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Magnetization scaling in the paramagnetic phase of Mn1-xFexSi solid solutions

The magnetization field and temperature dependences in the paramagnetic phase of Mn1-xFexSi solid solutions with x<0.3 are investigated in the range B<5 T and T<60 K. It is found that field dependences of the magnetization M(B,T=const) exhibit scaling behavior of the form B\partial M/\partial B-M=F(B/(T-Ts)), where Ts denotes an empirically determined temperature of the transition into the magnetic phase with fluctuation driven short-range magnetic order and F(\c{hi}) is a universal scaling function for given composition. The scaling relation allowed concluding that the magnetization in the paramagnetic phase of Mn1-xFexSi is represented by the sum of two terms. The first term is saturated by the scaling variable \c{hi}=B/(T-Ts), whereas the second is linearly dependent on the magnetic field. A simple analytical formula describing the magnetization is derived and applied to estimates of the parameters characterizing localized magnetic moments in the studied system. The obtained data may be qualitatively interpreted assuming magnetic inhomogeneity of the paramagnetic phase on the nanoscale.


I. INTRODUCTION
Spiral magnets based on manganese monosilicide, MnSi, continue attracting attention mainly due to various aspects of skyrmion physics [1][2][3][4][5]. As long as skyrmions are expected developing in the so-called A-phase, this region on the magnetic phase diagram is the focus of researchers, whereas another magnetic phases are often considered as more simple and less interesting. From the general point of view, the simplest magnetic phase is the paramagnetic (PM) phase where strong correlation effects are not expected and which is believed to be well described by Moriya theory of spin fluctuations [6]. However, some anomalies of the magnetic and transport properties of the PM phase in MnSi have been reported recently [7][8][9]. For example, the formation of spin-polaron states of the ferron type at temperatures well above the Curie temperature T C was suggested for the explanation of the µSR experiments [7]. The analysis of the transport and magnetic resonance data also favors the explanation of magnetic properties of MnSi by spin polarons rather than within standard itinerant magnetism approach [8,9]. Thus, the physical idea, combining the results of the above works [7][8][9], is the concept of magnetic inhomogeneity on the nanometer scale, which should be manifested in the magnetic properties of manganese monosilicide.
In this connection, the study of Mn 1−x Fe x Si solid solutions may be prospective for better understanding of the magnetic inhomogeneities problem. According to Refs. 10-12 the increase of the iron content leads to the suppression of the transition into the helical phase with long-range magnetic order and formation of the quantum critical point at x = x * ≈ 0.12−0.15 for which T C (x * ) = 0 [10][11][12]. Some experimental [12,13] and theoretical works [14,15] suggests that an intermediate spiral-based phase with short-range magnetic order may be formed either in the vicinity of T C (x) or in the range x > x * due to strong spiral fluctuations extending into the paramagnetic phase [1,2,12,13]. Consequently certain regions of the magnetic phase diagram of Mn 1−x Fe x Si solid solutions may correspond to the formation of spatially inhomogeneous magnetic state with specific magnetization.
Magnetization and magnetic susceptibility studies are often aimed at obtaining information about phase boundaries on the magnetic phase diagrams of Mn 1−x Fe x Si [12,13,16].
Therefore the regions, where the magnetization field and temperature dependences M(B, T ) undergo abrupt changes due to magnetic transitions appears in the focus of interest, whereas the paramagnetic phase characterized by smooth M(B, T ) curves attracts less attention.
In the present work the magnetization in the paramagnetic phase of Mn 1−x Fe x Si solid solutions with x < 0.3 is investigated in the range B < 6 T and T < 60 K. We show that M(B, T ) function in the studied system may be represented as a sum of two terms, one of which shows universal scaling behavior. A simple analytical expression describing experimental M(B, T ) data in the studied B − T domain is suggested. The analysis of the approximation parameters concentration dependences indicates possible magnetic inhomogeneity of the paramagnetic phase on the nanoscale.

II. EXPERIMENTAL DETAILS
Single crystals of Mn 1−x Fe x Si solid solutions were synthesized by both Czochralski and Bridgeman methods. The crystal structure of the samples was controlled by the X-ray Laue diffraction. EPMA was applied for determination of the samples chemical composition.
Assuming formula (Mn 1−x Fe x ) 1+y Si 1−y we found that stoichiometry of crystals was kept at the level y ∼ 0.01 − 0.005 comparable with the absolute error of our EPMA measurements.
The parameters x characterizing studied samples de facto were found to deviate from the nominal ones calculated for the initial ingot and the discrepancy could be as big as ∼ 0.05.
Below we will present only real iron concentration in the samples. The third digit of the x number is used as a reference and corresponds to average Fe content obtained by several scans along the sample surface. The magnetization and magnetic susceptibility data in the magnetic fields up to 5 T for temperatures in the range 1.8-60 K were obtained with the help of SQUID magnetometer (Quantum Design).

III. EXPERIMENTAL DATA AND MAGNETIZATION SCALING IN THE PARA-
Before discussing magnetic properties in the paramagnetic phase it worth considering location of the paramagnetic phase on the T − x magnetic phase diagram. At present, it is widely accepted [12,13] that in Mn 1−x Fe x Si the transition into spiral phase with long-range magnetic order at T C may be preceded by formation of a fluctuation driven intermediate short-range ordered magnetic phase with the transition temperature T s > T C . These empirical findings [12,13] are supported by theoretical analysis [14,15]. Nevertheless at a moment it is not clear whether transition temperature T s is a true sharp phase boundary or this characteristic temperature merely marks a crossover region where spiral fluctuations in Mn 1−x Fe x Si system slow down and freeze. Consequently for x < x * the region of the paramagnetic phase may correspond either to T > T s (x) or to T > T C (x). As long as diapason x > x * is characterized by the absence of the long-range magnetic order [10][11][12], the area of the paramagnetic phase may be defined as T > T s (x) or T > 0 in this concentration range.
Detailed comparison of polarized neutron scattering data and temperature dependences of magnetic susceptibility χ(T ) in Mn 1−x Fe x Si system was carried out in [13]. The analysis of the data concerning magnetic structure at various temperatures allowed concluding that transitions into the phases with short-range and long-range magnetic orders unambiguously correspond to the inflection points of the χ(T ) curve [13]. For example, the magnetic susceptibility peak in Fig. 1,a does not occur at T C (x) as would be naive to assume, and real transition temperature into spiral phase with long-range magnetic order is located below mentioned peculiarity. Therefore for determination of the magnetic transition temperatures it is instructive to analyze temperature dependences of the magnetic susceptibility derivatives ∂χ/∂T = f (T ) (Fig. 1,b). According to Ref. 13 the broad minima of the ∂χ/∂T should mark transition into fluctuation driven spiral phase with short-range magnetic order at T s (x), whereas narrow maximum of the magnetic susceptibility derivative denotes formation of the magnetic phase with long-range spiral magnetic order at T C (x).
The experimental temperature dependences of the magnetic susceptibility and its derivative for Mn 1−x Fe x Si are shown in Fig. 1,a,b. It is visible that for any x the inequality T s (x) > T C (x) is valid and increase in iron concentration suppresses magnetic transition temperatures. The transition into magnetic phase with long-range order may be detected for x < x * ≈ 0.11 only. In agreement with the previous experimental results and theoretical models [11,12,14,15] it is reasonable to assume from the data in Fig. 1 that T C (x > x * ≈ 0.11) = 0. However, the transition into the phase with short-range magnetic order lasts up to x ∼ 0.24, i.e. this phase exists even for x > x * (Fig. 1,b). For iron concentrations exceeding the latter value the transition temperature T s (x) is not observed, which means that this parameter is either less than the lowest temperature used in our experiments or turns to zero.
The examples of raw magnetization data for Mn 1−x Fe x Si are presented in Fig. 1,c,d. The area of the paramagnetic phase corresponds to smooth M(B) curves, whereas those demon- strating a kink belongs to magnetically ordered phase and this type of the magnetization field dependences will be excluded from further analysis.
When discussing possible scaling for magnetization we suppose that it can be presented In Eq. (1) function φ(ξ) satisfy conditions φ(ξ ≪ 1) ∼ ξ and φ(ξ ≫ 1) = 1, argument Eq. (2) is valid for any form of A(T ) and scaling function F (ξ) is given by This transformation allows eliminating any linear term in magnetization (including linear term in φ(ξ)) and therefore rendering the field dependences of magnetization M(B, T = const) of the coordinates (2) it is possible to reveal unmasked scaling behavior if any.
The aforementioned analysis was performed for various forms of scaling variable ξ. It if found that for ξ = B/(T −T s ), where T s is defined from experiment as shown in Fig. 1 reasonable to use the model expression for φ(ξ). Namely we assume that and accordingly It is visible from Figs. 2,3 that model expression for scaling function (5)  It is visible that the model form (6) (Fig. 6,a) corresponds to T s ≈ 0 (x = 0.244) and T s ≈ −0.6 K (Fig. 7,a). Comparing this finding with the consequence of scaling function analysis it is possible to expect that T s may change sign in diapason x > x c ∼ 0.24.
The similar consideration was carried out in the case of the A(T ) temperature dependences obtained from fitting of the M(B, T = const) curves (Figs. 4,5). The data shown in Fig. 6,b suggest that this coefficient may be presented in the form where temperature T * = T s strongly depends on x changing sign in the vicinity of x * ∼ 0.11 (Figs. 6,b and 7,a). Thus in Mn 1−x Fe x Si two terms of magnetization (Eq. (1)) are described by two characteristic temperatures differ from one another and showing specific concentration dependences.
The effective magnetic moment µ * and parameter a 0 in Eq. (7) were calculated from the slopes ∂m −1 /∂T = k B /µ * and ∂A −1 /∂T = 1/a 0 of the corresponding lines in Fig. 6,a,b. The resulting concentration dependences of these quantities are shown in Fig. 7,b and Fig. 7,c respectively. We wish to mark that these two parameters vary weakly around average values µ * ∼ 6 µ B and a 0 ∼ 0.4 µ B K/T (Fig. 7,b,c). At the same time (Fig. 7 (1), which for x < 0.3 and B < 5 T can be well approximated as The concentration dependences of the parameters found in the present work suggest that µ * (x) ≈ const and a 0 (x) ≈ const, whereas M 0 (x), T s (x) and T * (x) strongly depend on iron content (Fig. 7,a-c). It is worth noting that none of the above parameters is a function of temperature and temperature dependence of magnetization in Mn 1−x Fe x Si may be completely described by Eq. (8).
As a summary of experimental facts, the empirical formula (8) leads to several consequences important in practical and theoretical sense. First of all, it is necessary to point out that temperature dependence of magnetic susceptibility at low temperatures (T < 60 K) in Mn 1−x Fe x Si system will always deviate from Curie-Weiss law owing to inequality of T s and T * . Therefore any estimates of effective on-site magnetic moment magnitude and (or) concentration of the magnetic dipoles based on Curie-Weiss analysis could be misleading for the case studied. At the same time, Eq. (8) may be applied to model calculations of magnetocaloric effect in Mn 1−x Fe x Si essential, for example, in the case of pulsed magnetic field measurements [17].
Secondly, assuming that classical itinerant picture of MnSi magnetism [6] is valid, it is not possible to suggest any physical mechanism for existence of two magnetic contributions.
Therefore, if itinerant ansatz is taken for granted, Eqs. (1) and (8) are nothing but some mathematical approximations of experimental data, which does not have direct physical sense. However, even in the considered approach, the singularity of denominator at T s in the first term of (8) suggests that the transition between paramagnetic phase and phase with short-range magnetic order [13] occurs at sharp boundary and can not be treated as a simple crossover phenomenon.
In addition, it is necessary to mark that any of the magnetic contributions in Mn 1−x Fe x Si obtained in the present work could not be associated with the effect of substitution of manganese with iron as long as both terms are present in pure MnSi. Moreover, the total magnetic moment of the sample decreases with x (Figs. 4,5), and therefore it is reasonable to suppose that the observed phenomena including magnetization scaling are due to Mn magnetic subsystem.
In order to explain the observed magnetization structure (Eq. (8)) it is possible to consider an alternative description of Mn 1−x Fe x Si magnetism based on Heisenberg-type localized magnetic moments. At present this type of models for MnSi magnetism are often considered as a simplified approximation of itinerant case. However, the experimental study of magnetic resonance and magnetoresistance [8,9] and LDA calculations [18] indicate real existence of localized magnetic moments (LMM) on Mn sites. In order to interpret the reduction of the saturated magnetic moment [6], realization of Yosida mechanism of magnetic scattering [9], presence of strong spin fluctuations in neutron scattering [19] and electron spin resonance [8,9] data in the Heisenberg paradigm, it is necessary to imply specific mechanism of screening of localized magnetic moments [8,9]. According to the hypothesis formulated in Ref.  [8,9]. This quasibound state may be viewed as an analogue of spin polaron, and it is possible to show that the model based on these specific states allows explaining peculiarities of magnetic scattering and magnetic resonance data better than traditional itinerant approach [8,9].
Therefore it is interesting to consider the results on the magnetization structure obtained in the present study in the framework of spin polaron model [8,9]. At first glance in the Heisenberg paradigm without any spin polarons, there are LMMs on Mn sites and free electrons. Therefore it is possible to speculate that saturating part of magnetization is due to LMMs subsystem, whereas linear part is nothing but enhanced Pauli term. However, the estimates of contributions to magnetization from LMMs and free electrons show that magnitude of the possible Pauli term is about three orders of magnitude less than that of LMMs [9]. Consequently the aforementioned explanation must imply an increase of the effective mass by a factor ∼ 10 2 − 10 3 , which does not meet the experimental situation in MnSi [20].
In our opinion, this problem may be resolved when possible composite nature of LMM in short-range magnetic order [13] and thus this parameter reflects characteristics of interaction between spin polarons in the mean field approximation. However, a difference between T s (x) and T * (x) (for x > x * these parameters have even opposite signs) suggest the presence of another type of magnetic interaction different from that describing ordering of spin polarons.
Moreover, the Curie-Weiss form for the additional term in Eq. (8) may suggest that this additional interaction can be also described by some mean field. Apparently it is not the case for the simple on-site model [8,9], however it may become possible, when the existence of a small cluster, where the electron spins are oriented opposite to the localized magnetic moments of Mn and all manganese LMMs are aligned in the same direction due to strong ferromagnetic coupling, is assumed.
If LMMs and electrons may be treated in quasiclassical approximation this cluster is a kind of "elementary"ferrimagnet. It is worth noting that in some ferrimagnets like Mn[FeCr]O 4 the magnetization field dependence contains both saturating and non-saturating linear parts [21]. In this case the characteristic temperature T * (x) corresponds to the mean field inside the ferrimagnetic cluster and the iron doping induced change of the interaction parameters between band electrons and manganese LMMs may alter and even change the sign of the effective paramagnetic temperature in the corresponding Curie-Weiss law as in conventional ferrimagnets [21]. At the same time, T s (x) will characterize interaction between the ferromagnetic spin clusters.