Josephson effect in point contacts between "f -wave" superconductors

A stationary Josephson effect in point contacts between triplet superconductors is analyzed theoretically for most probable models of the order parameter in UPt₃ and Sr₂RuO₄. The consequence of misorientation of crystals in the superconducting banks on this effect is considered. We show that differe...

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Дата:	2002
Автори:	Mahmoodi, R., Kolesnichenko, Yu.A., Shevchenko, S.N.
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Мова:	English
Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2002
Назва видання:	Физика низких температур
Теми:	Свеpхпpоводимость, в том числе высокотемпеpатуpная
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/130162
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Цитувати:	Josephson effect in point contacts between "f -wave" superconductors / R.Mahmoodi, Yu.A.Kolesnichenko, S.N. Shevchenko // Физика низких температур. — 2002. — Т. 28, № 3. — С. 262-269. — Бібліогр.: 28 назв. — англ.

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spelling	irk-123456789-1301622018-02-09T03:03:30Z Josephson effect in point contacts between "f -wave" superconductors Mahmoodi, R. Kolesnichenko, Yu.A. Shevchenko, S.N. Свеpхпpоводимость, в том числе высокотемпеpатуpная A stationary Josephson effect in point contacts between triplet superconductors is analyzed theoretically for most probable models of the order parameter in UPt₃ and Sr₂RuO₄. The consequence of misorientation of crystals in the superconducting banks on this effect is considered. We show that different models for the order parameter lead to quit different current-phase relations. For certain angles of misorientation a boundary between superconductors can generate a spontaneous current parallel to the surface. In a number of cases the state with a zero Josephson current and minimum of the free energy corresponds to a spontaneous phase difference. This phase difference depends on the misorientation angle and may possess any value. We conclude that experimental investigations of the current-phase relations of small junctions can be used for determination of the order parameter symmetry in the superconductors mentioned above. 2002 Article Josephson effect in point contacts between "f -wave" superconductors / R.Mahmoodi, Yu.A.Kolesnichenko, S.N. Shevchenko // Физика низких температур. — 2002. — Т. 28, № 3. — С. 262-269. — Бібліогр.: 28 назв. — англ. 0132-6414 PACS: 74, 70.Tx, 74.80.Fp http://dspace.nbuv.gov.ua/handle/123456789/130162 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная
spellingShingle	Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Mahmoodi, R. Kolesnichenko, Yu.A. Shevchenko, S.N. Josephson effect in point contacts between "f -wave" superconductors Физика низких температур
description	A stationary Josephson effect in point contacts between triplet superconductors is analyzed theoretically for most probable models of the order parameter in UPt₃ and Sr₂RuO₄. The consequence of misorientation of crystals in the superconducting banks on this effect is considered. We show that different models for the order parameter lead to quit different current-phase relations. For certain angles of misorientation a boundary between superconductors can generate a spontaneous current parallel to the surface. In a number of cases the state with a zero Josephson current and minimum of the free energy corresponds to a spontaneous phase difference. This phase difference depends on the misorientation angle and may possess any value. We conclude that experimental investigations of the current-phase relations of small junctions can be used for determination of the order parameter symmetry in the superconductors mentioned above.
format	Article
author	Mahmoodi, R. Kolesnichenko, Yu.A. Shevchenko, S.N.
author_facet	Mahmoodi, R. Kolesnichenko, Yu.A. Shevchenko, S.N.
author_sort	Mahmoodi, R.
title	Josephson effect in point contacts between "f -wave" superconductors
title_short	Josephson effect in point contacts between "f -wave" superconductors
title_full	Josephson effect in point contacts between "f -wave" superconductors
title_fullStr	Josephson effect in point contacts between "f -wave" superconductors
title_full_unstemmed	Josephson effect in point contacts between "f -wave" superconductors
title_sort	josephson effect in point contacts between "f -wave" superconductors
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2002
topic_facet	Свеpхпpоводимость, в том числе высокотемпеpатуpная
url	http://dspace.nbuv.gov.ua/handle/123456789/130162
citation_txt	Josephson effect in point contacts between "f -wave" superconductors / R.Mahmoodi, Yu.A.Kolesnichenko, S.N. Shevchenko // Физика низких температур. — 2002. — Т. 28, № 3. — С. 262-269. — Бібліогр.: 28 назв. — англ.
series	Физика низких температур
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first_indexed	2025-07-09T13:00:00Z
last_indexed	2025-07-09T13:00:00Z
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fulltext	Fizika Nizkikh Temperatur, 2002, v. 28, No. 3, p. 262–269Mahmoodi R., Shevchenko S. N., and Kolesnichenko Yu. A.Josephson effect in point contacts between «f-wave» superconductorsMahmoodi R., Shevchenko S. N., and Kolesnichenko Yu. A.Josephson effect in point contacts between «f-wave» superconductors Josephson effect in point contacts between «f-wave» superconductors R. Mahmoodi1, S. N. Shevchenko2, and Yu. A. Kolesnichenko1,2 1 Institute for Advanced Studies in Basic Sciences, 45195-159, Gava Zang, Zanjan, Iran 2 B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Lenin Ave., 61103 Kharkov, Ukraine E-mail: kolesnichenko@ilt.kharkov.ua Received October 25, 2001 A stationary Josephson effect in point contacts between triplet superconductors is analyzed theoretically for most probable models of the order parameter in UPt3 and Sr2RuO4 . The consequence of misorientation of crystals in the superconducting banks on this effect is considered. We show that different models for the order parameter lead to quite different current–phase relations. For certain angles of misorientation a boundary between superconductors can generate a spontaneous current parallel to the surface. In a number of cases the state with a zero Josephson current and minimum of the free energy corresponds to a spontaneous phase difference. This phase difference depends on the misorientation angle and may possess any value. We conclude that experimental investigations of the current–phase relations of small junctions can be used for determination of the order parameter symmetry in the superconductors mentioned above. PACS: 74.50.+r, 74 70.Tx, 74.80.Fp 1. Introduction Triplet superconductivity, which is an analog of superfluidity in 3He, was first discovered in the heavy-fermion compound UPt3 more than ten years ago [1,2]. Recently, a novel triplet superconductor Sr2RuO4 was found [3,4]. In these compounds, the triplet pairing can be reliably determined, for exam- ple, by Knight shift experiments [5,6], but the identification of a symmetry of the order parameter is much more difficult task. A large number of experimental and theoretical investigations done on UPt3 and Sr2RuO4 are concerned with different thermodynamic and transport properties, but the precise order parameter symmetry remains still to be worked out (see, for example, [7,10–12], and original references therein). Calculations of the order parameter ∆̂(k̂) in UPt3 and Sr2RuO4 as a function of the momentum direction k̂ on the Fermi surface is a very complex problem. Some general information about ∆̂(k̂) can be obtained from the symmetry of the normal state: Gspin−orbit × τ × U(1), where Gspin−orbit represents the point group with inversion, τ is the time-inversion operator, and U(1) is a gauge transformation group. A superconducting state breaks one or more symme- tries. In particular, a transition to the supercon- ducting state implies the appearance of a phase coherence corresponding to breaking of the gauge symmetry. According to the Landau theory [13] of second-order phase transitions, the order parameter transforms only according to irreducible repre- sentations of the symmetry group of the normal state. Conventional superconducting states have the total point symmetry of the crystal and belong to the even unitary representation A1g . In unconven- tional superconductors this symmetry is broken. The parity of a superconductor with inversion sym- metry can be specified using the Pauli principle. Because for triplet pairing the spin part of the ∆̂ is a symmetric second-rank spinor, the orbital part has to belong to an odd representation. In the general case the triplet pairing is described by an order parameter of the form ∆∆∆∆̂(k̂) = id(k̂)σσσσ̂σ̂2, where the vector σσσσ̂ = (σ̂ 1 ,σ̂ 2 ,σ̂ 3 ), and σ̂ i are Pauli matrices in © R. Mahmoodi, S. N. Shevchenko, and Yu. A. Kolesnichenko, 2002 the spin space. A vector d(k̂) = − d(−k̂) in spin space is frequently referred to as an order parameter or a gap vector of the triplet superconductor. This vec- tor defines the axis along which the Cooper pairs have zero spin projection. If d is complex, the spin components of the order parameter spontaneously break time-reversal symmetry. Symmetry considerations reserve for the order parameter considerable freedom in the selection of irreducible representation and its basis functions. Therefore in many papers (see, for example, [7,10– 12,14–16]) authors consider different models (so- called scenarios) of superconductivity in UPt3 and Sr2RuO4 , which are based on possible repre- sentations of crystallographic point groups. The subsequent comparison of theoretical results with experimental data makes it possible to draw conclu- sions about the symmetry of the order parameter. In real crystalline superconductors there is no classification of Cooper pairing by angular momen- tum (s-wave, p-wave, d-wave, f-wave pairing, etc.). However, these terms are often used for unconventional superconductors in the meaning that the point symmetry of the order parameter is the same as one for the corresponding repre- sentation of the SO3 symmetry group of an isotropic conductor. In this terminology conventional super- conductors can be referred to as s-wave. For exam- ple, the «p-wave» pairing corresponds to the odd two-dimensional representation E1u of the point group D6h or the Eu representation of the point group D4h . The order parameter for these repre- sentations has the same symmetry as for the super- conducting state with angular momentum l = 1 of Cooper pairs in an isotropic conductor. If the sym- metry of ∆̂ can not be formally related to any irreducible representation of the SO3 group, these states are usually referred to as hybrid states. Apparently, in crystalline triplet superconduc- tors the order parameter has more complex depend- ence on k̂ in comparison with well known p-wave order parameter for superfluid phases of 3He. The heavy-fermion superconductor UPt3 belongs to the hexagonal crystallographic point group (D6h), and it is most likely that the pairing state belongs to the E2u («f-wave» state) representation. The layered perovskite material Sr2RuO4 belongs to the tetrago- nal crystallographic point group (D4h). Initially the simplest «p-wave» model based on the Eu repre- sentation was proposed for the superconducting state in this compound [8,9]. However, this model was inconsistent with available experimental data, and later [10,11] other «f-wave» models of the pairing state were proposed. Theoretical studies of the specific heat, thermal conductivity, and ultrasound absorption for dif- ferent models of triplet superconductivity show considerable quantitative differences between cal- culated dependences [7,10,11,16]. The Josephson effect is much more sensitive to dependence of ∆̂ on the momentum direction on the Fermi surface. One of the possibilities for forming a Josephson junction is to create a point contact between two massive superconductors. A microscopic theory of the sta- tionary Josephson effect in ballistic point contacts between conventional superconductors was devel- oped in Ref. 17. Later this theory was generalized for a pinhole model in 3He [18,19] and for point contacts between «d-wave» high-Tc superconduc- tors [20,21]. It was shown that current–phase rela- tions for the Josephson current in such systems are quite different from those of conventional supercon- ductors, and states with a spontaneous phase differ- ence become possible. Theoretical and experimental investigations of this effect in novel triplet super- conductors seem to be interesting and enable one to distinguish among different candidates for the su- perconducting state. In Ref. 22 the authors study the interface An- dreev bound states and their influence on the Josephson current between clean «f-wave» super- conductors both self-consistently (numerically) and non-self-consistently (analytically). The tempera- ture dependence of the critical current is presented. However, in that paper there is no detailed analysis of the current–phase relations for different orienta- tions of the crystals in the superconducting banks. In this paper we theoretically investigate the stationary Josephson effect in a small ballistic junc- tion between two bulk triplet superconductors with different orientations of the crystallographic axes with respect to the junction normal. In Sec. 2 we describe our model of the junction and present the full set of equations. In Sec. 3 the current density in the junction plane is calculated analytically for a non-self-consistent model of the order parameter. In Sec. 4 the current–phase relations for most likely models of «f-wave» superconductivity in UPt3 and Sr2RuO4 are analyzed for different mutual orienta- tions of the banks. We end in Sec. 5 with some conclusions. 2. Model of the contact and formulation of the problem We consider a model of a ballistic point contact as an orifice of diameter d in a partition impenetra- ble to electrons, between two superconducting half spaces (Fig. 1). We assume that the contact diame- Josephson effect in point contacts between «f-wave» superconductors Fizika Nizkikh Temperatur, 2002, v. 28, No. 3 263 ter d is much larger than the Fermi wavelength and use the quasiclassical approach. In order to calcu- late the stationary Josephson current in point con- tact we use «transport-like» equations for ξ-inte- grated Green functions g v (k̂,r,εm) [23]   iεm τv3 − ∆ v , g v   + ivFk̂∇ g v = 0 , (1) and the normalization condition g v g v = − 1 . (2) Here εm = πT(2m + 1) are discrete Matsubara ener- gies, vF is the Fermi velocity, k̂ is a unit vector along the electron velocity, and τv3 = τ̂3 ⊗ Î; τ̂i (i = 1, 2, 3) are Pauli matrices in a particle–hole space. The Matsubara propagator g v can be written in the form [24]: g v =      g1 + g1σ̂ iσ̂2(g3 + g3σ̂) (g2 + g2σ̂2)iσ̂2 g4 − σ̂2g4σσσσ̂σ̂2      ; (3) as can be done for an arbitrary Nambu matrix. Matrix structure of the off-diagonal self-energy ∆ v in Nambu space is ∆ v =      0 iσ̂2d ∗ σσσσ̂ idσσσσ̂σ̂2 0      . (4) Below we consider so-called unitary states, for which d × d∗ = 0. The gap vector d has to be determined from the self-consistency equation: d(k̂,r) = πTN(0) ∑ m 〈V(k̂,k̂′)g2(k̂′,r,εm)〉 , (5) where V(k̂,k̂′) is a pairing interaction potential; 〈...〉 stands for averaging over directions of an elec- tron momentum on the Fermi surface; N(0) is the electron density of states. Solutions of Eqs. (1), (5) must satisfy the condi- tions for the Green functions and vector d in the banks of superconductors far from the orifice: g v (+−∞) = iεmτv3 − ∆ v 1,2 √εm 2 + \|d1,2\| 2 ; (6) d(+−∞) = d1,2(k̂) exp    +− iφ 2    , (7) where φ is the external phase difference. Equations (1) and (5) have to be supplemented by the boun- dary continuity conditions at the contact plane and conditions of reflection at the interface between superconductors. Below we assume that this inter- face is smooth and that electron scattering is negli- gible. 3. Calculation of the current density The solution of Eqs. (1) and (5) allows us to calculate the current density: j(r) = 2πeTvF N(0) ∑ m 〈k̂g1(k̂,r,εm)〉 . (8) We consider the simple model of a constant order parameter up to the surface. The pair breaking and the scattering on the partition and in the junction are ignored. This model can be rigorously found for calculations of the current density (8) in ballistic point contacts between conventional superconduc- tors in the zero approximation in the small parame- ter d/ξ0 (ξ0 is the coherence length) [17]. In an- isotropically paired superconductors the order parameter changes at distances of the order of ξ0 even near a specular surface [25,26]. Thus for calcu- lations of the current (8) in the leading approxima- tion in the parameter d/ξ0 it is necessary to solve Eq. (5) near a surface of the semi-infinite supercon- ductor. It can be done only numerically and will be the subject of our future investigations. Below we assume that the order parameter does not depend on coordinates and in each half space is equal to its value (7) far form the point contact. For this non-self-consistent model the current–phase rela- tion of a Josephson junction can be calculated Fig. 1. Scheme of a contact in the form of an orifice be- tween two superconducting banks, which are misorien- tated by an angle α. R. Mahmoodi, S. N. Shevchenko, and Yu. A. Kolesnichenko 264 Fizika Nizkikh Temperatur, 2002, v. 28, No. 3 analytically. This makes it possible to analyze the main features of the current–phase relations for different scenarios of «f-wave» superconductivity. We believe that under this strong assumption our results describe the real situation qualitatively, as has been justified for point contacts between «d- wave» superconductors [20] and pinholes in 3He [27]. It was also shown in Ref. 22 that for the contact between «f-wave» superconductors there is also good qualitative agreement between self-consis- tent and non-self-consistent solutions (although, of course, quantitative distinctions take place). In a ballistic case the system of 16 equations for functions gi and gi can be decomposed into inde- pendent blocks of equations. The set of equations which enables us to find the Green function g1 is ivFk̂∇ g1 + (g3d − g2d ∗ ) = 0 ; (9) ivFk̂∇ g− + 2i(d × g3 + d∗ × g2) = 0 ; (10) ivFk̂∇ g3 − 2iεmg3 − 2g1d ∗ − id∗ × g− = 0 ; (11) ivFk̂∇ g2 + 2iεmg2 + 2g1d − id × g− = 0 ; (12) where g− = g1 − g4 . Equations (9)–(12) can be solved by integrating over ballistic trajectories of electrons in the right and left half spaces. The general solution satisfying the boundary conditions (6) at infinity is g1 (n) = iεm Ωn + iCn exp (− 2sΩnt) ; (13) g− (n) = Cn exp (− 2sΩnt) ; (14) g2 (n) = − 2Cndn − dn × Cn − 2sηΩn + 2εm exp (− 2sΩnt) − dn Ωn ; (15) g3 (n) = − 2Cndn ∗ + dn ∗ × Cn 2sηΩn + 2εm exp (− 2sΩnt) − dn ∗ Ωn ; (16) where t is the time of flight along the trajectory, sgn (t) = sgn (z) = s; η = sgn (vz); Ωn = √ εm2 + \|dn\| 2. By matching the solutions (13)– (16) at the orifice plane (t = 0), we find the constants Cn and Cn . Index n numbers the left (n = 1) and right (n = 2) half spaces. The function g1(0) = g1 (1)(−0) = g1 (2)(+0), which determines the current density in the con- tact, is g1(0) = = iεm (Ω1 + Ω2) cos ζ + η (εm 2 + Ω1Ω2) sin ζ ∆ → 1∆ → 2 + (εm 2 + Ω1Ω2) cos ζ − iεmη(Ω1 + Ω2) sin ζ . (17) In formula (17) we have taken into account that for unitary states the vectors d1,2 can be written as dn = ∆ → n exp iψn , (18) where ∆ → 1,2 are real vectors. Knowing the function g1(0), one can calculate the current density at the orifice plane j(0): j(0) = 4πeN(0)vFT ∑ m=0 ∞ ∫ dk̂ k̂ Re g1(0) , (19) where Re g1(0) = = [∆1 2∆2 2 cos ζ + (εm 2 + Ω1Ω2)∆ → 1∆ → 2] sin ζ [∆ → 1∆ → 2 + (εm 2 + Ω1Ω2) cos ζ] 2 + εm 2 (Ω1 + Ω2) 2 sin2 ζ (20) or, alternatively, Re g1(0) = ∆1∆2 2 ∑ ± sin (ζ ± θ) εm 2 + Ω1Ω2 + ∆1∆2 cos (ζ ± θ) , (21) where θ is defined by ∆ → 1(k̂)∆ → 2(k̂) = ∆1(k̂)∆2(k̂) cos θ, and ζ(k̂) = ψ2(k̂) − ψ1(k̂) + φ. Misorientation of the crystals would generally result in the appearance of current along the inter- face [20,22], as can be calculated by projecting the vector j on the corresponding direction. We consider a rotation R only in the right- hand superconductor (see Fig. 1), (i.e., d2(k̂) = = Rd1(R −1k̂)). The c axis in the left half space is chosen along the partition between superconductors (along the z axis in Fig. 1). To illustrate results obtained by computing Eq. (19), we plot the cur- rent–phase relation for different below-mentioned scenarios of «f-wave» superconductivity for two different geometries corresponding to different ori- entations of the crystals to the right and to the left at the interface (see Fig. 1): Josephson effect in point contacts between «f-wave» superconductors Fizika Nizkikh Temperatur, 2002, v. 28, No. 3 265 (i) The basal ab plane to the right is rotated about c axis by the angle α; ĉ1 \|\| ĉ2 . (ii) The c axis to the right is rotated about the contact axis (y axis in Fig. 1) by the angle α; b̂1 \|\| b̂2 . Further calculations require a certain model of the vector order parameter d. 4. Current–phase relation for different scenarios of «f-wave» superconductivity The model which has been successful to explain properties of the superconducting phases in UPt3 is based on the odd-parity E2u representation of the hexagonal point group D6h for strong spin–orbital coupling with vector d locked along the c axis of the lattice [10]: d = ∆0ẑ[η1Y1 + η2Y2], where Y1 = kz(kx 2 − ky 2) and Y2 = 2kxkykz are the basis function of the representation. The coordinate axes x, y, z here and below are chosen along the crystal- lographic axes â, b̂, ĉ as at the left in Fig. 1. This model describes the hexagonal analog of spin-triplet «f-wave» pairing. For the high-temperature A-phase (η2 = 0) the order parameter has an equatorial line node and two longitudinal line nodes. In the low- temperature B-phase (η2 = i) or the axial state d = ∆0ẑkz(kx + iky) 2 (22) the longitudinal line nodes are closed and there is a pair of point nodes. The B-phase (22) breaks the time-reversal symmetry. The function ∆0 = ∆0(T) in Eq. (22) and below describes the dependence of the order parameter d on temperature T (in carrying out numerical calculations we assume T = 0). Other candidates for describing the orbital states, which imply that the effective spin–orbital coupling in UPt3 is weak, are the unitary planar state d = ∆0kz[x̂(kx 2 − ky 2) + ŷ2kxky] (23) (or d = ∆0(Y1,Y2,0)) and the non-unitary bipolar state d = ∆0(Y1,iY2,0) [7]. In Fig. 2 we plot the Josephson current–phase relation jJ(φ) = jy(y = 0) calculated from Eq. (19) for both the axial (with the order parameter given by Eq. (22)) and the planar (Eq. (23)) states for a particular value of α under the rotation of the basal ab plane to the right (the geometry (i)). For simplicity we use a spheri- cal model of the Fermi surface. For the axial state the current–phase relation is just a slanted sinusoid and for the planar state it shows a «π-state». The appearance of the π-state at low temperatures is due to the fact that different quasiparticle trajectories contribute to the current with different effective phase differences ζ(k̂) (see Eqs. (19) and (21)) [19]. Such a different behavior can be a criterion for distinguishing between the axial and the planar states, taking advantage of the phase-sensitive Josephson effect. Note that for the axial model the Josephson current formally does not equal zero at φ = 0. This state is unstable (does not correspond to a minimum of the Josephson energy), and a state with a spontaneous phase difference (value φ0 in Fig. 2), which depends on the misorientation angle α, is realized. The remarkable influence of the misorientation angle α on the current–phase relation is shown in Fig. 3 for the axial state in the geometry (ii). For some values of α (in Fig. 3 it is α = π/3) there are more than one state, which correspond to minima of the Josephson energy (jJ = 0 and djJ/dφ > 0). Calculated x and z components of the current, which are parallel to the surface, js(φ) are shown in Fig. 4 for the same axial state in the geometry (ii). Note that the current tangential to the surface as a function of φ is not zero when the Josephson current (Fig. 3) is zero. This spontaneous tangential cur- rent (see also Ref. 22) is due to a specific «pro- ximity effect» similar to the spontaneous cur- Strictly speaking, in crystals with a strong spin–orbit coupling the spin is a «bad» quantum number, but the electronic states are twofold degenerate and can be characterized by pseudospins. Fig. 2. Josephson current densities versus phase φ for axial (22) and planar (23) states in the geometry (i); misorientation angle α = π/4; the current is given in units of j0 = (π/2)eN(0)vF∆0(0). R. Mahmoodi, S. N. Shevchenko, and Yu. A. Kolesnichenko 266 Fizika Nizkikh Temperatur, 2002, v. 28, No. 3 rent in contacts between «d-wave» superconduc- tors [20,28]. The total current is determined by the Green function, which depends on the order pa- rameters in both superconductors. As a result of this, for nonzero misorientation angles a current parallel to the surface can be generated. In the geometry (i) the tangential current for both the axial and planar states at T = 0 is absent. The first candidate for the superconducting state in Sr2RuO4 was the «p-wave» model [8] d = ∆0ẑ(k̂x + ik̂y) . (24) Recently [11,12] it was shown that the pairing state in Sr2RuO4 most likely has lines of nodes. It was suggested that this can occur if the spin-triplet state belongs to a nontrivial realization of the Eu representation of the group D4h , with either B2g ⊗ Eu [12] or B1g ⊗ Eu [11] symmetry: d = ∆0ẑk̂xk̂y(k̂x+ ik̂y), for B2g ⊗ Eu symmetry; (25) d = ∆0ẑ(k̂x 2− k̂y 2)(k̂x+ ik̂y), for B1g ⊗ Eu symmetry. (26) Note that models (24)–(26) of the order parameter spontaneously break time-reversal symmetry. Taking into account a quasi-two-dimensional electron energy spectrum in Sr2RuO4 , we calculate the current (19) numerically using the model of a cylindrical Fermi surface. The Josephson current for the hybrid «f-wave» model of the order parameter Eq. (26) is compared to the «p-wave» model (Eq. (24)) in Fig. 5 (for α = π/4). Note that the critical current for the «f-wave» model is several times smaller (for the same value of ∆0) than for the «p-wave» model. This different character of the current–phase relations enables us to distinguish between the two states. In Figs. 6 and 7 we present the Josephson cur- rent and the tangential current for the hybrid «f- wave» model for different misorientation angles α (for the «p-wave» model it equals zero). Just as in Fig. 2 for the «f-wave» order parameter (22), in Fig. 3. Josephson current versus phase φ for the axial (22) state in the geometry (ii) for different α. Fig. 4. The x (a) and z (b) components of the tangen- tial current versus phase φ for the axial state (22) in the geometry (ii) for different α. Fig. 5. Josephson current versus phase φ for hybrid «f-wave» and «p-wave» states in the geometry (i); α = π/4. Josephson effect in point contacts between «f-wave» superconductors Fizika Nizkikh Temperatur, 2002, v. 28, No. 3 267 Fig. 6 for the hybrid «f-wave» model (25) the steady state of the junction with zero Josephson current corresponds to a nonzero spontaneous phase difference if the misorientation angle α ≠ 0. Conclusion We have considered the stationary Josephson effect in point contacts between triplet supercon- ductors. Our analysis is based on models with «f-wave» symmetry of the order parameter belong- ing to the two-dimensional representations of the crystallographic symmetry groups. It is shown that the current–phase relations are quite different for different models of the order parameter. Because the order parameter phase depends on the momen- tum direction on the Fermi surface, misorientation of the superconductors leads to a spontaneous phase difference that corresponds to zero Josephson cur- rent and to the minimum of the weak link energy. This phase difference depends on the misorientation angle and can possess any values. We have found that in contrast to the «p-wave» model, in the «f-wave» models the spontaneous current may be generated in a direction which is tangential to the orifice plane. Generally speaking this current is not equal to zero in the absence of the Josephson cur- rent. We demonstrate that the study of the current– phase relation of a small Josephson junction for different crystallographic orientations of the banks enables one to assess the applicability of different models to the triplet superconductors UPt3 and Sr2RuO4 . 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Josephson effect in point contacts between "f -wave" superconductors

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