arXiv:2108.03847v1 [cond-mat.supr-con] 9 Aug 2021 agns-ae opud MnO compounds the manganese-based as known to angle constant plane a one angle by from helical plane rotates direc- neighboring magnetization the a while plane plane, of lattice crystal tion some on remain parallel spins a.k.a. systems all helical-ordered order, In spin from helical exceptions order. the do- spin-spiral of is One at alignment spin type. occurs parallel Neel struc- alignment the or domain Bloch spin of of a walls rotation main of the are formation where to spins definitions ture attributed such when usually from deviation are ordering, a while spin co-aligned, counter-aligned, are anti-ferromagnetic lattice mo- crystal spin or the when within ordering, localized spin menta ferromagnetic either with EuFe ]adspratcs7,mtlslcd aiyMS and EuCo MnSi family europium-based silicide [8–10], metal FeCoSi superlattices[7], alloys[4– a metals, and for earth confirmed 6] rare is materials: order different spin of helical variety the Nowadays, 3]. [2, 1,1] h eia pnodri trbtdt either to attributed com- to is or order peting 18], spin interaction[17, helical Dzyalosinsky-Moriya The the 16]. [14, sage[2 ob eae oceitnebtenthe between coexistence to ob- related order be spin to helical argued[22] the is EuRbFe Also, in non- served helical junction. the Josephson superconductor/ferromagnet that spin-triplet in long-range reported pairing the was superconducting promotes it order spin 21] collinear [20, hybrids. inter- Refs. superconductor/ferromagnetic he- gaining of In the field are the behind materials in interaction est helical-ordered exchange order, the lical of details liar 5,adEuRbFe and 15], antzto yaisi eia-ree pnsses A systems. spin helical-ordered in dynamics Magnetization rgnly h eia pnodrwsosre in observed was order spin helical the Originally, characterized are materials magnetic Conventional pr rmantrlfnaetlitrs npecu- in interest fundamental natural a from Apart 2 As J z 2 1 n EuNiGe and , 2 − .A Golovchanskiy A. I. n xeietlyuigmgeiainmaueet n f and EuFe measurements technique. magnetization using experimentally and opeey xeietlyfrEuFe for Experimentally dependenc ferromagne completely. clear and measurements demonstrate magnetization responses of of combination fie types molecular the Both systems: modes. helical-ordered for examined are eia nl saot2 about is angle helical ainlUiest fSineadTcnlg II,4Leni 4 MISIS, Technology and Science of University National Russia; J z nti ok antzto yaisi eia-ree sp helical-ordered in dynamics magnetization work, this In Russia; 2 ..Dzhumaev P.S. .INTRODUCTION I. xhneitrcin[,19]. [4, interaction exchange 4 4 As As 5 4 4 ainlRsac ula nvriyMPI(ocwEngine (Moscow MEPhI University Nuclear Research National 4 ermgei superconductor[16] ferromagnetic 3 izugCne o ihTmeaueSprodciiyan Superconductivity Temperature High for Center Ginzburg ermgei superconductors ferromagnetic 3 eia nifroant [11– anti-ferromagnets helical ntttkypr,Dlordy ocwRgo,110,R 141700, Region, Moscow Dolgoprudny, per., Institutskiy 9 uhvRsac nttt fAtmtc VIA,175 Mo 127055 (VNIIA), Automatics of Institute Research Dukhov 1 2 As ..LbdvPyia nttt fteRS 191 Moscow, 119991, RAS, the of Institute Physical Lebedev P.N. ocwIsiueo hsc n ehooy tt Universi State Technology, and Physics of Institute Moscow 5 2 ..Emelianova O.V. , 1 scniee stecs td.Tertclytotpso r of types two Theoretically study. case the as considered is , 2 π/ 1KsisoeSos,150 ocw Russia; Moscow, 115409 Shosse, Kashirskoye 31 , 3 5. ..Abramov N.N. , 2 1,adMnAu and [1], 2 P 2 EuCo , 2 As 5 2 2 .Baranov D. , As 2 h oeua edrsos sosre.Tedefined The observed. is response field molecular the ..Vlasenko V.A. , 2 2 , fcytlpae eurdfracmlt 2 complete a for required planes crystal of antztos rwt h eia angle helical the with or magnetizations, ueet rvd l aaeeso h eia system. helical ferromag- the of the EuFe mea- parameters For of magnetization all combination provide with the surements spectroscopy that resonance show netic We ple. edsrbdwt h helicity can the degree with This described orders. be spin ferromag- conventional anti-ferromagnetic or to netic comparison in angle, [23–25] rotation fields bias marginal or zero magnonic high at ensure eigenfrequencies and modes in magnonic of interest variety rich a potential provide interactions of antiferromagnetic since be magnonics, can addition, In materials helical-ordered phases. ferromagnetic and superconducting eia-ree aeil sn EuFe exper- in using study dynamics we materials magnetization work helical-ordered this theoretically In helicity and the scattering. imentally for neutron technique the measurement is direct orientations. only magnetization The nearest-neighbor the tween rsa xs hc nue h hnfimgoer with geometry film thin the ensures which 30-50 axis, a crystal about c and planes, crystal ab hc r xoitdfo ssnhszdbl crystals. bulk as-synthesized from EuFe samples Cleaved flake exfoliated composition, of are orientation the the which confirm and structure EBSD crystal microscopy, and the cleav- Optical XRD by EDX, crystals. SEM, obtained bulk as-synthesized were of age spectroscopy resonance ferromag- and 27]. netic measurements 26, [14, magnetization works for previous Samples with analogy by method, flux I XEIETLDTISADRESULTS AND DETAILS EXPERIMENTAL II. h eia pnodrpse nadtoa ere the degree, additional an posses order spin helical The EuFe 1 ..Pudalov V.M. , drsos n olciesi resonance spin collective and response ld 2 i eoac tde en h helicity the define studies resonance tic As roantcrsnnemeasurement resonance erromagnetic 2 nhlct.I sdmntae that demonstrated is It helicity. on e 4 sypop,Mso,194,Russia; 119049, Moscow, prosp., nsky As .Pervakov K. , nssesi tde theoretically studied is systems in 2 2 igecytl eegonuigteself- the using grown were crystals single 2 .Fbiaino samples of Fabrication A. As edfiehelicity define we 2 ape eeo e mi iealong size in mm few a of were samples unu Materials, Quantum d rn hsc Institute), Physics ering 4 ..Stolyarov V.S. , aesuyo EuFe of study case ussia; 4 ..Shchetinin I.V. , ty, snneresponses esonance scow, N N h hc stenumber the is which , h ≈ µ 2 1 hc ln the along thick m As 5. , 3 φ 2 h 2 π , sa exam- an as 2 = oainof rotation π/N 2 As h 2 be- . 2 defined crystal orientation. Cleaved EuFe2As2 samples a closed-cycle cryostat (Oxford Instruments Triton, base contain a minor amount of EuRbFe4As4 phase. See de- temperature 1.2 K). Magnetic field is applied in-plane tails in Ref. [14]. along the direction of the waveguide. The response of experimental samples is studied by analyzing the trans- mitted microwave signal S21(f,H) with the VNA Rohde B. Magnetization measurements & Schwarz ZVB20.
25
linear fit
0.4
H is in-plane, H || ab
H is out-of-plane, H || c 20
0.2 , GHz , , emu , 15 f M
1.0
1.0 0.0
10
1.0 Frequency Frequency
1.0 Magnetization Magnetization 5
-0.2
1.0
1.0
0.0 0.2 0.4 0.6 0.8 1.0
-0.4
Magnetic field H, T
-3 -2 -1 0 1 2 3 0
Magnetic field H, T
0 FIG. 2. FMR absorption spectra S21(f,H) of EuFe2As2 mea- FIG. 1. The dependence of magnetization of EuFe2As2 on sured at magnetic field along applied in-plane along ab layers magnetic field M(H) at T =5 K. and at T = 5 K. Red line is the linear fit of the FMR absorp- tion line. Blue dash indicates the absorption “patch”.
Magnetization measurements were performed using the Quantum Design PPMS-9 EverCool-II system. Fig- Figure 2 shows FMR absorption spectrum of EuFe2As2 ure 1 show a typical magnetization curves for EuFe2As2 sample measured at magnetic field applied in-plane along sample measured at both the in-plane orientation of mag- ab layers at T = 5 K. The spectrum contains a resonance netic field along ab crystal layers (red curve), and the line at µ0H . 0.55 T and f > 10 GHz with linear field de- out-of-plane orientation of magnetic field along the c-axis pendence (indicated with the red line), and an absorption (black curve). Following Refs. [14, 28–31], these magneti- “patch” at 0.55 . µ0H . 0.58 T and f > 20 GHz (con- zation curves indicate the helical spin order for the zero- fined schematically with blue dashed curve). A ferromag- field ground state with the helicity Nh > 4.5 and the he- netic resonance line with the negative slope is commonly lix axis along the c crystal axis. The magnetization jump attributed to some antiferromagnetic response. As will at in-plane field µ0H ≈ 0.6 T, which is about a half of become clear in the next section, the spectrum line and x the saturation field along ab-planes µ0HS ≈ 1.25 T, cor- the patch are signatures of the ferromagnetic resonance of responds to the meta-magnetic helix-to-fan phase transi- Eu layers in helical phase and in fan phase, respectively, tion of the first order. These statements will be confirmed while the transition between absorption features is due in section IV. to the meta-magnetic helix-to-fan phase transition.
C. Ferromagnetic resonance measurements III. THEORY Ferromagnetic resonance spectroscopy was performed using the VNA-FMR flip-chip approach[32, 33]. Cleaved A. Magnetization of helical spin chains EuFe2As2 sample was glued on top of the transmis- sion line of coplanar waveguide. The waveguide with The spin configuration in the helical anti-ferromagnet impedance 50 Ohm and the width of the transmission line and its dependence on magnetic field can be obtained by 0.5 mm is patterned on a Arlon AD1000 copper board employing the macrospin approximation for sub-lattice and is equipped with SMP rf connectors. The board magnetizations with the Jz1 − Jz2 Heisenberg model, as with the sample is installed in a brass sample holder. A was done in a number of previous works[4, 28–31, 34– thermometer and a heater are attached directly to the 36], and by minimizing the free energy of spin configura- holder for precise temperature control. The holder is tions. Each macrospin characterizes coherent magnetiza- placed in a home-made superconducting solenoid inside tion of an individual layer. The total free energy of the 3
1D spin chains) and the ground state configuration for the spin-system depends on exchange field parameters Hz1 and Hz2. In case when both Hz1 < 0 and Hz2 < 0 the ground state is ferromagnetic. In case when Hz1 > 0 and Hz2 < 0 the ground state is anti-ferromagnetic. He- lical configurations correspond to Hz1 < 0& Hz2 > 0, or Hz1 > 0 & Hz2 > 0. From Eq. 2 it follows that the min- imum of energy for the helical spin configuration at zero field is obtained with helicity Nh and the helical angle φh = φi − φi+1 =2π/Nh, which satisfies the relation H +4H cos(φ )=0. (4) FIG. 3. Coordinate system z1 z2 h This relation implies that the stable helical spin configu- ration can be obtained when |Hz1/4Hz2| < 1. In case of the saturation of the spin system along the x- macrospin configuration with N macrospin Eu-layers is direction (in Fig. 3 φH = 0 and θH = π/2) minimization N of the free energy of spin chain provides the saturation 1 2 field [28] F = [M~ iH~ + M~ iNM~ i − Ku cos (θi)+ X 2 (1) i=1 x HS =2Hz1(1 − cos(φh))+2Hz2(1 − cos(2φh)). (5) + Jz1M~ iM~ i+1 + Jz2M~ iM~ i+2], x From magnetization data in Fig. 1 it follows that µ0HS = where the first term is the Zeeman energy with the exter- 1.25 T. In case of the saturation of the spin system along nal field H~ , the second term is the demagnetizing energy the z-direction (in Fig. 3 θH = 0) minimization of the free of individualr Eu-layers with N being the demagnetiz- energy of spin chain provides the saturation field [35] ing factor, Ku is the out-of-plane uniaxial magnetic ani- Hz =2H (1 − cos(φ ))+2H (1 − cos(2φ )) + M . zotropy of the macrospin layer, Jz1 and Jz2 are the ex- S z1 h z2 h eff change interaction coefficients between the nearest neigh- (6) z bor and the next-nearest neighbor layers, respectively. In From magnetization data in Fig. 1 it follows that µ0HS = spherical coordinates (the film sample is in x-y plane, θ 1.95 T. and φ are the polar and the azimuthal angles, see Fig. 3)
1.0
N = 5 the general expression for energy of the particular layer h
N = 6
h
0.8 i is reduced to s M / M
1 0.6 f = F /M = − H cos(α ) − M cos2(θ )+ i i s i,H 2 eff i
0.4
1 1 Magnetization + Hz1 cos(αi−1,i)+ Hz1 cos(αi,i+1)+ 2 2 0.2 1 1
+ Hz2 cos(αi−2,i)+ Hz2 cos(αi,i+2), 0.0 2 2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Magnetic field H, T (2) 0 where Ms is the saturation magnetization of the Eu layer, Meff = Ms − 2Ku/Ms is the effective satura- FIG. 4. Theoretical dependencies M(H) for helical magnets tion magnetization, which incorporates the out-of-plane with Nh = 5 and Nh = 6. anisotropy Ku, Hz1(z2) = Jz1(z2)Ms are the correspond- ing exchange fields, cos αi,H and cos αi,j are cosines be- x tween the macrospin vector i and external field, and be- According to Eqs. 4 and 5 the saturation field HS de- tween macrospin vectors i and j, respectively. In spheri- fines both parameters of the exchange interaction Hz1 cal coordinates the cosine between two vectors is and Hz2 in their relation with the helicity Nh. According to Eq. 5 and 6 the difference in saturation fields provides cos(αi,j ) = sin(θi) sin(θj ) cos(φi − φj )+ the effective saturation magnetization µ (Hz − Hx) = (3) 0 S S + cos(θi)cos(θj ). µ0Meff =0.72 T, which recovers the result of the Stoner- Wohlfarth model for thin films. Thus, the system of three Characteristic parameters of the spin system can equations (Eqs. 4, 5, and 6) contains four independent be obtained by considering energy minima ∂f/∂θi = parameters. ∂f/∂φi = 0 of particular spin configurations. In case Following Ref. [14], we employ the Nelder-Mead sim- when the out-of-plane z-direction represents the hard plex numerical algorithm for search of local minima of axis, which is the case of this work, at zero field the free energy (Eq. 2) for a spin configuration employ- macrospin magnetizations are locked to x-y plane (a.k.a. ing periodic boundary conditions. We use the leap-frog 4
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
N =5 N =6 1000
80
50 h h
3,4
3 1,3,4,6
70
2,5 , GHz , f
40 GHz ,
2,4 f
2,5 , GHz , 2,5 60 f
3,4 1,6
1
50
2 30 2,5
1,6
3 100 40
3,4
1
1,5 20
30 1
2 2
1 4
3 5
Resonance frequency frequency Resonance 20
MF resonances Resonance frequency frequency Resonance
3 MF resonances 10
Collective optical spin wave mode with =N
Collective optical spin wave mode with =N h 4 Natural resonance frequency frequency resonance Natural 10 h
4 Collective acoustic mode
Collective acoustic mode 6 5 5
0 0 10
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 2 3 4 5 6 7 8 9 10 11
Helicity N
Magnetic field H, T Magnetic field H, T h (c) (a) (b)
0 0
FIG. 5. a,b) The dependence of ferromagnetic resonance frequencies fr on magnetic field for Nh = 5 (a) and Nh = 6 (b). Black solid curves show molecular field resonances. Red dashed curve shows the collective spin-wave response with the wavelength λ = Nh (optical mode). Blue dotted line shows the collective acoustic spin response. Pictograms in (a,b) illustrate the helix and fan phases, as well as the saturated phase, and attribute magnetization orientations with the MFL resonance curves. c) The dependence of the natural MF resonance frequency on helicity fr(Nh,H = 0). Red dashed line in (c) correlates the experimentally-obtained natural resonance frequency in Fig. 2 with the helicity.
numerical scheme while sweeping the magnetic field up determine static or molecular field (MF) interactions be- x x from 0 to 1.3HS and down from 1.3HS to 0, and calcu- tween layers, while off-diagonal elements fθiθj , fφiφj , and late the free energy at each field step and each field-sweep fθiφj (i 6= j) determine dynamic collective interactions. direction. The resulting spin configuration is selected by Figure 5a,b shows the dependence of ferromagnetic res- the absolute minimum of free energy at a given field. Em- onance frequencies fr on magnetic field for Nh = 5 and ployment of the leap-from scheme and initial helix or fan Nh = 6. Black curves in Fig. 5a,b show the MF resonance conditions[31] speeds up the convergence considerably. when off-diagonal elements fθiθj and fθiφj in Eq. 7 are As example, below we consider Nh = 5 and Nh = 6. Fig- set to 0. As follows from Fig. 5a,b several MF resonance ure 4 shows the dependence of magnetization on magnetic curves are expected at magnetic fields below and above x field for Nh = 5 and Nh = 6 and experimentally defined the the spin-flop transition Hsf ≈ H /2. Below Hsf the x z S saturation fields µ0HS =1.25 T, µ0HS =1.95 T. dependence of the resonance frequency on magnetic field shows a negative slope for spins which are counter-aligned with the external field and a positive slope for spins which B. Ferromagnetic resonance in helical spin chains are aligned with the magnetic field (see pictograms in Fig. 5a,b). Above Hsf all MF resonance curves show Once the dependence of orientations of Eu macrospin positive slope. At H = Hsf the abrupt change in the layers on the bias field is determined for an arbitrary resonance spectrum is due to the helix-to-fan transition. helicity, a single step required for derivation of ferro- By comparing the MF resonance spectrum for two he- magnetic resonance properties is application of the Suhl- licities in Fig. 5a,b the dependence of the natural MF res- Smit-Beljers approach[37, 38] extended for the case of onance frequency (i.e., fr(H = 0)) on Nh should be no- magnetization dynamics in coupled magnetic multilayers ticed. The dependence fr(Nh,H = 0) can be expressed [39–41]. With this approach the following set of equa- analytically from Eqs. 2, 4, 5, 6, and 7 (see Appendix) tions of motion for magnetization vector in each Eu layer and is shown in Fig. 5c. defines the collective dispersion of the spin system: Next, we consider collective modes when off-diagonal
i+2 elements in Eq. 7 are included. In general, the eigenvalue iω problem with the 2N × 2N matrix provides 2N eigenfre- γ sin(θi)δθi = fφiθj δθj + fφiφj δφj , j=Pi−2 quencies [ω] and eigenvectors [δθ ] and [δφ ] with a half (7) i i i+2 of eigenfrequencies being positive. The sum of eigenvec- − iω sin(θ )δφ = f δθ + f δφ , γ i i θiθj j θiφj j tor elements is equivalent to the dynamical susceptibil- j=Pi−2 ity. Calculations show that at H (see Fig. 3), fθiθj and fθiφj are corresponding second- lent to the helicity Nh (red dashed curves in Fig. 5a,b). order partial derivatives of the free energy (Eq. 2), ω Intuitively, this result should be understood as follows. is the eigen-frequency of magnetization precession, and Conventionally in saturated spin systems even collective γ = 28 GHz/T is the gyromagnetic ratio. The combina- modes show zero susceptibility because a half of spins tion of equations 7 for each of N considered layers can precess in phase and the other half precess in anti-phase. be represented as the eigenvalue problem with 2N × 2N However, in helical phase at H Fig. 5a and 2-5 in Fig. 5b) and the other part (spins 1,5 two types of resonance lines is expected: the individual in Fig. 5a and 1,6 in Fig. 5b) is aligned against the bias molecular-field-like response and the collective spin wave field. Then these fractions precess in opposite phases response with optical mode at fields below the metamag- their partial susceptibilities add up. For the same reason netic transition, and acoustic mode above the metamag- the acoustic mode is not observed at H IV. DISCUSSIONS This work was supported by the Russian Science Foundation and by the Ministry of Science and Higher By collating the experimental resonance spectrum in Education of the Russian Federation. Fig. 2 with theoretical plots in Fig. 5a,b we conclude that the spectral line at µ0H < 0.5 T with the negative VII. APPENDIX: EXPRESSIONS FOR MF field-slope (indicated with red line) is molecular-field-like NATURAL RESONANCE FREQUENCY resonance response of Eu layers in the helical phase. The the absorption “patch” in Fig. 2 at 0.5 <µ0H < 0.8 T is The dependence fr(φh,H = 0) with φh = 2π/Nh can also the molecular field resonance response, which ap- be expressed analytically from Eqs. 2, 4, 5, 6, and 7 using pears due to the helix-to-fan transition. The natural the following set of recurrent expressions: resonance frequency in experimental resonance spectrum (red line Fig. 2 at H = 0) is fr(H = 0) ≈ 28 GHz. In x Fig. 5c this value corresponds to helicity Nh ≈ 4.9, which 2HS cos(φh) Hz1 = 2 (8) is consistent with the previous report [14]. 4 cos(φh) − 4cos (φh) + cos(2φh) − 1 x V. 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