Introduction

Layered systems featuring a Kagome lattice provide an ideal playground for discovering and understanding the topological electronic states in strongly correlated materials1,2,3,4,5,6,7,8,9,10,11,12,13,14. The crystal structure of the material Co3Sn2S215,16,17,18 is layered featuring a kagome lattice of CoSn. It was shown to have an out-of plane ferromagnetic ground state (Curie temperature of TC ≃ 177 K) with a magnetization arising mainly from the cobalt moments. This ferromagnetic semimetal has been shown to possess both a large anomalous Hall conductivity of 1130 Ω−1 cm−119,20 and a giant anomalous Hall angle of 20%20,21,22 in the 3D bulk. Density functional theory (DFT) calculations have predicted several pairs of Weyl points located only 60 meV above the Fermi level23. Recently, using the combination of ARPES and μSR experiments we established Co3Sn2S2 as a material that hosts a topological electronic structure and frustrated magnetism6,24. Namely, we showed that there is volume wise competition between ferromagnetic order of Co spins along the c-axis, and 120° antiferromagnetic order of Co spins within the kagome plane6. Moreover, we find a near-perfect correlation between the topological Hall conductivity and the ferromagnetically ordered volume fraction as a function of temperature. Theoretical modelling, considering both localized and itinerant electrons, was recently shown11 to reproduce the out-of-plane ferromagnetism and 120° antiferromagnetic ordering in the kagome plane, as we observed experimentally6. Frustration in the magnetic kagome lattice was also shown to drive an exchange biased anomalous Hall effect25. However, despite knowing the remarkable thermodynamic response of the magnetic and topological states in Co3Sn2S2, a possible quantum phase transition (QPT), as shown from standard magnetization measurements19,26, is less studied, despite being crucial to get full insight into the link between magnetic and topological transitions in this system. A QPT is a transition between a quantum ordered phase and a quantum disordered phase, occuring at zero temperature as function of pressure, magnetic field, or doping and is driven by quantum rather than thermal fluctuations27,28,29,30,31,32. Since the bulk magnetization19 is a very indirect probe for the magnetic structure, it is essential to explore the temperature-doping phase diagram using a microscopic magnetic probe. Namely, to study how the phase competition between the AFM state and the topological FM structure (which exhibits a strong AHC response in undoped Co3Sn2S2) evolves as a function of In-doping. Furthermore, understanding the nature of the magnetic to paramagnetic quantum phase transition (QPT) is crucial, e.g., whether it is first-order or second-order (continuous) in nature. There is quantum criticality associated with a second-order QPT, that may cause intriguing electronic phases. In case of a first-order QPT28,29,30,31,32, the system is expected to exhibit phase coexistence and abrupt changes in the ground state.

We have employed high-resolution muon spin relaxation/rotation (μSR) and transport experiments to systematically characterize the temperature-doping phase diagram in Co3Sn2−xInxS2. μSR is extremely sensitive local magnetic probe. Namely, using μSR the magnetic volume fraction and the ordered moment size can be measured independently. Therefore it is ideally suited to distinguish between the first-order or second-order behavior at a QPT. The current results indicate that the out-of-plane FM ground state transitions into mixed (FM+AFM)-states or (FM+HM)-states 5 and 10% of In-doping, respectively (see Fig. 1). Finally, the putative in-plane HM state is established at the critical doping level of xcr,1 ≃ 0.3, above which the intrinsic anomalous Hall conductivity becomes negligibly small. Interestingly, the QPT between the FM and HM states is first order. In contrast, the quantum evolution of the helimagnet–paramagnet transition at zero temperature happens as a second-order transition with xcr,2 ≃ 1. These results demonstrate not only the effective In-doping tuning of the competition between FM and in-plane AFM magnetic correlations, but that the competing magnetic correlations determine the anomalous Hall response of this system. Moreover, the current results uncover the first-order QPT transition as well as an AFM quantum critical point (QCP) in the phase diagram of the kagome magnet Co3Sn2−xInxS2.

Fig. 1: (Color online) Schematic temperature-doping magnetic phase diagram of CoSn2−xInxS2.
figure 1

FM, AFM, and HM denote the out-of-plane ferromagnetic, in-plane antiferromagnetic and in-plane helimagnetic orders, respectively (see the text for the explanation).

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Results

Temperature and doping dependence of microscopic magnetic response

Zero-field (ZF) μSR signals for Co3Sn2−xInxS2 recorded at 5 K for different indium dopant concentrations are shown in Fig. 2a. The frequency and amplitude of ZF oscillation are the measures of the ordered moment size and the magnetic volume fraction, respectively. ZF μSR spectra for Co3Sn2−xInxS2 reveal clear oscillations for In-doping levels x ≤ 0.9. This points to the presence of the long-range magnetic order and a well-defined internal field, associated to it. For the x = 1 sample, we observe relaxation with low amplitude and with no well-defined oscillations. Longitudinal field μSR (see Supplementary Figs. 1 and 2) experiments indicate that this relaxation arises from dynamic spin fluctuations within a small fraction of the sample. The majority of the fraction shows the weak Gaussian μSR depolarization33, evidencing the dominant paramagnetic state for x = 1 sample (see Supplementary Fig. 1 and the Supplementary Note 1). For x = 1.1, the entire sample volume shows paramagnetic response. To better visualize the effect of doping on the magnetic response, Fourier transform (FFT) amplitudes of the μSR time spectra are shown at T = 5 K for various In-doping (Fig. 2b), which is a measure for the the internal field distribution. For the undoped sample, the FFT amplitude recorded at 165 K is also shown. For x = 0 sample, a well-defined single internal field is observed with a narrow width of the field distribution in the sample, which was explained by the presence of a homogeneous out-of-plane ferromagnetic ground state6. However, at temperatures above \({T}_{{\rm{C}}}^{* }\ \sim\) 90 K, a second lower internal field is observed in the μSR spectra (165 K spectrum is shown as an example), which was interpreted as the appearance of the high-temperature in-plane AFM state. It was also shown that the in-plane AFM component with a larger width of the field distribution (component II) appears above \({T}_{{\rm{C}}}^{* }\ \sim\) 90 K, causing the reduction of the volume fraction of the FM component6. It is interesting that for the low In-doped x = 0.05 and 0.1 samples, two distinct internal fields appear in the μSR spectra already at the base temperature. This suggests the presence of spatially separated antiferromagnetically and ferromagnetically ordered regions already in the ground state of the x = 0.05 and 0.1 samples, implying that \({T}_{{\rm{C}}}^{* }\ \simeq\) 0. But, as the temperature increases, a transition from a two-field to a single-field is observed across 100 K (see Fig. 2c) and only the high-field narrow FM component remains at high temperatures. So, the results of x = 0.05 and 0.1 samples show that the FM component becomes dominant and occupies the full sample volume at high temperatures, while the ground state is characterized by spatially separated FM and AFM regions. For the samples with x ≥ 0.3, we observed the field distribution (Eq. 3) which is characterized by minimum (\({B}_{\min }\)) and maximum (\({B}_{\max }\)) cutoff fields (see Fig. 2b), which is consistent with the incommensurate helimagnetic order, observed in MnSi34,35,36 and MnP37,38. The cutoff fields \({B}_{\min }\) and \({B}_{\max }\) are measures of the magnetic order parameter of the helimagnetic state. The difference between \({B}_{\min }\) and \({B}_{\max }\) increases with increasing In-content. For x = 0.15 and 0.2 samples, both the commensurate FM peak and incommensurate HM field distribution are observed simultaneously. These results show that within a narrow region of In-concentration, the magnetic ground state changes from FM to a mixed FM and AFM/HM and finally to a fully HM state.

Fig. 2: (Color online) Zero-field (ZF) μSR time spectra for CoSn2−xInxS2.
figure 2

Zero-field (ZF) μSR time spectra (a) and the corresponding Fourier transform amplitudes (b) of the oscillating components for undoped and various In-doped CoSn2−xInxS2, recorded at 5 K. FFT amplitude for the x = 0 sample, recorded at 165 K, is also shown. The error bars represent the s.d. of the fit parameters. c Fourier transform amplitudes of the oscillating components of the μSR time spectra for various temperatures for the x = 0.1 sample. FFT amplitudes were normalized to its maximum value to make the signals well visible especially for samples for high In-doping. FM, AFM, and HM denote the out-of-plane ferromagnetic, in-plane antiferromagnetic and helical orders, respectively (see the text for the explanation).

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In the following, we provide the quantitative information about the muon spin relaxation rate, the magnetic volume fraction, and the static internal field size at the muon site (data analysis procedure is described in the methods section). The temperature dependences of the internal fields (\({B}_{{\rm{int}}}=\omega /{\gamma }_{\mu }^{-1}\)) for the two components for x = 0.05 and 0.1 samples are shown in the Fig. 3a, d, respectively. Both order parameters show a monotonous decrease. Remarkably, the two components have significantly different transition temperatures, with the AFM component having an onset at TAFM ≃ 120 K (x = 0.05) and 100 K (x = 0.1), and the FM component at TFM ≃ 175 K (x = 0.05) and 170 K (x = 0.1). Remarkably, there is a volume-wise interplay between FM and AFM states similar to that observed in the undoped sample, since the volume fraction of the AFM component decreases with increasing temperature and above 120 and 100 K the full volume of the samples x = 0.05 and 0.1 becomes ferromagnetically ordered (see Fig. 3b, e). So, the value of the AFM order temperatures TAFM for x = 0.05 and 0.1 samples are reduced significantly compared to the x = 0 sample. However, the AFM state extends down to the base-T unlike in the undoped sample6. Figure 3c, f show the temperature dependences of the transverse depolarization rates λT,AFM and λT,FM for the x = 0.05 and 0.1 samples for the AFM and the FM components, respectively. In general, muon spin depolarization may arise from the finite width of the static internal field distribution and/or by dynamic spin fluctuations. Due to the small value of the longitudinal relaxation rate λL in the present case, the damping is predominantly caused by former effect. The value of λT,AFM is at least a factor of two higher than λT,FM, which is similar to the undoped sample, indicating a more disordered nature of the AFM state in Co3Sn2−xInxS2. λT,AFM monotonously decreases with increasing temperature, similar to the internal field and becomes zero at TAFM. On the other hand, λT,FM shows increase upon increasing temperature and shows a maximum near the transition temperature TFM with a higher value as compared to the one at 5 K. Such an unconventional temperature evolution of the depolarization rate was also observed in the x = 0 sample6, and implies that the magnetism becomes more disordered upon approaching the transition. The domain wall motion near the critical temperature, as recently reported39, may explain such a behavior. Figure 4a, b show the temperature dependences of the internal fields for the FM and HM components for x = 0.15 and 0.2 samples, respectively. We note that for simplicity we define the internal field in the helical phase Bint,Hel as the average value of the minimum and the maximum cutoff fields Bint,Hel = (\({B}_{\min }\) + \({B}_{\max }\))/2. Both order parameters show a monotonous decrease. Both FM and HM regions exhibit similar transition temperatures. However, the FM fraction is only 30 and 15% for x = 0.15 and 0.2 samples, respectively. For x ≥ 0.3, only HM order is observed. The temperature dependences of the corresponding internal fields Bint,Hel and the relaxation rates λT,HM are displayed in Fig. 4d, f, respectively, revealing a gradual increase in the internal field as the temperature is lowered. This suggests a second-order-like behavior of the thermal phase transition and reveals the continuous development of the ordered moment size, as is the case for the undoped sample6. The value of Bint,Hel at the lowest measured temperature of T = 2 K is continuously reduced with In-doping. Bint,Hel becomes zero for the semiconducting sample of x = 119,40. The temperature dependence of the magnetic volume fraction extracted from the weak-TF data for all measured In-doped samples is displayed in Fig. 4e. Interestingly, the samples with x ≤ 0.9, in which the long-range magnetism is observed, are fully ordered at low temperatures. The semiconducting sample x = 1 shows a significantly reduced ordered volume fraction, which is suppressed nearly to zero for x = 1.1. To characterize the changes in magnetic critical temperature, we identify the temperatures TFM, TAFM, and THM at which the internal fields of the FM, AFM, and HM components start to appear, which coincides with the temperature at which the total magnetic fraction is 50%.

Fig. 3: (Color online) The temperature dependence of various fit parameters of the μSR experiments.
figure 3

The temperature dependences of the internal magnetic fields (a, d), the relative volume fractions (b, e) and the transverse relaxation rates (c, f) of the two magnetically ordered regions in the samples x = 0.05 and 0.1. The error bars represent the s.d. of the fit parameters. Arrows mark the critical temperatures TAFM and TFM for AFM and FM components.

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Fig. 4: (Color online) The temperature dependence of various fit parameters of the μSR experiments.
figure 4

The temperature dependences of the internal magnetic fields of the two magnetically ordered regions in the samples x = 0.15 (a) and 0.2 (b). c The temperature dependence of the transverse relaxation rates of the two magnetically ordered regions in the sample x = 0.2. Arrows mark the critical temperatures TFM and THel for FM and helical magnetic components, respectively. The temperature dependences of the average value of magnetic fields (d) and the transverse relaxation rates (f) for the helical magnetic component shown for various In-doped samples. e The temperature dependence of the total magnetic volume fraction for various In-doped samples. The error bars represent the s.d. of the fit parameters. Arrows in e mark the temperature at which the magnetic fraction is 50%.

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Temperature-doping magnetic phase diagram

The results are summarized in Fig. 5, showing a complex temperature-doping phase diagram for Co3Sn2−xInxS2. Figure 5a displays the evolution of the FM, AFM, and HM ordering temperatures as a function of In-content. In our previous work6, based on the theoretical group analysis and the local field simulations at the muon site, we showed that the x = 0 sample exhibits a phase separation between in-plane AFM (Fig. 5d) and out-of-plane FM states (Fig. 5e) in the temperature range between \({T}_{{\rm{C}}}^{* }\ \simeq\) 90 K and TAFM ≃ 172 K, while below \({T}_{{\rm{C}}}^{* }\ \simeq\) 90 K a homogeneous FM structure is observed. Here, we uncover an incommensurate HM phase and in the temperature-doping phase diagram, we identify five different magnetic phases: a ferromagnetic (FM), phase separation between ferromagnetic and antiferromagnetic (FM + AFM), phase separation between ferromagnetic and helimagnetic (FM + HM), helimagnetic (HM), and a paramagnetic phase (PM). The determination of the corresponding transition temperatures is described above. The present experiments suggest that a small amount of In-doping tunes the ground state of the Co3Sn2−xInxS2 from FM via a mixed FM + AFM and FM + HM states to a HM structure. The critical In-content at which the transition into an incommensurate HM state takes place is xcr,1 ≃ 0.3. The suppression of the out-of-plane FM state is also supported by the strong reduction of the c-axis coercive field, extracted from bulk magnetization measurements (Fig. 5c). Small c-axis coercive field for x ≥ 0.3 also points to the fact that the moments in the HM state are mostly in-plane. Thus, a helical version of the in-plane AFM state is anticipated at the high In-doping region of the phase diagram. It is important to note that there is a volume wise competition between FM and AFM/HM states, meaning that the FM volume fraction is heavily reduced upon approaching xcr,1 (Fig. 5b), while the ordered moment size (Fig. 5b) is only slightly reduced between x = 0 and 0.2 and is abruptly suppressed at xcr,1. This demonstrates that the change from FM to HM happens through a first order transition. Upon further increasing the In-content, the HM ordering temperature THel gradually decreases and is suppressed towards the critical doping xcr,2 ≃ 1, at which the system exhibits a metal to semiconductor transition (see the Supplementary Fig. 3 and the Supplementary Note 2). Indium doping also causes a smooth reduction of the internal field Bint,Hel, i.e. the ordered moment size of the HM state, to zero at xcr,2 ≃ 1 (Fig. 5b), suggesting a continuous QPT across xcr,2. Moreover, the magnetically ordered fraction Vm is unchanged as the In-content is increased from 0 to 0.9, indicating a fully ordered volume for all samples, exhibiting long-range magnetic order. Vm is reduced drastically for xcr,2 ≃ 1. For x = 1.1, Vm approaches nearly zero. The continuous suppression of the ordered moment size and the sharp reduction of Vm from 1 to ~0 across xcr,2 classiffies the transition from long-range magnetic metal to a paramagnetic semiconductor as a second-order quantum phase transition. This provides the direct evidence of a QCP at xcr,2 in Co3Sn2−xInxS2. This is to the best of our knowledge the first microscopic experimental observation of a QCP in a topological kagome magnet. The presence of the QCP in Co3Sn2−xInxS2 suggests that the physical properties of this material at low temperatures are controlled by the QCP. Our high pressure μSR measurements (see the details in the Supplementary Fig. 4 and the Supplementary Note 3) show strong and unusual reduction of the pressure induced suppression rate of the magnetic ordering temperature for the sample x = 0.9 as compared to the suppression rate for the samples x ≤ 0.8, which might also be related to the proximity of the x = 0.9 sample to the putative quantum critical point at x ≃ 1. The presence of small dynamic magnetic regions in the x = 1 sample may also be linked to the QCP.

Fig. 5: (Color online) Phase diagrams for CoSn2−xInxS2.
figure 5

a Temperature-doping phase diagram. TFM, TAFM and THM indicates the FM, AFM, and putative HM transition temperatures. \({T}_{{\rm{C}}}^{* }\) denotes the temperature below which only the FM state exists. b The doping dependence of the base-temperature value of the internal internal magnetic fields, the FM ordered fraction and the total magnetic fraction. The black dashed line marks the region with weak disordered magnetic response. c The In-doping dependence of the in-plane anomalous Hall conductivity σxy,int and the coercive field at T = 2 K. d, e Spin structures of Co3Sn2S2, i.e., the c-axis aligned FM structure (R-3m\(^{\prime}\) subgroup) and the in-plane AFM structure (R-3m subgroup). HM state is most likely a helical variation of the in-plane AFM structure.

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Discussion

According to our previous experiments and DFT calculations6 the out-of-plane ferromagnetic interactions and in-plane antiferromagnetic interactions within the kagome plane of Co3Sn2S2 have similar energy scales. The HM structure in this system may be interpreted by a competition between dominant Heisenberg-type (FM and AFM) interactions and a weaker antisymmetric Dzyaloshinskii–Moriya (DM) interaction. It seems that In-doping affects the competing interactions such that it promotes the HM state. We note that hydrostatic pressure also causes a suppression of both FM and AFM states41, but a pressure as high as 20 GPa is needed at which both orders are suppressed simultaneously. No HM phase was induced by pressure. In Co3Sn2−xInxS2 however, only a small amount of In is sufficient to push the system towards the HM state. Indium substitution introduces holes to the system and at the same time increases the separation of the kagome layers, while hydrostatic pressure shrinks the lattice and no doping is expected. This suggests that lattice expansion and/or hole doping disfavors the out-of-plane FM state, and leads to an incommensurate HM state. The present μSR experiments along with the AHC and c-axis magnetic hysteresis measurements suggest the in-plane HM state (moments rotate within the ab-plane, see the discussion below) for x ≥ 0.3. A helical variation of the in-plane AFM structure (see Fig. 5e) is possible. However, additional experiments such as polarized neutron scattering is necessary to determine the precise nature of this magnetic phase. The advantage of μSR is its extreme sensitivity to low-moment magnetism as it is the case for Co3Sn2S2.

It is well established that the system Co3Sn2S2 has a large Berry curvature driven anomalous Hall conductivity (AHC) σxy,int20. Recently, it was shown that σxy,int features a maximum at x ≃ 0.15 above which it sharply decreases and becomes negligibly small for x ≥ 0.3, well below the magnetic to paramagnetic critical doping xcr,2 ≃ 1. This indicates that the doping induced disappearance of the σxy,int is closely related to the depression of the FM state and that around the critical doping xcr,1 ≃ 0.3, both the ferromagnetism and the AHE are strongly suppressed simultaneously. This can be understood by the DFT calculations, showing that the c-axis FM order has the dominant contribution into the AHC6 (see the Supplementary Fig. 5 and the Supplementary note 4). Thus, the data provide direct experimental evidence for the quantum tuning (In-doping) of topological Hall conductivity driven by the competition between the FM, AFM, and HM structures in the kagome lattice. It is remarkable that the phase diagram (Fig. 5a) exhibits a doping region where different magnetically ordered phases, each of which may possess distinct topological responses, coexist at zero temperature. The non-zero but small value of σxy,int and the coercive field for x ≥ 0.3 (Fig. 5c) may indicate, either the presence of an experimentally non-resolvable small FM fraction in these high In-doped samples or the presence of some canting within the in-plane HM structure, giving rise to a small c-axis net FM moment and a related AHC component. Note that the AHC in the undoped system scales with the FM volume fraction as a function of temperature6. Since the FM fraction is gradually reduced by In-doping, one would expect the gradual suppression of σxy,int as well. Instead, we see a sharp peak of σxy,int at x ≃ 0.15. We currently do not provide a microscopic mechanism explaining this effect. However, we discuss several possible reasons for this behavior: (1) According to first principle calculations, the anomalous Hall conductivity σxy,int strongly depends on the electron density ne (which does not change much with temperature). In-doping causes the reduction of the FM fraction, but it may modify ne in the FM volumes of the sample in such a way that σxy increases. Once the FM fraction becomes negligible, σxy,int also becomes negligibly small. (2) The maximum of σxy,int occurs in the doping range where FM and HM regions coexist. Accordingly, a possible electron/hole transfer between FM and HM phases appears possible, thereby providing a higher effective doping for the FM region and therefore contributing to the increase of the σxy,int. (3) Indium doping might modify the band structure leading to an enhancement of σxy,int. The detailed electronic structure for this system Co3Sn2−xInxS2, considering here provided magnetic phase diagram, should be done and this is a subject of the separate studies.

In summary, the transition metal based kagome material Co3Sn2−xInxS2 is an ideal platform to study two intriguing, and in this case intertwined, properties of quantum materials, namely magnetism and topological electronic band structure. Our key finding is the possibility of an effective quantum tuning of the competition between FM, in-plane AFM and mostly in-plane HM orders within the kagome plane of Co3Sn2−xInxS2 by In-doping and the presence of multiple quantum phase transitions of different nature in the phase diagram of this system. Our experiments show the presence of at least two zero temperature phase transitions in the phase diagram of this system. One at the low In-doping of xcr,1 ≃ 0.3, where a QPT from a FM to a HM state through mixed states is observed and a second one at xcr,2 ≃ 1, with a transition from a helical metallic to a paramagnetic semiconducting state. Remarkably, the QPT from FM to incommensurate HM state is first order, while the transition from HM metal to a PM semiconductor is a continuous/second-order transition. Thus, we have uncovered a quantum critical point (QCP) in Co3Sn2−xInxS2 at the doping level of xcr,2 ≃ 1. The interplay between the competing magnetic states, the charge carrier density within the FM regions and the spin–orbit coupled band structure further seem to induce non-trivial variations of the topological properties of Co3Sn2−xInxS2. This is evidenced by a non-monotonous In-doping dependence of the anomalous Hall conductivity with the nearly full suppression of σxy,int above xcr,1 ≃ 0.3, where the FM state disappears. Therefore the exciting perspective arises of a magnetic system in which the topological response can be controlled, and thus explored, over a wide range of parameters.

Methods

Homogeneous macroscopic crystal structure

The crystal structure of the samples Co3Sn2−xInxS2 was examined by powder X-ray diffraction measurements, revealing a homogeneous rhombohedral macroscopic lattice structure. The obtained lattice parameters a and c show smooth linear dependence on the In-content x, as shown in our previous work19. Thus, we do not see any lattice irregularities or distortions on the macroscopic scale.

μSR experiment

Positive muons μ+, which are 100% spin-polarized when generated, are implanted one at a time in the sample. Thermalization of the positively charged μ+ at interstitial lattice sites occurs almost instanteously. They then can be used as magnetic microprobes, precessing around the local Bμ field felt at their stopping site. They precess with the Larmor frequency ω = 2πνμ = γμ/(2π)Bμ (muon gyromagnetic ratio γμ/(2π) = 135.5 MHz T−1).

Single crystals of Co3Sn2−xInxS2 were studied at ambient pressure with the GPS (πM3 beamline)42 and the HAL-9500 (πE3 beamline) μSR instruments at the Paul Scherrer Institute, Switzerland. High-momentum (pμ = 100 MeV/c) beam of muons (μE1 beamline) is used to investigate the x = 0.8 and 0.9 crystals as a function of hydrostatic pressure.

Analysis of weak TF-μSR data

The following function is used to analyze43 the asymmetry spectra collected in the weak TF setup:

$${A}_{{\mathrm{S}}}(t)={A}_{p}\exp (-\lambda t)\cos (\omega t+\phi ).$$
(1)

Here, A(t) and Ap is the time-dependent asymmetry and the amplitude of the oscillation, respectively. Ap is measure of the paramagnetic volume fraction. λ is the muon spin depolarization rate, arising from the paramagnetic spin fluctuations and/or nuclear dipolar moments. ω and ϕ is the Larmor precession frequency and a phase offset, respectively. When refining A(t), the zero was allowed to vary for each temperature in order to account for the asymmetry baseline shift occurring for magnetically ordered samples. The magnetically ordered volume fraction is then obtained at each temperature T by 1 − Ap(T)/Ap(Tmax), where Ap(Tmax) is the amplitude of the paramagnetic phase at high temperature.

Analysis of ZF-μSR data

The ZF-μSR spectra for x < 0.3 were analyzed using the following commensurate model:

$${A}_{{\rm{ZF}}}(t)=F\left(\mathop{\sum }\limits_{j=1}^{2}\left({f}_{{\rm{j}}}\cos (2\pi {\nu }_{{\rm{j}}}t+\phi ){e}^{-{\lambda }_{{\rm{j}}}t}\right)+{f}_{{\rm{L}}}{e}^{-{\lambda }_{{\rm{L}}}t}\right)+(1-F)\left(\frac{1}{3}+\frac{2}{3}\left(1-{(\sigma t)}^{2}\right){e}^{-\frac{1}{2}{(\sigma t)}^{2}}\right).$$
(2)

The model in (2) is composed of two components: a slowly relaxing “longitudinal” component, and an anisotropic magnetic contribution with an oscillating “transverse” component. The longitudinal component is the result of a local field component experienced by the muon that is parallel to the initial muon spin polarization. This results in the so-called “one-third tail” with \({f}_{{\rm{L}}}=\frac{1}{3}\) due to the orientational averaging of the randomly oriented fields in polycrystalline samples. As the orientation between field and polarization changes from being perpendicular to parallel, fL varies between zero and unity for single crystalline samples. It is important to note that the whole volume of the sample is magnetically ordered nearly up to TC. Only very close to the transition there is a paramagnetic component in addition to the magnetically ordered contribution. This is characterized by the width σ of the field distribution at the muon site created by the densely distributed nuclear moments. The temperature-dependent magnetic ordering fraction 0 ≤ F ≤ 1 governs the trade-off between magnetically-ordered and paramagnetic behaviors.

The field distribution for the samples x ≥ 0.3 presented in Fig. 2b is characterized by a minimum (\({B}_{\min }\)) and a maximum (\({B}_{\max }\)) cutoff field, which is consistent with the incommensurate helimagnetic order, and is well described by the field distribution given by34,35,36,37,38:

$$P(B)=\frac{2}{\pi }\frac{B}{\sqrt{({B}^{2}-{B}_{{\rm{min}}}^{2})({B}_{{\rm{max}}}^{2}-{B}^{2})}}$$
(3)

Analysis of ZF-μSR data under pressure

Since the large fraction (~50%) of the μSR signal originates from muons landing in the MP35N pressure cell, the data were modeled by weighted sum of a contribution of the sample and a contribution of the pressure cell:

$$A(t)={A}_{{\rm{S}}}(0){P}_{{\rm{S}}}(t)+{A}_{{\rm{PC}}}(0){P}_{{\rm{PC}}}(t),$$
(4)

where AS(0) and APC(0) are the initial asymmetries and PS(t) and PPC(t) are the time-dependent muon-spin polarizations of the muons stopping in the sample and the pressure cell, respectively. A damped Kubo–Toyabe function33,44 was used to analyze the signal of the pressure cell.

Pressure cell

Similar to previous studies44,45, hydrostatic pressures up to 1.9 GPa were generated in a double wall clamp cell made of CuBe/MP35N material44,45,46 and by using Daphne oil as a pressure transmitting medium. The precise value of the applied pressure was obtained by following the superconducting transition of an indium plate by AC susceptibility.