Coherent current states in a two-band superconductor

Coherent current states in a two-band superconductor

Homogeneous current states in thin films and Josephson current in superconducting microbridges are studied within the frame of a two-band Ginzburg–Landau theory. By solving the coupled system of equations for two order parameters the depairing current curves and Josephson current-phase relation ar...

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Дата:2007
Автори: Yerin, Y.S., Omelyanchouk, A.N.
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Мова:English
Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
Назва видання:Физика низких температур
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Свеpхпpоводимость, в том числе высокотемпеpатуpная
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Цитувати:Coherent current states in a two-band superconductor / Y.S. Yerin, A.N. Omelyanchouk // Физика низких температур. — 2007. — Т. 33, № 5. — С. 538-545. — Бібліогр.: 36 назв. — англ.

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spelling irk-123456789-1278112017-12-29T03:02:55Z Coherent current states in a two-band superconductor Yerin, Y.S. Omelyanchouk, A.N. Свеpхпpоводимость, в том числе высокотемпеpатуpная Homogeneous current states in thin films and Josephson current in superconducting microbridges are studied within the frame of a two-band Ginzburg–Landau theory. By solving the coupled system of equations for two order parameters the depairing current curves and Josephson current-phase relation are calculated for different values of phenomenological parameters and . Coefficients and describe the coupling of order parameters (proximity effect) and their gradients (drag effect), respectively. For definite values of parameters the dependence of current j on superfluid momentum q contains local minimum and corresponding bi-stable states. It is shown that the Josephson microbridge from two-band superconductor can demonstrate -junction behavior. 2007 Article Coherent current states in a two-band superconductor / Y.S. Yerin, A.N. Omelyanchouk // Физика низких температур. — 2007. — Т. 33, № 5. — С. 538-545. — Бібліогр.: 36 назв. — англ. 0132-6414 PACS: 74.25.Fy, 74.81.Fa http://dspace.nbuv.gov.ua/handle/123456789/127811 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
Yerin, Y.S.
Omelyanchouk, A.N.
Coherent current states in a two-band superconductor
Физика низких температур
description Homogeneous current states in thin films and Josephson current in superconducting microbridges are studied within the frame of a two-band Ginzburg–Landau theory. By solving the coupled system of equations for two order parameters the depairing current curves and Josephson current-phase relation are calculated for different values of phenomenological parameters and . Coefficients and describe the coupling of order parameters (proximity effect) and their gradients (drag effect), respectively. For definite values of parameters the dependence of current j on superfluid momentum q contains local minimum and corresponding bi-stable states. It is shown that the Josephson microbridge from two-band superconductor can demonstrate -junction behavior.
format Article
author Yerin, Y.S.
Omelyanchouk, A.N.
author_facet Yerin, Y.S.
Omelyanchouk, A.N.
author_sort Yerin, Y.S.
title Coherent current states in a two-band superconductor
title_short Coherent current states in a two-band superconductor
title_full Coherent current states in a two-band superconductor
title_fullStr Coherent current states in a two-band superconductor
title_full_unstemmed Coherent current states in a two-band superconductor
title_sort coherent current states in a two-band superconductor
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2007
topic_facet Свеpхпpоводимость, в том числе высокотемпеpатуpная
url http://dspace.nbuv.gov.ua/handle/123456789/127811
citation_txt Coherent current states in a two-band superconductor / Y.S. Yerin, A.N. Omelyanchouk // Физика низких температур. — 2007. — Т. 33, № 5. — С. 538-545. — Бібліогр.: 36 назв. — англ.
series Физика низких температур
work_keys_str_mv AT yerinys coherentcurrentstatesinatwobandsuperconductor
AT omelyanchoukan coherentcurrentstatesinatwobandsuperconductor
first_indexed 2025-07-09T07:47:08Z
last_indexed 2025-07-09T07:47:08Z
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fulltext Fizika Nizkikh Temperatur, 2007, v. 33, No. 5, p. 538–545 Coherent current states in a two-band superconductor Y.S. Yerin and A.N. Omelyanchouk B. Verkin Istitute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: omelyanchouk@ilt.kharkov.ua Received October 11, 2006, revised January 11, 2007 Homogeneous current states in thin films and Josephson current in superconducting microbridges are studied within the frame of a two-band Ginzburg–Landau theory. By solving the coupled system of equa- tions for two order parameters the depairing current curves and Josephson current-phase relation are calcu- lated for different values of phenomenological parameters � and �. Coefficients � and �describe the coupling of order parameters (proximity effect) and their gradients (drag effect), respectively. For definite values of parameters the dependence of current j on superfluid momentum q contains local minimum and correspond- ing bi-stable states. It is shown that the Josephson microbridge from two-band superconductor can demon- strate �-junction behavior. PACS: 74.25.Fy Transport properties (electric and thermal conductivity, thermoelectric effects, etc.); 74.81.Fa Josephson junction arrays and wire networks. Keywords: current states, two-band superconductivity, proximity effect, drag effect, Josephson microbridge, �-junction. 1. Introduction To present day overwhelming majority works on the- ory of superconductivity were devoted to single gap su- perconductors. More than 40 years ago the possibility of superconductors with two superconducting order parame- ters were considered by V. Moskalenko [1] and H. Suhl, B. Matthias, and L. Walker [2]. In the model of supercon- ductor with the overlapping energy bands on Fermi sur- face V. Moskalenko has theoretically investigated the thermodynamic and electromagnetic properties of two-band superconductors. The real boom in investigation of mul- ti-gap superconductivity started after the discovery of two gaps in MgB2 [3] by the scanning tunneling [4,5] and point contact spectroscopy [6–8]. The compound MgB2 has the highest critical temperature Tc � 39 K among su- perconductors with phonon mechanism of the pairing and two energy gaps � 1 � 7 meV and � 2 � 2,5 meV at T � 0. At this time two-band superconductivity is studied also in another systems, e.g. in heavy fermion compounds [9,10], borocarbides [11] and liquid metallic hydrogen [12–14]. Various thermodynamic and transport properties of MgB2 were studied in the framework of two-band BCS model [15–22]. Ginzburg–Landau (GL) functional for two-gap superconductors was derived within the weak-coupling BCS theory in dirty [23] and clean [24] superconductors. Within the Ginzburg–Landau scheme the magnetic pro- perties [25–27] and peculiar vortices [28–30] were studied. The aim of this article is to present Ginzburg–Landau theory of the current carrying states in superconductors with two order parameters. In the case of several order pa- rameters the qualitatively new features in superconduct- ing current state are related to mutual influence of the modules of complex order parameters as well of the gradi- ents of their phases. We study the manifestations of these effects in the current-momentum dependence and in the Josephson current-phase relation. In Sec. 2 the general phenomenological description of two-band superconduc- tors within Ginzburg–Landau theory is given. The Ginz- burg–Landau equations for two coupled superconducting order parameters include the proximity and drag effects. In Sec. 3 the peculiarities of homogeneous current states in multi-gap superconductors are studied. The depend- ence of current on superfluid momentum for different val- ues of parameters is calculated. We demonstrate that for definite values of parameters it contains local minima and corresponding bi-stable states in GL free energy. In Sec. 4 the Josephson effect in simple model of weak supercon- ducting link (generalization of Aslamazov–Larkin theory [31] on two-band superconductor) is considered and pos- sibility of �-junction behavior is demonstrated. © Y.S. Yerin and A.N. Omelyanchouk, 2007 2. Ginzburg–Landau equations for two-band super- conductivity The phenomenological Ginzburg–Landau free energy density functional for two coupled superconducting order parameters � 1 and � 2 can be written as F F F FGL � � � �1 2 12 2 8 ( )rot A � , where F m i e c 1 1 1 2 1 1 4 1 1 2 1 2 1 2 2 � � � � � � � � � � � � � �( )� A , (1) F m i e c 2 2 2 2 2 2 4 2 2 2 1 2 1 2 2 � � � � � � � � � � � � � �( )� A , (2) and F i e c 12 1 2 1 2 1 2 � � � � � � � � � � � �� �� � � � � � �( )* * � A � � � � � � � � � � � � � � � � � � � � � � �i e c i e c i e c � � � 2 2 2 1 2A A A� �* � �� . (3) The terms F1 and F2 are conventional contributions from � 1 and � 2, term F12 describes without the loss of generality the interband coupling of order parameters. The coefficients � and �describe the coupling of two or- der parameters (proximity effect) and their gradients (drag effect) [25–27], respectively. By minimization of the free energy F = � � � � � � � � � � � �� F F F H d r1 2 12 2 3 8� with respect to� 1,� 2 and A we obtain the differential GL equations for two-band su- perconductor 1 2 2 2 1 2 1 1 1 1 1 2 1 2 m i e c i e c � � � � � � � � � � � � �A A� � � � � �� � � � � � � � � � � � � � � � � � 2 2 2 2 2 2 2 2 2 2 2 0 1 2 2 � � � � � � �� , m i e c � A 1 2 1 2 0� � � � � � � � � � � � � � � � �i e c � A , (4) and expression for the supercurrent j � � � � � ie m ie m � � 1 1 1 1 1 2 2 2 2 2( ) ( )* * * *� � � � � � � � � � � � � 2 1 2 2 1 1 2 2 1ie�� � � � � � � � �( )* * * * � � � � � � � � � � � 4 4 82 1 1 2 2 2 2 2 2 1 2 2 1 e m c e m c e c � � � � � � �( )* * A. (5) In the absence of currents and gradients of order pa- rameters modules the equilibrium values of order parame- ters � � ! 1 2 1 2 0 1 2 , , ( ) ,� e i are determined by the set of coupled equations � � � �� ! ! 1 1 0 1 1 0 3 2 0 2 1 0 ( ) ( ) ( ) ( )� � e i , � � � �� ! ! 2 2 0 2 2 0 3 1 0 1 2 0 ( ) ( ) ( ) ( )� � e i . (6) For the case of two order parameters the question arises about the phase difference " ! !� 1 2 between � 1 and � 2. In homogeneous no-current state, by analyzing the free energy term F12 (3), one can obtain that for � # 0 phase shift " � 0 and for � $ 0" �� . The statement, that " can have only values 0 or � takes place also in a current carrying state, but for coefficient � % 0 the criterion for " equals 0 or � depends now on the value of the current (see below). If the interband interaction is ignored, the Eqs. (4) are decoupled into two ordinary GL equations with two dif- ferent critical temperatures Tc1 and Tc2 . In general, inde- pendently of the sign of �, the superconducting phase tran- sition results at a well-defined temperature exceeding both Tc1 and Tc2 , which is determined from the equation: �1 2 2( ) ( )T Tc c � . (7) Let the first order parameter is stronger then second one, i.e., T Tc c1 2 # . Following [24] we represent tempera- ture dependent coefficients as 1 1 11( ) ( ),T a T/Tc� 2 20 2 11( ) ( ).T a a T/Tc� (8) Phenomenological constants a a1 2 20, , and� �1 2, , can be re- lated to microscopic parameters in two-band BCS model. From (7) and (8) we obtain for critical temperature Tc : T T a a a a a a c c� � � � �� � � �� � � � � �� � � � �� 1 20 2 2 2 1 2 20 2 1 2 2 � . (9) For arbitrary value of the interband coupling � Eq. (6) can be solved numerically. For � � 0 , T Tc c� 1 and for tempe- Coherent current states in a two-band superconductor Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 539 rature close to Tc (hence for T T Tc c2 $ & ) equilibrium values of the order parameters are � 2 0 0 ( ) ( )T � , � � 1 0 1 11 ( ) ( ) ( )T a T/T /c� . Considering in the following weak interband coupling, we have from Eqs. (6)–(9) cor- rections ~ � 2 to these values: � � � �1 0 2 1 1 2 1 20 2 1 1 1 ( ) ( ) ( ) T a T T a a T T T Tc c c � � � �� � � �� � 1 20a � � � � � � � � � � � � , � � � 2 0 2 1 1 2 20 2 2 1 1 ( ) ( ) ( ( )) .T a T T a a T T c c � � � �� � � �� (10) Expanding Eq. (9) over 1 1 � � �� � � �� $$ T Tc we have conven- tional temperature dependence of equilibrium order pa- rameters in weak interband coupling limit � � � 1 0 1 1 20 2 20 2 1 21 1 2 1 ( ) ( ) ,T a a a a a T Tc � � �� � � � � � � � � � � 2 0 1 1 20 1 ( ) ( )T a a T Tc � . (11) Considered above case (Eqs. (9)–(11)) corresponds to different critical temperatures T Tc c1 2 # in the absence of interband coupling �. Order parameter in the second band � 2 0( ) arises from the «proximity effect» of stronger � 1 0( ) and is proportional to value of � (11). Consider now an- other situation. Suppose for simplicity that two conden- sates in current zero state are identical. In this case for ar- bitrary value of � we have � � �'1 2 1 21( ) ( ) ( ) ,T T T a T Tc � ( � � � �� � � �� � ( (12) � � � �1 0 2 0( ) ( ) .� � (13) 3. Homogeneous current states and Ginzburg–Landau depairing current In this Section we will consider the homogeneous current states in thin wire or film with transverse dimension d T T$$ ) *1 2 1 2, ,( ), ( ) (see Fig. 1), where )1 2, ( )T and*1 2, ( )T are coherence lengths and London penetration depths for each order parameter correspondingly without interband in- teraction. This condition leads to one-dimensional problem and permits us to neglect self-magnetic field of the system. The current density j and modules of the order parame- ters do not depend on the longitudinal direction x. Writing � 1 2, ( )x as � � !1 2 1 2 1 2, , ,exp( ( ))� i x and introducing the difference and weighted sum phases: " ! ! + ! ! � � � � � 1 2 1 1 2 2 , ,c c for the free energy density (1)–(3) obtain F m m � � � � � � � � � � � � � � � ��1 1 2 2 2 2 1 1 4 2 2 4 2 1 2 1 2 2 2 1 1 2 1 2 2 2 2� � " + 2 2 cos ) � � � � � � � � � � � � � � � d dx � � � � � � � � � � � � 2 2 2 1 2 1 1 2 2 2 2 1 2 1 2 2 2 2c m c m c c d dx � � � � � " " cos � � � � � 2 1 22�� � "cos , (14) where c m m m 1 1 2 1 1 2 1 2 1 2 2 2 1 2 2 4 � � � � � �� � " � � �� � " cos cos , c m m m 2 2 2 2 1 2 1 2 1 2 2 2 1 2 2 4 � � � � � �� � " � � �� � " cos cos . (15) The current density j in terms of phases + and " has the following form j e m m d dx � � � � � � � � � � � 2 4 1 2 1 2 2 2 1 2� � � �� � " + cos (16) and includes the partial inputs j1 2, and proportional to � the drag current j12. In contrast to the case of single order parameter [32], the condition div j � 0 does not fix the constancy of 540 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 Y.S. Yerin and A.N. Omelyanchouk d x j Fig. 1. Geometry of the system. superfluid velocity. In appendix we present the Eu- ler–Lagrange equations for +( )x and "( )x . They are com- plicated coupled nonlinear equations, which generally permit the soliton like solutions (in the case � � 0 they were considered in [33]). The possibility of states with inhomogeneous phase "( )x is needed in separate investi- gation. Here, we restrict our consideration by the homoge- neous phase difference between order parameters " � const. For " � const from Eqs. (A.4) (see Appendix) follows that +( )x qx� (q is total superfluid momentum) and sin" � 0, i.e.," equals 0 or �. Minimization of free en- ergy for " gives cos ( )" � �� sign � 2 2q . (17) Note, that now the value of ", in principle, depends on q, thus, on current density j. Finally, the Eqs. (14), (16) take the form: F m q m � � � � � � � � � � � � � � �1 1 2 1 1 4 2 1 1 2 2 2 2 2 2 2 4 2 2 2 21 2 2 1 2 2 � � q q q2 2 2 1 2 2 22 ( ) ( )� � � � � �� �sign , (18) j e m m q q� � � � � � � � � � � 2 4 1 2 1 2 2 2 1 2 2 2 � � � � �� � � �sign( ) . (19) We will parameterize the current states by the value of superfluid momentum q, which for given value of j is de- termined by Eq. (19). The dependence of the order pa- rameter modules on q determines by GL equations: � � � �1 1 1 1 3 2 1 1 2 2 � � � m q �� � � � �2 2 2 2 2 0( ) ( )� �q qsign , (20) � � � �2 2 2 2 3 2 2 2 2 2 � � � m q �� � � � �1 2 2 2 2 0( ) ( )� �q qsign . (21) At the beginning we consider the case of small values of interband coupling � and dragging coefficient �. In the same manner as for q � 0 (Sec. 2) instead expression (11), for � 1 ( )q and � 2 ( )q we obtain: � � � � �1 2 1 1 2 1 1 2 2 20 1 1 2 ( ) ( ) q a T T m q a T T c c � � � � � ( ) ( ( ) ) � � � � � 2 2 2 1 20 2 2 2 21 2 q a a T T m q c , (22) � � �1 2 1 1 2 1 1 2 1 2 ( ) ( ) q a T T m qc� � � � � � � � � � � � � � � � � ( ) ( ( ) ) � � � � � 2 2 2 1 20 2 2 2 21 2 q a a T T m q c . (23) The system of Eqs. (19), (22), (23) describes the depairing curve j q T( , ) and the dependences � 1 and � 2 on the current j and the temperature T. It can be solved nu- merically for given superconductor with concrete values of phenomenological parameters. In order to study the specific effects produced by interband coupling and dragging consider now the model case when order parameters coincide at j � 0 (Eqs. (12), (13)) but gradient terms in Eq. (4) are different. Eqs. (19)–(21) in this case take the form f f f q f q q1 1 2 1 2 2 2 21 1 0( ( ~) ) (~ ~ ) (~ ~ ) � � �� � � � �sign , (24) f f kf q f q q2 2 2 2 2 1 2 21 1 0( ( ~) ) (~ ~ ) (~ ~ ) � � �� � � � �sign , (25) j f q kf q f f q q� � � 1 2 2 2 1 2 22~ (~ ~ )� � �sign . (26) Here we normalize � 1 2, on the value of the order parame- ters at j � 0 (13), j is measured in units of 2 2 1 e m � � � , q is measured in units of � 2 12m , ~� � � , ~� �� 2 1m , k m m � 1 2 . If k �1 order parameters coin- cides also in current-carrying state f f f1 2� � and from Eqs. (24)–(26) we have the expressions f q q q 2 2 21 1 ( ) ~ ~ ~� � � � � , (27) j q f q q( ) ( ~ (~ ~ ))� 2 12 2� � �sign , (28) which for ~ ~� �� � 0 are conventional dependences for one-band superconductor [32] (see Fig. 2). Coherent current states in a two-band superconductor Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 541 For k % 1depairing curve j q( ) can contain two increas- ing with q stable branches, which corresponds to possibil- ity of bistable state. In Fig. 3 the numerically calculated from Eqs. (24)–(26) the curve j q( ) and dependences f j f j1 2( ), ( ) are shown for k � 5 and ~ ~� �� � 0. The interband scattering (~� � 0) smears the second peak in j q( ), see Fig. 4 If dragging effect (~� � 0) is taking into account the depairing curve j q( ) can contain the jump at definite value of q (for k �1 see Eq. (28)), see Fig. 5. This jump corresponds to the switching of relative phase difference from 0 to �. 4. Josephson effect in two-band superconducting microconstriction In the previous section GL theory of two-band super- conductors was applied for filament’s length L , -. Op- posite case of the strongly inhomogeneous current state is the Josephson microbridge geometry, which we model as narrow channel connecting two massive superconductors (banks). The length L and the diameter d of the channel (see Fig. 6) are assumed to be small as compared to the or- der parameters coherence lengths ) )1 2, . For d L$$ we can solve one-dimensional GL equations (4) inside the channel with the rigid boundary conditions for order parameters at the ends of the channel [34]. In the case L $$ ) )1 2, we can neglect in Eqs. (4) all terms except the gradient ones and solve equations: 542 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 Y.S. Yerin and A.N. Omelyanchouk 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 j q 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1.0 j f , f1 2 a b Fig. 2. Depairing current curve (a) and dependence of the order parameter modules vs current (b) for coincident order parameters. 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 q 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 j j f , f1 2 a b f2 f1 j 0 Fig. 3. Dependence of the current j on superfluid momentum q. For value of the current j = j0 stable states (�) and unstable states (�) are shown (a). Dependences of the order parameters on current j, k = 5 and ~ ~� �� � 0 ( b). 0.2 0.4 0.6 0.8 1.0 0 0.4 0.8 1.2 1.6 q j Fig. 4. Depairing current curves for different values of the interband interaction: ~� � 0 (solid line), ~ .� � 01(dotted line) and ~� �1 (dashed line). Ratio of the effective masses equals k = 5, ~� � 0. d dx d dx 2 1 2 2 2 2 0 0 � � � � � � � � � � , , (29) with the boundary conditions: � � !1 01 10( ) exp( )� i , � � !2 02 30( ) exp( )� i , � � !1 01 2( ) exp( )L i� , � � !2 02 4( ) exp( )L i� . (30) Calculating the current density j in the channel we obtain: j j j j� � �1 2 12, (31) j e Lm 1 1 01 2 2 1 2 � � � ! !sin( ), (32) j e Lm 2 2 02 2 4 3 2 � � � ! !sin( ), (33) j e L 12 01 02 2 3 4 1 4 � � � �� � ! ! ! !(sin( ) sin( )). (34) Let ! ! !2 1 � . The difference between two order param- eter phases in the banks equals 0 or �, depending on the sign of the constant interband interaction �. Therefore, if � # 0 ! !3 1� and ! !4 2� and if � $ 0 then ! ! �3 1 � , ! ! �4 2 � . Thus the current-phase relation j( )! in gen- eral case of arbitrary values of phenomenological con- stants � and � for two-band superconducting microbridge has the form: j j e L m m ( � � � � � � � � � � �0 01 2 1 02 2 2 01 02 2 4sin ( )! � � �� � � � sign sin!. (35) The value of j 0 in (35) can be both positive and negative: j m m 0 1 01 02 2 02 01 0 1 4 1 4 # # � � � �� � � ��if ( )� � � � � � sign , (36), j m m 0 1 01 02 2 02 01 0 1 4 1 4 $ $ � � � �� � � ��if ( )� � � � � � sign . (37) When the condition (37) for set of parameters for two-band superconductor is satisfied the microbridge be- haves as the so-called �-junction (see Ref. 35). 5. Conclusions We have investigated the current carrying states in two-band superconductors within phenomenological Ginzburg–Landau theory. Two limiting situations were considered, homogeneous current state in long film or channel and Josephson effect in short superconducting microconstriction. We used the GL functional for two or- der parameters which includes the interband coupling (proximity effect) and the effect of dragging in current state of two-band system. For the case of two order param- eters the question arises about the phase difference " ! !� 1 2 between � � ! 1 1 1� e i and � � ! 2 2 2� e i . In homogeneous no-current state the value of" equals to 0 or � depending on the sign of interband coupling constant � [36]. The statement, that " can have only values 0 or � takes place also in a current carrying state, but for non- zero drag coefficient � the criterion for " equals 0 or � de- pends now on the value of the superfluid momentum q, namely cos ( )" � �� sign � 2 2q . The system of coupled GL equations is analyzed for different values of pheno- menological parameters. The depairing current expres- sion contains the term cos" and, in general, depending on parameters � and � the increasing of momentum q can switch the value of" from 0 to �. In current driven regime it leads to existence of two growing branches of j(q), which both are stable. This bistability is intrinsic property of two-band superconductor. It is interesting to study the effects of relative phase switching in magnetic flux driven Coherent current states in a two-band superconductor Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 543 q 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1,0 j 1.2 Fig. 5. Depairing current curves for different values of the ef- fective masses ratio k �1 (solid line), k �15. (dotted line) and k � 2 (dashed line). Interband interaction coefficient ~ .� � 01 and drag effect coefficient ~ .� � 05. ( ) ( ) 1 01 1 ø 0 ø exp i÷= ( ) ( ) 2 02 3 ø 0 ø exp ÷i= ( ) ( ) 1 01 2 ø L ø exp ÷i= ( ) ( ) 2 02 4 ø L ø exp ÷i= d L >> d Fig. 6. Geometry of S–C–S contact as narrow superconducting channel in contact with bulk two-band superconductors. The values of the order parameters at the banks are indicated. regime in multivalued geometry. The Josephson cur- rent-phase relation for two band superconducting weak link j(!) also contains the difference of order parameters phases " in the banks, j j� 0( ) sin" !. The value of j0 may be as positive as negative. In the last case we have what is called the �-junction, again due to intrinsic properties of two-band superconductivity. In Sec. 2 we restrict our con- sideration by the homogeneous phase difference between two order parameters ". The general Eqs. (A4) permit the possibility of inhomogeneous, soliton-like distributions "( )x , which will be subject of separate publication. The authors would like to acknowledge S.V. Kuple- vakhsky for useful discussions. APPENDIX: Free energy transformation Instead of the phases !1 and ! 2 introduce new vari- ables " and +: ! ! " ! ! + 1 2 1 1 2 2 � � � � � , ,c c (A.1) where coefficients c1 and c2 are chosen as c m m m 1 1 2 1 1 2 1 2 1 2 2 2 1 2 2 4 � � � � � �� � " � � �� � " cos cos , c m m m 2 2 2 2 1 2 1 2 1 2 2 2 1 2 2 4 � � � � � �� � " � � �� � " cos cos . (A.2) Expression for the free energy density in new variables takes a quadratic form on derivatives of + and ": F A B d dx C d dx D� � � � � � � � � � � � � � � + " " 2 2 cos . 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