Effect of impurities on superconductors with helical ordering of localized spins L. N. Bulaevskiland S. V. Panyukov P. N. Lebedeu Physics Institute, USSR Academy of Sciences (Submitted23 March 1981, resubmitted 3 July 1981) Zh. Eksp. Teor. Fiz. 82, 192-202 (January 1982) Cooper pairing of electrons in a crystal with regularly distributed localized spins is considered. The case is studied in which exchange interaction between the electrons and localized spins leads to helical magnetic ordering of the localized spins in the superconducting state, whereas in the absence of superconductivity ferromagnetic order would occur. It is shown that scattering of conducting electrons by nonmagnetic impurities narrows the region of existence of the superconducting phase with spin ordering (HS phase).The narrowing, however, is not very pronounced even in dirty crystals. Thus the requirement of the purity of the crystals in which the H S phase may be observed is not very rigid. PACS numbers: 74.30.Ci, 75.10.Jm,75.30.Et 1. INTRODUCTION In connection with experimental work on the compound ErRh,B,, Kulic, Rusinov and one of the present authors1 considered a system of regularly distributed localized spins and conduction electrons with an exchange inter- action of the spins and electrons, and a Cooper pairing of the conducting electrons. It was assumed that in the absence of superconductive pairing the indirect ex- change interaction of the localized spins via the conduc- tion electrons (RKKY interaction) would lead to ferro- magnetic ordering below a Curie temperature 8 << T,,, where T,, is the critical temperature for superconduc- tive pairing of the electrons in the absence of interac- tion between spins and electrons. To describe the sys- tem Kulic et al.' used the BCS model and the self-con- sistent field approximation for the magnetic impurities. It was also assumed that the electron energy spectrum was isotropic, and that there was no magnetic aniso- tropy or electron scattering by impurities. Anderson and Suh12 showed that in the system consid- ered ordering of the spins in the superconducting phase appears at a temperature TM - 8 in the form of inhomo- geneous magnetic ordering with a characteristic wave vector Q,~ -(k2,5,1)1'3, where 5, is the superconducting correlation length. In previous work1 the region of ex- istence of a superconducting phase I th a helicoidal type of ordering of the localized spins was found. There it was shown that for the HS [Helicoidal-spin] phase persists down to zero temperature, while for systems with 8> O,, lowering of the temperature leads to a first-order transition from the HS phase to a nonsuperconducting phase with ferro- magnetic ordering of the spins (F phase). Since Ref. 1 used the simple self-consistent field approximation, which neglects the scattering of the electrons by spin excitations, the results obtained there are valid only for a system with 8 << T, and at temperatures T which are not close to the critical magnetic point T, Thus, Ref. 1 found the region of existence of the HS-phase in an isotropic system without impurities in the region of the variables (T, 8) where T<< 8 << T, and (&,)lh >> T,,. Both impurity scattering and the effects of magnetic anisotropy have a substantial effect on the HS-phase and lead to a narrowing of its region of existence. In the present paper we investigate helicoidal ordering of the spins in a superconductor in the presence of nonmag- netic impurities and find the region of existence of the HS phase and the character of the quasiparticle spec- trum in this phase as a function of the electron mean free path. 2. BASIC EQUATIONS FOR A SUPERCONDUCTOR WITH A HELICOIDAL EXCHANGE FIELD IN THE PRESENCE OF IMPURITIES The Hamiltonian of a system of electrons in the pres- ence of an exchange interaction with localized spins, Cooper pairing, and scattering by impurities has the form %=%Bcs+ jd3r$=+(r) [E r(r-rl)aaB] *(I) d , r) V(r-ra)qIu(r), +C J as*. ( h'VE aBC.= d3r[1D.*(I) (- x) $=(I) +A (r) Q+(r) (ia.)~qI~+(r) +A'(r)$=(r) (b)=t.$~(r) 1, where j(r - ri) is the exchange interaction integral of the conduction electrons with spins St localized on lat- tice sites with coordinates ri, oi are the Pauli matrices and V(r - ri) is the potential of an impurity situated at the point r,. The localized spins are assumed to be ordered in a helicoidal structure: (S.,)=So cos Qr, <S,,)=Sosin Qrl, (Szr>=O, (2) where the parameter a(0 Qa GI) characterizes the heli- coid amplitude and the xy plane is the easy plane of the magnetic anisotropy. Of all the Fourier components in (2), as was shown previously,' we need to keep only the field with wave vector Q: h (I) = (h cos Qr, h sin Qr, 0) , h=Z(O) Sna, where I(0) is the Fourier component of the function q r ) with zero wave vector and n is the concentration of 115 Sov. Phys. JETP 55(1), Jan. 1982 0038-5646/82/010115-06$04.00 O 1982 American Institute of Physics 115