arXiv subjects

B. Büchner

Publications and source records attributed to B. Büchner.

At least 37 records · Page 2Linked to original sources

arXiv2024

Ubiquitous order-disorder transition in the Mn antisite sublattice of the (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ magnetic topological insulators

Magnetic topological insulators (TIs) herald a wealth of applications in spin-based technologies, relying on the novel quantum phenomena provided by their topological properties. Particularly promising is the (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ layered family of established intrinsic magnetic TIs that can flexibly realize various magnetic orders and topological states. High tunability of this material platform is enabled by manganese-pnictogen intermixing, whose amounts and distribution patterns are controlled by synthetic conditions. Positive implication of the strong intermixing in MnSb$_2$Te$_4$ is the interlayer exchange coupling switching from antiferromagnetic to ferromagnetic, and the increasing magnetic critical temperature. On the other side, intermixing also implies atomic disorder which may be detrimental for applications. Here, we employ nuclear magnetic resonance and muon spin spectroscopy, sensitive local probe techniques, to scrutinize the impact of the intermixing on the magnetic properties of (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ and MnSb$_2$Te$_4$. Our measurements not only confirm the opposite alignment between the Mn magnetic moments on native sites and antisites in the ground state of MnSb$_2$Te$_4$, but for the first time directly show the same alignment in (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ with n = 0, 1 and 2. Moreover, for all compounds, we find the static magnetic moment of the Mn antisite sublattice to disappear well below the intrinsic magnetic transition temperature, leaving a homogeneous magnetic structure undisturbed by the intermixing. Our findings provide a microscopic understanding of the crucial role played by Mn-Bi intermixing in (MnBi$_2$Te$_4$)(Bi$_2$Te$_3$)$_n$ and offer pathways to optimizing the magnetic gap in its surface states.

cond-mat.mtrl-sci↗

arXiv2024

Transport of Intensity Phase Retrieval in the Presence of Intensity Variations and Unknown Boundary Conditions

The so-called Transport of Intensity Equation (TIE) phase retrieval technique is widely applied in light, x-ray and electron optics to reconstruct, e.g., refractive indices, electric and magnetic fields in solids. Here, we present a largely improved TIE reconstruction algorithm, which properly considers intensity variations as well as unknown boundary conditions in a finite difference implementation of the Transport of Intensity partial differential equation. That largely removes reconstruction artifacts encountered in state-of-the-art Poisson solvers of the TIE, and hence significantly increases the applicability of the technique.

cond-mat.mtrl-sci↗

arXiv2023

Interplay of topology and antiferromagnetic order in two-dimensional van der Waals crystals of (NixFe1-x)2P2S6

Mermin-Wagner theorem forbid spontaneous symmetry breaking of spins in one/two-dimensional systems at finite temperature and rules out the stabilization of this ordered state. However, it does not apply to all types of phase transitions in low dimensions such as topologically ordered phase rigorously shown by Berezinskii-Kosterlitz-Thouless (BKT) and experimentally realized in very limited systems such as superfluids, superconducting thin films. Quasi 2D van der Waals magnets provide an ideal platform to investigate the fundamentals of low-dimensional magnetism. We explored the 2D honeycomb antiferromagnetic single crystals of (NixFe1-x)2P2S6 with varying spins ( ) using in depth temperature dependent Raman measurements supported by first-principles calculations of the phonon frequencies. As a function of doping, a tunable transition from paramagnetic to antiferromagnetic ordering is shown via phonons reflected in the strong renormalization of the self-energy parameters of the Raman active phonon modes. An anomalously broad magnetic continuum attributed to two-magnon excitations is observed and its coupling with the phonons is revealed in the observation a Fano line asymmetry. Interestingly, the two-magnon continuum is observed only for the finite doping understood invoking underlying nature of insulator these materials belongs to, i.e. exchange interaction between transition metals via surrounding ligands (sulphur) and the resonance involving phonon modes associated with the (P2S6) cage. Quite surprisingly, we also observed renormalization of the phonon modes much below the long-range magnetic ordered temperature attributed to the topological ordered state, namely the BKT phase, which is also found to change as a function of doping. The extracted critical exponent of the order-parameter evince the signature of topologically active state driven by vortex-antivortex excitations.

cond-mat.mtrl-sci↗

arXiv2023

Large magnetocaloric effect in the kagome ferromagnet Li$_9$Cr$_3$(P$_2$O$_7$)$_3$(PO$_4$)$_2$

Single-crystal growth, magnetic properties, and magnetocaloric effect of the $S = 3/2$ kagome ferromagnet Li$_9$Cr$_3$(P$_2$O$_7$)$_3$(PO$_4$)$_2$ (trigonal, space group: $P\bar{3}c1$) are reported. Magnetization data suggest dominant ferromagnetic intra-plane coupling with a weak anisotropy and the onset of ferromagnetic ordering at $T_{\rm C} \simeq 2.6$ K. Microscopic analysis reveals a very small ratio of interlayer to intralayer ferromagnetic couplings ($J_{\perp}/J \simeq 0.02$). Electron spin resonance data suggest the presence of short-range correlations above $T_{\rm C}$ and confirms quasi-two-dimensional character of the spin system. A large magnetocaloric effect characterized by isothermal entropy change of $-ΔS_{\rm m}\simeq 31$ J kg$^{-1}$ K$^{-1}$ and adiabatic temperature change of $-ΔT_{\rm ad}\simeq 9$ K upon a field sweep of 7 T is observed around $T_{\rm C}$. This leads to a large relative cooling power of $RCP \simeq 284$ J kg$^{-1}$. The large magnetocaloric effect, together with negligible hysteresis render Li$_9$Cr$_3$(P$_2$O$_7$)$_3$(PO$_4$)$_2$ a promising material for magnetic refrigeration at low temperatures. The magnetocrystalline anisotropy constant $K \simeq -7.42 \times 10^4$ erg cm$^{-3}$ implies that the compound is an easy-plane type ferromagnet with the hard axis normal to the $ab$-plane, consistent with the magnetization data.

cond-mat.mtrl-sci↗

arXiv2023

Peculiarities of electron transport and resistive switching in point contacts on TiSe2, TiSeS and CuxTiSe2

TiSe2 has received much attention among the transition metals chalcogenides because of its thrilling physical properties concerning atypical resistivity behavior, emerging of charge density wave (CDW) state, induced superconductivity etc. Here, we report discovery of new feature of TiSe2, namely, observation of resistive switching in voltage biased point contacts (PCs) based on TiSe2 and its derivatives doped by S and Cu (TiSeS, CuxTiSe2). The switching is taking place between a low resistive mainly metallic-type state and a high resistive semiconducting-type state by applying bias voltage (usually below 0.5V), while reverse switching takes place by applying voltage of opposite polarity (usually below 0.5V). The difference in resistance between these two states can reach up to two orders of magnitude at the room temperature. The origin of the effect can be attributed to the variation of stoichiometry in PC core due to drift/displacement of Se/Ti vacancies under high electric field. Additionally, we demonstrated, that heating takes place in PC core, which can facilitate the electric field induced effect. At the same time, we did not found any evidence for CDW spectral features in our PC spectra for TiSe2. The observed resistive switching allows to propose TiSe2 and their derivatives as the promising materials, e.g., for non-volatile resistive random access memory (ReRAM) engineering.

cond-mat.mes-hall↗

arXiv2023

Crystal growth, characterization and electronic band structure of TiSeS

Layered semimetallic van der Waals materials TiSe2 has attracted a lot of attention because of interplay of a charge density wave (CDW) state and superconductivity. Its sister compound TiS2, being isovalent to TiSe2 and having the same crystal structure, shows a semiconducting behavior. The natural rises what happens at the transition point in TiSe2-xSx, which is expected for x close to 1. Here we report the growth and characterization of TiSeS single crystals and the study of the electronic structure using density functional theory (DFT) and angle-resolved photoemission (ARPES). We show that TiSeS single crystals have the same morphology as TiSe2. Transport measurements reveal a metallic state, no evidence of CDW was found. DFT calculations suggest that the electronic band structure in TiSeS is similar to that of TiSe2, but the electron and hole pockets in TiSeS are much smaller. The ARPES results are in good agreement with the calculations.

cond-mat.str-el↗

arXiv2023

Magnetic anisotropy and low-energy spin dynamics in the van der Waals compounds Mn$_{2}$P$_{2}$S$_{6}$ and MnNiP$_{2}$S$_{6}$

We report the detailed high-field and high-frequency electron spin resonance (HF-ESR) spectroscopic study of the single-crystalline van der Waals compounds Mn$_{2}$P$_{2}$S$_{6}$ and MnNiP$_{2}$S$_{6}$. Analysis of magnetic excitations shows that in comparison to Mn$_{2}$P$_{2}$S$_{6}$ increasing the Ni content yields a larger magnon gap in the ordered state and a larger g-factor value and its anisotropy in the paramagnetic state. The studied compounds are found to be strongly anisotropic having each the unique ground state and type of magnetic order. Stronger deviation of the g-factor from the free electron value in the samples containing Ni suggests that the anisotropy of the exchange is an important contributor to the stabilization of a certain type of magnetic order with particular anisotropy. At temperatures above the magnetic order, we have analyzed the spin-spin correlations resulting in a development of slowly fluctuating short-range order. They are much stronger pronounced in MnNiP$_{2}$S$_{6}$ compared to Mn$_{2}$P$_{2}$S$_{6}$. The enhanced spin fluctuations in MnNiP$_{2}$S$_{6}$ are attributed to the competition of different types of magnetic order. Finally, the analysis of the temperature dependent critical behavior of the magnon gaps below the ordering temperature in Mn$_{2}$P$_{2}$S$_{6}$ suggests that the character of the spin wave excitations in this compound undergoes a field induced crossover from a 3D-like towards 2D XY regime.

cond-mat.str-el↗

arXiv2022

Evolution of the spin dynamics in the van der Waals system $M_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$ ($\boldsymbol{M}_{\text{2}}$ = Mn$_{\text{2}}$, MnNi, Ni$_{\text{2}}$) series probed by electron spin resonance spectroscopy

In this work we report a detailed ESR spectroscopic study of the single-crystalline samples of the van der Waals compounds $M_{\text{2}}$P$_{\text{2}}$S$_{\text{6}}$ ($M_{\text{2}}$ = Mn$_{\text{2}}$, MnNi, Ni$_{\text{2}}$), performed at an excitation frequency of 9.56 GHz, in a broad range of temperatures above the magnetic order, and at different orientations of the magnetic field with respect to the sample. Analyzing temperature and angular dependences of the resonance field and of the linewidth of the Mn$_2$P$_2$S$_6$ compound we have observed a significant change of the spin dynamics from the dominance of the 3D-like fluctuations close to the magnetic order to a relative increase of the 2D-like spin fluctuations at higher temperatures. Such a behavior, which is opposite to the development of the low-D signatures in the previously studied Cr$_{\text{2}}$Ge$_{\text{2}}$Te$_{\text{6}}$ compound, can be explained by the difference in the type of magnetic order in Mn$_2$P$_2$S$_6$ and Cr$_{\text{2}}$Ge$_{\text{2}}$Te$_{\text{6}}$. On the other hand, MnNiP$_2$S$_6$ compound exhibits angular dependences of the linewidth typical for the system with 3D-like spin correlations in the whole measurement temperature range, however the 2D-like correlations can be seen in the temperature dependences of the resonance field and the linewidth. Ni$_2$P$_2$S$_6$, in turn, does not show any 2D signatures. This suggests that varying the Ni content in (Mn$_{1-x}$Ni$_x$)$_2$P$_2$S$_6$ one can control the exchange interaction, possibly also in the third dimension.

cond-mat.str-el↗

arXiv2022

Two-dimensional ferromagnetic extension of a topological insulator

Inducing a magnetic gap at the Dirac point of the topological surface state (TSS) in a 3D topological insulator (TI) is a route to dissipationless charge and spin currents. Ideally, magnetic order is present only at the surface and not in the bulk, e.g. through proximity of a ferromagnetic (FM) layer. However, such a proximity-induced Dirac mass gap has not been observed, likely due to insufficient overlap of TSS and the FM subsystem. Here, we take a different approach, namely FM extension, using a thin film of the 3D TI Bi$_2$Te$_3$, interfaced with a monolayer of the lattice-matched van der Waals ferromagnet MnBi$_2$Te$_4$. Robust 2D ferromagnetism with out-of-plane anisotropy and a critical temperature of $\text{T}_\text{c}\approx$~15 K is demonstrated by X-ray magnetic dichroism and electrical transport measurements. Using angle-resolved photoelectron spectroscopy, we observe the opening of a sizable magnetic gap in the 2D FM phase, while the surface remains gapless in the paramagnetic phase above T$_c$. This sizable gap indicates a relocation of the TSS to the FM ordered Mn moments near the surface, which leads to a large mutual overlap.

cond-mat.mtrl-sci↗

arXiv2022

Strong uniaxial pressure dependencies evidencing spin-lattice coupling and spin fluctuations in Cr$_2$Ge$_2$Te$_6$

Single crystals of Cr$_2$Ge$_2$Te$_6$ were studied by high-resolution capacitance dilatometry to obtain in-plane ($B\parallel ab$) and out-of-plane ($B\parallel c$) thermal expansion and magnetostriction at temperatures between 2 and 300 K and in magnetic fields up to 15 T. The anomalies in both response functions lead to the 'magnetoelastic' phase diagrams and separate the paramagnetic (PM), ferromagnetic low-temperature/low-field (LTF) and aligned ferromagnetic (FM) phases. The presence of two distinct thermal expansion anomalies at small fields $B\parallel ab$ of different magnetic field dependence clearly supports the scenario of an intermediate region separating PM and LTF phases and is indicative of a tricritical point. Simulations of the magnetostriction using the Stoner-Wohlfarth model for uniaxial anisotropy demonstrate that the observed quadratic-in-field behavior in the LTF phase is in line with a rotation of the spins from the preferred $c$ direction into the $ab$ plane. Both the LTF and the PM phase close to T$_{\rm C}$ exhibit very strong pressure dependencies of the magnetization, ${\partial}\ln{M_{\rm ab}}/{\partial}p_{\rm ab}$, of several hundred %/GPa and also the transition from the LTF to the FM phase strongly depends on $p_{\rm ab}$ ($\sim -$280%/GPa), indicating a strong decrease of the uniaxial anisotropy under applied in-plane pressure. Our data clearly demonstrate the relevance of critical fluctuations and magnetoelastic coupling in Cr$_2$Ge$_2$Te$_6$.

cond-mat.str-el↗

arXiv2022

Combined experimental and theoretical study of hydrostatic (He-gas) pressure effects in $α$-RuCl$_3$

We report a detailed experimental and theoretical study on the effect of hydrostatic pressure on the structural and magnetic aspects of the layered honeycomb antiferromagent $α$-RuCl$_{3}$. Magnetic susceptibility measurements performed under almost ideal hydrostatic-pressure conditions yield that the phase transition to zigzag-type antiferromagnetic order at $T_N$ = 7.3 K can be rapidly suppressed to about 6.1 K. A further suppression with increasing pressure is impeded due to the occurrence of a pressure-induced structural transition at $p \geq$ 104 MPa, accompanied by a strong dimerization of Ru-Ru bonds, which gives rise to a collapse of the magnetic susceptibility. Whereas the dimerization transition is strongly first order, as reflected by large discontinuous changes in $χ$ and pronounced hysteresis effects, the magnetic transition under varying pressure and magnetic field also reveals indications for a weakly first-order transition. We assign this observation to a strong magnetoelastic coupling in this system. Measurements of $χ$ under varying pressure in the paramagnetic regime ($T > T_N$) and before dimerization ($p <$ 100 MPa) reveal a considerable increase of $χ$ with pressure. These experimental observations are consistent with the results of ab-initio Density Functional Theory (DFT) calculations on the pressure-dependent structure and the corresponding pressure-dependent magnetic model. Comparative susceptibility measurements on a second crystal showing two consecutive magnetic transitions instead of one, indicating the influence of stacking faults. Using different temperature-pressure protocols the effect of these stacking faults can be temporarily overcome, transforming the magnetic state from a multiple-$T_N$ into a single-$T_N$ state.

cond-mat.str-el↗

arXiv2022

Low-energy excitations and magnetic anisotropy of the layered van der Waals antiferromagnet Ni$_2$P$_2$S$_6$

The quasi-two-dimensional antiferromagnet Ni$_2$P$_2$S$_6$ belongs to the family of magnetic van der Waals compounds that provides a rich material base for the realization of fundamental models of quantum magnetism in low dimensions. Here, we report high-frequency/high-magnetic field electron spin resonance measurements on single crystals of Ni$_2$P$_2$S$_6$. The results enable one to reliably determine the positive, "easy"-plane type of the single ion anisotropy of the Ni$^{2+}$ ions with the upper limit of its magnitude $\lesssim 0.7$ meV. The resonance response reveals strongly anisotropic spin fluctuations setting in shortly above the Néel temperature $T_{\rm N} = 158$ K and extending in the antiferromagnetically ordered state down to low temperatures. There, a low-energy magnon excitation gapped from the ground state by only $\sim 1$ meV was found. This magnon mode may explain unusual low temperature relaxation processes observed in a latest nuclear magnetic resonance experiments and together with the estimate of the single ion anisotropy they should enable setting up a realistic spin model for an accurate description of magnetic properties of Ni$_2$P$_2$S$_6$.

cond-mat.str-el↗

arXiv2022

Strong effects of uniaxial pressure and short-range correlations in Cr2Ge2Te6

Cr$_2$Ge$_2$Te$_6$ is a quasi-2D semiconducting van der Waals ferromagnet down to the bilayer with great potential for technological applications. Engineering the critical temperature to achieve room temperature applications is one of the critical next steps on this path. Here, we report high-resolution capacitance dilatometry studies on Cr$_2$Ge$_2$Te$_6$ single crystals which directly prove significant magnetoelastic coupling and provide quantitative values of the large uniaxial pressure effects on long-range magnetic order (${\partial}T_{\mathrm{C}}/{\partial}p_{\mathrm{c}}$ = 24.7 K/GPa and ${\partial}T_{\mathrm{C}}/{\partial}p_{\mathrm{ab}}$ = $-$15.6 K/GPa) derived from thermodynamic relations. Moderate in-plane strain is thus sufficient to strongly enhance ferromagnetism in Cr$_2$Ge$_2$Te$_6$ up to room temperature. Moreover, unambiguous signs for short-range magnetic order up to 200~K are found.

cond-mat.str-el↗

arXiv2022

Electron-phonon interaction and point contact enhanced superconductivity in trigonal PtBi2

PtBi2 is a Weyl semimetal, which demonstrates superconductivity with low critical temperature Tc ~ 0.6 K in the bulk. Here, we report our study of electron-phonon interaction (EPI) in trigonal PtBi2 by the Yanson point contact (PC) spectroscopy and presenting the observation of PC enhanced superconductivity. We show, that the Yansons PC spectra display a broad maximum around 15 meV, indicating, apparently, EPI mechanism of Cooper pairing in PtBi2. Moreover, we discovered a substantial increase of Tc up to ~ 3.5 K in PCs. The observed Tc is sufficiently higher than the bulk value, as well as detected at hydrostatic pressure. We calculated the phonon density of states and Eliashberg EPI function in PtBi2 within the framework of the density functional theory. A comparison of experimental data with theoretical calculations showed acceptable agreement. The theoretical Tc is 3.5 K, which corresponds to the experimental value.

cond-mat.supr-con↗

arXiv2022

Frustration model and spin excitations in the helimagnet FeP

The metallic compound FeP belongs to the class of materials that feature a complex noncollinear spin order driven by magnetic frustration. While its double-helix magnetic structure with a period $λ_{\text{s}} \approx 5c$, where $c$ is the lattice constant, was previously well determined, the relevant spin-spin interactions that lead to that ground state remain unknown. By performing extensive inelastic neutron scattering measurements, we obtained the spin-excitation spectra in a large part of the momentum-energy space. The spectra show that the magnons are gapped with a gap energy of $\sim$5 meV. Despite the 3D crystal structure, the magnon modes display strongly anisotropic dispersions, revealing a quasi-one-dimensional character of the magnetic interactions in FeP. The physics of the material, however, is not determined by the dominating exchange, which is ferromagnetic. Instead, the weaker two-dimensional antiferromagnetic interactions between the rigid ferromagnetic spin chains drive the magnetic frustration. Using linear spin-wave theory, we were able to construct an effective Heisenberg Hamiltonian with an anisotropy term capable of reproducing the observed spectra. This enabled us to quantify the exchange interactions in FeP and determine the mechanism of its magnetic frustration.

cond-mat.str-el↗

arXiv2021

${}^{31}$P NMR investigation of quasi-two-dimensional magnetic correlations in $T_2$P$_2$S$_6$ ($T$ = Mn & Ni)

We report the anomalous breakdown in the scaling of the microscopic magnetic susceptibility, as measured via the ${}^{31}$P nuclear magnetic resonance (NMR) shift $K$, with the bulk magnetic susceptibility $χ$ in the paramagnetic state of Mn$_2$P$_2$S$_6$. This anomaly occurs near $T_\mathrm{max} \sim 117$ K the maximum in $χ(T)$ and is therefore associated with the onset of quasi-two-dimensional (quasi-2D) magnetic correlations. The spin-lattice relaxation rate divided by temperature $(T_1T)^{-1}$ in Mn$_2$P$_2$S$_6$ exhibits broad peak-like behavior as a function of temperature, qualitatively following $χ$, but displaying no evidence of critical slowing down above the Néel temperature $T_N$. In the magnetic state of Mn$_2$P$_2$S$_6$, NMR spectra provide good evidence for 60 degree rotation of stacking-fault-induced magnetic domains, as well as observation of the spin-flop transition that onsets at 4 T. The temperature-dependent critical behavior of the internal hyperfine field at the P site in Mn$_2$P$_2$S$_6$ is consistent with previous measurements and the two-dimensional anisotropic Heisenberg model. In a sample of Ni$_2$P$_2$S$_6$, we observe only two magnetically split resonances in the magnetic state, demonstrating that the multiple-peaked NMR spectra previously associated with 60 degree rotation of stacking faults is sample dependent. Finally, we report the observation of a spin-flop-induced splitting of the NMR spectra in Ni$_2$P$_2$S$_6$, with an onset spin-flop field of $H_\mathrm{sf} = 14$ T.

cond-mat.str-el↗

arXiv2021

Frustration enhanced by Kitaev exchange in a $\boldsymbol{\tilde{j}_{\text{eff}}=\frac12}$ triangular antiferromagnet

Triangular Heisenberg antiferromagnets are prototypes of geometric frustration, even if for nearest-neighbor interactions quantum fluctuations are not usually strong enough to destroy magnetic ordering: stronger frustration is required to stabilize a spin-liquid phase. On the basis of static magnetization and electron spin resonance measurements, we demonstrate the emergence of ${\tilde{j}_{\text{eff}}=\frac12}$ moments in the triangular-lattice magnet Na$_2$BaCo(PO$_4$)$_2$. These moments are subject to an extra source of frustration that causes magnetic correlations to set in far above both the magnetic ordering and Weiss temperatures. Corroborating the $\tilde{j}_{\text{eff}}=\frac12$ ground state, theory identifies ferromagnetic Kitaev exchange anisotropy as additional frustrating agent, altogether putting forward Na$_2$BaCo(PO$_4$)$_2$ as a promising Kitaev spin-liquid material.

cond-mat.str-el↗

arXiv2021

Reduced crystal symmetry as origin of the ferroelectric polarization within the incommensurate magnetic phase of TbMn2O5

The precise crystal symmetry and hence the emergence of the electric polarization still remains an open question in the multiferroic materials $R$Mn$_2$O$_5$ ($R$ = rare-earth, Bi, Y). While previous diffraction studies have indicated that $R$Mn$_2$O$_5$ possesses the centro-symmetric space group P$bam$, an atomic displacement allowing for the electric polarization would require a non-centrosymmetric crystal symmetry. Our single crystal neutron diffraction experiments on TbMn$_2$O$_5$ provide direct evidence of a reduced crystallographic symmetry already above the magnetic and ferroelectric phase transitions and a change in magnetic order upon entering the ferroelectric phase. This is indicated through the presence of additional nuclear Bragg reflections that are otherwise forbidden for the space group P$bam$ but are in good agreement with the polar space group P$12_11$. It implies that the exchange-striction, which arises from a symmetric $S_i \cdot S_j$ spin coupling, is the dominating mechanism for the generation of the electric polarization in the commensurate magnetic phase of TbMn$_2$O$_5$. Furthermore, the commensurate magnetic reflections are in accordance with a quartile step spin-spiral along the $c$-axis. Therefore, the antisymmetric $S_i \times S_j$ exchange via the inverse Dzyaloshinskii-Moriya interaction contributes as well and becomes the leading term in the low temperature incommensurate spin-spiral magnetic phase. These new findings provide important information for the understanding of the complex interplay between the magnetic and the structural order throughout the $R$Mn$_2$O$_5$ series of type-II multiferroics.

cond-mat.str-el↗