Pis'ma v ZhETF, vol.91, iss.8, pp.451-456
© 2010 April 25
Order parameter sign-reversal near s±-superconductor surface
A. M. Bobkov, I. V. Bobkova1*!
Institute of Solid State Physics, 142432 Chernogolovka, Moscow reg., Russia
Submitted 1 February 2010 Resubmitted 12 March 2010
The superconducting order parameter and LDOS spectra near an impenetrable surface are studied on the basis of selfconsistent calculations for a two band superconductor with nodeless extended s-wave order parameter symmetry, as possibly realized in Fe-based high-temperature superconductors. It is found that for a wide range of parameters the spatial behavior of the order parameter at a surface is not reduced to a trivial suppression. If the interband scattering at a surface is of the order of the intraband one or dominates it, it can be energetically favorable to change the symmetry of the superconducting state near the surface from s± to conventional s-wave. The range of existing of this surface conventional superconductivity is very sensitive to the relative values of interband and intraband pairing potentials. It is shown that the LDOS spectra near the surface can qualitatively differ upon calculating with and without taking into account the selfconsistency of the order parameter.
The discovery of a new family of iron-based high-temperature superconductors with distinct multi-orbital band structure [1 - 3] has renewed interest to the problem of multi-band superconductivity, firstly discussed fifty years ago [4, 5]. It was proposed theoretically [6, 7] that the Fe-based superconductors represent the first example of multigap superconductivity with a phase difference between the superconducting condensates belonging to different bands. This state was discussed previously [8, 9], but not yet observed in nature. In the most simple case there is the phase difference ir between the superconducting condensates arising on the hole Fermi surfaces around F point and the electron Fermi surfaces around M point. This so-called s± (or extended s-wave) state has been favored by a variety of models within random phase approximation (RPA) [7, 10, 11] and renor-malization group techniques [12-14]. Currently the s±-state is viewed to be the most plausible candidate for the role of the superconducting order parameter in these compounds.
Surface and interface phenomena in s-t-superconductors have attracted considerable recent attention. The formation of bound states at a free surface of an s±-superconductor [15-18], at an S±/N interface [19-22], an N/S/S± junction [23] and at Josephson junctions including s±-superconductors [24, 22] was investigated theoretically. In particular, the finite energy subgap bound states (depending on the interface parameters) were found and their influence on the conductance spectra and Josephson current was investigated.
However almost all these calculations (except for a few numerical results [19]) assume non-selfconsistent su-
1) e-mail: bobkova0issp.ac.ru
perconducting order parameter (OP). In the present paper we focus on the study of the OP at a surface of s±-superconductor. We have found that for a wide range of parameters the spatial behavior of the OP at a surface can not be reduced to a trivial suppression. If the interband scattering at a surface R\2 is of the order of the intraband one Rq or dominates it, it can be energetically favorable to change the symmetry of the superconducting state near the surface from s± to conventional s-wave. The range of existing of this surface conventional superconductivity is very sensitive to the relative values of interband and intraband pairing potentials. We demonstrate that the selfconsistent OP behavior affects the surface local density of states (LDOS) profiles, and, consequently, should be taking into account when interpreting experimental results. It is worth to note here that, while there is a wide parameter range of existing complex OP at the surface region [25], in this paper we only discuss the case when the surface OP is of conventional s-wave type.
We consider an impenetrable surface of a clean two-band superconductor. The OP is assumed to be of s±-symmetry in the bulk of the superconductor, that is the phase difference between the OP's in the two bands (called 1 and 2) is n. It is supposed that an incoming quasiparticle from band 1,2 can be scattered by the surface as into the same band (intraband scattering), so as into the other band (interband scattering).
We make use of the quasiclassical theory of superconductivity, where all the relevant physical information is contained in the quasiclassical Green function gi(e,Pf,x) for a given quasiparticle trajectory. Here e is the quasiparticle energy measured from the chemical potential, p/ is the momentum on the Fermi surface
(that can have several branches), corresponding to the considered trajectory, x is the spatial coordinate along the normal to the surface and í = 1,2 is the band index. Quasiclassical Green function is a 2 x 2 matrix in particle-hole space, that is denoted by the symbol The equation of motion for ¿¿(e, p/, a;) is the Eilenberger equation subject to the normalization condition [26, 27]. For superconductivity of .? . -type, when the pairing of electrons from different bands is absent, the Eilenberger equations corresponding to the bands 1 and 2 are independent. The trajectories belonging to the different bands can only be entangled by the surface, which enters the quasiclassical theory in the form of effective boundary conditions connecting the incident and outgoing trajectories.
However, owing to the normalization condition for the quasiclassical propagator, the boundary conditions for the quasiclassical Green functions are formulated as non-linear equations [28-30]. Furthermore, they contain unphysical, spurious solutions, so their practical use is limited. For this reason in the present work we make use of the quasiclassical formalism in term of so-called Riccati amplitudes [31, 32], that allows an explicit formulation of boundary conditions [32-36]. The retarded Green function <)»(£, p/, a;), which is enough for a complete description of an equilibrium system, can be parametrized via two Riccati amplitudes (coherence functions) 7¿(e, p/, a;) and 7¿(e, p/, a;) (in the present paper we follow the notations of Refs. [32, 36]). The coherence functions obey the Riccati-type transport equations. In the considered here case of two-band clean «¿-superconductor the equations for the two bands are independent and read as follows
ivixdx-yi + 2e7 i = ^A*7 ?
îi(e,Pf,x) = 7,*(—£, —pf, x).
(1)
(2)
Here ViX is the normal to the surface Fermi velocity component for the quasiparticle belonging to band i. A* stands for the OP in the i-th band, which should be found self-consistently.
Let us suppose that the surface is located at x = 0 and the superconductor occupies the halfspace x > 0. For the sake of simplicity we assume that the surface is atomically clean and, consequently, conserves parallel momentum component. Then there are four quasiparticle trajectories, which are involved in each surface scattering event. These are two incoming trajectories belonging to the bands 1,2 (with i>ix < 0) and two outgoing ones (with ViX >0). It can be shown [32, 36] that the coherence function 7»(£, p/,a;), corresponding
to the incoming trajectory can be unambiguously calculated making use of Eq. (1) up to the surface starting from its asymptotic value in the bulk
7? =
Afsg:
ne
Aj
(3)
where Af is the bulk value of the OP in the appropriate band, 5 > 0 is an infinitesimal. As for the coherence function 7i(e, pf, x), it is determined unambiguously by the asymptotic conditions for the outgoing trajectories and can be obtained according to Eqs. (1),(2).
Otherwise, the coherence functions 7»(£, p/,a;) for the outgoing trajectories and, correspondingly, 7i(e,p/,a;) for the incoming ones should be calculated from Eq. (1) supplemented by the boundary conditions at the surface and Eq. (2). The surface is described by the normal state scattering matrix for particle-like excitations, denoted by S and for hole-like excitations, denoted by S, that connect outgoing with incoming quasiparticles. The scattering matrix S have elements Skip,, which connect outgoing quasiparticles from band i with momentum k, to the incoming ones belonging to band j with momentum pHere and below all the momenta corresponding to the incoming trajectories are denoted by letter p and all the momenta for the outgoing quasiparticles are denoted by k. For the model we consider S is a 2 x 2-matrix (for the particular value of the momentum parallel to the surface) in the trajectory space. It obeys the unitary condition SSt = 1 and without loss of generality can be parameterized by three quantities JZ12, 0 and a as follows
•^kipi Sk2pi
5ÍciP2 $k2p2
(4)
where J2o and R12 are coefficients of intraband and interband reflection, respectively. They obey the constraint Ro + R12 = 1- The phase factors a = ±1 and 0 appear to be unimportant for further consideration. While in general the scattering matrix elements are functions of the momentum parallel to the surface p||, we disregard this dependence in order to simplify the analysis. The scattering matrix S for hole-like excitations are connected to S by the relation S(p||) = Str(^p||). In the absence of spin-orbit interaction the S-matrix elements are only functions of |p|| |, that is in the case we consider S = S.
From the general boundary conditions [36], which are also valid for a multiband system, one can obtain the explicit values of the coherence functions 7i(e,k, x = 0)
and ji(e,p,x = 0) via the scattering matrix elements and the values of the coherence functions 7,-(e, p, x = 0) and 7,-(e,k, x = 0) at the surface. They read as follows
7lk = -RoTlp + -Rl272p
7lp = Д071к + -Rl272k
Д0Д1272к(71р - 72p)2 1 + 72k (-Rl27lp + Ло72р) '
(5)
Д0Д1272р(71к - 72k)2
+ 72p (-Rl27lk + До72к) '
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