F Porcher

F Damay

S Petit

Isabelle Mirebeau

M Deutsch

P Bonville

A V Tsvyashchenko

L N Fomicheva

Stress-induced magnetic textures and fluctuating chiral phase in MnGe chiral magnet

Keywords: number(s): 75.30.Kz, 62.50.-p, 75.25.-j, 76.80.+y

Stress-induced magnetic textures and fluctuating chiral phase in MnGe chiral magnet

I. INTRODUCTION

Chiral itinerant magnets of the B20 family now attract increasing attention due to their complex and highly tunable spin structures. Long wavelength helical modulations of the local magnetic moment result from the competition between ferromagnetism and weaker order anisotropy terms, since the lack of inversion symmetry of the cubic P 2 1 3 space group leads to nonvanishing chiral interactions. In some cases, peculiar spin textures called Skyrmions (SK) are stabilized, behaving as topologically protected objects much larger than the magnetic unit cell. Skyrmions, studied both in thin films and in the bulk, are now considered as possible building blocks for future spin-based electronics [1,2]. In thin films, where surface anisotropy plays the main role, SK were observed by electron microscopy [3,4] and manipulated with ultra low current densities, yielding spin transfer torque and topological Hall effect (THE) [5,6], with promising applications to spintronics [2,7].

In the bulk, helical structures are induced by the anisotropy of the spin-relativistic Dzyaloshinskii-Moriya (DM) interaction. The smaller crystalline field anisotropy pins the propagation vector of the helices along the crystal axes. In bulk MnSi and FeGe, a triangular skyrmion lattice was revealed by single crystal neutron diffraction in applied field [6,8,9]. It was analyzed as a long range ordered multi-k structure built from in-phase helices propagating along different directions [10]. This so-called A phase is induced by a magnetic field and is limited to a narrow temperature range just below the Néel temperature T N , being favored by thermal fluctuations. In MnSi, the A phase extends under pressure as T N decreases, approaching a quantum phase transition (QPT) reached at a critical pressure P c of 1.4 GPa. Chiral fluctuations of isotropic character recalling skyrmions were reported either in the * isabelle.mirebeau@cea.fr non-Fermi liquid phase above P c [11] or in the paramagnetic phase [12,13].

Within the B20 family, MnGe stands apart due to its peculiar properties. Cubic MnGe exists in metastable and powdered state only, and must be synthesized under high temperature and high pressure conditions. It shows a strong exchange interaction, revealed by a high transition temperature T N of about 170 K (instead of 29.5 K in MnSi), and a large ordered Mn moment of about 1.8μ B at 2 K (compared to 0.4μ B at 2 K in MnSi). The strong spin orbit coupling results in the shortest helix pitch of the series of B20 metals (29 Å instead of 180 Å in MnSi and 700 Å in FeGe) [14,15]. MnGe also exhibits a giant THE [14] and a topological Nernst effect [16], which makes it very promising for spintronic applications. Very recently, by applying high hydrostatic pressures, we observed the onset of a pressure induced collapse of the magnetic order in MnGe [17]. This collapse occurs in two steps, and may be associated with the specific band structure of MnGe, opening the possibility of a high spin-low spin transition [17,18].

In MnGe, a skyrmion lattice was postulated to exist in the bulk from small angle neutron scattering (SANS) and resistivity measurements, and the measured giant THE was tentatively related to a high skyrmion density [14,19]. By contrast with the two-dimensional (2D) triangular SK lattice observed in MnSi and in all other B20 magnets, the proposed SK lattice in MnGe is 3D and cubic. Moreover, as a striking difference with other B20 magnets, it would extend in the whole phase diagram below T N and down to T = 0, even without magnetic field. In zero field, alternating skyrmions and antiskyrmions have been proposed, realizing a complex texture with no net magnetization [7]. However, in the absence of visualization by single crystal neutron diffraction, the existence of a SK lattice in bulk MnGe remains hypothetical.

In this paper, we first investigate the transition region in MnGe by combining magnetic susceptibility, magnetization, neutron scattering, and 57 Fe Mössbauer spectroscopy in a Mn 0.98 57 Fe 0.02 Ge sample. We observe the coexistence of two types of magnetic phases below T N , in a large temperature range down to about 100 K, namely rapidly fluctuating Mn moments and long range ordered "frozen" Mn moments. We determine the thermal evolution of the relative fraction and spin correlations of both phases. The analysis of the Mössbauer spectra also suggests the onset of a distortion at the Néel transition, in agreement with previous neutron diffraction data [15]. We also studied the influence of a uniaxial pressure component on the magnetic order, as directly probed by neutron diffraction. We show that a small uniaxial pressure is sufficient to induce a strong magnetic texture, clearly observed on a 2D multidetector, which persists up to T N . The anisotropy of the intensity distribution is readily interpreted by a reorientation of the helical axes along the axis of the pressure cell, and numerical integrations show that the great majority of the helices reorient under stress. Our results strongly suggest that the zero field ground state at ambient pressure is a multidomain state, consisting of helical domains with random orientations, rather than a 3D SK lattice.

Altogether, our results support the existence in MnGe of a fluctuating chiral state, extending over a large temperature range from about 100 K up to well above T N . Below T N , this state is inhomogeneous, with coexistence of fluctuating and frozen magnetic phases, possibly favored by magnetoelastic interactions. We discuss its possible nature in the scope of the available data on MnGe.

II. SAMPLE SYNTHESIS AND MAGNETIC MEASUREMENTS

Polycrystalline MnGe was synthesized under 8 GPa in a toroidal high-pressure apparatus by melting reaction with Mn and Ge. The purity of the constituents was 99.9% and 99.999% for Mn and Ge, respectively. The pellets of well-mixed powdered constituents were placed in rocksalt pipe ampoules and then directly electrically heated to T 1600 • C. Then the samples were quenched to room temperature before releasing the applied pressure [20]. Two samples (called 1 and 2) doped with 2% at. 57 Fe were also synthesized for the Mössbauer experiments. The amount of impurity phases in these samples is estimated from x ray diffraction to less than 2%.

The magnetic susceptibility was measured versus temperature in a field of 0.01 T with zero field cooling (ZFC) procedure. In all samples, the susceptibility curves (Fig. 1, upper panel) show a broad and high maximum at the Néel transition, namely T N = 170(5) K for the neutron sample and the Mössbauer sample 1, and T N = 150(5) K for the Mössbauer sample 2.

A second anomaly is observed at lower temperature near T c 40-50 K, which is strongly enhanced in the sample 1. The difference between samples arises from an unknown parameter in the synthesis, likely linked to the high-temperature high-pressure quench. Comparison between the susceptibility curves in samples 1 and 2 shows that their differences are not related to the Fe doping. These curves also differ in two pure MnGe samples [21]. As for the anomaly at T c , it is strongly enhanced under pressure for a given sample [17]. This suggests relating the amplitude of this second anomaly to the mobility of the helical domain walls, and to the pinning by defects induced by thermal quench, pressure, or stress. The temperature of this anomaly roughly coincides with the lock-in of the helical modulation, the neutron and Mössbauer results not being affected. The unusual width of the Néel transition, namely T N ± 70 K, is observed in all samples, showing that it is a genuine feature of MnGe. This unusually broad transition region is identified as the range of existence of the fluctuating moment phase as shown below.

Isothermal magnetization curves m(H ) were measured up to 14 T. The lower panel of Fig. 1 shows the curves in the neutron MnGe sample (the magnetization in the Fe-doped sample shows the same features). Our data are in good agreement with the measurements reported in Ref. [14]. These curves can be divided into two sets: (i) a low temperature set below 70 K, where the m(H ) curves increase slowly with the applied field, then show an inflection point near 2-3 T, and finally present an approach to saturation above a field of about 12 T at the lowest temperature; (ii) a high temperature set above 120 K where the m(H ) curves are quite different, increasing steadily with the field without any inflection point and presenting a Brillouin-like shape. The temperature at which occurs this remarkable change of shape of the magnetization curves coincides with the onset of the rapidly fluctuating state as seen by Mössbauer spectroscopy and neutron diffraction (see below). At the highest field and lowest temperature, the system is close to a polarized ferromagnetic state with a moment value (1.65μ B /Mn) close to the spontaneous saturated moment of 1.80μ B /Mn measured by neutron diffraction (see below). The saturation field and saturated magnetic moments are much higher in MnGe than in MnSi (0.5 T and 0.4μ B /Mn, respectively).

The magnetization curves of the polycrystalline MnGe sample at low temperature show a hysteretic behavior (see Fig. 2): the first ZFC magnetization curve performed at 2.5 K presents a dip when increasing the field, which disappears when the field is decreased. This is considered as a signature of the helicoidal magnetic structure [14] with a significant in-plane anisotropy: as long as the field is sufficiently low, the Mn moments tend to remain close to the (001) planes resulting in a reduced susceptibility. On further increase of the field, the moments reorient towards the field direction adopting a conical structure ending in a ferromagnetically induced state, as we have seen, above about 12 T. When the field is decreased, the dip is not observed since the moments recover smoothly their original positions. The irreversibility field for which the upward and downward magnetization curves coincide (about 7 T) is quite high, pointing to a very strong planar anisotropy as compared to that in MnSi, showing an irreversibility field of 0.1 T [22,23]. The hysteresis has almost disappeared at 20 K.

III. M ÖSSBAUER EXPERIMENTS

The Mössbauer spectra were recorded in zero applied field in the temperature range from 4.2 K to 170 K in samples 1 and 2. The spectra show qualitatively similar features in both samples. Since T N in sample 1 is identical to T N in the neutron sample, we show and discuss in the following only the spectra in sample 1, for comparison with the neutron data. 57 Fe Mössbauer absorption spectra in 57 Fe-doped MnGe sample 1 at selected temperatures. At 4.2 and 77 K, the lines are fits to a standard six-line magnetic hyperfine pattern. At 150 K, the fit corresponds to a superposition of a single narrow line (in red) and of a six-line hyperfine pattern (in blue). At 170 K, the spectrum consists of a single narrow line.

Below 100 K, the spectra are six-line slightly broadened magnetic hyperfine patterns. Above 110 K, a single line appears in the spectra, with the same isomer shift as the magnetic spectrum, i.e., pertaining to the same crystallographic phase. The relative weight of this single line grows with increasing temperature at the expense of that of the magnetic pattern. At 170 K, the single line alone is present (see Fig. 3). This single line corresponds to Fe nuclei which are either in a paramagnetic phase or which are submitted to a hyperfine field fluctuating rapidly with respect to the hyperfine Larmor time for 57 Fe: 10 -8 s. In Fig. 4 are represented the thermal variations of the hyperfine field (left scale) and of the fluctuating fraction (right scale). The presence of the narrow single line alone at 170 K means that, at this temperature, the whole sample is paramagnetic and that the electric field gradient (EFG) is actually zero in the paramagnetic phase. Comparing the thermal variation of the fluctuating fraction with that of the ZFC susceptibility (inset in Fig. 4), it is clear that the appearance of the phase with rapidly fluctuating moments around 110 K corresponds to the onset of the unusually broad anomaly of the susceptibility peaking at T N . So this anomaly is related to the progressive extension of the "fluctuating" phase throughout the sample as temperature increases; its maximum occurs when all the sample is in the paramagnetic phase. As to the thermal variation of the hyperfine field, which abruptly drops from 8 T at 160 K to zero at 170 K, it strongly suggests that the transition has a pronounced first order character. The coexistence of ordered and fluctuating/paramagnetic states is also a typical feature of a first order transition. However, it usually occurs over a much smaller temperature interval than observed in MnGe.

The hyperfine field at 4.2 K is unusually small (16.1 T) compared to that in metallic Fe (34 T) or in insulating oxides (50 T) at the same temperature. This rather low value suggests that the hyperfine field is a transferred field, i.e., it is due to the spin polarization of the Mn conduction bands at the Fe site. Since the latter is proportional to the Mn moment, the hyperfine field should show the same behavior. When fitting the spectra, in a first step, to a standard six-line pattern with Lorentzian-shaped lines, one observes the following: (i) some small misfits in the wings, hinting at the presence of a narrow non-Lorentzian distribution of hyperfine fields; (ii) a misfit at the left-hand inner line (marked by green arrows in Fig. 3): this line is too intense when compared to the right-hand inner line; (iii) a vanishingly small effective quadrupolar parameter which, in first analysis, can be due either to an actually zero electric field gradient (EFG) at the nucleus [although the noncubic Fe (Mn) site symmetry allows for a nonzero EFG] or to an effective zero EFG, due for instance to the fact that the hyperfine field is directed at the "magic angle"

θ m = a cos(1/ √ 3) = 54.7
• with respect to the principal axis of the EFG at the Mn site, which is the [111] axis.

In a second fitting step, the helical nature of the magnetic ordering in MnGe must be taken into account. The neutron diffraction data [15] show that the Mn moments lie in the (001) plane according to an incommensurate helix (becoming commensurate below 40 K) with propagation vector along [001]. Assuming that the Fe moment "follows" the Mn moment, the former is then statistically homogeneously distributed according to a helical order in the (001) plane. We have computed the corresponding spectrum by diagonalizing the full hyperfine Hamiltonian, with the EFG axis along [111] and the hyperfine field homogeneously distributed in the (001) plane. In case the EFG in the ordered phase is zero, like in the paramagnetic phase, then the spectrum is a regular broadened six-line pattern and the asymmetry of the two inner lines cannot be reproduced. In order to obtain such an asymmetry, one must assume the presence of a nonzero EFG; then the angle between the EFG axis [111] and the hyperfine field in the (001) plane comes into play in defining the hyperfine energies. Due to the helicoidal arrangement, this angle varies, giving rise to a small distribution of hyperfine energies resulting in an asymmetry of the two inner lines of the spectrum. Furthermore, in order to account for the non-Lorentzian shape of the hyperfine field distribution, it is convenient to fit the spectrum with a histogram of hyperfine fields. The corresponding fit, together with the obtained histogram, is represented in Fig. 5. The correct asymmetry of the inner lines was obtained with a small negative quadrupolar parameter E Q = -0.38 mm/s, with

E Q = eQV ZZ 2 ,
where Q is the electric quadrupole moment of the excited nuclear state and V ZZ is the component of the EFG along OZ = [111]. The fit shown in Fig. 5 represents the experimental data satisfactorily; in particular it yields a vanishing mean spectral EFG. The obtained narrow hyperfine field distribution probably arises from small inhomogeneities in the sample.

It must be noted that the presence of a nonzero EFG in the magnetic phase, whereas the EFG is zero in the paramagnetic phase, can only occur if a distortion takes place in the magnetically ordered phase, in agreement with the neutron diffraction findings [15].

IV. NEUTRON DIFFRACTION

A. Ambient pressure results

Neutron diffraction at ambient pressure was carried out on the diffractometer G4.1 (LLB), in similar conditions as in Ref. [24], but with increased statistics. To study the fluctuating chiral phase, many patterns were taken in the transition region. We focus here on the low angle region. Ferromagnetic correlations are observed well above T N up to about 250 K, and their intensity is maximum at T N (Fig. 6, upper panel), where a magnetic satellite of the q = 0 Bragg peak appears (Fig. 6, middle panel). When temperature decreases below T N the satellite moves towards higher angles and its intensity increases. Ferromagnetic correlations persist below T N down to about 100 K, as shown by the presence of an angular dependent background below the satellite. The intensity and linewidth of these correlations show a critical behavior, with a dynamical narrowing and maximum of intensity at T N (see Fig. 6, lower panel). Unexpectedly, the ferromagnetic correlations are present in a very large temperature range (T N ± 70 K), which corresponds to the region of spin fluctuations identified from Mössbauer spectroscopy and susceptibility. Good fits of the correlation peaks were obtained with a Gaussian line shape centered at 2θ = 0 • , allowing one to follow the temperature dependence of the intensity and to extract a typical length scale D(T ). It is identified as the domain size, using Scherrer's formula [START_REF] Guinier | Théorie et technique de la radiocristallographie: par A. Guinier[END_REF]:

D = 2 ln 2 π λ (2θ ) G cos θ , (1)
where (2θ ) G is the full width at half maximum (FWHM) of the Gaussian line, of the correlations probed in the experimental 2θ window (Fig. 6). The small signal measured at 10 K and 330 K, namely well below or well above T N , is attributed to nuclear diffraction of the powder grains. Subtracting this signal from the data allows the purely magnetic contribution to be extracted, yielding a slight increase of the domain size without changing its temperature dependence. However, this magnetic contribution cannot be fitted to a Lorentzian line shape, by contrast to what usually holds for critical phenomena.

Below T N , the ferromagnetic contribution from the fluctuating Mn moments coexists with an intense satellite of the q = 0 Bragg peak (hereafter referred to as "zero satellite"), arising from the moments involved in the static helical order. The average ordered moment m av (T ) is deduced from FULLPROF refinement (Fig. 7, upper panel), assuming the same magnetic and crystal structure as in Ref. [15]. Its value at low temperature (m = 1.80μ B at 2 K) agrees rather well with the moment value deduced from the high field magnetization data in the almost fully polarized ferromagnetic state (m = 1.65μ B at 2 K and 14 T). Its temperature dependence is plotted in the lower panel of Fig. 7 together with that of the scaled hyperfine field. The two quantities are proportional at low temperature up to 100 K, but they start to deviate in the temperature range 100-170 K where the fluctuating chiral phase is stabilized. In this region, the magnetic moment deduced from neutron diffraction decreases with heating more steeply that the hyperfine field. We attribute this deviation to the presence of the fraction F (T ) of rapidly fluctuating correlated spins evaluated from the Mössbauer spectra (see Fig. 3). The moment m hel (T ) associated with the helical order alone is then evaluated as m hel (T ) = m av (T ) √ 1-F (T ) . As shown in Fig. 7 (lower panel), this quantity scales very well with the hyperfine field in the whole temperature range up to T N , which supports our previous assumption of a transferred hyperfine field at the Fe site.

B. Stress-induced magnetic textures

Neutron experiments under nonhydrostatic pressure were carried out at the Laboratoire Léon Brillouin on the G6.1 diffractometer (incident neutron wavelength 4.745 Å), using focusing devices and a new 2D multidetector, made of 16 horizontal tubular detectors. The sample was inserted in a Kurchatov-LLB sapphire anvil cell [START_REF] Goncharenko | [END_REF], with NaCl as transmitting medium. The pressure was calibrated by ruby fluorescence. With this setup, the uniaxial pressure component or stress can be estimated to 0.3(1) GPa [START_REF] Goncharenko | [END_REF]27]. The stress component is oriented along the principal axis of the pressure cell, which corresponds to the Z axis of the detector (Fig. 8).

We measured patterns at four pressures between 0.8 and 3.2 GPa. The cell was inserted in a helium cryostat and, for each pressure, we measured at several temperatures down to 5 K. Measurements at ambient pressure were also performed with the same multidetector for comparison.

Below T N , the zero satellite appears as an intensity ring on the 2D multidetector [Fig. 8(e)]. We notice that its intensity seems to be reduced in the vertical direction due to the anisotropic pixel shape of the multitube detector, with limited resolution along the Z axis (Fig. 8). This instrumental anisotropy is taken into account in the simulation by applying a filter to an incident isotropic signal.

Under pressure the distribution of intensity over the satellite ring changes dramatically [Fig. 8(f)]. Even at the smallest pressure of 0.8 GPa, the intensity is strongly enhanced along the Z axis. This corresponds to a magnetic texture effect, which practically does not depend on the value of the applied pressure. It is induced by the uniaxial component of the applied pressure, evaluated to 0.3(1) GPa and independent of pressure within our experimental accuracy.

This magnetic texture can be readily associated with a preferential orientation of the helical axes along the Z axis of the pressure cell. The anisotropic distribution of intensity in the detector plane (Fig. 9) can be accounted for by a Gaussian distribution of the helical axes around the Z axis, in the simple assumption of a multidomain state.

The magnetic cross section is written as as

dσ M d (Q)=N (2π ) 3 v 0 k τ |F M⊥ (Q)| 2 δ(Q -k -τ ), (2)
F M⊥ = F M -[F M • Q] Q, (3)
F M = γ e 2 2m e c 2 i,j m i,j f (Q)e iQ•R i,j • e -W i,j , (4)
with f (Q) the magnetic form factor, and m i,j and R i,j the magnetic moment and position of atom j of lattice unit i. W i,j is the Debye-Waller factor. For single helix the moment of atom j of lattice unit i can be written as

m i,j = m 0 [cos(k h • R i,j )e 1 + sin(k h • R i,j )e 2 ], (5)
where e 1 and e 2 are two orthogonal unit vectors perpendicular to the helix axis k h . Equations ( 5) and (2) show that helical spin structures yield satellites of the nuclear peaks (τ ) located at Q = τ ± k h . At ambient pressure, the random orientation of the helical axes in the randomly oriented powder grains yields an isotropic distribution of intensity in the detector plane. The magnetic texture induced under stress is readily linked with a preferential orientation of the helical axes. As a single domain with helical axis along Z yields only two satellites (±k h ) in the detector plane, the distribution of intensity measured in Fig. 9, which obeys a Gaussian line shape with a maximum of intensity along Z, is then readily associated with a Gaussian distribution of orientations of the helical axes along the Z axis. This magnetic texture persists in the whole pressure-temperature range up to T N , and the Gaussian width [FWHM 45 (5) • ] is independent of pressure and temperature. With the assumption of a strong magnetic texture, where all helical axes contribute in the same way to the scattering in the detector plane, this means that 95% of them belong to a cone of vertical axis and 76 • solid angle and 98% are situated in a similar cone of 90 • aperture.

V. DISCUSSION

A. Magnetic ground state

In bulk MnGe powder below T N , applying a stress induces a uniaxial magnetic anisotropy. This reveals a magnetoelastic coupling, also observed for instance in thin films of MnSi under epitaxial strains [28,29]. Here we find that the anisotropy axis is a hard axis, so that the majority of the helical axes reorients along the stress axis. The resulting magnetic texture is very similar to that induced by a magnetic field in bulk MnGe powder [19]. However, whereas in Ref. [19] the field induced reorientation occurs through grain reorientation, in our heavily compressed sample, the isotropic pressure component is much greater than the uniaxial one (P iso /P u ) ∼ 3-10 and the grains cannot reorient.

Therefore, the most natural explanation for the stress induced magnetic texture is the growth of magnetic domains within the ensemble of randomly oriented grains. Recalling that, in MnGe, the helical axes are oriented along 001 crystal axes, each grain can select the [001] axis closest to the Z axis as the preferential axis to minimize the magnetoelastic energy. This process occurs in powder samples when nonhydrostatic pressures are applied below a magnetostructural transition. It naturally explains the value of the Gaussian linewidth, yielding 98% helical axes within a cone of 90 • aperture, since whatever the orientation of a given grain, it will possess a [001] axis in such a cone.

We notice that when the effect of textures is averaged over the 2D detector, we obtain an excellent agreement between these data and those obtained under hydrostatic pressure, as concerns the pressure dependence of the Néel transition, propagation vector, and magnetic moment (see Ref. [17] and its Supplemental Material). This confirms our analysis and shows that there is no deep change in the magnetic structure in this pressure range.

As a natural consequence, our results suggest that the ambient pressure and zero field ground state is a multidomain state made of single-k helical domains of random orientations, and not a lattice of skyrmions and antiskyrmions, namely a complex multi-k structure where the selection of single-k single-domain state under stress would be energetically less favorable. Similarly in single crystals, applying uniaxial stress is a usual way to distinguish single-k from multi-k magnetic structures [START_REF] Rossat-Mignod | Handbook on the Physics and Chemistry of the Actinides[END_REF].

Our simple description of the bulk MnGe ground state is supported by several observations and calculations in the B20 family. In bulk B20 samples allowing a direct observation of the SK lattice by single crystal neutron diffraction [8,[START_REF] Adams | [END_REF]32], this lattice was observed only in the vicinity of T N and under applied field, but not in the zero field ground state, so that MnGe would stand as an exception. Thermal fluctuations [8] and/or a reduced stiffness of the magnetization modulus [28,33] are needed to stabilize the SK phase in the bulk state, which explains its small extension with temperature. Taking into account the high values of the anisotropy and saturation fields of MnGe (7 T and 12 T, respectively, as compared to 0.1 T and 0.6 T in MnSi), if the involved planar and DM anisotropies were strong enough to favor a condensed skyrmionic phase down to T = 0 K, this phase should exist and even be favored by large fields in contrast with experimental observation using neutrons [19].

On the other hand, in MnGe at low temperature, the evolution of the magnetization and small angle neutron scattering with the magnetic field can be easily explained assuming a multidomain ground state in zero field, rather than the SK lattice previously proposed [19]. The field irreversibilities can naturally be related to the persistence of small helical domains blocked along a [001] axis at a large angle with the field axis. If the grains align along the magnetic field, such domains would be oriented at 90 • of the field and yield a neutron intensity in the direction Q ⊥ H , as experimentally observed [19].

Then what could be the origin of the giant THE of MnGe? Although helical or conical structures contribute to the magnetoresistance, they do not yield THE since the scattered electrons do not carry a Berry phase. The fluctuating chiral phase could play a role just below T N , but the strong temperature independent value of the transverse Hall resistivity at very low temperature remains to be explained. Other contributions to the THE related to a multicarrier effect [START_REF] Ziman | Principles of the Theory of Solids[END_REF] could perhaps be at play, as recently proposed for Mn 1-x Fe x Si thin films [START_REF] Yokouchi | [END_REF]. A THE has also been reported in fanlike noncoplanar helimagnet MnP [36], where the spin structure has no net skyrmion number.

B. Transition region

We have evidenced a specific feature of MnGe, namely a broad transition temperature range extending in our sample up to 70 K on each side of T N , and we have determined its characteristics by combining magnetization, Mössbauer, and neutron data. This transition range is magnetically inhomogeneous, but the different magnetic regions belong to the same crystal phase. It presents some analogies with the non-Fermi liquid phase identified in MnSi and FeGe, but in MnGe it extends in a much larger temperature range above T N and persists well below in the ordered state. Below T N , ferromagnetically correlated rapidly fluctuating spins coexist with frozen moments involved in the helices. We have measured the relative fraction of the Mn moments involved in each type of state thanks to a quantitative comparison between neutron and Mössbauer data. We notice that the correlation length of the fluctuating moments (10-30 Å in our experimental range) and that of the helices (limited by the instrumental resolution, namely greater than 500 Å) differ by at least an order of magnitude, suggesting that the system segregates into different subphases. Above T N , the diffraction signal from the long range ordered helices vanishes and the Mössbauer spectrum consists of a single line. Susceptibility and neutron data show that spin correlations persist up to about 240 K, and field cooling irreversibilities extend even above. It suggests that above T N chiral fluctuations such as short range ordered helices [37,38] still coexist with a few large scale quasistatic inhomogeneities. This crossover region is also observed in the data obtained in Refs. [14,19]. In Fig. 1(a) of Ref. [14], a broad peak in the magnetic susceptibility is present, but it is rather asymmetric around a maximum at T N 180 K: it increases sharply above 150 K and decreases smoothly above T N , following the behavior of the ferromagnetic correlation contribution seen on Fig. 2(g) of Ref. [19] which decreases smoothly above T N but abruptly disappears at 142 K. This means that this crossover region is an intrinsic feature of MnGe but its temperature expansion and shape are sample dependent and can be probed by susceptibility measurements.

Mössbauer experiments have shown that the transition is first order in MnGe. In chiral magnets of the B20 structure, the onset of helical order through a first order transition is expected from theory [39]. The DM interaction can be expressed as a socalled Lifschitz invariant in the Landau free energy [33,40] so that no second order transition is expected. Going beyond the Bak-Jensen model, in MnSi the features of the transition just above T N have been analyzed as an example of the Brazovskii mechanism [41,42]. But the large crossover region around T N and the persistence of spin fluctuations well below T N is specific to MnGe. Another specificity of MnGe is the onset at T N of a lattice distortion, probed by neutron diffraction and Mössbauer spectroscopy, which has not been reported in other B20 magnets.

This crossover regime and the magnetostructural anomalies are possibly related to the peculiar band structure of MnGe which enables a high-spin (HS) to low-spin (LS) transition [18]. Such a transition could be in principle encompassed either with temperature or pressure [18]. Our recent measurements combined with refined calculations [17] suggest that it indeed occurs in the ground state under very high pressures, yielding a two step collapse of the helical order.

With varying temperature, the observation of a HS-LS transition is not obvious, and we cannot ascertain it presently. We however notice that an inhomogeneous magnetic state, a crossover regime at the transition, and the occurrence of a structural anomaly are in qualitative agreement with a HS-LS transition. HS and LS regions are expected to have very different specific volumes, so that thermal or quantum fluctuations are needed to accommodate the volume mismatch. This yields an intermediate fluctuating magnetoelastic state, in a process which recalls that of invar alloys [43,44]. We speculate that this intermediate state is realized through a dynamical phase separation above T N , and through a phase segregation between static and fluctuating regions below T N .

FIG. 2 .

 2 FIG. 2. Hysteretic behavior of MnGe magnetization at 2.5 K.

FIG. 3 .

 3 FIG. 3. (Color online)57 Fe Mössbauer absorption spectra in 57 Fe-doped MnGe sample 1 at selected temperatures. At 4.2 and 77 K, the lines are fits to a standard six-line magnetic hyperfine pattern. At 150 K, the fit corresponds to a superposition of a single narrow line (in red) and of a six-line hyperfine pattern (in blue). At 170 K, the spectrum consists of a single narrow line.

FIG. 4 .

 4 FIG. 4. (Color online) Thermal variations of the hyperfine field (black solid squares) and of the fluctuating fraction (red solid circles) deduced from the Mössbauer spectra in Fe-doped MnGe sample 1. The red dashed line is a guide for the eye. The inset shows the susceptibility vs temperature in Fe-doped MnGe sample 1.

FIG. 5 .

 5 FIG.5.57 Fe Mössbauer spectrum (right panel) and hyperfine field distribution (right panel) at 77 K in Fe-doped MnGe sample 1. The line is a fit to a line shape taking into account the spiral magnetic structure of the Mn ions and a nonvanishing EFG at the Fe site (see text).

FIG. 6 .

 6 FIG. 6. (Color online) Upper panel: diffraction pattern at low angle vs temperature above T N . Middle panel: diffraction pattern at low angle vs temperature below T N : the zero satellite is superimposed on a low angle signal fitted with a Gaussian line shape centered at 2θ = 0 • . Lower panel: temperature dependence of the maximum of the Gaussian line used to fit the low angle signal and of the correlated domain size calculated by Scherrer's formula [25].

FIG. 7 .

 7 FIG. 7. (Color online) Upper panel: experimental powder diffraction pattern of MnGe at 1.5 K (red dots) and FULLPROF refinement (black line). The difference pattern is the blue line; upper (lower) tick marks show the positions of the nuclear (magnetic) Bragg peaks. Inset: zoom around the (011) and (111) Bragg peaks showing the small magnetic satellites at high angles. Lower panel: comparison of the temperature dependence of the average Mn ordered magnetic moment obtained by neutron diffraction (blue diamonds) and by Mössbauer measurements in sample 1 (red squares). Green dots show the scaled Mössbauer moments taking into account the paramagnetic fraction in order to compare to neutron data. Lines are guides for eyes. Inset: fluctuating fraction vs temperature for the Mössbauer sample 1.

FIG. 8 .

 8 FIG. 8. (Color online) Illustration of a diffraction process for a powder without texture (upper panel) and with texture (lower panel) in MnGe. Upper panel: magnetic zero satellite without texture (isotropic distribution): (a) simulated on an isotropic 2D detector; (c) simulated on a multitubes detector as installed on the G6.1 diffractometer (LLB); (e) measured on G6.1 at 5 K and ambient pressure. Lower panel: magnetic zero satellite with texture oriented along the vertical direction with a Gaussian distribution of orientations (ODF): (b) simulated on an isotropic 2D detector; (d) simulated on a multitubes detector as installed on G6.1; (f) measured on G6.1 at 5 K and 3.2 GPa. Under pressure the intensity is enhanced along the axis of the pressure cell, due to the reorientation of helical domains.where τ are reciprocal lattice vectors, k are the wave vectors associated with the Fourier component of the magnetic distribution, and F M is the magnetic structure factor defined

ACKNOWLEDGMENTS

We are deeply thankful to O. L. Makarova for her help in the high-pressure experiments. We thank U. K. Rößler, C. Lacroix, K. Pappas, S. V. Grigoriev, and S. M. Stishov for many useful discussions and comments. The post doc training of M.D. is funded by the ANR (DYMAGE) and the Triangle de la Physique. The stays of O. L. Makarova at LLB in 2012-2013 are funded by the Triangle de la Physique and Labex PALM. A public grant from the "Laboratoire d'Excellence Physics Atom Light Mater" (Labex PALM) was overseen by the French National Research Agency (ANR) as part of the "Investissements d'Avenir" program (Grant No. ANR-10-LABX-0039-PALM). A.V.T. acknowledges the support of the Russian Foundation for Basic Research (Grant No. 14-02-00001).


Arvind Maurya

P Bonville

A Thamizhavel

S K Dhar

Phys. Rev. B Manfrinetti

R K Das

A Kulkarni

S K Thamizhavel

P Dhar

J Bonville

Phys

EuNiGe 3 , an anisotropic antiferromagnet

Keywords: numbers: 75.50.Ee, 81.10.Fq, 81.10.-h, 75.30.Gw, 76.80.+y, 75.25.-j, 72.15.Gd EuNiGe 3, non-centrosymmetric, antiferromagnetism, Mössbauer spectra, Magnetic anisotropy

Single crystals of EuNiGe 3 crystallizing in the non-centrosymmetric BaNiSn 3 -type structure have been grown using In flux, enabling us to explore the anisotropic magnetic properties which was not possible with previously reported polycrystalline samples. The EuNiGe 3 single crystalline sample is found to order antiferromagnetically at 13.2 K as revealed from the magnetic susceptibility, heat capacity and electrical resistivity data. The low temperature magnetization M(H) is distinctly different for field parallel to ab-plane and c-axis; the ab-plane magnetization varies nearly linearly with field before the occurrence of an induced ferromagnetic phase (spin-flip) at 6.2 Tesla; on the other hand M(H) along the c-axis is accompanied by two metamagnetic transitions followed by a spin-flip at 4.1 T. A model including anisotropic exchange and dipole-dipole interactions reproduces the main features of magnetization plots but falls short of full representation. (H,T) phase diagrams have been constructed for the field applied along the principal directions. From the 151 Eu Mössbauer spectra, we determine that the 13.2 K transition leads to an incommensurate antiferromagnetic intermediate phase followed by a transition near 10.5 K to a commensurate antiferromagnetic configuration.

I. INTRODUCTION

Determining the magnetic structure of divalent Eu compounds remains a challenge since neutron diffraction is not easy to perform in such materials due to the large absorption of neutrons by Eu isotopes. One way of obtaining information about the spin arrangement is to perform single crystal magnetization measurements. Mössbauer spectroscopy on the isotope 151 Eu in polycrystalline samples can also yield information about the commensurability of the magnetic structure with the lattice. Recently, we studied single crystals of two members of the EuMX 3 family, where M = Pt and X is Ge or Si, 1,2 which crystallize in the body-centered tetragonal BaNiSn 3 -type structure which is noncentrosymmetric (space group I4mm). The properties of a polycrystalline sample of a third compound of the family, EuNiGe 3 , have recently been reported in the literature. 3 EuNiGe 3 orders antiferromagnetically at T N =13.6 K. Assuming a collinear A-type antiferromagnetic (AFM) structure in which ferromagnetic layers of Eu magnetic moments in the ab-plane alternate along the c-axis with AFM coupling, the in-plane J 1 and interplane J c nearest neighbor exchange integrals were derived in Ref. 3.

In the present work, we report on the synthesis of a single crystal of EuNiGe 3 . We have measured the magnetization along the principal directions of the crystal and applied 151 Eu Mössbauer spectroscopy to study the hyperfine interaction. The heat capacities in zero and applied magnetic field, electrical resistivity and magnetoresistivity of the single crystal have also been measured. The prominent feature of our data is the magnetization curves at 1.8 K, which are strongly anisotropic, the curve along the [001] crystal axis showing a metamagnetic staircase-like behavior. We present a model, including the dipole-dipole interaction and exchange anisotropy, to interpret these findings.

II. EXPERIMENTAL DETAILS

The single crystal of EuNiGe 3 was grown by using a high temperature solution growth method with In as a solvent. Initially we used Sn flux as it had resulted earlier in the successful growth of the single crystals of EuPtX 3 (X = Si or Ge), 1,2 but we failed to obtain the single crystals of the Ni compound. However, we were successful in growing the single crystals of EuNiGe 3 from In solvent. Arc melted polycrystalline EuNiGe 3 and excess temperature of the ampoule was uniformly raised to 1100 • C/h in 24 hours and held at that value for another 24 hours to ensure homogeneity of the solution. After this a cooling rate of 2 • C/h was employed upto 600 • C followed by relatively faster cooling (60 • C/h) to room temperature. The excess indium flux was removed by centrifugation process. The quality of the single crystals was confirmed by recording the Laue patterns which showed sharp spots conforming to the tetragonal symmetry. A portion of the crystal was powdered for x-ray diffraction; all the peaks of the spectrum could be indexed to the tetragonal structure with space group I4mm (#107). The absence of any extra peaks confirms the phase purity of the grown crystal. The single crystal was cut along the desired directions by electric spark discharge under a suitable dielectric and the orientation of the desired plane(s) confirmed by Laue patterns. Magnetization was measured in a Quantum Design VSM magnetometer, and the heat capacity and electrical resistivity were measured in a Quantum Design PPMS.

151 Eu Mössbauer spectra were recorded at a few selected temperatures using a constant acceleration spectrometer with a 151 Sm * F 3 source.

III. RESULTS

A. Magnetic properties

The Rietveld analysis of the powder x-ray diffraction spectrum yielded the lattice parameters a = 0.4338 (8) nm and c = 0.9895 (9) nm, in good agreement with the previously reported values.

The inverse susceptibility, χ -1 , of EuNiGe 3 is practically isotropic above 20 K for the AFM nearest neighbor exchange, J 1 , and a ferromagnetic (F) next neighbor exchange J 2 with J 2 > |J 1 |. The susceptibility M/H below 20 K along the three directions is shown in Fig. 1. The antiferromagnetic transition is marked by a peak at T N = 13.2 K. The transition temperature is slightly lower than that reported on the polycrystalline sample.

The susceptibility is isotropic in the ab-plane but its magnitude at T N is distinctly higher for H

[001]. This difference, reflected in the higher value of θ p along [001], persists for T > T N but decreases with temperature and eventually disappears above 45 K. It is due to the crystal field and exchange anisotropy to be introduced below in section IV. The susceptibility along the c-axis [001] at 1.8 K is 0.27 emu/mol and is field-independent for applied fields of H = 0.005, 0.02 and 0.1 T, in the temperature range below 20 K. The ab-plane susceptibility at 1.8 K in a field of 0.1 T is about twice larger with a value of 0.47 emu/mol. The evolution of magnetic susceptibility of EuNiGe 3 with magnetic field applied along [001] and [100] at low temperatures is shown in Fig. 2. There is a broad hump in the ab-plane susceptibility centered near T m =10.5 K and the temperature dependent magnetization shows a field dependence even at low fields (see Fig. 2), unlike the data for H [001]. However, the field dependence of the susceptibility is seen even for H [001] at higher fields as inferred from Fig. 2)b. The broad hump is presumably the phase boundary between commensurate and incommensurate modulated antiferromagnetic structures, which is elaborated in the Mössbauer spectroscopy section.The T N decreases with increase in the strength of the magnetic field as expected for an antiferromagnetic compound. Ref. 3 reports a cusp in the susceptibility at 5 K in low fields (< 500 Oe), but such a feature is

B. Heat Capacity

The heat capacity between 1.8 and 30 K in zero magnetic field and with a field of 3 T is plotted in Fig. 5. A single sharp peak at 13.2 K in the zero-field data is representative of the AFM transition; a broad anomaly centered around 10.5 K correlates well with the anomaly in the susceptibility for H [100] observed near the same temperature. The correlation holds in an applied magnetic field as well, at least upto 3 T. For this field value, the narrow peak has shifted down to 11.6 K and the broad anomaly occurs near 8 K. For the sake of comparison, the M/H data measured at the same value of the field for H

[100] are also plotted in Fig. 5. There is a very good correspondence between the two sets of the data. The zero-field 151 Eu Mössbauer spectra, to be described in section III D, reveal that the transition at T N =13.2 K leads to an incommensurate (ICM) AFM phase, and that the anomaly near 10.5 K marks a transition to a commensurate AFM phase. This latter transition temperature will be referred to in the following as T IC . are however comparable: 3.5 for the single crystal compared to 7 for the polycrystalline sample. It may be recalled that there is an anomaly near 5 K in the susceptibility of the polycrystalline sample, which is absent in the single crystal. Though the resistivity value at 300 K in the two samples is nearly the same, the polycrystalline sample has a resistivity of 1.5 µΩcm at 1.8 K, compared to 11 µΩcm for the single crystal. In the presence of the The inset shows the field derivative of the MR ratio at 2 K. magnetic field, the magnetic transition temperature can be easily identified as the resistivity decreases rapidly below T N . As seen in Fig. 6, application of a field decreases T N , which is consistent with the magnetization and the heat capacity data discussed above.

0 M (µ B /Eu) 7 6 5 4 3 2 1 0 H (T) EuNiGe 3 H // [001] 1.8 K 4 K 8 K 12 K
At H ≥ 3.5 T the anomaly in the resistivity due to the magnetic transition is no longer discernible. For H ≥ 4 T the resistivity is less than the corresponding zero-field values in the range 1.8-40 K shown in the figure. Typically the magneto-resistance ratio MR, defined as [R(H)-R(0)]/R(0), of an ordinary polycrystalline metal is positive and of the order of a few %. MR can be appreciable for metals and single crystals with large residual resistivity ratio which is not the case with our sample. A negative MR suggests a magnetic origin The sign transition occurs in the field region where the magnetization along [100] shows a minor anomaly followed by the spin-flip transition at 6 T (see Fig. 3). As the temperature increases the field at which the anomaly occurs decreases and the MR also changes sign at lower fields. At 12 and 20 K, the MR is negative in the entire range of applied fields, as expected in the paramagnetic phase under field. The label PM is for paramagnetic phase, and the labels AF1, AF2 ... for different AFM phases.

Lines are guide to the eyes.

D. Mössbauer spectroscopy

151 Eu Mössbauer spectra have been recorded at 4.2, 10, 11, 12, 13 and 14 K in EuNiGe 3 (see Fig. 11). The spectra below 10 K show a magnetic hyperfine spectrum due to a single hyperfine field of 30.6(1) T at 4.2 K and 25.7(1) T at 10 K, with an Isomer Shift with respect to Sm * F 3 of -10.6 mm/s. All these values are characteristic of a divalent Eu ion. Above 10 K and up to 13 K, the spectra change shape and become characteristic of a distribution due to an incommensurate modulation of collinear hyperfine fields. 4 At 14 K, the spectrum is a broad featureless line characteristic of the paramagnetic phase.

This hyperfine field modulation arises from a modulation of moments since the hyperfine field is proportional to the Eu 2+ magnetic moment to a good approximation when the former has the standard value of ≃30 T. The change in spectral shape occurs between 10 and 11 K, and thus we think that it correlates with the broad anomaly in specific heat and susceptibility observed near T IC =10.5 K. The AFM moment modulation along the direction For the 4.2 K spectrum, the line is a fit to a single hyperfine field pattern. For the 12 and 13 K spectra, the line is a fit to a distribution of hyperfine fields arising from an incommensurate moment modulation. The insets show the corresponding hyperfine field modulations.

of the propagation vector k was described by the expression:

m(kx) = 4 l=0 m 2l+1 sin (2l + 1)kx (1)
where the m n are the odd Fourier coefficients of the modulation and are fitted to the shape of the spectrum. The shape of the modulation is represented in the insets of Fig. 11.

It tends to a sine-wave shape close to T N and squares up when approaching T IC . The situation in EuNiGe 3 is therefore similar to that in EuPtSi 3 1 , where a cascade of transitions is also present, the intermediate phase being an amplitude modulated magnetic phase. One difference is that the specific heat anomaly marking the transition to the commensurate AFM phase is much sharper in EuPtSi 3 than in EuNiGe 3 . One can also notice that the presence of an ICM phase just below T N in the EuMT 3 series seems to be correlated with the presence of anisotropy: in EuPtGe 3 , where the magnetisation is rather isotropic, no ICM phase is present. 2

IV. MODEL FOR COMPUTING THE SINGLE CRYSTAL MAGNETIZATION

In the centered tetragonal lattice of the Eu 2+ ions (without however inversion symmetry due to different arrangements of the Ni and Ge ions above and below a given plane), one can consider three exchange integrals: nearest neighbour intra-plane integral J 1 (ion separation a), nearest neighbour interplane integral J c (ion separation 1/2 √ (c 2 + 2a 2 ), both considered in Ref. 3 and using the notations therein, to which one can add a next-nearest neighbour integral J 2 between ions in two non-adjacent planes (ion separation c). 5 With the lattice parameters in EuNiGe 3 , these ion separations are respectively about 0.43 nm, 0.6 nm and 1 nm. Thus we restrict ourselves to exchange J 1 and J c , and neglect J 2 since the corresponding ion separation is much larger. According to the relative signs and magnitudes of J 1 and J c (our convention is a negative integral corresponds to an AFM interaction), the two simplest AFM structures are:

k B T N = S(S + 1) 3 j J ij cos φ ij (2) k B θ p = S(S + 1) 3 j J ij (3)
where J is the exchange tensor, the sum runs over the neighbours of ion i and φ ij is the angle between moments at sites i and j, one can obtain the exchange integrals for each of the magnetic structures S1 and S2. Using an isotropic mean value θ p = 4 K and T N = 13 K, as sketched in Fig. 12. When describing an S1 type structure, sublattices 1 and 2 on the one hand, 3 and 4 on the other hand, are identical. As to the S2 structure, sublattices 1 and 4 on the one hand, 2 and 3 on the other hand, are identical. The computation of the magnetization is performed treating the exchange and dipole-dipole interactions in mean field involving the 4 sublattices, in a self-consistent way. We have first simulated the magnetization curves at 1.8 K for the S1 and S2 structures relevant to EuNiGe 3 using the aforementioned exchange integrals. The value of the D parameter was taken negative and adjusted so that the spin-flop field value, for H [001], be 2 T, as experimentally observed.

For the S1 structure, |D| has to be taken rather large lest the structure lies in the (ab) plane, which is an effect of the dipolar field. The calculated magnetization curves along [100] and

[001] are represented in Fig. 13. The obtained spin-flip fields along [001] and [100], 3.3 and 4.8 T, respectively, for S1, and close to 4.5 T for S2, do not precisely match the experimental values 4.1 and 6.2 T, respectively (see Fig. 3), and neither the initial non-zero slope of M(H) nor its staircase-like behavior when H [001] are reproduced. For the S1 case, the large difference in spin-flip fields is due to the large value of |D|.

Another type of possible anisotropy is that of the exchange integrals (see Appendix B for details). In a simple model, we assume that, for each Eu-Eu bond, the symmetric part of the exchange tensor is diagonal and has axial symmetry in a frame linked with the Eu-Eu bond, i.e. it possesses two components J and J ⊥ . Since we consider the nearest neighbor intraplane and interplane integrals, we introduce the four parameters, J 1 , J ⊥ 1 , J c and J ⊥ c . For each type of AFM structure (S1 and S2), it is then possible to find a set of section.

V. DISCUSSION

One can readily think of two mechanisms leading to a magnetic structure canted off the c axis. The first one is a crystallographic distortion that would occur in the AFM phase, such that the Eu site no longer has fourfold point symmetry. This would imply a loss of the isotropy of the magnetic properties in the (ab) plane, which contradicts observations since the magnetization and susceptibilty along [100] and [110] are identical. So a low temperature symmetry breaking distortion can be discarded.

The second mechanism is the presence of an asymmetric term of the exchange tensor, namely the Dzyaloshinski-Moriya interaction. It can be seen to be non-zero for the two exchange paths considered here since none of the mid points of the intraplane and interplane exchange bonds is an inversion center for the crystal structure of EuNiGe 3 . Considering only nearest neighbour exchange, i.e. intraplane exchange, application of the Moriya rules 9

shows that the associated Dzyaloshinski-Moriya vector D DM lies in the (ab) plane and is perpendicular to the Eu-Eu bond. This can lead to a canted zero field magnetic structure.

However, in order to test this hypothesis, a more sophisticated sublattice decomposition is needed, because convergence cannot be reached in our present in-field calculation in the presence of asymmetric exchange. This suggests that the 4 sublattices used here are not adequate to reproduce the actual magnetic structure and its evolution with the field, and one must await the determination of the magnetic structure of EuNiGe 3 to check for the presence and magnitude of the Dzyaloshinski-Moriya exchange.

One can also note that the absence of mirror symmetry with respect to the (ab) plane implies that there should be two different interplane nearest neighbour exchange integrals J u c and J l c , according to the upper or lower position of the Eu-Eu bond with respect to the (ab) plane. This asymmetry is a priori not expected to lead to a transverse spin component since it does not break the fourfold symmetry around the c axis. With our present definition of the 4 sublattices, which are invariant by a translation of vector c, introduction of these different integrals amounts to replacing J c by (J u c + J l c ) and therefore this kind of asymmetry is not relevant within our model.

Finally, an anisotropic magnetic behavior has also been observed in EuPtSi 3 1 and, for the case of the other S-state ion Gd 3+ , in several intermetallics: GdRu 2 T 2 (T = Ge or Si), 10

Gd 2 PdSi 3 11 and Gd 5 Ge 4 , 12 where a metamagnetic behavior of the magnetization is reported along some particular direction. This behavior was attributed in these works to exchange anisotropy.

VI. CONCLUSION

As a conclusion, single crystals of the intermetallic material EuNiGe 3 have been synthesized and studied with a variety of techniques: magnetization, specific heat, transport and Mössbauer spectroscopy. EuNiGe 3 contains divalent Eu, its Néel temperature is found to be T N = 13.2 K, in agreement with Ref. 3. The Mössbauer spectra show that there occurs a cascade of transitions, as often observed in intermetallic Eu materials, from paramagnetic to incommensurate AFM to commensurate AFM as temperature decreases. The magnetization and magnetoresistance in the AFM phase are found to be strongly anisotropic, which is a priori surprising for a spin only ion like Eu 2+ . In particular, the magnetization along the c axis at 2 K shows an unusual staircase-like behavior. With the aim of obtaining the exchange integrals in EuNiGe 3 , we applied a model with anisotropic exchange and dipole-dipole interactions which reproduces the main features of the magnetization and susceptibility curves. However, we could not reach a fully satisfying reproduction of the magnetization curves along c, which suggests that the magnetic structure is more complex than the AFM collinear arrangement assumed in our model. To go further, one must await the determination of the magnetic structure by neutron diffraction. We emphasize that the dipole-dipole interaction must be taken into account for a full understanding of the magnetic exchange Hamiltonian writes, in a frame linked with the Eu-Eu bond:

H ex = -J α S z i S z j -J ⊥ α (S x i S x j + S y i S y j ). (B1)

For instance, considering the intraplane exchange bond between sublattice 1 and 2 along a (see Fig. 12), the frame Oxyz linked to this bond is taken as: z a, x c and y -b.

The diagonal axial exchange tensor with components J ⊥ 1 and J 1 is then transformed into the (abc) frame and, taking into account 4 such intraplane bonds, one obtains the total intraplane exchange matrix between sublattices 1 and 2, or 3 and 4, to which one can add an antisymmetric Dzyaloshinski-Moriya term with vector D DM which lies in the ab-plane and is perpendicular to the bond:

J t 1 = 2      J ⊥ 1 + J 1 0 -D DM 0 J ⊥ 1 + J 1 -D DM D DM D DM 2J ⊥ 1      . ( B2
)
Repeating this procedure for the interplane exchange (without Dzyaloshinski-Moriya term), one finds that the total interplane exchange matrix between sublattices 1 and 3 or 1 and 4, 2 and 3 or 2 and 4, reads:

J t c = 2      J ⊥ c + δ δ 0 δ J ⊥ c + δ 0 0 0 J c -2δ      , ( B3
)
where δ = J c -J ⊥ c 2+y 2 and y = c/a.

  FIG. 1: (Color online) Low temperature part of the magnetic susceptibility χ=M/H in EuNiGe 3 for a field of 0.1 T applied along [001], [100] and [110].

  FIG. 2: (Color online) Low temperature susceptibility for (a) H [100] and (b) H [001] in EuNiGe 3 for different values of the applied field.

  FIG. 3: (Color online) Isothermal magnetization curves at 1.8 K in EuNiGe 3 for field along [100], [110] and [001].

FIG. 4 :

 4 FIG. 4: (Color online) Magnetization curves along [001] in EuNiGe 3 at different temperatures.

FIG. 6 :

 6 FIG. 5: (Color online) Heat capacity (left scale) and magnetic susceptibility (right scale) in EuNiGe 3 for fields of 0 and 3 T. The 3 T heat capacity curve is shifted down by 3 J/mol.K for clarity.

FIG. 7 :

 7 FIG. 7: (Color online) Magnetoresistance with H[001] in EuNiGe 3 at various temperatures.

FIG. 8 :

 8 FIG. 8: (Color online) Magnetoresistance with H[100] in EuNiGe 3 at various temperatures.The inset shows the field derivative of the MR ratio at 2 K.

Fig. 9 FIG. 10 :

 910 FIG. 10: (Color online) Magnetic phase diagram of EuNiGe 3 for (a) H [001] and (b) H [100].

FIG. 11 :

 11 FIG. 11: (Color online) 151 Eu Mössbauer absorption spectra at 4.2, 12 and 13 K in EuNiGe 3 .

  (i) ferromagnetic (ab) planes with alternating moment directions along c (S1 structure wih k = [001]) (ii) ferromagnetic stripes along [110] in the (ab) planes with alternating moment directions in parallel stripes (S2 structure with k = [1/2,1/2,0]). Using the molecular field relations for the Néel temperature T N and the paramagnetic Curie temperature θ p quoted in Ref. 3:

FIG. 12 :

 12 FIG. 12: (Color online) Decomposition of the EuNiGe 3 tetragonal lattice into 4 sublattices, in projection onto the (ab) plane. The full circles represent Eu atoms in the z = 0 plane, the open circles those in the z = c/2 plane.

  FIG. 14: (Color online) Calculated magnetization curves at T = 1.8 K (a) and susceptibility curves at H = 0.1 T (b) for the S1 structure in the presence of exchange anisotropy and dipolar interaction.

  FIG.15:(Color online) Self-consistent computation of the magnetization curves at 1.8 K for J 1 = 0.40 K, J c = -0.11 K and D = -0.32 K, for two directions of the applied magnetic field, in the absence (a) and in the presence (b) of the dipolar field.

one obtains 1. for structure S1: J 1 = 0.40 K and J c = -0.11 K, 2. for structure S2: J 1 = -0.62 K and J c = 0.40 K.

In order to compute the magnetization, and in view of its anisotropic behavior, one must add two interactions:

(i) the single ion crystalline anisotropy E CF , which is small for Eu 2+ but plays a role in defining the easy plane/axis. Since the local symmetry at the Eu site is tetragonal (4mm), we consider only the leading second order term E CF = DS 2 z , where D is a parameter and Oz the local fourfold axis, i.e. the c crystal axis. Usually, |D| amounts to a few 0.05 K. For negative D, the easy axis is the fourfold c axis, while for positive D the ab-plane is the easy plane.

(ii) the standard dipole-dipole interaction H dip which is taken to be of infinite range (see Appendix A for details about the summation over lattice sites. 6,7 ) This interaction is also weak, but it is of the same order as the crystalline anisotropy and must be taken into account. The importance of the dipole-dipole interaction in determining magnetic structures for Gd compounds has been examined, using a different approach, in Ref. 8. An example of magnetization single crystal curves computed with and without the dipolar interaction is provided in Appendix A. The total hamiltonian acting on an Eu ion is thus:

where the different terms are, from left to right, the crystal field, the Zeeman interaction (g=2 for Eu 2+ ), the exchange coupling restricted to intra-plane and interplane nearest neighbours, and the infinite range dipolar interaction with the field H dip acting on the given ion.

The sublattice decomposition of the zero-field magnetic structure is essential for the calculation of the single crystal magnetization. In the absence of knowledge of this magnetic structure, and in order to be able to examine the two simple types of aforementioned structures S1 and S2 with the same formalism, we have used a decomposition into 4 sublattices, for this field direction. However, it is still not possible to obtain a non-zero initial slope, nor a staircase like behavior, for M (H) along [001] since the easy axis in both S1 and S2

structures is the c axis. This suggests that the zero field structure is a "canted" structure, i.e. it possesses a spin component in the (ab) plane, and one must find mechanisms able to induce such a deviation from the easy axis. This is the subject of discussion in the next We show here that it is important to take into account the dipole-dipole interaction in computing the single crystal magnetization curves. Indeed, Eu 2+ has a rather large saturated moment (7 µ B ), and the dipolar field is of the order of 0.05 T. This value is small with respect to the characteristic exchange fields in EuNiGe 3 (a few T), but it is of the same magnitude as the anisotropy field in Eu 2+ compounds. Therefore, it must be taken into account for determining both the zero field magnetic structure, since it can compete with the crystalline anisotropy in defining the easy magnetic axis, and the low field magnetization since it plays a role in defining the spin-flop field.

In order to obtain the dipolar field on a given Eu 2+ ion on a given sublattice, we use the Ewald summation method to compute the infinite lattice sums 6,7 , and we write the dipolar field acting on an ion in the l sublattice as:

where the Q l k matrices contain lattice sums. For instance, the self-sublattice matrix Q l l is diagonal and writes, in units of the Lorentz field parameter H L = 4π 3V µ B , where V = 2a 2 c is the volume of the primitive cell, and a = 0.4388 nm and c = 0.9895 nm:

and the matrix Q 1 3 writes, in the same units:

To illustrate the importance of taking into account the dipolar interaction, we consider the S1 magnetic structure (ferromagnetic planes), with J 1 = 0.40 K and J c = -0.11 K. For D < 0, in the absence of dipolar field, the Eu moments lie along the c axis. However, when

Appendix B: The anisotropic exchange tensor

We consider a simple implementation of anisotropy of the exchange interaction, by considering that the exchange tensor J ij has axial symmetry in a frame where the Eu-Eu bond is taken as the z -axis, i.e. it has a longitudinal component J and two identical transverse components J ⊥ . Then, for each considered exchange bond α (intraplane or interplane), the