Josephson currents in point contacts between dirty two-band superconductors

We developed microscopic theory of Josephson effect in point contacts between dirty two-band superconductors. The general expression for the Josephson current, which is valid for arbitrary temperatures, is obtained. This expression was used for calculation of current-phase relations and temperature...

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Datum:	2010
Hauptverfasser:	Yerin, Y.S., Omelyanchouk, A.N.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2010
Schriftenreihe:	Физика низких температур
Schlagworte:	Quantum coherent effects in superconductors and normal metals
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spelling	nasplib_isofts_kiev_ua-123456789-1175222025-02-09T18:04:00Z Josephson currents in point contacts between dirty two-band superconductors Yerin, Y.S. Omelyanchouk, A.N. Quantum coherent effects in superconductors and normal metals We developed microscopic theory of Josephson effect in point contacts between dirty two-band superconductors. The general expression for the Josephson current, which is valid for arbitrary temperatures, is obtained. This expression was used for calculation of current-phase relations and temperature dependences of critical current with application to MgB₂ superconductor. Also we have considered influence on contact characteristics interband scattering effect appeared in case of dirty superconductors. It is shown that the correction to Josephson current due to the interband scattering depends on phase shift in the banks (i.e.,s- or s±-wave symmetry of order parameters). 2010 Article Josephson currents in point contacts between dirty two-band superconductors / Y.S. Yerin, A.N. Omelyanchouk // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1204–1208. — Бібліогр.: 17 назв. — англ. 0132-6414 PACS: 74.50.+r, 74.20.Mn https://nasplib.isofts.kiev.ua/handle/123456789/117522 en Физика низких температур application/pdf 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	Quantum coherent effects in superconductors and normal metals Quantum coherent effects in superconductors and normal metals
spellingShingle	Quantum coherent effects in superconductors and normal metals Quantum coherent effects in superconductors and normal metals Yerin, Y.S. Omelyanchouk, A.N. Josephson currents in point contacts between dirty two-band superconductors Физика низких температур
description	We developed microscopic theory of Josephson effect in point contacts between dirty two-band superconductors. The general expression for the Josephson current, which is valid for arbitrary temperatures, is obtained. This expression was used for calculation of current-phase relations and temperature dependences of critical current with application to MgB₂ superconductor. Also we have considered influence on contact characteristics interband scattering effect appeared in case of dirty superconductors. It is shown that the correction to Josephson current due to the interband scattering depends on phase shift in the banks (i.e.,s- or s±-wave symmetry of order parameters).
format	Article
author	Yerin, Y.S. Omelyanchouk, A.N.
author_facet	Yerin, Y.S. Omelyanchouk, A.N.
author_sort	Yerin, Y.S.
title	Josephson currents in point contacts between dirty two-band superconductors
title_short	Josephson currents in point contacts between dirty two-band superconductors
title_full	Josephson currents in point contacts between dirty two-band superconductors
title_fullStr	Josephson currents in point contacts between dirty two-band superconductors
title_full_unstemmed	Josephson currents in point contacts between dirty two-band superconductors
title_sort	josephson currents in point contacts between dirty two-band superconductors
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2010
topic_facet	Quantum coherent effects in superconductors and normal metals
url	https://nasplib.isofts.kiev.ua/handle/123456789/117522
citation_txt	Josephson currents in point contacts between dirty two-band superconductors / Y.S. Yerin, A.N. Omelyanchouk // Физика низких температур. — 2010. — Т. 36, № 10-11. — С. 1204–1208. — Бібліогр.: 17 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT yerinys josephsoncurrentsinpointcontactsbetweendirtytwobandsuperconductors AT omelyanchoukan josephsoncurrentsinpointcontactsbetweendirtytwobandsuperconductors
first_indexed	2025-11-29T07:02:12Z
last_indexed	2025-11-29T07:02:12Z
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fulltext	© Y.S. Yerin and A.N. Omelyanchouk, 2010 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11, p. 1204–1208 Josephson currents in point contacts between dirty two-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., Kharkov 61103, Ukraine E-mail: yerin@ilt.kharkov.ua Received April 27, 2010 We developed microscopic theory of Josephson effect in point contacts between dirty two-band super- conductors. The general expression for the Josephson current, which is valid for arbitrary temperatures, is ob- tained. This expression was used for calculation of current-phase relations and temperature dependences of criti- cal current with application to MgB2 superconductor. Also we have considered influence on contact characteristics interband scattering effect appeared in case of dirty superconductors. It is shown that the correc- tion to Josephson current due to the interband scattering depends on phase shift in the banks (i.e., s- or s±-wave symmetry of order parameters). PACS: 74.50.+r Tunneling phenomena; point contacts, weak links, Josephson effects; 74.20.Mn Nonconventional mechanisms. Keywords: two-band superconductor, MgB2, Usadel equations, Josephson effect, current-phase relation, critical current. 1. Introduction The first mentioning about multiband superconductivity has appeared in theoretical works of Matthis, Suhl and Walker [1] and Moskalenko and Palistrant [2] more than 50 years ago. At that time their papers were considered as attempts to fit for BSC-theory some characteristics of su- perconducting materials (refinement of the ratio of an en- ergy gap to critical temperature, heat capacity jump, lon- don penetration depth etc.). Really multiband supercon- ductivity became hot topic in condensed matter physics in 2001, when Nagamatsu has discovered two-band super- conductivity in MgB2 with anomalous high Tc = 39 K [3]. It’s striking that pairing mechanism had electron–phonon origin in magnesium diboride and order parameters, which attribute to superconducting energy gaps, have s-wave symmetry. Iron-based superconductors, which have discovered not long ago [4], most probably can be adding on to multiband systems. For example, ARPES specifies on existence of two full gaps in Ba0.6K0.4Fe2As2 [5]. Moreover there is an assumption that in this iron-based superconductor π-shift between phases of order parameters and thus so called ex- tended with sign-reversal of order parameter or s±-wave symmetry is realized [6]. Existence of such phase-shift can lead to new fundamental phenomena and effects in these superconducting systems. Unfortunately the most of mo- dern experimental methods (measuring magnetic penetra- tion length, calorimetric method, Knight shift and spin- relaxation velocity in NMR and above mentioned ARPES) cannot give unique answer about gap pairing symmetry. However, recently new phase-sensitive technique based on proximity effect between niobium wire and massive NdFeAsO0.88F0.12 plate for probing unconventional pair- ing symmetry in the iron pnictides was reported [7]. Therefore it’s interesting to research phase coherent ef- fects in two-band superconductors which are sensible to shift of phase. Josephson effects are one of such sensitive to the phase phenomena. At the present time there are ma- ny theoretical publications, devoted to this problem [8–12]. In these papers the Josephson effect in tunnel junctions between one-band and two-band superconductors is con- sidered. In present paper we study the stationary Josephson effect in superconductor–constriction–superconductor (S–C–S) contact, which behavior even in the case of one-band su- perconductors, as was revealed in Kulik and Omelyan- chouk papers [13,14] (KO theory), has the qualitative dif- ferences with respect to superconductor–insulator–super- conductor (S–I–S) tunnel junctions. We built the Josephson currents in point contacts between dirty two-band superconductors Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1205 microscopic theory of the «dirty» S–C–S contact for two- band superconductors, which generalizes the KO theory for this case. We show that the interband scattering effects occurring in dirty superconductors result in mixing of Jo- sephson currents between different bands. 2. Model and basic equations Consider the weak superconducting link as a thing filament of length L and diameter d, connecting two bulk superconductors (banks) (Fig. 1). Such model describes the S–C–S contacts with direct conductivity (point contacts, microbridges), which qualitatively differ from the tunnel S–I–S junctions. On condition that d L and ( ) ( )1 2min [ 0 , 0 ]d ξ ξ ( ( )i Tξ — coherence lengths) we can solve inside the filament ( 0 x L≤ ≤ ) a one-di- mensional problem with «rigid» boundary conditions. At 0,x L= all functions are assumed equal to the values in homogeneous no-current state of corresponding bank. We investigate the case of dirty two-band superconduc- tor with strong impurity intraband scattering rates (dirty limit) and weak interband scattering. In the dirty limit su- perconductor is described by the Usadel equations for normal and anomalous Green’s functions g and f , which for two-band superconductor take the form [15]: ( ) ( )2 2 1 1 1 1 1 1 1 1 12 1 2 2 1 ,f D g f f g g g f g fω − ∇ − ∇ = Δ + γ − (1) ( ) ( )2 2 2 2 2 2 2 2 2 2 21 2 1 1 2 .f D g f f g g g f g fω − ∇ − ∇ = Δ + γ − (2) Usadel equations are supplemented with self-consistency equations for order parameters iΔ : 0 2 D i ij j j T f ω ω> Δ = π λ∑∑ , (3) and with expression for the current density ( )* * i i i i i i i j ie T N D f f f f ω = − π ∇ − ∇∑∑ . (4) Index i = 1, 2 numerates the first and the second bands. Normal and anomalous Green’s functions ig and if , which are connected by normalization condition 22 1i ig f+ = , are functions of x and the Matsubara fre- quency ( )2 1n Tω = + π . iD are the intraband diffusivities due to nonmagnetic impurity scattering, iN are the density of states on the Fermi surface of the ith band, electron– phonon constants ijλ take into account Coulomb pseudo- potentials and ijγ are the interband scattering rates. There are the symmetry relations 12 1 21 2N Nλ = λ and 12 1Nγ = 21 2N= γ . In considered case of short weak link we can neglect all terms in the equations (1), (2) except the gradient one. Using the normalization condition we have equations for 1,2f 2 2 2 2 1 1 1 12 21 1 0,d df f f f dx dx − − − = (5) 2 2 2 2 2 2 2 22 21 1 0.d df f f f dx dx − − − = (6) The boundary conditions for Eqs. (5) and (6) are deter- mined by solutions of equations for Green’s functions in left (right) banks: ____________________________________________________ 2 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 12 1 2 2 1 2 2 2( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 21 2 1 1 2 1 1 1 , 1 1 1 . L R L R L R L R L R L R L R L R L R L R L R L R L R L R f f f f f f f f f f f f ⎧ ⎛ ⎞ ω = Δ − + γ − − −⎪ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎨ ⎛ ⎞⎪ ω = Δ − + γ − − −⎜ ⎟⎪ ⎜ ⎟ ⎝ ⎠⎩ (7) _______________________________________________ Note, that solutions 1f and 2f of Eqs. (5), (6) are coupled due to the interband scattering in the banks through the boundary conditions (7). Introducing the phases of order parameters in banks ( ) ( )( ) ( ) 1 1 2 21 2exp ( ), exp ( )L R L RL R L Ri iΔ = Δ φ Δ = Δ φ , (8) and writing ( )if x in Eqs. (5) and (6) as ( ) ( )i if x f x= × exp ( ( ))ii x× χ we have (0) ( ) , (0) , ( ) ,L R i i i i i i if f L f L= = χ = χ χ = χ (9) where if and ( )L R iχ are connected with iΔ and ( )L R iφ through Eq. (7). The solution of Eqs. (5)–(9) determines the Josephson current in the system. It depends on the phase difference on the contact, which we define as 1 1 2 2 R L R Lφ ≡ φ − φ = φ − φ , and from possible phase shift in each banks 2 1 2 1 L L R Rδ = φ − φ = φ − φ . The phase shift δ between the Fig. 1. Model of S–C–S contact. The right and left banks are bulk two-band superconductors connected by the thing filament of length L and diameter d. \|\| \| exp (i\| exp (i ))�� 11 11 LL \|\| \| exp (i\| exp (i ))�� 11 11 RR \|\| \| exp (i\| exp (i ))�� 22 22 RR \|\| \| exp (i\| exp (i ))�� 22 22 LL LL dd SSLL SSRR Y.S. Yerin and A.N. Omelyanchouk 1206 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 phases of the two order parameters in two-band supercon- ductor can be 0 or π , depending on the sign on the inter- band coupling constants, the values of the interband scat- tering rates and the temperature of the system (see Appendix). Equations (5) and (6) with boundary conditions (9) ad- mit analytical solution, and for the current density (4) we obtain: 1 1 1 1 2 2 21 1 1 1 1 1 1 1 2 2 21 1 1 1 1 2 2 2 2 2 2 22 2 2 2 2 2 2 2 cos2 2 (1 ) sin cos 2 2 sin 2 arctan (1 ) sin cos 2 2 cos2 2 + (1 ) sin cos 2 2 sin arctan L R L R L RN L R L R L R L R L R L RN L R fTI eR f f f fT eR f f ω ω χ −χ π = × χ − χ χ −χ − + χ − χ × + χ −χ χ −χ − + χ −χ π × χ −χ χ −χ − + χ −χ × ∑ ∑ 2 2 22 2 2 2 2 2 . (1 ) sin cos 2 2 L R L R f χ −χ χ −χ − + (10) Here 1NR , 2NR — contributions to normal resistance of each band, where 1 2(2 / )Ni i iR Se L N D− = ( 2 /4S d= π is cross section area). The general expression (10) together with Eqs. (7)–(9) describes the Josephson current as function of gaps in the banks iΔ and phase difference on the contact φ . 3. Josephson currents 3.1. Josephson current without interband scattering When interband scattering is vanished, from system of equations (7) we obtain: ( ) 1,2( ) 1,2 2( ) 2 1,2 ( ) 1,2 2( ) 2 1,2 L R L R L R L R L R f g ⎧ Δ ⎪ = ⎪ Δ +ω⎪⎪ ⎨ ω⎪ =⎪ ⎪ Δ +ω⎪⎩ (11) Taking into account (11) we rewrite expression for the current (10) in terms of order parameters: 1 1 2 21 2 2 2 2 1 1 cos sin2 2 2arctan cos cos 2 2 N TI eR ω φ φ Δ Δπ = + φ φ ω + Δ ω + Δ ∑ 2 2 2 22 2 2 2 2 2 2 cos sin2 2 2+ arc tan . cos cos 2 2 N T eR ω φ φ Δ Δπ φ φ ω + Δ ω + Δ ∑ (12) Thus, if to neglect interband scattering rates ikγ Josephson current (10) decomposes on two parts: current flows inde- pendently from the first band to the first one and from the second band to the second one. For zero temperature 0T = Eq. (12) takes the form 1 2 1 2 cos Arctanhsin cos Arctanhsin . 2 2 2 2 2 2N N I eR eR π Δ π Δφ φ φ φ = + (13) Equation (12) is straightforward generalization of Josephson current for one-band superconductor [13]. The partial inputs of bands currents (1 1)I → and (2 2)I → are determined by the ratio of normal resistances 1 2/N Nr R R= . Introducing the total resistance NR = 1 2 1 2/ ( )N N N NR R R R= + and normalizing the current on the value 0 (2 / )N cI eR T= π we plot on Fig. 2 the current– phase relation (12) for different values of r and tempera- ture T. The current-phase relation (12) determines the tempera- Fig. 2. Current-phase relations of S–C–S MgB2\|MgB2 for dif- ferent temperatures / cT Tτ = : 0 (1); 0.5 (2); 0.9 (3); and ratio of resistances 1 2/N Nr R R= . 0.50.5 0.50.5 1.01.0 1.01.0 1.51.5 1.51.5 2.02.0 2.02.0 2.52.5 2.52.5 3.03.0 3.03.0 0.050.05 0.100.10 0.150.15 0.200.20 0.250.25 0.300.30 11 22 33 r = 0.1r = 0.1 00 00 0.020.02 0.040.04 0.060.06 0.080.08 0.100.10 0.120.12 r = 10r = 10 11 22 33 �� I/ I I/ I 00 I/ I I/ I 00 aa bb Josephson currents in point contacts between dirty two-band superconductors Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 1207 ture dependence of critical current cI , which is plotted on Fig. 3. In calculations of ( )I φ and ( )cI T (Figs. 2, 3) we use the parameters for superconductor MgB2 with 0ijγ = . In the case of zero interband scattering the Josephson current (12) does not depend on possible phase shift δ (i.e., in the case of s±-wave symmetry). 3.2. Josephson current with interband scattering Now our aim is investigation of effects of interband scattering on S–C–S contact behavior. In general, the sys- tem (7) has no analytical solution. The case of tempera- tures near critical temperature cT and arbitrary strength of interband scattering was considered in [16]. Here, in the case of arbitrary temperature 0 cT T≤ ≤ we consider the interband scattering ijγ by the theory of perturbations. In the first approximation for Green’s functions in each bank we obtain: ( )( ) ( ) ( ) ( )( ) ( ) ( ) 22 2 * * 1 2 1 1 2 11 1 122 32 2 22 21 1 2 22 2 * * 1 1 2 2 1 22 2 212 32 2 22 22 1 2 2 , 2 2 . 2 f f ⎧ ω + Δ Δ −Δ −Δ Δ −ΔΔ⎪ = +γ⎪ Δ +ω⎪ Δ +ω Δ +ω ⎪ ⎨ ⎪ ω + Δ Δ −Δ −Δ Δ −ΔΔ⎪ = +γ ⎪ Δ +ω⎪ Δ +ω Δ +ω ⎩ (14) When the interband scattering is taken into account and if phase shift 0δ ≠ , the phases of Green functions if not coincide with phases of order parameters iΔ . Using (14) we obtain expressions for the corrections to the current (12): 1 2I I Iδ = δ + δ , (15) ____________________________________________________ ( ) ( ) 2 2 1 112 1 3 21 2 22 22 2 2 11 2 2 1 2 1 2 2 22 2 2 2 1 1 2 e cos sin2 2 2arctan coscos 22 e sin1 , 2 cos 2 i N i T I eR δ ω δ ⎛ φ⎛ ⎞⎜ φω Δ − Δ Δ⎜ ⎟⎜π γ ⎝ ⎠δ = +⎜ φ⎜ ⎛ φ ⎞⎛ ⎞ ω + Δ⎜ Δ +ω Δ +ω⎜ ⎟⎜ ⎟⎜ ⎝ ⎠⎝ ⎠⎝ ⎞ ⎟ω Δ Δ − Δ φ ⎟+ ⎟⎛ φ ⎞⎛ ⎞ ⎛ ⎞ ⎟Δ +ω Δ +ω Δ +ω⎜ ⎟⎜ ⎟⎜ ⎟ ⎟⎝ ⎠⎝ ⎠⎝ ⎠ ⎠ ∑ (16) ( ) ( ) 2 1 2 221 2 3 22 2 22 22 2 2 22 1 2 2 1 2 2 2 22 2 2 2 2 2 1 e cos sin2 2 2arctan coscos 22 e sin1 . 2 cos 2 i N i T I eR δ ω δ ⎛ φ⎛ ⎞⎜ φω Δ − Δ Δ⎜ ⎟⎜π γ ⎝ ⎠δ = +⎜ φ⎜ ⎛ φ ⎞⎛ ⎞ ω + Δ⎜ Δ +ω Δ +ω⎜ ⎟⎜ ⎟⎜ ⎝ ⎠⎝ ⎠⎝ ⎞ ⎟ω Δ Δ − Δ φ ⎟+ ⎟⎛ φ ⎞⎛ ⎞ ⎛ ⎞ ⎟Δ +ω Δ +ω Δ +ω⎜ ⎟⎜ ⎟⎜ ⎟ ⎟⎝ ⎠⎝ ⎠⎝ ⎠ ⎠ ∑ (17) _______________________________________________ The correction to Josephson current due to the inter- band scattering (15)–(17) depends on phase shift in the banks 0 or δ = π . This reflects the mixing of Josephson currents between different bands (see [16]). Fig. 3. Temperature dependences of critical current ( )cI T for different values of 1 2/N Nr R R= : 0.1 (1); 1 (2); 10 (3). 00 0.20.2 0.40.4 0.60.6 0.80.8 1.01.0 0.100.10 0.200.20 0.300.30 11 22 33 T/TT/Tcc II /I/I cc 00 Y.S. Yerin and A.N. Omelyanchouk 1208 Fizika Nizkikh Temperatur, 2010, v. 36, Nos. 10/11 Conclusions We developed microscopic theory of Josephson effect in point contacts between dirty two-band superconductors. The general expression for the Josephson current, which is valid for arbitrary temperatures, is obtained. We consid- ered the dirty superconductors with interband scattering effect. This effect in the contacting superconductors pro- duces the coupling of the currents between different bands. With taking into account the interband scattering the ob- servable characteristics ( )I φ and ( )cI T depend on the phase shift of the order parameters in different bands. It permits to distinguish between the s- and s±-wave symme- try by studying the Josephson effect in point contacts of two band superconductors. This work was supported by FRSF (grant F28.2/019). Appendix Free energy within microscopic consideration is given by [17]: 2 * 1 1 2 int 1 2 8i j i ij ij BF N F F F−= Δ Δ λ + + + + π∑ , where (A.1) ( ) ( ) ( ) * 0 2* 0 0 2 1 Re 1 2 2 , 4 i i i i i i i i i F T N g f i iD f f g ω> ⎡ = π − − Δ +⎢ ⎢⎣ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞π π + ∇ + ∇ − + ∇ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟Φ Φ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠⎦ ∑ A A (A.2) ( )( )* int 1 12 2 21 1 2 1 2 0 2 Re 1F T N N g g f f ω> = π γ + γ + −∑ . (A.3) Representing order parameters as ( )( ) exp ( )L RL R i i iiΔ = Δ φ and anomalous Green functions ( ) ( ) exp ( ( ))i i if x f x i x= χ and simultaneously taking into account that 1 12N λ = 2 21N= λ and 1 12 2 21N Nγ = γ we extract from (A.1) terms, which contain the phase difference ( ) ( ) 2 1 L R L Rδ = φ − φ : 12 1 2 12 1 2 11 22 12 21 0 cos 4 T f f ω> λ Δ Δ − δ + π γ λ λ −λ λ ∑ . (A.4) Substitution of 1f and 2f into (A.4) assuming 12 0γ → yields ( )( ) 12 12 2 22 211 22 12 21 0 1 2 14 cos .T ω> ⎛ ⎞ ⎜ ⎟λ − + π γ δ⎜ ⎟ λ λ −λ λ⎜ ⎟ω + Δ ω + Δ⎜ ⎟ ⎝ ⎠ ∑ (A.5) For 0δ = minimum of free energy is satisfied if bracketed expression is less than zero and vice versa for δ = π the minimum of (A.1) takes place if bracketed expression is greater than 0. On the basis of this we obtain existence criterion of π-shift between phases of order parameters: 12 11 22 12 21 12 2 22 20 1 2 sgn 14 1. ( )( ) T ω> ⎛ λ − +⎜ λ λ −λ λ⎝ ⎞ ⎟+ π γ = ±⎟⎟ω + Δ ω + Δ ⎠ ∑ (A.6) In this condition «+1» correspond to δ = π and «–1» takes place if 0δ = . The derived criterion can be simplified at 0T = : ( ) 2 12 12 1 2 11 22 12 21 2 2 2 \| \| (0) sgn 1 1, \| 0 \| \| \| (0) ⎛ ⎞⎛ ⎞λ γ Δ⎜ ⎟⎜ ⎟− + Κ − = ± ⎜ ⎟⎜ ⎟λ λ −λ λ Δ Δ⎝ ⎠⎝ ⎠ (A.7) where 2 1 2 2 \| \| (0) 1 \| \| (0) ⎛ ⎞Δ⎜ ⎟Κ − ⎜ ⎟Δ⎝ ⎠ is full elliptic integral of the first kind, ( )1 0Δ and ( )2 0Δ are values of the order pa- rameters at 0T = . In the vicinity of cT condition (A.6) transforms to: 12 12 11 22 12 21 sgn 1 2 cT ⎛ ⎞λ π − + γ = ±⎜ ⎟λ λ −λ λ⎝ ⎠ . (A.8) 1. H. Suhl, B.T. Matthias, and L.R. Walker, Phys. Rev. Lett. 3, 552 (1959). 2. V.A. Moskalenko, Fiz. Met. Metallov. 8, 503 (1959); Phys. Met. and Metallov. 8, 25 (1959). 3. J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and J. Akimitsu, Nature 410, 63 (2001). 4. Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. 130, 3296 (2008). 5. H. Ding, P. Richard, K. Nakayama, K. Sugawara, T. Araka- ne, Y.X. Dai, Z. Fang, G.F. Chen, J.L. Luo, and N.L. Wang, EPL 83, 47001 (2008). 6. I.I. Mazin and J. Schmalian, Physica C469, 614 (2009). 7. C.-T. Chen, C.C. Tsuei, M.B. Ketchen, Z.-A. Ren, and Z.X. Zhao, Nature Phys. 6, 260 (2010). 8. D.F. Agterberg, E. Demler, and B. Janko, Phys. Rev. B66, 214507 (2002). 9. Y. Ota, M. Machida, T. Koyama, and H. Matsumoto, Phys. Rev. Lett. 102, 237003 (2009). 10. K. Ng and N. Nagaosa, EPL 87, 17003 (2009). 11. I.B. Sperstad, J. Linder, and A. Sudbø, Phys. Rev. B80, 144507 (2009). 12. J. Linder, I.B. Sperstad, and A. Sudbø, Phys. Rev. B80, 020503(R) (2009). 13. I.О. Kulik, A.N. Omelyanchouk, Pis’ma Zh. Eksp. Teor. Fiz. 21, 216 (1975). 14. I.О. Kulik and A.N. Omelyanchouk, Fiz. Nizk. Temp. 4, 296 (1978) [Sov. J. Low Temp. Phys. 4, 142 (1978)]. 15. A. Gurevich, Phys. Rev. B67, 184515 (2003). 16. A.N. Omelyanchouk and Y.S. Yerin, arXiv:0910.1429. 17. A. Gurevich and V.M. Vinokur, Phys. Rev. Lett. 97, 137003 (2006).

Josephson currents in point contacts between dirty two-band superconductors

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