Incommensurate Wave Vector Research Articles

Incommensurate Magnetic Order in Hole‐Doped Infinite‐Layer Nickelate Superconductors due to Competing Magnetic Interactions

AbstractThe discovery of superconductivity in hole‐doped infinite‐layer nickelates has fueled intense research to identify the critical factor responsible for high‐Tc superconductivity. Magnetism and superconductivity are closely entangled, and elucidating the magnetic interactions in hole‐doped nickelates is critical for understanding the pairing mechanism. Here, these calculations based on the generalized Bloch theorem (GBT) and magnetic force theorem (MFT) consistently reveal that hole doping stabilizes an incommensurate (IC) spin state and increases the IC wave vector continuously, in a way strikingly similar to hole‐doped cuprates. Going further, a nonlinear Heisenberg model including first‐neighbor and third‐neighbor in‐plane magnetic interactions is developed. The analytical solutions successfully reproduce GBT and MFT results and reveal that the competition between the two magnetic interactions is the decisive factor for the IC magnetic transition. Eventually, by analyzing the doping‐controlled spin splitting of band and orbital‐contributed exchange interactions, direct links between hole doping, magnetization, exchange constants, and magnetic order are established. This discovery of the IC spin state, new understanding of its electronic origin, and establishment of direct connection with the paring electrons radically change the current understanding of the magnetic properties in hole‐doped NdNiO2 and open new perspectives for the superconducting mechanism in nickelates superconductors.

Detection of a pair density wave state in UTe2

Spin-triplet topological superconductors should exhibit many unprecedented electronic properties, including fractionalized electronic states relevant to quantum information processing. Although UTe2 may embody such bulk topological superconductivity1–11, its superconductive order parameter Δ(k) remains unknown12. Many diverse forms for Δ(k) are physically possible12 in such heavy fermion materials13. Moreover, intertwined14,15 density waves of spin (SDW), charge (CDW) and pair (PDW) may interpose, with the latter exhibiting spatially modulating14,15 superconductive order parameter Δ(r), electron-pair density16–19 and pairing energy gap17,20–23. Hence, the newly discovered CDW state24 in UTe2 motivates the prospect that a PDW state may exist in this material24,25. To search for it, we visualize the pairing energy gap with μeV-scale energy resolution using superconductive scanning tunnelling microscopy (STM) tips26–31. We detect three PDWs, each with peak-to-peak gap modulations of around 10 μeV and at incommensurate wavevectors Pi=1,2,3 that are indistinguishable from the wavevectors Qi=1,2,3 of the prevenient24 CDW. Concurrent visualization of the UTe2 superconductive PDWs and the non-superconductive CDWs shows that every Pi:Qi pair exhibits a relative spatial phase δϕ ≈ π. From these observations, and given UTe2 as a spin-triplet superconductor12, this PDW state should be a spin-triplet PDW24,25. Although such states do exist32 in superfluid 3He, for superconductors, they are unprecedented.

Incommensurate charge density wave in multiband intermetallic systems exhibiting competing orders

The appearance of an incommensurate charge density wave vector $\textbf{Q} = (Q_x,Q_y)$ on multiband intermetallic systems presenting commensurate charge density wave (CDW) and superconductivity (SC) orders is investigated. We consider a two-band model in a square lattice, where the bands have distinct effective masses. The incommensurate CDW (inCDW) and CDW phases arise from an interband Coulomb repulsive interaction, while the SC emerges due to a local intraband attractive interaction. For simplicity, all the interactions, the order parameters and hybridization between bands are considered $\textbf{k}$-independent. The multiband systems that we are interested are intermetallic systems with a $d$-band coexisting with a large $c$-band, for which a mean-field approach has proved suitable. We obtain the eigenvalues and eigenvectors of the Hamiltonian numerically and minimize the free energy density with respect to the diverse parameters of the model by means of the Hellmann-Feynman theorem. We investigate the system in real as well as momentum space and we find an inCDW phase with wave vector $\textbf{Q} = (\pi, Q_y) = (Q_x, \pi)$. Our numerical results show that the arising of an inCDW state depends on parameters, such as the magnitude of the inCDW and CDW interactions, band filling, hybridization and the relative depth of the bands. In general, inCDW tends to emerge at low temperatures, away from half-filling. We also show that, whether the CDW ordering is commensurate or incommensurate, large values of the relative depth between bands may suppress it. We discuss how each parameter of the model affects the emergence of an inCDW phase.

Spin fluctuations in Sr1.8La0.2RuO4

We use inelastic neutron scattering to study spin fluctuations in Sr$_{1.8}$La$_{0.2}$RuO$_4$, where Lanthanum doping triggers a Lifshitz transition by pushing the van Hove singularity in the $\gamma$ band to the Fermi energy. Strong spin fluctuations emerge at an incommensurate wave vector $\mathbf{Q}_{ic} = (0.3,0.3)$, corresponding to the nesting vector between $\alpha$ and $\beta$ Fermi sheets. The incommensurate antiferromagnetic fluctuations shift toward $(0.25,0.25)$ with increasing energy up to ${\sim}110$ meV. By contrast, scatterings near the ferromagnetic wave vectors $\mathbf{Q} = (1,0)$ and $(1,1)$ remain featureless at all energies. This contradicts the weak-coupling perspective that suggests a sharp enhancement of ferromagnetic susceptibility due to the divergence of density of states in the associated $\gamma$ band. Our findings imply that ferromagnetic fluctuations in Sr$_2$RuO$_4$ and related materials do not fit into the weak-coupling paradigm, but instead are quasi-local fluctuations induced by Hund's coupling. This imposes significant constraints for the pairing mechanism involving spin fluctuations.

Evolution of static charge density wave order, amplitude mode dynamics, and suppression of Kohn anomalies at the hysteretic transition in EuTe4

Charge density wave (CDW) induces periodic spatial modulation of the charge density that is commensurate or incommensurate with the host lattice periodicity, and leads to partial or complete electronic band-gap opening at the Fermi level (${E}_{\mathrm{F}}$). The recent finding of unconventional hysteresis within the CDW phase of ${\mathrm{EuTe}}_{4}$, not observable in other rare-earth tellurides $R{\mathrm{Te}}_{n}$ ($n=2$, 3), has highlighted the role of the relative phase of CDW distortion in weakly coupled Te layers. However, detailed structural and dynamical characterization of CDW distortion on the hysteretic transition is lacking. Here we report on the static CDW order, dynamics of the amplitude mode, and their evolution on the hysteretic transition using meV resolution elastic and inelastic x-ray scattering. We discover previously unidentified multiple commensurate and incommensurate CDW wave vectors ${\mathbf{q}}_{\mathrm{CDW}}$ along all three crystallographic axes. Importantly, we find that the previously reported $b$-axis CDW peak is coupled with the interlayer CDW phase and consequently co-occurs with the doubling of the unit cell along the $c$ axis. We confirm the presence of the competing $a$-axis CDW order but found it to be four orders of magnitude weaker than the $b$-axis CDW. Furthermore, we observe multiple Kohn anomalies at ${\mathbf{q}}_{\mathrm{CDW}}$ driven by Fermi surface nesting and hidden nesting, confirming earlier reports based on electronic and lattice susceptibility simulations. The amplitude mode and Kohn anomalies are found to suppress on unconventional hysteretic transition, suggesting the presence of nondegenerate metastable states, which we identify from the x-ray scattering measurements and simulations.

Charge and spin correlations in insulating and incoherent metal states of twisted bilayer graphene

We study electronic, charge, and magnetic properties of twisted bilayer graphene with fillings $2\leq n\leq 6$ per moire unit cell within the recently introduced formulation of extended dynamical mean-field theory (E-DMFT) for two-sublattice systems. We use previously obtained hopping parameters between the states, described by Wannier functions, centered at the lattice spots of AB and BA stacking, and the long-range Coulomb interaction, obtained within cRPA analysis. We show that account of spin exchange between AB and BA nearest neighbour spots is crucial to introduce charge and spin correlations between these spots. Account of this exchange yields preferable concentration of electrons in the same valley, with the tendency of parallel spin alignment of electrons in AB and BA spots, in agreement with earlier results of the strong coupling analysis, suggested SU(2)$\times$SU(2) emergent spin-valley symmetry. The local spectral functions show almost gapped state at the fillings $n=2,4,6$, and incoherent metal state for the other fillings. We find that in both cases the local states of electrons have rather long lifetime. At the same time, the non-local charge- and spin susceptibilities, obtained within the ladder approximation, are peaked at incommensurate wave vectors, which implies that the above discussed ordering tendencies are characterised by an incommensurate pattern.

Moiré-like Superlattice Generated van Hove Singularities in a Strained CuO2 Double Layer

While it is known that the double-layer Bi2Sr2CaCu2O8+y (BSCCO) cuprate superconductor exhibits a one-dimensional (1D) incommensurate superlattice (IS), the effect of IS on the electronic structure remains elusive. Following the recent shift of interest from an underdoped phase to optimum and overdoped phases in BSCCO by increasing the hole doping x, controlled by the oxygen interstitials concentration y, here we focus on the multiple splitting of the density of states (DOS) peaks and emergence of higher order van Hove singularities (VHS) due to the 1D incommensurate superlattice. It is known that the 1D incommensurate wave vector q=ϵb (where b is the reciprocal lattice vector of the orthorhombic lattice) is controlled by the misfit strain between different atomic layers in the range 0.209–0.215 in BSCCO and in the range 0.209–0.25 in Bi2Sr2Ca1−xYxCu2O8+y (BSCYCO). This work reports the theoretical calculation of a complex pattern of VHS due to the 1D incommensurate superlattice with large 1D quasi-commensurate supercells with the wave vector ϵ=9/η in the range 36>η>43. The similarity of the complex VHS splitting and appearing of higher order VHS in a mismatched CuO2 bilayer with VHS due to the moiré lattice in strained twisted bilayer graphene is discussed. This makes a mismatched CuO2 bilayer quite promising for constructing quantum devices with tuned physical characteristics.

Peculiarities of incommensurate modulation behavior during superstructure creation in case of potential symmetry n = 4

Type or paste your abstract here as prescribed by the journal’s instructions for authors. Type or paste your abstract here as prescribed by the journal’s instructions for authors. Type or paste your abstract here as prescribed by the journal’s instructions for authors. Type or paste your abstract here. This paper deals with dynamics of incommensurate superstructure at the moment of its formation. Lyapunov exponent value spectra, phase portraits, incommensurate modulation wave vector behavior, Fourier spectra and maps of incommensurate phase dynamic modes under changed parameters of long-range interaction (Т) and anisotropic interaction (K) in interval 0.0 ÷ 1.0 were analyzed. The calculations were made in Python environment applying Skipy, JiTCODE libraries. Under low values of Т and K parameters (0.0015 ÷ 0.03) a chaotic state occurs in this system. It is determined by the interaction between different spatial areas of the correlated motion of tetrahedral groups. The interaction of spatial areas of the correlated motion of tetrahedral groups violates the modulation pattern of tetrahedral group motion causing the superstructure chaotic state. The study of Fourier spectra has shown that interaction of spatial areas of tetrahedral group correlated motion of causes the chaotic state that is non-stable and is destroyed by long-range interaction. The impact of surface energy on the interaction of spatial areas of tetrahedral group correlated motion is determined by decreased amplitude of vibrations of tetrahedral groups causing the disappearance of the chaotic state.

A broken translational symmetry state in an infinite-layer nickelate

A defining signature of strongly correlated electronic systems is the existence of competing phases with similar ground state energies, resulting in a rich phase diagram. While in the recently discovered nickelate superconductors, a high antiferromagnetic exchange energy has been reported, which implies the existence of strong electronic correlations, signatures of competing phases have not yet been observed. Here, we uncover a charge order (CO) in infinite-layer nickelates La1-xSrxNiO2 using resonant x-ray scattering across the Ni L-edge. In the parent compound, the CO arranges along the Ni-O bond direction with an incommensurate wave vector (0.344+/-0.002, 0) r.l.u., distinct from the stripe order in other nickelates which propagates along a direction 45 degree to the Ni-O bond. The CO resonance profile indicates that CO originates from the Ni 3d states and induces a parasitic charge modulation of La electrons. Upon doping, the CO diminishes and the ordering wave vector shifts toward a commensurate value of 1/3 r.l.u., indicating that the CO likely arises from strong correlation effects and not from Fermi surface nesting.

Spin stripe fluctuations in antiferromagnetic Pr2−xSrxNiO4+δ

We report inelastic neutron scattering study of the antiferromagnetic spin stripe fluctuations above the spin stripe melting temperature ${\mathrm{T}}_{so}\ensuremath{\approx}190\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ in the hole doped ${\mathrm{Pr}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{NiO}}_{4+\ensuremath{\delta}}$ with stripe incommensurability $\ensuremath{\epsilon}=0.33$. The fluctuations are nondispersive and detected upto 10 meV at the incommensurate wave vector indicating a persisting instantaneous spin and charge stripe correlation above ${\mathrm{T}}_{so}$, while they are strongly diminished already below the charge stripe melting temperature ${\mathrm{T}}_{co}\ensuremath{\approx}255\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, which indicates that static charge stripe order is essential for the dynamical spin stripe correlation to exist. Furthermore, it also suggests that the presence of spin stripe fluctuations is not a prerequisite for the formation of static charge stripes.

Possible quadrupole-order-driven commensurate-incommensurate phase transition in B20 CoGe

The B20-type cobalt germanide CoGe was investigated by measuring the specific heat, resistivity, and $^{59}$Co nuclear magnetic resonance (NMR). We observed a phase transition at $T_Q=13.7$ K, evidenced by a very narrow peak of the specific heat and sharp changes of the nuclear spin-spin ($T_2^{-1}$) and spin-lattice ($T_1^{-1}$) relaxation rates. The fact that the entropy release is extremely small and the Knight shift is almost independent of temperature down to low temperatures as anticipated in a paramagnetic metal indicates that the $T_Q$ transition is of non-magnetic origin. In addition, we detected a crossover scale $T_0\sim30$ K below which the resistivity and the NMR linewidth increase, and $T_1^{-1}$ is progressively distributed in space, that is, a static and dynamical spatial inhomogeneity develops. While the order parameter for the $T_Q$ transition remains an open question, a group-theoretical analysis suggests that the finite electric quadrupole density arising from the low local site symmetry at cobalt sites could drive the crystal symmetry lowering from the P2$_1$3 symmetry that is commensurate to the R3 symmetry with an incommensurate wavevector, which fairly well accounts for the $T_Q$ transition. The quadrupole-order-driven commensurate-incommensurate phase transition may be another remarkable phenomenon arising from the structural chirality inherent in the noncentrosymmetric B20 family.

Coupled Hubbard ladders at weak coupling: Pairing and spin excitations

The Hubbard model provides a simple framework in which one can study how certain aspects of the electronic structure of strongly interacting systems can be tuned to optimize the superconducting pairing correlations and how these changes affect the mechanisms giving rise to them. Here we use a weak-coupling random phase approximation to study a two-dimensional Hubbard model with a unidirectional modulation of the hopping amplitudes as the system evolves from the uniform square lattice to an array of weakly coupled two-leg ladders. We find that the pairing correlations retain their dominant $d_{x^2-y^2}$-wave like structure and that they are significantly enhanced for a slightly modulated lattice. This enhancement is traced backed to an increase in the strength of the spin-fluctuation pairing interacting due to favorable Fermi surface nesting in the modulated system. We then use a random-phase approximation BCS framework to examine the evolution of the neutron resonance in the superconducting state. We find that it changes only weakly for moderate modulations, but breaks up into two distinct resonances at incommensurate wave-vectors in the limit of weakly coupled ladders.

Low-dimensional antiferromagnetic fluctuations in the heavy-fermion paramagnetic ladder compound UTe2

Inelastic-neutron-scattering measurements were performed on a single crystal of the heavy-fermion paramagnet UTe$_2$ above its superconducting temperature. We confirm the presence of antiferromagnetic fluctuations with the incommensurate wavevector $\mathbf{k}_1=(0,0.57,0)$. A quasielastic signal is found, whose momentum-transfer dependence is compatible with fluctuations of magnetic moments $\mu\parallel\mathbf{a}$, with a sine-wave modulation of wavevector $\mathbf{k}_1$ and in-phase moments on the nearest U atoms. Low dimensionality of the magnetic fluctuations, consequence of the ladder structure, is indicated by weak correlations along the direction $\mathbf{c}$. These fluctuations saturate below the temperature $T_1^*\simeq15$~K, in possible relation with anomalies observed in thermodynamic, electrical-transport and nuclear-magnetic-resonance measurements. The absence or weakness of ferromagnetic fluctuations, in our data collected at temperatures down to 2.1 K and energy transfers from 0.6 to 7.5 meV, is emphasized. These results constitute constraints for models of magnetically-mediated superconductivity in UTe$_2$.

Model of the low temperature magnetic phases of gadolinium gallium garnet

The magnetic behaviour of gadolinium gallium garnet in an external magnetic field at zero temperature is considered. For high fields a classical spin model of the gadolinium ions predicts a spin configuration that is periodic at the level of the smallest repeating unit cell. The quantum version of the model is treated via a truncated Holstein–Primakoff transformation with axes defined by the classical spin configuration, and the magnon excitation bands are calculated. The model predicts a transition in the field range of 1.9–2.1 T, sensitive to the direction of the applied field, which is caused by one or more magnon modes becoming soft as the field is decreased. In general the soft modes occur at incommensurate wavevectors and therefore break the periodicity of the spin configuration below the transition. One exception occurs when the field aligns with one of the principle crystal axes, in which case periodicity of the spin configuration is found to be maintained on a larger crystallographic cubic cell even below the transition. This simple case is studied in more detail. Comparisons are drawn with existing experimental data, and further experimental tests of the model are suggested.

Mn-induced Fermi-surface reconstruction in the SmFeAsO parent compound

The electronic ground state of iron-based materials is unusually sensitive to electronic correlations. Among others, its delicate balance is profoundly affected by the insertion of magnetic impurities in the FeAs layers. Here, we address the effects of Fe-to-Mn substitution in the non-superconducting Sm-1111 pnictide parent compound via a comparative study of SmFe_{1-x}Mn_{x}AsO samples with x(text{Mn})= 0.05 and 0.10. Magnetization, Hall effect, and muon-spin spectroscopy data provide a coherent picture, indicating a weakening of the commensurate Fe spin-density-wave (SDW) order, as shown by the lowering of the SDW transition temperature T_text{SDW} with increasing Mn content, and the unexpected appearance of another magnetic order, occurring at T^{*} approx 10 and 20 K for x=0.05 and 0.10, respectively. We attribute the new magnetic transition at T^{*}, occurring well inside the SDW phase, to a reorganization of the Fermi surface due to Fe-to-Mn substitutions. These give rise to enhanced magnetic fluctuations along the incommensurate wavevector varvec{Q}_2 =(pi pm delta ,pi pm delta ), further increased by the RKKY interactions among Mn impurities.

Multiple fermion scattering in the weakly coupled spin-chain compound YbAlO3

The Heisenberg antiferromagnetic spin-1/2 chain, originally introduced almost a century ago, is one of the best studied models in quantum mechanics due to its exact solution, but nevertheless it continues to present new discoveries. Its low-energy physics is described by the Tomonaga-Luttinger liquid of spinless fermions, similar to the conduction electrons in one-dimensional metals. In this work we investigate the Heisenberg spin-chain compound YbAlO3 and show that the weak interchain coupling causes Umklapp scattering between the left- and right-moving fermions and stabilizes an incommensurate spin-density wave order at q = 2kF under finite magnetic fields. These Umklapp processes open a route to multiple coherent scattering of fermions, which results in the formation of satellites at integer multiples of the incommensurate fundamental wavevector Q = nq. Our work provides surprising and profound insight into bandstructure control for emergent fermions in quantum materials, and shows how neutron diffraction can be applied to investigate the phenomenon of coherent multiple scattering in metals through the proxy of quantum magnetic systems.

Magnetic order and fluctuations in the quasi-two-dimensional planar magnet .

We use neutron scattering to investigate spin excitations in , which has a -axis incommensurate helical structure of the two-dimensional (2D) in-plane ferromagnetic (FM) ordered layers for . By comparing the wave vector and energy dependent spin excitations in helical ordered and paramagnetic , we find that Ni doping, while increasing lattice disorder in , enhances quasi-2D FM spin fluctuations. However, our band structure calculations within the combined density functional theory and dynamic mean field theory (DFT+DMFT) failed to generate a correct incommensurate wave vector for the observed helical order from nested Fermi surfaces. Since transport measurements reveal increased in-plane and -axis electrical resistivity with increasing Ni doping and associated lattice disorder, we conclude that the helical magnetic order in may arise from a quantum order-by-disorder mechanism through the itinerant electron mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions.

Magnetic‐Field‐Controlled Quantum Critical Points in the Triangular Antiferromagnetic Cs2CuCl4‐xBrx Mixed System

AbstractThe triangular antiferromagnetic Cs2CuCl4‐xBrx mixed system is studied by neutron single‐crystal diffraction in magnetic field. It shows a rich magnetic phase diagram consisting of four regimes depending on the Br concentration and is characterized by different exchange coupling mechanisms. For the investigated compositions from regime I (0 < x ≤ 1.5), a critical magnetic field Bc is found for a Br concentration x = 0.8 at Bc = 8.10(1) T and for x = 1.1 at Bc = 7.73(1) T and from regime IV (3.2 < x < 4) for x = 3.3 at Bc = 0.99(3) T. For magnetic fields larger than the respective Bc, magnetic superlattice reflections of these compounds are not found. The incommensurate magnetic wave vector q = (0, 0.470, 0) appears below the ordering temperature TN = 0.51(1) K for Cs2CuCl3.2Br0.8, and q = (0, 0.418, 0) below TN = 1.00(6) K for Cs2CuCl0.3Br3.7. Neutron diffraction experiments at around 60 mK for x = 3.7 in a magnetic field show the critical magnetic field at Bc = 7.94(16) T and the formation of the second magnetic phase at around 8.5 T depending on the temperature. Inelastic neutron scattering experiments for the compound from regime III (2 < x ≤ 3.2) with x = 2.2 show dynamical correlations at a temperature around 50 mK giving evidence for a spin liquid phase.

Charge carrier drop at the onset of pseudogap behavior in the two-dimensional Hubbard model

We show that antiferromagnetic spin-density wave order in the two-dimensional Hubbard model yields a drop of the charge carrier density as observed in recent transport measurements for cuprate superconductors in high magnetic fields upon entering the pseudogap regime. The amplitude and the (generally incommensurate) wave vector of the spin-density wave is obtained from dynamical mean-field theory (DMFT). An extrapolation of the finite temperature results to zero temperature yields an approximately linear doping dependence of the magnetic gap $\Delta(p) \propto p^*-p$ in a broad doping range below the critical doping $p^*$. The magnetic order leads to a Fermi surface reconstruction with electron and hole pockets, where electron pockets exist only in a restricted doping range below $p^*$. DC charge transport properties are computed by combining the renormalized band structure as obtained from the DMFT with a doping-independent phenomenological scattering rate. A pronounced drop of the longitudinal conductivity and the Hall number in a narrow doping range below $p^*$ is obtained.

Complex magnetic order in the decorated spin-chain system Rb2Mn3(MoO4)3(OH)2

Macroscopic magnetic properties and microscopic magnetic structure of Rb$_2$Mn$_3$(MoO$_4$)$_3$(OH)$_2$ (space group $Pnma$) are investigated by magnetization, heat capacity and single-crystal neutron diffraction measurements. The compound's crystal structure contains bond-alternating [Mn$_3$O$_{11}$]$^{\infty}$ chains along the $b$-axis, formed by isosceles triangles of Mn ions occupying two crystallographically nonequivalent sites (Mn1 site on the base and Mn2 site on the vertex). These chains are only weakly linked to each other by nonmagnetic oxyanions. Both SQUID magnetometry and neutron diffraction experiments show two successive magnetic transitions as a function of temperature. On cooling, it transitions from a paramagnetic phase into an incommensurate phase below 4.5~K with a magnetic wavevector near ${\bf k}_{1} = (0,~0.46,~0)$. An additional commensurate antiferromagnetically ordered component arises with ${\bf k}_{2} = (0,~0,~0)$, forming a complex magnetic structure below 3.5~K with two different propagation vectors of different stars. On further cooling, the incommensurate wavevector undergoes a lock-in transition below 2.3~K. The experimental results suggest that the magnetic superspace group is $Pnma.1'(0b0)s0ss$ for the single-${\bf k}$ incommensurate phase and is $Pn'ma(0b0)00s$ for the 2-${\bf k}$ magnetic phase. We propose a simplified magnetic structure model taking into account the major ordered contributions, where the commensurate ${\bf k}_{2}$ defines the ordering of the $c$-axis component of Mn1 magnetic moment, while the incommensurate ${\bf k}_{1}$ describes the ordering of the $ab$-plane components of both Mn1 and Mn2 moments into elliptical cycloids