Spin waves and high-frequency response in layered superconductors with helical magnetic structure
SSpin waves and high-frequency response in layered superconductors with helicalmagnetic structure
A. E. Koshelev
Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 (Dated: February 25, 2021)We evaluate the spin-wave spectrum and dynamic susceptibility in a layered superconductors withhelical interlayer magnetic structure. We especially focus on the structure in which the momentsrotate 90 ◦ from layer to layer realized in the iron pnictide RbEuFe As . The spin-wave spectrum insuperconductors is strongly renormalized due to the long-range electromagnetic interactions betweenthe oscillating magnetic moments. This leads to strong enhancement of the frequency of the modecoupled with uniform field and this enhancement exists only within a narrow range of the c-axiswave vectors of the order of the inverse London penetration depth. The key feature of materialslike RbEuFe As is that this uniform mode corresponds to the maximum frequency of the spin-wave spectrum with respect to c-axis wave vector. As a consequence, the high-frequency surfaceresistance acquires a very distinct asymmetric feature spreading between the bare and renormalizedfrequencies. We also consider excitation of spin waves with Josephson effect in a tunneling contactbetween helical-magnetic and conventional superconductors and study the interplay between thespin-wave features and geometrical cavity resonances in the current-voltage characteristics. I. INTRODUCTION
Experimental realization, characterization, and under-standing quantum materials have emerged as central top-ics in the modern physics research. Quantum materialshave the potential to offer new functionalities enablingnovel applications and therefore provide a fundamen-tal basis for future technological advances. Supercon-ductors supporting long-range magnetic order representa rare class of quantum materials with unique proper-ties caused the interplay between magnetic and super-conducting subsystems[1–4]. As singlet superconductiv-ity and ferromagnetism are strongly incompatible states,the ground-state configurations are always characterizedby nonuniform structures of either magnetic moments orsuperconducting gap parameter. The nature of nonuni-form configurations ultimately determines transport andthermodynamic properties of these materials. In thecase of strong superconductivity and soft magnetism, itwas theoretically predicted that the exchange interactionbetween two subsystems favors a nonuniform magneticstate either in the form of small-size domains [5] or heli-cal structure [6] for strong and weak magnetic anisotropy,respectively.Several classes of magnetic singlet superconductorshave been discovered and thoroughly characterized.The magnetism in these materials is hosted in rare-earth-elements sublattice spatially separated from theconduction-electrons sublattice. In spite of high densityof rare-earth ( RE ) local moments, the superconductivitysurvives because the exchange interaction between twosublattices is relatively weak. Various nonuniform mag-netic structures have been revealed in the coexistenceregions.The first two groups of magnetic superconductors dis-covered half century ago are ternary molybdenum chalco-genides (Chevrel phases), such as HoMo S with su-perconducting transition at T c ≈ . B with T c ≈ . RE Ni B C, seereviews[12–14]. In contrast to the nearly cubic ternarycompounds, these are layered materials composed ofmagnetic RE C layers and conducting Ni layers. Thesuperconductivity coexists with different kinds of mag-netic order in four compounds with RE → Tm, Er, Ho,and Dy. The magnetic moments typically order ferro-magnetically within RE C layers and alternate from layerto layer (A-type antiferromagnets). This basic configu-ration, however, is perturbed in some compounds. Par-ticularly interesting are Er and Ho compounds wherethe magnetic transition takes place inside the supercon-ducting state at temperatures comparable with the su-perconducting transition temperature (10.5 K and 8 K,for Er and Ho, respectively). Magnetic structure in theErNi B C is characterized by additional in-plane modu-lation, which is probably caused by interaction with thesuperconducting sublattice. In addition, a peculiar weakferromagnetic state appears below 2.3 K, and, contraryto HoMo S and ErRh B , it coexists with superconduc-tivity at lower temperatures. In the Ho compound themagnetic phase diagram is also very rich: the transitionto the low-temperature A-type antiferromagnetic stateoccurs via two intermediate incommensurate spiral con-figurations, one with helix direction along the c axis andanother with additional in-plane modulation.Contrary to singlet Cooper pairing, a rare triplet su- a r X i v : . [ c ond - m a t . s up r- c on ] F e b perconducting state may coexist with uniform ferromag-netism. Such triplet state is realized in uranium-basedcompounds UGe , URhGe, and UCoGe, which becomesuperconducting at sub-Kelvin temperature range, insidethe ferromagnetic state, see reviews [15–17]. In spite oflow transition temperatures, due to triplet pairing, thesuperconducting state survives up to remarkably highmagnetic field, 10-25 teslas. The triplet state is also likelyrealized in the recently discovered compound UTe , eventhough this material is not magnetic [18].Interest to the physics of magnetic singlet supercon-ductors has been recently reinvigorated by the discoveryof the magnetically-ordered iron pnictides, in particular,europium-based 122 compounds, see review [19]. Thelayered structure of these materials is similar to borocar-bides: they are composed of the magnetic Eu and con-ducting FeAs layers. The parent material EuFe As isa nonsuperconducting compensated multiple-band metalwhich has the spin-density-wave transition in the FeAslayers at 189 K and the A-type antiferromagnetic tran-sition in the Eu layers at 19 K with the magnetic mo-ments aligned along the layers [20–22]. Superconductingstate emerges under pressure with the maximum transi-tion temperature reaching 30 K at 2.6 GPa exceeding themagnetic transition temperature in the Eu sublattice[23].Superconducting compounds with Eu magnetic orderalso have been obtained by numerous chemical substi-tutions on different atomic sites of the parent compoundincluding isovalent substitutions of P on As site[24–29]and of Ru on Fe site[30], electron doping via substitutionsof Co [31] or Ir[32] on Fe site, and hole-doping via sub-stitutions of K[33] or Na[34] on Eu site. The maximumsuperconducting transition temperature for different sub-stitution series ranges from 22 to 35 K exceeding themagnetic-transition temperature in Eu layers. Thereforethe key feature of these materials is that they have mag-netic transition in the Eu sublattice at the temperaturescale, comparable with the superconducting transition inFeAs sublattice. The most studied substituted supercon-ductor in this family is EuFe(As − x P x ) . The supercon-ducting transition temperature reaches maximum of 26K for x ≈ .
3. followed by the ferromagnetic transition at19 K. Contrary to the parent compound, the Eu momentsalign ferromagnetically along the c axis at 19 K[29]. Atlower temperatures, coexistence of ferromagnetism withsuperconductivity leads to the formation of the com-posite domain and vortex-antivortex structure visualizedby the decorations [35] and magnetic-force microscopy[36]. This structure has been explained assuming purelyelectromagnetic coupling between the magnetic momentsand superconducting order parameter [37].The recent addition to the family of Eu-basediron-pnictides is the stoichiometric 1144 compounds A EuFe As with A =Rb [38–42] and Cs [39, 43] in whichevery second layer of Eu in the parent material is replacedwith the layer of nonmagnetic Rb or Cs. These materialshave the superconducting transition temperature of 36.5K, higher than the doped 122 Eu compounds. Such high transition temperature is achieved because of close-to-optimal hole concentration and the absence of disordercaused by random substitutions. On the other hand, themagnetic transition temperature 15 K is 4 K lower thanin the parent 122 compound, probably due to the weakerinteraction between the magnetic layers. These materi-als are characterized by highly-anisotropic easy-axis Eumagnetism [41, 44, 45]. With increasing pressure thesuperconducting transition temperature decreases andthe magnetic transition temperature increases so that atpressures larger that ∼ ◦ from layerto layer [48, 49].New materials frequently host new physical phenom-ena. In this paper we investigate spin waves and re-lated properties for layered superconductors with helicalmagnetic structure with the modulation perpendicularto the layer direction. Spin waves is the most impor-tant dynamic characteristic of magnetic materials [50–52] and their properties are essential for the emergingspintronics[53, 54] and magnonics[55–57] applications.As the ground-state configuration, the spin-wave spec-trum is determined by the exchange and electromag-netic interactions between the moments and by magneticanisotropy. A key feature of superconducting materialsis that the long-wave part of the spin-wave spectrum isrenormalized in a nontrivial way by long-range electro-magnetic interactions between the oscillating magneticmoments. In the case of a ferromagnetic triplet super-conductor with uniform magnetization, spectrum of spinwaves, their excitation by the external electromagneticwaves, and related features in the surface impedancehave been considered in Refs. [58, 59]. Here we ex-tend this consideration to superconductors with helicalmagnetic structure. While some of our results are validfor a general modulation period, we mostly focus on thecase relevant for RbEuFe As , namely, the structure inwhich the moments rotate 90 ◦ from layer to layer andthe easy-plane anisotropy exceeding the interlayer ex-change interaction. We evaluate the spin-wave spectrumas a function of the c-axis wave vector and find that themode having c-axis uniform component of the oscillat-ing spins corresponds to the spectrum maximum. Thismode is strongly renormalized by the long-range elec-tromagnetic interactions, its frequency increases by thefactor of square root of the magnetic permeability withrespect to the bare value determined only by local inter-actions. This enhancement rapidly drops when the c-axiswave-vector shift exceeds the inverse London penetrationdepth. This behavior is qualitatively different from thecase of ferromagnetic alignment[58, 59], where the fre-quency of the uniform mode is the smallest frequency ofthe spectrum. We evaluate the high-frequency surfaceresistance and find that it acquires a very asymmetricfeature with sharp maximum at the bare uniform-modefrequency and a tail extending up to the renormalizedfrequency.We also investigate excitation of spin waves with ACJosephson effect in a tunneling contact between helical-magnetic and conventional superconductors and studythe interplay between spin-wave features and geometri-cal Fiske resonances in the current-voltage characteris-tics. This consideration is somewhat related to the ex-citation of the spin waves by the Josephson effect in theferromagnetic interlayer in SFS junctions [60, 61]. In ourcase, however, the spin-wave feature in current-voltagecharacteristic has a very distinct shape due to the un-usual spectrum in helical-magnetic superconductor, sim-ilar to the feature in the frequency dependence of thesurface resistance. Namely, the current is sharply en-hanced when the Josephson frequency matches the bareuniform-mode frequency and at higher frequencies thisexcess current has a long tail extending up to the renor-malized frequency.The paper is organized as follows. In Sec. II, we in-troduce the model and write general relations determin-ing the spin-wave spectrum via the dynamic magneticsusceptibility. In Sec. III, we consider the helical mag-netic ground state. The bare spin-wave spectrum due tothe short-range interactions is derived in SectionIV. InSec. V, we investigate the response to nonuniform mag-netic field and the nonlocal dynamic susceptibility arederived. Electromagnetic renormalization of spectrumis considered in Sec. VI. In Sec. VII, we consider dy-namic equation for smooth magnetization, derive mag-netic boundary condition, and evaluate the frequencydependence of surface impedance. In Sec. VIII, we in-vestigate the excitation of spin waves by the Josephsoneffect. II. MODEL AND GENERAL EQUATIONS
We consider a layered magnetic superconductor de-scribed by the energy functional E = E m + E s + (cid:90) d r (cid:18) B π − BM + 2 π M − H e B π (cid:19) , (1)where the term E s = (cid:90) d r (cid:88) i = x,y,z πλ i (cid:18) A i − Φ π ∇ i ϕ (cid:19) (2)is the kinetic energy of the superconducting subsystem inLondon approximation determined by the components ofthe penetration depth λ i and A is vector potential deter-mining the local magnetic induction, B = ∇ × A . In thefollowing, we consider Meissner state and drop the phaseof the superconducting order parameter ϕ . We assume that the magnetic subsystem is described by the classicalquasi-two-dimensional easy-plane Heisenberg model E m = −J (cid:88) (cid:104) i , j (cid:105) ,n S i ,n S j ,n + K (cid:88) i ,n (2 S z, i ,n − − (cid:88) i ,n,(cid:96)> J z,(cid:96) S i ,n S i ,n + (cid:96) , (3)where S i ,n is the spin at the site i and in the layer n with the absolute value equal S , J is the in-plane ex-change constant, K is the easy-plane anisotropy, J z,(cid:96) inthe interlayer exchange constants. The exchange con-stants likely have a substantial RKKY contribution. Thebehavior of J z,(cid:96) for (cid:96) > ξ c /d is strongly affected bysuperconductivity[6, 62], where d is the separation be-tween the magnetic layers and ξ c is the c-axis coher-ence length. Local spins determine local magnetic mo-ments m i ,n = gµ B S i ,n where µ B is the Bohr magne-ton. Therefore, the bulk magnetization M ( r ) in Eq. (1)is related with the coarse-grained spin distribution as M ( r ) = n M gµ B S ( r ), where n M is the bulk density ofspins. S ( r ) in this relation is obtained by averaging of S i ,n over distances much larger than neighboring spinseparations.Slowly-varying in space oscillating magnetization gen-erates macroscopic magnetic fields which couple with thismagnetization. This effect is especially important in su-perconductors where it leads to significant renormaliza-tion of spin-wave spectrum [58, 59]. We will assumethat the supercurrent response to the slowly oscillatingmagnetization can be treated quasistatically. The corre-sponding equation is obtained by variation of the energywith respect to the vector potential A , (cid:16) ˆ λ − − (cid:52) (cid:17) A − π ∇× M = 0 . (4)We can transform this equation into the equation con-necting the local magnetic field strength H = B − π M and magnetization H + ∇ × ˆ λ ∇ × H = − π M . (5)For time-dependent fields, this equation is modified byquasiparticle currents. We neglect this contribution as-suming that the time variations are slow. On the otherhand, the oscillating magnetic field generates oscillatingmagnetization due to dynamic magnetic response and re-lation between their Fourier components is determined bythe dynamic magnetic susceptibility ˆ χ ( k , ω ), M ( k , ω ) = ˆ χ ( k , ω ) H ( k , ω ) . (6)Note that the poles of ˆ χ ( k , ω ) give the bare spin-wavespectrum due to local interactions unrenormalized bylong-range fields. From Eqs. (5) and (6), we obtain gen-eral linear equation for H ( q, ω ) which determines thespectrum of spin waves H − k × ˆ λ k × H = − π ˆ χ ( k , ω ) H . (7)In the following, we consider a simple geometry of thewave vector oriented along z direction and isotropic in-plane case, λ x = λ y ≡ λ . In this case, Eq. (7) becomes (cid:0) λ k z (cid:1) H α ( k z , ω ) = − πχ αβ ( k z , ω ) H β ( k z , ω ) . with α = x, y . Note that the off-diagonal susceptibility χ xy ( k z , ω ) is finite in the helical magnetic state. Since χ yx ( k z , ω ) = − χ xy ( k z , ω ), we obtain the following equa-tion for the spin-wave spectrum renormalized by long-range electromagnetic interactions1+ 4 π [ χ xx ( k z , ω ) ± iχ xy ( k z , ω )]1 + λ k z = 0 . (8)The dynamic susceptibility χ αβ ( k , ω ) can be evaluatedby solving the Landau-Lifshitz equation d M dt = − γ (cid:20) M × δ E m δ M (cid:21) + γ [ M × H ] (9)in the linear order with respect to small deviations of themagnetization from the equilibrium configuration. Here γ = gµ B / (cid:126) is the gyromagnetic factor. We neglected thedamping terms. III. MAGNETIC GROUND-STATECONFIGURATIONA. Arbitrary modulation wave vector
We start with consideration of helical interlayer mag-netic ground state determined by the energy in Eq. (3).In the classical description, it is convenient to introducethe unit vectors s i ,n = (cos φ i ,n cos θ i ,n , sin φ i ,n cos θ i ,n , sin θ i ,n ) along the direction of S i ,n , S i ,n = S s i ,n . Thenwe can rewrite the energy in Eq. (3) as E m = − J (cid:88) (cid:104) i , j (cid:105) ,n s i ,n s j ,n + K (cid:88) i ,n (2 s z, i ,n − − (cid:88) i ,n,(cid:96)> J z,(cid:96) s i ,n s i ,n + (cid:96) . (10)with new parameters J = J S , K = K S , and J z,(cid:96) = J z,(cid:96) S . The advantage of the constants K , J and J z,(cid:96) isthat they immediately represent the energy scales of thecorresponding interactions. Frustrated interlayer interac-tions may lead to the helical ground state correspondingto φ (0) i ,n = qn and θ (0) i ,n = 0 [63, 64]. The energy per spinfor such a state is given by E ( q ) = − J z ( q ) / J z ( q ) = 2 ∞ (cid:88) (cid:96) =1 J z,(cid:96) cos( q(cid:96) ) (12) is the discrete Fourier transform of the interlayer inter-actions.The total energy also has electromagnetic (dipole) con-tribution, which is substantially affected by superconduc-tivity. As follows from Eq. (5), the magnetic field gener-ated by uniformly polarized layers with arbitrary in-planeorientation of magnetization, M ( z ) = (cid:80) n M n δ ( z − z n ),is given by H ( z ) = − (cid:88) n (2 π M n /λ ) exp ( −| z − z n | /λ ) , (13)where M n = dn M m n is the two-dimensional momentdensities, d is the separation between the magnetic lay-ers, and z n = nd . Note that superconducting environ-ment leads to the finite magnetic field outside a uniformlypolarized layer, contrary to normal-state case, in whichsuch field is absent. The corresponding magnetic induc-tion and vector-potential are B ( z ) = 4 π (cid:88) n M n [ δ ( z − z n ) − (1 / λ ) exp ( −| z − z n | /λ )] , A ( z ) = − π (cid:88) n n z × M n sign( z − z n ) exp ( −| z − z n | /λ ) . Substituting these distribution into superconducting andmagnetic energy terms in Eq. (1), we derive the bulkelectromagnetic energy density F em = πλL z (cid:88) n,m M n M m exp ( −| z n − z m | /λ ) . (14)Therefore, for the helical order, M x,n = M cos ( Qn ), M y,n = M sin ( Qn ), the bulk electromagnetic-energycost is F em ( Q ) = πλd M sinh ( d/λ )cosh ( d/λ ) − cos Q ≈ πM λ (1 − cos Q ) /d , (15)where M = M /d = n M m is the bulk saturation magne-tization. The corresponding energy per spin E em ( Q ) = F em ( Q ) /n M has to be compared with the exchange en-ergy in Eq. (11). In the range 2 λ (1 − cos Q ) /d (cid:29)
1, thisamounts to comparison of the typical dipole energy scale E d = πd n M m /λ with the interlayer exchange con-stants J z,(cid:96) . Typically the dipole interactions are muchweaker than exchange ones. For example, for parame-ters of RbEuFe As , d = 1 . n M = 5 · cm − , m = 7 µ B , and λ = 100nm, we estimate E d ≈ − K,while the typical magnitude of J z,(cid:96) is 0 . . Q . Equation (11) determines theminimum-energy condition at q = Q (cid:88) (cid:96) (cid:96)J z,(cid:96) sin( Q(cid:96) ) = 0 . (16)If we keep only three nearest neighbors, this equationbecomes J z, + 4 J z, cos Q + 3 J z, (cid:0) Q − (cid:1) = 0 . (17)The energy has minimum at q = Q if E (cid:48)(cid:48) ( Q ) = ∞ (cid:88) (cid:96) =1 (cid:96) J z,(cid:96) cos( Q(cid:96) ) > . (18)The last two equations determine the optimal modula-tion vector in the case of frustrating interlayer exchangeinteractions. B. Case Q = π/ In the following, we will pay a special attention to thecase of commensurate modulation with Q = π/ As . In this case, assuming J z,(cid:96) = 0 for (cid:96) > J z, = 3 J z, and J z ( q ) = 2 J z, (cid:2) cos( q ) + cos(3 q ) (cid:3) + 2 J z, cos(2 q ) . (19)The condition for minimum, Eq. (18), simply gives J z, <
0, i.e., antiferromagnetic next-neighbor interaction. Thecase Q = π/
2, however, is special, because within the sim-plest exchange model, the energy is degenerate with re-spect to relative rotation of the two sublattices composedof odd and even layers. Adding interactions with moreremote layers does not resolve this issue. The continuousdegeneracy is eliminated by the 4-fold crystal anisotropyterm, − K ( s x, i ,n + s y, i ,n ). In addition, such anisotropylocks Q = π/ J z, = 3 J z, .Such 4-fold anisotropy, however, does not completelyeliminate the ground-state degeneracy, because the he-lical state, φ (0) n = πn/
2, still has the same energy asthe double-periodic state with φ (0) n = (0 , , π, π, , , . . . ).The simplest term eliminating the latter degeneracy isthe nearest-neighbor biquadratic term J z,b ( s i ,n s i ,n +1 ) with J z,b >
0. Without the 4-fold anisotropy term, thisyields the modified energy E ( q ) = − ∞ (cid:88) (cid:96) =1 J z,(cid:96) cos( q(cid:96) ) + J z,b cos ( q )and the modified ground-state condition ∞ (cid:88) (cid:96) =1 (cid:96)J z,(cid:96) sin( Q(cid:96) ) − J z,b cos( Q ) sin( Q ) = 0 . For three nearest neighbors this gives J z, +4 J z, cos Q +3 J z, (cid:0) Q − (cid:1) − J z,b cos Q = 0 . For Q = π/ J z, = 3 J z, remains un-changed, while the condition for minimum becomes2 J z, − J z,b <
0. In the following analysis, we will as-sume the hierarchy of the energy constants J z,b , K (cid:28) J z,(cid:96) < K (cid:28) J . In this case, the degeneracy-breakingterms ∝ J z,b , K select the Q = π/ n n m m n n m m Q(m-n)
FIG. 1. Local coordinate system ( ς , ξ , η ) used for computa-tion of the spin-wave spectrum. IV. BARE SPIN-WAVE SPECTRUMA. Arbitrary modulation wave vector
In this section, we investigate a bare spectrum of spinwaves due to the local exchange interactions neglectingcoupling to macroscopic fields. We consider spin wavespropagating along the direction of helical modulation ( z axis) assuming that spin oscillations are uniform in thelayer direction. In the following derivations, we drop thein-plane index i , S i ,n → S n . A useful trick allowingfor analytical solution is to introduce a local coordinatesystem ς , ξ , η following local equilibrium spin orienta-tion [63]. We assume that the ς axis coincides with theequilibrium spin direction at each layer, the ξ axis is per-pendicular to this direction in the layer xy plane, andthe η axis is parallel to the z axis, as illustrated in Fig. 1.Then the ς , ξ axes at the layer m are rotated with re-spected to those at the layer n by an angle of Q ( m − n )corresponding to the coordinate transformation ζ m = ζ n cos [ Q ( m − n )] + ξ n sin [ Q ( m − n )] ,ξ m = − ζ n sin [ Q ( m − n )] + ξ n cos [ Q ( m − n )] . To fix the global coordinate system, we assume ( x, y ) =( ζ , ξ ) meaning that ζ n = x cos ( Qn ) + y sin ( Qn ) ,ξ n = − x sin ( Qn ) + y cos ( Qn ) . Correspondingly, the spin components in the rotated andglobal coordinates are related as S ζn = S xn cos ( Qn ) + S yn sin ( Qn ) , (20a) S ξn = − S xn sin ( Qn ) + S yn cos ( Qn ) . (20b)This and inverse transformations can also be presentedin the complex form S ζn + iδS ξn = ( S xn + iδS xn ) exp ( − iδQn ) , (21a) S xn + iδS yn = ( S ζn + iδS ξn ) exp ( iδQn ) (21b)with δ = ± S ξn = S ηn h ζn − S ζn h ηn , (22a)˙ S ηn = S ζn h ξn − S ξn h ζn , (22b)where h n = − ∂ E m /∂ S n is the local reduced magneticfield acting on spins in the layer n , which, according toEq. (3), has the components h ζn = (cid:88) m J z,n − m (cid:8) S ζm cos [ Q ( n − m )] − S ξm sin [ Q ( n − m )] (cid:9) , (23a) h ξn = (cid:88) m J z,n − m (cid:8) S ζm sin [ Q ( n − m )]+ S ξm cos [ Q ( n − m )] (cid:9) , (23b) h ηn = (cid:88) m J z,n − m S ηm − K S ηn . (23c) For small spin oscillations, the local ζ component of eachspin can be taken as a constant, S ζm → S . Substituting h jn into Eqs. (22a) and (22b), we obtain equations forlinear oscillations, S un ( t ) = S un exp( iωt ) with u = ξ, η , iωS ξn = S (cid:88) m J z,n − m { cos [ Q ( n − m )] S ηn − S ηm } + 4 S K S ηn , (24a) iωS ηn = S (cid:88) m J z,n − m cos [ Q ( n − m )] ( S ξm − S ξn ) . (24b)We can see that, in spite of the helical structure, in the ro-tating coordinates this system is uniform. Fourier trans-formation S u q = (cid:80) n S un exp ( − i q n ) yields the 2 × iωS ξ q = S [ J z ( Q ) − J z ( q ) + 4 K ] S η q , (25a) iωS η q = S (cid:20) J z ( Q + q )+ J z ( Q − q )2 −J z ( Q ) (cid:21) S ξ q , (25b)from which we obtain the spectrum ω s ( q ) = S (cid:115) [4 K + J z ( Q ) −J z ( q )] (cid:20) J z ( Q ) − J z ( Q + q )+ J z ( Q − q )2 (cid:21) (26)in terms of the reduced wave vector q . Since Q is theground-state modulation wave vector, J z ( q ) has maxi-mum at q = Q , as discussed in Sec. III. This propertyinfluences the spectrum shape near q = 0 and Q . Spinoscillations in the propagating wave have both in-planeand out-of-plane components. Substituting ω s ( q ) intoEq. (25b), we derive the relation between the spin com-ponents in the mode S η q = i (cid:115) J z ( Q ) − J z ( Q + q )+ J z ( Q − q )2 K + J z ( Q ) − J z ( q ) S ξ q . (27)From Eq. (21b), we obtain the in-plane oscillating spincomponents in real space (cid:18) S xn S yn (cid:19) = S ξ q exp ( i q n ) (cid:18) − sin ( Qn )cos ( Qn ) (cid:19) . (28)We should emphasize that, as q represents the wave vec-tor in the rotating-coordinates basis, the real-space spincomponents S x,y do not behave as exp( i q n ). In partic-ular, the mode with q = 0 corresponding to the uniformhelix rotation does not generate spin variations uniformin real space.The mode with q = Q will play a key role in the fol-lowing consideration. For this mode, the in-plane spin components (cid:18) S xn S yn (cid:19) = S ξQ (cid:18) − i + i exp (2 iQn )1 + exp (2 iQn ) (cid:19) . (29)are superposition of the uniform and 2 Q terms. Thepresence of the uniform n -independent component in the q = Q mode implies that it can be excited by the os-cillating uniform field. The frequency of the mode for q = Q is given by the geometrical average of the easy-plane anisotropy and combination of the interlayer ex-change constants, ω s ( Q ) = S (cid:112) K [2 J z ( Q ) −J z (2 Q ) −J z (0)] . (30)From Eq. (27), we also obtain the z-axis component ofthis mode S ηQ = i (cid:114) J z ( Q ) −J z (2 Q ) −J z (0)8 K S ξQ . (31)We see that it decreases with increase of the easy-planeanisotropy. B. Case Q = π/ In the case Q = π/ J z ( q ) = J z ( q ) S , we obtain the spectrum ω s ( q ) = 2 S (cid:113)(cid:8) K−J z, (cid:2) cos( q )+ cos(3 q ) (cid:3) + |J z, | [1+cos (2 q )] (cid:9) |J z, | [1 − cos (2 q )] . (32)This frequency vanishes at q = 0 and π . The q = 0 modecorresponds to uniform helix rotation. Zero frequencyat q = π is a consequence of the degeneracy with re-spect to the relative rotation of two sublattices, whichis the property of the exchange model in Eq. (3) for Q = π/
2. These degeneracies are eliminated by the ad-ditional terms considered at the end of section III: thein-plane 4-fold anisotropy and the nearest-neighbor bi-quadratic term. The former term generates spin-wavegaps at both q = 0 and π , while the latter term only gen-erates a gap at q = π . We assume that both these termsare small. We mostly focus on the mode with q = π/ q = π/
2. Expansion ofthe frequency in Eq. (32) near this wave vector yields ω s (cid:16) π q (cid:17) ≈ S (cid:113) K|J z, | (cid:20) − (cid:18) − |J z, |K (cid:19) q J z, K q (cid:21) . (33)We see that the frequency has maximum at q = π/ K > |J z, | . Moreover, one can check that in this case ω s ( π/
2) is the largest frequency in the spectrum. Wewill focus on this case because it is likely realized inRbEuFe As .
1. Transformation to magnetic unit cell and foldedBrillouin zone
For the modulation vector π/
2, the magnetic unit cellcontains four layers. Correspondingly, the folded mag-netic Brillouin zone is four times smaller than the crys-talline Brillouin zone. It is therefore useful to presentspin-wave spectrum in the folded Brillouin zone, whichbetter corresponds to a standard crystallographic de-scription. Introducing the index j numbering magneticunit cells, we present the layer index as n = 4 j + ν with ν = 1 , , ,
4. Correspondingly, the spins can be repre-sented as S j,ν = A ν ( k ) exp ( ikj ) , where k is the wave vector within the folded Bril-louin zone. Using presentation in Eq. (21b) for S ξn = S exp ( i q n ), we write S x,j,ν + iδS y,j,ν = iδS exp (cid:104) i (cid:16) q + δ π (cid:17) (4 j + ν ) (cid:105) , meaning that k = 4 (cid:0) π m + q (cid:1) and A ( m ) x,ν ( k ) + iδA ( m ) y,ν ( k ) = iδS exp (cid:20) i (cid:18) k mπ + δ π (cid:19) ν (cid:21) . The integer m should be selected to reduce k to the range[ − π, π ]. This means that the four modes within suchfolded Brillouin zone correspond to the frequencies ω ( k ) = ω s ( k/ , (34a) ω ( k ) = ω s ( π/ k/ , (34b) ω ( k ) = ω s ( − π/ k/ , (34c) ω ( k ) = ω s ( π + k/ , (34d)where ω s ( q ) is the spectrum for vector q within the orig-inal crystalline Brillouin zone, Eq. (32). Note that while ω and ω are symmetric with respect to k = 0, ω and ω do not have this symmetry and are related as ω ( − k ) = ω ( k ). In addition, the boundary values of thefrequencies are connected as ω ( ± π ) = ω ( − π ) = ω ( π )and ω ( ± π ) = ω ( π ) = ω ( − π ).At the center of the folded Brillouin zone, k = 0, thefirst mode corresponds to the uniform spin rotations, A (0) x,ν (0) = − S sin( πν/ , A (0) y,ν (0) = S cos( πν/ k = 0 cor-respond to the modes at q = ± π/ ω (0) = ω (0) = ω s ( π/ A ( ± x,ν (0) = ± i ( − ν − S , A ( ± y,ν (0) = ( − ν +12 S . The forth mode at k = 0 corresponds to mutual rotation of odd and even sub-lattices, A (2) x,ν (0) = S sin( πν/ , A (2) y,ν (0) = S cos( πν/ V. RESPONSE TO NONUNIFORMOSCILLATING MAGNETIC FIELD ANDDYNAMIC SUSCEPTIBILITYA. Arbitrary modulation wave vector
In this section, we consider the response to the exter-nal oscillating nonuniform magnetic field ˜ h n = gµ B ˜ H n .The real-space components (˜ h xn , ˜ h yn , ˜ h zn ) correspond torotating-coordinates components (˜ h ξn , ˜ h ζn , ˜ h ηn ) with˜ h ζn = ˜ h xn cos ( Qn ) + ˜ h yn sin ( Qn ) , (35a)˜ h ξn = − ˜ h xn sin ( Qn ) + ˜ h yn cos ( Qn ) , (35b)and ˜ h ηn = ˜ h zn . In the presence of such external field,equations for the linear spin oscillations, Eqs. (25), be-come iωS ξ q − S [ J z ( Q ) − J z ( q ) + 4 K ] S η q = − S ˜ h η q , (36a) iωS η q + S (cid:20) J z ( Q ) − J z ( Q + q )+ J z ( Q − q )2 (cid:21) S ξ q = S ˜ h ξ q , (36b)where ˜ h α q is the Fourier transform of ˜ h αn . Solution ofthese equations can be presented as S ξ q = χ Sξξ ( q , ω ) ˜ h ξ q + χ Sξη ( q , ω ) ˜ h η q , (37a) S η q = χ Sηξ ( q , ω ) ˜ h ξ q + χ Sηη ( q , ω ) ˜ h η q , (37b)where we defined the susceptibility components in therotating-coordinates basis χ Sξξ ( q , ω ) = − S [ J z ( Q ) −J z ( q )+4 K ] ω − ω s ( q ) , (38a) χ Sηη ( q , ω ) = − S (cid:104) J z ( Q ) − J z ( Q + q )+ J z ( Q − q )2 (cid:105) ω − ω s ( q ) , (38b) χ Sξη ( q , ω ) = χ S ∗ ηξ ( q , ω ) = iωSω − ω s ( q ) , (38c)and ω s ( q ) is given by Eq. (26).Equations (38) give the susceptibility components inthe helically-rotating coordinates. To study interactionswith macroscopic fields, however, we need the suscepti-bility in real space. As follows from Eq. (28), the spinFourier components in real coordinates are given by S xq = − S ξ,q + Q − S ξ,q − Q i ,S yq = S ξ,q + Q + S ξ,q − Q . We emphasize that here and below the wave vector q cor-responding to real space distinguishes from the ’fractur’wave vector q in the rotated basis which we used above.We use the result for S ξ q from Eq. (37a), in which wesubstitute the field Fourier components˜ h ξq = 12 (cid:88) δ = ± (cid:104) iδ ˜ h x,q + δQ + ˜ h y,q + δQ (cid:105) , following from Eq. (35b). This yields the spin responsein real coordinates S xq = χ Sxx ( q, ω ) ˜ h x,q + χ Sxy ( q, ω ) ˜ h yq + (cid:88) δ = ± (cid:40) χ Sξξ ( q + δQ, ω )4 (cid:104) ˜ h x,q +2 δQ + iδ ˜ h y,q +2 δQ (cid:105) − δχ Sξη ( q + δQ, ω )2 i ˜ h z,q + δQ (cid:41) , (39a) S yq = χ Syx ( q, ω ) ˜ h x,q + χ Syy ( q, ω ) ˜ h yq + (cid:88) δ = ± (cid:40) χ Sξξ ( q + δQ, ω )4 (cid:104) iδ ˜ h x,q +2 δQ +˜ h y,q +2 δQ (cid:105) + χ Sξη ( q + δQ, ω )2 ˜ h z,q + δQ (cid:41) (39b) with the real-space spin susceptibility components χ Sxx ( q, ω ) = χ Syy ( q, ω ) = χ Sξξ ( Q + q, ω )+ χ Sξξ ( Q − q, ω )4 , (40a) χ Sxy ( q, ω ) = − χ Syx ( q, ω ) = − χ Sξξ ( Q + q, ω ) − χ Sξξ ( Q − q, ω )4 i . (40b)As expected, in addition to the usual diagonal response atthe same wave vector, the helical magnetic structure alsogenerates nondiagonal susceptibility and responses at thewave vectors shifted by the modulation wave vector Q .Note that the bulk magnetic susceptibility χ αβ ( k z , ω ) inEq. (6) is related to the dynamic spin susceptibility as χ αβ ( k z , ω ) = n M ( gµ B ) χ Sαβ ( dk z , ω ) . (41)We will be mostly interested in the smooth spin re-sponse to smooth field with the wave vectors muchsmaller than Q . In this case, we can drop the short-wave length terms ˜ h x ( q ± mQ ) with m (cid:54) = 0, i.e., keep onlythe first lines in Eqs. (39a) and (39b). In addition, toobtain the long-wave length response, we use the small- q expansion (remind that J (cid:48)(cid:48) z ( Q ) < χ Sxx ( q, ω ) ≈ (cid:88) δ = ± − S (cid:20) | J (cid:48)(cid:48) z ( Q ) | q + 4 K (cid:21) ω − ω s ( Q ) + δa s q + c s q , (42)with a s = 2 S KJ (cid:48) z (2 Q ) and c s = S (cid:40) K [ J (cid:48)(cid:48) z (0) + J (cid:48)(cid:48) z (2 Q )] . − |J (cid:48)(cid:48) z ( Q ) | (cid:20) J z ( Q ) − J z (0)+ J z (2 Q )2 (cid:21)(cid:41) . Note that for incommensurate-modulation wave vector Q , the linear coefficient a s is finite meaning that the fre-quency ω s ( q ) does not have an extremum at q = Q . Theimportant particular cases of this result include the re-sponse to the uniform oscillating field χ Sxx (0 , ω ) = − S K ω − S K [2 J z ( Q ) −J z (0) −J z (2 Q )] . (43)and the static uniform susceptibility χ Sxx (0 , ≈ J z ( Q ) −J z (0) −J z (2 Q ) , (44)which only depends on the interlayer exchange constants.The long-range off-diagonal component χ Sxy ( q, ω ) is given In our notations, k z and q in Eq. (40a) are the dimensional anddimensionless c-axis wave vectors, respectively, with q = dk z by χ Sxy ( q, ω ) ≈ − i S (cid:20) | J (cid:48)(cid:48) z ( Q ) | q + 4 K (cid:21) a s q [ ω − ω s ( Q )+ c s q ] − a s q . (45)It vanishes for q → qχ Sxy ( q, ω ) ≈ − iS K a s q { ω − S K [2 J z ( Q ) −J z (0) −J z (2 Q )] } , (46)meaning that the transverse spin response is proportionalto the field gradient S x ∝ ∂ ˜ h/∂z . B. Case Q = π/ For the commensurate modulation with Q = π/
2, thespin response, Eq. (39a), simplifies as S xq = χ Sxx ( q, ω ) (cid:104) ˜ h x,q +˜ h x,π − q (cid:105) + χ Sxy ( q, ω ) (cid:104) ˜ h yq +˜ h y,π − q (cid:105) − (cid:88) δ = ± δ i χ Sξη (cid:16) q + δ π , ω (cid:17) ˜ h z,q + δ π , (47)where the diagonal susceptibility, Eq, (40a), explicitly isgiven by χ Sxx ( q, ω ) = 14 (cid:88) δ = ± − S [ J z ( π/ − J z ( δq + π/
2) + 4 K ] ω − S (cid:2) J z (cid:0) π (cid:1) −J z (cid:0) δq + π (cid:1) +4 K (cid:3) (cid:104) J z (cid:0) π (cid:1) − J z ( q )+ J z ( π − q )2 (cid:105) , (48)and we used the relation χ Sxx ( π − q ) = χ Sxx ( q ).In the small- q expansion, Eq. (42), the linear term inthe denominator ∝ a s vanishes , since J (cid:48) z (2 Q ) ≡ J (cid:48) z ( π ) =0 and the quadratic-term coefficient becomes c s = S (cid:34) K ( J (cid:48)(cid:48) z (0)+ J (cid:48)(cid:48) z ( π )) . −|J (cid:48)(cid:48) z ( π/ | (cid:18) J z ( π/ − J z (0)+ J z ( π )2 (cid:19)(cid:35) . (49)For three-neighbor model, Eq. (19), this coefficient ac-quires a simple form c s = 16 S |J z, | ( K−|J z, | ) . (50)The behavior of the off-diagonal component is very dif-ferent from the case of incommensurate modulation. Itvanishes in static case and for finite frequency in the small q limit it behaves as χ Sxy ( q, ω ) ≈ iS J (cid:48)(cid:48)(cid:48) z ( π/ ω q (cid:0) ω − ω s (cid:0) π (cid:1)(cid:1) ≈ − iS J z, ω q ω − S K|J z, | ) , i. e., it vanishes ∝ q for q →
0. This behavior allowsus to neglect the off-diagonal component in the furtherphenomenological considerations.
VI. ELECTROMAGNETICRENORMALIZATION OF SPECTRUM INSUPERCONDUCTING STATEA. Arbitrary modulation vector
In this section, we consider the renormalization of thespin-wave spectrum in the superconducting state due tothe long-range electromagnetic interactions between thelocal moments using Eq. (8) in terms of the reduced wavevector q = dk z . Using Eq. (41) connecting the spin andmagnetic susceptibilities and the relation χ Sxx ( q ) ± iχ Sxy ( q ) = 12 χ Sξξ ( Q ∓ q )= − S [ J z ( Q ) −J z ( Q ∓ q )+4 K ] ω − ω s ( Q ∓ q )following from Eqs. (40a) and (40b), we obtain the equa-tion1+( λ/d ) q − π n M m [ J z ( Q ) −J z ( Q ∓ q )+4 K ] ω − ω s ( Q ∓ q ) = 0(51)for the renormalized spin-wave spectrum, ω = Ω s ( Q ∓ q ).The solution of this equation isΩ s ( Q + q ) = ω s ( Q + q )+ 2 πn M m [ J z ( Q ) −J z ( Q + q )+4 K ]1 + ( λ/d ) q . (52)The second term gives the spin-wave frequency enhance-ment due to the long-range electromagnetic interactions.The maximum enhancement is realized near q = 0 corre-sponding to the uniform mode, Eqs. (29) and (30). Us-ing the presentation for the static magnetic susceptibility0 k mode coupled to uniform field Renormalized spectrum ~ ‐ f f f f f [ G H z ] Renormalized spectrum
Original
BZ Folded BZ mode coupled to uniform field (a) (b) q FIG. 2. The representative spectrum of spin waves for the helical structure with Q = π/ As . The dashed lines in both plots show the bare spectra obtained without takinginto account the renormalization caused by the coupling to macroscopic magnetic field. χ xx (0 ,
0) = 2 n M m K /ω s ( Q ) following from Eqs. (41),(44), and (30), we can rewrite Eq. (52) for q = 0 asΩ s ( Q ) = [1+4 πχ xx (0 , ω s ( Q ) meaning that the renor-malized frequency of the uniform mode isΩ s ( Q ) = √ µ x ω s ( Q ) , (53)where µ x = 1 + 4 πχ xx (0 ,
0) is the static magnetic per-meability. Neglecting a weak q dependence in the nom-inator of the second term in Eq. (52), we can rewritethe frequency renormalization for q (cid:28) s ( Q + q ) ≈ ω s ( Q + q ) +4 πχ xx (0 , / [1+( λ/d ) q ]. B. Case Q = π/ The key features of the commensurate state with Q = π/ ω s ( q ) has maximum at q = Q (for K > |J z, | ) and (ii)the off-diagonal spin sus-ceptibility vanishes as q for q → χ xx ( k z , ω ) ≈ − χ xx (0 , ω s ( Q ) ω − ω s ( Q )+ c s d k z (54)with the static susceptibility χ xx (0 ,
0) = n M ( gµ B ) / (8 |J z, | ) and c s is given by Eqs. (49)and (50). The key difference from the ferromagneticstate[58, 59] is the opposite sign of the quadratic coefficient, since in our case the spin-wave frequency hasmaximum at k z = 0 (corresponding to q = π/ q = Q in term of the reduced wave vector q = dk z Ω s ( Q + q ) = ω s ( Q ) (cid:20)
1+ 4 πχ xx (0 , λ/d ) q (cid:21) − c s q . (55)In particular, the renormalization of the uniform modeis again given by Eq. (53). In the folded Brillouin zonediscussed in Sec. IV B 1 this mode corresponds to secondand third modes at k = 0, Eqs. (34b) and (34c).Figure 2 shows spectrum of spin waves in both originaland folded Brillouin zone for the parameters typical forRbEuFe As . Namely, we took S = 7 / J z, = 0 . J z, = − . K = 0 . λ = 70nm, and µ x =3. For this parameters the bare maximum frequencyis ∼ λ/d ) | q − π/ | , λ/d ) k >
1. We deliberately took some-what large value of J z, to enhance the difference between f ( k ) and f ( k ). For more realistic choice J z, (cid:46) |J z, | these frequencies become indistinguishable.1 VII. DYNAMIC EQUATION FOR SMOOTHMAGNETIZATION, MAGNETIC BOUNDARYCONDITION, AND SURFACE IMPEDANCE
In this section we consider magnetization response tothe alternating magnetic field at the surface and derivethe magnetic boundary condition. Here and below, welimit ourselves to the commensurate case Q = π/
2, forwhich the frequency of the z -axis uniform mode is max-imal. As follows from the shape of the susceptibility,Eq. (54), a phenomenological equation for the in-planemagnetization in the case of uniform in-plane field is χ − (cid:18) ω − ∂ ∂t + ζ ∇ z (cid:19) M = H (56)with χ = χ xx (0 , ω = ω s ( Q ), and ζ = c s d /ω s ( Q ).This equation is only valid for smoothly varying magne-tization, i. e. for ζ |∇ z M | (cid:28)
1. On the other hand, thelocal magnetic field H is connected with the magnetiza-tion as (cid:0) − λ ∇ z (cid:1) H ≈ − π M . (57)The magnetic length scale ζ is much smaller than theLondon penetration depth λ . We find the magnetiza-tion response to the external oscillating magnetic field.This corresponds to boundary condition for H ( z, t ) atthe surface, z = 0, H (0 , t ) = H exp ( iωt ) , (58)This condition has to be supplemented by the bound-ary condition for the magnetization, which is usually as- sumed as ∇ z M (0 , t ) = 0 . (59)We look for the oscillating magnetization and field atthe semispace z > M ( z, t ) = (cid:88) α M α exp ( iωt − κ α ( ω ) z ) , (60a) H ( z, t ) = (cid:88) α H α exp ( iωt − κ α ( ω ) z ) . (60b)In absence of internal dissipation mechanisms, the pa-rameters κ α ( ω ) may be either purely real or purely imag-inary. It is clear that in the former case κ α ( ω ) have tobe positive. The care should taken to select the correctsign for purely imaginary κ α ( ω ). Since for the spectrumdescribed by Eq. (56) the group velocity has the oppo-site sign with respect to the wave vector q = Im [ κ α ( ω )],the energy flows away from the surface for negativeIm [ κ α ( ω )]. Substituting the above distributions intoEqs. (56) and (57), we obtain equations connecting thevectors M α and H α (cid:0) − λ κ α (cid:1) H α ≈ − π M α , (61a) (cid:0) − ω /ω + ζ κ α (cid:1) M α = χ H α , (61b)which give the quadratic equation for κ α ( ω ) (cid:0) − ω /ω + ζ κ α (cid:1) (cid:0) − λ κ α (cid:1) + µ x − . Solution of this equation is κ α = λ − + (cid:0) ω /ω − (cid:1) ζ − δ α (cid:115) (cid:2) λ − +( ω /ω − ζ − (cid:3) ζ − λ − ( µ x − ω /ω )= λ − + (cid:0) ω /ω − (cid:1) ζ − δ α (cid:115) (cid:2) λ − − ( ω /ω − ζ − (cid:3) ζ − λ − ( µ x − . (62)We select δ = sign (cid:2) λ − + (cid:0) ω /ω − (cid:1) ζ − (cid:3) and δ = − δ .Such choice implies that | κ ( ω ) | > | κ ( ω ) | in the wholefrequency range. Note that this solution is only formallyvalid in the frequency range where ζ | κ ( ω ) | (cid:28) κ ( ω ) is not valid for static case at ω = 0.Consider important special cases of Eq. (62). At thebare uniform frequency, ω = ω , we obtain κ α ( ω ) = λ − ± λ − (cid:113) λ − +4 ζ − ( µ x − ≈ ± ζ − λ − (cid:112) µ x − , (63) while at renormalized frequency ω = √ µ x ω , we have κ ( √ µ x ω ) = λ − + ( µ x − ζ − ,κ ( √ µ x ω ) = 0 . However, in the latter case the value of κ is alreadybeyond the applicability range of Eq. (56). Since ζ (cid:28) λ ,the inequality λ − (cid:28) (cid:12)(cid:12) ω /ω − (cid:12)(cid:12) ζ − is satisfied almosteverywhere, except a narrow region where the frequencyis very close to ω . Away from this region, we can expand2 κ α ( ω ) with respect to (cid:0) ω /ω − (cid:1) − ζ /λ , which yields κ ≈ (cid:0) ω /ω − (cid:1) ζ − + ( µ x − λ − ω /ω − , (64a) κ ≈ λ − ω /ω − µ x ω /ω − κ and κ mostly describemagnetic and superconducting decay, respectively. Theapproximation is valid until the second term in κ issmall with respect to the first one giving a somewhatmore accurate condition for the expansion (cid:12)(cid:12) ω /ω − (cid:12)(cid:12) (cid:29)√ µ x − ζ /λ . In addition, the condition ζ | κ | (cid:28) κ is only valid for (cid:12)(cid:12) ω /ω − (cid:12)(cid:12) (cid:28)
1. However, the result for κ in Eq. (64b) correspondsto the approximation of local magnetic response, ζ → ζ | κ | (cid:28) ω →
0. In the immediatevicinity of the frequency ω , in the range (cid:12)(cid:12) ω /ω − (cid:12)(cid:12) (cid:28)√ µ x − ζ /λ , the parameters κ α can be evaluated as κ α ≈ ± ζ − λ − (cid:112) µ x − λ − + (cid:0) ω /ω − (cid:1) ζ − ± (cid:2) λ − − (cid:0) ω /ω − (cid:1) ζ − (cid:3) ζ − λ − √ µ x − . This region is characterized by a very strong mixing ofspin and supercurrent oscillations. The key observation isthat, in contrast to nonmagnetic superconductors, wherelow-frequency magnetic field decays on the distance ofthe order of the London penetration depth, in our casefor frequency smaller than √ µ x ω one of the parame-ters κ α is complex meaning that the oscillating magneticfield penetrates at much larger distance limited by exter-nal dissipation mechanisms .We now proceed with evaluation of the vector coeffi-cients M α and H α from Eqs. (61) using the bound-ary conditions in Eqs. (58) and (59). Substituting therelation H α ≈ − − λ κ α π M α into the boundary con-dition for H , we obtain a 2 × κ M + κ M = 0 , (65a) − − λ κ M − − λ κ M = H π , (65b)which yields the solution (cid:18) M M (cid:19) = (cid:0) − λ κ (cid:1) (cid:0) − λ κ (cid:1) H / π ( κ − κ ) [1 − λ ( κ + κ κ + κ )] (cid:18) κ − κ (cid:19) . (66)The corresponding field components are (cid:18) H H (cid:19) = H ( κ − κ ) [1 − λ ( κ + κ κ + κ )] . × (cid:18) − κ (cid:0) − λ κ (cid:1) κ (cid:0) − λ κ (cid:1) (cid:19) (67) -Im ] / Re ] x -Im ] -Im ]Re ] FIG. 3. The frequency dependences of the real and imaginarypart of the parameter η ω , Eq. (68b), which determines the dy-namic magnetic boundary condition, Eq. (68a). The dashedlines show the approximate result in Eq. (69) valid away fromthe frequency ω . The upper right inset shows zoom in the re-gion near the frequency ω . The navy and wine dotted lines inthis inset show the approximate scaling result in Eq. (71).Thelower left inset shows the logarithmic plot of − Im[ η ω ] to illus-trate that it remains finite down to zero frequency. The plotsare made for ζ = 0 . λ and µ x = 3. The interaction between the magnetic superconduc-tor and outside world can be conveniently formulatedin terms the boundary condition connecting the gradi-ent ∇ z H with the field at the surface. From Eq. (67),we obtain ∇ z H | z =0 = − κ H − κ H = − κ κ ( κ + κ ) κ + κ κ + κ − λ − H Using the relations κ κ = − iζ − λ − (cid:112) µ x − ω /ω and κ + κ − λ − = (cid:0) ω /ω − (cid:1) ζ − , we can rewrite this bound-ary condition as ∇ z H = − η ω H /λ (68a)with η ω = − iζ (cid:112) µ x − ω /ω ( κ + κ ) ω /ω − − iζ /λ (cid:112) µ x − ω /ω . (68b)In the range (cid:12)(cid:12) ω /ω − (cid:12)(cid:12) (cid:29) √ µ x ζ /λ the parameters κ α are given by Eqs. (64a) and (64b). In this case | κ | (cid:29)| κ | , λ − and we obtain a simple approximate result η ω ≈ λκ ≈ − i (cid:115) µ x − ω /ω ω /ω − . (69)Note that this result corresponds to the approximation oflocal magnetic response and it remains valid even in the3regime where ζ | κ | >
1. In particular, it gives correctlythe static-case result η ω =0 = √ µ x . On the other hand,at ω = ω , using Eq. (63), we obtain η ω ≈ (1 − i ) ( µ x − / (cid:112) λ/ζ . (70)In the range ω /ω − (cid:28) µ x − η ω ≈ (cid:112) λ/ζ ( µ x − / v (cid:18) ω /ω − ζ /λ √ µ x − (cid:19) , (71) v ( u ) = 11+ iu (cid:115)(cid:114) u u − i (cid:115)(cid:114) u − u . The real and imaginary parts of the complex func-tion v ( u ) are connected by the relation Re [ v ( − u )] = − Im [ v ( u )]. The real part reaches the maximum valueequal to 1 .
162 at u ≈ − . v ( u ) in the range u (cid:29) v ( u ) (cid:39) − i/ √ u yielding η ω ≈ − i √ µ x − / (cid:112) ω /ω −
1. This matches the resultin Eq. (69) in the range ω /ω − (cid:28)
1. In the large neg-ative region, u < | u | (cid:29)
1, the imaginary part of v ( u )decays as Im[ v ( u )] (cid:39) −| u | − / .Figure 3 shows plots of the real and imaginary partof the parameter η ω , Eq. (68b), computed using typicalparameters ζ = 0 . λ and µ x = 3. We also show in thefigure the approximate result, Eq. (69), valid for frequen-cies not very close to ω , and, in the upper right inset,the approximate scaling result in Eq. (71) describing be-havior near ω . The frequency dependence of η ω can besummarized as follows. In the range ω < ω , the real partof η ω is much larger then − Im( η ω ). Both parts increasefor ω → ω and become equal in absolute value at ω = ω .The real part reaches maximum 1 . (cid:112) λ/ζ ( µ x − / slightly below ω , at ω ≈ ω (1 − . ζ /λ √ µ x − − Im( η ω ) reaches the same maximum slightly above ω , at ω ≈ ω (1 + 0 . ζ /λ √ µ x − ω < ω < √ µ x ω , the real part of η ω is much smallerthen − Im( η ω ). Finally, in the region ω > √ µ x ω theimaginary part is zero, while the real part monotonicallyincreases asymptotically approaching unity.The parameter η ω is directly connected with the con-ventional parameter characterizing the microwave re-sponse, surface impedance Z = E x (cid:82) ∞ j x ( z ) dz = 4 π c E x H y . (72)To establish this connection, we have to relate the tan-gential electric field with the normal gradient of the mag-netic field. At small frequencies, we can use the Londonrelation π c ∂j x ∂t ≈ cλ − E x neglecting a small contributionfrom the quasiparticle current and the Maxwell equation −∇ z H y = π c j x omitting the displacement current. Thisgives −∇ z H y = c iωλ E x and from Eq. (68a) we obtain Z = 4 πiωη ω λ/ c . (73) R s [ oh m ] frequency [GHz] frequency [GHz] FIG. 4. The frequency dependence of the surface resistivityusing the same parameters as in Figs. 2 and 3 correspondingto RbEuFe As . The inset shows the same plot in logarithmicscale for the better presentation of the low-frequency behav-ior. The vertical dashed lines show locations of the frequencies f = ω / π and √ µ x f . The real part of this equation, R s = Re( Z ), can beconverted to the practical formula for surface resistivity R s [ohm] = − π − Im [ η ω ] f [GHz] λ [ µ m].Figure 4 shows the frequency dependence of the sur-face resistivity using the same parameters as in Figs. 2and 3. We can see that the surface resistivity has avery distinct shape. It is very small at small frequen-cies f < f = ω / π and starts to increase sharplywhen the frequency approaches f . After reaching a peakvalue ∼ .
06 ohm slightly above f , it slowly decreaseswithin extended frequency range f < f < √ µ x f , andabruptly vanishes at √ µ x f . VIII. EXCITATION OF SPIN WAVES WITH ACJOSEPHSON EFFECT
The presence of magnetic order inside superconduct-ing material provides a unique possibility to generate andmanipulate magnons using AC Josephson effect. In thissection, we consider the excitation of spin waves in atunneling contact between a conventional superconduc-tor marked by the index 1 and a superconductor withhelical magnetic structure marked by the index 2, as il-lustrated in Fig. 5. We assume that the system is uniformalong the y direction and the interlayer with thickness t is insulating and nonmagnetic. The magnetic and con-ventional superconductors occupy the regions z > z < − t , respectively.4 J x z y B FIG. 5. Illustration of a planar tunneling contact betweena conventional superconductor (1) and magnetic supercon-ductor with helical magnetic structure (2). Purple arrowsillustrate orientation of the magnetic moments.
A. Dynamic equation for the Josephson phase
We follow the standard derivation of the dynamic equa-tion for the gauge-invariant phase difference between twosuperconductors θ = φ − φ − π t Φ A z taking into thedynamic magnetization response. The starting point ofderivation is the z component of the Maxwell equation ∇ x H y = 4 π c j z + ε c ∂E z ∂t , (74)in which the total current density across the junction iscomposed of the superconducting and quasiparticle con-tributions, j z = j s,z + j n,z , where the superconductingcontribution is given by the DC Josephson relation, j s,z = j J sin θ, (75)and the quasiparticle contribution is determined by tun-neling conductivity σ , j n,z = σE z . The electric field isrelated to the phase by the AC Josephson relation E z = Φ πcd ∂θ∂t . (76)To relate ∇ x H y in Eq. (74) with the phase gradient, weuse the x component of the Maxwell equations −∇ z H y = π c j x and the London relation for supercurrents along thejunction π c j x ≈ λ − i (cid:0) Φ π ∇ x φ − A x (cid:1) . Here, we neglectedthe displacement current assuming small frequencies andquasiparticle current inside the superconductors. Thisleads to the relation between the in-plane phase gradientand magnetic fields ∇ x θ = 8 π cΦ (cid:0) λ j x, − λ j x, (cid:1) + 2 π t Φ B y = − π Φ (cid:0) λ ∇ z H y, − λ ∇ z H y, (cid:1) + 2 π t Φ B y , (77)where j x,i and H y,i are the current densities and the mag-netic fields at the surfaces of two superconductors and B y is the magnetic induction inside the junction. We assumenonmagnetic interlayer meaning that B y = H y . Also, forthe nonmagnetic superconductor in the Meissner stateat z < − t , we have ∇ z H y, = B y /λ . To obtain theclose system, we need the boundary condition connect-ing ∇ z H y, with H y = B y at the surface of magneticsuperconductor at z = 0. Note that H y ( z ) is continuous,while B y ( z ) has a jump at z = 0 . Due to magnetizationdynamics, this boundary condition is frequency depen-dent. At fixed frequency, such boundary condition hasbeen derived in Sec. VII and is given by Eq. (68a), whichin our case becomes ∇ z H y, = − η ω B y /λ . The com-plex parameter η ω is determined by the general resultin Eq. (68b). In the approximation of local magneticresponse valid for frequencies not too close to the bareuniform-mode frequency ω , it has much simpler approx-imate presentation in Eq. (69). Therefore, Eq. (77) atfinite frequency becomes ∇ x θ = 2 π ( t + λ + η ω λ )Φ H y (78)Applying ∇ x to both sides, substituting ∇ x H y fromEq. (74), and using the Josephson relations for currentand electric field, Eqs. (75) and (76), we obtain the dy-namic phase equation at finite frequency in the form1 t + λ + η ω λ ∇ x θ = 8 π cΦ j J [sin θ ] ω − ε ω ω t c θ (79)where ε ω ≡ ε − πiσ/ω and [sin θ ] ω notates the timeFourier transform of sin [ θ ( x, t )]. The only differencefrom the standard phase-dynamics Sine-Gordon equation[65, 66] is the presence of the complex factor η ω with com-plicated frequency dependence, see Fig. 3. In the staticcase, the phase equation is1 t + λ + √ µ x λ ∇ x θ = 8 π cΦ j J sin θ. (80)Therefore, the effective junction interlayer width ˜ t = t + λ + √ µ x λ is enlarged by the magnetic response. Fromthe last equation, we can evaluate the static Josephsonlength λ J = (cid:40) cΦ (cid:2) π (cid:0) t + λ + √ µ x λ (cid:1) j J (cid:3) (cid:41) / . (81)In the next subsection we consider the influence of mag-netic response on the spectrum and damping of electro-magnetic wave propagating through the Josephson junc-tion. B. Spectrum and damping of the Josephsonplasma mode
The superconductor-insulator-superconductor sand-wich structure with sufficiently large width is a waveg-uide capable to support a traveling electromagnetic wave5 w , r [ p ] J k w , i [ p ] =190nm, = 70nm x0 = 3, = 0.02 p = 0.5 = 0.01 x FIG. 6. Spectrum and damping of the electromagnetic waveinside a Josephson junction between conventional and helicalmagnetic superconductors, Eq. (84). In the lower plot, thedashed and dash-dotted lines show spectra corresponding tolow-frequency and high-frequency limits, respectively. [66, 67] with the phase θ ( x, t ) ∝ exp [ i ( ω w t ± kx )]. Sucha wave can be resonantly excited by the AC Joseph-son effect. For the fixed real wave vector k , Eq. (79)gives the following equation for the complex frequency ω w ( k ) = ω w,r ( k ) + iω w,i ( k ), with the real and imaginarypart giving the wave spectrum and its damping, respec-tively, ω w − πσε iω w = ω p + tt + λ + η ω λ c ε k , (82)where ω p = (cid:115) π c t ε Φ j J (83)is the Josephson plasma frequency. Note that the mag-netic response does not modify this parameter. It is con-venient to rewrite Eq. (82) in the reduced form ω w ω p − iν σ ω w ω p = 1 + λ + √ µ x λ λ + η ω λ λ J k (84)with the static Josephson length λ J , Eq. (81), and thedumping parameter ν σ = 4 πσεω p . (85)The parameter η ω has the strongest feature around ω = ω . Therefore, the spectrum of the Josephson plasmon issubstantially affected only if ω p < ω . Figure 6 shows the spectrum and damping of thepropagating wave computed from Eq. (84) for param-eters corresponding to the contact between NbN andRbEuFe As , λ = 190nm, λ = 70nm, µ x = 3, and ς =0 . λ . We also assume ω p = 0 . ω and ν σ = 0 .
01. Onecan distinguish three regions with qualitatively differentbehavior. In the low-frequency region ω w,r ( k ) < ω ,the spectrum is approximately ω w,r ( k ) (cid:39) (cid:113) ω p + c s k ,where c s is the low-frequency Swihart velocity c s = λ J ω p = (cid:115) t λ + √ µ x λ c √ ε . (86)In this region the spin waves give a small contribu-tion to the mode damping. The intermediate region ω < ω w,r ( k ) < √ µ x ω is characterized by a sharp en-hancement of the damping caused by excitation of spinwave. Finally, in the high-frequency region ω w,r ( k ) > √ µ x ω the damping caused by spin waves is absentand the spectrum approaches the high-frequency limit ω w,r ( k ) (cid:39) (cid:113) ω p + c s k (cid:39) c s k , where c s is the high-frequency mode velocity, c s = (cid:114) dλ + λ c √ ε = (cid:115) λ + √ µ x λ λ + λ c s . (87)In this limit the influence of magnetism is weak. C. Current-voltage characteristics and Fiskeresonances in finite magnetic field
Transport properties of a Josephson junction in themagnetic field directly probe its dynamic response [65,66, 68, 69]. In particular, one can directly excite collec-tive modes in superconducting materials and spectrum ofthese modes can be inferred from the dynamic featuresin the current-voltage characteristics [70]. In this sec-tion, we evaluate the current-voltage characteristics forour system using the standard approach of the expan-sion with respect to the Josephson current [71]. Considera junction in finite magnetic field B y and in the resis-tive state with finite voltage drop across the junction, V = t E z . In this state, in the zeroth order with respectto the Josephson current, the phase has the shape of atraveling wave θ ( x, t ) = k B x + ωt (88)with the wave vector k B = 2 π Φ ( t + λ + √ µ x λ ) B y , (89)and the Josephson frequency ω = 2 π cΦ V. (90)6Representing sin θ ( x, t ) = Re [ − i exp ( ik B x + iωt )], we ob-tain from Eq. (79) equation for the first-order correc-tion to the dynamic phase, ˜ θ ( x, t ) = Re (cid:104) ˜ θ ( x ) exp ( iωt ) (cid:105) ,which we present as ∇ x ˜ θ + p ω ˜ θ = − ir ω λ − J exp ( ik B x ) (91)with r ω ≡ t + λ + η ω λ t + λ + √ µ x λ , (92a) p ω ≡ ε ω ω c t + λ + η ω λ t = ω − (4 πσ/ε ) iωc s r ω , (92b) where c s is the low-frequency Swihart velocity, Eq. (86).We look for the solution of Eq. (91) in the form˜ θ ( x ) = r ω λ − J k B − p ω [ A c cos ( p ω x )+ A s sin ( p ω x )+ i exp ( ik B x )] . (93)Assuming the nonradiative boundary conditions, ∇ x ˜ θ = 0for x = 0 , L , we find the coefficients A c and A s , A s = k B /p ω , (94a) A c sin ( p ω L ) = k B p ω [cos ( p ω L ) − exp ( ik B L )] (94b)and substitute them into Eq. (93). This yields the oscil-lating phase˜ θ ( x ) = r ω λ − J k B − p ω (cid:20) k B p ω cos [ p ω ( L − x )] − exp ( ik B L ) cos ( p ω x )sin ( p ω L ) + i exp ( ik B x ) (cid:21) . (95)The average Josephson current density is given by δj = j J L L (cid:90) (cid:68) sin (cid:16) k B x + ωt +Re (cid:104) ˜ θ ( x ) exp ( iωt ) (cid:105)(cid:17)(cid:69) t dx ≈ j J L L (cid:90) Re (cid:104) ˜ θ ( x ) exp ( − ik B x ) (cid:105) dx. (96)Substituting ˜ θ ( x ) from Eq. (95), we obtain δj ≈ j J λ − J × Im (cid:26)(cid:20)
1+ cos ( p ω L ) − cos ( k B L ) p ω L sin ( p ω L ) 2 k B p ω − k B (cid:21) r ω p ω − k B (cid:27) . The key difference from the standard result [71] is thepresence of the complex factor η ω in the parameters p ω and r ω in Eqs. (92a) and (92b) from the magnetic bound-ary condition describing the excitation of spin waves inthe magnetic superconductor. The location of the Fiskepeaks corresponding to excitation of the standing elec-tromagnetic waves inside the junction is determined bythe condition Re[ p ω ] L = πn . In the regions ω < ω and ω > √ µ x ω this condition gives equation for the reso-nance frequencies ω n = (cid:115) t + λ + √ µ x λ t + λ +Re( η ω ) λ πnλ J L ω p , (98)where ω p is the Josephson plasma frequency, Eq. (83).To facilitate numerical calculations, we rewriteEq. (97) in the reduced form. We define the dimension- less size ˜ L = L/λ J and frequency ˜ ω = ω/ω p . We also in-troduce the reduced wave-vector parameters ˜ k B = λ J k B and ˜ p ω = λ J p ω = (cid:112) (˜ ω − iν σ ˜ ω ) r ω , where ν σ is the dimensionless damping parameter,Eq. (85). With these variables, we rewrite Eq. (97) as δjj J = Im (cid:40)(cid:34)
1+ cos(˜ p ω ˜ L ) − cos(˜ k B ˜ L )˜ p ω ˜ L sin(˜ p ω ˜ L ) 2˜ k B ˜ p ω − ˜ k B (cid:35) r ω ˜ p ω − ˜ k B (cid:41) . (99)The product ˜ k B ˜ L here may be related with the magneticfield as ˜ k B ˜ L = πB y /B L = 2 π Φ y / Φ , where B L = Φ L (cid:0) t + λ + √ µ x λ (cid:1) (100)is the size-dependent scale determining periodicity ofmagnetic oscillations of the Fiske resonances and Φ y = L (cid:0) t + λ + √ µ x λ (cid:1) B y is the magnetic flux through thejunction. For frequency in Eq. (98), the strongest res-onance is realized at B = nB L . For other Fiske res-onances, odd peaks with n = 2 m + 1 are maximal for B y = 2 jB L (Φ y / Φ = j ) while even peaks with n = 2 m aremaximal for B y = (2 j +1) B L (Φ y / Φ = j +1 /
2) [65, 66, 71].Adding the tunnel quasiparticle current, j n = σE z , weobtain the total current in the reduced form jj J = ν σ ˜ ω + δjj J . (101)This equation together with Eq. (99) determines thecurrent-voltage characteristic in the reduced form in thesecond order with respect to the Josephson current.7 j [ j J ] E z [ E p ] L = J p = = B y = B L j [ j J ] L = 2 J j [ j J ] L = 3 J Spin-wave feature
FIG. 7. Representative current-voltage characteristics forjunctions with different lateral sizes L . The horizontal-axisscale E p is the electric field at which the Josephson fre-quency equals ω p , E p = Φ ω p / (2 πc t ). For comparison, thedashed lines show the current-voltage characteristics withoutdynamic magnetic response using static parameters c so and λ J . They display a usual sequence of the Fiske resonances.The inset in the bottom plot zooms into the spin-wave feature. The shape of current-voltage characteristic mostly de-pends on the relation between the Josephson plasmafrequency ω p , the location of the first Fiske resonance ω = πc s /L , and the two typical spin-wave frequencies ω and √ µ x ω . As the resistive state is stable until theJosephson frequency exceeds the plasma frequency ω p ,Eq. (83), spin waves can be excited only if ω p is at leastsmaller than √ µ x ω . The clearest spin-wave featurescan be observed if ω p (cid:46) ω . In addition, the behavior isalso very sensitive to the junction size L . For junctionsnarrower than the typical size L c = c s π/ ( √ µ x ω ), thewhole spin-wave region ω < ω < √ µ x ω is located be-low the Fiske resonances allowing for its clear resolution.For wider junctions the behavior is more complicated, be-cause in this case the Fiske resonances are located bothabove and below the spin-wave region and some of them may fall inside this region. A very special situation is re-alized for the particular junction size L res , at which thethe first Fiske resonance is very close to ω . To estimatethis junction size, we substitute the maximum value ofRe ( η ω ) ∼ (cid:112) λ/ζ ( µ x − / to Eq. (98) at n = 1 yield-ing L res = πλ J ω p ω (cid:115) t + λ + √ µ x λ t + λ +( µ x − / λ / / √ ζ . (102)For this size, at the Josephson frequency slightly below ω the excited cavity mode generates the strongest spinwave inside the magnetic superconductor.Figure 7 shows the representative current-voltage char-acteristics computed for the parameters ω p = ω , ν σ =0 .
05, three different sizes,
L/λ J =1, 2, and 3, and themagnetic field B y = B L . For reference, we also show bythe dashed lines the current-voltage characteristics com-puted without dynamic magnetic response using staticjunction parameters. Note that (i) only the ascendingleft-side parts of the peaks are usually observed exper-imentally and (ii) the used linear approximation breaksin the middle of resonances meaning that the approxi-mation overestimates the peak heights. We can see thatthere are substantial qualitative differences between thethree shown cases. The junction size for the smallestjunction is smaller than L c and therefore the spin-waveregion is well below the Fiske resonances. The spin-wavefeature has the same asymmetric shape as the surface re-sistivity in Fig. 4, it has a sharp peak when the Josephsonfrequency matches ω following by an extended tail up tofrequency √ µ x ω , see the inset in the bottom plot. Thejunction size 2 λ J (middle plot) is very close to the reso-nance value L res , Eq. (102), meaning that the spin-waveresonance at ω = ω coincides with the first Fiske reso-nance leading to the very strong peak. A very peculiarfeature of this case is that, due to strongly nonmonotonicbehavior of Re( η ω ) near the frequency ω , the conditionfor the first resonance in Eq. (98) is satisfied at two fre-quencies, slightly below ω and slightly above √ µ x ω .Correspondingly, two strong peaks are realized at bothfrequencies. The largest size 3 λ J exceeds both L c and L res (top plot). The first Fiske resonance in this caseis located below ω and is slightly separated from thepeak marking the onset of the spin-wave region. Cor-respondingly, the spin-wave region is located in betweenthe first and second Fiske resonances. We also observelarger amplitude of the spin-wave feature in the region ω > ω . The reason is that the condition for the first res-onance in Eq. (98) is also formally satisfied in the range ω < ω < √ µ x ω where the absolute value of Im( η ω ) islarge marking very strong spin-wave damping of the res-onance. As the resonance takes place in the overdampedregion, it is seen as a shallow maximum.The amplitudes of the Fiske resonances have oscillatingdependence on the magnetic field [65, 66, 71]. Figure 8shows the magnetic-field evolution of the current-voltagecharacteristics for the junction with L = 2 λ J . We see8 E z [ E p ] B y [ B L ] L = 2 J , p = = 0.05 j [ j J ] Spin-waveregion
FIG. 8. The magnetic-field evolution of the current-voltage characteristics for junction with the parameters shown in the plot.The spin-wave feature is located in the region 1 (cid:46) E z /E p (cid:46) .
7. As the Fiske resonances, it is strongly modulated by themagnetic field. the familiar modulation of the resonances with magneticfield but with specific features. We see that the firsttwo peak have similar dependence on the magnetic field,since they both represent the first Fiske resonance, whilethe third peak representing the second Fiske resonanceis shifted by half period. Note that the maximums of thefirst two peaks at B = B L and maximum of the thirdpeak at B = 2 B L are out of this general trend becausethey correspond to Eck resonance, ω = c si k B . IX. SUMMARY AND DISCUSSION
In summary, in this paper we consider spin waves andrelated observable effects in superconductors with heli-cal magnetic order. Most computed specific results cor-respond to the structure realized in the iron pnictideRbEuFe As , in which the moments rotate 90 ◦ from layerto layer, Fig. 1. The key feature of such materials is thatthe mode coupled with uniform field corresponds to themaximum frequency of the spin-wave spectrum with re-spect to the c-axis wave vector. The frequency of thismode is strongly enlarged by the long-range electromag-netic interactions between the oscillating magnetic mo-ments and this enlargement rapidly vanishes when the c-axis wave-vector mismatch exceeds the inverse Londonpenetration depth, see Fig. 2. For the parameters ofRbEuFe As , we estimate the bare uniform-mode fre-quency f as ∼
11 GHz and renormalized one as ∼ ω . In addition, the featuresin the current-voltage characteristics are very sensitiveto the junction size due to the interplay between thespin-wave excitation and Fiske resonances. The sim-plest behavior is realized in small-size junctions, whenthe renormalized frequency √ µ x ω is below the lowestFiske resonance. In this case, the whole spin-wave regionis separated from the Fiske resonances and has a stronglyasymmetric shape resembling the feature in the surfaceresistivity, see the inset in Fig. 7(bottom). In larger junc-tions, the Fiske resonances may fall inside the spin-wave9region leading to more complicated behavior, see Fig.7(top and middle). The strongest excitation of the spinwave can be achieved in the situation when the Fiske res-onance frequency is slightly below ω corresponding tothe junction size in Eq. (102). As the Fiske resonances,the shape and amplitude of the spin-wave feature is mod- ulated by magnetic field, see Fig. 8. We conclude thatthe AC Josephson effect provides a unique way to exciteand manipulate spin waves in magnetic superconductors.This work was supported by the US Department of En-ergy, Office of Science, Basic Energy Sciences, MaterialsSciences and Engineering Division. [1] L. Bulaevskii, A. Buzdin, M. Kuli´c, and S. Panjukov,Coexistence of superconductivity and magnetism theo-retical predictions and experimental results, Adv. Phys. , 175 (1985).[2] M. Kuli´c and A. I. Buzdin, Superconductivity , edited byK. H. Bennemann and J. B. Ketterson (Springer, Berlin,2008) Chap. 4. Coexistence of Singlet SuperconductivityandMagnetic Order in Bulk Magnetic Superconductorsand SF Heterostructures, p. 163.[3] C. T. Wolowiec, B. D. White, and M. B. Maple, Con-ventional magnetic superconductors, Physica C , 113(2015).[4] M. B. Maple and Ø. Fischer, eds.,
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