
models assumed the presence of two components of the
order parameter (a two-band superconductor). However,
the latest experimental data give clear evidences about the
presence of at least three energy gaps in the spectrum of
quasiparticle excitations in iron-based superconductors
[9,10]. The presence of three interacting parameters also
leads to the BTRS state [11–20].
In this regard, a reasonable question arises about the
possible experimental techniques for the creating and sub-
sequent detection of this phenomenon in iron-based super-
conductors. Currently in this sense the most prominent
candidate among known superconducting iron oxypnic-
tides and chalcogenides is Ba1–xKxFe2As2, in which BTRS
can be achieved by the controlling the level of doping [20].
In turn, methods that have been proposed to reveal this phe-
nomenon use the detection of Legget modes [21], observa-
tion of the unusual behavior of the magnetization of the
sample in the process of the fast quench cooling [22] and
the detection of kinks on the current versus applied mag-
netic flux dependencies in a doubly-connected mesoscopic
sample [23].
We believe that another useful way to detect the state
with BTRS in iron-based superconductors is the investiga-
tion of the Josephson effect, which is traditionally consid-
ered as the most powerful tool for detecting the manifesta-
tion of the phase of the order parameter in single and
© Y.S. Yerin and A.N. Omelyanchouk, 2014
Frustration phenomena in Josephson point contacts between single-band and three-band superconductors
multiband superconductors [24,25]. Attempt to understand
how the frustration of phases of the order parameters ef-
fects on current-phase relations of a contact between con-
ventional (s-wave) single-band and three-band supercon-
ductor with BTRS one has been undertaken already in
[26]. However this investigation was done for the ballistic
regime which is difficultly to achieve in real experimental
conditions.
In the present paper within the formalizm of the Usadel
equations [27], generalized for the case of three energy
gaps [28], we investigated a dirty point contact between
the s-wave single-band superconductor and the three-band
one. We found qualitative differences in the structure of
the current-phase relations of the point contact for cases of
the three-band superconductor with the presence of BTRS
and without of this state.
2. Ground states of a homogeneous equilibrium
three-band superconductor
At the beginning we investigate a homogeneous equi-
librium three-band superconductor with strong impurity
intraband scattering rates (dirty limit) and without
interband scattering in order to find all possible frustrated
and nonfrustrated ground states. In this limit the three-band
superconductor is described by the Usadel equations for
normal and anomalous Green’s functions ig and :if
2 21 ( ) , 1, 2, 3
2i i i i i i i if D g f f g g iω − ∇ − ∇ = ∆ = . (1)
Equations (1) must be supplemented with self-consistency
equation for order parameters :i∆
0
0
2i ij j
j
T f
〈ω 〉
ω>
∆ = π λ∑ ∑ , (2)
and the expression for the current density
* *
0
2 ( )i i i i i i
i
j ie T N D f f f f
ω>
= − π ∇ − ∇∑∑ . (3)
Normal and anomalous Green’s functions ig and ,if
which are connected by the normalization condition
2 2| | 1,i ig f+ = are functions of coordinates r and the
Matsubara frequency (2 1) .n Tω = + π iD are the intraband
diffusivities due to nonmagnetic intraband impurity scat-
tering, iN are the densities of states on the Fermi surface
of the ith band, ijλ are BCS interaction constants and 0〈ω 〉
is the cut-off frequency.
For the equilibrium homogeneous state Usadel equa-
tions (1) have solutions
2 2
| | exp ( )
| |
i i
i
i
i
f
∆ ϕ
=
ω + ∆
, (4)
where iϕ are phases of each order parameter.
Ground states of a three-band superconductor can be
found from the minimization of the free energy density in
respect to phase differences of the order parameters
1 2φ = ϕ −ϕ and 1 3 :θ = ϕ −ϕ
11
2 i j i ij i
ij i
F N F∗ −= ∆ ∆ λ +∑ ∑ , (5)
where 1
ij
−λ is the inverse matrix of interaction constants
ijλ and
0
*
0
2 [ (1 ) Re( )i i i i iF T N g f
〈ω 〉
ω>
= π ω − − ∆ +∑
1 ( )]
4 i i i i iD f f g g∗+ ∇ ∇ +∇ ∇ (6)
represents intraband energies of the three-band supercon-
ductor.
For 1, 2i = we obtain the free energy density of a two-
band superconductor, which was used for the prediction of
phase textures in multiband and multiband-like supercon-
ducting systems [29]. The first term in (5) contains three
interband (Josephson-like) interaction energies 12 cos ,γ φ
13 cos ,γ θ and 23 cos ( ),γ θ − φ where 1
ij ji jN−γ = −λ (usual-
ly 1 1
ij i ji jN N− −λ = λ , i j≠ ) are interband interaction coeffi-
cients, which are used in Ginzburg–Landau approach; for
0ijγ > attractive interband interactions are took place,
while for 0ijγ < interactions are repulsive.
The first variation of (5) on φ and θ gives
1 1
12 1 21 2 1 2( ) | || | sinN N− −− λ + λ ∆ ∆ φ+
1 1
23 2 32 3 2 3( ) | || | sin ( ) 0N N− −+ λ + λ ∆ ∆ θ−φ = , (7)
1 1
13 1 31 3 1 3( )| || | sinN N− −− λ + λ ∆ ∆ θ−
1 1
23 2 32 3 2 3( )| || | sin ( ) 0N N− −− λ + λ ∆ ∆ θ−φ = . (8)
Solutions of (7) and (8) for φ and ,θ which determine
the points of extremum, depend from their arrangement in
quadrants.
Introducing
22 2 2 2 2 2 2 2 2
3 2 3 1 3 1 1 2 2
2
1 2 3 1 2
| | | | | |
1 ,
2 | || |
G G G G G G
G G G
∆ − ∆ − ∆
Ω = − ∆ ∆
where 1 1
1 12 1 21 2 ,G N N− −= λ + λ 1 1
2 23 2 32 3G N N− −= λ + λ and
1 1
3 13 1 31 3G N N− −= λ + λ
for ,
2 2
π π φ∈ −
and ,
2 2
π π θ∈ −
we have
1 2
3 3
arcsin ,
| |
arcsin ,
| |
G
G
φ = ± Ω
∆θ = Ω ∆
0,
0,
φ =
θ =
(9)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 10 1207
Y.S. Yerin and A.N. Omelyanchouk
for 3,
2 2
π π φ∈
and ,
2 2
π π θ∈ −
1 2
3 3
arcsin ,
| |
arcsin ,
| |
G
G
φ = π± Ω
∆θ = ± Ω ∆
,
0,
φ = π
θ =
(10)
for ,
2 2
π π φ∈ −
and 3,
2 2
π π θ∈
1 2
3 3
arcsin ,
| |
arcsin ,
| |
G
G
φ = ± Ω
∆θ = π± Ω ∆
0,
,
φ =
θ = π
(11)
and for 3,
2 2
π π φ∈
and 3,
2 2
π π θ∈
1 2
3 3
arcsin ,
| |
arcsin ,
| |
G
G
φ = π± Ω
∆θ = π Ω ∆
,
.
φ = π
θ = π
(12)
For given ijλ and computed for these values | |i∆ by
Eqs. (2) and (4) we have eight possible solutions for φ and
θ (9)–(12). Selection of the proper solution, which corre-
sponds to the ground state, is provided by the condition for
the minimum of ( , ),F θ φ following from the second varia-
tion of the free energy density (5). Final form of the ex-
pression for the second variation also depends on the ar-
rangement in quadrants of φ and .θ
3. Josephson current between single-band and three-
band superconductors
The point contact can be considered as a weak super-
conducting link in the form of thin filament of the length
L and diameter ,d connecting two superconducting bulk
banks (Fig. 1).
On conditions that d L and min ( )id Tξ ( ( )i Tξ
is the coherence lengths in the ith band) we can solve a
one-dimensional problem inside the filament (0 )x L≤ ≤
and neglect all terms in Usadel equations (1) except the
gradients ones. Using the normalization condition we have
equations for if
2 2
2 2
2 21 | | 1 | | 0,i i i i
d df f f f
dx dx
− − − = 1, 2, 3i = . (13)
The boundary conditions for Eqs. (13) at 0, x L= are de-
termined by the values of if in banks:
0
2 2
0
| |
(0)
| |
if
∆
=
∆ + ω
, (14)
1
1 2 2
1
| | exp ( )
( )
| |
if L ∆ χ
=
∆ + ω
, 2
2 2 2
2
| | exp ( )
( )
| |
i if L ∆ χ + φ
=
∆ + ω
,
3
3 2 2
3
| | exp ( )
( )
| |
i i
f L
∆ χ + θ
=
∆ + ω
,
(15)
where χ is the phase difference between the first order
parameter of the three-band superconductor and the order
parameter of the single-band one and where φ and θ de-
termine phase differences (ground state) in the bulk three-
band superconductor.
Equations (13) admit analytical solution with boundary
conditions (14), (15). Taking into account expression for the
current density (3) we get for the Josephson current between
the single-band and the three-band superconductor
i
i
I I=∑ , (16)
where
0
0
2 1 arctan arctan .i i i i i i i
i i
Ni i i i
d a b d a bTI b
eR p p pω>
∆ − ∆ +π
= +
∑
(17)
Here NiR are partial contributions to the point-contact
resistance. Also notations
2 2 2( 1)i i ip b a= + + ω , 1 0
1
1 0
| | | |
cot
| | | | 2
a
∆ − ∆ χ
=
∆ + ∆
,
2 0
2
2 0
| | | |
cot
| | | | 2
a
∆ − ∆ χ + φ
=
∆ + ∆
, 3 0
3
3 0
| | | |
cot
| | | | 2
a
∆ − ∆ χ + θ
=
∆ + ∆
,
0 1
1
0 1
2 | || |
cos
| | | | 2
b
∆ ∆ χ
=
∆ + ∆
, 0 2
2
0 2
2 | || |
cos
| | | | 2
b
∆ ∆ χ + φ
=
∆ + ∆
,
0 3
3
0 3
2 | || |
cos
| | | | 2
b
∆ ∆ χ + θ
=
∆ + ∆
, 2
1 1( 1)sin
2
d a χ
= + ,
2
2 2( 1)sin
2
d a χ + φ
= + and 2
3 3( 1)sin
2
d a χ + θ
= + were used.
For 1i = we get expression for the Josephson current be-
tween single-band superconductors [30,31], which for co-
inciding values of energy gaps turns into Kulik–
Omelyanchouk theory for dirty point contacts [32].
3.1. Josephson current for T = 0
For 0T = in the expression (17) we can turn from the
summation over Matsubara frequencies to the integration
and get for the total current
Fig. 1. The model of the point contact between bulk single-band
and three-band superconductors as two banks connected by the
thing filament of a length L and a diameter d.
1208 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 10
Frustration phenomena in Josephson point contacts between single-band and three-band superconductors
0 30 1 0 2
2 2 32 2 22 0 1 2 0 2 3 0 31 2 3
| || | | || |cos ( /2) cos [( )/2] cos [( )/2]ln ln ln .
| | | | | | | |1 1 1N N N
I Q Q Q
eR eR eRa a a
∆ ∆∆ ∆ ∆ ∆π χ π χ + φ π χ + θ
= + +
∆ + ∆ ∆ + ∆ ∆ + ∆+ + +
(18)
Here
2 2 2 2
0 0
2
| | (| | ) | | (| | )i i i i i i i i i i i i i i i i
i
i
d a b d a b b d a b d a b b
Q
b
∆ − + ∆ − + ∆ + + ∆ + + = . (19)
In the following we consider for the simplicity the case of coinciding energy gaps 0 1 2 3| | | | | | | | | |∆ = ∆ = ∆ = ∆ = ∆ and
obtain for the total current
1 2 3
| | | | | |cos arctanh sin cos arctanh sin cos arctanh sin .
2 2 2 2 2 2N N N
I
eR eR eR
π ∆ χ χ π ∆ χ + φ χ + φ π ∆ χ + θ χ + θ
= + + (20)
Integrating (20) over χ we obtain the expression for the Josephson energy of the point contact
2 20 0
1 2
20
3
| | | |
2sin arctanh sin ln cos 2sin arctanh sin ln cos
2 2 2 2 2 2 2 2
| |
2sin arctanh sin ln cos .
2 2 2 2
N N
N
E
eR eR
eR
∆ Φ ∆ Φχ χ χ χ + φ χ + φ χ + φ = + + + +
∆ Φ χ + θ χ + θ χ + θ + +
(21)
_______________________________________________
We can select arbitrary values of φ and θ because it’s
possible to match appropriate values of ijλ to satisfy the
self-consistency equation (2) and expressions (9)–(12). In
other words, we have only five equations, three of which
follow from the self-consistency equation (2) and two from
expressions for ,φ ,θ for the determination of nine varia-
bles .ijλ Based on these arguments we assume that for the
frustrated three-band superconductor one of the ground
state can be, for instance, 0.6 ,φ = π 1.2 .θ = π Since these
phase differences were chosen in the second and in the
third quadrants, respectively, according to the solution (12)
another ground state should correspond to such values:
1.4 ,φ = π 0.8 .θ = π
So when one of contacting banks is the three-band su-
perconductor with BTRS state we observe complicated
current-phase relations with the behavior of Josephson
energies, corresponding to ϕ-contact (Fig. 2), following to
the terminology after [33]. Here and hereinafter we will
call ϕ-contact as a type of the Josephson junction with an
arbitrary phase shift ϕ in the ground state.
So during experimental measurements for the same
BTRS three-band superconductor we can observe different
current-phase relations. It depends from the “prehistory” of
the three-band superconductor, i.e., how was the ground
state for this superconducting system achieved.
Also we consider the simplest case, which is often cited
for the illustration of BTRS in three-band superconductors,
when such systems have the odd number of repulsive
interband interactions ijγ with equal modules. In this case
two ground states are possible: ( , ) ( /3, /3)φ θ = −π π and
( , ) ( /3, /3),φ θ = π − π if we have only one repulsive in-
terband interaction and two attractive ones and ( , )φ θ =
( 2 /3,2 /3)= − π π and ( , ) (2 /3, 2 /3),φ θ = π − π if all interband
interactions are repulsive.
Firstly we found that for these two ground-states cur-
rent-phase relations and Josephson energies coincide (see
Fig. 3) in comparison with above considered case (Fig. 2).
This fact can be easily understood from expressions (20)
and (21) bearing in mind that the cosine is even function
and the inverse hyperbolic tangent and the sine are odd
ones. Secondly, despite the presence of the BTRS state in
the three-band superconductor the most remarkable feature
for ( , ) ( /3, /3)φ θ = −π π and ( , ) ( /3, /3),φ θ = π − π is that
there is no frustration of the point contact (Fig. 3(a)) with
inflection points in the middle of the current-phase relation
curve. Thirdly, for ( , ) ( 2 /3,2 /3)φ θ = − π π and ( , )φ θ =
(2 /3, 2 /3)= π − π we have current-phase relation with triply
degenerates states (Fig. 3(b)), i.e., frustration with three
ground states of the point contact is occurred.
The current-phase relation (20) and the Josephson ener-
gy (21) for the point contact between single-band super-
conductor and non-BTRS three-band one are shown on the
Fig. 4. Here the peculiarity of such point contacts are the
occurrence of a ϕ-contact with frustration of ground states
(Figs. 4(b) and 4(c)) as for the BTRS case if the three-band
superconductor has the ground state for 0,φ = θ = π or
.φ = θ = π
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 10 1209
Y.S. Yerin and A.N. Omelyanchouk
Thus at 0T = in both cases of BTRS and non-BTRS
three-band superconductors we can have frustration phe-
nomenon in point contacts. In order to distinguish three-
band superconductors with and without BTRS state in the
next section we consider point contacts in the vicinity of
the critical temperature cT (it can be the critical tempera-
ture of the single- or the three-band superconductor in de-
pendence on what is the value lower).
3.2. Josephson current in the vicinity of the critical
temperature
By the linearization of the expression (17) we get for
total current
2 2
1 2
| | | |sin sin ( )
4 4c N c N
I
eT R eT R
π ∆ π ∆
= χ + χ + φ +
2
3
| | sin ( )
4 c NeT R
π ∆
+ χ + θ , (22)
and after integrating over χ for the Josephson energy
2 2
0 0
1 2
| | | |
cos cos ( )
8 8c N c N
E
eT R eT R
∆ Φ ∆ Φ
= − χ − χ + φ −
2
0
3
| |
cos ( )
8 c NeT R
∆ Φ
− χ + θ . (23)
For the three-band superconductor with BTRS the intri-
cate behavior of ( )I ϕ and ( )E ϕ dependencies (Fig. 2)
turns into simple sinusoidal forms but nevertheless with
the conservation of a φ-contact feature (Fig. 5).
For the point contact when one of the bank is the three-
band superconductor with one repulsive interband interac-
tion and with equal modules of ijγ current-phase relations
continue to coincide, but now lost inflection points, which
are took place for the very low temperature (Fig. 3(a)) and
transform to clear sinusoidal dependence (Fig. 6). At the
same time the ground state is not varied and the point con-
tact remains conventional.
Fig. 2. Current-phase relations (solid lines) and Josephson ener-
gies (dashed lines) of point contacts between single-band and
three-band superconductors with BTRS in the case of coinciding
energy gaps for φ = 0.6π, θ = 1.2π (a) and φ = 1.4π, θ = 0.8π (b).
Ratios RN1/RN2 = RN1/RN3 = 1.
Fig. 3. The same as in Fig. 2 for φ = π/3, θ = –π/3; φ = –π/3, θ =
π/3 (a) and φ = 2π/3, θ = –2π/3; φ = –2π/3, θ = 2π/3 (b). Ratios
RN1/RN2 = RN1/RN3 = 1.
1210 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 10
Frustration phenomena in Josephson point contacts between single-band and three-band superconductors
If one of the contacting banks is the three-band super-
conductor with all repulsive interband interaction and
again with equal modules of ijγ the point contact is char-
acterized by the zero Josephson current for both ground
states of such three-band superconductor. It seems some-
Fig. 4. Current-phase relations (solid lines) and Josephson ener-
gies (dashed lines) of point contacts between single-band and
three-band superconductors without BTRS in the case of coincid-
ing energy gaps for φ = θ = 0 (a) and φ = 0, θ = π (b, for φ = π
and θ = 0 it will be the same dependence) and φ = θ = π (c). Rati-
os RN1/RN2 = RN1/RN3 = 1.
Fig. 5. Current-phase relations (solid lines) and Josephson energies
(dashed lines) of point contacts between single-band and three-band
superconductors with BTRS in the case of coinciding energy gaps
and in the vicinity of the critical temperature for φ = 0.6π, θ =
= 1.2π (a) and φ = 1.4π, θ = 0.8π (b). Ratios RN1/RN2 = RN1/RN3 = 1.
Fig. 6. The same as in Fig. 5 for φ = π/3, θ = –π/3 and φ = –π/3, θ =
= π/3. For φ = 2π/3, θ = –2π/3 and φ = –2π/3, θ = 2π/3 absence of
the Josephson current is took place (see explanation in the text).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 10 1211
Y.S. Yerin and A.N. Omelyanchouk
thing unexpectedly but if we substitute φ = –2π/3, θ = 2π/3
(for φ = 2π/3, θ = –2π/3 it would be the same) into the ex-
pression for the current (22) after the simplification we get
zero current.
Current-phase relations (22) and Josephson energies
(23) in the case of the three-band superconductor without
BTRS are shown on the Fig. 7. Figure 7 demonstrates that
for the nonfrustrated three-band superconductor with
0,φ = θ = π and φ = θ = π the proximity of the critical
temperature removes a degeneracy of the ground state of
the point contact transforming φ-contact to conventional
one (if 0,φ = )θ = π or to π-contact (if ).φ = θ = π
Comparing current-phase relations of point contacts we
can definitely claim the difference between three-band su-
perconductors with the BTRS state and without one. From
the experimental point of view the identification procedure
can be done in the following way: if a point contact demon-
strates two different current-phase relations with the proper-
ties of a φ-contact at very low temperature and in the vicini-
ty of cT during several repeating measurements, unam-
biguously this three-band superconductor has the state with
BTRS. Otherwise even if we observe a φ-contact at the tem-
perature close to the zero but conventional or π-contact near
cT a three-band superconductor has no BTRS state.
For the special case of a three-band superconductor
with odd number of repulsive interband interactions and
equal modules of ijγ the detection procedure undergoes
changes. During measurements we will observe conven-
tional current-phase relations with inflection points at very
low temperature, which disappear near cT if a three-band
superconductor has one repulsive interband interaction and
two attractive, or we will observe current-phase relations
with triply degenerate ground states at 0T = and zero
Josephson current in the vicinity of the critical temperature
if all interband interactions are repulsive.
Conclusions
Based on the microscopic approach, we have obtained
general analytical expressions for phase differences of or-
der parameters, corresponding to the ground state of a ho-
mogeneous equilibrium three-band superconductor. We
have developed microscopic theory of the Josephson effect
in dirty point contacts between single-band and three-band
superconductors. For a BTRS three-band superconductor
we have revealed frustration phenomenon of the point con-
tact with different current-phase relations. By analyzing
the Josephson energy we have found that the contact has
shown the property of a φ-contact for whole temperature
interval from zero to the critical temperature. For a three-
band superconductor, which is characterized by the ab-
sence of the BTRS state, with the increasing of the temper-
ature we have observed the evolution of the contact behav-
ior from the frustrated φ-contact to conventional or π-
contact in dependence on the values of phase differences in
a three-band superconductor. We stress that our theoretical
results can be useful in experiments on the detection of
BTRS states in multiband superconductors.
Fig. 7. Current-phase relations (solid lines) and Josephson energies
(dashed lines) of point contacts between single-band and three-
band superconductors without BTRS in the case of coinciding en-
ergy gaps and in the vicinity of the critical temperature for φ = θ =
= 0 (a) and φ = 0, θ = π (b, for φ = π and θ = 0 it will be the same
dependence) and φ = θ = π (c). Ratios RN1/RN2 = RN1/RN3 = 1.
1212 Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 10
Frustration phenomena in Josephson point contacts between single-band and three-band superconductors
Acknowledgments
This work was supported by DKNII (Project No.
M/231-2013) and by BMBF (UKR-2012-028).
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2014, v. 40, No. 10 1213
1. Introduction
2. Ground states of a homogeneous equilibrium three-band superconductor
3. Josephson current between single-band and three-band superconductors
3.1. Josephson current for T = 0
3.2. Josephson current in the vicinity of the critical temperature
Conclusions
Acknowledgments
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