Existence of complex magnetic ground state and topological Hall effect in centrosymmetric silicide DyScSi

Topological Hall effect (THE) originating from non-trivial spin arrangements in magnetic materials has been extensively investigated in recent years. In this context, a centrosymmetric ternary silicide, DyScSi, is explored. Here we show that, a complex magnetic ground state drives THE in a centrosymmetric system. Temperature dependent magnetisation and neutron diffraction results establish the presence of commensurate antiferromagnetic (AFM) phase around 92 K, followed by an incommensurate AFM phase below 40 K. Additionally, two cluster glass transitions near 20 and 8 K, are also noted. These observed features arise due competing AFM and FM interactions. In conjunction with this, a finite contribution of THE is also observed in the intermediate field regime (8–30 kOe), at low temperature in DyScSi. The behaviour of this silicide appears to be fascinating in terms of interplay between complex magnetic ground state and THE in centrosymmetric structure.


Introduction
Interplay between complex magnetic ground states and various novel quantum phenomena in magnetic systems have recently sparked interest among researchers.Rare earth based ternary intermetallics with competing magnetic exchange interactions and suitable lattice geometry provide fertile playground to get insight into factors aiding such phenomena.In this context, a prospective class of materials exhibiting wide range of physical properties are equiatomic ternary intermetallic, RTX (R is rare earth element, T is transition metal and X is p block element) [1][2][3].Depending upon the constituent R, T and X, these intermetallics crystallize in different crystal structures owing to which they exhibit versatile magnetic and electrical properties such as crystal field splitting, exchange bias effect, metamagnetic behaviour, magnetic polaron effect etc. Further, some members of this family also display giant magnetocaloric effect (MCE), along with a sign change, suggesting a complicated magnetic ground state [4].
Most of these systems crystallize either in tetragonal or hexagonal structure.In hexagonal type of structure, mostly triangular arrangement of the magnetic R ions in the crystal lattice is observed.In such cases, the magnetic exchange interaction between these R ions is of Rudermen-Kittel-Kasuya-Yosida (RKKY) type.Interestingly, in some special cases, the interplay between RKKY interaction and this type of geometrical arrangement of R ions may compliment frustration resulting in complex magnetic ground state along with a glassy magnetic state [5].The existence of complex magnetic ground states in these systems can considerably alter the electronic transport properties.Literature reports suggest that several RTX intermetallics show large magnetoresistance (MR), positive or negative, in the magnetically ordered regime.For instance, GdMnSi exhibits positive MR at low magnetic field (H) which also changes the sign with increase in H and shows negative MR [6].GdCuSi shows a positive MR of magnitude 25% at 1.5 K and a large negative MR of magnitude 27% near the ordering T of 15 K [7].
One of the key transport phenomena in terms of extensive technical applicability is the Hall effect.In ferromagnets, anomalous Hall effect (AHE), which arises due to spin-orbit coupling, is observed [8].This type of effect has been observed in intermetallic like Co 2 FeSi [9], Fe 3 Ge [10], Co 2 FeAl [11].Apart from this, an additional contribution to ρ xy known as topological Hall effect (THE) is recently reported in some systems.It arises due to the crystallisation of spin textures (such as magnetic skyrmions) in the crystal lattice [12,13].Initially, skyrmions were observed in non-centrosymmetric materials such as MnSi [14] and Fe 0.5 Co 0.5 Si [15] and were explained in terms of Dzyaloshinskii-Moriya interactions.These interactions are induced by spin orbit coupling, due to the absence of inversion symmetry in crystal lattice.However, it was recently discovered that the skyrmions can also be stabilized in centrosymmetric materials by competing magnetic interactions.Till date, a very small number of centrosymmetric skyrmion materials have been explored.Among them, Gd 2 PdSi 3 is a well-known system [16].Experimental investigation reveals the coexistence of FM and AFM correlations in Gd 2 PdSi 3 .Presence of Kondo-like resistivity minimum before the onset of long-range magnetic ordering is also observed.Interestingly, H induced first order like transitions in the magnetically ordered state are noted, which results in large negative MR and modification of Fermi surface.Recently, Kurumaji et al also observed a skyrmion lattice phase in this system, where the presence of magnetic frustration helps in stabilizing this phase [17].THE has been observed in some centrosymmetric materials such as EuGa 2 Al 2 [18], EuAl 4 [19] and thin films of Mn 5 Si 3 [20].The later system crystallizes in centrosymmetric hexagonal P6 3 /mcm structure and undergoes orthorhombic distortion.Here, the additional contribution to Hall effect is observed in the non-collinear antiferromagnetic (AFM) regime which has a topological origin.However, it is absent in the collinear AFM regime [20].Despite the observation of such effects, the mechanism of its origin is still under debate.Therefore, it is important to investigate other centrosymmetric materials which exhibit THE.
In this context, ternary silicides (RScSi, R = Dy, Ho, Er, and Lu) belonging to RTX series appears intriguing.These materials crystallize in centrosymmetric hexagonal structure (space group-P6 3 /mcm) and the R ions form a triangular arrangement in the crystal lattice.The structural analysis of RScSi (R = Ce, Nd, Gd-Tm) indicates that unlike other silicides, GdScSi and CeScSi crystallize in a tetragonal structure rather than in hexagonal one [21,22].GdScSi and CeScSi exhibits FM and AFM ordering at 354 K and 26 K, respectively.The study of critical behaviour of GdScSi reveals that the magnetic interactions are short ranged, and the values of the critical parameters obtained are consistent with the 3D-Heisenberg class [23].The critical parameters obtained from a similar study of NdScSi are in excellent agreement with the 3D XY universality class [24].Unfortunately, certain members of this series have not received adequate attention from the viewpoint of magnetism and transport properties.
Hence, with this primary motivation, here, we have reported a systematic investigation of structural, magnetic, and electrical transport properties of DyScSi.Our magnetic and neutron diffraction (ND) studies clearly reveal that DyScSi undergoes phase transition from paramagnetic to commensurate AFM state near T 1 ∼ 92 K. Furthermore, incommensurate AFM state is established below T 2 ∼ 40 K. Additionally, a double cluster glass (CG) phases are is observed around 20 and 8 K. Further, a finite contribution of THE is observed in intermediate field region which coincides with the two step first order metamagnetic transitions noted in isothermal magnetisation data.

Methodology
Polycrystalline DyScSi is synthesized by arc melting (in argon atmosphere) the stoichiometric ratio of high-purity constituent elements (>99.9%).The ingot is re-melted several times to ensure the homogeneity and is subsequently subjected to annealing at 1073 K for 200 h.To confirm the crystal structure and structural transition (if any), powder x-ray diffraction (XRD) patterns at different temperatures (below 300 K) are obtained using rotating anode Rigaku Smart lab diffractometer (Cu-Kα; λ = 1.5406Å).Powder ND data of DyScSi is measured as function of d-spacing at different temperatures to study the evolution of the magnetic structure.The neutron data is taken using the super-high-resolution diffractometer Super HRPD installed at the beam line BL-08 of the Materials and Life Science Experimental Facility of the J-PARC center of KEK/JAEA, Tokai, Japan.Rietveld refinement of the neutron data taken by the QA detector of Super HRPD is carried out using computer program Z-code [25].T and H dependent magnetisation (M) measurements are carried using magnetic property measurement system from Quantum Design, U.S.A.Heat capacity (C) is measured using the heat capacity option of physical property measurement system (PPMS) from Quantum design, U.S.A. DC electrical and hall resistivity, and MR are measured using four probe method, using PPMS.In case of Hall resistivity (ρ xy ), measurements are performed in both positive and negative magnetic fields.Also, longitudinal contribution to ρ xy is removed by an antisymmetrisation procedure i.e. ρ xy (H) = [ρ xy (+H)-ρ xy (−H)]/2.

XRD
Room temperature (T) XRD pattern reveals that DyScSi is formed in single-phase, without any trace of the constituent elements or any other impurities.The observed peak positions are well indexed with the hexagonal structure (space group: P6 3 /mcm) (shown in the figure 1(a)).The lattice parameters and Wyckoff positions obtained from Rietveld refinement of XRD pattern are summarized in supplementary material (table S1) [26].The crystal structure obtained using VESTA software [60] is shown in figure 1(b).Interestingly, in this crystal lattice, it noted that each Dy atom forms an equilateral triangle with other two Dy atoms, which may result in geometrical frustration in case of AFM arrangement of spins (as shown in figure 1(c)).
To understand the effect of T on the crystal structure and positions of atoms, XRD patterns at different temperatures (below 300 K) are obtained and are analysed using Rietveld refinement method.The structural parameters and lattice parameters (a and c) at selected temperatures are listed in table S1 of [26]. Figure S1 depicts the variation of a, c, and X Dy (Wyckoff position of Dy ions) located at 6 g position as function of T. It is observed that there is drastic decrement in magnitude of a and c near 92 K, resulting in shrinking of the unit cell.An enhancement in lattice parameters is noted in the vicinity of 40 K, implying expansion of the unit cell.This indicates that, the unit cell parameters vary significantly with decrease in T. Similar trend is noted in T response of X Dy .This anomalous change in the structural parameters (a and c) and unit cell volume near 92 and 40 K can be explained in terms of Dy 3+ quadrupole moment interaction with the crystal lattice (section S1 of [26]).
In this type of systems, RKKY interactions are dominant, and strength of exchange interaction (J ex ) depends on the Fermi wave vector (k F ) and inter-ionic distances (d) i.e.J ex ∼ (−cos 2k F .d)/d 3 .In the present case, the nearest neighbour (NN) rare-earth ionic distance d NN ∼ 3.389 Å is found to be comparable with next nearest neighbor (NNN) distance d NNN ∼ 3.583 Å.As a result, the NN interaction (J NN ) and NNN interactions (J NNN ) are found to be of comparable strength.Since the third NN is placed far away, generally its contribution to J ex is quite small.If the nature of interaction between Dy atoms is AFM, then the geometry formed by Dy atoms may compliment frustration among Dy moments.Additionally, as the distance between NN and NNN is comparable, it can enhance frustration in this system.In RKKY model, k F strongly depends on a/c ratio and the number of free electrons Z per magnetic ion and relation is given by [33]: In the present case, Z = 3.The above equation signifies that change in a/c ratio can alter the sign of exchange interaction resulting in either FM or AFM coupling among magnetic ions.For DyScSi, a/c is calculated and listed in table S1 of [26].From the table, it can be inferred that a/c ratio does not change substantially near 92 and 50 K, indicating that it will not significantly alter the k F .Hence, the inter-ionic distances will affect the nature and strength of RKKY interactions.

Temperature and magnetic field dependent magnetisation studies
T dependent DC susceptibility (χ DC ) obtained at various H under zero field cooled (ZFC) and field cooled (FC) conditions, is shown in figure 2. From the ZFC curve at 100 Oe, it is noted that with decrease in T, χ ZFC increases and a weak hump centred around 93 K (T 1 ) is observed.With further decrement in T, a peak is observed around 40 K (T 2 ).Also, nature of magnetic interactions near T 1 and T 2 can be different due to contrasting variation of the lattice parameters.As the T is reduced further, a decrease in χ ZFC is followed by a peak around 20 K (T 3 ).In contrary to ZFC curve, χ FC increases below T 2 , followed by a peak around T 3 (magnetic phase below T 3 will be discussed in detail in subsequent sections).A bifurcation between ZFC and FC curves is observed below 40 K which hints towards the possibility of a phase with competing magnetic interactions or glassy magnetic phase [61].Inverse χ DC at 1 kOe is plotted as a function of T (shown in inset (a) of figure 2).The curve is fitted with modified Curie-Weiss (CW) law of the form.
where χ 0 is the temperature independent susceptibility, C is Curie's constant and θ P is CW temperature.The values of obtained parameters are χ 0 = 1.30* 10 −5 emu/mole, θ P = 51.28 ± 0.29 K and C = 0.034 ± 0.002 emu K mole −1 .In addition to this, the effective magnetic moment (µ eff ) is found to be ∼8.37 µ B /Dy 3+ using the formula 2.83√C.This value is quite lower as compared to theoretically calculated free moment of Dy 3+ ion (∼10.65 µ B ), probably due to the presence of competing exchange interactions.Interestingly, the inverse χ DC curve shows an upward deviation from CW law around T 1 (as shown by the arrow in inset (a) of figure 2).However, this behaviour is not in accordance with the formation of Griffith's phase in the paramagnetic regime.In the latter case, usually a downward deviation in inverse χ DC curve is observed [62].Hence, the observed feature in our case can possibly be ascribed to AFM interactions, resulting in suppression of χ DC [63].To show the magnified view of the effect of H on the feature at T 1 ; normalized χ DC is plotted as a function of normalized T in the range 70-120 K.Here both χ DC and T are normalized with respect to their highest magnitude (as shown in inset (b) of figure 2).It is noted that with increase in H the hump becomes broader and shifts towards lower T side (as shown by the arrow).Coming back to the main panel of figure 2, it is observed that peak at T 2 is suppressed with increasing H and disappears at 5 kOe.Also, it is noted that with the application of H, the bifurcation between ZFC and FC magnetisation curves is suppressed to lower temperatures, indicating the presence of complex magnetic ground state below T 2 .In addition to this, T dependent heat capacity (C (T)) measurements at different applied H have also been carried out.The obtained data is shown in figure S3 of supplementary material [26].A careful analysis of C (T) under different applied H, indicates the dominance of AFM correlations near T 1 which vanishes under high applied fields.Also, no anomaly is noted near T 2 in C (T). Magnetic isotherm M(H) measurements are also performed at different temperatures, as shown in figures 3(a)-(f).From figure 3(a), it is observed that M(H) curve obtained at 300 K shows a linear behaviour, as is expected in a paramagnetic region.As the T is reduced to 80 K, a deviation from linearity is observed, which can be due to the presence of AFM correlations (figure 3(b)).This is also in accordance with the inference drawn from the χ DC (T) curves.Below T 2 , the change in slope become more prominent and no significant magnetic hysteresis is noted (figure 3(c)).However, magnetic hysteresis is observed below T 3 (figure 3(d)).In all three cases, no saturation in magnetisation is noted up to 70 kOe.As seen from figure 3(e), a significant hysteresis is observed at 2 K, which reflects the dominance of FM correlations under applied H.It implies that as the temperature is reduced below T 3 , there is a development of FM correlations on the application of H. Additionally, two metamagnetic transitions are noted near 10 and 30 kOe.This can be clearly seen from derivative (dM/dH vs H curve), as indicated by the arrows in figure 3(f).Hence, the systematic analysis of χ DC (T) and M (H) curves indicate towards the coexistence of FM and AFM interactions in DyScSi.Such features can be attributed to presence of GF (as discussed in previous section).Also, the change in lattice parameters in the vicinity of phase transitions can alter the nature of RKKY interaction between Dy ions, resulting in enhancement of frustration in lattice.

Temperature dependent ND studies
To study the change in magnetic structure across these transitions, powder ND measurements as function of d-spacing are performed at few selected temperatures.Due to the very high absorption cross-section of Dy for neutrons, the obtained data is little bit noisy.At all temperatures, ND data is refined (using Z-Rietveld software) using P6 3 /mcm space group, where Dy occupy single 6 g crystallographic site.The obtained structural parameters at 300 K are summarized in table S4 of [26], which is found to be consistent with XRD results.At room T, no magnetic peaks are observed.Only fundamental nuclear Bragg peaks are noted as shown in figure 4(a).Here, the propagation vector cannot be defined due to non-existence of magnetic Bragg peaks.This is consistent with the magnetisation results, where it is noted that DyScSi is in paramagnetic state at room T. As the T is decreased to 85 K, no additional Bragg peaks are noted.In systems, where there is no structural transition near the magnetic phase transition, the difference between two ND spectra recorded below and above the transition is purely magnetic, and it gives the information about the nature of magnetic interactions.In our system, no structural transition (as discussed in section A) is noted near T 1 .Therefore, ND spectra obtained at 300, 100 and 85 K are carefully compared, and it is noted that there is an increment in intensity of (2 0 0) and (2 −1 1) peaks (shown in figure 4(b)).This indicates the presence of magnetic component on the nuclear Bragg peaks.It gives us a direct estimation of the magnetic propagation vector i.e. k 1 = (0, 0, 0).Moreover, in the present case, magnetic hysteresis and residual magnetisation are not observed in M(H) curves obtained at 80 K (figure 3(b)), which rules out the possibility of FM ordering near T 1 .The presence of rational propagational vector reflects the presence of commensurate AFM-type ordering, which is in analogy with C(T) analysis.It is also important to mention that the width of (2 0 0) and (2 −1 1) magnetic peaks is comparable with nuclear peaks.This indicates towards the presence of long-range magnetic ordering around T 1 .Hence, based on ND, M and C studies, it can be said that there is presence of long-range commensurate AFM ordering near T 1 .
As the T is further lowered below 40 K, two additional Bragg peaks in the spectra are observed, as shown in figure 4(c).These peaks are not located at d value where there is a nuclear Bragg peak; indicating that these are magnetic Bragg peaks.This reflects to the existence of non-zero propagation vector describing a magnetic ordering i.e., an AFM-like structure.The peaks are identified as (0 −1 2) and (2 −2 2) with corresponding magnetic propagation vector k 2 = (0, 0, 0.28).It can be noted that magnetic unit cell is irrational multiple of nuclear unit cell.It indicates towards the evolution of incommensurate spin ordering between Dy ions.Apart from this, FM component is not observed in M (H) curves.This suggests that there is a change over from commensurate to incommensurate AFM near T 2 .These magnetic peaks are persistent till 5 K.It is also observed that intensity of these peaks does not vary significantly with decrease in temperature (as shown in figure 4(d)).However, below 25 K, a slight enhancement in intensity of other peaks (2 0 0) and (2 −1 1) is seen, which reflects the existence of weak FM interactions.This observation is in analogy with the M(H), where S shaped non-saturating curve is observed with magnetic hysteresis and remanent magnetisation.Hence, it can be said that there is presence of incommensurate AFM phase with weak FM correlations below T 3 .Additionally, no extra peaks, apart from those mentioned above are observed down to 5 K. Also, as observed from low T XRD, below 25 K, lattice parameter a is constant but a slight increment in c is observed.This can further affect the strength and nature of exchange interactions between Dy ions.

Nature of low temperature magnetic phase
In some systems exhibiting competing magnetic interactions, glassy magnetic phase is reported.Additionally, recent reports on glassy magnetic systems [64,65] indicate that frustration due to disorder and competing interactions can significantly enhance the MCE.In order to explore the presence MCE in DyScSi, isothermal entropy change (∆S M ) is determined.Figure S4 of [26] shows ∆S M (T) curves plotted at different magnetic fields.A smooth crossover from direct MCE at low T to inverse MCE below T 3 is noted.This crossover can be ascribed to the development of FM correlation under applied magnetic field below T 3 , which is also in accordance with the ND analysis.Remarkably, a significant magnitude of ∆S M ∼ −20 J kg −1 −K is noted at lowest T. Also, at low T, the geometry formed by Dy atoms, and the coexistence of FM and AFM interactions can facilitate the formation of metastable frozen states.Hence, to understand the nature of the low T magnetic phase AC susceptibility measurements are done.

AC susceptibility
AC susceptibility (χ AC ) is measured as a function of T at different frequencies 13-931 Hz at H AC = 1 Oe, after cooling the sample in zero DC field.The real part of χ AC (χ ′ ) is plotted as function of T and is shown in figure 5(a).It is found that χ ′ exhibits a hump like feature near 93 K (T 1 ) and a peak around 20 K (T 3 ).It is seen that hump does not exhibit any shift in temperature with frequency whereas the peak shifts towards higher T with increase in frequency (inset of figure 5(a)).No peak is observed around T 2 ; however, a weak change of curvature is noted.Imaginary part of χ AC (χ ′′ ) is also plotted as a function of T (figure 5(b)).Detectable signal in χ ′′ is only observed below 25 K, whereas the value of χ ′′ is negligible above this.In analogy with χ ′ , a frequency dependent peak is noted at T 3 .Interestingly, an additional peak around 8 K (T 4 ) is also observed.Both peaks shift towards high T with the increment of frequency.Such type of behaviour is generally seen in case of freezing or blocking of spins or due to domain wall motion [66].This latter effect is usually noted in case of long ranged ordered ferromagnets or ferrimagnets.
Therefore, to characterize the nature of transition exhibited by DyScSi near T 3 , and T 4 , Mydosh parameter (δT f ) is calculated which is often used to distinguish various magnetic systems.Mydosh parameter is expressed as [67] where ∆T f = (T f ) ν 1 −(T f ) ν 2 and ∆log 10 (ν) = log 10 (ν 1 ) −log 10 (ν 2 ).For our system, this value is calculated to be δT 3 ∼ 0.024 and δT 4 ∼ 0.084.It is found that the values are in the range of those observed for typical CG systems [68][69][70][71][72]. Therefore, the observed peaks in χ AC seems to be associated with interacting magnetic clusters, leading to a CG state.The behaviour of magnetic clusters is further investigated, using standard critical slowing down model [73] where τ * is the microscopic flipping time, T g is the true glass transition temperature, z is the dynamic critical exponent and v is the critical exponent of the correlation length.Also, the T maxima are fitted with Vogel-Fulcher (V-F) law (which considers the interaction between spins) of the form [74] Figure 6.Critical slowing down model fit of relaxation time (τ) as function of reduced temperature T f using equation ( 4) for (a) transition at T3 and (c) transition at T4. V-F law fit of relaxation time (τ ) as function of reduced T = 1/(T f −T0) using equation ( 5) for (b) transition at T3 and (d) transition at T4. Table 1.Parameters obtained from critical slowing down model and V-F law using equations ( 4) and (5).
where T 0 is the V-F temperature, which represents the strength of interactions between clusters.The fittings using equations ( 4) and ( 5) are shown in figures 6(a)-(d).The values of parameters obtained after fitting are summarized in table 1. Non-zero value of T 0 rules out the presence of superparamagnetic state.The obtained values of τ * corresponding to peak at T 3 and T 4 are 2.2 × 10 −5 s and 1.74 × 10 −8 s respectively.These values are large compared to the typical values of canonical spin glass systems (10 −13 s).This suggests a slow spin dynamic in DyScSi, which is due to presence of strongly interacting clusters [75].Moreover, the difference in τ * for two cases clearly suggest a considerable difference in size and distribution of clusters at these temperatures.The change in lattice parameters around 40 K is believed to change the nature of RKKY interaction and the arrangement of Dy atoms compliments GF.This can lead to variation in size and distribution of clusters, resulting in the observed spin dynamics.Additionally, non-equilibrium dynamical measurements like magnetic relaxation and memory effects are performed and discussed in detail [26].The value of the stretched exponent (β) obtained from thermoremanent magnetisation measurements fall in the range of that observed for other CG systems.Both these measurements signify that the system evolves through various metastable states.Also, both ZFC and FC memory effects provide clear signatures of memory effect in DyScSi.Negative and positive T-cycle relaxation memory measurements indicate that our system follows the hierarchical model of metastable states.This also suggests that there is finite interaction between spins leading to cluster formation, thereby, resulting in free energy valleys, which is in analogy χ AC with results.
From above discussions, it can be concluded that DyScSi undergoes multiple magnetic transitions with change in T. Mainly five different magnetic regions are observed.Above T 1 , a paramagnetic phase is noted.Around T 1 , a transition to long ranged commensurate AFM phase is observed.As T is further reduced an incommensurate AFM state is established below T 2 .It is followed by double CG transition near T 3 and T 4 .This indicates towards the presence of complex magnetic ground state in DyScSi.Further, literature reports on RTX intermetallics reveal a strong relationship between magnetic phases and other physical properties.Hence, the electrical transport properties of DyScSi are also investigated.

Electrical transport properties
The crystal geometry and competing interactions in DyScSi (as discussed in previous sections) can make it a potential candidate for the realisation of THE.With this motivation, we have explored the H and T dependent electrical transport properties.

Temperature dependence of electrical resistivity
Figure 7 shows the T dependence of electrical resistivity (ρ xx ) measured at various applied fields.The schematic diagram in figure 7 shows the technique of measurement used for Hall voltage (V xy ) and longitudinal voltage (V xx ).At 0 Oe, above 50 K, ρ xx linearly increases with T, indicating a metallic behaviour.However, no anomaly is noted around T 1 , T 2 and T 3 .Below 30 K, a slope change in the curve is noted which becomes more prominent (in form of a minima) under applied H. Resistivity minima in such system are generally attributed either to Kondo effect or magnetic superzone energy gap or to weak localisation (WL) effects [76,77].The Kondo effect is ruled out because in this system the 4f orbital lies deep inside Fermi level.Hence, to get some insight about the observed low T behaviour (below T 3 ), the nature of variation of ρ xx with T is analysed.Interestingly, it is noted that ρ xx does not follow either quadratic or T 5 or T 1/2 dependence (as shown in the inset (a) of figure 7).Szlawska et al had reported that the contribution to ρ xx due to WL effect is described by a T n/2 term where n ⩾ 1.5 [78].Also, in glassy magnetic systems any spin flip excitation is unable to propagate like a magnon.This excitation perishes away with some diffusion constant Λ.Rivier and Adkins have shown that the initial T dependence of ρ xx in glassy systems as (T/Λ) 3/2 [79].The range of this dependence in such case is well below glass freezing T.However, our analysis of zero-H ρ xx reveals that it follows a T 3/2 dependence upto ∼20 K i.e. in the CG regime (shown in the inset of figure 7).This could possibly arise from WL or from the diffusive modes of excitations in clusters as discussed by Fischer [80].Similar type of dependence has also been reported in other CG systems [75].
To shed some light on the low temperature ρ xx behaviour, we have measured MR.MR (which is defined as ρ(H)−ρ(0)/ρ(0)) as a function of H at different temperatures is shown in figure 8. Below T 1 , a negative MR is noted throughout the measured H range. Generally, the systems which show large MCE also show large MR.Furthermore, since both these properties are related to the change in the magnetic state brought by H, the signs of MCE and MR are also correlated.However, we have observed a crossover in MCE around 20 K, but such feature is absent in MR.Usually, in case of ideal antiferromagnet a positive MR is expected.The observed negative MR suggest to the complexity of magnetic structure in this system.Additionally, this also rules out the contribution of magnon scattering and spin fluctuations [81].As noted from figure 8(b), above 100 K, very negligible MR is observed.Further, as the temperature is decreased, the value of MR is enhanced, and it remains constant below T 3 .At 3 K, a value of ∼−12.5% is noted for an applied H of 70 kOe.Additionally, a hysteresis is observed (figure 8(a)).
According to Lee and Ramakrishnan, a H 1/2 dependence of MR is expected for 3D disordered systems due to WL effects [82].But, in our system MR does not follow this dependence (as shown by the blue solid line in inset of figure 8(a)), which rules out WL effect.Also, as inferred earlier, an incommensurate AFM state is observed below T 2 .Therefore, the observed upturn in ρ xx can be attributed to magnetic super zone gap formation.In this scenario, magnetic sub-lattice distorts the Fermi surface resulting in a gap opening in the conduction band, which in turn is responsible for the upturn observed in ρ xx .The application of strong H results in disappearance of magnetic super zone gaps in some portions of the Fermi surface causing a decrease in ρ xx ; giving rise to negative MR. Similar type of behaviour is not unusual has also been reported in Gd 2 PdSi 3 and Dy 2 NiSi 3 [83,84].

Hall effect
In systems with complex magnetic ground states, Hall is effect is often found to exhibit non-linear curvature.This behaviour is believed to arise from the additional contributions to the ordinary Hall effect (OHE).As discussed earlier, the complex magnetic ground state in DyScSi provides an excellent platform for the realisation of additional contributions to OHE.Hence, ρ xy is measured as function of H at different temperatures.Figure 9 shows the H dependence (±70 kOe) of ρ xy at 3 K.In order to eliminate the offset voltage due to misalignment, the ρ xy has been anti-symmetrized using the relation ρ xy = (ρ xy (+H)−ρ xy (−H))/2.In such systems, the total ρ xy is generally made up of three components: (6) where the first and second terms are the normal and anomalous ρ xy proportional to H and M, respectively.The third term represents the topological component.Generally, the first two terms can be determined from magnetisation measurements.Hence, ρ T xy can be extracted and can be considered as a good probe for the existence of non-collinear magnetic state, non-uniform magnetisation, and skyrmions or related topological spin states.From inset (a) of figure 9, it can be inferred that at 3 K, ρ xy decreases with increase in H upto 0.9 kOe.However, above 0.9 kOe, ρ xy increases linearly till 2 kOe.This type of behaviour can be attributed to the contribution from OHE.Also, above 2 kOe, a non-linear increment in xy (∼2.98 µΩ cm at 8 kOe) is observed; followed by a plateau till 30 kOe (as by the arrows in figure 9).This plateau region the two metamagnetic transitions, noted in dM(H)/dH curve.The non-linear H dependence of ρ xy might arise due to simultaneous contributions from different type of charge carriers or from AHE.Generally, AHE is reported in conventional ferromagnet, where it is proportional to the net magnetisation.In DyScSi, there is a co-existence of FM and AFM correlations at lower T and the shape of ρ xy curve is different from M(H) curve.Hence, the non-linearity cannot be attributed to typical AHE, observed in ferromagnet.This indicates to the presence of some extra contribution to ρ xy .As also mentioned earlier, in our case, M(H) does not saturate till 70 kOe.Hence, it is difficult to extract the exact contribution of AHE using equation (6).Additionally, the plateau region is persistent upto 15 K (inset (b) of figure 9).Therefore, it can be concluded that only OHE and AHE cannot account for the observed ρ xy .The observed behaviour strongly suggests the existence of THE which is manifested in the form of strong enhancement in ρ xy observed at low H at T < 15 K. Similar features have been reported in some systems where THE is ascribed to the presence of complex magnetic state [85].Therefore, in DyScSi, the presence of incommensurate AFM phase with weak FM correlations followed by double CG transitions at low T results in the observation of THE.

Conclusions
To conclude, DyScSi crystallizes in a geometrically frustrated centrosymmetric structure.Nature of RKKY exchange interactions and geometrical frustration between Dy ions plays a crucial role in observation of the complex magnetic state.Systematic analysis of magnetisation and ND results reveal that DyScSi undergoes transition from commensurate AFM (∼92 K) to incommensurate AFM state (∼40 K) followed by two CG transitions around 20 and 8 K respectively.Existence of competing FM and AFM correlations facilitates the unusual multiple CG transitions.A negative MR is noted up to 100 K, which is in contrast with MCE behaviour.Interestingly, in Hall Effect, a finite contribution from THE is detected below 15 K in intermediate field regime.It is interesting to note that, THE does not develop immediately at the onset of AFM order.Instead, it develops well below T N .Our results indicate that incommensurate AFM phase with weak FM correlations at low temperature drive THE in a centrosymmetric system.Therefore, our studies on DyScSi may offer an excellent platform to investigate the correlation between frustrated magnetism and topological Hall effect.

Figure 1 .
Figure 1.Structural analysis of DyScSi.(a) Indexed room temperature XRD pattern (b) crystal structure of DyScSi obtained using VESTA software.Dy (red circle); Sc (green circle); Si (pink circle) atoms.(c) Triangular arrangement of Dy atoms in crystal lattice.

Figure 2 .
Figure 2. T dependence of χDC under ZFC and FC conditions in the range 1.8-300 K. Inset (a): inverse χDC as function of T at 1 kOe; red line represents the fit using equation (2) and the arrow shows the upward deviation from CW law.Inset (b) normalized χDC under ZFC and FC conditions as function of normalized T at different H (in the range 70-120 K) and the arrow shows the shift of broad hump towards low T.

Figure 3 .
Figure 3. H dependence of M. (a)-(e) M (H) isotherms in the range 0-70 kOe obtained at few selected temperatures (f) dM/dH vs H at 2 K, the arrows indicate the change in slope.

Figure 4 .
Figure 4. Neutron diffraction patterns at different T. (a) Rietveld refinement analysis of room T ND pattern of DyScSi.(b) Magnified ND pattern as a function of d spacing (3.2-3.7Å ) obtained at T = 300, 100 and 85 K. (c) Magnified ND pattern as a function of d spacing (2.4-3.4Å) obtained at T = 85 and 35 K (d) magnified ND pattern as a function of d spacing (2.9-3.2Å) obtained at T = 35, 25 and 5 K.

Figure 5 .
Figure 5. χAC as a function of T. (a) T response of χ ′ component of χAC measured at different frequencies.Inset shows the shift of peak T towards higher T with increment in frequency.(b) T response of χ ′′ component of χAC.

Figure 7 .
Figure 7. T dependence of ρxx at different applied H with a schematic drawing of the technique of measurement used for Hall voltage (Vxy) and longitudinal voltage (Vxx) measurements.Inset: (a) T dependence of zero-field ρxx in the range 3-30 K. Solid red line corresponds to the T 1/2 dependence, blue line corresponds to T 2 dependence and green line corresponds to T 5 dependence.(b) T dependence of zero-field ρxx in the T range 3-30 K. Solid red line corresponds to the T 3/2 fit.

Figure 8 .
Figure 8. MR as a function of H (a) At 3 K.Inset: Same plot in H range (0-70 kOe).Blue line corresponds to H 1/2 dependence.(b) MR as function of H at few selected temperatures.

Figure 9 .
Figure 9. Field dependence of ρxy at 3 K.The arrows indicate the fields between which the plateau is observed.Inset (a): same plot in the H range 0-4 kOe.Inset (b): same plot at different temperatures.